The ﬁrst argument, d, is either the empty string or a subderivation that has already been constructed.
Probabilistic Parsing Strategies
In the former case, d must be ǫ if X ∈ Σ and d must be a subderivation from nonterminal X otherwise.
Probabilistic Parsing Strategies
Depending on the question whether we encounter ⊣ as the immediately following symbol of the output string, we return the derivation d′ and the remainder v ′ of the output string, or call SLC recursively once more to obtain a larger subderivation.
Probabilistic Parsing Strategies
The produced transitions do what was deferred by the left-corner part of the strategy: they construct subderivations for the epsilon-generating nonterminals in strings µ.
Probabilistic Parsing Strategies
The main difference is that now subderivations deriving ǫ for the ﬁrst m nonterminals in the righthand side of a rule are obtained by calls of the function fǫ-TD.
Probabilistic Parsing Strategies
According to Lemma 2.15, for η ∈ L0 ++ (G ) and bounded away from zero, we have ˜v ′ (η) = η E [ˆh(η)I (ˆh(η)) | G ], noting the relationship between the subderivative and Gˆateaux derivative.
On the Stability of Utility Maximization Problems
The linear form lx is called a subderivative of f at x.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
Use (14) again, we conclude that if (xj , vj ) is a minimizer for Am,n (xm , xn ), the subderivative of Am,m+1 (xm , xm+1 ) in the ﬁrst component is also a subderivative of Am,n (xm , xn ) in the ﬁrst component.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
The subderivative of a semi-concave function is upper semi-continuous as a set function.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
In particular, if the subderivative is unique at x0 , then any subderivative at xn → x0 must converge to the derivative at x0 .
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
To prove the second statement, note that the subderivative ∂xψ− (x, 0) is upper semi-continuous as a set function.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
For any c ∈ RM , if x(c) ∈ X (c) := arg minx{ψ(x) + V (x, c)}, then ∂cV (x(c), c) is a subderivative of G at c.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
Conversely, if lc is a subderivative of G at c, then lc ∈ convx∈X (c){∂cV (x, c)}.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
It follows that G(c) is semi-concave and ∂cV (x(c)) is a subderivative of G(c) at c.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
Since a semi-concave function is almost everywhere differentiable, for almost every c, the subderivative of G(c) is unique.
Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations
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