We introduce a generalization of Ar-type Toda theory based on a non-abelian group G, which we call the (Ar, G)-Toda theory, and its affine extensions in terms of gauged Wess-Zumino-Witten actions with deformation terms. In particular, the affine (A1, SU(2))Toda theory describes the integrable deformation of the minimal conformal theory for the critical Ising model by the operator Φ(2,1). We der...

Let AN be the symmetric operator given by the restriction of A toN , where A is a self-adjoint operator on the Hilbert space H and N is a linear dense set which is closed with respect to the graph norm on D(A), the operator domain of A. We show that any self-adjoint extension AΘ of AN such that D(AΘ)∩D(A) = N can be additively decomposed by the sum AΘ = Ā + TΘ, where both the operators Ā and TΘ...

We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n-by-d matrix Ā, n-vector b̄, and d-vector c̄ satisfying ∥∥Ā, b̄, c̄∥∥ F ≤ 1 and every σ ≤ 1, E A,b,c [logC(A, b, c)] = O(log(nd/σ)), where A, b and c ar...

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

Lemma 17 Suppose that Θ(ā, h̄ ≥ θ > 0 for all ā > 0. (i) For any initial conditions F (w) and F ′(w) the HJB equation (3) has a unique solution F in any interval [w, w̄] ⊂ R. (ii) F is twice continuously differentiable and (F, F ′) depends continuously on the initial conditions. (iii) F ′ is monotone with respect to F ′(w). That is, if F1 and F2 are two solutions of the HJB equation in an interva...

A computable structure A is decidable if, given a formula φ(x̄) of elementary first-order logic, and a tuple ā ∈ A, we have a decision procedure to decide whether φ holds of ā. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is Σ1-complete. This result holds ...

Let L be a Lie pseudoalgebra, a ∈ L. We show that, if a generates a (finite) solvable subalgebra S = 〈a〉 ⊂ L, then one may find a lifting ā ∈ S of [a] ∈ S/S such that 〈ā〉 is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U⋉N , where U is a subalgebra of V whose underlying Lie conforma...

We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of theirs null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie...