Related papers: Thirty-Six Unsolved Problems in Number Theory
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value…
Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase…
Supersymmetry remains compelling theory over 30 years in spite of lack of its discovery. It could be already near the corner our days, therefore present and upcoming experiments are crucial for constraining or even discovery of the…
In this talk we introduce several topics in combinatorial number theory which are related to groups; the topics include combinatorial aspects of covers of groups by cosets, and also restricted sumsets and zero-sum problems on abelian…
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…
We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in $\Rd$, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in $\Rd$, decide…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
In backgrounds with compact dimensions there may exist several phases of black objects including the black-hole and the black-string. The phase transition between them raises puzzles and touches fundamental issues such as topology change,…
Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We consider also a special categorical invariant $K_\Theta (H)$ of this geometry. The…
We summarize the arguments that space and time are likely to be emergent notions; i.e. they are not present in the fundamental formulation of the theory, but appear as approximate macroscopic concepts. Along the way we briefly review…
The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system…
String theory has transformed our understanding of geometry, topology and spacetime. Thus, for this special issue of Foundations of Physics commemorating "Forty Years of String Theory", it seems appropriate to step back and ask what we do…