text:
#### Good job! You’re all caught up

Topcoder Copilot Recruitment Challenge (3)
TOSCA Editor - Web Application Wireframe Challenge (1)
Earlier Dismiss All
Topcoder Copilot Recruitment Challenge (2)

## July 19, 2019 Intoduction to Deep Learning using Pytorch

This main focus of the blog post is to introduce readers to Deep Learning and how Pytorch, a popular deep learning framework, is being used in the Deep Learning era.

So first things first, let’s get introduced to Deep Learning.

What is Deep Learning:

Are Machine Learning and Deep Learning the same or different? Let’s first understand what these two terms are exactly. In simple layman’s terms Machine Learning is a process where a machine tries to learn by itself with experience and without human intervention. How will the computer/machine learn? The machine will learn by feeding the data with features or attributes, choosing an algorithm to construct a classifier/model, training the model, and finally the machine will make predictions. The feature extraction part is done manually in traditional Machine Learning.

Example: Suppose you want to identify whether an image is of a cat or not. Using the Machine Learning approach you can classify the image.

Deep Learning, on the other hand, is a subset of Machine Learning in which the learning happens through a neural network/Artificial Neural Network (ANN). The structure of a neural network is inspired by the human brain. Like a human brain ANN consists of artificial neurons . An artificial neuron is the basic unit, there are many neurons in an ANN organized in layers and all of the neurons are densely connected to each other.

A Simple Artificial Neural Network

In the above image the network consists of an input layer, a hidden layer with 4 neurons, and an output layer with a single output. The term deep indicates the number of hidden layers in the network, i.e the more hidden layers in a neural network, the more Deep Learning it will do to solve complex problems. A simple neural network can consist of 2-3 layers whereas a deep neural network can consist of hundreds of hidden layers. In Deep Learning models are being trained by providing a large set of labeled data, and the neural network architecture learns features directly from data without explicitly doing manual feature extraction.

Popular Deep Learning methods

• Convolutional Neural Networks
• Recurrent Neural Networks
• Recursive Neural Networks
• Long short -term memory

So now we can say that both Machine learning and Deep Learning fall under the umbrella of Artificial Intelligence and even though Deep Learning is a subset of Machine Learning, Deep Learning is specific to problems which are more complex and have a more diverse and unstructured dataset. Feature extraction is automatic in Deep Learning, whereas Machine Learning requires manual feature extraction. Also, DL performs better for larger datasets, whereas Machine Learning is good for small or medium sized datasets.

The analogy to deep learning is that the rocket engine is the deep learning model and the fuel is the huge amounts of data we can feed to these algorithms.

– source (“Andrew NG”)

How Deep Learning works:

The basic building block of a neural network is an artificial neuron also known as perceptron. A Deep neural network consists of the following:

1. An Input Layer X
2. One or more hidden layers
3. An Output Layer (y)
4. A set of weights and biases between each layer denoted as W ,b
5. An Activation function for each hidden layer .

All the nodes/neurons in the input layer have some associated input weight, and the weight signifies the importance of the feature in the output prediction. Multiplication of the input feature and associated weight is summed up (this whole process is Matrix Multiplication) and passed to an Activation function to get the predicted output. Activation functions are used to introduce non-linearity in neural networks. In real world problems the data is most often nonlinear so in order to make neurons learn we use Activation functions.

Mathematical notation of the same is as follows:

In the above image the first layer contains 2 input units/neurons (x1,x2) with weight (w1,w2). The arrows are known as synapses, which take input and multiply it by weight, sum it together and assign it to hidden unit h. After that the activation function is applied to get the predicted output.

Let’s understand the computation of Neural Network. We have 2 input values (x1,x2) with their associated weights (w11,w21,w31) and (w12,w22,w32), 3 nodes in the hidden layer (h1,h2,h3) nd target output value as 0.

Sum of products

h1 = 1*0.8+2*0.2 = 0.8+0.4 = 1.2

h2 = 1*0.4 + 2*0.9 = 0.4 +1.8 = 2.2

h3 = 1*0.3 + 2*0.1 = 0.3 + 0.2 = 0.5

Now apply Activation functionS(x) on these sums. There are many Activation functions. For this example let’s take sigmoid activation function.

Equation of Sigmoid func:

S(1.2) = 0.76852 rounded to 0.77

S(2.2) = 0.90025 rounded to 0.90

S(0.5) = 0.62246 rounded to 0.62

Now we sum all the values of hidden layer after taking dot product with the second set of weights.

0.77*0.3 + 0.90*0.5 + 0.62*0.9 = 1.239

Finally apply activation function to get target output:

S(1.239) = 0.77539

What is Pytorch:

Pytorch is a popular Deep Learning library. It is a DL research platform which provides maximum speed and flexibility. Pytorch was developed as an open source library by the Facebook research team in October, 2016 and was publicly released in January, 2017. Pytorch gives us the power of scientific computation similar to NumPy with strong GPU acceleration.

Pytorch implements imperative programming. The key feature of Python is dynamic computation graph. This capability is also known as Define by Run. Dynamic computation graph gives us the flexibility to change how the network behaves on the fly, which in turn makes debugging a lot easier.

Pytorch: Tensor

Tensor is the primary data structure in Pytorch which is similar to arrays in Numpy. Pytorch tensors run on high computational unit GPU. Tensor can be scalar, vector or nd-array.

• A scalar is a 0-dimensional tensor
• A vector is a 1 dimensional tensor
• A matrix is a 2 dimensional tensor
• A nd-array is an n dimensional tensor

Building Neural Networks using Pytorch

#importing pytorch library
import torch
#Define Activation function, we are using Sigmoid activation function in this example
def activation(x):
""" Sigmoid activation function
Argument :
x = torch.tensor"""
return 1/(1+torch.exp(-x))
#Initializing features
# Features are 3 random normal variables
features = torch.randn((1, 3))
"""torch.randn creates tensor with shape[1,3] one row and 3 columns"""
#initializing the size of input layer,hidden layer and output layer
ninput = features[1]
nhidden = 2
noutput = 1
# weights for input to hidden layer
W1 = torch.randn(ninput,nhidden)
#Weights for hidden layer to output Layer
W2 = torch.randn(nhidden,noutput)
#bias terms for hidden and output layer
B1 = torch.randn(1,nhidden)
B2 = torch.randn(1,noutput)
#Calculating the output by applying activation function on dot product
#torch.mm will do the matrix multiplication/dot product
h = activation(torch.mm(features,W1) + B1) #torch.mm will do matric
Predicted_output = activation(torch.mm(h,W2) + B2)
print(output)

The predicted output is tensor([[0.3171]])

Our Target Output was 0 and the predicted output we got is 0.3171.

In the above code the weights are initialized randomly. The above neural network consists of 3 input units, 2 hidden units and an output unit.

Conclusion:

I hope you enjoyed learning about the difference between traditional Machine Learning and Deep Learning, what Neural Network is and how it works, why we call Deep Learning “Deep” and this gentle introduction to Pytorch.

rkhemka

Guest Blogger

categories & Tags

UNLEASH THE GIG ECONOMY. START A PROJECT OR TALK TO SALES
Close


text:
Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 14 Feb 2016, 04:55

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Is My GPA Too Low?

Author Message
Joined: 21 Jun 2010
Posts: 572
Location: CA, USA
Schools: Harvard, Wharton, Stanford, Columbia, Chicago, MIT, NYU, Yale, +
Followers: 9

Kudos [?]: 23 [0], given: 1

Is My GPA Too Low? [#permalink]  09 Jul 2012, 10:52
Expert's post
From Dr. Shel (Shelly Watts), http://www.mbaadmit.com

MBA Admit.com: Proudly, one of the most affordable top-quality MBA admissions consulting companies, with prices nearly 50% lower than our direct competitors.

Is My GPA Too Low?

Many candidates write to me worried about their undergraduate GPAs and ask, “Do you think my GPA is too low?”

There is no stock answer to this question. Certainly, stats about each business school’s average GPA for matriculating students can help you gain a sense of your chances for admission. But beyond this, specifics matter a great deal. For example:

Where did you attend college? If your school was a top-ranked college in its country, your GPA likely can be lower than the GPA of a candidate who attended a much lower-ranked college.

What was your major? Some majors are known to be very difficult, and so what might look like a low GPA in a much “softer” major may be perceived by the admissions committee as a relatively high GPA for your major. (If your major was known for its academic rigor, you should point that out in your essays and recommendations.)

What is your gender? Sorry guys – gals do sometimes have an easier time because, for many business schools, fewer women apply. As a result, women sometimes receive a little more leeway on their GPA.

Did you participate in a varsity-level sport in college? If so, most admissions committees will realize that you were diverting a lot of time to the sport, and they might cut you a little slack on your GPA. Representing a school in a varsity-level sport is generally seen as very admirable and an indication of multifaceted talent.

Many MBA candidates might be hopeful that this same reasoning applies to other types of extracurricular activities, yet on the whole it does not. For most other types of extracurricular activities, the committee will feel you had more control over how much time you devoted and won’t see your choice to spend time on extracurricular activities rather than on your studies as a good enough reason for a lower GPA.

Did you have to work your way through school financially? That can matter, because the committee will realize you were juggling work with your academics and may be more understanding if your GPA is slightly lower.

Other factors beyond these also affect the GPA assessment. If you are concerned about your GPA, try to evaluate how these and other mitigating factors might affect the way admission committees will view your GPA. These simple methods can help you understand whether your GPA will be seen as a strength or weakness of your MBA application.

Best wishes,
Dr. Shel (Shelly Watts)

_________________

MBA Admit.com offers comprehensive service for approx. $1495-$1795 for the first application and approximately $875 for subsequent MBA applications. Basic editing available for$925 for 1 application. Restrictions apply. Contact Dr. Watts for details.

Dr. Shel Watts, the Founder and CEO of MBA Admit,.com, is a graduate of Harvard and Oxford who has Harvard admissions experience. Clients can opt to work directly with Dr. Shel in the admissions process. At MBAAdmit.com, "Dr. Shel" provides--at nearly 50% lower pricing than many comparable competitors--excellent MBA essay editing services and comprehensive admissions consulting, including one-to-one strategizing, essay development, school selection, GMAT advice, interview preparation, and recommendation advice. http://www.mbaadmit.com

Is My GPA Too Low?   [#permalink] 09 Jul 2012, 10:52
Similar topics Replies Last post
Similar
Topics:
Please evaluate my profile - low GPA, engineer 3 02 Feb 2016, 09:56
Career Switcher/Low GPA/Giving it my all 0 17 Jun 2013, 16:40
Low GPA 1 23 Sep 2008, 19:24
Low undergrad GPA - my odds of getting in? 3 09 Jul 2007, 20:37
Low GPA, overseas - what are my options? 1 23 Apr 2007, 05:59
Display posts from previous: Sort by

format: plain


text:
# Deduction of the entropy balance equation with input and output of matter by convection (mass flow)

In many books of Engineering thermodynamics, the entropy balance equation is wrritten as

$$\frac{dS}{dt}=\Pi_S+I_S$$

$$I_S=\sum_{j=1}\frac{\dot{Q}_{j}}{T_j}+\sum_{i=1}\dot{m}_is_i-\sum_{e=1}\dot{m}_es_e$$

$$\frac{dS}{dt}=\Pi_S+\sum_{j=1}\frac{\dot{Q}_{j}}{T_j}+\sum_{i=1}\dot{m}_is_i-\sum_{e=1}\dot{m}_es_e$$

Where the term "$$\Pi_S$$" represents the entropy rate production, which is always postive or equal to zero, $$\Pi_S\ge0$$ , and the term $$I_S$$ means the entropy exchange rate. $$\sum_{i=1}\dot{m}_is_i$$, $$\sum_{e=1}\dot{m}_es_e$$ are the incoming and outgoing entropy respectively due to mass flow. Where $$s_i$$, $$s_e$$ are the intensive entropy per unit mass.

Where does the $$\sum_{i=1}\dot{m}_is_i-\sum_{e=1}\dot{m}_es_e$$ come from? How to derived it in a formal way? Other authors like Ansermet, can derived that in fact $$I_S=\frac{\dot{Q}}{T}$$ but no the entropy exchanged by mass flows

In the book "Principles of thermodynamics" by Jean Phillipe Ansermet, says that the entropy evolution in a macroscopic scale, can be described as follows:

$$\frac{dS}{dt}=\Pi_S+I_S$$

A few pages back, the author says that the internal energy evolution for a system can be described as

$$\dot{U}=T\dot{S}-P\dot{V}+\sum_{i=1}^C\mu_i\dot{N_i}=P_Q+P_W+P_C$$

Where $$P_Q , P_W, P_C$$ are the heat power, mechanical power and chemical power respectively. The heat power $$P_Q$$ is the same variable that $$\dot{Q}_j$$, so $$P_Q=\dot{Q}_j$$. We can re write the internal energy evolution equation in terms of $$\dot{S}$$.

$$\dot{S}=\frac{1}{T}(P_Q+P_W+P_C+P\dot{V}-\sum_{i=1}^C\mu_i\dot{N_i})$$

Entropy exchanges happens when exist heat transfer. Due to the above, we can rewrite the equation as follows:

$$\dot{S}=\frac{1}{T}(P_W+P_C+P\dot{V}-\sum_{i=1}^C\mu_i\dot{N_i})+\frac{P_Q}{T}$$

Therefore, when the process is reversible there is no entropy production $$\Pi_S=0$$ and there is no entropy exchage $$I_S=0$$. Then, we can deduce the reversible heat, mechanical work and chemical work

$$\frac{1}{T}(P_W+P_C+P\dot{V}-\sum_{i=1}^C\mu_i\dot{N_i})=\Pi_S=0$$ y $$\frac{P_Q}{T}=I_S=0$$

$$P_W=-P\dot{V}\Rightarrow\delta W=P_Wdt=-PdV\Rightarrow W_{if}=\int_i^f\delta W=-\int_{V_i}^{V_f}PdV$$

$$P_C=\sum_{i=1}^C\mu_i\dot{N_i}\Rightarrow\delta C=P_Cdt=\sum_{i=1}^C\mu_idN_i\Rightarrow C_{if}=\int_i^f\delta C=\int_{N_{i_i}}^{N_{i_f}}\sum_{i=1}^C\mu_idN_i$$

$$P_Q=TI_S$$

In reversible process, we can rewrite the internal energy equation in order to obtain another expression for $$P_Q$$

$$\dot{U}=T\dot{S}-P\dot{V}+\sum_{i=1}^C\mu_i\dot{N_i}=P_Q+P_W+P_C\Rightarrow T\dot{S}-P\dot{V}+\sum_{i=1}^C\mu_i\dot{N_i}=P_Q-P\dot{V}+\sum_{i=1}^C\mu_i\dot{N_i} \Rightarrow P_Q=T\dot{S}$$

$$P_Q=T\dot{S}\Rightarrow \delta Q=P_Qdt=TdS\Rightarrow Q_{if}=\int_i^f\delta Q=\int_{S_i}^{S_f}TdS$$

But, Ansermet cannot show where does the $$\sum_{i=1}\dot{m}_is_i-\sum_{e=1}\dot{m}_es_e$$ come from.

Also, how can we fit the term $$\sum_{i=1}\dot{m}_is_i-\sum_{e=1}\dot{m}_es_e$$ when we take the differential form of entropy $$dS$$?

• Are you familiar with the open system (control volume) version of the 1st law of thermodynamics? If not, please look that over and get back to us. If so, do you have any issues with the convective terms in that equation? Jul 2 '21 at 11:50

$$\sum_i \dot{S}_i - \sum_e \dot{S}_e.$$
But by definition, the relationship between an entropy, a mass and an entropy per unit mass is $$S_i = m_i s_i$$. Making this substitution yields the terms about which I believe you're inquiring.
There's one last issue. In the case of entropy, we know that flows of heat themselves carry entropy. So in addition to the internal entropy production rate, we must include heat flows in addition to mass flows, yielding the $$\dot{Q}_j/T_j$$ terms. As to why these correspond to entropies well: the way I teach the subject, at least, this serves as the definition for entropy and the consistency of this definition follows from the Second Law of Thermodynamics.


text:
# Based on Crow Manufacturing's data in Exercise 24-20, assume that a transfer price of $110 has been... 1 answer below » Ex 24-21 Based on Crow Manufacturing's data in Exercise 24-20, assume that a transfer price of$110 has been established and that 40,000 units of materials are transferred, with no reduction in the Materials Division's current sales.
a) How much would the Crow Manufacturing's total income from operations increase?
b) How much would the Industrial Division's income from operations increase?
c) How much would the Materials Division's income from operations increase?
d) If the negotiated price approach is used, what would be the range of acceptable transfer prices and why?


text:
# Let's Barter!

Algebra Level 1

Three pencils cost the same as two paper clips.
Four paper clips cost the same as one ruler.
Pencils are more expensive than rulers.

If the first two statements are true, the third statement is $\text{\_\_\_\_\_\_\_\_\_}$.

Note: Assume that each item costs some positive amount.

×


text:
Browse Questions
–1 vote

# The probability that a contractor will get a plumbing contract is $\large\frac{2}{3}$ and an electric contract is $\large\frac{4}{9}$.If the probability of getting at least one contract is $\large\frac{4}{5}$,find the probability that he will get both the contracts.

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

+1 vote


text:
#### Required:

If you didn’t have a chance to read it yet, have a look at Chapter 21 in ‘R for Data Science’ by Hadley Wickham & Garrett Grolemund

#### Other resources:

We will be working through this tutorial developed by the Ocean Health Index Data Science Team

## Today’s learning objectives

By the end of today’s class, you should be able to:

• Write a for loop to repeat operations on different input
• Implement if and if else statements for conditional execution of code

Acknowledgements: Today’s tutorial is adapted (with permission) from the excellent Ocean Health Index Data Science Training.

We’ll first finish where we left off in lesson 14 and work through the rest of that lesson. Then we’ll look at some other ways to write for loops by working through these Data Carpentry’s notes.

We can also apply this to the gapminder data.

## Looping with an index and storing results

In the gapminder example we’ve been using to build a for loop together, we’ve been iterating over a list of countries (in turn assigning each of these to our cntry object). You may often see for loops iterating over a numerical index, often using i as the object that in turn gets assigned each number from a sequence. Here is an example:

for (i in 1:10) {
print(paste("Part_", i, sep = ""))
}

As another example, last class, we needed to calculate the product of gdpPercap and population size for each year and each country. We did this efficiently in a single step for all years and countries with a mutate(), prior to defining our loop or function.

gap_europe <- gapminder_est %>%  # Here we use the gapminder_est that includes information on whether data were estimated
filter(continent == "Europe") %>%
mutate(gdpTot = gdpPercap * pop)

A (not very computationally efficient) alternative would be to do this calculation for a specific country with a for loop and using square bracket indexing to select the i’th element of a vector.

gapminder$gdpTot <- vector(length = nrow(gapminder)) for (i in 1:nrow(gapminder)) { gapminder$gdpTot[i] <- gapminder$gdpPercap[i] * gapminder$pop[i]
}

To understand how this loop is working exactly the same way as our previous loop, have a look of the list of elements 1:nrow(gapminder) that we loop over.

1:nrow(gapminder)`

You see that this just gives a vector of integers from 1 to the number of rows in the gapminder data. Each of these numbers in turn get assigned to i as we run through the loop.


text:
0
Research Papers

# Multiscale Simulations for Impact Load–Induced Vibration: Assessing a Structure's Vulnerability

[+] Author and Article Information
Jeongwon Park

Department of Mechanical Engineering,
Hanyang University,
17 Haengdang-dong,
Seongdong-gu, Seoul 133-791, Korea

Man Hoi Koo

Agency for Defense Development,
111 Sunam dong,
Yoseong, Daejeon 305-152, Korea

Junhong Park

e-mail: parkj@hanyang.ac.kr
Department of Mechanical Engineering,
Hanyang University,
17 Haengdang-dong,
Seongdong-gu, Seoul 133-791, Korea

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received June 1, 2012; final manuscript received January 4, 2013; published online March 25, 2013. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 135(2), 021006 (Mar 25, 2013) (7 pages) Paper No: MATS-12-1120; doi: 10.1115/1.4023774 History: Received June 01, 2012; Revised January 04, 2013

## Abstract

Vibration resulting from high-velocity projectiles impacting a structure was simulated at multiple scales. Local impact simulations were performed to predict the material deformation and penetration phenomena at the location of impact. The resulting penetration behavior of a steel panel was analyzed for various projectile velocities, sizes, and panel thicknesses. Three-layer panels with Kevlar as the core material were simulated to understand the effects of structural layering on the reduction of the impact force. The forces acting on the panel in the longitudinal and transverse directions were calculated from the obtained stress distribution in the local deformation model. Using the estimated force input, transient longitudinal and flexural wave propagations were calculated to analyze the radiation of the impact energy along the structural span. Vulnerable positions with high possibilities of damage to crucial components due to impact loading were identified from the resulting vibration responses.

<>

## Figures

Fig. 1

A localized impact simulation model to predict material deformation and the resulting stress distribution at the impacted location in an infinitely long armored panel impacted by a high velocity projectile: (a) single and (b) three-layer panels

Fig. 2

Deformation shapes of the homogeneous steel panel impacted by projectiles and the consequent transient variation of the (a) normal and (b) shear stress (a = 37 mm, vi = 900 m/s, h = 50 mm)

Fig. 3

Impact forces variation acting on the steel panel for 0.35 milliseconds: (a) normal force and (b) shear force and its interpolation. Fast fluctuations were minimized during interpolation, since those components do not propagate due to resonance in the thickness direction.

Fig. 4

Vibration analysis on the beam-structure model subjected to impact forces in the longitudinal and transverse directions

Fig. 5

Effects of projectile velocity on the interpolated (a) normal and (b) shear forces acting on the homogeneous steel panel (a = 37 mm, b = 105 mm, and h = 50 mm)

Fig. 6

Effects of projectile velocity on (a) the maximum normal and shear forces and (b) their duration obtained from the results in Fig. 5

Fig. 7

Effects of panel thickness (h) on (a) normal and (b) shear forces acting on a steel panel and effects of projectile size (a × b) on (c) normal and (d) shear forces. The maximum normal and shear forces increased with increasing panel thickness. The duration of force generation and the total transmitted impact energy depended on panel thickness, projectile size, and occurrence of perforation.

Fig. 8

Comparison of interpolated (a) normal and (b) shear force variations for single (hc = 0 mm) and three-layered panels utilizing Kevlar as the core material

Fig. 9

(a) Transverse and (b) longitudinal displacement responses of a beam subjected to impact forces in the transverse and longitudinal directions.

Fig. 10

Maximum vibration amplitude on the component located at x = L/3 for a beam impacted by projectiles at different locations. The location that induces the maximum displacement, velocity, or acceleration responses corresponds to sections of beam with higher probability of damage when impacted.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections


text:
## Nonsymmetric potential-reduction methods for general cones

In this paper we propose two new nonsymmetric primal-dual potential-reduction methods for conic problems. Both methods are based on {\em primal-dual lifting}. This procedure allows to construct a strictly feasible primal-dual pair linked by an exact {\em scaling} relation even if the cones are not symmetric. It is important that all necessary elements of our … Read more

## Towards nonsymmetric conic optimization

In this paper we propose a new interior-point method, which is based on an extension of the ideas of self-scaled optimization to the general cones. We suggest using the primal correction process to find a {\em scaling point}. This point is used to compute a strictly feasible primal-dual pair by simple projection. Then, we define … Read more

## ON THE LIMITING PROPERTIES OF THE AFFINE-SCALING DIRECTIONS

We study the limiting properties of the affine-scaling directions for linear programming problems. The worst-case angle between the affine-scaling directions and the objective function vector provides an interesting measure that has been very helpful in convergence analyses and in understanding the behaviour of various interior-point algorithms. We establish new relations between this measure and some … Read more


text:
### Qualified's blog

By Qualified, history, 7 weeks ago,

I was solving this problem and I got AC. While I was solving the problem, I had some cerr's and I commented them out when submitting. After getting AC, I wanted to experiment if uncommenting those cerr's would have a difference in runtime. Lo and behold! The difference was huge. Take a look at these two submissions. 100116835 is the AC and 100116299 is the TLE. The AC runs in 62 ms and the TLE is over 1000 ms. Now I have questions. Does Codeforces use the standard error stream? Why does this affect my runtime? All the comments are print statements and even if they were outputted to standard output stream, this wouldn't cause TLE. Why, why, why?

• +6

 » 7 weeks ago, # |   0 i think it was for this reason trav(i, hld) { //cerr << i << ' '; //}
•  » » 7 weeks ago, # ^ |   0 100120172. I commented out the trav(i, hld) { cerr << i << ' '; }. But it is still TLE. That's not the problem.
•  » » » 7 weeks ago, # ^ |   0 idk then
 » 7 weeks ago, # |   0 I'm not sure what's surprising about this. cerr is a stream that exists during grading but isn't used to determine your verdict. You can just think about it as cout but not looked at by the grader. And when you flush an out stream 3e5 times (you cerr endl every time in your two main loops), I wouldn't be too surprised with a TLE.
•  » » 7 weeks ago, # ^ |   0 But it ain't endl it is "\n".
•  » » » 7 weeks ago, # ^ |   0 Oh oops. Maybe I should've read macros more carefully.Anyways, cerr is flushed whenever you use the << operator, as unitbuf is set. You can probably google for more information.


text:
## Cryptology ePrint Archive: Report 2019/105

Non-Malleable Secret Sharing in the Computational Setting: Adaptive Tampering, Noisy-Leakage Resilience, and Improved Rate

Antonio Faonio and Daniele Venturi

Abstract: We revisit the concept of *non-malleable* secret sharing (Goyal and Kumar, STOC 2018) in the computational setting. In particular, under the assumption of one-to-one one-way functions, we exhibit a *computationally* private, *threshold* secret sharing scheme satisfying all of the following properties.

-) Continuous non-malleability: No computationally-bounded adversary tampering independently with all the shares can produce mauled shares that reconstruct to a value related to the original secret. This holds even in case the adversary can tamper *continuously*, for an *unbounded* polynomial number of times, with the same target secret sharing, where the next sequence of tampering functions, as well as the subset of shares used for reconstruction, can be chosen *adaptively* based on the outcome of previous reconstructions.

-) Resilience to noisy leakage: Non-malleability holds even if the adversary can additionally leak information independently from all the shares. There is no bound on the length of leaked information, as long as the overall leakage does not decrease the min-entropy of each share by too much.

-) Improved rate: The information rate of our final scheme, defined as the ratio between the size of the message and the maximal size of a share, asymptotically approaches 1 when the message length goes to infinity.

Previous constructions achieved information-theoretic security, sometimes even for arbitrary access structures, at the price of *at least one* of the following limitations: (i) Non-malleability only holds against one-time tampering attacks; (ii) Non-malleability holds against a bounded number of tampering attacks, but both the choice of the tampering functions and of the sets used for reconstruction is non-adaptive; (iii) Information rate asymptotically approaching zero; (iv) No security guarantee in the presence of leakage.

Category / Keywords: foundations / secret sharing, non-malleability, leakage resilience, computational security

Date: received 2 Feb 2019, last revised 13 Feb 2019

Contact author: antonio faonio at imdea org,venturi@di uniroma1 it

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2019/105

[ Cryptology ePrint archive ]


text:
# Fenchel conjugate with indicator function

How to show the following: Let $C$ be a compact set in some Hilbert space $H$ and $I_C(x)$ the indicator function (i.e. $0$ for $x\in C$ and $\infty$ otherwise). Consider the following optimization problem,

$$\min\limits_{x\in H} f(x) + I_C(x)$$

Show that the dual of this problem is,

$$\max\limits_{u\in H} -f^*(u) - I^*_C(-u)$$

where the $*$ denotes the Fenchel conjugate/dual. I understand how to find the dual problem when there are inequality or equality constraints but how do you handle "set constraints"?

Edit: Here is a complete solution based off the accepted answer. Using the definition of the conjugate, $g^*(u) = \sup\limits_{x\in H}\langle x, u\rangle -g(x)$, we have the following,

$$\min\limits_{x\in H} f(x) + I_C(x) = -(f + I_C)^* (0)$$

Now, the conjugate of a sum of functions is the infimal convolutions of the conjugates. The infimal convolution, denoted $f\square g$, is defined to be,

$$(f\square g)(u) = \inf\limits_{v \in H}\left(f(u-v) + g(v)\right)$$

Applying this to our problem we get,

$$-(f+I_C)^*(0) =-(f^* \square I_C^*)(0)= -\inf\limits_{v \in H}\left(f^*(0-v) + I_C^*(v)\right)$$

And then renaming $-v$ to be $u$, which doesn't matter because the $\inf$ is over the whole space, we get,

$$\min\limits_{x\in H} f(x) + I_C(x)= \max\limits_{u\in H}-f^*(u) - I_C^*(-u)$$

Let $g$ be a convex function, and $h$ be a concave function. By Fenchel's duality theorem: $$\min_{x \in H} g(x)-h(x) = \max_{u \in H} h_*(u)-g^*(u)$$ Taking $h(x) = 0$ gives: $$\min_{x \in H} g(x) = -g^*(0)$$ Taking $g(x) = f(x) + I_C(x)$ and using the well known theorem that the conjugate of the sum is the infimum convolution gives the desired result.
• Should the first line have $h^*(u) - g^*(u)$ or am I missing something? – TSF Feb 26 '18 at 16:07
• It doesn't make sense to be taking the max over $u$ if the functions have argument $x$? – TSF Feb 26 '18 at 16:16
• The max in the last line is unnecessary right, you only wrote it because of the Fenchel Duality theorem? Because $-g^*(0)$ does not depend on $u$? – TSF Feb 26 '18 at 16:28


text:
# Chapter 13 - Newton's Theory of Gravity - Exercises and Problems - Page 373: 18

The mass of the sun is $2.01\times 10^{30}~kg$.

#### Work Step by Step

We can convert the earth's orbital period to units of seconds. $T = (1.0~year)[\frac{(365~days)(24~hours/day)(3600~s/hr)}{1~year}]$ $T = 3.154\times 10^7~s$ We then use the earth's orbital period $T$ and the earth's orbital radius $R$ to find the mass of the sun $M_s$. $T^2 = \frac{4\pi^2~R^3}{G~M_s}$ $M_s = \frac{4\pi^2~R^3}{G~T^2}$ $M_s = \frac{(4\pi^2)(1.50\times 10^{11}~m)^3}{(6.67\times 10^{-11}~m^3/kg~s^2)(3.154\times 10^7~s)^2}$ $M_s = 2.01\times 10^{30}~kg$ The mass of the sun is $2.01\times 10^{30}~kg$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
Skip to main content

# 3.1: Introduction

We turn to a problem that is dual to the overconstrained estimation problems considered so far. Let $$A$$ denote an array of $$m$$ vectors, $$A=\left[a_{1}|\cdots| a_{m}\right]$$, where the $$a_{i}$$ are vectors from any space on which an inner product is defined. The space is allowed to be infinite dimensional, e.g. the space $$\mathcal{L}^{2}$$ of square integrable functions mentioned in Chapter 2 . We are interested in the vector $$x$$, of minimum length, that satisfy the equation

$y=\prec A, x \succ \ \tag{1a}$

where we have used the Gram product notation introduced in Chapter 2.

Example 3.1

Let y[0] denote the output at time 0 of a noncausal FIR filter whose input is the sequence $$x[k]$$, with

$y[0]=\sum_{i=-N}^{N} h_{i} x[-i]\nonumber$

Describe the set of input values that yield $$y[0] = 0$$; repeat for $$y[0] = 7$$. The solution of minimum energy (or RMS value) is the one that minimizes $$\sum_{i=-N}^{N} x^{2}[i]$$.

• Was this article helpful?


text:
# that a b < a + c b + d < c d where a , b , c , and d are positive numbers.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.7, Problem 104E
To determine

## To show:that ab<a+cb+d<cd where a,b,c, and d are positive numbers.

Expert Solution

### Explanation of Solution

Given:

The given condition is a,b,c,and d are positive numbers such that ab<cd .

Calculation:

According to the question, the given inequality is

ab<cd .

The inequality is expressed as

Also, the expression is expressed as


text:
> >
> Through Naples to Sorrento or Pompei?
Notices

# Through Naples to Sorrento or Pompei?

Nov 8th, 2010, 02:02 AM
#1
Original Poster

Join Date: Oct 2010
Posts: 1,254
Through Naples to Sorrento or Pompei?

This may be useful if you plan to come down from Rome etc...

Since the completion of its extensive makeover, changing trains at the main station in Naples - Napoli Centrale - is now simpler than ever; essentially, all you need do is look for signs saying "Circumvesuviana" and follow them!

You'll first go down the new main escalator...
http://www.pbase.com/isolaverde/image/123069561

... then along an underground passage...
http://www.pbase.com/isolaverde/image/128343808

... and eventually you'll arrive at the barriers...
http://www.pbase.com/isolaverde/image/125760466

Once there, turn around and you'll see the new ticket office.
Both "Pompei - Scavi" and Sorrento are served by trains that terminate at Sorrento, and the "BIN" column on the display boards indicates the platform (binario) you'll need go down to...
http://www.pbase.com/isolaverde/image/130148215

You'll probably find hordes of people making the same journey, but a route map may help: this one has all of the CV's lines...
http://www.pbase.com/isolaverde/image/124987094

... and this shows just the stops between Naples and Sorrento...
http://www.pbase.com/isolaverde/image/128288647

Peter

Nov 8th, 2010, 07:07 AM
#2

Join Date: Mar 2003
Posts: 2,553
Brilliant! You document pretty much what it looked like when I passed through in early May, only this
http://www.pbase.com/isolaverde/image/128799399
was a mass of construction and had to navigate around it (of course with our large American wheelie luggage!).

I hadn't been to Naples in decades, so imagine my surprise at such a dramatic change to the station. The funniest thing was an outpost of the Hudson News newsstand, a chain that originated here in New York.
http://www.flickr.com/photos/[email protected]/5157792461/

Nov 8th, 2010, 08:25 AM
#3

Join Date: Sep 2007
Posts: 6,503
Thanks. Looks much improved. Have they upgraded the trains too?

Nov 8th, 2010, 10:56 AM
#4
Original Poster

Join Date: Oct 2010
Posts: 1,254
The Circumvesuviana?

They've got some shiny new ones - and plenty of the old as well!

http://www.pbase.com/isolaverde/image/125740296

But it's generally all going forwards, step by step....

Peter

Nov 8th, 2010, 10:19 PM
#5

Join Date: Oct 2010
Posts: 68
Peter

Thank you so much for posting this. Not only directions but pictures too! I will be going to Sorrento in Jan it is nice to have a picture guide of where to go and what do to, etc... WONDERFUL!!

THANK YOU!

Nov 9th, 2010, 06:10 AM
#6
Original Poster

Join Date: Oct 2010
Posts: 1,254
My pleasure!

There are so many tales around of how difficult it had been (and certainly WAS, during the renovations) that an up-to-date report seemed worthwhile.

Have a nice time in January!
Peter

Nov 10th, 2010, 03:09 PM
#7

Join Date: Feb 2003
Posts: 6,523
Thank you so much. We will need all the help we can get!! I am thinking we will do the trip in reverse, but I'm sure coming into the station from Pompei , to head back to Rome will be made so much easier with our wonderful information.

Dec 6th, 2010, 09:59 PM
#8

Join Date: Oct 2010
Posts: 6
I agree--the new Circumvesuviana station layout is *much* less confusing than the old one. However, if you take it, be sure to take measures to avoid pickpockets; some of them are collected here:
http://www.wikinapoli.com/transport#...ng-Pickpockets

Mar 29th, 2011, 07:39 AM
#9

Join Date: Mar 2011
Posts: 148
bookmarking

Mar 29th, 2011, 12:58 PM
#10

Join Date: Nov 2003
Posts: 585
Thank you so much, Peter. This will help us a lot when we are there in May.

Apr 28th, 2011, 01:30 PM
#11

Join Date: May 2004
Posts: 4,146
Wow!! Peter, you are a treasure!!! Thank you so much!!!

Apr 28th, 2011, 04:16 PM
#12

Join Date: Jan 2007
Posts: 1,135
Bookmarking

May 10th, 2011, 05:54 AM
#13

Join Date: Jan 2011
Posts: 180
bookmarking

Thanks Peter!

May 10th, 2011, 09:04 AM
#14

Join Date: Sep 2005
Posts: 3,926
This was the most helpful picture for me - http://www.pbase.com/isolaverde/image/130148215 - I might have searched for Pompei under destination, so now I know exactly what to look for in July! Thanks Peter!!!

May 10th, 2011, 09:32 AM
#15

Join Date: May 2011
Posts: 65
Perfect timing with this post as we are researching this trip right now. One question, do you know if it's secure to store your luggage in Naples or Pompei? We wanted to visit Pompei on the trip down from Rome to Sorrento, rather than make a separate trip back up.

May 10th, 2011, 10:23 AM
#16
Original Poster

Join Date: Oct 2010
Posts: 1,254

There's an LLO in the train station...

http://www.grandistazioni.it/cms/v/i...003f16f90aRCRD

... and a free one at Pompei...
http://www.pompeiviva.it/pv/en/serviziofferti.htm

I know of no reason not to use either, and would be happy to do so myself if the circumstance arose!

For you though, Pompei's the better choice as you could then continue on by train to Sorrento.

Peter

Aug 11th, 2011, 08:50 PM
#17

Join Date: Mar 2011
Posts: 148
Thank you so much Peter - this is wonderful !

Oct 4th, 2011, 06:03 PM
#18

Join Date: Aug 2011
Posts: 178
Great directions. Exactly what I was looking for

Oct 4th, 2011, 07:46 PM
#19

Join Date: Oct 2003
Posts: 30,500
>>>You'll first go down the new main escalator...
http://www.pbase.com/isolaverde/image/123069561<<<

The new escalator wasn't working when I was there early Sept.

Oct 23rd, 2011, 11:41 PM
#20

Join Date: Jan 2006
Posts: 1,328
TTT

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off


text:
# zbMATH — the first resource for mathematics

Integrals with a large parameter: Legendre functions of large degree and fixed order. (English) Zbl 0539.33005
The Legendre functions $$P_ n^{-m}(\cosh z), Q_ n^{-m}(\cosh z)$$ are considered for large values of n, m fixed, and z in a domain containing $$z=0$$. It is known that the asymptotic expansion has modified Bessel functions $$I_ m(uz), K_ m(uz)$$ as approximants $$(u=n+1/2).$$ The author gives a new approach for deriving the expansion (with as starting point integral representation). Moreover he gives a new method (based on the maximum-modulus theorem for analytic functions) for obtaining information on the remainders in the asymptotic expansions.
Reviewer: N.M.Temme

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 30E15 Asymptotic representations in the complex plane
Full Text:
##### References:
 [1] Watson, Bessel Functions (1944) [2] Ursell, Proc. Cambridge Philos. Soc. 72 pp 49.65– (1972) [3] Ursell, Proc. Cambridge Philos. Soc. 61 pp 113– (1965) [4] A., Higher Transcendental Functions: Bateman Manuscript Project (1953) · Zbl 0052.29502 [5] DOI: 10.1112/plms/s2-36.1.427 · Zbl 0008.16301 [6] Olver, Asymptotics and Special Functions (1974) [7] Martin, Proc. Cambridge Philos. Soc. 76 pp 211– (1974) [8] Titchmarsh, Theory of Functions (1939)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Math Help - Show that f is not differentiable at 0

1. ## Show that f is not differentiable at 0

Hey guys, I need help with this question that has been bugging me...
Let f(x) = x3/5. Show that f is not differentiable at zero. Show that y = x3/5 has a vertical tangent line at (0,0)

I don't know how to prove the statements... Any tips and pointers (plus the method) will help alot...

Thanks!

2. ## Re: Show that f is not differentiable at 0

There are several different ways to do this, depending on what you can or want to use.

Of course, a function of one variable is "differentiable" at x= a if and only if $\lim_{x\to h} \frac{f(a+h)- f(a)}{h}$ exists. It is NOT differentiable if that limit does not exist. So you could show that $\lim_{h\to 0}\frac{h^{3/5}- 0}{h}$ does not exist.

Or you could use the fact that, while the derivative of a function is not necessarily continuous, it does have the "intermediate value property". In particular, if the derivative at x= a exists, then $\lim_{x\to a}f'(x)= a$. So what is the derivative of $f(x)= x^{3/5}$? What is the limit of that derivative as x goes to 0?


text:
# Two blackbodies at two foci inside an ellipsoidal shell

• I
Let's consider two spherical blackbodies at two foci inside an ellipsoidal heat-reflecting shell. Consider the situation that they both have different radii and that their temperatures are such that they emit the same power. Thus, the surface power flux density of each is inversely proportional to the square of their radius. Therefore, when these spherical blackbodies are emitting the same power, they are at different temperatures.

Conversely, if their blackbody temperatures were to become the same, they would be emitting different powers, as the power emitted would be proportional to the surface of the object.

So should I expect these temperatures to come together, or not?

256bits
Gold Member
Let's consider two spherical blackbodies at two foci inside an ellipsoidal heat-reflecting shell. Consider the situation that they both have different radii and that their temperatures are such that they emit the same power. Thus, the surface power flux density of each is inversely proportional to the square of their radius. Therefore, when these spherical blackbodies are emitting the same power, they are at different temperatures.

Conversely, if their blackbody temperatures were to become the same, they would be emitting different powers, as the power emitted would be proportional to the surface of the object.

So should I expect these temperatures to come together, or not?
Good question.
My vote is that the two bodies will come to thermal equilibrium.

( Would the two blackbodies have all of their emission transmitted to the other m even if they are situated at the loci? )

Simon Bridge


text:
## Channing Arndt and Finn Tarp

Print publication date: 2016

Print ISBN-13: 9780198744801

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780198744801.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see www.oxfordscholarship.com/page/privacy-policy).date: 19 November 2018

# Multidimensional First-Order Dominance Comparisons of Population Wellbeing

Chapter:
(p.24) 3 Multidimensional First-Order Dominance Comparisons of Population Wellbeing
Source:
Measuring Poverty and Wellbeing in Developing Countries
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198744801.003.0003

# Abstract and Keywords

This chapter conveys the concept of first-order dominance (FOD) with particular focus on applications to multidimensional population welfare comparisons. It gives an account of the fundamental equivalent definitions of FOD both in the one-dimensional and multidimensional setting, illustrated with simple numerical examples. An implementable method for detecting dominances that relies on linear programming is explained along with a bootstrapping procedure that yields additional information relative to what can be obtained from dominance comparisons alone. The chapter discusses strengths and weaknesses of FOD compared to other multidimensional population comparison concepts, and describes practical tools that enable the reader to easily use it.

# 3.1 Introduction

A central question in applied welfare economics is how to make comparisons of population wellbeing across groups or over time. Appropriate comparison concepts have many potential uses. For example, if a study is able to detect that one population group is clearly worse off than another (i.e. is overall poorer or has less social welfare), society might wish to undertake policies aimed at narrowing this gap. Also, since reducing poverty or improving social welfare over time is often a key objective for public policies and reforms, the ability to make relevant comparisons over time is crucial for the formulation of meaningful goals and for policy evaluation.

The traditional approach to comparing population wellbeing is the use of a social welfare (or poverty) measure based on a one-dimensional individual wellbeing indicator, typically a monetary variable such as income or wealth. However, it has long been recognized that poverty and wellbeing are multidimensional phenomena, which are not adequately represented by a single income variable. As Sen (1976) points out, there is good reason to think that sometimes a richer person may have lower wellbeing than a poorer person; e.g. if he is disabled. This has given rise to numerous proposals of appropriate dimensions to include in multidimensional welfare analyses, including (but not restricted to) health and education (World Bank 1990) as well as standards of living (Sen 1988) to name a few.

Multidimensional welfare is often measured by aggregating multiple dimensions and weighting each dimension (see e.g. Alkire and Foster 2011; Roelen and Gassmann 2008; Rippin 2010). (p.25) The method is covered in-depth in Alkire et al. (2015) in a comprehensive representation of multidimensional poverty measurement and analysis, which the reader is encouraged to consult. The weighting is primarily made in order to reflect societal judgements about different dimensions as well as to be able to obtain a single measure of the welfare for a given population. This aggregation procedure enables the analyst to rank the populations. Furthermore, the approach is very convenient and can easily be justified when there exists a reasonably high degree of consensus about which weights should be applied. There is, however, no natural and generally agreed methodology to obtaining these weights. Often, it is not easy to say if one dimension is more important than another, and even when it is, quantifying by how much is often very difficult and perhaps not even meaningful to people.

The challenges described above have motivated the development of methods for comparing population wellbeing, poverty, or inequality with multidimensional indicators that are methodologically ‘robust’ in the sense that the conclusions obtained do not rely on predetermined weights on each dimension. In the context of applied welfare economics, such methods were popularized by Atkinson and Bourguignon (1982) who showed how stochastic dominance techniques for comparisons of probability distributions can be used to make comparisons of populations across broad classes of underlying social welfare functions. Such techniques have been further refined and applied by, e.g., Atkinson and Bourguignon (1987), Bourguignon (1989), Atkinson (1992), Bourguignon and Chakravarty (2003), Duclos et al. (2006, 2007), Gravel et al. (2009), Gravel and Mukhopadhyay (2010), Muller and Trannoy (2011), Gravel and Moyes (2012), and many others.

These contributions apply dominance concepts, which rely on assumptions that are typically formulated in terms of a specified sign on the second- (and higher-)order partial- or cross-derivatives of the underlying individual utility function considered by a utilitarian planner. This leads to so-called lower- or upper-orthant dominance (or even more demanding concepts). For example, Duclos and Échevin (2011) assume substitutability between health and income, i.e. an underlying utility function with a negative cross-partial derivative between health and income.

These concepts, while considerably more robust than methods relying on given weights, do not apply to ordinal data, where only the ranking of outcomes along each dimension is known to the analyst (based on a more-is-better assumption) but no information is available regarding, for example, the complementarity/substitutability relationship across the dimensions. However, welfare indicators are often ordinal in nature. For example, a higher educational attainment (e.g. a university degree) is considered to be better (p.26) than a lower (e.g. primary school), but quantifying by how much is not easily done and perhaps not even meaningful.

A natural concept for making comparisons of population distributions with multidimensional ordinal data is first-order dominance (FOD), also known as the usual (stochastic) order in the probability theory literature (see e.g. Shaked and Shanthikumar 2007). A finite (population, probability) distribution A first-order dominates distribution B if one can obtain distribution B from A by shifting (population, probability) mass within A from preferred to less preferred outcomes (where a less preferred outcome is not better in any dimension and is strictly worse in at least one dimension). Hence, if one distribution first-order dominates another, it is unambiguously better than the other. Thus, under the assumption that outcomes within each distribution can be ranked—e.g. we prefer the child attending school as opposed to not—the FOD approach provides a maximally robust way of making comparisons of multidimensional welfare. Technically, it does so without making any assumptions on utility functions and/or social welfare functions other than a more-is-better assumption. No additional assumptions are required about the strength of preferences for each dimension, nor about the relative desirability of changes between levels within or between dimensions (Arndt et al. 2012).

The absence of restrictive assumptions in the FOD approach makes the concept not only robust but also intuitively appealing. However, robustness comes at a cost. First, the result of comparing two distributions may be indeterminate. In other words, it may happen that distribution A does not dominate B and B does not dominate A. This makes the analyst unable to distinguish groups A and B according to wellbeing based on the selected indicators. Second, the FOD approach provides no information about whether a dominating distribution is slightly or substantially better than a dominated distribution. This chapter will discuss a way of mitigating these costs by applying a bootstrapping approach that provides a measure for the probability of observing dominances under resampling. This can serve both as a robustness check for the magnitude of the dominances observed and for the probability of observing dominance. Furthermore, if one is willing to accept the likelihood of performing well in head-to-head comparisons with other groups as an indicator of the relative wellbeing of a group, a full ranking of the groups can be calculated (Arndt et al. 2016).

The remainder of this paper is structured as follows: section 3.2 provides an overview of the theory of FOD with definitions and intuitive explanations using examples. Section 3.3 discusses faster checking algorithms and an alternative dominance criterion than FOD; and lastly, section 3.4 sums up and concludes.

# (p.27) 3.2 Theory and Examples

This section provides the basic definitions and theory of the FOD approach, illustrated with some simple examples. Furthermore, a practical linear programming method for detecting dominances is described and the bootstrapping procedure is explained.

## 3.2.1 One-Dimensional FOD

Suppose first that the outcome of interest is one-dimensional. The outcome could, for example, be individual income (or wealth). In this case, there is a natural ordering of outcomes (assuming that a higher income is better), but only this single dimension is taken into account.

### 3.2.1.1 Notations and Definitions

Let X denote a finite set of real-valued outcomes. Let the distribution of wellbeing of population A be described by a probability mass function1 f over X, i.e. $∑f(x)=1$ and $∑f(x)≥0$ for all $x∈X$. Similarly, let the distributions of populations B and C be described by the probability mass functions g and h respectively.

Table 3.1. Distributions f, g, and h (per cent), one-dimensional

Population A

f

Total

Income

0 (deprived)

35

1 (not deprived)

65

Total

100

Population B

g

Total

Income

0 (deprived)

50

1 (not deprived)

50

Total

100

Population C

h

Total

Income

0 (deprived)

40

1 (not deprived)

60

Total

100

Source: Authors’ hypothetical example

As a very simple example, suppose that there are only two possible outcomes, $X={0,1}$. We will always assume that higher numbers are better, so 0 is the bad outcome (‘income-deprived’), and 1 is the good outcome (‘not income-deprived’). In this situation, a population distribution is completely described by its share of individuals being income-deprived. Table 3.1 shows distributions for three hypothetical populations.

In the one-dimensional case, f first-order dominates g if and only if any of the following (equivalent) conditions hold:2

1. (a) g can be obtained from f by a finite number of shifts of probability mass in f from one outcome to another that is worse.

2. (b) Social welfare is at least as high for f as for g for any non-decreasing additively separable social welfare function, i.e. $∑x∈Xf(x)w(x)≥∑x∈Xg(x)w(x)$ for any weakly increasing real function w(∙).

3. (c) $F(x)≤G(x)$ for all $x∈X$, where F(∙) and G(∙) are the cumulative distribution functions (CDFs)3 corresponding to f and g.

(p.28) Condition (a) provides perhaps the most intuitive definition of FOD. It provides a natural criterion for the case where one distribution is unambiguously better than another. Condition (b) is a robustness property in relation to social welfare comparisons and thus provides a link to welfare economics (and to expected utility theory in the case of a probability distribution). If there is FOD, social welfare will be at least as high for the dominating population no matter the functional form of the social welfare function as long as w(∙) is weakly increasing. For ordinal data, this condition on w(∙) simply means that outcomes can be ranked from worse to better. As noted in the Introduction, no additional assumptions are required. Condition (c) turns out to be equivalent to the first two conditions and is useful for checking FOD.

### 3.2.1.2 Checking One-Dimensional FOD

Figure 3.1. Multidimensional first-order dominance comparisons of population wellbeing

Note: The black line illustrates F(x), the gray line illustrates G(x), and the dotted line illustrates H(x).

In the one-dimensional case, FOD can be checked in a simple and effective way with direct application of condition (c). To illustrate, consider the CDFs F(∙), G(∙), and H(∙) corresponding to the three probability mass functions, f, g, and h, respectively; cf. Table 3.1. We have $F(0)=0.35$, $G(0)=0.50$, $H(0)=0.40$, and of course $F(1)=G(1)=H(1)=1$. These are shown in Figure 3.1 (the black line illustrates F(x), the gray line illustrates G(x), and the dotted line illustrates H(x)).

(p.29) As can be seen from the graph, $F(x)≤H(x)≤G(x)$ for all $x∈X$. More precisely, $F(0) whereas $F(1)=G(1)=H(1)=1$. Hence f dominates both g and h. Since $H(x)≤G(x)$ for all $x∈X$, h dominates g.

Condition (a) also provides an intuitive way of explaining dominances. For example, it can be seen from Table 3.1 that f dominates g since g can be obtained from f by shifting probability mass from one outcome to another that is worse. More precisely, shifting fifteen percentage points from (1) to (0) in f yields exactly g.

## 3.2.2 Multidimensional FOD

Now suppose that the outcome is multidimensional. In the case of two dimensions, these could, for example, be income and health. In the case of three dimensions, one may wish to add educational attainment, and so on.

### 3.2.2.1 Notations and Definitions

Let Y be a finite set of (multidimensional) outcomes. A distribution of wellbeing of population A is described by a probability mass function f over Y, i.e. $∑f(y)=1$ and $f(y)≥0$ for all $y∈Y$. Similarly, let the distributions of populations B and C be described by the probability mass functions g and h respectively.

To illustrate, suppose that the two dimensions each have two possible outcomes, 0 or 1. Thus, $Y={(0,0),(0,1),(1,0),(1,1)}$. One dimension could be income (dimension I) and the other dimension could be health (dimension II). Then the outcome (0,0) for a person means that she is deprived in both dimensions, while the outcome (1,0) means that she is not deprived in the first dimension (I) but deprived in the second dimension (II), and so on. (p.30)

Table 3.2. Distributions f, g, and h (per cent), two-dimensional

 Population A

f

II (health)

Total

1 (not deprived: good)

I (income)

0 (deprived: poor)

10

25

35

1 (not deprived: rich)

25

40

65

Total

35

65

100

Population B

g

II (health)

Total

1 (not deprived: good)

I (income)

0 (deprived: poor)

25

25

50

1 (not deprived: rich)

25

25

50

Total

50

50

100

Population C

h

II (health)

Total

1 (not deprived: good)

I (income)

0 (deprived: poor)

30

10

40

1 (not deprived: rich)

10

50

60

Total

40

60

100

Source: Authors’ hypothetical example

Suppose that three probability mass functions f, g, and h are distributed as shown in Table 3.2. Note that distributions over dimension I (income) in the rightmost column are identical to those in Table 3.1 representing a situation where the same populations are considered but now an additional dimension is taken into consideration.

In the case of multidimensional outcomes, f first-order dominates g if and only if any of the following (equivalent) conditions hold:4

1. (A) g can be obtained from f by a finite number of shifts of probability mass from one outcome to another that is worse.

2. (B) Social welfare is at least as high for f as for g for any non-decreasing additively separable social welfare function, i.e. $∑y∈Yf(y)w(y)≥∑y∈Yg(y)w(y)$ for any weakly increasing real function w(∙).

3. (C) $∑y∈Zg(y)≥∑y∈Zf(y)$ for any lower comprehensive set $Z⊆Y$.5

(p.31) Each condition is the natural multidimensional extension of its counterpart for the one-dimensional case. Again, note that condition (A) provides an intuitive criterion for one distribution being unambiguously better than another, (B) provides a foundation in welfare economics but one that is not conveniently amenable to testing, while (C) provides a directly testable condition that may not be particularly intuitive.

### 3.2.2.2 Checking Multidimensional FOD

First, due to the intuitive nature, we appeal to condition (A). It can be seen from Table 3.2 that f dominates g, since it is possible to obtain g from f by shifting probability mass in f from better to worse outcomes. More precisely, shifting fifteen percentage points of probability mass from (1,1) to (0,0) in f yields exactly g, which implies that f dominates g. The distribution f is thus unambiguously preferred to the distribution g.

However, consider f and h. Neither f dominates h, nor does h dominate f. Intuitively, this is the case since f would be better if what matters most is minimization of the share of the population who are deprived in dimensions II (health) since $f(0,0)+f(1,0)=35. On the contrary, h would be better if what matters most is maximization of the share of the population not deprived in neither dimension since $h(1,1)=50>f(1,1)=40$. Consequently, no dominances are detected since no assumptions are made about the relative importance of the different dimensions. Note that the conclusion that f does not dominate h is in contrast to the one-dimensional case. This illustrates that the conclusions might change when more dimensions are added to the analysis. It is therefore important to bear in mind that an analysis applying few welfare indicators may conclude that one population dominates another whereas a multidimensional FOD analysis of the same populations with more indicators may be indeterminate. Attention should therefore be given to include important dimensions that cover overall wellbeing reasonably well.

As mentioned, condition (C) provides a direct method for checking multidimensional FOD. In our example, f dominates g if and only if the following four inequalities are jointly satisfied:6

• (i) $g(0,0)≥f(0,0)$

• (ii) $g(0,0)+g(0,1)≥f(0,0)+f(0,1)$

• (iii) $g(0,0)+g(1,0)≥f(0,0)+f(1,0)$

• (iv) $g(0,0)+g(1,0)+g(0,1)≥f(0,0)+f(1,0)+f(0,1)$.

(p.32) Considering the distributions in Table 3.2 in relation to the four inequalities above, it can be seen that, when comparing f and g, each of the four inequalities (i)–(iv) are (strictly) satisfied; (i) $g(0,0)≥f(0,0)$ since $0.25>0.10$, (ii) $g(0,0)+g(0,1)≥f(0,0)+f(0,1)$ since $0.25+0.25>0.10+0.25$, (iii) $g(0,0)+g(1,0)≥f(0,0)+f(1,0)$ since $0.25+0.25>0.10+0.25$, and (iv) $g(0,0)+g(1,0)+g(0,1)≥f(0,0)+f(1,0)+f(0,1)$ since $0.25+0.25+0.25>$ $0.10+0.25+0.25$. Hence, f dominates g, as already observed. When comparing each of the other distribution pairs, it can easily be seen that at least one of the inequalities is violated. For example, when comparing f and h, inequalities (i)–(iii) are satisfied whereas inequality (iv) is not.

### 3.2.2.3 Detecting FOD in Practice

Criterion (C) provides a simple method for detecting dominance, which can be visually perceived in cases with few outcomes such as in the example with two binary indicators giving four different outcomes in total. However, the number of inequalities to be checked increases drastically when more dimensions and levels are added. For real-world applications, computationally efficient algorithms for checking dominance are required (Range and Østerdal 2013). Mosler and Scarsini (1991) and Dyckerhoff and Mosler (1997) show that, appealing to definition (A), checking FOD corresponds to determining if a certain linear program has a feasible solution. FOD can thus be determined using a linear programming package. The first empirical implementation of this approach was provided by Arndt et al. (2012) in a study of child poverty in Mozambique and Vietnam.

Let A and B be two populations characterized by probability mass functions f and g respectively. For outcomes y and y′ with $y′≤y$, let ty,y′ be the amount of probability mass transferred from outcome y to y′. Note that the first subscript denotes the source of the transfer whereas the second denotes the destination.

Given the conditions outlined above, population A dominates population B if and only if there exists a feasible solution to the following linear program:7

$Display mathematics$
(3.1)

To provide an example, let us return to Table 3.2. There, f dominates g since it is possible to obtain g from f by shifting fifteen percentage points of probability mass from outcome (1,1) to (0,0); cf. condition (A). In terms of the linear program in equation (3.1), this implies that for $y=(0,0)$ and $y′=(1,1)$, $ty′,y=0.15$ in (p.33) order for the equality to be fulfilled for $y=(0,0)$. Furthermore, for $y=(1,1)$ and $y′=(0,0)$, $ty,y′=0.15$ in order for the equality to be fulfilled for $y=(1,1)$. It is thus possible to fulfil all the constraints for all $y∈Y$ whereby f dominates g. An implementation of the FOD approach using linear programming in GAMS is reviewed in Chapter 4.

## 3.2.3 Mitigating the Limitations of FOD

As mentioned previously, due to the absence of strong assumptions such as predetermined weights, there are some inherent limitations to the FOD approach. First, when comparing two groups, it may be the case that no dominations are found and hence the FOD approach might yield an indeterminate result. This provides little information about the populations’ relative wellbeing, as was the case when considering f and h with multiple dimensions in section 3.2.2.2. Second, the FOD approach provides no information about the strength of dominance. For example, if population A dominates population B, the FOD check itself provides no information as to whether A is marginally or substantially better than B. Both of these limitations can be mitigated using a bootstrapping approach as described below.

### 3.2.3.1 Bootstrapping

To mitigate the limitations mentioned in section 3.2.3, a bootstrapping procedure can be applied (Arndt et al. 2012). In general, bootstrapping is a procedure that relies on random sampling with replacement from the original dataset. When comparing populations A and B, J samples of size K are drawn with replacement for each population group where $K≤N$, N being the number of individuals in that population in the original sample. In the bootstrap procedure shown in Chapter 4, the samples are drawn in clusters from each stratum with $K=N$. When a cluster is drawn, all households in that cluster are drawn. Due to the drawing with replacement, each cluster (and thus household) may appear more than once. The FOD approach is then applied to each of the J bootstrap samples. When these repeated bootstrap samples are compared using the FOD approach, the final output can be interpreted as an empirical probability that population A dominates population B since the original sample is a subsample from a larger population. These probabilities yield significantly more information than only applying FOD to the original data where, for example, an indeterminate result will make it impossible to draw further conclusions about the comparative wellbeing of the two populations. The bootstrapping procedure thus enables the analyst to extract some information about the strength of conclusions based on the probability of dominance under resampling.

(p.34) For example, with bootstrapping, we may find that occasionally A dominates B and occasionally the inverse is true, but most of the time the results are indeterminate, i.e. rough equality of A and B. Alternatively, we may find that the probability that A dominates B is fairly high, the probability that B dominates A is very low or zero, and the probability of an indeterminate result is somewhat low, i.e. likely dominance of A over B, or we might find that A dominates B almost always, i.e. solid dominance of A over B.

As a concrete example, say that population A dominates B in 995 of the $J=1,000$ bootstrap samples and that A dominates C in 870 of the bootstrap samples. This corresponds to a 99.5 per cent chance of A dominating B and an 87 per cent change of A dominating C. In this example, A is thus better than C (likely dominance of A over C) and considerably better than B (solid dominance of A over B). If no dominations between B and C are obtained in the original sample, the ranking of these is ambiguous. However, if, for example, C dominates B in four (0.4 per cent) of the bootstrap samples whereas B dominates C in eighty (8 per cent) of the bootstrap samples, B is seemingly better than C (though rough equality of B and C).

Furthermore, if one is willing to accept the tendency to outperform other groups as an overall relative indicator of population wellbeing, it is possible to provide an intuitive ranking of all population groups via the Copeland (1951) method, which is analogous to the way in which teams are ranked by assigning points to wins, draws, and losses from matchups in a sports tournament. For instance, for each population group (n population groups in total), one can count how many of the $(n−1)$ other population groups it dominates and from that subtract the number of times it is dominated by these other groups. This yields a score in the interval $[−(n−1),n−1]$ which can then be normalized to the interval $[−1,1]$ (see e.g. Arndt et al. 2016).

# 3.3 Further Considerations

## 3.3.1 Faster Solution Algorithms

The linear programming approach presented provides a practical method for checking FOD for many applied problems. For most applications, the method is computationally fast enough to allow for a great number of pairwise comparisons as well as, if desired, hundreds of bootstrap repetitions for each pair of distributions compared. In applications with multiple binary indicators, the linear programming method is particularly suitable.8

(p.35) However, the linear programming approach is not the fastest possible way to make FOD comparisons. Using a network flow formulation of the problem, as outlined in Preston (1974) or Hansel and Troallic (1978), it is possible to check FOD via computation of the maximum flow. As discussed in Range and Østerdal (2013), the problem of checking FOD for multidimensional distributions can also be formulated as a special bipartite network problem related to the classical transportation problem. Generally, these formulations are computationally faster than the linear programming method. In particular, for the bivariate case, Range and Østerdal (2013) provide an algorithm for checking FOD where the worst-case computational complexity grows linearly in the size of the problem (determined by the total number of outcomes).

## 3.3.2 Alternative Dominance Criterion

There are many other dominance criteria in the literature than FOD. In general, the alternative dominance criteria all impose stronger underlying assumptions on the underlying utility/social welfare functions. A comprehensive overview of alternative dominance criteria is outside the scope of this chapter—see e.g. Shaked and Shanthikumar (2007) for an extensive review. However, we will compare FOD with the lower-orthant dominance ordering, which is one of the most frequently used alternative dominance criteria in welfare economics.

As mentioned in the Introduction, the FOD approach differs from the criteria for robust welfare comparisons of the Atkinson–Bourguignon type (see Atkinson and Bourguignon 1982; Atkinson and Bourguignon 1987; Bourguignon 1989; Atkinson 1992). These are variations of orthant stochastic orderings (see Dyckerhoff and Mosler 1997) even though the name first-order dominance has sometimes been used synonymously with orthant stochastic orderings in the welfare economics literature (e.g. Atkinson and Bourguignon 1982). Orthant dominance is not suitable to ordinal data. However, if one assumes substitutability between dimensions (as e.g. Duclos and Échevin 2011, where substitutability between health and income is assumed, i.e. an underlying utility function with a negative cross-partial derivative), a criterion less restrictive than FOD can be used. In particular, f orthant dominates g if and only if:

$Display mathematics$

The label (C0) is used to indicate that this condition relates to condition (C) in the case of multidimensional FOD in section 3.2.2.1. However, condition (C0) is less restrictive than (C). This implies that condition (C0) may be satisfied even though conditions (A), (B), and (C) are not. For a two-dimensional comparison with binary indicators (as in Table 3.2), f orthant dominates g if and only if each of the following three conditions are satisfied:

1. (p.36) (i0) $g(0,0)≥f(0,0)$

2. (ii0) $g(0,0)+g(0,1)≥f(0,0)+f(0,1)$

3. (iii0) $g(0,0)+g(1,0)≥f(0,0)+f(1,0)$.

The labels (i0)(iii0) are used to indicate that these conditions relate to conditions (i)–(iii) in section 3.2.2.2. Note that the fourth inequality (iv) need not be satisfied for orthant dominance. Returning to f and h in Table 3.2, recall that neither f first-order dominates h, nor does h first-order dominate f. However, in the case of orthant dominance, it can be seen that: (i0) $h(0,0)≥f(0,0)$ since $0.30>0.10$, (ii0) $h(0,0)+h(0,1)≥f(0,0)+f(0,1)$ since $0.30+0.10>0.10+0.25$, and (iii0) $h(0,0)+h(1,0)≥f(0,0)+f(1,0)$ since $0.30+0.10>0.10+0.25$. Hence f orthant dominates h even though f does not first-order dominate h.

# 3.4 Conclusion

Population wellbeing is increasingly recognized as a multidimensional phenomenon that is not adequately described by a single dimension (e.g. by income only). Several methods of measuring and comparing welfare have been proposed where application of a weighting or counting scheme to different dimensions is used. It is often, however, difficult to determine these weights. Due to the sensitivity of the outcome to the weights applied, different conclusions about welfare rankings are likely to occur if the weighting scheme differs from one analysis to another. While comparisons using, for example, lower orthant (stochastic) orderings following Atkinson and Bourguignon (1982) are considerably more ‘robust’ than applying weighting schemes, they typically apply conditions formulated in terms of the second- (or higher-)order cross-partial derivative and do not apply to ordinal data.

The first-order (stochastic) dominance (FOD) approach requires only that the outcomes in each dimension can be ranked from worse to better. The FOD approach can be applied to ordinal multidimensional data, enabling the analyst to perform wellbeing comparisons across population groups with a minimum of assumptions imposed. FOD is thus robust across all possible weighting schemes. This advantage is accompanied by limitations in that the FOD approach can yield indeterminate outcomes and does not directly provide information with respect to degree of dominance. A bootstrapping approach can be used to obtain more information, thus mitigating these limitations. Moreover, a Copeland approach can be used to obtain a ranking (i.e. a complete and transitive ordering) of all groups being compared.

Finally, it is worth mentioning that even though the FOD approach and bootstrapping procedure enable the analyst to rank population welfare (p.37) without assumptions about weights, the analysis should ideally be performed together with alternative welfare measurements, which provide cardinal information about the relative size of wellbeing differences under fixed weights assumptions. As Ferreira (2011) puts it, looking at a few core, truly irreducible, dimensions and applying dominance analysis (as well as a number of indices) is likely to contribute to the design and targeting of policy actions. Thus, FOD comparisons should form part of a broader population wellbeing analysis strategy.

References

Bibliography references:

Alkire, S. and J. Foster (2011). ‘Counting and Multidimensional Poverty Measurement’, Journal of Public Economics, 95(7–8): 476–87.

Alkire, S., J. Foster, S. Seth, M. E. Santos, J. M. Roche, and P. Ballon (2015). Multidimensional Poverty Measurement and Analysis. Oxford: Oxford University Press.

Arndt, C., R. Distante, M. A. Hussain, L. P. Østerdal, P. L. Huong, and M. Ibraimo (2012). ‘Ordinal Welfare Comparisons with Multiple Discrete Indicators: A First Order Dominance Approach and Application to Child Poverty’, World Development, 40(11): 2290–301.

Arndt, C., M. A. Hussain, S. Vincenzo, F. Tarp, and L. P. Østerdal (2016). ‘Poverty Mapping Based on First-Order Dominance with an Example from Mozambique’, Journal of International Development, 28(1): 3–21.

Atkinson, A. B. (1992). ‘Measuring Poverty and Differences in Family Composition’, Economica, 59(233): 1–16.

Atkinson, A. B. and F. Bourguignon (1982). ‘The Comparison of Multi-Dimensioned Distributions of Economic Status’, Review of Economic Studies, 42(2): 183–201.

Atkinson, A. B. and F. Bourguignon (1987). ‘Income Distribution and Differences in Need’, in G. R. Feiwek (ed.), Arrow and the Foundations of the Theory of Economic Policy. New York: New York University Press, 350–70.

Bourguignon, F. (1989). ‘Family Size and Social Utility: Income Distribution Dominance Criteria’, Journal of Econometrics, 42(1): 67–80.

Bourguignon, F. and S. R. Chakravarty (2003). ‘The Measurement of Multidimensional Poverty’, Journal of Economic Inequality, 1(1): 25–49.

Copeland, A. H. (1951). ‘A “Reasonable” Social Welfare Function’, University of Michigan Seminar on Applications of Mathematics to the Social Sciences.

Duclos, J.-Y. and D. Échevin (2011). ‘Health and Income: A Robust Comparison of Canada and the US’, Journal of Health Economics, 30(2): 293–302.

Duclos, J.-Y., D. E. Sahn, and S. D. Younger (2006). ‘Robust Multidimensional Poverty Comparisons’, Economic Journal, 116(514): 943–68.

Duclos, J.-Y., D. E. Sahn, and S. D. Younger (2007). ‘Robust Multidimensional Poverty Comparisons with Discrete Indicators of Well-Being’, in S. P. Jenkins and J. Micklewright (eds), Inequality and Poverty Re-Examined. Oxford: Oxford University Press, 185–206.

(p.38) Dyckerhoff, R. and K. Mosler (1997). ‘Orthant Orderings of Discrete Random Vectors’, Journal of Statistical Planning and Inference, 62(2): 193–205.

Ferreira, F. H. G. (2011). ‘Poverty Is Multidimensional. But What Are We Going to Do About It?’, Journal of Economic Inequality, 9(3): 493–5.

Gravel, N. and P. Moyes (2012). ‘Ethically Robust Comparisons of Bidimensional Distributions with an Ordinal Attribute’, Journal of Economic Theory, 147(4): 1384–1426.

Gravel, N., P. Moyes, and B. Tarroux (2009). ‘Robust International Comparisons of Distributions of Disposable Income and Regional Public Goods’, Economica, 76(303): 432–61.

Gravel, N. and A. Mukhopadhyay (2010). ‘Is India Better Off Today than 15 Years Ago? A Robust Multidimensional Answer’, Journal of Economic Inequality, 8(2): 173–95.

Hansel, G. and J. Troallic (1978). ‘Measures marginales et théorème de Ford-Fulkerson’, Zeitschrift für Wahrsheinlichkeitstheorie und verwandte Gebiete, 43(3): 245–51.

Hussain, M. A. (2016). ‘EU Country Rankings’ Sensitivity to the Choice of Welfare Indicators’, Social Indicators Research, 125(1): 1–17.

Kamae, T., U. Krengel, and G. L. O’Brien (1977). ‘Stochastic Inequalities on Partially Ordered Spaces’, Annals of Probability, 5(6): 899–912.

Lehmann, E. L. (1955). ‘Ordered Families of Distributions’, Annals of Mathematical Statistics, 26(3): 399–419.

Levhari, D., J. Paroush, and B. Peleg (1975). ‘Efficiency Analysis for Multivariate Distributions’, Review of Economic Studies, 42(1): 87–91.

Mosler, K. C. and M. R. Scarsini (1991). Stochastic Orders and Decision under Risk. Heyward, CA: Institute of Mathematical Statistics.

Muller, C. and A. Trannoy (2011). ‘A Dominance Approach to the Appraisal of the Distribution of Well-Being across Countries’, Journal of Public Economics, 95(3–4): 239–46.

Preston, C. (1974). ‘A Generalization of the FKG Inequalities’, Communications in Mathematical Physics, 36(3): 233–41.

Range, T. M. and L. P. Østerdal (2013). ‘Checking Bivariate First Order Dominance’, Discussion Papers on Business and Economics No. 9/2013. Odense: Department of Business and Economics, University of Southern Denmark.

Rippin, N. (2010). ‘Poverty Severity in a Multidimensional Framework: The Issue of Inequality between Dimensions’, Courant Research Centre, Poverty, Equity and Growth Discussion Paper 47. Göttingen: University of Göttingen.

Roelen, K. and F. Gassmann (2008). ‘Measuring Child Poverty and Wellbeing: A Literature Review’, Maastricht Graduate School of Governance Working Paper Series 2008/WP001. Maastricht: Maastricht University.

Sen, A. K. (1976). ‘Poverty: An Ordinal Approach to Measurement’, Econometrica, 44(2): 219–31.

Sen, A. K. (1988). The Standard of Living. Cambridge: Cambridge University Press.

Shaked, M. and J. G. Shanthikumar (2007). Stochastic Orders. New York: Springer Science and Business Media.

(p.39) Strassen, V. (1965). ‘The Existence of Probability Measures with Given Marginals’, Annals of Mathematical Statistics, 36(2): 423–39.

World Bank (1990). World Development Report 1990: Poverty. New York: Oxford University Press for the World Bank.

Østerdal, L. P. (2010). ‘The Mass Transfer Approach to Multivariate Discrete First Order Stochastic Dominance: Direct Proof and Implications’, Journal of Mathematical Economics, 46(6): 1222–8.

## Notes:

(1) A probability mass function is a function that to each outcome assigns the probability of that outcome. In the context of population comparisons, it assigns the share of the population in that outcome.

(2) Note that FOD is conventionally defined in the weak sense, i.e. a distribution always dominates itself.

(3) CDFs express the probability that the real-valued outcomes X will have a value less than or equal to x.

(4) The equivalence between (B) and (C) was shown by Lehmann (1955). It was also proved independently by Levhari et al. (1975). Kamae et al. (1977) observed that the equivalence between (A) and (C) is a consequence of Strassen’s Theorem (Strassen 1965). See also Østerdal (2010).

(5) A set Z ⊆ Y is lower comprehensive if y ∈ Z, z ∈ Y, and z ≤ y implies z ∈ Z.

(6) Note that the fifth inequality $g(0,0)+g(1,0)+g(0,1)+g(1,1)≥f(0,0)+f(1,0)+f(0,1)+f(1,1)$ is always satisfied with equality by the definition of the probability mass functions since $∑f(y)=∑g(y)=1$.

(7) The approach differs slightly from that outlined by Mosler and Scarsini (1991) and Dyckerhoff and Mosler (1997). In particular, the transfers here are absolute and not relative. Note furthermore that most linear programming packages require the specification of an objective function. This can be defined as an arbitrary constant function.

(8) Even with binary indicators, the linear programming approach might actually be computationally challenged, but that is only so if the number of dimensions is large (Hussain et al. 2016).


text:
# Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.14

14. Show that $[2x] + [2y] >= [x] + [y] + [x + y]$ whenever x and y are real numbers.
Solution: We have to show that
$[2x]+[2y]>=[x]+[y]+[x+y]$
Where $x,y$ are real numbers
We write $x,y$ as
$x=n+\epsilon$
$y=m+\delta$
Where $n,m$ are integers and $\epsilon,\delta$ are non-negative real numbers less than 1.
There are two possibilities for the right hand side.
$n+m+(n+m)=2n+2m$
Or
$n+m+(n+m+1)=2n+2m+1$
If
$\epsilon+\delta>=1$
There are two possibilities for the left hand side.
$\lceil2x\rceil=\begin{cases} 2n & \epsilon<1/2\\ 2n+1 & otherwise \end{cases}$
And
$\lceil2y\rceil=\begin{cases} 2m &\delta<1/2\\ 2m+1 & otherwise \end{cases}$
We are trying to prove that the right hand side is always less than or equal to the left hand side. We construct a proof by contradiction and assume that right hand side is greater than left hand side. So the right hand side must be $2n+2m+1$ and the left hand side must be $2n+2m$
But then,
$\epsilon+\delta>=1$
So at least one of them must be equal to or greater than $1/2$
So the left hand side cannot equal $2n+2m$
Hence we arrive at a contradiction and end the proof.

## 17 Replies to “Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.14”

1. Hi my friend! I want to say that this post is amazing, nice written and include approximately all important infos. I would like to see more posts like this.

2. Having read this I thought it was rather informative.

I appreciate you taking the time and energy to put this article together.

I once again find myself spending a significant amount of time both reading and leaving comments.
But so what, it was still worth it!

3. An impressive share! I have just forwarded this onto a co-worker who has been doing a little research on this.
And he actually ordered me dinner simply because I found it for him…
lol. So let me reword this…. Thanks for the meal!!
But yeah, thanks for spending some time to discuss this topic here on your
web site.

4. I’m not that much of a online reader to be honest but your blogs
really nice, keep it up! I’ll go ahead and bookmark your site
to come back down the road. Cheers

5. Thanks for finally writing about >Elementary Number Theory and Its Application, 6th Edition by Kenneth H.

Rosen Exercises 1.1.14 – Student Clan <Liked it!

6. Hello Dear, are you actually visiting this website on a regular basis, if so
after that you will absolutely take pleasant knowledge.

7. Wow that was unusual. I just wrote an extremely long comment
but after I clicked submit my comment didn’t appear.
Grrrr… well I’m not writing all that over again. Anyways, just wanted to say superb blog!

8. I regard something truly special in this web site .

9. Wow, fantastic blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your website is excellent, let alone the content!

10. Having read this I believed it was rather enlightening. I appreciate you taking the time and energy to put this information together. I once again find myself spending a significant amount of time both reading and commenting. But so what, it was still worth it!

11. Hello i am kavin, its my first time to commenting anyplace, when i read this paragraph i thought i could
also create comment due to this sensible post.

12. But wanna tell that this is very useful , Thanks for taking your time to write this.

13. Thank you very much for the content. I wish you continued success.

14. Everything is very open with a clear description of the challenges.
It was definitely informative. Your site is very helpful.
Thanks for sharing!

15. Appreciate this post. Let me try it out.


text:
# Talk:Compound Poisson distribution

WikiProject Mathematics (Rated Start-class, Low-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Low Importance
Field:  Probability and statistics
WikiProject Statistics (Rated Start-class, Low-importance)

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

Start  This article has been rated as Start-Class on the quality scale.
Low  This article has been rated as Low-importance on the importance scale.

I assume that E[Y] = λ * E[X]

What is Var[Y] in terms of the distribution of X? Say, if X has a gamma distribution.

## Some properties

${\displaystyle E[Y]=E[E[Y|N]]=\lambda E[X]}$

${\displaystyle Var[Y]=Var[E[Y|N]]+E[Var[Y|N]]=\lambda \{E^{2}[X]+Var[X]\}}$

The cumulant generating function ${\displaystyle K_{Y}(t)={\mbox{ln}}E[e^{tY}]={\mbox{ln}}E[E[e^{tY}|N]]={\mbox{ln}}E[e^{NK_{X}(t)}]=K_{N}(K_{X}(t))}$

One could add to the above, that if N has a Poisson distribution with expected value 1, then the moments of X are the cumulants of Y. Michael Hardy 20:39, 23 Apr 2005 (UTC)
The cumulant generating function treatment above is now in the article. Melcombe (talk) 15:26, 6 August 2008 (UTC)

## Why is ${\displaystyle Y(0)\neq 0}$?

I would have thought that the process should be started in zero? Just thinking in terms of (shudder) actuarial science, a claims process would make very little sense if it started with a claim at time zero? What I'm proposing is to change the definition to the one given on the page for 'Compound Poisson Process'. — Preceding unsigned comment added by Fladnaese (talkcontribs) 18:54, 26 May 2011 (UTC)

Fixed-up for this point. JA(000)Davidson (talk) 08:40, 27 May 2011 (UTC)

I see that a citation is needed for the relationship between the cumulants of the compound Poisson distribution Y, and the moments for the random variables Xi. Back in 1976, I proved this result, that is:

For j > 0, K(j) = lambda * m(j), where:

- K(j) are the cumulants of Y
- m(j) are the moments for the Xi
- lambda is the parameter of the Poisson distribution

I made use of the characteristic function in thís proof. Is my proof of interest as a citation? If so, I can send the reference number and a pdf of the paper I wrote.

David LeCorney (talk) 12:03, 8 June 2012 (UTC)