Interaction Uncertainty
Last edited: August 8, 2025The interaction of multiple agents/decision makers causes additional uncertainty
Interactive Proof
Last edited: August 8, 2025We have prover \(P\) and randomized verifier \(V\). The \(V\) asks \(P\) for membership statements, and \(P\) responds with statements. These proofs can be used to prove membership in very powerful languages.
Languages \(L\) with a \(k\) round interactive proof system, where the verifier \(V\) is poly randomized machine and its interacting with an all-powerful prover \(P\).
- \(x \in L \implies \exists_{ \text{prover}}\) such that \(V\qty(x_1, \dots, y_{k})\) accepts with probability \(\geq \frac{2}{3}\)
- \(x \not \in L \implies \forall _{\text{prover}}\) such that \(V\qty(x_1, \dots, y_{k})\) accepts with probability \(\leq \frac{1}{3}\)
interpolation
Last edited: August 8, 2025nyquist limit is great and all, but I really don’t want to wait for all \(T\) to be able to sample all the necessary terms to solve for every \(a_{j},b_{j}\) before we can reconstruct our signal.
So, even if we got our sequence of \(\frac{1}{2B}\) length of points, we need an alternative way to reconstruct the signal as we go.
One way to reconstruction via interpolation is just to connect the dots; however, this is bad because it creates sharp corners.
Intersession 2023
Last edited: August 8, 2025invariant subspace
Last edited: August 8, 2025invariant subspaces are a property of operators; it is a subspace for which the operator in question on the overall space is also an operator of the subspace.
constituents
requirements
\(U\) is considered invariant on \(T\) if \(u \in U \implies Tu \in U\)
(i.e. \(U\) is invariant under \(T\) if \(T |_{U}\) is an operator on \(U\))
additional information
nontrivial invariant subspace
(i.e. eigenstuff)
A proof is not given yet, but \(T \in \mathcal{L}(V)\) has an invariant subspace that’s not \(V\) nor \(\{0\}\) if \(\dim V > 1\) for complex number vector spaces and \(\dim V > 2\) for real number vector spaces.