Search results for "Applied Mathematics"

Optical Routing of Uniform Instances in Cayley Graphs

2001

Abstract Abstract We consider the problem of routing uniform communication instances in Cayley graphs. Such instances consist of all pairs of nodes whose distance is included in a specified set U. We give bounds on the load induced by these instances on the links and for the wavelength assignment problem as well. For some classes of Cayley graphs that have special symmetry property (rotational graphs), we are able to construct routings for uniform instances such that the load is the same for each link of the graph.

Mappings of finite distortion: discreteness and openness for quasi-light mappings

2005

Abstract Let f ∈ W 1 , n ( Ω , R n ) be a continuous mapping so that the components of the preimage of each y ∈ R n are compact. We show that f is open and discrete if | D f ( x ) | n ⩽ K ( x ) J f ( x ) a.e. where K ( x ) ⩾ 1 and K n − 1 / Φ ( log ( e + K ) ) ∈ L 1 ( Ω ) for a function Φ that satisfies ∫ 1 ∞ 1 / Φ ( t ) d t = ∞ and some technical conditions. This divergence condition on Φ is shown to be sharp.

Number of Sylow subgroups in $p$-solvable groups

2003

If G is a finite group and p is a prime number, let vp(G) be the number of Sylow p-subgroups of G. If H is a subgroup of a p-solvable group G, we prove that v p (H) divides v p (G).

ℓ-distant Hamiltonian walks in Cartesian product graphs

2009

Abstract We introduce and study a generalisation of Hamiltonian cycles: an l-distant Hamiltonian walk in a graph G of order n is a cyclic ordering of its vertices in which consecutive vertices are at distance l. Conditions for a Cartesian product graph to possess such an l-distant Hamiltonian walk are given and more specific results are presented concerning toroidal grids.

Group algebras whose units satisfy a group identity

1997

Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the group of units of F G FG . We prove that if U U satisfies a group identity, then F G FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.

The Ptolemy and Zbăganu constants of normed spaces

2010

Abstract In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖ x − y ‖ ‖ z − w ‖ ≤ ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ for any points w , x , y , z in H . It is known that for each normed space ( X , ‖ ⋅ ‖ ) , there exists a constant C such that for any w , x , y , z ∈ X , we have ‖ x − y ‖ ‖ z − w ‖ ≤ C ( ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ ) . The smallest such C is called the Ptolemy constant of X and is denoted by C P ( X ) . We study the relationships between this constant and the geometry of the space X , and hence with metric fix…

Residual 𝑝 properties of mapping class groups and surface groups

2008

Let M ( Σ , P ) \mathcal {M}(\Sigma , \mathcal {P}) be the mapping class group of a punctured oriented surface ( Σ , P ) (\Sigma ,\mathcal {P}) (where P \mathcal {P} may be empty), and let T p ( Σ , P ) \mathcal {T}_p(\Sigma ,\mathcal {P}) be the kernel of the action of M ( Σ , P ) \mathcal {M} (\Sigma , \mathcal {P}) on H 1 ( Σ ∖ P , F p ) H_1(\Sigma \setminus \mathcal {P}, \mathbb {F}_p) . We prove that T p ( Σ , P ) \mathcal {T}_p( \Sigma ,\mathcal {P}) is residually p p . In particular, this shows that M ( Σ , P ) \mathcal {M} (\Sigma ,\mathcal {P}) is virtually residually p p . For a group G G we denote by I p ( G ) \mathcal {I}_p(G) the kernel of the natural action of Out ⁡ ( G ) \ope…

Nullstellen bei Lösungen der Differentialgleichung y(n) + gy(n−1)+ fy = 0

1990

Commensurators of parabolic subgroups of Coxeter groups

1996

Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$ such that $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$. The subgroup $W_X$ can be decomposed in the form $W_X = W_{X^0} \cdot W_{X^\infty} \simeq W_{X^0} \times W_{X^\infty}$ where $W_{X^0}$ is finite and all the irreducible components of $W_{X^\infty}$" > are infinite. Let $Y^\infty$ be the set of $t$ in $S$ such that $m_{s,t}=2$" > for all $s\in X^\i…

The case of equality in the dichotomy of Mohammadi–Oh

2019

If $n \geq 3$ and $\Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of $\mathbf{SO}^o(1,n+1)$ such that $\delta_\Gamma=n-m$ where $m$ is an integer, $1 \leq m \leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $\Gamma$ on the quotient of the frame bundle is $U$-recurrent.