under the initial conditions (4.2). where, matrix, is a continuous matrix, and.
(f_{1}: R^{n}rightarrow R^{n}) is a continuous vector function and (f_{2}: R^{n}rightarrow R^{ntimes m}) is a continuous matrix function.
where is fuzzy coefficient, is the set of all upper semicontinuously convex fuzzy numbers on with, and are nonlinear regular fuzzy functions, is a nonlinear continuous function, is an continuous matrix such that is continuous for and with,, is a control function, is an initial value and are bounded functions,, where and represent the left and right limits of at, respectively.
Let ((A_{t})_{t in Omega }) be a continuous field of positive definite matrices in (mathbb {M}_{k}) such that (operatorname {Sp}(A_{t}) subseteq[m,M] subseteq 0,infty)) for each (t in Omega ).
Let (f cdot)=bigl ( {scriptsizebegin{matrix} f_{1}(cdot cr f_{2}(cdot) end{matrix}} bigr )) be a continuous vector-valued function.
Let (F cdot)= left[{scriptsizebegin{matrix} f cdot) cr w_{1}cr w_{2} end{matrix}} right] ), where (f cdot)= bigl[ {scriptsizebegin{matrix} f_{1}(cdot) cr f_{2}(cdot) end{matrix}} bigr]), be a continuous vector-valued function.
Each of these diverse data types can be represented as a continuous matrix of genes and conditions (e.g. cell types, tissues, pathways, experimental conditions).
(1) where (T>0), (A t)) is a continuous symmetric matrix of order N and (F: [0,T]times{mathbb{R}}^{N}rightarrow mathbb{R}) is locally Lipschitz continuous in x and (partial F t,x)) denotes the Clarke subdifferential of F for x.
In the special case, the half-linear system becomes the linear system (mathbf{x}' = A t mathbf {x}) where (A t)) is a (2 times2) continuous matrix and x is a two-dimensional vector.
