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10. Prove the uniqueness of the Dirichlet problem Δι-f in D, 11-g on bdy D by...
Hint: assume that there are two solutions u and v and consider a point on the boundary of D where...
Hint: assume that there are two solutions u and v and consider a
point on the boundary of D where u - v is maximal.
11. Prove the uniqueness of the Robin problem au Ll an where D is any domain in three dimensions and where a is a positive constant.
11. Prove the uniqueness of the Robin problem au Ll an where D is any domain in three dimensions and where a is a positive constant.
#2 Problem 2 . If the Wronskian of f and g is tcost - sint and...
#2
Problem 2 . If the Wronskian of f and g is tcost - sint and if u f+3g z-f-g find the Wronskian of u and z. 0, find a fundamental set of . Given that i(t) is a solution of 2t2y(2) +3ty1) -y 0;t> solution. Problem 3 . Find the L-11 . Using power series method provide solution for the d.e. Problem 4 . Provide the Convolution Theorem and its prove. ve using Laplace transform y2 +2y+5y 0, y(0)...
solve problem #1 depending on the given information Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
Two questions,please! 7. Assume C is a linear code. Prove that G is a generator matrix for C if and only if the columns of G form a basis of C 8. Let V. W U be vector spaces over F of finite dimen...
Two questions,please!
7. Assume C is a linear code. Prove that G is a generator matrix for C if and only if the columns of G form a basis of C 8. Let V. W U be vector spaces over F of finite dimension and φ: V → W, t : W → U linear maps. Prove that Im(φ)-ker( ) holds if and only if ψφ-0 and dimF1m(φ)-dimF kere).
7. Assume C is a linear code. Prove that G is...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four c...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a sta...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a starting vertex DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s) 1: RELAX (u, v, w) 1: if dlv] > dlu (u, v) then 2d[v] <- d[u] +w(u, v) 3 4: end if 4: while Q φ do 5: uExTRACT-MIN Q) for each vertex v...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
Hint: assume that there are two solutions u and v and consider a
point on the boundary of D where u - v is maximal.
11. Prove the uniqueness of the Robin problem au Ll an where D is any domain in three dimensions and where a is a positive constant.
11. Prove the uniqueness of the Robin problem au Ll an where D is any domain in three dimensions and where a is a positive constant.
#2
Problem 2 . If the Wronskian of f and g is tcost - sint and if u f+3g z-f-g find the Wronskian of u and z. 0, find a fundamental set of . Given that i(t) is a solution of 2t2y(2) +3ty1) -y 0;t> solution. Problem 3 . Find the L-11 . Using power series method provide solution for the d.e. Problem 4 . Provide the Convolution Theorem and its prove. ve using Laplace transform y2 +2y+5y 0, y(0)...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
Two questions,please!
7. Assume C is a linear code. Prove that G is a generator matrix for C if and only if the columns of G form a basis of C 8. Let V. W U be vector spaces over F of finite dimension and φ: V → W, t : W → U linear maps. Prove that Im(φ)-ker( ) holds if and only if ψφ-0 and dimF1m(φ)-dimF kere).
7. Assume C is a linear code. Prove that G is...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a starting vertex DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s) 1: RELAX (u, v, w) 1: if dlv] > dlu (u, v) then 2d[v] <- d[u] +w(u, v) 3 4: end if 4: while Q φ do 5: uExTRACT-MIN Q) for each vertex v...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
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**86 at 40 not 88. please show how got Tr in calculations thank
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