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1. CMB 2011 (vol 56 pp. 265)
Embedding Distributions of Generalized Fan Graphs Total embedding distributions have been known for a few classes of graphs.
Chen, Gross, and Rieper
computed it for necklaces, close-end ladders and cobblestone
paths. Kwak and Shim computed it for bouquets of circles and
dipoles. In this paper, a splitting theorem is generalized
and the embedding distributions of
generalized fan graphs are obtained.
Keywords:total embedding distribution, splitting theorem, generalized fan graphs Category:05C10 |
2. CMB 2008 (vol 51 pp. 140)
First Variations of the Best Sobolev Trace Constant with Respect to the Domain In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 |
3. CMB 2006 (vol 49 pp. 82)
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$
Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality Categories:26D10, 46E20 |
4. CMB 2005 (vol 48 pp. 622)
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any
hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.
Keywords:variables, Abhyankar--Sathaye Embedding Problem Categories:14R10, 14R25 |
5. CMB 2002 (vol 45 pp. 349)
Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some
algebraically closed field $k$ of characteristic $0$. For an integer
$t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre
twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the
dimension of the space of global sections of $O(t)$, and let $k$ be an
integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$
be general points on $\mathbf{P}^n$ and let $\pi \colon X \to
\mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points.
Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional
divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 -
\cdots -E_k)$ is a very ample invertible sheaf on $X$.
Keywords:blowing-up, projective space, very ample linear system, embeddings, Veronese map Categories:14E25, 14N05, 14N15 |