Sentence examples for the unique constant from inspiring English sources

It is obvious that w ∗ is the unique constant positive solution of (5.6) for any t ∈ [ 0, 1 ].

It is obvious that (mathbf{w}_{ast}) is the unique constant positive solution of (5.8) for any (0 leq t leq1).

By [19], if (max{1,d}>M/mu_{1}), where (d=min{d_{1},d_{2}}), then every solution of system (1.2) converges exponentially to the unique constant equilibrium solution ((u_,v_)).

For (lambda=0), it is easy to calculate that ((S^{ast}, I^{ast})) is the unique constant periodic solution in U, which implies that (2.2) has only one periodic solution in U. □.

If, additionally, (max{1,d}>M/lambda_{1}), then every solution of system (1.2) converges exponentially to the unique constant equilibrium solution ((u_,v_)), where (mu_{1}) is the smallest positive eigenvalue of −Δ on Ω subject to homogeneous Neumann boundary conditions, and M:=maxbiggl{ lambda_{1}alpha_{1}+1, frac{lambda_{2}^{2}(1+alpha _{2}{a{2}+1biggrda_{1})}{a}+1biggr}.

end{aligned}Since the speed along the curve (Tmapsto X _t Y) has the constant value ({Vert }log (X^{-1/2}Y X^{-1/2}){Vert }_2), this, together with the uniqueness in Theorem C.3, shows that for all (t_0 < t_1) in ({mathord {{mathbb {R}}}}), the restriction of (tmapsto X _t Y) to ([t_0,t_1]) is the unique constant speed geodesic running from (X#_{t_0} Y) to (X#_{t_1} Y) in time (t_1- t_0).

He said the boys had sometimes chaotic lives in which an affection for the game was a unique constant.

Note that the value of β may be a unique constant (system parameter) at that time, although during the relaxation the system consists of the many small regions with different β. Figure 4 Populations for the negative temperature distribution are plotted separately inside and outside the clumps.

Implementation of either approach is greatly simplified if each electrode current ik can be described as the product ak*g(t) i.e. each electrode's current is the product of a (possibly unique) constant scale factor and a common time course.

For the three-phase experiments, each core was waterflooded and simultaneously depleted by a fixed depletion rate to a unique constant back pressure below the initial bubble point of the binary mixture.

Note that (1.2) has the following three non-negative constant solutions: (i) the trivial solution ((0,0)); (ii) the semi-trivial solution ((0,K)); (iii) the unique positive constant solution (mathbf{w}_{ast }=:(u_{ast},v_{ast})), where u_{ast}=frac{er}{K ec-ab)^{2}} bigl[K ec-ab)-a bigr],qquad v_{ast}= frac{a}{ec-ab}.