Let G = SpecA be an affine K-group scheme and à = {w ∈ A : dimK A ∗ · w · A < ∞}. Let 〈−,−〉 : A × Ã → K, 〈w, w̃〉 := tr(ww̃), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A.

The recent cavity ringdown (CRD) measurement of the forbidden à 2E ← X̃ A2 transition of the nitrate radical NO3 reveals a rich, well-resolved spectrum in the near-infrared. The spectroscopic detail provides a new window onto the Jahn–Teller (JT) and pseudo-Jahn–Teller (PJT) effects in NO3. This paper reviews the current experimental evidence for vibronic coupling in the à state and discusses th...

Electronic spectra of LiNH(3) and its partially and fully deuterated analogues are reported for the first time. The spectra have been recorded in the near-infrared and are consistent with two electronic transitions in close proximity, the Ã(2)E-X(2)A(1) and B(2)A(1)-X(2)A(1) systems. Vibrational structure is seen in both systems, with the Li-N-H bending vibration (ν(6)) dominant in the Ã(2)E-X(...

An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them if the answer is positive. In this paper we introduce effective probabilistic linear programming (LP) heuristics to solve the graph isomorphism problem. We m...

ÐPhotographic volumes present a unique, interesting challenge for volume rendering. In photographic volumes, voxel color is predetermined, making color selection through transfer functions unnecessary. However, photographic data does not contain a clear mapping from the multivalued color values to a scalar density or opacity, making projection and compositing much more difficult than with tradi...

Combining (35) with (34), and the fact that 1nb = 0, we have that P(A,θ,π0) = P(Ã,θ,π0). Since à has identical (n−1)-th and n-th columns, and fθn−1 = fθn we have that P(Ã,θ,π0) = P(Ā,θ̄,π̄0). Thus H ′ = (Ā, θ̄, π̄) is an equivalent (n−1)-state HMM and H is not minimal, proving the claim. We prove (35) by induction on the sequence length k ≥ 2. First note that since δ = 0, by Lemma 1 we have that

Let à be a self-adjoint extension in K̃ of a fixed symmetric operator A in K ⊆ K̃. An analytic characterization of the eigenvalues of à is given in terms of the Q-function and the parameter function in the Krein–Naimark formula. Here K and K̃ are Krein spaces and it is assumed that à locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are ...

Let A be a domain of the boundary of a strictly pseudoconvex domain Ω of Cn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain à of Ω, depending only on Ω and A, and a complex variety W ⊂ à such that bW ∩A = M . Moreover, a generalization to analytic sets of depth at least 4 is given.

Sections 3, 4 and 5 give results under Lp assumptions without any further assumptions on the decay rates. Section 3 deals with the case r ∈ Lp, p > 2. Section 4 generalises to p > 1. Section 5 covers the case p = 1 for mean zero observables. There is still the general p = 1 case to consider! The optimal result in the stretched exponential case is given in Section 6. Section 7 gives an alternati...