text:
## Introduction

The SharePoint Design is a beautiful web site that provides design guidance on beautiful and fast sites, pages, and web parts with SharePoint in Office 365.

Unfortunately, the SharePoint Design site does not tell you how to create the beautiful web parts they show you.

This series is intended as a companion to the SharePoint Design site, providing you with code samples and detailed how-to information for every design topic. It should help you create web parts that look exactly like the ones on the SharePoint Design site.

In our last post we discussed layout patterns and showed how to create a Grid Layout web part.

In today’s post, we’ll continue our discussion about the web part layout patterns.

## The Filmstrip layout

Like the grid layout, the filmstrip displays cards to display content. You can use any other rectangular content though. For example, you can use images of kittens, or event cards (as I did with my Calendar Feed web part).

However, unlike the grid layout, the filmstrip displays items on a single row. As the screen resizes, the filmstrip layout resizes the items to keep the same number of items displayed at once. If there are more items than can be shown at once, the filmstrip will change to a carousel and break items into "pages".

If you’re a web designer, the word carousel may make you shudder. In fact, if you search online, you’ll find over fifty-four million search results explaining why carousels are bad.

### Banner blindness

You see, most people have issues with the giant photo carousels on web sites. For most users, the giant photos trigger our banner blindness because we automatically assume it is an advertisement, and we ignore it.

Since the carousel in the filmstrip layout does not show a giant image, but a series of smaller images (usually a preview of documents), this isn’t something we have to worry about (although we’ll have to have another conversation when we discuss the Carousel layout).

### Mobile issues

Another issue is that giant carousels are notoriously bad on mobile devices.

One of the issues with carousels on mobile devices is that because of limited screen real-estate, it shows only one item at a time. This forces users to use sequential access: to view items in the order that is dictated by the control, not by the user’s choice.

Research shows that — with sequential access — users will stop looking at items after the third or fourth item in a carousel.

Another issue with carousels that people have is with discoverability. If someone is in a hurry and they quickly glance at a typical carousel, they may not know that there are more items to see. Most carousels will use a teeny tiny little dot (which are often a bad design choice) to indicate that there are more items.

Fortunately, the mobile version of the filmstrip control uses bigger dots that are as big as the main text with a lot of colour contrast to help.

If you plan on using the filmstrip layout, follow these guidelines:

• Limit your items to less than 5: Otherwise, people may not see the other items.
• Prioritize your items: Keep in mind that, depending on the screen size, some users may only see one item at a time. Make sure that you place the most important items first.
• Make sure items are related: Because people may not see all items in your filmstrip, make sure that the items you show on the filmstrip are related so that users can predict/easily guess what items are hidden. In other words, don’t put apples and oranges on the filmstrip.

### Accessibility

For me, the biggest concern about most carousels has to do with accessibility.

You see, most evil carousels break a lot of accessibility guidelines:

• The dots at the bottom are often too small to see and/or click
• The dots often have a poor colour contrast
• The slides often change automatically, without giving users a chance to read the content or slow it down
• The left/right arrows often rely on the use of a mouse, with no keyboard alternatives
• Left/right arrows often have poor colour contrasts
• Carousels often do not provide alternative text

Luckily for us, the SharePoint filmstrip (and the carousel layout) doesn’t have many of the typical carousel accessibility issues:

• The slides do not change automatically
• Users can navigate through the slides using the and keys on their keyboard and use TAB to cycle through the slides.
• The left/right navigation arrows use a high contrast so that users with colour blindness and/or various forms of visual impairment can see them
• The left/right navigation arrows aren’t too small (but they could be bigger)
• The filmstrip provides alternative text for the entire filmstrip (e.g.: "Highlighted content web part, showing Most recent Documents., Use right and left arrow keys to navigate between cards in the film strip."), as well as alternative text for every item in the filmstrip.
• The bullets at the bottom of the filmstrip are still pretty small, but they have high contrast (21:1 on the theme I tested) exceeding the minimum required contrast (4.5:1)

It still isn’t perfect, but it isn’t as bad as most carousel controls out there.

## How the filmstrip layout is implemented

Behind the scenes, the out-of-the-box SharePoint filmstrip and carousel layouts use Ken Wheeler’s awesome Slick carousel.

They simply made it more "Fabric UI" and fixed some accessibility issues.

Unlike the grid layout, which calculates the optimal card size as you resize the page, the filmstrip layout uses breakpoints to use pre-determined settings depending on the dimensions of the filmstrip control.

## How to use the filmstrip layout

Unfortunately — as was the case with the grid layout — there aren’t any ready-to-use controls to create a filmstrip layout web part.

I had to create my own filmstrip layout when I created the Calendar Feed web part for the SharePoint/sp-dev-fx-webparts repo.

And since I’m the world’s laziest developer and I hate to write things more than once, I had created the filmstrip component as a re-usable component. (Ok, I had called it CarouselContainer back then, but it was really a filmstrip layout control).

For the rest of this post, we’ll use my filmstrip layout control. If you don’t like the one I created, feel free to use your own.

As we did for the grid layout, I’ll show you how to insert a copy of my control into your own web parts — that way, you can customize it to suit your own needs.

(Don’t worry, I’ll show you how to move the controls into a component library in a later post, so you won’t have to always copy and paste my code into your solutions)

1. Create a web part solution with the Yeoman generator. For this sample, I’ll call my web part filmstrip
2. Copy the src\components\filmstripLayout folder from my project to yours
3. In your project’s terminal, install the slick-carousel and react-slick dependencies by using the following command:
npm i slick-carousel react-slick
4. In your web part’s component, located at src\webparts\[YourWebPartName]\components\[YourWebPartName].tsx, add an import for the filmstripLayout control:
import { FilmstripLayout } from '../../../components/filmstripLayout';
5. In your code, load the items you wish to display. You can load an array of any items you wish, but for my sample, I used items with thumbnail, title, name, profileImageSrc, location, and activity props because I want to use the DocumentCard control to render my list items. For my sample, I initialized the state with a static list of items in my component’s constructor:

constructor(props: IFilmstripProps) {
super(props);

this.state = {
items: [{
thumbnail: "https://lorempixel.com/400/200/technics/1/",
name: "Perry Losselyong",
location: "SharePoint",
activity: "3/13/2019"
}, {
thumbnail: "https://lorempixel.com/400/200/technics/2",
title: "The Wild, Untold Story of SharePoint!",
name: "Ebonee Gallyhaock",
profileImageSrc: "https://robohash.org/delectusetcorporis.bmp?size=50x50&set=set1",
location: "SharePoint",
activity: "6/29/2019"
}, {
thumbnail: "https://lorempixel.com/400/200/technics/3",
title: "Low Code Solutions: PowerApps",
name: "Seward Keith",
profileImageSrc: "https://robohash.org/asperioresautquasi.jpg?size=50x50&set=set1",
location: "PowerApps",
activity: "12/31/2018"
}, {
thumbnail: "https://lorempixel.com/400/200/technics/4",
name: "Sharona Selkirk",
profileImageSrc: "https://robohash.org/velnammolestiae.png?size=50x50&set=set1",
location: "SharePoint",
activity: "11/20/2018"
}, {
thumbnail: "https://lorempixel.com/400/200/technics/5/",
title: "Get with the Flow",
name: "Boyce Batstone",
location: "Flow",
activity: "5/26/2019"
}]
};
}
6. In your render method, add a filmstripLayout component. You should set the ariaLabel prop, but it is optional.

public render(): React.ReactElement<IFilmstripProps> {
return (

<FilmstripLayout
ariaLabel={"Sample filmstrip layout web part, showing sample items., Use right and left arrow keys to navigate between cards in the film strip."}

>

</FilmstripLayout>

);
}
7. To render elements within your filmstrip layout, you simply add rectangular elements as children of the FilmstripLayout component. In my sample, I’ll use a DocumentCard to render each item. To do so, you’ll need to add the following imports:
// Used to render document cards
import {
DocumentCard,
DocumentCardActivity,
DocumentCardPreview,
DocumentCardDetails,
DocumentCardTitle,
IDocumentCardPreviewProps,
DocumentCardLocation,
DocumentCardType
} from 'office-ui-fabric-react/lib/DocumentCard';
import { ImageFit } from 'office-ui-fabric-react/lib/Image';
8. Then, loop through your items and render a DocumentCard in your render method:

public render(): React.ReactElement<IFilmstripProps> {
return (

<FilmstripLayout
ariaLabel={"Sample filmstrip layout web part, showing sample items., Use right and left arrow keys to navigate between cards in the film strip."}
>
{this.state.items.map((item: any, _index: number) => {
const previewProps: IDocumentCardPreviewProps = {
previewImages: [
{
previewImageSrc: item.thumbnail,
imageFit: ImageFit.cover,
height: 130
}
]
};

return

<DocumentCard
type={DocumentCardType.normal}
onClick={(ev: React.SyntheticEvent<HTMLElement>) => alert("You clicked on an item")}
>
<DocumentCardPreview {...previewProps} />
<DocumentCardLocation location={item.location} />
<DocumentCardDetails>
<DocumentCardTitle
title={item.title}
shouldTruncate={true}
/>
<DocumentCardActivity
activity={item.activity}
people={[{ name: item.name, profileImageSrc: item.profileImageSrc }]}
/>
</DocumentCardDetails>
</DocumentCard>

;
})}
</FilmstripLayout>

);
}

If I didn’t forget anything, you should see a web part that looks like this:

## Conclusion

The filmstrip layout is a great way to show more items than available screen, but keep in mind that there are accessibility/usability issues with it.

If you want to build a filmstrip layout web part, you can use my code sample to get started.

In our next post, we’ll continue exploring the web part layout design patterns.

## Photo Credits

Microsoft MVP and PnP Team Member. Independent consultant. Certified SCRUM Master. SharePoint, Office 365 and Dynamics 365 are his favourite toys.


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# Math Help - One more group Question

1. ## One more group Question

Yesterday I posted a qn. on subgroups. The answers really helped me.
I have one more ..

I didn't understand this qn. at all

What does it mean and how to do it?

2. Originally Posted by Amy
Yesterday I posted a qn. on subgroups. The answers really helped me.
I have one more ..

I didn't understand this qn. at all

What does it mean and how to do it?
Sketch an equilateral triangle and plot a point at the center of it.

Note that if you rotate the triangle 120 degrees about this point the triangle looks the same as it initially did. This is the $R_{120}$ symmetry. Similarly for $R_{240}$. ( $R_0$ would be the "identity" transformation. Obviously if you rotate the triangle by 0 degrees it looks the same as it did before you "rotated" it.)

For the "flips" sketch a line from the center of the triangle through vertex A. Note that we have reflection symmetry over this line: if we reflect the triangle over this line it looks the same as it did before. This is the element $F_A$. Obviously we have the same kind of symmetry no matter which vertex we pick, so we also have $F_B$ and $F_C$. (The $F_H$ and $F_V$ are symmetries about "horizontal" and "vertical" lines. These symmetries depend on the orientation of the square, obviously.)

See what you can do with these rotations and reflections for a pentagon. (Hint: You'll have 5 rotations and 5 reflections.)

-Dan

So, is it some thing like
s={R0, R72,R154,R226,R298, FA, FB, FC,FD,FE}??

4. Originally Posted by Amy
So, is it some thing like
s={R0, R72,R154,R226,R298, FA, FB, FC,FD,FE}??
That'll do it!

Another one to watch out for (not present here) is "inversion" symmetry. The best way to describe this one is to let the center point of your figure be the origin. A group has inversion symmetry if you perform $x \to -x$ and $y \to -y$ and the figure looks the same as it did before the operation. The typical symbol for inversion is "i."

-Dan

5. In general, the symmetry group of a regular n-gon will consist of n rotations and n flips. The rotations will be through a mulitple of $\frac{360}{n}$ degrees. For the flips, it depends on whether n is odd or even. If n is odd, all the flips will be about axes passing from one vertex to the midpoint of the opposite side. If n is even, half the flips will be about axes passing from one vertex to the opposite vertex and the other half will be about axes passing from the midpoint of one side to the midpoint of the opposite side.

The symmetry group of the regular n-gon (n ≥ 3) is called the dihedral group of order 2n. Also, flips are more usually called “reflections”.

So, for the regular pentagon:

Originally Posted by Amy

format: plain


text:
# MATLAB: Selection of Matrix Elements without for loop

arrayimage processingMATLAB and Simulink Student Suitematrix

Hi.
I have a matrix of cooardinates like this :
A = [1 3;2 4;7 9];
And a Matrix 9×9:
B = rand(9);
How can I select B's elements in A cooardinest (for example : B(1,3) & B(2,4) & B(7,9)) without using a for loop.
Assuming that in my original code A has too many rows, ( more than 10K) and B is an image with too many points either.
As a matter of fact, I have some pixels' coordinates, and want to just update their's value in the original image by some formula. But I don't want to use for loop for selecting each pixel.

• B(sub2ind(size(B),A(:,1),A(:,2)))


text:
# [.net] Getting started with WPF

This topic is 3265 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi! I've been wanting to learn WPF since some time and now that I have a week off, I've decided to attempt to create a basic hexadecimal viewer and editor. I've been trying to accomplish something yesterday but I realized that WPF is indeed quite different from Windows Forms. I've managed to create a menu in my window and that's about it :). I don't really know which control (or Element, whatever) to use in order to create the main [Address Offset, Byte 0, Byte 1, ...] table that you usually see in hex editors. I've tried Grid, UniformGrid and attempted to use TextBlocks and FlowDocuments without satisfying results. I need a control that will space the text between each byte columns but allow text selection across them. Any ideas what I should use? Also, are there any must-read or must-see resources out there on getting started with WPF? Any suggestion will be helpful, thanks!

##### Share on other sites
With a Grid, you can specify rows and/or columns with whatever dimensions that you want:

<Grid>   <Grid.ColumnDefinitions>      <ColumnDefinition Width="32" /> // Creates a column of width 32      <ColumnDefinition Width="auto" /> // Creates a column of whatever width is required by its child elements      <ColumnDefinition Width="*" /> // Creates a column of width of whatever width is left   </Grid.ColumnDefinitions></Grid>

However, ListBox is most likely what you want, since you're probably going to want to specify any number of rows. You can set up a template for how each new object inserted into the row will be formatted. I don't know how, off hand, but you can just google it.

Alternatively, if you don't want all of the extra stuff that comes with a ListBox (selection, hover-over color changes, scrollbar, etc.), you can just use a StackPanel, and popuplate it with a set of user controls that are formatted how you want for the display. Then, in code, you just create these user controls, and add them to the StackPanel's Children property. That's probably what I'd do, though data binding might be a better fit.

I hope that helps!

##### Share on other sites
Quote:
 Also, are there any must-read or must-see resources out there on getting started with WPF?

As far as that goes.... play! You can do a lot of cool stuff with WPF, and when you get good, it's really easy to create a great looking UI. My suggestion, once you get the layout basics down is to look at animation (System.Windows.Media.Animation, I believe). It's really easy to animate any number of types. Most often you'll be using DoubleAnimations, but you can do any number of others, and it's even possible to roll your own (though I've never done it). Once you get the hang of that, you'll be animating anything and everything on your UI that has any sort of transition, and it'll be super easy to do.

And, as an added bonus, once you know WPF, you pretty much also know Silverlight. I entered my current job never touching Silverlight, but within a couple of days I'd essentially completely taken over the all Silverlight UI just because of what I knew of WPF. They're very similar (though WPF is nicer, obviously).

##### Share on other sites
Thanks a lot for your feedback, CyberSlag,

I'm going to try the ListBox and StackPanel right now to see if they fit my use case. There's only one thing that's still worrying me: if I use one of those "layout containers" (I still have to learn proper WPF terminology), and embed a number of Labels (I think that's what you're suggesting), how will I make it possible to select text from multiple adjacent labels simultaneously? For example, if I use a StackPanel and have Labels for each byte value (as is usual in hex editors), will I be able to make a selection that spans an arbitrary number of bytes even if they are individual labels?

##### Share on other sites
Well, you could do it a number of ways. You could just use a text box, and modify its control template using Expression Blend. I don't know enough to tell you exactly how to do it, or even that it's fully possible, but I think it would be.

Alternatively, you could handle the mouse events to facilitate selection. Clicking the mouse selects the byte that's currently under the mouse as your FirstSelectedByte, and, more importantly, sets a Selecting flag, such that every time you MouseEnter over a byte, you select everything from the FirstSelectedByte to the sender byte. You'll have to manually facilitate what that means, whether it's a rectangle of selected bytes, or selection as through it was a 1-dimensional array of bytes (even though it's represented 2-dimensionally).

##### Share on other sites
Quote:
Original post by CyberSlag5k
Quote:
 Also, are there any must-read or must-see resources out there on getting started with WPF?

As far as that goes.... play! You can do a lot of cool stuff with WPF, and when you get good, it's really easy to create a great looking UI. My suggestion, once you get the layout basics down is to look at animation (System.Windows.Media.Animation, I believe). It's really easy to animate any number of types. Most often you'll be using DoubleAnimations, but you can do any number of others, and it's even possible to roll your own (though I've never done it). Once you get the hang of that, you'll be animating anything and everything on your UI that has any sort of transition, and it'll be super easy to do.

And, as an added bonus, once you know WPF, you pretty much also know Silverlight. I entered my current job never touching Silverlight, but within a couple of days I'd essentially completely taken over the all Silverlight UI just because of what I knew of WPF. They're very similar (though WPF is nicer, obviously).

If you already know C# and are familiar with the old .Net controls XAML is really the only major change from winforms so it shouldn't be too different.
Petzold latest Programming for Windows book covers all the nitty gritty detail you need for WPF programming but is lacking in pictures compared to other books.
Actually, now that I think about it he probably has the program you are trying to make in that book since I know the first program Petzold likes to write for any new language is a hex editor [lol]
If you are more of a visual person like myself though they got plenty of WPF videos here.

• 9
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• 21
• 10
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## No-Regret Exploration in Goal-Oriented Reinforcement Learning

Dec 07, 2019

Many popular reinforcement learning problems (e.g., navigation in a maze, some Atari games, mountain car) are instances of the so-called episodic setting or stochastic shortest path (SSP) problem, where an agent has to achieve a predefined goal state (e.g., the top of the hill) while maximizing the cumulative reward or minimizing the cumulative cost. Despite its popularity, most of the literature studying the exploration-exploitation dilemma either focused on different problems (i.e., fixed-horizon and infinite-horizon) or made the restrictive loop-free assumption (which implies that no same state can be visited twice during any episode). In this paper, we study the general SSP setting and introduce the algorithm UC-SSP whose regret scales as $\displaystyle \widetilde{O}(c_{\max}^{3/2} c_{\min}^{-1/2} D S \sqrt{ A D K})$ after $K$ episodes for any unknown SSP with $S$ non-terminal states, $A$ actions, an SSP-diameter of $D$ and positive costs in $[c_{\min}, c_{\max}]$. UC-SSP is thus the first learning algorithm with vanishing regret in the theoretically challenging setting of episodic RL.


text:
# Singularity of potential

1. Oct 19, 2016

### spaghetti3451

1. The problem statement, all variables and given/known data

Determine the electric field and the charge density associated with the potential $\displaystyle{V(r)=A\frac{\exp{-\lambda r}}{r}}$, where $A$ and $\lambda$ are constants.

2. Relevant equations

3. The attempt at a solution

The electric field is easy to determine:

$\displaystyle{{\bf{E}} = -\nabla V}$
$\displaystyle{= -A\ \nabla\left(\frac{\exp\left(-\lambda r\right)}{r}\right)}$
$\displaystyle{= -A\ \left(\frac{\partial}{\partial r}\left(\frac{\exp\left(-\lambda r\right)}{r}\right),0,0\right)}$
$\displaystyle{= A\ \left(\frac{(1+\lambda r)\exp\left(-\lambda r\right)}{r},0,0\right)}$

But the electric field is singular at the origin.

How does this affect the calculation of the charge density?

2. Oct 19, 2016

### Fightfish

Well, that means you have a point charge at the origin - recall that $\nabla \cdot \left(\frac{\vec{r}}{r^2}\right) = 4 \pi \delta(r)$.

3. Oct 19, 2016

### spaghetti3451

In which step have I made the mistake?

$\rho = \epsilon_{0}\ \nabla\cdot{{\bf{E}}}$

$= \epsilon_{0}\ A\ \nabla\cdot{\left((1+\lambda r)\frac{\exp\left(-\lambda r\right)}{r},0,0\right)}$

$= \epsilon_{0}\ \frac{A}{r^{2}}\ \frac{\partial}{\partial r} \left(r^{2}(1+\lambda r)\left(\frac{\exp\left(-\lambda r\right)}{r}\right)\right)$

$= \epsilon_{0}\ \frac{A}{r^{2}}\ \frac{\partial}{\partial r} \left[(r+\lambda r^{2})\exp\left(-\lambda r\right)\right]$

$= \epsilon_{0}\ \frac{A}{r^{2}}\ \left[(1+2\lambda r)\exp\left(-\lambda r\right)-\lambda(r+\lambda r^{2})\exp\left(-\lambda r\right)\right]$

$= \epsilon_{0}\ \frac{A}{r^{2}}\ \left[(1+2\lambda r)-\lambda(r+\lambda r^{2})\right]\exp\left(-\lambda r\right)$

$= \epsilon_{0}\ A\ \left[1+\lambda r-(\lambda r)^{2})\right]\frac{\exp\left(-\lambda r\right)}{r^{2}}$

4. Oct 19, 2016

### Fightfish

Here:

5. Oct 20, 2016

### spaghetti3451

No, the signs are right; its the denominator that should be $r^2$, no?


text:
# Homotopy limits in type theory

@article{Avigad2015HomotopyLI,
title={Homotopy limits in type theory},
author={Jeremy Avigad and Krzysztof Kapulkin and Peter LeFanu Lumsdaine},
journal={Mathematical Structures in Computer Science},
year={2015},
volume={25},
pages={1040 - 1070}
}
• Published 2 April 2013
• Mathematics
• Mathematical Structures in Computer Science
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

### Nilpotent types and fracture squares in homotopy type theory

• Luis Scoccola
• Mathematics
Mathematical Structures in Computer Science
• 2020
The basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory is developed and general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven.

### On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory

This dissertation presents various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.

### Internal Languages of Finitely Complete (∞, 1)-categories

• Mathematics
• 2017
We prove that the homotopy theory of Joyal’s tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Löf Type Theory with

### Non-wellfounded trees in Homotopy Type Theory

• Mathematics
TLCA
• 2015
This work constructs coinductive types in a subsystem of Homotopy Type Theory given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom.

### Joyal's Conjecture in Homotopy Type Theory

Joyal's Conjecture asserts, in a mathematically precise way, that Martin--Lof dependent type theory gives rise to locally cartesian closed quasicategories. We prove this conjecture.

### Internal Language of Finitely Complete $(\infty, 1)$-categories

• Mathematics
• 2017
We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Lof Type Theory with

### Model structures on categories of models of type theories

• V. Isaev
• Mathematics
Mathematical Structures in Computer Science
• 2017
If a theory T has enough structure, then the category T-Mod of its models carries the structure of a model category, and it is proved that if T has Σ-types, then weak equivalences can be characterized in terms of homotopy categories of models.

### Formalizing Cartan Geometry in Modal Homotopy Type Theory

Both, the category of smooth manifolds and the category of schemes may be faithfully embedded in categories of (higher) sheaves on appropriate sites. Homotopy Type Theory is used to reason about

## References

SHOWING 1-10 OF 47 REFERENCES

### Inductive Types in Homotopy Type Theory

• Mathematics
2012 27th Annual IEEE Symposium on Logic in Computer Science
• 2012
Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined.

### Pull-Backs in Homotopy Theory

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no

### Homotopy Type Theory : Univalent Foundations of Mathematics

• Mathematics
• 2014
These lecture notes are based on and partly contain material from the HoTT book and are licensed under Creative Commons Attribution-ShareAlike 3.0.

### Homotopy Type Theory: Univalent Foundations of Mathematics

• Mathematics
• 2013
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy the- ory

### Cofibrations in Homotopy Theory

We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categories

### Homotopy Limit Functors on Model Categories and Homotopical Categories

Model categories: An overview Model categories and their homotopy categories Quillen functors Homotopical cocompleteness and completeness of model categories Homotopical categories: Summary of part

### Univalence for inverse diagrams and homotopy canonicity

A homotopical version of the relational and gluing models of type theory that uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory.

### Homotopy theoretic models of identity types

• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2009
Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success

### Higher Topos Theory

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the


text:
# #6 - Basic Operations with Polynomials

### Sample Question:

6. What is the sum of the polynomials $$3a^{2}b + 2a^{2}b^{2}$$ and $$-ab^{2} + a^{2}b^{2}$$ ?

A. $$3a^{2}b -ab^{2} + 3a^{2}b^{2}$$

B. $$3a^{2}b -ab^{2} + 2a^{2}b^{2}$$

C. $$2a^{2}b + 3a^{2}b^{2}$$

D. $$2a^{2}b^{3} + 2a^{4}b^{4}$$

E. $$-3a^{3}b^{3} + 2a^{4}b$$


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

 It is currently 26 Sep 2016, 19:48

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# When are you quitting?

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Manager
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18 Apr 2007, 11:52
Guys... I need help...

Let's say I want to finish working on day X. Now, I know I will most likely only need to give 2 weeks notice. but suppose my manager wants me to stay longer, then I should tell him earlier.

but if I tell him too early, and he says "get the f*ck out tomorrow" then I am equally screwed.

what should I do? I want to end on May 25 :D
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18 Apr 2007, 12:16
In my opinion, if you are unsure how your manager will react, look out for yourself. You are only required to give two weeks so that time frame is what 'i would suggest. Business is business, you do not think that they tell people 1.5 months in advance when they fire someone, so why should u have a conscious to do anything else. This is all being said with the assumption you really think your boss would say "f@#$you leave tomorrow". If you are close to your boss I would give him/her a heads up. Director Joined: 28 Jun 2006 Posts: 958 Followers: 1 Kudos [?]: 78 [0], given: 0 [#permalink] ### Show Tags 18 Apr 2007, 12:17 If you think it's likely that they would say, "Get out of here tomorrow," then I would only give two weeks notice. GMAT Club Legend Affiliations: HHonors Diamond, BGS Honor Society Joined: 05 Apr 2006 Posts: 5926 Schools: Chicago (Booth) - Class of 2009 GMAT 1: 730 Q45 V45 WE: Business Development (Consumer Products) Followers: 307 Kudos [?]: 1987 [0], given: 7 [#permalink] ### Show Tags 18 Apr 2007, 13:00 Id look out for yourself first. If you think theres even a remote chance of him saying "get out" ... I'd give him 2 to 3 weeks notice. Manager Joined: 25 Sep 2006 Posts: 248 Followers: 1 Kudos [?]: 0 [0], given: 0 [#permalink] ### Show Tags 18 Apr 2007, 13:58 ok Well, I don't think there is a likely chance for "get out now". But along the same lines, I was thinking I could give more notice (say 1 month) to be nice and courteous. But what if they say "oh how nice of you, we only need 2 weeks, see ya" then I'd be short 1 pay check. which I most likely need to cover my vacation plans One more thing I should mention, our company just announced that we will cut$500MM in spending this year, which means there will be many layoffs. Tho I doubt I would be lucky enough be chosen for a layoff.... it would be PERFECT to get a 1 or 2 month severance right now
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18 Apr 2007, 17:11
fluffydot wrote:
ok Well, I don't think there is a likely chance for "get out now".
But along the same lines, I was thinking I could give more notice (say 1 month) to be nice and courteous.

But what if they say "oh how nice of you, we only need 2 weeks, see ya" then I'd be short 1 pay check. which I most likely need to cover my vacation plans

One more thing I should mention, our company just announced that we will cut \$500MM in spending this year, which means there will be many layoffs. Tho I doubt I would be lucky enough be chosen for a layoff.... it would be PERFECT to get a 1 or 2 month severance right now

If you want to get fired, you can
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18 Apr 2007, 18:24
I am planning on telling my boss when I am leaving as soon as I pick a school that accepts me next year. They wont be letting me go and by giving them a lot of notice I will be able to train up a co-work on my areas of expertise. The people I work with are great overall so I am not going to screw them over by only giving two weeks of notice and dropping all my work in someones lap. Most likely I will attempt to get ahead on my work too...I know how pissed I have been in the past when people have left me with all their work even if its just for a vacation. After figuring out my saved up vacation I will probably leave about a month before the start date of school, which is about how much time I should have saved up. I would rather not miss the paycheck by quitting early but I wouldn't have to worry about health insurance since I am on my wifes.

As to the person who thinks they may get fired...that seems like something straight out of the Office if your boss would do that. Remember when the two branches merged and Tony was trying to quit but Michael fired him. Jan then called up and yelled at Michael cause now they had to give Tony severance.
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24 Apr 2007, 17:26
This is exactly what I do these days!
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27 Apr 2007, 14:22
i'm not matriculating in fall 2007 (aiming for fall 2008). but i do plan on quitting sometime in may before school. then i'm going to travel, travel, travel - possible locations include egypt/jordan and china/tibet. of course this'll put a dent in my wallet, but hey - life is too short!

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# Distributed Submodular Maximization with Parallel Execution

The submodular maximization problem is widely applicable in many engineering problems where objectives exhibit diminishing returns. While this problem is known to be NP-hard for certain subclasses of objective functions, there is a greedy algorithm which guarantees approximation at least 1/2 of the optimal solution. This greedy algorithm can be implemented with a set of agents, each making a decision sequentially based on the choices of all prior agents. In this paper, we consider a generalization of the greedy algorithm in which agents can make decisions in parallel, rather than strictly in sequence. In particular, we are interested in partitioning the agents, where a set of agents in the partition all make a decision simultaneously based on the choices of prior agents, so that the algorithm terminates in limited iterations. We provide bounds on the performance of this parallelized version of the greedy algorithm and show that dividing the agents evenly among the sets in the partition yields an optimal structure. We additionally show that this optimal structure is still near-optimal when the objective function exhibits a certain monotone property. Lastly, we show that the same performance guarantees can be achieved in the parallelized greedy algorithm even when agents can only observe the decisions of a subset of prior agents.

## Authors

• 3 publications
• 6 publications
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• ### The Cost of Denied Observation in Multiagent Submodular Optimization

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• ### Probably Approximately Correct Greedy Maximization

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## I Introduction

Submodular maximization is an important topic with relevance to many fields and applications, including sensor placement [1], outbreak detection in networks [2], maximizing and inferring influence in a social network [3, 4]

[5], clustering [6], assigning satellites to targets [7], path planning for multiple robots [8], and leader selection and resource allocation in multiagent systems [9, 10]. An important similarity among these applications is the presence of an objective function which exhibits a “diminishing returns” property. For instance, a group of leaders can impose some influence on a social network, but the marginal gain in influence achieved by adding a new leader to the group decreases as the size of the group increases. Objective functions (such as influence) satisfying this property are submodular.

The submodular maximization problem is to choose a set of elements (such as leaders) which maximize the submodular objective function, according to some constraints. This problem is known to be NP-Hard for certain subclasses of submodular functions [11]. Therefore, much research has focused on how to approximate the optimal solution [12, 13, 14, 15]. The overall message of this research is that simple algorithms can perform well by providing solutions which are guaranteed to be within some factor of optimal.

One such algorithm is the greedy algorithm, first proposed in [16]. It was shown in this seminal work that for certain classes of constraints the solution provided by the greedy algorithm is guaranteed to be within of the optimal, and within of the optimal for the more general case [17]. Since then, more sophisticated algorithms have been developed to show that there are many instances of the submodular maximixation problem which can be solved efficiently within the guarantee [12, 18]. It has also been shown that progress beyond this level of optimality is not possible using a polynomial-time algorithm [19].

More recent research has focused on distributed algorithms, since in many cases having a centralized agent with access to all the relevant data is untenable [20, 6, 21]. In this case, the greedy algorithm can be generalized using a set of agents, each with its own decision set. The combined set of decisions by the agents is evaluated by the submodular objective function, which they seek to maximize. Each agent chooses sequentially, maximizing its marginal contribution compared to the actions of the prior agents. In this setting, the greedy algorithm has been shown to provide a solution within of the optimal.

Observe that with this distributed greedy algorithm, each agent must have access to all prior agents’ decisions, meaning that the algorithm requires iterations to terminate. For large , this may be unacceptable. Therefore, researchers have explored how to parallelize the greedy algorithm [5, 22, 23]. While much has been done, many of these parallelization techniques still require some entity to have access to all decisions – in essence, part of the parallelization technique is how to assign decision sets to agents.

A natural extension is to consider the case where no such centralized authority is present, i.e., the agents’ decision sets are determined a priori and cannot be modified. Terminating the greedy algorithm in iterations thus requires a partitioning of the agents into sets , where now each agent in set simultaneously chooses an action which maximizes its marginal contribution relative to the actions chosen by all the agents in sets . In this setting, one can ask the following questions:

1. What is the best way to partition the agents? Should the agents be spread evenly across the sets in the partition, or should set 1 or set be larger than the others?

2. If some additional structure is known about the submodular objective function, how does that affect the partitioning strategy?

In the non-parallelized setting, recent research has explored how the performance of the greedy algorithm degrades as we relax the information sharing constraint that each agent must have access to the decisions of all prior agents [24, 25]. Therefore, it is a natural extension to ask in the parallelized setting:

1. If we relax the requirement that an agent in partition set must have access to the decisions of all agents in sets , can the same level of performance be maintained by the greedy algorithm?

In response to the questions listed above, the contributions of this paper are the following:

1. Theorem 1 shows that partitioning the agents into equally-sized sets yields the highest performance guarantee of the greedy algorithm, given a time constraint . The performance guarantee is also shown, in terms of and .

2. Theorem 2 shows that if we know some additional structure about the submodular function, i.e., a lower bound on each agent’s relative marginal contribution, then a near-optimal partitioning strategy is the same as that of Theorem 1. Increased performance guarantees are also shown.

3. Theorems 3 and 4 prove that the above performance guarantees on the greedy algorithm can be preserved even with less information sharing among the agents.

## Ii Model

Consider a base set and a set function . Given , we define the marginal contribution of with respect to as

 f(A∣B)=f(A∪B)−f(B) (1)

In this paper, we are interested in distributed algorithms for maximization of submodular functions, where there is a collection of decision-making agents, named as , and a set of decisions . And each decision-making agent is associated with a set of permissible decisions so that form a partition of . For rest of the paper, we refer to decision-making agents as simply agents. Then the collection of decisions by all agents is called an action, and we denote the set of all actions as the action profile . An action is evaluated by an objective function as . Furthermore, we restrict our attention to ’s that are submodular functions, which satisfy the following properties:

• Normalized .

• Monotonic for all and .

• Submodular for all and .

For simplicity, we will refer to the set of submodular functions as . The goal of the submodular maximization problem is to find:

 xopt∈argmaxx∈Xf(x) (2)

For convenience, we overload to accept elements: for , and allow multiple inputs: for . We also denote for any nonnegative integer and for any nonnegative integers . For any and , we will write with the understanding that we mean . For instance .

A distributed greedy algorithm can be used to approximate the problem as stated in (2). In the greedy algorithm, the agents make decision in sequential order according to their names and each agent attempts to maximize its marginal contribution with respect to the choices of prior agents:

 xsoli∈argmaxxi∈Xif(xi∣x1:i−1) (3)

Note that the greedy solution may not be unique. In the context of this paper, we assume that is the worst solution among all possible greedy solutions.

Let be from the greedy algorithm subject to objective function and action profile . Then, to measure the quality of the greedy algorithm, we consider its competitive ratio:

 γ=inff∈F,Xγ(f,X) (4)

The well known result from [17] states that , which means that (3) guarantees that the performance of the greedy solution is always within 1/2 of optimal. Furthermore, only one agent can make a decision at any given time and thus a solution will be found in precisely iterations. The central focus of this works is on characterizing the achievable competitive ratio for situations where a system-design does not have the luxury of have iterations to construct a decision. To this end, we introduce the parallelized greedy algorithm to allow multiple agents make decisions at the same time. More formally, we consider a situation where the system is given limited number of iterations and needs to come up with an iteration assignment so that each agent makes decision only based off the choices made by agents from earlier iterations:

 xsoli∈argmaxxi∈Xif(xi∣xNi) (5)

where . Note, that to maintain consistency with (3), must preserve the implicit ordering of the greedy algorithm: whenever .

We are interested in the performance of (5) given a fixed number of agents and number of iterations. We define the competitive ratio of a time step assignment as:

 γ(P)=inff∈F,Xγ(f,X,P) (6)

where is from the parallelized greedy algorithm subject to objective function , action profile and iteration assignment . Denote the set of all possible iteration assignments with agents and at most iterations as:

 Pn,q={P:[n]→[q]} (7)

So we seek to find the best possible competitive ratio, and the iteration assignment which achieves the optimal competitive ratio (if such assignment exists):

 ρ(n,q)=supP∈Pn,qγ(P) (8)
 P∗n,q∈argmaxP∈Pn,qγ(P) (9)

Additionally, we want to know on average how many agents are making decisions in the same iteration, so we define the concurrency as . Later we will find to be heavily dependent on this measure.

An example of the submodular maximization problem is the weighted set cover problem [18]. Given a target set and a mapping from the decisions to subsets of . The value of an action profile is determined by a weight function :

 f(x)=∑y∈∪iT(xi)w(y) (10)

Intuitively, this problem aims to “cover” as much of the target set as possible. An instance of this problem is illustrated by Figure 1.

## Iii Optimal Parallel Structures

In this section, we present the best possible competitive ratio for the parallelized greedy algorithm and optimal iteration assignment which achieves such a bound. We perform this analysis for both general submodular objective functions and submodular objective functions with an additional property which we call -strict monotonicity.

Note that according to (5), a pair of agents do not utilize each other’s decisions (in either direction) when they are in the same iteration. This intuitively represent some “blindspots” in the parallelized greedy algorithm compared to the original greedy algorithm. One way to reduce those “blindspots” is to divide the agents evenly among the available iterations, in other words, we want the number of agents deciding in parallel to be as close to the concurrency as possible. Theorems 1 and 3 illustrate that this idea yields the best possible iteration assignment.

###### Theorem 1

Given a parallelized submodular maximization problem with agents and iterations, the competitive ratio is:

 ρ(n,q)={1rif n≡1(modq)1r+1otherwise (11)

In particular, when ,

 P∗n,q={⌈ir−1⌉if i

Otherwise,

 P∗n,q=⌈ir⌉ (13)

The above optimal iteration assignment is illustrated by Figure 2. This theorem shows that the competitive ratio is inversely proportional to concurrency. We will present a more general version of Theorem 1 and its proof in Section IV-C.

Recent research, such as [21], has been focused on whether imposing additional structure on the objective function and action profile would help the underlying greedy algorithm to yield better performing solutions. Here, we introduce a property that enables a parallelized greedy algorithm to yield higher competitive ratio.

###### Definition 1

Objective function is said to be -strictly monotonic for some real value if for all and . For simplicity, we will refer to the set of functions that are both submodular and -strictly monotone as .

We can extend competitive ratio defined in (6) and (8) to when they are subject to -strict monotonicity:

 γ′(P)=inff∈F′,Xγ(f,X,P) (14)
 ρ′(n,q)=supP∈Pn,qγ′(P) (15)

The theorem below presents an upper bound on the best possible competitive ratio subject to -strict monotonicity and demonstrates that the iteration assignment (13) nearly achieves this bound.

###### Theorem 2

Given a parallelized submodular and -strictly monotone maximization problem with agents and iterations, then:

 (r−1)β+1r≥ρ′(n,q)≥(r−1)β+1r−β+1 (16)

Furthermore, the time step assignment in (13) achieves the lower bound .

Note that as , (16) converges to (11), and as , converges to 1. This confirms the intuition that for higher , actions have less “overlap” and as a result the greedy algorithm can perform closer to the optimal. For more detail, Figure 3 illustrates how the lower bound changes with respect to at different . Also, as both the lower and upper bounds converge to . So the lower and upper bounds are close to each other when concurrency is high. We will present a more general version of Theorem 2 and its proof in Section IV-D.

## Iv Parallelization as Information Exchange

### Iv-a Preliminaries

In this section, we will employ several concepts from graph theory. Throughout this section, we will assume that we have an undirected graph :

###### Definition 2

Nodes form a clique if for all distinct , . A clique cover of is a partition of so that each set in the partition forms a clique. And the clique cover number is the least number of cliques necessary to form a clique cover.

###### Definition 3

Nodes form an independent set if for all distinct , . The maximum independent set is an independent set with the largest possible number of nodes. And the independence number is the size of the maximum independent set.

### Iv-B Parallization and Information

In both (3) and (5), each agent uses the choices of prior agents to make its own decision. Alternatively, we can view this process as an agent communicating its decision to other agents who depend on this piece of information. We can thus model this information exchange as an undirected graph , where represents the agents and each edge represents information being exchanged between two agents. Since the greedy algorithm has an implicit ordering induced by the agents’ names, for every pair so that , agent requires knowing the choice of agent , but agents does not access agent ’s decision. Hence, we can use , which we will call the information graph, to determine the set of prior agents which agent accesses as . From here, it is natural to consider the generalized greedy algorithm proposed in [24]:

 xsoli∈argmaxxi∈Xif(xi∣xNi) (17)

First we noticed that the parallelized greedy algorithm as defined in (5) is a special case of (17). In the parallelized greedy algorithm, each agent requires information from all agents executed in prior iterations. Therefore, an iteration assignment it induces a corresponding information graph so that , . Figure 4 illustrates how the iteration assignments from Figure 2 induce corresponding information graphs. For rest of this section, we mention an iteration assignment with the understanding that we are interested in its induced information graph.

Conversely, an information graph induces an ordering to parallelize the generalized greedy algorithm. If all agents in have made their decisions but agent is still undecided, then we can let agents and to make their decisions in the same iteration without affecting the algorithm. Intuitively, we can think of information being relayed through some paths in and the earliest iteration for which agent can decide depends on the longest path in leading to node . Formally, a simple induction argument shows that the function determines the earliest iteration in which an agent could decide:

 ¯¯¯¯P(G;i)={1if Ni=∅1+maxj∈Ni¯¯¯¯P(G;j)otherwise (18)

This function can be used to construct a parallelization of the greedy algorithm in which agent makes its decision in iteration and for every , . From here we can define as the set of graphs which induces a parallelization with at most iterations:

 Gn,q={G=(V,E):|V|=n,maxi¯¯¯¯P(G;i)≤q} (19)

From the the definitions, it is clear that . Hence we can adopt competitive ratios for information graphs so that (6), (8) and (9) become:

 γ(G)=inff,Xγ(f,X,G) (20)
 η(n,q)=supG∈Gn,qγ(G) (21)
 G∗n,q∈argmaxG∈Gn,qγ(G) (22)

and (14) and (15) become:

 γ′(G)=inff∈F′,Xγ(f,X,G) (23)
 η′(n,q)=supG∈Gn,qγ′(G) (24)

where is from the generalized greedy algorithm subject to objective function , action profile and information graph .

It is natural to ask whether the above generalization of parallelized greedy algorithm yields higher competitive ratio under the same number of agents and iterations. Also, in some applications, the information exchange may incur some costs, and therefore having a information graph with many edges is undesirable. So we wish to answer a second question, whether we can achieve the same competitive ratio in (11) and (16) with information graphs that have fewer edges than that of (12) and (13). Theorems 3 and 4, as presented below, show that generalizing to information graphs does not yield higher competitive ratio but allows us to achieve the same optimal competitive ratio with fewer edges.

### Iv-C Extension of Theorem 1 to Information Graphs

We will present and prove a version of Theorem 1 generalized to information graphs, and Theorem 1 follows from the same logic.

###### Theorem 3

Given a parallelized submodular maximization problem with agents and iterations, the competitive ratio is:

 η(n,q)={1rif n≡1(modq)1r+1otherwise (25)

In particular, when , for the following edge set , achieves the equality :

 E={(i,j) for all i

Otherwise, for the following edge set , achieves the equality :

 E={(i,j)∣i≠j,i≡j(modr)} (27)

Figure 5 illustrates the aforementioned optimal information graphs. Note that (26) and (27) are subgraphs of the information graph induced by (12) and (13), respectively. Furthermore, (26) has approximately times as many edges as (12) and (27) has approximately times as many edges as (13). This is true because every agent (except agent when ) requires information from just one agent from each prior time step. Additonally, (27) is “delightfully parallel” so that we can partition the agents into threads according to the clique in which they reside, and no information need to be exchanged among threads.

To prove Theorem 3, we utilize results from [25] that relate the competitive ratio of a graph to its independence number and clique cover number:

###### Lemma 1

Given any information graph ,

 1α(G)≥γ(G)≥1θ(G)+1 (28)

Additionally, if there exists and a maximum independent set s.t. for some , then

 γ(G)≤1α(G)+1 (29)

To better leverage Lemma 1, it is also useful to show that is related to the concurrency:

###### Lemma 2

For any , .

Let be the set of agents executed in the th iteration. Note that must be an independent set for any . By the pigeonhole principle, ; hence .

If , combining Lemma 1 and 2 yields . Now let be the graph described in (26) and we can show that . First note that for any , and if , . Hence this graph is in . Let , and note that for all , . Then we have:

 f(xopt) ≤f(xopt,xsolT) (30a) =f(xsolT)+n∑i=1f(xopti∣xopt1:i−1,xsolT) (30b) ≤f(xsolT)+n∑i=1f(xopti∣xsolNi) (30c) =f(xsolT)+r−1∑j=1q−1∑i=0f(xopti(r−1)+j∣xsolNi) (30d) ==+f(xoptn∣xsolT) ≤f(xsolT)+r−1∑j=1q−1∑i=0f(xsoli(r−1)+j∣xsolNi) (30e) ==+f(xsoln∣xsolT) =r−1∑j=1f(xsol{j,j+(r−1),…,j+(r−1)(q−1)}) (30f) ==+f(xsoln,xsolT) ≤rf(xsol) (30g)

where (30a) and (30g) follow from monotonicity, (30b) and (30f) follow from telescoping sums, (30c) follows from submodularity, (30d) follows from the graph structure as defined in (26), and (30e) follows from the definition of the greedy algorithm as stated in (17). From here we can conclude that when .

Now we consider the case where . If , then by Lemma 1, . If , then and apply the pigeonhole principle on , there exists some , so that . The above is true because the condition ensures that . Therefore there must exist some so that . Then, Lemma 1 implies that . Now let be the graph described by (27) For any , , hence this graph is in . Also note that it consists of disjoint cliques, therefore . By Lemma 1, we have , hence the equality is achieved.

### Iv-D Extension of Theorem 2 to Information Graphs

We will prove a version of Theorem 2 generalized to information graphs, and the original theorem follows from the same logic.

###### Theorem 4

Given a parallelized submodular and -strictly monotone maximization problem with agents and iterations, then:

 (r−1)β+1r≥η′(n,q)≥(r−1)β+1r−β+1 (31)

Furthermore, the graph as defined in (27) achieves the lower bound .

To prove Theorem 4, we present a lower and an upper bound on the competitive ratio subject to -strict monotoncity in a similar fashion as Lemma 1. The following lemma is proven in the Appendix.

###### Lemma 3

Given any information graph ,

 (α(G)−1)β+1α(G)≥γ′(G)≥(θ(G)−1)β+1θ(G)−β+1 (32)

Now we come back to the proof of Theorem 4. The upper bound in (31) follows from combining Lemma 2 and the upper bound in Lemma 3. Consider the graph described by (27); as we already argued previously in Section IV-C, and . Hence, from the lower bound in Lemma 3, we conclude that this achieves the lower bound in (31), so we are done.

## V Conclusion

In this paper, we derived bounds on the competitive ratio of the parallelized greedy algorithm for both submodular objective functions and those with an additional property of -strictly monotone. We also provided the optimal design which achieves such bound and showed that a graph theoretic approach yields more effective parallelization that still achieves the same competitive ratio.

There are several directions of future research. One possibility is to consider whether employing other structural properties on the objective functions can also improve the competitive ratio of greedy algorithm. In particular, we are interested in properties that consider fixed number actions at once because -strict monotonicty considers arbitrarily many actions at once. Another possible direction is to consider applying something other than the marginal contribution in making the greedy decisions.

[Proof of Lemma 3]

We first show the lower bound on . In this proof, we consider some minimal clique covering of . Let denotes the clique containing the node, and . We will upper bound in terms of ’s. Then we express as a concave function in and using convexity to derive the final lower bound.

First, we need to have an appropriate objective function and action profile . Given , we can assume that without affecting the competitive ratio. We want for all , which ensures that all decisions are distinct. Suppose not and for some , , then we transform to so that , for any , and . Also define as:

 ⎧⎪ ⎪⎨⎪ ⎪⎩¯¯¯f(u,v,A)=f(u,A)+βf(u)¯¯¯f(u,A)=¯¯¯f(v,A)=f(u,A)¯¯¯f(A)=f(A) (33)

where . Note that the above transformation does not affect the competitive ratio because and . Also, from some simple algebra:

 ¯¯¯f(u∣v,A)=¯¯¯f(v∣u,A) =βf(u) (34a) ¯¯¯f(y∣u,v,A) =f(y∣u,A) (34b)

where , . From (34), it is easy to verify that satisfies all properties of submodular functions and -strict monotonicty; but for brevity, we will not explicitly show them here. Hence, for rest of the proof, we can safely assume that for all .

We bound through strict monotonicity:

 f(xopt)+βθ∑j=1σj ≤f(xopt)+θ∑j=1∑i∈Kjβf(xsoli) (35a) =f(xopt)+n∑i=1βf(xsoli) (35b) ≤f(xopt)+n∑i=1f(xsoli∣xsol1:i−1,xopt) (35c) =f(xopt,xsol) (35d)

where (35a) follows from submodulairty, (35b) follows from the fact that are disjoint, (35c) follows from -strict monotonicity, and (35d) is a telescoping sum.

Then we bound using properties of submodular functions and the greedy algorithm.

 f(xopt,xsol) =f(xsol)+n∑i=1f(xopti∣xopt1:i−1,xsol) (36a) ≤f(xsol)+n∑i=1f(xopti∣xsolNi) (36b) ≤f(xsol)+n∑i=1f(xsoli∣xsolNi) (36c) ≤f(xsol)+n∑i=1f(xsoli∣xsolNi∩K(i)) (36d) =f(xsol)+θ∑j=1∑i∈Kjf(xsoli∣xsolNi∩Kj) (36e) =f(xsol)+θ∑j=1σj (36f)

where (36a) and (36f) are telescoping sums, (36b) and (36d) follow from submodularity, (36e) follows from the fact that ’s are disjoint, and (36c) follows from the greedy algorithm as defined in (17).

Combining (35) and (36), we have that

 f(xopt)f(xsol)≤1+(1−β)θ∑j=1σjf(xsol) (37)

And we can upper bound ’s using strict monotonicity in a manner same as that of (35).

 σj+∑ℓ≠jβσℓ≤ σJ+∑i∉Kjβf(xsoli) (38a) ≤ σj+∑i∉Kjf(xsoli∣xsol1:i−1∪xsolKj) (38b) = f(xsol) (38c)

Hence for any ,

 σjf(xsol)≤σjβ∑θℓ=1σℓ+(1−β)σj=σjβθ+(1−β)σj (39)

where we WLOG impose the normalization that . Note that the RHS is concave with respect to . Recall Jensen’s inequality states that given a convex function and numbers , then is minimized when . Therefore, substituting into (37) yields:

 1γ′(G)=f(xopt)f(xsol)≤1+(1−β)θβ(θ−1)+1=θ−β+1β(θ−1)+1 (40)

Hence we derived the lower bound.

Now we show the upper bound on . Let be a maximum independent set of and . Consider and action profile determined by:

 Xi={{ui.vi}if i∈I∅otherwise (41)

Define the objective function so that for any ,

 f(x)=min(1−β,∑ui∈x1−β)+∑ui∈xβ+∑vi∈x1 (42)

Note that through this construction, is a submodular function and is also -strictly monotonic. also has the following properties for any :

1. .

2. for any because by definition, we have .

3. for any

From these properties, the agents in are equally incentivized to pick either option. In the greedy solution, the actions are the ’s. And in the optimal solution, the actions are the ’s. This results in and , hence .

## References

• [1] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg, “Near-optimal sensor placements: Maximizing information while minimizing communication cost,” in Proceedings of the International Conference on Information Processing in Sensor Networks.   ACM, 2006, pp. 2–10.
• [2] J. Leskovec, A. Krause, C. Guestrin, C. Faloutsos, C. Faloutsos, J. VanBriesen, and N. Glance, “Cost-effective outbreak detection in networks,” in Proceedings of the ACM SIGKDD international conference on Knowledge discovery and data mining.   ACM, 2007, pp. 420–429.
• [3] D. Kempe, J. Kleinberg, and É. Tardos, “Maximizing the spread of influence through a social network,” in Proceedings of the ACM SIGKDD international conference on Knowledge discovery and data mining.   ACM, 2003, pp. 137–146.
• [4] M. Gomez-Rodriguez, J. Leskovec, and A. Krause, “Inferring networks of diffusion and influence,” ACM Transactions on Knowledge Discovery from Data (TKDD), vol. 5, no. 4, p. 21, 2012.
• [5] H. Lin and J. Bilmes, “A class of submodular functions for document summarization,” in Proceedings of the Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1.   Association for Computational Linguistics, 2011, pp. 510–520.
• [6] B. Mirzasoleiman, A. Karbasi, R. Sarkar, and A. Krause, “Distributed submodular maximization: Identifying representative elements in massive data,” in Advances in Neural Information Processing Systems, 2013, pp. 2049–2057.
• [7] G. Qu, D. Brown, and N. Li, “Distributed greedy algorithm for multi-agent task assignment problem with submodular utility functions,” Automatica, vol. 105, pp. 206–215, 2019.
• [8] A. Singh, A. Krause, C. Guestrin, W. J. Kaiser, and M. A. Batalin, “Efficient planning of informative paths for multiple robots.” in

International Joint Conferences on Artificial Intelligence

, vol. 7, 2007, pp. 2204–2211.
• [9] A. Clark and R. Poovendran, “A submodular optimization framework for leader selection in linear multi-agent systems,” in Conference on Decision and Control and European Control Conference.   IEEE, 2011, pp. 3614–3621.
• [10] J. R. Marden, “The role of information in distributed resource allocation,” IEEE TCNS, vol. 4, no. 3, pp. 654–664, Sept 2017.
• [11] L. Lovász, “Submodular functions and convexity,” in Mathematical Programming The State of the Art.   Springer, 1983, pp. 235–257.
• [12] G. Calinescu, C. Chekuri, M. Pál, and J. Vondrák, “Maximizing a submodular set function subject to a matroid constraint,” in

International Conference on Integer Programming and Combinatorial Optimization

.   Springer, 2007, pp. 182–196.
• [13] M. Minoux, “Accelerated greedy algorithms for maximizing submodular set functions,” in Optimization Techniques.   Springer, 1978, pp. 234–243.
• [14] N. Buchbinder, M. Feldman, J. Seffi, and R. Schwartz, “A tight linear time (1/2)-approximation for unconstrained submodular maximization,” SIAM Journal on Computing, vol. 44, no. 5, pp. 1384–1402, 2015.
• [15] J. Vondrák, “Optimal approximation for the submodular welfare problem in the value oracle model,” in

ACM Symposium on Theory of Computing

.   ACM, 2008, pp. 67–74.
• [16] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions I,” Mathematical Programming, vol. 14, no. 1, pp. 265–294, 1978.
• [17] M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, “An analysis of approximations for maximizing submodular set functions—ii,” in Polyhedral combinatorics.   Springer, 1978, pp. 73–87.
• [18] M. Gairing, “Covering games: Approximation through non-cooperation,” in International Workshop on Internet and Network Economics.   Springer, 2009, pp. 184–195.
• [19] U. Feige, “A threshold of ln for approximating set cover,” Journal of the ACM (JACM), vol. 45, no. 4, pp. 634–652, 1998.
• [20] A. Clark, B. Alomair, L. Bushnell, and R. Poovendran, Submodularity in dynamics and control of networked systems.   Springer, 2016.
• [21] M. Corah and N. Michael, “Distributed submodular maximization on partition matroids for planning on large sensor networks,” in Conference on Decision and Control.   IEEE, 2018, pp. 6792–6799.
• [22] X. Pan, S. Jegelka, J. E. Gonzalez, J. K. Bradley, and M. I. Jordan, “Parallel double greedy submodular maximization,” in Advances in Neural Information Processing Systems, 2014, pp. 118–126.
• [23] A. Ene, H. L. Nguyen, and A. Vladu, “A parallel double greedy algorithm for submodular maximization,” arXiv preprint arXiv:1812.01591, 2018.
• [24] B. Gharesifard and S. L. Smith, “Distributed submodular maximization with limited information,” Trans. on Control of Network Systems, vol. 5, no. 4, pp. 1635–1645, 2017.
• [25] D. Grimsman, M. S. Ali, J. P. Hespanha, and J. R. Marden, “The impact of information in greedy submodular maximization,” Trans. on Control of Network Systems, 2018.


text:
# Pricing of the compound coupon bond with PDE

I am now studying finance math using Steven E.Shereve's book. Using Interest Rate models, We can the price for zero-coupon with maturity price $$1$$ under Hull-White interest rate model[page 274] and Cox-Ingersoll-Ross positive constants model[page 275].

My question is that the formula for price of zero-coupon bond can be found in various literature, and I wonder why it is not covered in any book about a compound coupon bond pricing formula. Please propose the price of compound coupon in Hull-White interest rate model or let me know the relevant references. Compound coupon is a bond that does not pay interest before maturity and pays interest and principal at maturity, which is used in many financial sites, such as housing bonds.

My attempt to find price of compound coupon is as follows.

Statement Given probability space ($$\Omega$$, $$F$$, $$\mathbb{P}$$ ) and, under the Hull-White interest rate model for the short rate $$r$$, given by \begin{align*} dr_{t}=\left(b(t)-a r_{t} \right) dt + \sigma_{r} dW_{t}^{r} \end{align*} the no-arbitrage price at time $$t$$ of a compound bond expiring at time $$T$$ is as follows : $$B(t,T,r) = \mathbb{E}^{*}(e^{\int_{t}^{T}{r_{s}ds}} \, \vert \, r_t = r)$$ with the initial condition $$B(T,T,r) = 1$$. Then, $$B(t,T,r)$$ can be calculated as $$B(t,T,r) = e^{-C(t,T)r-A(r,T)}$$ where \begin{align*} C(t,T) &= \dfrac{1}{a} \left( {e^{a(t-T)-1}} \right) , \\ A(t,T) &= \dfrac{\sigma_{r}^{2}}{4 a^{3}} \left( - e^{-2a(T-t)} +4e^{-a(T-r)} +2a(T-t)-3 \right) + \int_{t}^{T}{\dfrac{b(s)}{a} \left( 1-e^{a(s-T)} \right) ds}. \end{align*}

Proof for Statement Under the Hull-White interest rate model given by \begin{align*} dr_{t}=\left(b(t)-a r_{t} \right) dt + \sigma_{r} dW_{t}^{r} \end{align*} the no-arbitrage price at time $$t$$ of a compound coupon bond with an expiry time $$T$$ is follows : \begin{align*} & B(t,T,r) = \mathbb{E}^{*}(e^{\int_{t}^{T}{r_{s}ds}} \, \vert \, r_t = r) \\ \end{align*} Then, by using Feynmann-Kac formula, the solution of $$B(t,T,r)$$ satisfies the following partial differential equation: \begin{align*} & \dfrac{\partial B}{\partial t} + \left( {b(t)-a r_t } \right) \dfrac{\partial B}{\partial r} + \dfrac{1}{2} \sigma_{r}^{2} \dfrac{\partial^{2}B}{\partial r^{2}} + r \, B = 0 \\ & B(T,T,r) = 1 \end{align*} and assuming the solution in the form of $$B(t,T,r) = e^{-C(t,T)r-A(r,T)}$$, We get two ordinary differential equations \begin{align*} & - \dfrac{d C(t,T)}{d t} + a C(t,T)+1 =0 \\ & - \dfrac{d A(t,T)}{d t}-b(t)C(t,T)+\dfrac{1}{2} \sigma_{r}^{2}{ \lbrace C(t,T) \rbrace } ^{2} = 0 \end{align*} $$C(t,T)$$ and $$A(t,T)$$ satisfying the above equation are as given in statement.


text:
# How to create a distribution and sample?

Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail.

So for example, a large data set we have n observations, for each n,

there is $$w_{i}$$ number of wires, voltage $$v_{i}$$ and $$f_{i}$$ of the wires fail.

And suppose we are given some of the information for example for the first few, ( note that each sample has increased voltage and we see increased proportion of failed wires). However we are not given the voltages, all we know is that it increases.

$$w_{1}=14$$ and $$f_{1}=4$$

$$w_{2}=13$$ and $$f_{2}=4$$

$$w_{3}=7$$ and $$f_{3}=3$$

$$w_{4}=10$$ and $$f_{4}=5$$

$$w_{5}=12$$ and $$f_{5}=7$$

$$w_{6}=20$$ and $$f_{6}=13$$

etc..

ie we have a parameter space such that ( $$t$$ is the proportion that fail) $$\{t_{i}: t_{1} \lt t_{2} \lt t_{3} \lt .. \lt t_{n} \le 1\}$$

My goal is to model this as a conditional distribution and sample so that I can make some statements about each $$t_{i}$$, such as the mean and deviations of each.(assuming flat prior)

Firstly I know about sampling, but I am wondering how from just the simple data, how I can accurately form the conditional distribution? Using rejection or transformations for example, and then Gibbs to make some conclusions on the individual failure proportions.

My thoughts:

Well it seems that the number of wires that fail is a function of the voltage. As voltage increases, so to does the proportion of failed wires.

Possibly I could use rejection method to sample from the distribution that is creating this?

So I would want to find some function $$g(x)$$ such that $$g(x) \ge f(x)$$ for all $$x$$ , then simulate uniform random variables and check the conditions.

However, as of now I don't have a distribution. I guess I could form a hand drawn using the points and x axis as 1,2,3,4,5,6... and y being the corresponding proportion rate of failure.

I know for a distribution, we need the probabilities to sum/integrate to 1.

The probabilities here I assume would be the probability that a certain proportion fail. So for n wires, we would have the probability that $$p_{1}=\frac{1}{n}$$ proportional fail, a probability for $$p_{2}=\frac{2}{n}$$proportion fail, all the way to the probability that all wires fail.

So it looks more like the form of a CDF, as voltage increases, ie if we write in the form of a function, $$F(v_{1})=\frac{4}{14}$$ , $$F(v_{2})=\frac{4}{13}$$, and so forth, so if we had an unlimited sample, as $$n \to \infty$$ , $$F(v_{n}) \to 1$$

and I suppose then $$F^{-1}=f$$ would be our density, but I am still not sure how to do this in finite case.

Issues: We are not told anything about underlying distribution, parameters or form. Only the data given. So do we take the data that is given to be the initialising values?

I was thinking I could possibly just assume that the failures follow a binomial distribution, with the binomial parameter following some other distribution such as a beta. How does that sound? Would we then need to also put some distribution on the $$w_{i}$$ ?

You seemed to have written a long post that mixes many issues together, in my opinion. My suggestion is to first think about the generative model, i.e. what exactly are the processes that generate your data.

In principle, you could come up with the generative model (i.e. probability distribution) for your whole data $$(f,v,w)$$. I'm guessing that you don't know what process generates $$w$$ and $$v$$. So, therefore, you need to generate a conditional distribution $$P(f|v,w)$$ and specify its parameters.

The meaningful distribution for f|v,w is a binomial distribution.

$$f|v,w \sim B(w,F(v))$$ where F is some kind of function that maps the voltage into [0,1] range. One example of such a function would be some kind of logistic function, i.e. $$F(v|a,b) = \frac{1}{1+\exp{(a+b v)}}$$

If you chose that function you will have the probability distribution $$P(f|v,w, a,b)$$ Assuming that individual $$f_i$$ are independent, you can write down the whole likelihood function that will depend on two parameters. You can then either sample this likelihood using a variety of methods (and after choosing a prior) or just optimize it to find best a,b.

• No knowledge on the generation of v and w. Maybe logistic will work better then beta.. But we are not given any of the values for $v$ only that it is increasing? Nov 29, 2018 at 4:59
• I didn't realize that you don't know voltages. It wasn't clear from the post. In that case, you can just use the index 'i' as 'voltage'. Also may want to change the logistic function to something else that better fits the data... Nov 29, 2018 at 5:55
• Sorry I should clarify. Everything given appears in the question above. Nov 29, 2018 at 6:03
• Yes I could use the index as voltage as that fits the criterion. Can you possibly expand at all on your comments? I will take note of them and work through to the best I can and update. Such as if I were to choose a beta or something diffirent then logistic Nov 29, 2018 at 6:27
• I am not sure how much I can expand more.First I didn't talke about beta at all. My B(n,p) distribution is the binomial distribution.Regarding using a different function then logistic,what I meant is that you somehow need to have a model for the failure fraction as a function of i.If your had voltages, that's easy and what I wrote in the text.If you don't,you need a model for the failure as a function of i.If you can assume that voltages linearly increase with i,then you can still use logistic function.If not you need either a different functional form or assume what process generates voltages Nov 29, 2018 at 16:07


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text:
# Junction to board thermal resistance

The Junction-to-case (bottom) thermal resistance of the TPS7A4701 LDO regulator is 1.7 °C/W. It is explained in the TI's app note "Semiconductor and IC package thermal metrics" that this metric refers to the thermal resistance between the junction and the exposed pad of the package.

On the other hand, the Junction-to-board thermal resistance of the regulator is 11.9 °C/W. I am not sure that I understand this metric. If the junction to the bottom case (exposed pad) is only 1.7 °C/W, and the junction to board is 11.9 °C/W, so there is a 10.2 °C/W thermal resistance between the exposed pad and the board. But the exposed pad is soldered to the board, so it doesn't make sense, the thermal resistance of the solder should be very low.

The reason is they are completely different measurement procedures.

$R_{ΘJB}$ Has nothing to do with a heatsink. It is a JEDEC standard where $R_{ΘJC}$ is not.

Historically the junction-to-case thermal resistance RθJC(top) of semiconductor devices has been determined by direct measurement of the temperature difference between junction and the case surface in contact with a water cooled heatsink as described in MIL Standard 883. The results are often not reproducible because the thermocouple measurement of the case temperature is prone to error.

$R_{ΘJB}$ measurement is done according to JEDEC JESD51-8, Integrated Circuit Thermal Test Method Environmental Conditions — Junction-to-Board

This standard specifies the environmental conditions necessary for determining the junction-to-board thermal resistance, RθJB, and defines this term. The RθJBthermal resistance is a figure of merit for comparing the thermal performance of surface mount packages mounted on a standard board.

The standard board used in JESD51-8 is defined in JEDEC JESD51-7 High Effective Thermal Conductivity Test Board for Leaded Surface Mount Packages

## $R_{ΘJB}$

There is a JEDEC procedure, not used by TI, for measuring $R_{ΘJC}$ for the application purpose of attaching a heatsink to the case, JESD51-14 TRANSIENT DUAL INTERFACE TEST METHOD FOR THE MEASUREMENT OF THE THERMAL RESISTANCE JUNCTION-TO-CASE OF SEMICONDUCTOR DEVICES WITH HEAT FLOW THROUGH A SINGLE PATH.

## $R_{ΘJC(top)}$

$R_{θJB}$ is used when no heat sink is on board.

If there is a heatsink, do not use above metric and instead use $R_{θJA}$ which is in parallel with $R_{θJC}+R_{θCS}+R_{θSA}~~~~$ as the net thermal resistance.

• $R_{θJB}$ Junction-to-board thermal resistance 11.9 °C/W
• $R_{θJA}$ Junction-to-ambient thermal resistance 32.5 °C/W
• $R_{θJC}$ (top) Junction-to-case (top) thermal resistance 27 °C/W
• $R_{θJC}$ (bot) Junction-to-case (bottom) thermal resistance 1.7 °C/W
• $ψ_{JT}$ Junction-to-top characterization parameter 0.3 °C/W
• $ψ_{JB}$ Junction-to-board characterization parameter 11.9 °C/W

Thus add 1.7 to your interfaces $R_{θCS}+R_{θSA}~$ then // with 32.5

For J= junction, C=case, S=sink, A=ambient, B=board and $R_θ$ means thermal resistance, although often just truncated to $R_{JC}$

The main revelation in the report is that the following analysis is old and obsolete, because the board does not dissipate 100% of the heat, with some cooling effect in the case to ambient.

$R_{θJA} = R_{θJC} + R_{θCA}~$ (obsolete)

• Actually, the RθJA that is given in the datasheet is calculated without heat sink. It is calculated when the device is mounted to a PCB that is designed in accordance with the JEDEC specifications. But I still don't understand the Junction-to-board thermal resistance RθJB. – Vadim Mar 20 '17 at 17:38
• @Vadim From how I read it, RθBA is heat resistance through the lid not the bottom. RθJB is a combination of RθBA and resistance through the bottom to a standard pad and board thickness to air. That is... RθBA is the total resistance to ambient through both the top and the bottom / board. See figure 6 of the metrics document. – Trevor_G Mar 20 '17 at 17:42
• Yes top + bot thermal resistance of case without heat sink computes board temp rise using $R_{θJB}$ sometimes called solder point to junction thermal resistance. Note location of thermocouple used for verification. – Sunnyskyguy EE75 Mar 20 '17 at 17:55
• Agreed @TonyStewart.EEsince'75, and the numbers quoted seem to make sense for that. aside.. 75 eh... four years more than me :) – Trevor_G Mar 20 '17 at 18:08


text:
Accelerating the pace of engineering and science

# ranksum

Wilcoxon rank sum test

## Description

example

p = ranksum(x,y) returns the p-value of a two-sided Wilcoxon rank sum test. ranksum tests the null hypothesis that data in x and y are samples from continuous distributions with equal medians, against the alternative that they are not. The test assumes that the two samples are independent. x and y can have different lengths.

This test is equivalent to a Mann-Whitney U-test.

example

[p,h] = ranksum(x,y) also returns a logical value indicating the test decision. The result h = 1 indicates a rejection of the null hypothesis, and h = 0 indicates a failure to reject the null hypothesis at the 5% significance level.

example

[p,h,stats] = ranksum(x,y) also returns the structure stats with information about the test statistic.

example

[___] = ranksum(x,y,Name,Value) returns any of the output arguments in the previous syntaxes, for a rank sum test with additional options specified by one or more Name,Value pair arguments.

## Examples

expand all

### Test for Equal Median of Two Populations

Test the hypothesis of equal medians for two independent unequal-sized samples.

Generate sample data.

```rng('default') % for reproducibility
x = unifrnd(0,1,10,1);
y = unifrnd(0.25,1.25,15,1);
```

These samples come from populations with identical distributions except for a shift of 0.25 in the location.

Test the equality of medians of x and y.

```p = ranksum(x,y)
```
```p =
0.0375
```

The p-value of 0.0375 indicates that ranksum rejects the null hypothesis of equal medians at the default 5% significance level.

### Statistics of the Test for Two Population Medians

Obtain the statistics of the test for the equality of two population medians.

`load mileage`

Test if the mileage per gallon is the same for the first and second type of cars.

`[p,h,stats] = ranksum(mileage(:,1),mileage(:,2))`
```p =
0.0043

h =
1

stats =
ranksum: 21.5000```

Both the p-value, 0.043, and h = 1 indicate the rejection of the null hypothesis of equal medians at the default 5% significance level. Because the sample sizes are small (six each), ranksum calculates the p-value using the exact method. The structure stats includes only the value of the rank sum test statistic.

### Increase in the Median

Test the hypothesis of an increase in the population median.

Navigate to a folder containing sample data.

```cd(matlabroot)
cd('help/toolbox/stats/examples')```

`load weather`

The weather data shows the daily high temperatures taken in the same month in two consecutive years.

Perform a left-sided test to assess the increase in the median at the 1% significance level.

```[p,h,stats] = ranksum(year1,year2,'alpha',0.01,...
'tail','left')```
```p =

0.1271

h =

0

stats =

zval: -1.1403
ranksum: 837.5000```

Both the p-value of 0.1271 and h = 0 indicate that there is not enough evidence to reject the null hypothesis and conclude that there is a positive shift in the median of observed high temperatures in the same month from year 1 to year 2 at the 1% significance level. Notice that ranksum uses the approximate method to calculate the p-value due to the large sample sizes.

Use the exact method to calculate the p-value.

```[p,h,stats] = ranksum(year1,year2,'alpha',0.01,...
'tail','left','method','exact')```
```p =

0.1273

h =

0

stats =

ranksum: 837.5000```

The results of the approximate and exact methods are consistent with each other.

## Input Arguments

expand all

### x — Sample datavector

Sample data, specified as a vector.

Data Types: single | double

### y — Sample datavector

Sample data, specified as a vector. The length of y does not have to be the same as the length of x.

Data Types: single | double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'alpha',0.01,'method','approximate','tail','right' specifies a right-tailed rank sum test with 1% significance level, which returns the approximate p-value.

### 'alpha' — Significance level0.05 (default) | scalar value in the range 0 to 1

Significance level of the decision of a hypothesis test, specified as the comma-separated pair consisting of 'alpha' and a scalar value in the range 0 to 1. The significance level of h is 100 * alpha%.

Example: 'alpha', 0.01

Data Types: double | single

### 'method' — Computation method of the p-value'exact' | 'approximate'

Computation method of the p-value, p, specified as the comma-separated pair consisting of 'method' and one of the following:

 'exact' Exact computation of the p-value, p. 'approximate' Normal approximation while computing the p-value, p.

When 'method' is unspecified, the default is:

• 'exact' if min(nx,ny) < 10 and nx + ny < 20

• 'approximate' otherwise

nx and ny are the sizes of the samples in x and y, respectively.

Example: 'method','exact'

Data Types: char

### 'tail' — Type of test'both' (default) | 'right' | 'left'

Type of test, specified as the comma-separated pair consisting of 'tail' and one of the following:

 'both' Two-sided hypothesis test, where the alternative hypothesis states that x and y have different medians. Default test type if 'tail' is not specified. 'right' Right-tailed hypothesis test, where the alternative hypothesis states that the median of x is greater than the median of y. 'left' Left-tailed hypothesis test, where the alternative hypothesis states that the median of x is less than the median of y.

Example: 'tail','left'

Data Types: char

## Output Arguments

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### p — p-value of the testnonnegative scalar

p-value of the test, returned as a positive scalar from 0 to 1. p is the probability of observing a test statistic as or more extreme than the observed value under the null hypothesis. ranksum computes the two-sided p-value by doubling the most significant one-sided value.

### h — Result of the hypothesis test1 | 0

Result of the hypothesis test, returned as a logical value.

• If h = 1, this indicates rejection of the null hypothesis at the 100 * alpha% significance level.

• If h = 0, this indicates a failure to reject the null hypothesis at the 100 * alpha% significance level.

### stats — Test statisticsstructure

Test statistics, returned as a structure. The test statistics stored in stats are:

• ranksum : Value of the rank sum test statistic

• zval: Value of the z-statistic (computed when 'method' is 'approximate')

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### Wilcoxon Rank Sum Test

The Wilcoxon rank sum test is a nonparametric test for two populations when samples are independent. If X and Y are independent samples with different sample sizes, the test statistic which ranksum returns is the rank sum of the first sample.

The Wilcoxon rank sum test is equivalent to the Mann-Whitney U-test. The Mann-Whitney U-test is a nonparametric test for equality of population medians of two independent samples X and Y.

The Mann-Whitney U-test statistic, U, is the number of times a y precedes an x in an ordered arrangement of the elements in the two independent samples X and Y. It is related to the Wilcoxon rank sum statistic in the following way: If X is a sample of size nX, then

$U=W-\frac{{n}_{X}\left({n}_{X}+1\right)}{2}.$

### z-Statistic

For large samples, ranksum uses a z-statistic to compute the approximate p-value of the test.

If X and Y are two independent samples of size nX and nY, where nX < nY the z-statistic is

$z=\frac{W-E\left(W\right)}{\sqrt{V\left(W\right)}}=\frac{W-\left[\frac{{n}_{X}{n}_{Y}+{n}_{X}\left({n}_{X}+1\right)}{2}\right]-0.5\ast sign\left(W-E\left(W\right)\right)}{\sqrt{\frac{{n}_{X}{n}_{Y}\left({n}_{X}+{n}_{Y}+1\right)-tiescor}{12}}},$

with continuity correction and tie adjustment. Here tiescor is given by

$tiescor=\frac{2\ast tieadj}{\left({n}_{X}+{n}_{Y}\right)\left({n}_{X}+{n}_{Y}-1\right)},$

where ranksum uses [ranks,tieadj] = tiedrank(x,y) to obtain tie adjustments. The standard normal distribution gives the p-value for this z-statistic.

### Algorithms

ranksum treats NaNs in x and y as missing values and ignores them.

For a two-sided test of medians with unequal sample sizes, the test statistic that ranksum returns is the rank sum of the first sample.

## References

[1] Gibbons, J. D., and S. Chakraborti. Nonparametric Statistical Inference, 5th Ed., Boca Raton, FL: Chapman & Hall/CRC Press, Taylor & Francis Group, 2011.

[2] Hollander, M., and D. A. Wolfe. Nonparametric Statistical Methods. Hoboken, NJ: John Wiley & Sons, Inc., 1999.