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A double bubble. Note that the surface separating the small lower bubble from the large bubble bulges into the large bubble. In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a ...
The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting. An equilibrium solution f e {\displaystyle f_{e}} to an autonomous system of first order ordinary differential equations is called:
The pressure (P) inside of the gas bubble continues to increase and the maximum value is obtained when the bubble has the completely hemispherical shape whose radius is exactly corresponding to the radius of the capillary. [3] Figure 2 shows each step of bubble formation and corresponding change of bubble radius and each step is described below.
A vacuum is defined as a space with as little energy in it as possible. Despite the name, the vacuum still has quantum fields.A true vacuum is stable because it is at a global minimum of energy, and is commonly assumed to coincide with the physical vacuum state we live in.
Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable.
For increased stability and persistence in the bloodstream, gases with high molecular weight as well as low solubility in the blood are attractive candidates for microbubble gas cores. [5] Microbubbles may be used for drug delivery, [6] biofilm removal, [7] membrane cleaning [8] [9] /biofilm control and water/waste water treatment purposes. [10]
The multivariate stable distribution defines linear relations between stable distribution marginals. [clarification needed] In the same way as for the univariate case, the distribution is defined in terms of its characteristic function. The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution.
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