Sentence examples for main theorem that from inspiring English sources

We now state our main theorem that gives a tight upper bound on the p-value at every location when {αi} are unknown and are estimated from the observations.

In that case, the sets (mathcal{A}_{i}) are n-simplices, and they will be used in the main theorem that follows.

The detailed proof of the second main theorem, that of the convergence of the network equations to the mean-field limit, is given in the Appendix.

But, we can see in the proof of the main theorem that if B z) is a meromorphic function having deficient value ∞ and f ≢ 0 is a meromorphic solution of Equation (1.1), then ρ(f) = ∞.

In this section we prove the main theorem, that the weights are determined by knowledge of N and the tree-average distances between members of X.

After recalling some definitions and results on triangulated and pretriangulated categories from  [1,13,16,17], we prove our first main theorem that equates filtered length of internal homs with the generation time of a given object.

Our main theorem states that those structural conditions guarantee that the network morphism is a kinetic emulation: Theorem (Emulation): If a morphisms (m S, m R ) is a reactant morphism and a stoichiomorphism, then it is an emulation.

Our main theorem shows that the functional F is in general a non-local one; this unexpected feature occurs even in very simple examples, when μ is the one-dimensional Hausdorff measure over a closed Lipschitz curve in the plane.

In this case our main theorem states that the solution of (1) is of the type u t,x =varepsilon vbigl varepsilon^{4}t,xbigr) and v(T,x =b(T cos(x)+mathcal{O}bigl varepsilon^{1-32kappa}bigr), where b is the solution of the amplitude equation of Itô type db=biggl[biggl(nu-frac{11}{18}rho_{0}^{2} biggr b-C_{0}biggr b-C_{,dT+rho_{0}b^{5}biggr]eta}_{0},dT+rho_{ere (rho_{0}bsqrt{frac{27}{38pi}}alpha_{0}).

In this section we present our main theorem, showing that the Pareto product always preserves Bellman's principle.

We do remark that our main Theorem 2 states that in the case a 2 + b 2 ≠ 0, the breaking of large ring neural networks extends the asymptotic stability domain in the parameter space providing a sufficiently large size.