Rubber mat ingestion

Sand – rubber mixtures ( large rubber chips )

Mixtures of small rigid sand particles Ds and large soft rubber particles Dr are prepared at different volume fractions and tested to investigate their small-strain and zero-lateral strain responses (Dr /Ds & 10). Both data sets are simultaneously gathered in an oedometer cell instrumented with bender elements. Data are analyzed in the context of mixture theory and with the aid of numerical sim...

Mat 201b Recap

4. Let (Ω,Σ, μ) be a measure space. If f : Ω → R, then we say that f is a measurable function if for every t ∈ R, the level set: Sf (t) = {x ∈ Ω|f(x) > t} is measurable. f : Ω→ C is measurable if and only if both Re(f) and Im(f) are measurable. 5. A real function f : Ω → R is lower semicontinuous on Ω if Sf (t) is open, and is upper semicontinuous if {x ∈ Ω|f(x) < t} is open. Equivalently, f is...

Mat 215a Recap

(a) D: α0 = 1, α1 = 1, α2 = 1. χ(D ) = 1. (b) S: α0 = 1, α1 = 0, α2 = 1. χ(S ) = 2. (c) Möbius band: α0 = 1, α1 = 2, α2 = 1. χ(M) = 0. (d) Mg of genus g: α0 = 1, α1 = 2g, α2 = 1. χ(Mg) = 2− 2g. 6. RP n = (R − {0})/{z ∼ λz} = S/{z ∼ −z} = S/Z2 RP n = σ ∪ RP n−1 implies RP n = σ ∪ σn−1 ∪ · · · ∪ σ χ(RP ) = 1 for n even and 0 for n odd. 7. CP n = (C − {0})/{z ∼ λz} = S/{z ∼ λz||λ| = 1} CP n = σ ∪ ...

Mat 300 Recitations – Week

MAT 544 Real

whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is eq...