Sentence examples for definite for all from inspiring English sources
Zermelo's original system included the assumption that, if a formula S x) is "definite" for all elements of a set A, then there exists a set the elements of which are precisely those elements x of A for which S x) holds.
As A ( t ) is positive definite for all t ∈ [ 0, T ], we have Lemma 2.1.
Recall that we assume that a t,x) is positive definite for all t and x.
The function (L(H_{1},H_{2},P)) is defined, continuous and positive definite for all (H_{1},H_{2},P>0).
end{gathered} Since S is symmetric positive definite, for all (chiinmathbb{R}^{n}), lambda_{min}(S |chi|^{2}leqchi^{T}Schileq lambda_{max}(S | chi|^{2}.
Assume that (B_{k}) is positive definite; for all (kgeq 1), we prove that (s_{k}^{T}y_{k}^>0) holds by the following three cases.
Since Q ˜ ( r ) is symmetric and positive-definite for all r > 0, one gets that B ( r ) is invertible.
Then ϕ is convex on D if and only if its Hessian matrix ((frac{partial^{2}f}{partial x_{i},partial y_{j}}(mathbf{x}))_{ntimes n}) is positive semi-definite for all (mathbf{x}in Dsubsetmathbb{R}^{n}).
A c 2 function f U⊂R n →R defined on a convex open set U is concave if and only if the Hessian matrix D 2 f(x) is negative semi-definite for all x∈U.
The function (hrightarrow varOmega _{i} phi_{h})) is n-exponentially convex in the Jensen sense on J and the matrix ([varOmega _{i} phi_{frac{h_{j}+h_{l}}{2}}) ]_{j,l=1}^{k}) is positive semi-definite for all (kinmathbb{N}), (kleq n), (h_{1},ldots,h_{k}in J).
In the process of proving Theorem 3.9, assume that (Phi Hrightarrowmathbb{R}) is a twice differentiable function on an open convex set H which contains the compact set (Delta_{n}= prod^{n}_{i=1}[m_{i},M_{i}]) such that its Hessian matrix ((frac{partial ^{2}f}{partial x_{i},partial y_{j}}(mathbf{x}))_{ntimes n}) is positive semi-definite for all (mathbf{x}inmathbb{R}^{n}).
