CONTROL OF FINGERING INSTABILITY BY VIBRATIONS

In applications involving the injection of a fluid in a porous medium to displace another fluid, a main objective is the maximization of the displacement efficiency. Displacement fronts moving in porous media are subjected to hydrodynamic instability when a liquid of low viscosity displaces a high-viscosity liquid and consequently finger-like structure forms along the interface. This finger instability is usually undesirable in technical applications and natural filtration processes. We discuss the external periodic forcing as one of the promising ways to control the instability and perform numerical simulation of an initially spherical drop in a porous media under vertical vibrations. The drop is favorable object to study since in this case one can observe the effect of vibrations on fluid interface domains inclined by different angles with respect to vibration axis. It is shown that under vibrations small-scale perturbations of interface are suppressed and in the case of vibrations of large enough intensity the drop becomes stable. The stability criterion is derived.

Path homology theory of edge-colored graphs

In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

On a Lévy process pinned at random time

In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.

Units and augmentation powers in integral group rings

The augmentation powers in an integral group ring {\mathbb{Z}G} induce a natural filtration of the unit group of {\mathbb{Z}G} analogous to the filtration of the group G given by its dimension series {\{D_{n}(G)\}_{n\geq 1}}. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

Green method of stemming the tide of invasive marine and freshwater organisms by natural filtration of shipping ballast water

Marine and freshwater pollution caused by transport of invasive species in shipping ballast water is a major global problem and will increase in magnitude as shipping of commodities increases in the future. An economical method to preclude biological organisms in the seawater used for ballast is to exclude them at the source port. Integrated natural filtration using onshore wells or seabed gallery systems has been thoroughly investigated for use as pretreatment for seawater desalination systems and has proven to be environmentally acceptable and economic. Thus, the use of this proven filtration technology to another issue, ballast water treatment, is an innovative method of providing marine organism free seawater by non-destructive means in port-based facilities. This method is ecosystem-friendly in that no chemicals or destructive processes are used. Design and construction of well or seabed gallery intake systems for production of ballast seawater are feasible in virtually all global port facilities.

Bridges with Random Length: Gamma Case

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.