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  1. M

    Length of an infinite square well?

    Actually, this is more of a general question relating to a homework problem I already did. I was given the initial wavefunction of a particle in an infinite square well: \Psi(x,0) = Ax if (0 \leq x \leq \frac{a}{2}), and =A(a-x) if (\frac{a}{2} \leq x \leq a) And of course \Psi(0,0) =...
  2. M

    Linear Algebra: Invariant Subspaces

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V. Assume that V is finite dimensional. The attempt at a solution I really think that I should be able to produce a counterexample, however...
  3. M

    Specific Linear Map Example

    Homework Statement Give a specific example of an operator T on R^4 such that, 1. dim(nullT) = dim(rangeT) and 2. dim(the intersection of nullT and rangeT) = 1 The attempt at a solution I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2. I also...
  4. M

    More linear maps

    Homework Statement Prove that if there exists a linear map on V whose null space and range are both finite dimensional, then V is finite dimensional. The attempt at a solution I *think* the following is true: For all v in V, T(v) is in range(T), otherwise T(v) = 0 which implies v is in...
  5. M

    Linear Algebra, Linear Maps

    1. Homework Statement Suppose that T is a linear map from V to F, where F is either R or C. Prove that if u is an element of V and u is not an element of null(T), then V = null(T) (direct sum) {au : a is in F}. 2. Relevant information null(T) is a subspace of V For all u in V, u is...
  6. M

    Epidemic models which incorporate disease evolution

    I was wondering if anyone knew anything about epidemic models which take into account the ability of a disease to mutate. Basically I’m curious if there are any existing models which could predict how a rapidly changing disease might affect the progression of an epidemic, or how slower...
  7. M

    LaTeX Including graphics in LaTeX Help!

    hi, I'm trying to put a graph generated in maple into a latex document, but I have no experience using either program. So far I've been able to save my maple plot in postscript format, and based on various online tutorials I've included the \usepackage{graphics} comand after...
  8. M

    Linear Algebra: Subspace Proof

    Prove that the intersection of any collection of subspaces of V is a subspace of V. Ok, I know I need to show that: 1. For all u and v in the intersection, it must imply that u+v is in the intersection, and 2. For all u in the intersection and c in some field, cu must be in the...
  9. M

    G.c.d.'s and PID's

    1. Homework Statement Prove: If a, b are nonzero elements in a PID, then there are elements s, t in the domain such that sa + tb = g.c.d.(a,b). 2. Homework Equations g.c.d.(a,b) = sa + tb if sa + tb is an element of the domain such that, (i) (sa + tb)|a and (sa + tb)|b and (ii) If...
  10. M

    Guided proof to the isomorphism theorems.

    1. Homework Statement Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient...
  11. M

    Infinite Dimensional Linear Algebra Proof

    Prove that $\mathbf{F}$^{\infty} is infinite dimensional. $\mathbf{F}$^{\infty} is the vector space consisting of all sequences of elements of $\mathbf{F}$, and $\mathbf{F}$ denotes the real or complex numbers. I was thinking of showing that no list spans $\mathbf{F}$^{\infty}, which would...
  12. M

    QM Value of an infinite sum

    hi, In the course of doing my quantum homework I ran into a bit of a snag. In one of my calculations I need to replace the sum from n = 1 to infinity of 1/n^2 (for odd n only) with its number value. My book instructs me to get the information from a table and actualy gives the value (for...
  13. M

    Are the real numbers compact?

    This is something that I think I should already know, but I am confused. It really seems to me that the set of all real numbers, \Re should be compact. However, this would require that \Re be closed and bounded, or equivalently, that every sequence of points in \Re have a limit...
  14. M

    Continuous functions, confusion with notation.

    hi, My question reads: Let f be defined and continuous on the interval D_1 = (0, 1), and g be defined and continuous on the interval D_2 = (1, 2). Define F(x) on the set D=D_1 \cup D_2 =(0, 2) \backslash \{1\} by the formula: F(x)=f(x), x\in (0, 1) F(x)=g(x), x\in (1, 2)...
  15. M

    Curious statement about operators in my QM book?

    hi, In a discussion of the historical motivations for a move from calculus to operators, my QM book says... "Many mathematicians were uncomfortable with the 'metaphysical implications' of a mathematics formulated in terms of infinitesimal quantities (like dx). This disquiet was the stimulus...
  16. M

    Sets & limit points and stuff

    The question says: Let A be a set and x a number. Show that x is a limit point of A if and only if there exists a sequence x_1 , x_2 , ... of distinct points in A that converge to x. Now I know from the if and only if statement that I need to prove this thing both ways. So, the...
  17. M

    Not so open minded open sets

    This is not a specific homework question so much as it is a general conceptual question. My analysis book includes a theorem that states: 1. The union of any number of open sets is an open set. 2. The intersection of a finite number of open sets is an open set. I follow the proof of...
  18. M

    How to prove stuff about linear algebra?

    How to prove stuff about linear algebra??? Question: Suppose (v_1, v_2, ..., v_n) is linearly independent in V and w\in V. Prove that if (v_1 +w, v_2 +w, ..., v_n +w) is linearly dependent, then w\in span(v_1, ...,v_n). To prove this I tried... If (v_1, v_2, ..., v_n) is linearly...
  19. M

    Do open sets stay open?

    Use the definition of an open set to show that if a finite number of points are removed, the remaining set is still open. Definition: A set is open if every point of the set lies in an open interval entirely contained in the set. I'm a bit lost, but I think that I somehow need to show...
  20. M

    Prove lim = inf(x)

    Question: Suppose there is a set E\subset \Re is bounded from below. Let x=inf(E) Prove there exists a sequence x_1, x_2,... \in E, such that x=lim(x_n). I am not sure but it seems like my x=lim(x_n) =liminf(x_n). In class we constructed a Cauchy sequence by bisection to find sup...
  21. M

    Compute sup,inf, and more

    Question: (I've got a few like this, so I'd like to know if I am doing them correctly.) Compute the sup, inf, limsup, liminf, and all the limit points of the following sequence x_1, x_2,... where x_n = 1/n + (-1)^n What I did was write down the first few terms to get an idea of the...
  22. M

    Inequality of Supremums

    Theorem: For every non empty set E of real numbers that is bounded above there exists a unique real number sup(E) such that 1. sup(E) is an upper bound for E. 2. if y is an upper bound for E then y \geq sup(E). Prove: sup(A\cap B)\leq sup(A) I can show a special case of...
  23. M

    More Cauchy equivalence

    Question: Prove that if a Cauchy sequence x_1, x_2,... of rationals is modified by changing a finite number of terms, the result is an equivalent Cauchy sequence. All the math classes I have taken previously were computational, and my textbook contains almost no definitions. So, I...
  24. M

    Prove Cauchy sequence & find bounds on limit

    Here's the problem statement: Prove that x_1,x_2,x_3,... is a Cauchy sequence if it has the property that |x_k-x_{k-1}|<10^{-k} for all k=2,3,4,.... If x_1=2, what are the bounds on the limit of the sequence? Someone suggested that I use the triangle inequality as follows: let n=m+l...
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