Math for Game Programmers: Dual Numbers
... Operations are similar to complex numbers, however since ε2 = 0, we have: ...
... Operations are similar to complex numbers, however since ε2 = 0, we have: ...
HW6 - Harvard Math Department
... and the median of the triangle are the three concurrent lines. In Yaglom's diagram, QD/AD = QC/AC, and Ceva's theorem says that AM = BM. The standard proof of Ceva's theorem uses the Euclidean law of sines, but it works also with the Galilean law of sines a/A = b/B = c/C. Just dualize it for this sp ...
... and the median of the triangle are the three concurrent lines. In Yaglom's diagram, QD/AD = QC/AC, and Ceva's theorem says that AM = BM. The standard proof of Ceva's theorem uses the Euclidean law of sines, but it works also with the Galilean law of sines a/A = b/B = c/C. Just dualize it for this sp ...
Exercise Sheet 4 - D-MATH
... ϕpxq “ x, and consider also the differentiable structure induced by the chart ψ : R Ñ R, ψpxq “ x3 . Show that the two differentiable structures are not equal, but that nevertheless the two differentiable manifolds thus defined are diffeomorphic. 4. (Review of Quaternions) Let Q denote the vector sp ...
... ϕpxq “ x, and consider also the differentiable structure induced by the chart ψ : R Ñ R, ψpxq “ x3 . Show that the two differentiable structures are not equal, but that nevertheless the two differentiable manifolds thus defined are diffeomorphic. 4. (Review of Quaternions) Let Q denote the vector sp ...
Solutions Sheet 3
... Hint: Play around with initial and final objects and products and coproducts. Solution: Any equivalence with its opposite category interchanges initial with final objects and products with coproducts, and any theorem involving these translates into a dual one. It therefore suffices to find a propert ...
... Hint: Play around with initial and final objects and products and coproducts. Solution: Any equivalence with its opposite category interchanges initial with final objects and products with coproducts, and any theorem involving these translates into a dual one. It therefore suffices to find a propert ...
PDF
... makes C (F) into a chain complex. The cohomology of this complex is denoted Ȟ i (X, F) and called the Čech cohomology of F with respect to the cover {Ui }. There is a natural map H i (X, F) → Ȟ i (X, F) which is an isomorphism for sufficiently fine covers. (A cover is sufficiently fine if H i (Uj ...
... makes C (F) into a chain complex. The cohomology of this complex is denoted Ȟ i (X, F) and called the Čech cohomology of F with respect to the cover {Ui }. There is a natural map H i (X, F) → Ȟ i (X, F) which is an isomorphism for sufficiently fine covers. (A cover is sufficiently fine if H i (Uj ...
The two reported types of graph theory duality.
... The outline of the talk 1. The two reported types of graph theory duality. 2. Duality between trusses and linkages and the ...
... The outline of the talk 1. The two reported types of graph theory duality. 2. Duality between trusses and linkages and the ...