Sentence examples for be a absolutely continuous from inspiring English sources

Let f T (t) be the pdf of a continuous random variable T defined on [a, b] and W be a absolutely continuous and monotonically non-decreasing function with W 0) → a and W(1) → b.

} end{aligned}The main theorem in this setting (originally proven in [3]) relates absolutely continuous curves in (mathbb W_p) with solutions of the continuity equation: Let ((mu _t)_{tin [0,1]}) be an absolutely continuous curve in (mathbb W_p(Omega )) (for (p>1) and (Omega subset mathbb R^d) an open domain).

Let (Phi: I rightarrowmathbb{C}) be an absolutely continuous functions on ([a,b]inmathring{I}), the interior of I.

Lemma 5.1 Let f : [ a, b ] → R be an absolutely continuous mapping for which f ′ ∈ L p [ a, b ], p > 1.

Let (f:[a,b]rightarrow mathbb{R}) be a Lebesgue integrable function and (h:[a,b]rightarrow mathbb{R}) be an absolutely continuous function with ((cdot-a)(b-cdot)[h']^{2} in L[a,b]).

Let (h : [a, b] tomathbb{R}) be a monotonic nondecreasing function and let (g : [a, b] tomathbb{R}) be an absolutely continuous function such that (g^{prime}in L_{infty}[a, b]).

Let (f:[a,b]rightarrowmathbb{R}) be a Lebesgue integrable function and (g:[a,b]rightarrowmathbb{R}) be an absolutely continuous function with ((cdot-a)(b-cdot)[g^{prime }]^{2}in L_{1}[a,b]).

Let (mathbb{F}_{1}: [delta_{1},delta_{2}] to mathbb {R}) be an absolutely continuous function with (mathbb{F}_{1}'in L_{infty}[delta_{1}, delta_{2}]) and (mathbb{F}_{2}: [delta_{1},delta_{2}] to mathbb {R}) be a monotonic nondecreasing function.

Let (g:[a,b]rightarrowmathbb{R}) be a Lebesgue integrable function and (h:[a,b]rightarrowmathbb{R}) be an absolutely continuous function with ((cdot-a)(b-cdot)[h^{prime}]^{2}in L[a,b]).

Let f : [a, b] × [c, d] → ℝ be an absolutely continuous function such that the partial derivative of order 2 exists and is bounded, i.e., ∂ 2 f ( t, s ) ∂ t ∂ s ∞ = sup ( x, y ) ∈ ( a, b ) × ( c, d ) ∂ 2 f ( t, s ) ∂ t ∂ s < ∞. for all (t, s) ∈ [a, b] × [c, d].

Theorem 2.3 Let f : [ a, b ] × [ c, d ] → R be an absolutely continuous function such that the second-order mixed partial derivative exists and there exist two functions γ ( x, y ), Γ ( x, y ) with γ ( x, y ) ≤ ∂ 2 f ( x, y ) ∂ x ∂ y ≤ Γ ( x, y ), x ∈ [ a, b ], y ∈ [ c, d ].