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On the impossibility of a partial mass violation in surface runoff systems
Author(s) -
Diskin M. H.,
Boneh A.,
Golan A.
Publication year - 1975
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr011i002p00236
Subject(s) - mathematics , bounded function , function (biology) , order (exchange) , series (stratigraphy) , type (biology) , set (abstract data type) , class (philosophy) , discrete mathematics , pure mathematics , mathematical analysis , computer science , paleontology , ecology , finance , evolutionary biology , artificial intelligence , economics , biology , programming language
The class of nonnegative, initially relaxed, and nonanticipating systems has many applications in engineering. In this paper a proof is given to a theorem stating that in this class of systems, if the input total mass is equal to the output total mass, then for any nonnegative input‐output pair, the system fulfills also a partial mass condition. In applying this theorem to systems expressed by the Volterra series it is concluded that the input functions must be bounded. Two such bounds on the input functions are considered: (1) bounds resulting from the requirement of a nonnegative output and (2) bounds resulting from the mass‐conserving property of the system. The theorem mentioned above implies that the set of input functions causing nonnegative output functions is a subset of the set of input functions that do not violate the mass‐conserving property of the system. It is therefore clear that the bounds of type 1 are the dominant among the two bounds for any nonnegative input function. In a system expressed by an N th order Volterra series the bounds on the input can be evaluated by solving a polynomial inequality of order N ‐ 1. An example is given for a system expressed by a third‐order Volterra series in which the bounds on the input form two regions. Explicit equations for the bounds of type 1 and 2 are derived for a second‐order system.