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...

Proof of Lemma A.1. (i) As Θa is open, we can choose a small neighborhood N (θ • a) whose closure N (θ• a) is contained in Θa. By continuity of a (·; ·), compactness of {z : |z| ≤ 1} × N (θ• a), and the condition a (z; θa) 6= 0 for |z| ≤ 1 and θa ∈ Θa, we can find a positive δ (θ• a) such that a (z; θa) 6= 0 for |z| ≤ 1 + 2δ (θ• a) and θa ∈ N (θ• a). That the inverse a (z; θa) −1 has the stated...

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...

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...

In this paper we study some classes of infinite words generalizing episturmian words, and analyse the relations occurring among such classes. In each case, the reversal operator R is replaced by an arbitrary involutory antimorphism θ of the free monoid A. In particular, we define the class of θ-words with seed, whose “standard” elements (θ-standard words with seed) are constructed by an iterati...

We present measurements of the extended dark halo profiles of bright early-type galaxies at redshifts 0.1 < z < 0.9 obtained via galaxy-galaxy lensing analysis of images taken at the CFHT using the UH8K CCD mosaic camera. Six 0.5 × 0.5 degree fields were observed for a total of 2 hours each in I and V , resulting in catalogs containing ∼ 20000 galaxies per field. We used V−I color and I magnitu...