422 research outputs found
On short time existence for the planar network flow
We prove the existence of the flow by curvature of regular planar networks
starting from an initial network which is non-regular. The proof relies on a
monotonicity formula for expanding solutions and a local regularity result for
the network flow in the spirit of B. White's local regularity theorem for mean
curvature flow. We also show a pseudolocality theorem for mean curvature flow
in any codimension, assuming only that the initial submanifold can be locally
written as a graph with sufficiently small Lipschitz constant.Comment: Final version, to appear in Journal of Differential Geometry. 51
page
Entropy and reduced distance for Ricci expanders
Perelman has discovered two integral quantities, the shrinker entropy \cW
and the (backward) reduced volume, that are monotone under the Ricci flow \pa
g_{ij}/\pa t=-2R_{ij} and constant on shrinking solitons. Tweaking some signs,
we find similar formulae corresponding to the expanding case. The {\it
expanding entropy} \ctW is monotone on any compact Ricci flow and constant
precisely on expanders; as in Perelman, it follows from a differential
inequality for a Harnack-like quantity for the conjugate heat equation, and
leads to functionals and . The {\it forward reduced volume}
is monotone in general and constant exactly on expanders.
A natural conjecture asserts that converges as to a
negative Einstein manifold in some weak sense (in particular ignoring
collapsing parts). If the limit is known a-priori to be smooth and compact,
this statement follows easily from any monotone quantity that is constant on
expanders; these include \Vol(g)/t^{n/2} (Hamilton) and
(Perelman), as well as our new quantities. In general, we show that if
\Vol(g) grows like (maximal volume growth) then \ctW,
and remain bounded (in their appropriate ways) for all time. We
attempt a sharp formulation of the conjecture
Ohutfilmihaiduttimen mallintaminen ionisen nesteen kierrÀttÀmiseksi
World demand for textiles is on the rise and there is a need for fiber source that does not require arable land. Ioncell-F is a novel method of producing fiber from pulp. It uses ionic liquid [DBNH][OAc] to dissolve cellulose. Ionic liquid is expensive so it has to be recycled to make the process economical. Ionic liquid is thermally unstable, therefore the recycling has to be done with moderate temperatures. One way to separate and recycle ionic liquid from water is thin film evaporation.
The purpose of this study was to model the evaporation of water from water/[DBNH][OAc] mixture in an agitated thin film evaporator in flowsheet simulator Aspen Plus. Accurate modeling of the evaporator is needed to design and optimize the recycling process. The study also studied the applicability of multiple-effect evaporation.
Various modeling approaches were studied to simulate the thin film evaporator and an Aspen Plus model was developed based on batch distillation theory. The performance of the model was compared to an earlier developed model based on flash drum model and experimental data. The batch model gave more accurate results than the often used flash model. The model did not include the hydrolysis product of the ionic liquid, implementation of which should be the focus in future works. A case study was conducted and the applicability of double-effect evaporation was tested with the model. A rapid boiling point elevation at low water concentrations made it harder to implement multiple-effect evaporation in recycling of the ionic liquid. It could be done with right pressure and temperature settings for most of the evaporation, with one additional evaporator to achieve desired water content. Multiple-effect evaporation proved useful in both achieving purer vapor and lowering the total required heating power.Maailmanlaajuinen tekstiilin kysyntÀ on kasvussa, ja tekstiilin lÀhteeksi tarvitaan raaka-aineita, jotka eivÀt vie tilaa viljeltÀvÀltÀ maalta. Ioncell-F on uusi prosessi, joka voi tuottaa kuitua sellusta. Se kÀyttÀÀ ionista nestettÀ [DBNH][OAc] liuottaakseen selluloosan. Ioninen neste on kallista, joten se tÀytyy erottaa vedestÀ ja kierrÀttÀÀ, jotta prosessi on kannattava. Se kuitenkin myös hajoaa korkeissa lÀmpötiloissa, joten erotus tÀytyy tehdÀ matalissa lÀmpötiloissa. Yksi keino tÀhÀn on ohutfilmihaihdutin.
TÀmÀn työn tarkoituksena oli mallintaa veden haihdutusta veden ja ionisen nesteen seoksesta ohutfilmihaihduttimessa Aspen-Plus ohjelmalla. Haihduttimen tarkka mallinnus on tÀrkeÀÀ kierrÀtysprosessin suunnittelemiseksi ja optimoinniksi. TyössÀ myös selvitettiin, olisiko monivaihelauhdutus mahdollista.
Erilaisia malllinnusvaihtoehtoja kÀsiteltiin ja panostislaukseen pohjautuva malli valittiin keskittymispohjaksi. Aspen Plus malli kehitettiin panostislausteorian pohjalta, ja sen tuloksia verrattiin aiemmin kehitettyyn flash-malliin sekÀ kokeellisiin tuloksiin. Panostislausmallin tulokset olivat lÀhempÀnÀ kokeellisia, kuin flash-malli. Kummastakin mallista kuitenkin puuttuu ionisen nesteen hydrolyysituote, mikÀ tuo epÀtarkkuutta mallinnukseen. Sen sisÀltÀminen mallinnukseen tulisi olla seuraavien tutkimusten kohde. Panostislausmallille tehtiin tapaustutkimus ja monivaihelauhdutuksen sisÀllyttÀmisen mahdollisuutta haihdutusprosessiin arvioitiin. Kiehumispisteen kohoama vÀhÀisissÀ vesipitoisuuksissa on nopeaa, mikÀ vaikeuttaa monivaihelauhdutuksen kÀyttÀmistÀ. Oikeilla paineilla ja lÀmpötiloilla se saatiin kuitenkin sisÀllytettyÀ, jos viimeisen haihduttimen lÀmmittÀmiseen kÀytettiin ulkoista energiaa. Monivaihelauhdutus sÀÀsti selvÀsti lÀmmitysenergiaa sekÀ auttoi kierrÀtystÀ jakamalla höyryn suureen virtaan erittÀin puhdasta vesihöyryÀ ja pienempÀÀn virtaan, jossa lÀhes kaikki höyrystynyt ioninen neste oli
Rigidity of generic singularities of mean curvature flow
Shrinkers are special solutions of mean curvature flow (MCF) that evolve by
rescaling and model the singularities. While there are infinitely many in each
dimension, [CM1] showed that the only generic are round cylinders \SS^k\times
\RR^{n-k}. We prove here that round cylinders are rigid in a very strong
sense. Namely, any other shrinker that is sufficiently close to one of them on
a large, but compact, set must itself be a round cylinder.
To our knowledge, this is the first general rigidity theorem for
singularities of a nonlinear geometric flow. We expect that the techniques and
ideas developed here have applications to other flows.
Our results hold in all dimensions and do not require any a priori
smoothness.Comment: revised after acceptance for Publications IHE
Topological Change in Mean Convex Mean Curvature Flow
Consider the mean curvature flow of an (n+1)-dimensional, compact, mean
convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We
prove that elements of the m-th homotopy group of the complementary region can
die only if there is a shrinking S^k x R^(n-k) singularity for some k less than
or equal to m. We also prove that for each m from 1 to n, there is a nonempty
open set of compact, mean convex regions K in R^(n+1) with smooth boundary for
which the resulting mean curvature flow has a shrinking S^m x R^(n-m)
singularity.Comment: 19 pages. This version includes a new section proving that certain
kinds of mean curvature flow singularities persist under arbitrary small
perturbations of the initial surface. Newest update (Oct 2013) fixes some
bibliographic reference
The round sphere minimizes entropy among closed self-shrinkers
The entropy of a hypersurface is a geometric invariant that measures
complexity and is invariant under rigid motions and dilations. It is given by
the supremum over all Gaussian integrals with varying centers and scales. It is
monotone under mean curvature flow, thus giving a Lyapunov functional.
Therefore, the entropy of the initial hypersurface bounds the entropy at all
future singularities. We show here that not only does the round sphere have the
lowest entropy of any closed singularity, but there is a gap to the second
lowest
Higher regularity of the inverse mean curvature flow
We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times, beginning from the first time where a surface in the evolution is star-shaped
The mean curvature at the first singular time of the mean curvature flow
Consider a family of smooth immersions
of closed hypersurfaces in moving by the mean curvature flow
, for .
We prove that the mean curvature blows up at the first singular time if all
singularities are of type I. In the case , regardless of the type of a
possibly forming singularity, we show that at the first singular time the mean
curvature necessarily blows up provided that either the Multiplicity One
Conjecture holds or the Gaussian density is less than two. We also establish
and give several applications of a local regularity theorem which is a
parabolic analogue of Choi-Schoen estimate for minimal submanifolds
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