On proper branched coverings and a question of Vuorinen

Polymers are classified as linear or branched on the basis of their chain architecture. The polymer chains in linear polymers have a long chain-like structure with minimal to no branching at all. Even if a polymer features large substituent groups on the monomer, which appear as branches to the skeleton, it is not considered a branched polymer. A branched polymer contains secondary polymer chains that arise from the main polymer chain. The branching occurs when the polymer growth shifts from...

The connective tissues have different properties and functions in the human body. They are broadly categorized into proper, supporting, or fluid connective tissues.
Connective Tissue Proper
Connective tissue proper is the most abundant class of connective tissues. As its name implies, it predominantly connects different tissues in the body. Depending on the cell types, ground substance, viscosity, and fiber types in the ECM, connective tissue proper is further categorized into loose and dense....

Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...

Bones have various surface features that help form joints and attach to other soft tissues. Depending on the function, bone markings are categorized into articulating projections, processes for attachment, depressions, and openings.
Articulating Projections
Articulating projections are found where two bones meet to form a joint. These structures are usually found at the ends of bones. The largest articulation is a rounded projection called the head, supported by a narrow neck at the ends of...

The protrusion of the cell surface is an initial step for several cellular processes, including cell migration, phagocytosis, and neurite outgrowth. These membrane protrusions are a result of cytoskeletal rearrangement. The most  widely observed cell protrusions include lamellipodia, pseudopodia, filopodia, microvilli, invadopodia, and podosomes. These protrusions can be of two types — static or dynamic.
The microvilli, an example of stable protrusions, are finger-like projections...

The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the...