Related papers: Pauli matrices and ring puzzles
We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on…
We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…
We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…
We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.
We are studying topological properties of the Julia set of the map $F(z, p)=((2z/(p+1)-1)^2, ((p-1)/(p+1))^2)$ of the complex projective plane $PC^2$ to itself. We show a relation of this rational function with an uncountable family of…
We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we…
We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in…
Family of replica matrices, related to general ultrametric spaces with general measures, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed. Replica symmetry breaking…
This paper studies linear classes of planar quasiregular mappings. We give a positive answer to a conjecture of K. Astala, T. Iwaniec, and G. Martin (2009) on reduced Beltrami equations. Moreover, we use it to prove a Wronsky-type theorem…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis.
We study the model theory of the ring of adeles of a number field. We obtain quantifier elimination results in the language of rings and some enrichments. We given consequences for definable subsets of the adeles, and their measures.
This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann-Beenker tilings as…
We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable…
This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A `locally acyclic cluster algebra' is a cluster algebra which admits a finite cover (in a geometric sense)…
``Quasi-elliptic'' functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared…
The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their…
We study a recursively defined two-parameter family of graphs which generalize Fibonacci cubes and Pell graphs and determine their basic structural and enumerative properties. In particular, we show that all of them are induced subgraphs of…
We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex…