I
... Seg(Rl , Rm , Rn ) = {A ∈ Rl×m×n | A = u ⊗ v ⊗ w} = {A ∈ Rl×m×n | ai1 i2 i3 aj1 j2 j3 = ak1 k2 k3 al1 l2 l3 , {iα , jα } = {kα , lα }} Hypermatrices that have rank > 1 are elements on the higher secant varieties of S = Seg(Rl , Rm , Rn ). E.g. a hypermatrix has rank 2 if it sits on a secant line thr ...
... Seg(Rl , Rm , Rn ) = {A ∈ Rl×m×n | A = u ⊗ v ⊗ w} = {A ∈ Rl×m×n | ai1 i2 i3 aj1 j2 j3 = ak1 k2 k3 al1 l2 l3 , {iα , jα } = {kα , lα }} Hypermatrices that have rank > 1 are elements on the higher secant varieties of S = Seg(Rl , Rm , Rn ). E.g. a hypermatrix has rank 2 if it sits on a secant line thr ...
ch7
... We obtain the transpose of a matrix by writing its rows as columns (or equivalently its columns as rows). This also applies to the transpose of vectors. Thus, a row vector becomes a column vector and vice versa. In addition, for square matrices, we can also “reflect” the elements along the main diag ...
... We obtain the transpose of a matrix by writing its rows as columns (or equivalently its columns as rows). This also applies to the transpose of vectors. Thus, a row vector becomes a column vector and vice versa. In addition, for square matrices, we can also “reflect” the elements along the main diag ...
Potential , Curls, and Electrical Energy
... construct potentials for a given E-field although it is clearly important in its own right. We begin by writing an expression for the change in T. The kinetic energy T only depends on the three components of a particle’s velocity v_x, v_y, and v_z. A change in T (called dT) can be due to a change in ...
... construct potentials for a given E-field although it is clearly important in its own right. We begin by writing an expression for the change in T. The kinetic energy T only depends on the three components of a particle’s velocity v_x, v_y, and v_z. A change in T (called dT) can be due to a change in ...
Advanced Classical Mechanics Lecture Notes
... so the conservation law follows from δ L − P 2 /2M = 0. This conservation law ensures that one can always choose the center of mass frame, RCM = 0, by a Galilei transformation combined with a spatial translation. Note on Relativity: In special relativity the relation between inertial frames is given ...
... so the conservation law follows from δ L − P 2 /2M = 0. This conservation law ensures that one can always choose the center of mass frame, RCM = 0, by a Galilei transformation combined with a spatial translation. Note on Relativity: In special relativity the relation between inertial frames is given ...
Linear Continuous Maps and Topological Duals
... Definition. Suppose X is a vector space, equipped with a linear topology T. A subset H ⊂ X is said to be a T-closed half-space in X , if there exist φ ∈ (X , T)∗ and s ∈ R, such that H = {x ∈ X : Re φ(x) ≤ s}. Note that, by continuity, H is indeed closed. Note also that, by linearity, H is also conv ...
... Definition. Suppose X is a vector space, equipped with a linear topology T. A subset H ⊂ X is said to be a T-closed half-space in X , if there exist φ ∈ (X , T)∗ and s ∈ R, such that H = {x ∈ X : Re φ(x) ≤ s}. Note that, by continuity, H is indeed closed. Note also that, by linearity, H is also conv ...
