217 research outputs found
THE PERIOD OF 2-STEP AND 3-STEP SEQUENCES IN DIRECT PRODUCT OF MONOIDS
Let M and N be two monoids consisting of idempotent elements. By the help of the presentation which defines Mx N, the period of 2-step sequences and 3-step sequences in MxN is given
Fibonacci lengths of all finite p-groups of exponent p²
The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent p and the lengths depend on the Wall number k(p). The p-groups of nilpotency class 3 and exponent p were studied in 2004 also by Dikici. In the paper, we study all p-groups of nilpotency class 3 and exponent p². Thus, we complete the study of Fibonacci lengths of all p-groups of order p⁴ by proving that the Fibonacci length is k(p²).Довжини Фібоначчі скінченних p-rpyn вивчалися Дікічі та співавторами з 1992 року. Всі групи, що розглядалися, були групами експоненти p, а всі довжини залежали від числа Уолла k(p). p-Групи класу нільпотентності 3 i експоненти p були також досліджені Дікічі у 2004 році. У даній статті ми вивчаємо всі p-групи класу нільпотентності 3 і експоненти p². Цим завершується дослідження довжини Фібоначчі всіх p-груп порядку p⁴; при цьому доведено, що довжина Фібоначчі дорівнює k(p²)
Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices
We study various aspects of the dynamics induced by integer matrices on the
invariant rational lattices of the torus in dimension 2 and greater. Firstly,
we investigate the orbit structure when the toral endomorphism is not
invertible on the lattice, characterising the pretails of eventually periodic
orbits. Next we study the nature of the symmetries and reversing symmetries of
toral automorphisms on a given lattice, which has particular relevance to
(quantum) cat maps.Comment: 29 pages, 3 figure
Heisenberg characters, unitriangular groups, and Fibonacci numbers
Let \UT_n(\FF_q) denote the group of unipotent upper triangular
matrices over a finite field with elements. We show that the Heisenberg
characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin
to the line using the steps , which are
labeled in a certain way by nonzero elements of \FF_q. In particular, we
prove for that the number of Heisenberg characters of
\UT_{n+1}(\FF_q) is a polynomial in with nonnegative integer
coefficients and degree , whose leading coefficient is the th Fibonacci
number. Similarly, we find that the number of Heisenberg supercharacters of
\UT_n(\FF_q) is a polynomial in whose coefficients are Delannoy numbers
and whose values give a -analogue for the Pell numbers. By counting the
fixed points of the action of a certain group of linear characters, we prove
that the numbers of supercharacters, irreducible supercharacters, Heisenberg
supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q)
consisting of matrices whose superdiagonal entries sum to zero are likewise all
polynomials in with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor
corrections, final versio
On Linear Differential Equations Involving a Para-Grassmann Variable
As a first step towards a theory of differential equations involving
para-Grassmann variables the linear equations with constant coefficients are
discussed and solutions for equations of low order are given explicitly. A
connection to n-generalized Fibonacci numbers is established. Several other
classes of differential equations (systems of first order, equations with
variable coefficients, nonlinear equations) are also considered and the
analogies or differences to the usual (''bosonic'') differential equations
discussed
On the Periods of 2-Step General Fibonacci Sequences in the Generalized Quaternion Groups
We study 2-step general Fibonacci sequences in the generalized quaternion groups Q4n. In cases where the sequences are proved to be simply periodic, we obtain the periods of 2-step general Fibonacci sequences
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