Notes on the Deficiency-One Theorem: multiple linkage classes

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...

Allosteric proteins have more than one ligand binding site; the binding of a ligand to any of these sites influences the binding of ligands to the other sites. When a protein is allosteric, its binding sites are called coupled or linked.  In the case of enzymes, the site that binds to the substrate is known as the active site and the other site is known as the regulatory site. When a ligand binds to the regulatory site, this leads to conformational changes in the protein that can influence the...

Allosteric proteins have more than one ligand binding site; the binding of a ligand to any of these sites influences the binding of ligands to the other sites. When a protein is allosteric, its binding sites are called coupled or linked.  In the case of enzymes, the site that binds to the substrate is known as the active site and the other site is known as the regulatory site. When a ligand binds to the regulatory site, this leads to conformational changes in the protein that can influence the...

In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...

The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete Factorization...

In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...