ME-week5 -NOTES

 

DL-CNN_Questions

 What happend when apply the input image convolution with the kernal of all elements are 1?

The image you've uploaded shows the Taj Mahal and a 3×3 convolution kernel consisting entirely of 1s:

$$\text{Kernel}=[ \begin{array}{ccc}\mathrm{1 }& \mathrm{1 }& 1\end{array}$$

$$\begin{array}{ccc}\mathrm{ \; \; \; \; \; \; \; 1 }& \mathrm{1 }& 1\end{array}$$

$$\begin{array}{ccc}\mathrm{ \; \; \; \; \; \; \; 1 }& \mathrm{1 }& 1\end{array}]\; 3x3$$

This is a convolution operation typically used in image processing.

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What happens during convolution?

The kernel slides over the image pixel by pixel. At each location, the element-wise multiplication of the kernel and the corresponding 3×3 patch of the image is computed, and all the results are summed up to produce the output pixel.

For a 3×3 kernel of all 1s:

• It sums all the 9 neighboring pixel values (including the center pixel).

• This value can be optionally normalized by dividing by 9.

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What is the result of this operation?

This is known as a mean filter (box blur) when you normalize the kernel.

• If you divide the kernel by 9:

  $$[\begin{array}{ccc}\frac{\mathrm{1/}}{\mathrm{9 }}& \frac{\mathrm{1/}}{\mathrm{9 }}& \frac{\mathrm{1/}}{9}\end{array}$$
• 
  
•  1/9 1/9 1/9
• 
  
•  1/9 1/9  1/9]

• You get a blurred image.

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Why does this happen?

• The kernel averages each pixel with its neighbors.

• Sharp details (like edges or textures) are smoothed out.

• This results in a blurring effect.

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Summary:

• Input: Image of the Taj Mahal.

• Kernel: 3×3 all-ones matrix.

• Operation: Convolution (possibly normalized).

• Output: A blurred version of the Taj Mahal image.

• Why: Each pixel is replaced by the sum (or average) of its neighboring pixels, reducing sharpness.

Testing notes

 

DeepLearning

TRY YOUR SELF THIS QUESTIONS

 1.Which architecture can approximate any continuous function on [0,1]?

• (A) Single hidden layer with ReLU and enough neurons

• (B) Deep network with arbitrary width but no nonlinearity

• (C) Only networks with sigmoid activations

• (D) None of the above

2. why deep networks can be more parameter-efficient than shallow ones when approximating      complex functions?

3. What core challenge arises when generalizing bump functions to two or more inputs?

4. Analyze how many bump functions you would need (and thus neurons) to approximate a          piecewise-constant function with 10 segments. What if you use two hidden layers—how          does "depth" help?

5. Construct a network using ReLUs that approximates the triangular waveform?

6. True or False: The “bump building” method from requires exponentially many neurons        as   dimension increases?

7.Derive a 1‑hidden-layer network that emulates a step function?

 Given ReLU activation, find weights and biases 

• $w_1, w_2, b_1, b_2, v_1, v_2$ such that the network’s output approximates a step at $x = a$

$f(x) = \sin(\pi x)$ on [0, 1]. Show how to approximate it using a finite sum of         bump functions, and write the corresponding neural network formulation?

ManagerialEconomics-week4-Notes

 ManagerialEconomics-week4-Notes

ManagerialEconomics-week3-Notes

 ManagerialEconomics-week3-Notes

ME-WEEK2-LECTURER 1

 ME-WEEK2-LECTURER 1

ME

 Mutual Industries owns 3 plants at which it produces exactly the same cars.

Plant 1 has cost function:

• $T{C}_1=300-10q+50q^2$ → (1)

Plant 2:

• $T{C}_2=50+10q^2$→ (2)

Plant 3:

• $T{C}_3=1000+20\mathrm{q3}$→ (3)

Mutual decides to produce 5 cars in the least costly way.
→ $Q=5$

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(1) If Mutual Industries decides to produce 5 cars in the most least costly manner in a single plant, which plant will be chosen?

Put $q=5$ in all 3 equations:

• $T{C}_1=300-10\times 5+50\times 5^2=300-50+1250=1500$

• $T{C}_2=50+10\times 5^2=50+250=300$

• $T{C}_3=1000+20\times 5=1100$

 Least cost → Plant 2

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(2) How many cars will be produced in each plant?

Let:

• $q_1$ cars produced in Plant 1

• $q_2$ cars produced in Plant 2

• $q_{\mathrm{3=}}$ cars produced in Plant 3

Then:

• $q_1+q_2+q_3=5$ → (i)

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Image 2:

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(ii) If the production takes place in different plants, how many cars will be produced in each plant? Find the cost?

No. of cars produced	Plant 1	Plant 2	Plant 3
1	240	60	1002
2	480	90	1040
3	720	140	1050
4	1500	210	1080
5	1500	300	1100

Cost of production is least when Plant 2 does it.

 If only no. of cars = 1 + x is scaled, 5 cars will be produced in Plant 2 only → at cost 300

→ 5th no. of cars produced in:

• $P_1,P_2,P_3=0,5,0$
  

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(3) Budget eqn is given by: $P_1{x}_1+P_2{x}_2=m$
The government decides to impose:

• a lump sum tax of t (on income),

• a unit tax on good 1 of τ (→ always add: + tax),

• and a per unit subsidy on good 2 of s (→ minus: – subsidy)

What is the formula for the new budget line?

New Budget Equation:

$$(P_1 + τ)x_1 + P_2x_2 = (m - t + s)$$