Gel filtration chromatography

Gel-filtration chromatography is a popular and versatile technique that permits the effective separation of proteins and other biological molecules in high yield. Here, the basis of the method is described and typical matrix types are contrasted. The selection of suitable operating conditions and applications of the method are also discussed

Linear-Size Approximations to the Vietoris-Rips Filtration

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees

Precoat filtration with body-feed and variable pressure. Part I: Mathematical modelling

The precoat filtration with body-feed is an unit operation of agricultural and food engineering. Mostly it is implemented by using centrifugal pump, which pump curve has a partial horizontal trend. Classically, in filtration theory, this prerogative of the centrifugal pumps leads to the simplifying assumption that filtration occurs with constant pressure. Because of this, it is easy to integrate the Darcy\u2019s differential equation [1, 2 and 3] for the precoat filtration with body-feed, obtaining the well known Carman equation [4]. This is the equation which relates the filtration time with the filtrate volume, the operating pressure, the filter area, and the solid-liquid suspension characteristics. The Carman equation is the start point for the subsequent optimization of the filtration cycles, e.g. by establishing the relationship between the filtration time and the filter cleaning time [5]. A better optimization of the precoat filtration with body-feed could be obtain, with some economic benefits, if an integration of the Darcy ODE was developed starting from actual trend of the pressure produced by the centrifugal pump, that is if a variable pressure was considered, as expected from the pump curve. In this sense a proposal was done by Tiller and Crump [6] many years ago in accordance with a graphic method of integration of the Darcy ODE. However the graphic procedure is tedious since it is iterative and not computerizable. For this reason the aim of this work was to find an analytical solution to the Darcy ODE for the filtration with variable pressure in order to obtain a quick and easy-to-use equation for the subsequent optimization calculations of filtration cycles, even if more complex of the Carman equation

Alternative versions of the Johnson homomorphisms and the LMO functor

Let $\Sigma$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $\Sigma$. By considering the action of $\mathcal{M}$ on the fundamental group of $\Sigma$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $\Sigma$. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of $3$-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.Comment: 62 pages, several figures. v_2 minor change