Sentence examples for given any varepsilon from inspiring English sources
Given any (varepsilon> 0).
More precisely, given any (varepsilon >0) and any subinterval I, there exists a (delta>0).
Then λ is continuous at (omega_{0} ), that is, given any (varepsilon >0 ), there exists (delta >0 ) such that Vert omega -omega_{0}Vert < delta for any (omega in Omega ), then biglvert lambda (omega -lambda (omega -lambdarvert < varepsilomega_{
Given any (varepsilon >0) and any finite set (mathcal {F}subset {mathcal A}), we may obtain from (2) a finite-dimensional linear subspace (Vsubset {mathcal A}) such that ( dim (V) ge frac{4 |mathcal {F}|}{varepsilon } ) and begin{aligned} frac{dim (a V +V)}{dim (V }le 1+ frac{varepsilon }{2}, quad mathrm {for~all}quad ain mathcal {F}.
The zero solution is said to be exponentially stable (or exponentially asymptotically stable); if there exists a (lambda> 0) and, given any (varepsilon> 0), there exists a (delta varepsilon) > 0) such that (t_{0} in I) and (Vertmathbf{x}_{0}Vert< delta varepsilon)) imply (Vertmathbf{x}(t t_{0},mathbf{x}_{0} Vertlevarepsilon e^{-lambda (t-t_{0})}) for all (t ge t_{0}).
Apart from this easy case, (L_mathbb {K}(E)) is always infinite-dimensional, so by Proposition 3.5, it suffices to show that, given any (varepsilon >0), any (N in mathbb {N}), and any finite subset (mathcal F) of (L_mathbb {K}(E)), we can find an ((mathcal {F}, varepsilon ))-Følner subspace W in (L_mathbb {K}(E)) with (dim (W) ge N).
So, given any (varepsilon>0), there is a positive number (M=M varepsilon)>0), such that (|x|geq M) imply bigl|psi'(x bigr|leqvarepsilon and bigl|psi(x bigr|leqvarepsilon|x| for (forall tin [0,2pi]).
To see this, we have to show that, given any (varepsilon>0), there exists an integer (n_{8}) such that for (m>n>n_{8}); we have bigglvert frac{(Ty)_{m}}{Q^{2}_{m}} - frac{(Ty)_{n}}{Q^{2}_{n}} biggrvert < varepsilon for any (y in S).
For (v: mathbb{T} tomathbb{R}) and (t in T_{kappa}), define the nabla derivative [14] of v at t, denoted by (v^{nabla_{t}}(t)), to be the number (provided it exists) with the property that given any (varepsilon>0), there is a neighborhood U of t such that (|v(rho (t))-v s -v^{nabla_{t}}(t)[rho(t -s]|leq varepsilon|rho(t -s]|leqr all (s in U).
If (x:mathbb{T}rightarrowmathbb{R} ) and (t inmathbb{T}^{kappa} ), then the delta derivative of x at t, denoted by (x^{Delta} (t) ), is the real number (provided it exists) with the property that given any (varepsilon>0 ), there is a neighborhood V of t such that (arrowvert[x sigma(t))-x s)]-x^{Delta} (t)(sigma(t -s) arrowvert -svarrowvertleqvarepsilonarrowvertsigmart -sor all (s in V ).
For (t>0), one can define (T_{alpha}(h)(t)) to be the number provided it exists with the property that, given any (varepsilon>0), there is a δ-neighborhood (V_{t}subsetmathbb{T}) of t such that biglvert bigl[ hbigl sigma(t bigr -h(s) bigr] t bigr -h1}-T_{alpha }(h) (t) bigl[ sigma(t)-s bigr] bigrvert leqvarepsilon biglvert^{alpha-1}-T_{alphat, (2.1) for all (sin V_{t}).
