Search results for "R0"
showing 10 items of 62 documents
Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups
2020
This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that…
Vertical versus horizontal Sobolev spaces
2020
Let $\alpha \geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2\alpha}(\mathbb{H}^{n})$, then $\varphi f$ belongs to the Euclidean Sobolev space $S^{p}_{\alpha}(\mathbb{R}^{2n + 1})$ for any test function $\varphi$. In short, $S^{p}_{2\alpha}(\mathbb{H}^{n}) \subset S^{p}_{\alpha,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2\alpha}^{p}(\mathbb{H}^{n})$ is continuously contained in the vertical Sobolev sp…
Embeddings of a family of Danielewski hypersurfaces and certain \C^+-actions on \C^3
2006
International audience; We consider the family of complex polynomials in \C[x,y,z] of the form x^2y-z^2-xq(x,z). Two such polynomials P_1 and P_2 are equivalent if there is an automorphism \varphi of \C[x,y,z] such that \varphi(P_1)=P_2. We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category.
Prognostic Implications of the Standardized Study of Resection Margins in Pancreatic Cancers
2013
Abstract Introduction Involvement of surgical resection margins is a fundamental prognostic factor in pancreatic oncological surgery. However, there is a lack of standardized histopathology definition. The aims of this study are to investigate the real rate of R1 resections when surgical specimens are evaluated according to a standardized protocol and to study its survival implications. Patients and methods One hundred consecutive surgically resected patients with pancreatic ductal adenocarcinoma were included in the study. They were further divided into 2 groups: pre-protocol, evaluated before the introduction of the standardized protocol and post-protocol, analyzed with the standardized p…
Treatment of Metachronous and Simultaneous Liver Metastases of Pancreatic Cancer
2009
<i>Aim:</i> Patients were analyzed who underwent treatment of liver metastases from pancreatic cancer. <i>Methods:</i> Selection criteria were the possibility of R0 resection of the primary and/or the liver metastases, no other sites of metastases, and the presentation of liver metastases. A comparison of treatment by surgery versus chemotherapy regarding overall survival and disease-free interval was performed. <i>Results:</i> Between 1996 and 2008, a total number of 23 patients were retrospectively identified from a prospective database of 193 cases of pancreatic cancer. In 14 cases, liver metastases were found simultaneously, and in 9 cases metachronou…
Local excision for more advanced rectal tumors
2008
Over the past 20 years, local excision (LE) of T1 rectal cancer was increasingly established and represents an oncologically established technique. In contrast, the situation for T2 tumors is less clear and has only been investigated in small patient collectives. LE for T2 tumors is thus discussed controversially.In addition to our own patients with T2 rectal cancer treated locally (n=40), we have analysed the local recurrence (LR) rates after LE alone (n=124), after immediate conventional radical reoperation (n=29), after adjuvant (chemo)-radiotherapy (n=294) and those after neoadjuvant chemoradiotherapy (nCRT) (n=269) using a PubMed search.LR rates of low-grade T2 tumors after R0 resectio…
Existence of dynamical low-rank approximations to parabolic problems
2021
The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions.
Energy dissipative solutions to the Kobayashi-Warren-Carter system
2017
In this paper we study a variational system of two parabolic PDEs, called the Kobayashi-Warren-Carter system, which models the grain boundary motion in a polycrystal. The focus of the study is the existence of solutions to this system which dissipate the associated energy functional. We obtain existence of this type of solutions via a suitable approximation of the energy functional with Laplacians and an extra regularization of the weighted total variation term of the energy. As a byproduct of this result, we also prove some $\Gamma$-convergence results concerning weighted total variations and the corresponding time-dependent cases. Finally, the regularity obtained for the solutions togethe…
Regular 1-harmonic flow
2017
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some …
A degenerating convection-diffusion system modelling froth flotation with drainage
2022
Abstract Froth flotation is a common unit operation used in mineral processing. It serves to separate valuable mineral particles from worthless gangue particles in finely ground ores. The valuable mineral particles are hydrophobic and attach to bubbles of air injected into the pulp. This creates bubble-particle aggregates that rise to the top of the flotation column where they accumulate to a froth or foam layer that is removed through a launder for further processing. At the same time, the hydrophilic gangue particles settle and are removed continuously. The drainage of liquid due to capillarity is essential for the formation of a stable froth layer. This effect is included into a previous…