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The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
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In pharmacokinetics, the elimination rate of a drug following a capacity-limited model is primarily controlled by two parameters: Vmax and KM. These parameters are crucial in how the drug behaves inside the body after administration.
Following the administration of a single intravenous (IV) bolus injection, we can determine the concentration of the drug in the plasma at any given time. This calculation is achieved using a specific equation that integrates the values of Vmax and KM.
We can also...
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Quadratic Equations01:29

Quadratic Equations

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A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
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Two-Compartment Open Model: IV Bolus Administration01:18

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The two-compartment model for intravenous (IV) bolus administration illustrates drug distribution in the body, subdividing it into central and peripheral compartments. This model operates on the concept of two-compartment kinetics. The drug's plasma concentration shows a bi-exponential decline following IV bolus administration, signaling the presence of two disposition processes: distribution and elimination.
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One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation01:24

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This lesson introduces two critical methods in pharmacokinetics, the Wagner-Nelson and Loo-Riegelman methods, used for estimating the absorption rate constant (ka) for drugs administered via non-intravenous routes. The Wagner-Nelson method relates ka to the plasma concentration derived from the slope of a semilog percent unabsorbed time plot. However, it is limited to drugs with one-compartment kinetics and can be impacted by factors like gastrointestinal motility or enzymatic degradation.
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Extraction: Effects of pH00:53

Extraction: Effects of pH

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Consider a neutral form of an amine, B, with a partition coefficient, K, in a liquid mixture containing organic and aqueous phases. The pH of the aqueous phase affects the charge on acidic and basic solutes, and the charged form is usually more soluble in the aqueous phase. Suppose the conjugate acid form of the amine is soluble only in the aqueous phase while the base form is soluble in both phases. Then the distribution coefficient, D, can be given as the ratio of amine concentration in the...
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KdV solves BKP.

Alexander Alexandrov1

  • 1Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Korea alexandrovsash@gmail.com.

Proceedings of the National Academy of Sciences of the United States of America
|June 16, 2021
PubMed
Summary
This summary is machine-generated.

Any tau-function from the Korteweg-de Vries (KdV) hierarchy can solve the type B Kadomtsev-Petviashvili (BKP) hierarchy. This is achieved through a straightforward rescaling of the time variables.

Keywords:
BKP hierarchyHirota bilinear identityKdV hierarchyτ-functions

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Area of Science:

  • Integrable systems
  • Mathematical physics
  • Nonlinear partial differential equations

Background:

  • The Korteweg-de Vries (KdV) hierarchy and the type B Kadomtsev-Petviashvili (BKP) hierarchy are significant integrable systems in mathematical physics.
  • Tau-functions are fundamental objects in the study of these hierarchies, encoding their solutions.

Purpose of the Study:

  • To establish a direct relationship between solutions of the KdV hierarchy and the BKP hierarchy.
  • To demonstrate that solutions of one system can be transformed into solutions of the other.

Main Methods:

  • Utilizing the properties of tau-functions within the framework of integrable hierarchies.
  • Applying a specific time rescaling transformation to KdV tau-functions.

Main Results:

  • We prove that any tau-function of the Korteweg-de Vries hierarchy is also a solution to the type B Kadomtsev-Petviashvili hierarchy.
  • The transformation involves a simple rescaling of the time variables.

Conclusions:

  • This finding provides a unifying perspective on the KdV and BKP hierarchies.
  • It simplifies the study of BKP solutions by leveraging known KdV solutions and transformations.