Search results for " function"

A bound on the p-length of p-solvable groups

2013

Let G be a finite p-solvable group and P a Sylow p-subgroup of G. Suppose that $\gamma_{l(p-1)}(P)\subseteq \gamma_r(P)^{p^s}$ for $l(p-1)<r+s(p-1)$, then the p-length is bounded by a function depending on l.

Hausdorff measures and dimension

1995

Hölder inequality for functions that are integrable with respect to bilinear maps

2008

Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1\le p&lt;\infty$, $X$ be a Banach space $X$ and $B:X\times Y \to Z$ be a bounded bilinear map. We say that an $X$-valued function $f$ is $p$-integrable with respect to $B$ whenever $\sup_{\|y\|=1} \int_\Omega \|B(f(w),y)\|^p\,d\mu&lt;\infty$. We get an analogue to Hölder's inequality in this setting.

Unions of identifiable classes of total recursive functions

1992

J.Barzdin [Bar74] has proved that there are classes of total recursive functions which are EX-identifiable but their union is not. We prove that there are no 3 classes U1, U2, U3 such that U1∪U2,U1∪U3 and U2∪U3 would be in EX but U1∪U2∪U3∉ EX. For FIN-identification there are 3 classes with the above-mentioned property and there are no 4 classes U1, U2, U3, U4 such that all 4 unions of triples of these classes would be identifiable but the union of all 4 classes would not. For identification with no more than p minchanges a (2p+2−1)-tuple of such classes do exist but there is no (2p+2)-tuple with the above-mentioned properly.

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…

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…

Size of Sets with Small Sensitivity: A Generalization of Simon’s Lemma

2015

We study the structure of sets \(S\subseteq \{0, 1\}^n\) with small sensitivity. The well-known Simon’s lemma says that any \(S\subseteq \{0, 1\}^n\) of sensitivity \(s\) must be of size at least \(2^{n-s}\). This result has been useful for proving lower bounds on the sensitivity of Boolean functions, with applications to the theory of parallel computing and the “sensitivity vs. block sensitivity” conjecture.

Operators on PIP-Spaces and Indexed PIP-Spaces

2009

As already mentioned, the basic idea of pip-spaces is that vectors should not be considered individually, but only in terms of the subspaces V r (r Є F), the building blocks of the structure. Correspondingly, an operator on a pipspace should be defined in terms of assaying subspaces only, with the proviso that only continuous or bounded operators are allowed. Thus an operator is a coherent collection of continuous operators. We recall that in a nondegenerate pip-space, every assaying subspace V r carries its Mackey topology \(\tau (V_r , V \bar{r})\) and thus its dual is \(V \bar{r}\). This applies in particular to \(V^{\#}\) and V itself. For simplicity, a continuous linear map between two…

Wedge filling and interface delocalization in finite Ising lattices with antisymmetric surface fields

2003

Theoretical predictions by Parry et al. for wetting phenomena in a wedge geometry are tested by Monte Carlo simulations. Simple cubic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ Ising lattices with nearest neighbor ferromagnetic exchange and four free $L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ surfaces, at which antisymmetric surface fields $\ifmmode\pm\else\textpm\fi{}{H}_{s}$ act, are studied for a wide range of linear dimensions $(4l~Ll~320,30l~{L}_{y}l~1000),$ in an attempt to clarify finite size effects on the wedge filling transition in this ``double-wedge'' geometry. Interpreting the Ising model as a lattice gas, the problem is equivalent to a li…

The complex of words and Nakaoka stability

2005

We give a new simple proof of the exactness of the complex of injective words and use it to prove Nakaoka's homology stability for symmetric groups. The methods are generalized to show acyclicity in low degrees for the complex of words in "general position". Hm(§ni1;Z) = Hm(§n;Z) for n=2 > m where §n denotes the permutation group of n elements. An elementary proof of this fact has not been available in the literature. In the first section the complex C⁄(m) of abelian groups is studied which in de- gree n is freely generated by injective words of length n. The alphabet consists of m letters. The complex C⁄(m) has the only non vanishing homology in degree m (Theorem 1). This is a result of F.…