Let M be a compact symplectic manifold on which a compact torus T acts Hamiltonialy with a moment map μ. Suppose there exists a symplectic involution θ : M → M , such that μ ◦ θ = −μ. Assuming that 0 is a regular value of μ, we calculate the character of the action of θ on the cohomology of M in terms of the character of the action of θ on the symplectic reduction μ(0)/T of M . This result gene...

in this paper, we first discuss about canonical dual of g-frameλp = {λip ∈ b(h, hi) : i ∈ i}, where λ = {λi ∈ b(h, hi) :i ∈ i} is a g-frame for a hilbert space h and p is the orthogonalprojection from h onto a closed subspace m. next, we provethat, if λ = {λi ∈ b(h, hi) : i ∈ i} and θ = {θi ∈ b(k, hi) :i ∈ i} be respective g-frames for non zero hilbert spaces hand k, and λ and θ are unitarily e...

Exercise 1. Consider the quiver Q1 with 2 nodes and r arrows 1→ 2. (1) Compute using either the Harder-Narasimhan filtrations or the Reineke formula I(~ d) = p(M(~ d, ~ θ))/(s− 1) for ~ θ = (1,−1) for dimension vectors ~ d = (2, 1), (1, 2). (2) Verify that p(M(~ d, ~ θ)) is a polynomial whose degree is given by the dimension formula. (3) Compute the Euler number χ(M(~ d, ~ θ)) = p(M(~ d, ~ θ),−...

Let M be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace τ . Let d be an injective positive measurable operator with respect to (M, τ ) such that d is also measurable. Define Lp(d) = {x ∈ L0(M) : dx+ xd ∈ Lp(M)} and ‖x‖Lp(d) = ‖dx+ xd‖p . We show that for 1 6 p0 < p1 6 ∞, 0 < θ < 1 and α0 > 0, α1 > 0 the interpolation equality (Lp0(d 0), Lp1(d α))θ = Lp(d ) hol...

Consider two orthogonal closed chains on a cylinder. These chains are monotone with respect to the tangential Θ direction. We wish to rigidly move one chain so that the total area between the two is minimized. This minimization is a geometric measure of similarity between two melodies proposed by Ó Maid́ın. The Θ direction represents time and the axial direction, z, represents pitch. Let the two...

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengthes of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M, θ). Any Sasakian manifold (M, θ) whose pseudohermitian sectional curvature Kθ(σ) is a p...

Given a space 〈X,J 〉 in an elementary submodel M of H(θ), define XM to be X ∩ M with the topology generated by {U ∩ M : U ∈ J ∩ M}. It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X = XM . Assuming CH+SCH (the Singular Cardinals Hypothesis...

1. Construct a generative or forward model M with parameters θ of how data D can be generated. We can think of this generative model as a stochastic simulator for the data, with parameters θ. We will assume for now that M , the structure or functional form of the model, is known, but that the parameters θ are unknown1. An example would be that M is a Gaussian (Normal) probability density functi...

Let X be a random variable from a normal distribution with unknown mean θ and known variance σ2. In many practical situations, θ is known in advance to lie in an interval, say [−m,m], for some m > 0. As the usual estimator of θ, i.e., X under the LINEX loss function is inadmissible, finding some competitors for X becomes worthwhile. The only study in the literature considered the problem of min...

3.5 Consistency of approximate M-estimators of ψ type. As in Sec. 3.3, let (X, A, P) be a probability space and Θ a locally compact separable metric space. Let ψ(θ, x) be a function of x in X and θ ∈ Θ with values in a Euclidean space R m. Let X 1 , X 2 ,. .. be independent with values in X and distribution P. A sequence of estimators T n := T n (X 1 ,. .. , X n) with values in Θ will be called...