In this problem a, b, c ≥ 0 are constants 1.1. (3) Suppose with the side condition w(n) = w(an) + w(bn) + cn, Find a solution to this equation. Hint: Use the ansatz: w(n) = d n lgn. a + b = 1.

Question 1 a) State the advantages of applying Karnaugh map instead of Boolean Laws to simplify a function. b) Let's say your ULAB student ID is 173014009. So, from the available distinct digits in the student ID, a function F can be expressed as F = E(0, 1,3,4,7,9). Now, express the following function F into a simplified Product-of-Sum (POS) form. Show step-by-step process to express the function F. F = E(digits in your student ID)

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Solve the following homogeneous recurrence equations and obtain the general solution and the solution based on the initial conditions

please show step by step solution. thank you

Theorem 1 (Master Theorem). Let a ≥ 1 and b> 1 be constants, let f(n) be a function f: N→ R+, and let T(n) be defined on the nonnegative integers by the recurrence T(n)= aT(n/b) + f(n). Then T(n) has the following asymptotic bounds: 1. if f(n) = O(nlogs a-E) for some constant e > 0, then T(n) = (nossa), 2. if f(n) = (nlos a) then T(n) = (nloga log n), and 3. if f(n) = N(n¹og a+s) for some e > 0 and if af (n/b) ≤df (n) for some constant d < 1 and all sufficiently large n, then T(n) = (f(n)). Use the Master Theorem above to solve the following recurrences when possible (note that not every blank needs to be filled in in every problem):

A manufacturer of programmable calculators is attempting to determine a reasonable free-service period for a model it will introduce shortly. The manager of product testing has indicated that the calculators have an expected life of 60 months. Assume product life can be described by an exponential distribution. T / MTBF -T / MTBF T| MTBF -т / МТВF T| MTBF -т / МТВF 0.10 0.9048 2.60 0.0743 5.10 0.0061 0.20 0.8187 2.70 0.0672 5.20 0.0055 0.30 0.7408 2.80 0.0608 5.30 0.0050 0.40 0.6703 2.90 0.0550 5.40 0.0045 0.50 0.6065 3.00 0.0498 5.50 0.0041 0.60 0.5488 3.10 0.0450 5.60 0.0037 0.70 0.4966 3.20 0.0408 5.70 0.0033 0.80 0.4493 3.30 0.0369 5.80 0.0030 0.90 0.4066 3.40 0.0334 5.90 0.0027 1.00 0.3679 3.50 0.0302 6.00 0.0025 1.10 0.3329 3.60 0.0273 6.10 0.0022 1.20 0.3012 3.70 0.0247 6.20 0.0020 1.30 0.2725 3.80 0.0224 6.30 0.0018 1.40 0.2466 3.90 0.0202 6.40 0.0017 1.50 0.2231 4.00 0.0183 6.50 0.0015 1.60 0.2019 4.10 0.0166 6.60 0.0014 1.70 0.1827 4.20 0.0150 6.70 0.0012 1.80 0.1653 4.30…