Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 58, Problem 8AR
Using a simple protractor, measure each of the angles, ∠l through ∠7, to the nearer degree.
It may be necessary to extend sides of angles.
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
6. [10 marks]
Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
T.
(a) How many vertices does BL(T) have?
(b) How many edges does BL(T) have?
Prove that your answers are correct.
Chapter 58 Solutions
Mathematics For Machine Technology
Ch. 58 - Add, subtract, multiply, or divide each of the...Ch. 58 - Determine A.Ch. 58 - Prob. 3ARCh. 58 - Express 68.85 as degrees and minutes.Ch. 58 - Express 64.1420 as degrees, minutes, and seconds.Ch. 58 - Express 3723' as decimal degrees to 2 decimal...Ch. 58 - Express 10338'43" as decimal degrees to 4 decimal...Ch. 58 - Using a simple protractor, measure each of the...Ch. 58 - Prob. 9ARCh. 58 - Write the complement of each of the following...
Ch. 58 - Write the complement of each of the following...Ch. 58 - Given: ABCD and FEGH . Determine the value of each...Ch. 58 - a. Determine: (1) 1 (2) Side a b. Determine: (1) 1...Ch. 58 - a. Given: a=8.400 and b=9.200 . Find c. b. Given:...Ch. 58 - Compute 1.Ch. 58 - Determine the circumference of a circle that has a...Ch. 58 - Determine the diameter of a circle that has a...Ch. 58 - a. Given: CD=184 mm and CE=118 mm. Determine CF...Ch. 58 - a. Given: EB=5.150. Determine AE . b. Given:...Ch. 58 - Given: Points A and E are tangent points. EB is a...Ch. 58 - a. Given: AC=110andr=4.700 Compute arc length AC...Ch. 58 - a. Given: Dia H=14.520 and d=8.300. Compute Dia M....Ch. 58 - Prob. 23ARCh. 58 - a. Given: x=360 inches and y=5.10 inches. Compute...Ch. 58 - Prob. 25ARCh. 58 - A flat is cut on a circular piece as shown....Ch. 58 - A spur gear is shown. Pitch circles of spur gears...Ch. 58 - Determine the arc length from point C to point D...Ch. 58 - Prob. 29ARCh. 58 - Determine dimension x to 3 decimal places.Ch. 58 - Refer to the drill jig shown. Determine 1.Ch. 58 - Prob. 32ARCh. 58 - Prob. 33ARCh. 58 - Lay out the template shown. Make the layout full...
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- Refer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward
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