We study a particular VOA which is a subVOA of the E8-lattice VOA and determine its automorphism group. Some of this group may be seen within the group E8(C), but not all of it. The automorphism group turns out to be the 3-transposition group O(10, 2) of order 2357.17.31 and it contains the simple group Ω(10, 2) with index 2. We use a recent theory of Miyamoto to get involutory automorphisms as...

Let A be a C∗-algebra. Let E and F be Hilbert A-modules with E being full. Suppose that θ : E → F is a linear map preserving orthogonality, i.e., 〈θ(x), θ(y)〉 = 0 whenever 〈x, y〉 = 0. We show in this article that if, in addition, A has real rank zero, and θ is an A-module map (not assumed to be bounded), then there exists a central positive multiplier u ∈M(A) such that 〈θ(x), θ(y)〉 = u〈x, y〉 (x...

We show that the Fourier transform on the Jacobian of a curve interchanges " δ functions " on the curve and the theta divisor. The Torelli theorem is an immediate consequence. 1. Statement of the theorem. 1.1. We live over an algebraically closed base field k. Let J be an abelian variety equipped with a principal polarization θ : J ∼ → J • = Pic 0 (J), so we have the corresponding Fourier trans...

Let K=Q(θ) be an algebraic number field with θ a root of irreducible trinomial f(x)=x6+ax+b belonging to Z[x]. In this paper, for each prime p we compute the highest power dividing discriminant K in terms powers a, b and f(x). An explicit p-integral basis is also given method described obtain integral from these bases which illustrated examples.

Let X be our data. Let θ̂ = T (X) be an estimator where T is some function. We say that θ̂ is unbiased for θ if Eθ[T (X)] = θ for all possible values of θ where Eθ denotes the expectation when θ is the true parameter value. Thus, the concept of unbiasness means that we are on average right. The bias of θ̂ is de ned by Bias(θ̂) = Eθ[θ̂] − θ. Thus, θ̂ is unbiased if and only if its bias equals 0. For e...

Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End(E)⊗ Ω X satisfying θ 2 = 0. Let h be a pluriharmonic metric of the Higgs bundle (E, θ). The tuple (E, θ, h) is called a harmonic bundle. Let X be a complex manifold, and D be a normal crossing divisor of X. In this paper, we study the harmonic bundle (E, θ, h) over X −D. We regard D as the singula...

Let m be the least cardinal θ such that MA θ fails. The only known model for " m is singular " was constructed by Kunen [K1]. In Kunen's model cof (m) = ω 1. It is unknown whether " ω 1 < cof (m) < m " is consistent. The purpose of this paper is to present a proof of Kunen's result and to identify the difficulties of generalizing this result to an arbitrary uncountable cofinality. The following...

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ be a homomorphism of [a, b] onto [c, d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extens...

Let M1 and M2 be closed, orientable 3–manifolds. Let Hi denote a Heegaard surface in Mi . We prove that if H1#H2 comes from stabilizing a lower genus splitting of M1#M2 then one of H1 or H2 comes from stabilizing a lower genus splitting. This answers a question of C Gordon [9, Problem 3.91]. We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of ...

We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We show that for any prior distribution, the Thompson Sampling strategy achieves a Bayesian regret bounded from above by 14 √ nK. This result is unimprovable in the sense that there exists a prior distribution such that any algorithm has a Bayesian regret bounded from below by 1 20 √ nK....