Search results for "Combinatorics"

Linear Diophantine Problems

1996

The Frobenius number g(A k ) Let A k \({A_k} = \{ {a_1},...,{a_k}\}\subset\) IN with gcd(A k ) = 1, n\( \in I{N_0}.\) If $$n = \sum\limits_{i = 1}^k {{x_i}{a_i},{x_i}}\in I{N_0}$$ (1) we call this a representation or a g-representation of n by Ak (in order to distinguish between several types of representations that will be considered in the sequel). Then the Frobenius number g(A k ) is the greatest integer with no g-representation.

Overlap free words on two symbols

1985

A Star-Variety With Almost Polynomial Growth

2000

Abstract Let F be a field of characteristic zero. In this paper we construct a finite dimensional F -algebra with involution M and we study its ∗ -polynomial identities; on one hand we determine a generator of the corresponding T -ideal of the free algebra with involution and on the other we give a complete description of the multilinear ∗ -identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the ∗ -variety generated by M , var( M , ∗ ) has almost polynomial growth, i.e., the sequence of ∗ -codimensions of M cannot be bounded by any polynomial function but any proper ∗ -subvariety of var( M , ∗ ) has polynomial growth. If G 2 is…

Involution codimensions and trace codimensions of matrices are asymptotically equal

1996

We calculate the asymptotic growth oft n (M p (F),*) andc n (M p (F),*), the trace and ordinary *-codimensions ofp×p matrices with involution. To do this we first calculate the asymptotic growth oft n and then show thatc n ⋍t n .

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…

Berechenbare Invarianten und Elementare Begründung: Kurt Reidemeister

1999

Nach dem ersten Weltkrieg muste der Faden der mathematischen Beschaftigung mit Knoten neu aufgenommen werden. Wie im letzten Kapitel bereits erwahnt, waren es dabei zunachst Dehns Arbeiten, die von den Mathematikern wahrgenommen wurden. Ein charakteristisches Zeugnis dafur, wie weitgehend unsichtbar die Wiener Beitrage zu geworden waren, gibt ein knapper Kommentar Oswald Veblens, der zu den ersten Mathematikern in den Vereinigten Staaten zahlte, die sich ernsthaft fur die Topologie zu interessieren begannen. In seinen 1922 als Buch erschienenen Cambridge Colloquium Lectures on Analysis Situs schrieb Oswald Veblen zum Thema der Knoten: „A large number of types of knots have been described by…

A knot without tritangent planes

1991

We show, with computations aided by a computer, that the (3,2)-curve on some standard torus (which topologically is the trefoil knot) has no tritangent planes, thus answering in the negative a conjecture of M. H. Freedman.

A knot without triple bitangency

1997

It is proved that certain trefoil knot has not triple bitangency, answering thus in the negative a conjecture of S. Izumiya and W. L. Marar.

ON THE INDEX OF VECTOR FIELDS TANGENT TO HYPERSURFACES WITH NON-ISOLATED SINGULARITIES

2002

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kahler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field $X:=\tilde{X}|_V$ on $V$ : \begin{equa…

On Fine and Wilf's theorem for bidimensional words

2003

AbstractGeneralizations of Fine and Wilf's Periodicity Theorem are obtained for the case of bidimensional words using geometric arguments. The domains considered constitute a large class of convex subsets of R2 which include most parallelograms. A complete discussion is provided for the parallelogram case.