The importance of collimator R&D with test beam facilities is demonstrated with the example of ILC collimator studies at SLAC End Station A. Related LHC collimator aspects and longer term plans are addressed. Presented at IDTB07, ILC Detector Test Beam workshop, Fermilab, USA, 2007.

We consider the multidimensional Newton-Einstein equation in static electromagnetic field ṗ = F (x, ẋ), F (x, ẋ) = −∇V (x) + 1 c B(x)ẋ, p = ẋ √ 1− |ẋ|2 c2 , ṗ = dp dt , ẋ = dx dt , x ∈ C(R,R), (∗) where V ∈ C2(R,R), B(x) is the d× d real antisymmetric matrix with elements Bi,k(x) = ∂ ∂xi Ak(x)− ∂ ∂xkAi(x), and |∂ j xAi(x)|+ |∂j xV (x)| ≤ β|j|(1 + |x|)−(α+|j|) for x ∈ R, |j| ≤ 2, i = 1..d and so...

In this paper, we characterize multiresolution analysis(MRA) Parseval frame multiwavelets in L^2(R^d) with matrix dilations of the form (D f )(x) = sqrt{2}f (Ax), where A is an arbitrary expanding dtimes d matrix with integer coefficients, such that |detA| =2. We study a class of generalized low pass matrix filters that allow us to define (and construct) the subclass of MRA tight frame multiwa...

Consider the Schrödinger equation: $$\mathrm{i}\partial _tu=Hu$$ over $$\mathbb {R}^n$$ , where H is a self-adjoint operator on $$L^2(\mathbb {R}^n)$$ which sum of $$-\Delta $$ and some potential. This paper aims to study observability for above equation, including observable sets time. We mention that measurable subset $$E\subset \mathbb called an set at time $$T>0$$ if there constant $$C>0$$ ...

A cube tiling of R is a family of pairwise disjoint cubes [0, 1) + T = {[0, 1) + t : t ∈ T} such that ⋃ t∈T ([0, 1) d + t) = R. Two cubes [0, 1) + t, [0, 1) + s are called a twin pair if |tj − sj | = 1 for some j ∈ [d] = {1, . . . , d} and ti = si for every i ∈ [d] \ {j}. In 1930, Keller conjectured that in every cube tiling of R there is a twin pair. Keller’s conjecture is true for dimensions ...

where x ∈ X where X = R, u ∈ U is the set of control inputs and d ∈ D is the set of disturbance inputs, where U andD are convex and compact subsets of R and R respectively; f is a vector field, assumed to be globally Lipschitz in x and continuous in u and d; and the initial state x(0) ∈ Init where Init ⊆ R. We will use U to denote the set of piecewise continuous functions from R to U and D to d...

For d ≥ 1, s ≥ 0 a (d,d + s)-graph is a graph whose degrees all lie in the interval {d,d +1, . . . ,d +s}. For r ≥ 1, a≥ 0 an (r,r+1)-factor of a graph G is a spanning (r,r+a)-subgraph of G. An (r,r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r,r +a)-factors. We prove a number of results about (r,r+a)-factorizations of (d,d+s)-bipartite multigraphs and of (d,d + s)...

In this note, a real algebraic curve X is a smooth proper geometrically integral scheme over R of dimension 1. A closed point P of X will be called a real point if the residue field at P is R, and a non-real point if the residue field at P is C. The set of real points, X(R), will always be assumed to be non empty. It decomposes into finitely many connected components, whose number will be denot...

In this paper we prove the following result. LetH be a real or complex Hilbert space, let L(H) be the algebra of all bounded linear operators on H and let A(H) ⊆ L(H) be a standard operator algebra. Suppose we have an additive mapping D : A(H) → L(H) satisfying the relation D(An) = D(A)A∗n−1 + AD(An−2)A∗ + An−1D(A) for all A ∈ A(H) and some fixed integer n > 1. In this case there exists a uniqu...