So far we have talked about the expectation of random variables, and moved on to the variance or the second moment. However, neither of these methods is strong enough to produce the tight bounds needed for some applications. We have already stated that if all you are given is the variance of a r.v., then Chebyshev’s Inequality is essentially the best you can do. Thus, more information is needed...

2. Query Uncertainty 4 2.1. Derivation of Special Case: Query Dependent on a Subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. Derivation of Special Case: Independent Representation Elements . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Linear Query with Independent Elements Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4. Empirical Ent...

for some constant Lσ,d ≥ L σ,d and are widely discussed in the literature (see e.g. [3, 9, 11]). A longstanding question is when (1.4) holds with Lσ,d = L cl σ,d. The most general result is due to Laptev and Weidl [10] who proved that Lσ,d = L cl σ,d for all σ ≥ 32 and d ≥ 1. Their proof is based on a dimensional reduction of Schrödinger operators with operator valued potentials which allows th...

In the framework of prediction with expert advice, we consider a recently introduced kind of regret bounds: the bounds that depend on the effective instead of nominal number of experts. In contrast to the NormalHedge bound, which mainly depends on the effective number of experts but also weakly depends on the nominal one, we obtain a bound that does not contain the nominal number of experts at ...

We introduce the Singleton bounds for codes over a finite commutative quasi-Frobenius ring.

We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems.

We prove sharp Lieb-Thirring inequalities for Schrödinger operators with potentials supported on a hyperplane and we show how these estimates are related to LiebThirring inequalities for relativistic Schrödinger operators.

We present a high-level approach to array bound check optimization that is neither hampered by recursive functions, nor disabled by the presence of partially redundant checks. Our approach combines a forward analysis to infer precise contextual constraint at designated program points, and a backward method for deriving a safety pre-condition for each bound check. Both analyses are formulated wi...