QUANTUM COMPUTING - EXAMPLE 32.7 : If | ± > = 0.707 ( | 0 > ± | 1 > ), prove that | Ψ (t = 0) > = | 0 > = 0.707 ( | + > + | - > ).

QUANTUM COMPUTING - EXAMPLE 32.8 : In quantum computing, a quantum state is given by S = a | 00 > + b | 01 > + g | 10 > + d | 11 >. (a) Find S in term of | 0 > and | 1 > etc. (b) The probability of getting x is P(x). For S = 0.5 | 00 > + 0.5 | 01 > + 0.5 | 10 > + 0.5 | 11 >, find P(0) and P(1). Hint : P(00) + P(01) = P(0) = a x a + b x b, P(10) + P(11) = P(1) = g x g + d x d.

QUANTUM COMPUTING - EXAMPLE 32.9 : In quantum computing, find the equations of S = (a | 0 > + b | 1 >) (g | 0 > + d | 1 >) in term of | 00 >, | 01 >, | 10 > and / or | 11 > when ad = 0.

QUANTUM COMPUTING - EXAMPLE 32.10 : In quantum computing, the conversion of Control Not (CNOT) gate in two input quantum bit gate could be decribed as : | 00 > --> | 00 >, | 01 > --> | 01 >, | 10 > --> | 11 >, | 11 > --> | 10 >. If | P > = 0.707 ( | 01 > - | 11 > ), find the value of CNOT | P >.

QUANTUM BIOLOGY - EXAMPLE 33.1 : In quantum biology, conditional entropy is given by H (S | O) = H (SO) - H (O). With reference to such equation, find another equation for H (SA).

QUANTUM BIOLOGY - EXAMPLE 33.2 : (a) In a biomolecule, ATP hydrolysis will produce 3 kcal / mol to 60 kcal / mol of energy, according to energy scale. Find the range of such energy generation if 2 moles of molecules involved. (b) A formula is given by ln P = ln a + b ln W where P = metabolic rate, W = body size, a = dependent of taxa, b = scaling exponent. If b is approximately 1 for plant, and b is approximately 0.75 for animal, find the relationship of P as a function of a and W in plant.

QUANTUM BIOLOGY - EXAMPLE 33.3 : In quantum biology, microtubule is used to store information in a cell. At temperature of T = 300 K, measured current I in a probe is directly proportional to the supplied voltage V, when passing through a microtubule with resistance R. (a) Form an equation of V as a function of I involving k as a constant. (b) If the microtubule has R = 1 ohm at such condition, find the value of V when I = 2 A. Hint : Ohm's law. (c) Find the relationship of k as a function of R.

QUANTUM BIOLOGY - EXAMPLE 33.4 : (a) According to Landauer (1986), the capacity of human memory is approximately X bits. Assume that a human retains 2 bits / second of visual, verbal, tactile and musical memory, find the value of X if a human lifetime is approximately 2.5 billion seconds. (b) The total power consumption of the human brain is about 25 Watts. The bread of 100 grams will produce 1000 kilojoules of energy. How much bread is needed to run a human brain for 1 day?

QUANTUM BIOLOGY - EXAMPLE 33.5 : (a) In the measurement of the number of photons received by a plant during photosynthesis, quantum meter is used. At 8 AM, 11 AM, 12 PM, 2 PM and 4 PM of the same day, the measured readings of the meter are 10, 70, 60, 120 and 120 units. Find the mode, median and min of the readings of the quantum meter. (b) Microtubule is used to carry information in a cell. If the cross-sectional area of the microtubule has a diameter of 25 nanometers, find the volume of the microtubule of 1 nanometer in height. State the assumption of calculation.

QUANTUM BIOLOGY - EXAMPLE 33.6 : The time to maintain quantum coherence in a biological system is t. Let t = k / (MT) where k = constant, M = mass, T = temperature. For living farm animals, normal rectal temperatures ranges for pig and goat are 38.7 - 39.8 and 38.5 - 39.7 in degree Celsius respectively. Let a pig has a mass of 100 kg and a goat has a mass of 300 pounds. (a) Find the mass of goat in the unit of kilogram, when 1 pound = 0.4536 kg. (b) Find t (p) / t (g) for living animals when t (p) = t for pig and t (g) = t for goat. (c) State the assumption of your calculation in question (b).

QUANTUM BIOLOGY - EXAMPLE 33.7 : (a) In a DNA of a living cell, the quantum information available in the bases guanine (G) and thymine (T) are | G > = | 110 > and | T > = | 010 > respectively. Calculate | G > - | T >. (b) In a living biological cell, the step time for random walk of an electron is t. The localization time of an electron is T. If i is the geometric average of T and t, find log T as a function of t and i.

QUANTUM BIOLOGY - EXAMPLE 33.8 : (a) Let ^ be the symbol of power where 1 ^ 2 = 1 x 1 = 1, 2 ^ 2 = 2 x 2 = 4. Let the number of electrons in a human body to be 10 ^ 28 = A, the number of all of the grains of sand on Earth planet to be 7 x (10 ^ 20) = B, the number of all the stars in the visible sky to be 8 x (10 ^ 3) = C. By assuming that every star in the visible sky has the same number of grains of sand as on Earth planet, prove by mathematical calculations that there are more electrons in one human body compared to the number of all of the grains of sand on the stars in the visible sky. (b) The incoming solar radiation to the Earth's surface is mainly from sun. Around 51 % of the radiation is absorbed by Earth's surface. Around 19 % is absorbed by atmosphere and clouds. In term of reflection, 4 % of the radiation is from surface of Earth, 6 % is reflected by atmosphere and the rest is reflected by clouds. Find the percentage of radiation absorbed by and reflected by biological beings on Earth, with reason for your response.

QUANTUM BIOLOGY - EXAMPLE 33.9 : (a) Let ^ be the symbol of power where 10 ^ 1 = 10, 10 ^ 2 = 100, 10 ^ 3 = 1000 etc. Total energy consumption of the brain is about 25 Watts, whereas a Blue Gene computer requires 1.5 Mega Watts. Blue Gene computer performs at 1 petaflop. In a human body, there are approximately 10 ^ 16 synapse operations per second i. e. at least 10 petaflops. Prove by calculations that a human brain is more energy efficient than a Blue Gene computer. (b) Quantum effects and quantum entanglement in the brain are identical to quantum gravity and string theory. If one is true, the other is true. What conclusion can be made if quantum effects in the brain and quantum gravity are true?

QUANTUM BIOLOGY - EXAMPLE 33.10 : The wavefunction starts out in a superposition of possible states in a closed black box like this : Ψ (kitty) = 0.7071 Ψ (alive) + 0.7071 Ψ (dead) where Ψ (kitty) = wavefunction of a kitten, Ψ (alive) = wavefunction of a living kitten, Ψ (dead) = wavefunction of a dead kitten. By prediction and calculation, find the probability of : (a) a living kitten inside the black box; (b) a dead kitten inside the black box.

ACCOUNTING AND FINANCIAL ENGINEERING - EXAMPLE 34.1 : (i) In the pricing of engineering bonds, 3 sets of data for Portfolio Value, Probability, Senior Tranche and Junior Tranche are : $2000, 81 %, $1000, $1000; $1000, 18 %, $1000, $0; $0, 1 %, $0, $0. By assuming independent defaults, find the price for : (a) Senior Tranche; (b) Junior Tranche. (ii) Assuming statistical independence of the values in the sample, the standard deviation of the mean (S) is related to the standard deviation of the distribution (s) by : N x S x S = s x s, where N is the number of observations in the sample used to estimate the mean. In a drug development project, let s = 1. Find the value of S if such a similar project is performed 100 times.