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In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity.An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than ().
Conversion can be separated into instantaneous conversion and overall conversion. For continuous processes the two are the same, for batch and semi-batch there are important differences. Furthermore, for multiple reactants, conversion can be defined overall or per reactant.
If is sequentially weakly lower semi-continuous, then for any given (,) the function (,,) is quasiconvex. The claim is true even when both m , n {\displaystyle m,n} are bigger than 1 {\displaystyle 1} and coincides with the previous claim when m = 1 {\displaystyle m=1} or n = 1 {\displaystyle n=1} , since then quasiconvexity is equivalent to ...
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Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let and ((,),) with (,) =.The Riesz-Markov-Kakutani representation theorem states that the dual space of () can be identified with the space of signed, finite Radon measures on it.
Continuous Catalytic reforming (CCR) unit. In addition to a gasoline blending stock, reformate is the main source of aromatic bulk chemicals such as benzene, toluene, xylene and ethylbenzene, which have diverse uses, most importantly as raw materials for conversion
A natural generalization from the above results gives sufficient local conditions for to be continuous and to be nonempty, compact-valued, and upper semi-continuous. If in addition to the conditions above, f {\displaystyle f} is quasiconcave in x {\displaystyle x} for each θ {\displaystyle \theta } and C {\displaystyle C} is convex-valued ...
Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy. [6]