PHASE PORTRAITS OF LINEAR SYSTEMS For our purposes phase
... 3.2. Zero eigenvalues. If one eigenvalue has zero real part, then there are two cases: If the eigenvalues are complex, then they form a conjugate pair and must hence both be imaginary. We just finished this case. If the eigenvalues are real, then having zero real part means being zero. The case wher ...
... 3.2. Zero eigenvalues. If one eigenvalue has zero real part, then there are two cases: If the eigenvalues are complex, then they form a conjugate pair and must hence both be imaginary. We just finished this case. If the eigenvalues are real, then having zero real part means being zero. The case wher ...
The graph planar algebra embedding
... • a multilinear map P(T ) : ⊗PTi → PT0 for each G-planar tangle T , such that the maps only depend on the G-planar tangle up to isotopy, and the usual associativity constraints for planar algebras hold. If all the vector spaces Pg , for g a length zero loop (that is, just a vertex), are 1-dimensiona ...
... • a multilinear map P(T ) : ⊗PTi → PT0 for each G-planar tangle T , such that the maps only depend on the G-planar tangle up to isotopy, and the usual associativity constraints for planar algebras hold. If all the vector spaces Pg , for g a length zero loop (that is, just a vertex), are 1-dimensiona ...
Vector Geometry - NUS School of Computing
... normal n on π. Let q denote a point whose projected point on π is not p. Show that the perpendicular distance d of q to π is given by d = (q − p) · n. 3. Solve Eq. 32 to obtain Eq. 33 for the intersecting line of two planes. 4. Show that for the n1 and n2 in Eq. 34, n1 · n2 = 0, and ...
... normal n on π. Let q denote a point whose projected point on π is not p. Show that the perpendicular distance d of q to π is given by d = (q − p) · n. 3. Solve Eq. 32 to obtain Eq. 33 for the intersecting line of two planes. 4. Show that for the n1 and n2 in Eq. 34, n1 · n2 = 0, and ...
![[Part 1]](http://s1.studyres.com/store/data/008797132_1-ed28b78ba857535a88b7a26b319a4fff-300x300.png)
[Part 1]
... which of course we recognize is the Binet form for the Pell sequence,, In fact, similarly we can find Binet forms for Fibonacci, Lucas, or any other Homogenous Linear Difference Equations where roots to S.A.x , the characteristic, are distinct. One more logical extension of Fibonacci sequence is the ...
... which of course we recognize is the Binet form for the Pell sequence,, In fact, similarly we can find Binet forms for Fibonacci, Lucas, or any other Homogenous Linear Difference Equations where roots to S.A.x , the characteristic, are distinct. One more logical extension of Fibonacci sequence is the ...
18.03 Differential Equations, Lecture Note 33
... the columns of the matrix weighted by the entries in the column vector. When is this product zero? One way is for x = 0 = y. If [a ; c] and [b ; d] point in different directions, this is the ONLY way. But if they lie along a single line, we can find x and y so that the sum cancels. Write A = [a b ; ...
... the columns of the matrix weighted by the entries in the column vector. When is this product zero? One way is for x = 0 = y. If [a ; c] and [b ; d] point in different directions, this is the ONLY way. But if they lie along a single line, we can find x and y so that the sum cancels. Write A = [a b ; ...
Solution 2
... Thus we would have ck+1 6= 0 and could solve this equation for y and find that y ∈ L(B), contradicting our assumption that y is not in L(B). Therefore, the set B 0 is independent but contains k + 1 elements. If L(B 0 ) = S, then B 0 is a basis of S. If L(B 0 ) 6= S, we can argue with B 0 as we did w ...
... Thus we would have ck+1 6= 0 and could solve this equation for y and find that y ∈ L(B), contradicting our assumption that y is not in L(B). Therefore, the set B 0 is independent but contains k + 1 elements. If L(B 0 ) = S, then B 0 is a basis of S. If L(B 0 ) 6= S, we can argue with B 0 as we did w ...
Whitney forms of higher degree
... q-moments, with p < q ≤ d, for finite p-elements of order k, are not integrals over domains of dimension q, but integrals of these p-elements of order k over suitable p-chains. This statement has a dual side. If one takes as dofs for a p-form u its integrals over small p-simplices, then Whitney p-for ...
... q-moments, with p < q ≤ d, for finite p-elements of order k, are not integrals over domains of dimension q, but integrals of these p-elements of order k over suitable p-chains. This statement has a dual side. If one takes as dofs for a p-form u its integrals over small p-simplices, then Whitney p-for ...
31 Semisimple Modules and the radical
... also contains nilpotent elements (since A3 ⊆ M3 (F )). However, M3 (F ) contains no nontrivial nilpotent two sided ideal. rA3 is a nilpotent ideal in A3 since the product of any three elements of rA3 is zero. Another example is F Z/3 where F is a field of characteristic 3. In that case we have (1 − ...
... also contains nilpotent elements (since A3 ⊆ M3 (F )). However, M3 (F ) contains no nontrivial nilpotent two sided ideal. rA3 is a nilpotent ideal in A3 since the product of any three elements of rA3 is zero. Another example is F Z/3 where F is a field of characteristic 3. In that case we have (1 − ...
Sample Paper - 2008 Mathematics
... One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me 10, I shall be six time as rich as you”. Tell me what is the amount of their capital? A and B each have certain nmber of oranges. A says to B, “if you give me 10 of your oranges, I wil ...
... One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me 10, I shall be six time as rich as you”. Tell me what is the amount of their capital? A and B each have certain nmber of oranges. A says to B, “if you give me 10 of your oranges, I wil ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.