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In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different ...
In a weak formulation, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.
A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. The Yamada–Watanabe theorem makes a connection between the two. An important example is the equation for geometric Brownian motion
Weak bases and weak acids are generally weak electrolytes. In an aqueous solution there will be some CH 3 COOH and some CH 3 COO − and H +. A strong electrolyte is a solute that exists in solution completely or nearly completely as ions. Again, the strength of an electrolyte is defined as the percentage of solute that is ions, rather than ...
A weak base is one which does not fully ionize in an aqueous solution, or in which protonation is incomplete. For example, ammonia transfers a proton to water according to the equation [11] NH 3 (aq) + H 2 O(l) → NH + 4 (aq) + OH − (aq)
Also acid ionization constant or acidity constant. A quantitative measure of the strength of an acid in solution expressed as an equilibrium constant for a chemical dissociation reaction in the context of acid-base reactions. It is often given as its base-10 cologarithm, p K a. acid–base extraction A chemical reaction in which chemical species are separated from other acids and bases. acid ...
A weak base is a base that, upon dissolution in water, does not dissociate completely, so that the resulting aqueous solution contains only a small proportion of hydroxide ions and the concerned basic radical, and a large proportion of undissociated molecules of the base.
Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. [1] This fact is called weak duality.