Hamiltonian opertaor problems
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9. Identify which of these problems are NP-complete and which can be exactly solved using a polynomial time algorithm (a) Finding the vertex cover in a line graph (b) Finding the maximum clique in a tree (c) Finding the independent set in complete graph (d) Finding the Hamiltonian cycle in a graph that has exactly one cycle
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete. Note:...
4. This problems deals with sudden approximation, please do not use perturbation theory. (a) (1 point) The system is described by the Hamiltonian t < 0, Ho+V, t>0 - and neither Ho nor V depends on time. Assuming that the system was in the eigenstate n) of Ho at t < 0, find the probability to find the system in the eigenstate V) of Ho V at t>0. (b) (5 points) Consider a particle in the ground state an infinite...
Hi,. the question is below: Help if you can.. Here is some background information/ an example: 9. Let k-Color be the following problem. Input: An undirected graph G. Question: Can the vertices of G be colored using k distinct colors, so that every pair of adjacent vertices are colored differently? Suppose that you were given a polynomial time algorithm for (k + 1)-Color. Use it to give a polynomial algorithm for k-Color. This means that you need to provide a...
Let G be a simple graph with at least four vertices. a) Give an example to show that G can contain a closed Eulerian trail, but not a Hamiltonian cycle. b) Give an example to show that G can contain a closed Hamiltonian cycle, but not a Eulerian trail.
Write down a generic molecular Hamiltonian (or, if you want a concrete example, the one for H2). Then using words and equations, show precisely what steps and approximations are made to ultimately come up with the harmonic oscillator Hamiltonian for vibration and the rigid rotator Hamiltonian for rotation of the molecule.
Guidelines for reduction problems. A reduction needs to be an accurate description of a polynomial-time process for converting instances of the input problem to instances of the target problem. You can use the examples from the slide deck as a basis for your descriptive level, while keeping to the following guidelines. Reduction descriptions: should be both clear and concise. should be composed of clearly enumerated steps that a well-practiced programmer could follow to construct an implementation. should include an analysis...
1. Problems can arise out of even the simplest properties of divisibility. For example, what are the possible common divisors of 12n and 2n +1? Does each occur infinitely often? 1. Problems can arise out of even the simplest properties of divisibility. For example, what are the possible common divisors of 12n and 2n +1? Does each occur infinitely often?
Discuss with an example one of the problems with standard regression analysis that can be solved by using instrumental variable(s). What are the conditions that the instrumental variable needs to satisfy? Discuss (you can refer to your example).
Example 8.3-3 (Companion Problems) (See Example 8.3-3 in the textbook for the solution to a similar problem.) 1-0 50 kQ 20 k2 24 V ( The switch in this circuit has been closed for a long time, and the circuit has reached steady state before the switch opens at time t-0. After the switch opens, the capacitor voltage is given by v)A+ Be v where A, B and a are constants. Determine the values of A, B and a. V...