Finite Voronoi decompositions of infinite vertex transitive graphs

In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of covering, which we use to show that s(G) is always at least two. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of the s(G) is. We show that all vertex transitive graphs with polynomial growth have a finite s(G); vertex transitive graphs with infinitely many ends have an infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has a finite s(G); and the lamplighter graph LL(Z^2), which is Liouville, has an infinite s(G)

Russia-Ukrainian War 2022: Battle of Hostomel

On February 24, 2022, the Russian Federation began the active phase of a full-scale invasion of the territory of sovereign Ukraine. This invasion was preceded by the recognition of the independence of the quasi-republics in the occupied territories of the Donetsk and Luhansk regions (located in the east of Ukraine) and the introduction of a large group of troops into their territory. The invasion began with a massive missile and air strike against critical infrastructure, military units, civilian facilities and air defense positions. At the same time, the ground troops of the Russian Federation crossed the border in all border areas in the north and east of Ukraine. The invasion army consisted of unofficial figures of approximately 200 - 250 thousand soldiers. The main task was blocking large cities, bypassing defense points and advancing into central Ukraine. Along with the ground operation, an airborne operation began. Elite units of the Airborne Forces and special operations forces carried out landings in the area of the Hostomel airport, which is 30 kilometers from Kyiv, the capital of Ukraine. However, the Russians came under massive fire, the landing force clashed with the Armed Forces of Ukraine, and Ukrainian planes appeared in the air. The situation was developing, but the ground troops from Belarus still managed to come to the aid of the landing party and consolidate their positions. Despite this, the aggressor lost precious time to cover Kyiv and seize it. Having demonstrated a series of tactical errors that almost led to the failure of the amphibious operation. In general, the battle for the Gostomel airport is rich in examples of troop landings and useful experience that can be used for further planning of similar operations. The final figures of the number of units involved are still unknown and are classified. However, it is subject to study and discussion regarding the analysis of the actions of the aggressor in the face of the Russian state

Transmission Robin problem for singular p(x)-Laplacian equation in a cone

We study the behavior near the boundary angular or conical point of weak solutions to the transmission Robin problem for an elliptic quasi-linear second-order equation with the variable p(x)-Laplacian

The Peculiarities of the Functioning of Publicism in the Conditions of the 20s – 30s of the 20th Century on the Territory of Brest Region

Окреслена роль періодичних видань у вихованні політичної і національної свідомості мешканців Берестейщини. Показано, що об’єктивне освітлення подій на території БРСР викликало у західняків різне ставлення до Радянської влади і до боротьби проти колоніального пригнічення. Береться до уваги релігія, яка сприяла знищенню духовної культури народів . A rewiew of the role of periodicals in fostering the political and national consciousness of Brest region residents. An objective interpretation of events on the territory of the BSSR which aroused various feelings towards the Soviet Government and struggle against colonial oppression is given. Special attention is devoted to religion which favoured the preservation of the integrity of the nations’ cultural wealth

Hope and Explanatory Style: A Study of Outcomes in Inpatient Headache Sufferers

Chronic daily headache (CDH) imposes significant burden on sufferers, among which are impaired quality of life, physical suffering, and financial cost. Psychological factors are known to play a considerable role in headache onset, maintenance, progression, and treatment outcome. Furthermore, headache-related disability cannot be accounted for by biological factors alone; both headache symptoms and psychological variables contribute to headache-related disability (Nicholson et al., 2007). Hope is an important predictor of improved health-related quality of life and has been linked with better outcomes for many diseases, disorders, and also in managing pain. Explanatory style, an individual’s style of making sense out of life events, has been associated with a variety of positive health outcomes. At this time, no research exists linking hope and explanatory style with treatment outcomes for CDH. Findings indicate that hope, as assessed through the Herth Hope Index (HHI), and an optimistic explanatory style, as assessed through the Attributional Style Questionnaire (ASQ), are closely related constructs. In this study greater hope predicted reduced headache impact for inpatient headache sufferers. Implementation of a hope instilling intervention could be helpful in reducing headache impact for those with CDH

The Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent

In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m(x)-Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere