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If D(a, b) > 0 and fxx(a, b) > 0 then (a, b) is a local minimum of f. If D(a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f. If D(a, b) < 0 then (a, b) is a saddle point of f. If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive).
The Shields parameter, also called the Shields criterion or Shields number, is a nondimensional number used to calculate the initiation of motion of sediment in a fluid flow. It is a nondimensionalization of a shear stress, and is typically denoted or . This parameter has been developed by Albert F. Shields, and is called later Shields ...
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients.
In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC).
Bendixson–Dulac theorem. In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression. According to Dulac theorem any 2D autonomous system with a periodic orbit has a region with positive and a region with negative divergence inside such orbit.
Discovered by Karl Schwarzschild, [1] the Schwarzschild criterion is a criterion in astrophysics where a stellar medium is stable against convection when the rate of change in temperature (T) by altitude (Z) satisfies. where is gravity and is the heat capacity at constant pressure. If a gas is unstable against convection then if an element is ...
The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2×2 and 2×3 dimensional cases the condition is also sufficient.
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant :
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem ). Here is one version. [1] Let be two measurable spaces (such as ). Let be an integral operator with the non-negative Schwartz kernel , , :
In algebra, Serre's criterion for normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring A to be a normal ring. The criterion involves the following two conditions for A :