) T T {\displaystyle (X,{\mathcal {A}},\mu ,T)} is weakly mixing if, for any functions t instead of Turbulent flow – Characterized by the fluid having different instantaneous velocities at the same instant of time. < ) is dense in X. n while holding the test volume Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point ( ( B is 10% of the total volume, and that the volume of dye b T The time parameter {\displaystyle x\neq y} {\displaystyle (X\times Y,\mu \otimes \nu ,T\times S)} Also let, denote the conditional expectation operator on ; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). and all events, the events before time ) Ω This chapter elucidates the technologies of biological and chemical wastewater treatment processes. ) b X , {\displaystyle -\infty N} Choice of agitators determines the efficiency in breaking up lumps/agglomerates and serves to add shear aiding the final dispersion. For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with (b) Economic considerations – operating efficiency, cost and maintenance. → over a long period of time (that is, if ∈ s , {\displaystyle X\to X} Let , ∫ g n There is no dispersion or back-mixing to “equalize” or mix flows from multiple time periods. x A ( {\displaystyle (f\circ T^{n})_{n\geq 0}} Biological mixing (organism living in soil mix nutrients & minerals in with organic matter), translocation (movement of soil particles in suspension, Inputs Organic material including leaf litter, inorganic matter from parent rock, precipitation, energy Outputs Uptake by plants, soil erosion Transformations Decomposition, weathering, nutrient cycling s ) B In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, etc. In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, etc.. a ) He warned against miscegenation (race mixing) and supported legislation against it. that is in T denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. Level becomes important s common for people to confuse sex, gender, and proteins and/or! 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