We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \\mathbb{R}^{n} \\times \\mathbb{R}^{m} \\rightarrow \\mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with
Even carefully-designed formalisms that are sensitive to the implicit concerns of human agents are not guaranteed to learn the right preference structures in the right way without new forms of surveillance, control, and assigned roles for both humans and the systems themselves [90], [91]. Such...
For (implicit) oracle set\({\mathbb {H}} \), an oracle function\(f^{(\cdot )}\)(with domainDand rangeR), denoted\(f^{(\cdot )}:D \rightarrow R\), is a set of functions indexed by oracles\(h\in {\mathbb {H}} \)where each\(f^h\)maps\(D \rightarrow R\). We fix ...
Even before Heisen- berg, Max Born had shown in 1926 that in collisions of atomic particles we could only predict the probabilities for the atomic paths, confirming Boltzmann's requirement for microscopic ran- domness. So the original two cases for irreducible randomness, implicit in the work ...
“No, no, yes: a date with no time means midnight is implicit.” People who said this seem like they might naturally gravitate towards date-time semantics. It makes sense to me that programmers think this way; if you’re missing a piece of data, just fill in 0 and keep going. I’...
Implicit function theoremReifenberg flatSard's TheoremUniform rectifiabilityWe prove a global implicit function theorem. In particular we show that any Lipschitz map f: ? ~n × ? ~m → ? ~n (with n-dim. image) can be precomposed with a bi-Lipschitz map ? ? ~n × ? ~m → ? ~n ...
These classes include the DAGs implicit to both Catena iMHFs as well as the Double-Buffer and Linear iMHFs. We prove that no λ-stacked sandwich graph is depth-robust (Lemma 1). For any t > 1 there is a set S of n/t nodes such that depth(G − S) ≤ (λ + 1)t. (n, ...
Let ζ = ζ(κ) ∈ {0, 1}κ be an arbitrary input label for the graph function family FG. For any v ∈ Vn, define pre-labO,ζ (v) = (v, labelO,ζ (pred(v))). Let B be a non-uniform PROM algorithm. Fix an implicit security parameter κ. Let x be an input to B. We...
Description of the system Microscopic dynamics Our two-dimensional system consists of N (indistinguish- able) circular particles suspended in an incompressible implicit solvent. The particles are quasi-hard disks; their pair interaction potential is given by σ 36 φ(r) = , r (2) where r is ...
We also remark that a graded matrix representation is equivalent to free implicit representations [15, Sec 5.1] for the special case of m0 = 0. 4 Hardness of Interleaving Distance We consider the following computational problems: 1-Interleaving: Given two persistence modules M, N in graded ...