We introduce a generalization of sequential compactness using barriers on $\omega $ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, Canad. Math. Bull., 66 (2023), 156–165]. We improve results from [C. Corral and O. Guzmán and C. López-Callejas, High dimensional sequential compactness, Fund. Math.] by building spaces that are ${\mathcal {B}}$-sequentially compact but not ${\mathcal {C}}$-sequentially compact when the barriers ${\mathcal {B}}$ and ${\mathcal {C}}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption ${\mathfrak {b}} ={\mathfrak {c}}$. We also exhibit some classes of spaces that are ${\mathcal {B}}$-sequentially compact for every barrier ${\mathcal {B}}$, including some classical classes of compact spaces from functional analysis, and as a byproduct, we obtain some results on angelic spaces. Finally, we introduce and compute some cardinal invariants naturally associated to barriers.

Geological disposal is the preferred option for the final storage of high-level nuclear waste and spent nuclear fuel in most countries. The selected host rock may be different in individual national programs for radioactive-waste management and the engineered barrier systems that protect and isolate the waste may also differ, but almost all programs are considering an engineered barrier. Clay is used as a buffer that surrounds and protects the individual waste packages and/or as tunnel seal that seals off the disposal galleries from the shafts leading to the surface.

Bentonite and bentonite/sand mixtures are selected primarily because of their low hydraulic permeability in a saturated state. This ensures that diffusion will be the dominant transport mechanism in the barrier. Another key advantage is the swelling pressure, which ensures a self-sealing ability and closes gaps in the installed barrier and the excavation-damaged zone around the emplacement tunnels. Bentonite is a natural geological material that has been stable over timescales of millions of years and this is important as the barriers need to retain their properties for up to 106 y.

In order to be able to license a final repository for high-level radioactive waste, a solid understanding of how the barriers evolve with time is needed. This understanding is based on scientific knowledge about the processes and boundary conditions acting on the barriers in the repository. These are often divided into thermal, hydraulic, mechanical, and (bio)chemical processes. Examples of areas that need to be evaluated are the evolution of temperature in the repository during the early stage due to the decay heat in the waste, re-saturation of the bentonite blocks installed, build-up of swelling pressure on the containers and the surrounding rock, and degradation of the montmorillonite component in the bentonite. Another important area of development is the engineering aspects: how can the barriers be manufactured, subjected to quality control, and installed?

Geological disposal programs for radioactive waste have generated a large body of information on the safety-relevant properties of clays used as engineered barriers. The major relevant findings of the past 35 y are reviewed here.

Bentonite colloids generated from the backfill barrier in nuclear waste repositories may act as radionuclide carriers, if they are stable and mobile. Repository scenarios with highly saline groundwater inhibit colloid stability as particles tend to aggregate but, in the time frame of repositories, groundwater conditions may evolve, promoting particle disaggregation and stabilization. The disaggregation of FEBEX bentonite colloids by fast dilution to lower ionic strength was analysed in this study. Time-resolved dynamic light-scattering experiments were carried out to evaluate the kinetics of bentonite colloid aggregation and disaggregation processes in Na+ and Na+-Ca2+ mixed electrolytes of low ionic strength. Attachment and detachment efficiencies were determined.

Aggregation is promoted by increasing ionic strength, being more efficient in the presence of divalent cations. Once bentonite colloids are aggregated, a decrease in ionic strength facilitates disaggregation, but the process is not fully reversible as the initial size of the stable bentonite colloids at low ionic strength is not fully recovered. Particle-size distribution and concentration in suspension were analysed on disaggregated samples by single particle-counting measurements. Small colloids were measured in the disaggregated samples but their population was smaller than in the initial stable sample, especially in the presence of Ca2+.

Lifelong learning is believed to have physical, social and emotional benefits for older adults. In recognition of this, numerous programmes encouraging learning in later life exist worldwide. One example is the University of the Third Age (U3A) – a lifelong learning co-operative rooted in peer-support and knowledge sharing. This article is based on a collaborative study conducted by university researchers and members of a U3A in North-East England (United Kingdom) investigating the social inclusivity of the group in light of low attendance levels among those from social housing and non-professional backgrounds. A qualitative approach comprising semi-structured interviews and focus groups was adopted to explore knowledge and experience of lifelong learning and the U3A. Sixty individuals aged 50+ were interviewed. The demographic profile of participants largely reflected the socio-economic make-up of the area, with the majority living in areas of high socio-economic deprivation. Several barriers to lifelong learning were revealed, including: poor health, insufficient transport and caring responsibilities. Regarding U3A participation, three exclusionary factors were outlined: lack of knowledge, organisational name and location. Poor comprehension of the purpose and remit of the U3A can result in the development of ‘middle-class' myths regarding membership, perpetuating poor participation rates among lower socio-economic groups. Such perceptions must be dispelled to allow the U3A to fulfil its potential as a highly inclusive organisation.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

Quercus suber L. has been investigated with special attention to the stem bark and its cork formation layer, but excluding the roots. Roots are the location of infection by pathogens such as Phytophthora cinnamomi responsible for the tree's sudden death. It is widely accepted that suberin establishes boundaries within tissues, serves as a barrier against free water and ion passage, and works as a shield against pathogen attacks. We followed the suberization of young secondary roots of cork oak. The first suberin deposition detectable by transmission electron microscopy (TEM) and neutral red (NR) was in the endoderm Casparian strips. Casparian strips are not detected by Sudan red 7B and Fluorol yellow (FY) that specifically stain lamellae suberin. Reaction to Sudan was verified in the endodermis and later on in phellem cells that resulted from the phellogen. Under TEM, the Sudan and FY-stained cells showed clear suberin lamellae while the newer formed phellem cells displayed a distinct NR signal compared to the outermost phellem cells. We concluded that suberin chemical components are arranged differently in the cell wall according to the physiological role or maturation stage of a given tissue.

In this paper we consider a general Lévy process X reflected at a downward periodic barrier At and a constant upper barrier K, giving a process VKt=Xt +LAt−LKt. We find the expression for a loss rate defined by lK=ELK1 and identify its asymptotics as K→∞ when X has light-tailed jumps and EX1<0.

In this work we prove the existence of a solution for a doubly reflected backward SDE with a continuous linearly increasing coefficient in the case where the barriers L and U are such that L < U on [0,T) and there exists a continuous semimartingale between L and U.