392 research outputs found
Combinatorics of multiboundary singularities B_n^l and Bernoulli-Euler numbers
Consider generalizations of the boundary singularities B_n of the functions
on the real line to the case where the boundary consists of a finite number of
l points. These singularities B_n^l could also arise in higher dimensional
case, when the boundary is an immersed hypersurface. We obtain a particular
recurrent equation on the numbers of connected components of very nice
M-morsification spaces of the multiboundary singularities B_n^l. This helps us
to express the numbers K_n^l (for l=2,3,4...) by Bernoulli-Euler numbers. We
also find the corresponding generating functions.Comment: 4 pages, 1 figur
Three examples of three-dimensional continued fractions in the sense of Klein
The problem of investigation of the simplest n-dimensional continued fraction
in the sense of Klein for n>2 was posed by V.Arnold. The answer for the case of
n=2 can be found in the works of E.Korkina and G.Lachaud. In present work we
study the case of n=3
Elementary notions of lattice trigonometry
In this paper we study properties of lattice trigonometric functions of
lattice angles in lattice geometry. We introduce the definition of sums of
lattice angles and establish a necessary and sufficient condition for three
angles to be the angles of some lattice triangle in terms of lattice tangents.
This condition is a version of the Euclidean condition: three angles are the
angles of some triangle iff their sum equals \pi. Further we find the necessary
and sufficient condition for an ordered n-tuple of angles to be the angles of
some convex lattice polygon. In conclusion we show applications to theory of
complex projective toric varieties, and a list of unsolved problems and
questions.Comment: 49 pages; 16 figure
Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
In this paper we develop an integer-affine classification of
three-dimensional multistory completely empty convex marked pyramids. We apply
it to obtain the complete lists of compact two-dimensional faces of
multidimensional continued fractions lying in planes with integer distances to
the origin equal 2, 3, 4 ... The faces are considered up to the action of the
group of integer-linear transformations. In conclusion we formulate some actual
unsolved problems associated with the generalizations for n-dimensional faces
and more complicated face configurations.Comment: Minor change
On tori triangulations associated with two-dimensional continued fractions of cubic irrationalities
We show several properties related to the structure of the family of classes
of two-dimensional periodic continued fractions. This approach to the study of
the family of classes of nonequivalent two dimexsional periodic continued
fractions leads to the visualization of special subfamilies of continued
fractions with torus triangulations (i.e. combinatorics of their fundamental
domains) that possess explicit regularities.Several cases of such subfamilies
are studied in detail; the method to construct other similar subfamilies is
given
On two-dimensional continued fractions for the integer hyperbolic matrices with small norm
In this note we classify two-dimensional continued fractions for cubic
irrationalities constructed by matrices with not large norm (). The
classification is based on the following new result: the class of matrices with
an irreducible characteristic polynomial over the field of rational numbers is
the class of matrices of frobenius type iff there exists an integer solution
for a certain equation with integer coefficients
- …