MATH 425, HOMEWORK 1, SOLUTIONS
MATH 425, HOMEWORK 1, SOLUTIONS 3 Again, we need to choose the functions h 1 and h 2 in such a way that the function u is di erentiable. b) Since the value of u is given on the y-axis, it follows that the solution is uniquely determined along the characteristic curves which intersect the y-axis. These includes the upwards and downwards
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Linear Systems of Di erential Equations Math 240 First order linear systems Solutions Beyond rst order systems Solutions to homogeneous linear systems As with linear systems, a homogeneous linear system of di erential equations is one in which b(t) = 0. Theorem If A(t) is an n n matrix function that is continuous on the
1. Geometric and telescoping series The geometric series is X1 n=0 a nr n = a + ar + ar2 + ar3 + = a 1 r provided jrj<1 (when jrj 1 the series diverges). We often use partial fractions to detect telescoping series, for which we can calculate explicitly the partial sums S n. D. DeTurck Math 104 002 2018A: Series 3/42
Eigenvectors Math 240 De nition Computation and Properties Chains Facts about generalized eigenvectors The aim of generalized eigenvectors was to enlarge a set of linearly independent eigenvectors to make a basis. Are there always enough generalized eigenvectors to do so? Fact If is an eigenvalue of Awith algebraic multiplicity k, then nullity ...
ating functions can be simple and easy to handle even in cases where exact ... Generating functions can give stunningly quick deriva-tions of various probabilistic aspects of the problem that is repre-sented by your unknown sequence. (d) …
Number theory is ﬁlled with questions of patterns and structure in whole numbers. One of the most important subsets of the natural numbers are the prime numbers, to which we now turn our attention. ... make up the matter of our universe. Unlike the periodic table of the elements, however, the list of prime numbers goes on indeﬁnitely.
g) The linear transformation T A: Rn!Rn de ned by Ais onto. h) The rank of Ais n. i) The adjoint, A, is invertible. j) detA6= 0. 14. Call a subset S of a vector space V a spanning set if Span(S) = V. Suppose that T: V !W is a linear map of vector spaces. a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets
5.3 Planar Graphs and Euler’s Formula Among the most ubiquitous graphs that arise in applications are those that can be drawn in the plane without edges crossing. For example, let’s revisit the example considered in Section 5.1 of the New York City subway system. We considered a graph in which vertices represent subway stops and edges represent
Find all of the eigenvalues and eigenvectors of A= 1 1 0 1 : The characteristic polynomial is ( 1)2, so we have a single eigenvalue = 1 with algebraic multiplicity 2. The matrix A I= 0 1 0 0 has a one-dimensional null space spanned by the vector (1;0). Thus, the geometric multiplicity of this eigenvalue is 1.
the chain rule gives df dx = @f @x + @f @y ·y0. (11.3) The notation really makes a di↵erence here. Both df /dx and @f/@x appear in the equation and they are not the same thing! Derivative along an explicitly parametrized curve One common application of the multivariate chain rule is when a point varies along
is the study of computational complexity. Naturally, we would expect that a computing problem for which millions of bits of input data are required would probably take longer than another problem that needs only a few items of input. So the time complexity of a calculation is measured by expressing the running time of the calculation as a ...
Jan 17 Homework Solutions Math 151, Winter 2012 Chapter 2 Problems (pages 50-54) Problem 2 In an experiment, a die is rolled continually until a 6 appears, at which point the experi-ment stops. What is the sample space of this experiment? Let E n denote the event that n rolls are necessary to complete the experiment. What points of the sample ...
Math 365: Elementary Statistics Homework and Problems (Solutions) Satya Mandal Spring 2019, Updated Spring 22, 6 March
Math 241 Homework 12 Solutions Section 6.1 Problem 1. The solid lies between planes perpendicular to the x-axis at x=0 and x=4. The cross-sections perpendicular to the axis on the interval 0 ≤x≤4 are squares whose diagonals run from the parabola y=− √ xto the parabola y= √ x Solution From the picture we have s2 +s2 =(2 √ x)2 ⇔ 2s2 ...
MATH 402A - Solutions for Homework Assignment 3 Problem 7, page 55: We wish to ﬁnd C(a) for each a ∈ S3. It is clear that C(i) = S3. For any group G and any a ∈ G, it is clear that every power of a commutes with a and therefore (a) ⊆ C(a) . Assume that a ∈ S3 and a 6= i. Then a has order 2 or 3. Thus, the subgroup (a) of S3 has order ...
SOLUTIONS OF SOME HOMEWORK PROBLEMS MATH 114 Problem set 1 4. Let D4 denote the group of symmetries of a square. Find the order of D4 and list all normal subgroups in D4. Solution. D4 has 8 elements: 1,r,r2,r3, d 1,d2,b1,b2, where r is the rotation on 90 , d 1,d2 are ﬂips about diagonals, b1,b2 are ﬂips about the lines joining the centersof opposite sides of a square.
SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5 Problem 1: For each of the following functions do the following: (i) Write the function as a piecewise function and sketch its graph, (ii) Write the function as a combination of terms of the form u a(t)k(t a) and compute the Laplace transform (a) f(t) = t(1 u
Solutions for homework assignment #4 Problem 1. Solve Laplace’s equation inside a rectangle 0 ≤ x ≤ L, 0 ≤ y ≤ H, with the following boundary conditions:
Homework 1 Solutions 1.1.4 (a) Prove that A ⊆ B iﬀ A∩B = A. Proof. First assume that A ⊆ B. If x ∈ A ∩ B, then x ∈ A and x ∈ B by deﬁnition, so in particular x ∈ A. This proves A ∩ B ⊆ A. Now if x ∈ A, then by assumption x ∈ B, too, so x ∈ A ∩ B. This proves A ⊆ A ∩ B. Together this implies A = A∩B.
Math 215 HW #8 Solutions 1. Problem 4.2.4. By applying row operations to produce an upper triangular U, compute det 1 2 −2 0 2 3 −4 1