Let A = ( H1 E ∗ E H2 ) and à = ( H1 O O H2 ) be Hermitian matrices with eigenvalues λ1 ≥ · · · ≥ λk and λ̃1 ≥ · · · ≥ λ̃k, respectively. Denote by ‖E‖ the spectral norm of the matrix E, and η the spectral gap between the spectra of H1 and H2. It is shown that |λi − λ̃i| ≤ 2‖E‖ η + √ η2 + 4‖E‖2 , which improves all the existing results. Similar bounds are obtained for singular values of matrices u...

Let Θ = (α, β) be a point in R, with 1, α, β linearly independent over Q. We attach to Θ a quadruple Ω(Θ) of exponents which measure the quality of approximation to Θ both by rational points and by rational lines. The two “uniform” components of Ω(Θ) are related by an equation, due to Jarńık, and the four exponents satisfy two inequalities which refine Khintchine’s transference principle. Conve...

Db(Coh(Aθ λ )) ∼ −→ D(Coh(Aλ)) for two different generic θ, θ′. These derived equivalences will play a crucial role in proving that ImCCv ⊂ Lω[ν]. They were introduced in [BPW, Section 6] under the name of twisting functors. Let us recall from Lecture 3 that abelian localization holds for (λ + nθ, θ) for all n sufficiently large, see Proposition 2.2 there. Also recall that Coh(Aλ) ∼ −→ Coh(Aλ+n...

In this lecture I want delve a bit more deeply into the problem of comparing performance of various tests. Among other things I will try to provide an exposition of Chernoff’s (1952) result employed in Lars Hansen’s talks here several years ago. The lecture is based on van der Vaart (1998, Chapter 14). Consider the problem of testing H0 : θ ∈ Θ0 vs H1 : θ ∈ Θ1, using a test statistic Tn that re...

Let Θ be a point in R. We split the classical Khintchine’s Transference Principle into n − 1 intermediate estimates which connect exponents ωd(Θ) measuring the sharpness of the approximation to Θ by linear rational varieties of dimension d, for 0 ≤ d ≤ n− 1. We also review old and recent results related to these n exponents.

For a real number x, we let ⌊x⌉ be the closest integer to x. In this paper, we look at the arithmetic properties of the integers ⌊θ⌉ when n ≥ 0, where θ > 1 is a fixed algebraic number.

and Applied Analysis 3 Let Γ(⋅) denote the gamma function. For any positive integer n and real number θ (n − 1 < θ < n), there are different definitions of fractional derivatives with order θ in [8]. During this paper, we consider the left, (right) Caputo derivative and left (right) Riemann-Liouville derivative defined as follows: (i) the left Caputo derivative: C 0 D θ t V (t) = 1 Γ (n − θ) ∫ ...

For (b), let Θ be any subset of Γ, x in its complement, U as in (a). The neighbourhood xU of x contains at most one element of Γ. There exists a neighbourhood of x contained in xU and not containing any element of Θ. For (c), let V = U ·U, which is compact. The intersection of Γ with V is compact, and covered by disjoint neighbourhoods of each of its points. This intersection must therefore be ...

Let θ(a1, a2, · · · , ak) denote the graph obtained by connecting two distinct vertices with k independent paths of lengths a1, a2, · · · , ak respectively. Assume that 2 ≤ a1 ≤ a2 ≤ · · · ≤ ak. We prove that the graph θ(a1, a2, · · · , ak) is chromatically unique if ak < a1 + a2, and find examples showing that θ(a1, a2, · · · , ak) may not be chromatically unique if ak = a1 + a2.

It turns out that H1 is a measure, now called one-dimensional Hausdorff measure because it was generalized by Hausdorff [10] to the whole family of measures Hα, where α is any positive number (integer or noninteger). The modern theory of “fractals” is largely based on the notion of the Hausdorff dimension dimH (F) of a set F , defined by dimH (F) = inf{α > 0 : Hα(F) = 0}. We recommend the book ...