Related papers: Algorithms and Geometric Constructions
According to mathematical constructivism, a mathematical object can exist only if there is a way to compute (or "construct") it; so, what is non-computable is non-constructive. In the example of the quantum model, whose Fock states are…
This article contains a short and entertaining list of unsolved problems in Plane Geometry. Their statement may seem naive and can be understood at an elementary level. But their solutions have refused to appear for forty years in the best…
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an…
The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the…
It has been a long entertained idea that self-bound gravitons, so-called geons, could be a dark matter candidate or form (primordial) black holes. The development of viable candidates for quantum gravity allows now to investigate these…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
A method is presented to construct a particular, non-minimally coupled scalar-tensor theory such that a given metric is an exact vacuum solution in that theory. In contrast to the standard approach in studies of gravitational dynamics,…
The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for…
This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
We propose a simple, geometrically-motivated construction of smooth random paths in the plane. The construction is such that, with probability one, the paths have finite curvature everywhere (and the realizations are visually pleasing when…
The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum…
If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and define computational methods that are…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
Properties of the logical reloading in the Euclidean geometry are considered. The logical reloading is a logical operation which replaces one system of basic concepts of a conception by another system of basic concepts of the same…
Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close…