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# Dynamics ?

1. Feb 10, 2009

### 2slowtogofast

When Vo=68 ; k=.027 ; theta = 39 degrees

(a) The American Football kicker lines up to attempt a field goal . Take the balls initial position to be the origin. For the field goal to be good it must pass between the upstairs, and over the crossbar which are 10 feet above ground level. Assuming the ball is kicked straight ( directly between the upright ), and the ball is given an initial velocity of Vo at an angle theta answer the following questions.

(a)What is the maximum field goal distance for this kicker if you ignore all drag forces.

I solved (a) to be 126.64 in the x direction.

(b) If we include a crude model for the drag force in which the net acceleration on the ball is given by:
a=(-k * vx ) i + (-g -k * Vy) j ( ft/s2 )

Derive the equations describing the horizontal and vertical positions as functions of time.

- I am lost when I get to (b) and also i couldt get the picture to load but its basicly a football x yards away from a goal post 10 ft in the air. im just confused on how to start this off my teacher said there would be a few integrals in this problem and i tried relating

a = dv/dt solving for v then v = ds/dt for each component but it didt work

the vx and vy are velocity in x nd veloctiy in y i just couldt get the subscipts on this computer im on

Last edited: Feb 10, 2009
2. Feb 10, 2009

### 2slowtogofast

anybody have any ideas

3. Feb 11, 2009

format: plain


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• # question_answer A body of mass 2 kg has an initial speed of 5 m/s. A force acts on it for some time in the direction of motion. The force-time graph is shown in figure. The final speed of body is A) 9.25 m/s.                      B) 5 m/s.   C)   14.31 m/s.        D)                4.25 m/s.

Correct Answer: C

Solution :

$F=\frac{dP}{dt}\Rightarrow dP=Fdt$ $\therefore$$\int_{{}}^{{}}{dP=\int_{{}}^{{}}{F\,dt}\Rightarrow \Delta \Pi }=\int_{{}}^{{}}{F\,dt}$ So the change in moment$\Delta P$is equal to the area under the curve $\therefore$$\Delta P=\frac{1}{2}\times 2\times 4+4\times 2+\frac{1}{2}(4+2.5)$$\times \,0.5+2.5\times 2$ $=4+8+\frac{6.5}{4}+5=18.625$ $\Rightarrow$${{P}_{2}}-{{P}_{1}}=18.625\,\Rightarrow \,{{P}_{2}}={{P}_{1}}+18.625$ $\therefore$$m{{v}_{2}}=m{{v}_{1}}+18.625$ $\Rightarrow$${{v}_{2}}={{v}_{1}}+\frac{18.625}{2}=5+\frac{18.625}{2}$ $=14.3125\,m/s.$ Hence, the correction option is (c).

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# Find the Mean (Arithmetic) 2 , 3 , 5

2 , 3 , 5
The mean of a set of numbers is the sum divided by the number of terms.
2+3+53
Simplify the numerator.
5+53


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# Distribution of Data

I have a question about glm model fitting of my data. The distribution shape is likely to follow a poisson distribution, but the response variable is not count/rate, but continuous decimals with both positive an negative values （as shown below).

Could you please give me some suggestions that which one should be more appropriate to my case?

• First fit an OLS model, then examine the residuals. In R you can get nice plots of the model by something like (pseudo code): m1 <- lm(DV~IV) plot(m1) – Peter Flom Feb 12 '14 at 15:34


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# Simulation Scenarios Reference¶

## Distributions¶

The ConstantDistribution, as the name suggests is used when we want to pass a custom parameter to the event, but this custom parameter is fixed for all events.

Distributions between 2 and 13 will generate uniform/normal/exponential distribution random numbers.

The SequentialDistributions are similar to the ConstantDistribution as they do not generate random numbers, but a predefined sequence. The BasicIntSequentialDistribution generates integers from the starting point provided in the constructor. For the GenIntSequentialDistribution, the whole sequence is provided in the constructor. Make sure you do not try to raise more events than the size of this sequence, as once all the numbers in the sequence have been drawn, null values will be returned.

In case these distributions to not fit your needs, you can define your own by extending the Distribution class

## Operations¶

Depending on the amount of customization you need for the simulation events, there are 6 types of operations allowing from 0 to 5 generated parameters. The parameters are generated using a distribution. You can choose among the distributions presented above or create your own parameter distribution that fits your needs better.

## StochasticProcesses¶

Event generation related methods:

Process order related methods start.., terminate


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Chapter 5 Dashboards

R Markdown is customizable and extensible. In Chapters 3 and 4, we have introduced basic document and presentation formats in the rmarkdown package, and explained how to customize them. From this chapter on, we will show several more existing extension packages that bring different styles, layouts, and applications to the R Markdown ecosystem. In this chapter, we introduce dashboards based on the flexdashboard package (Iannone, Allaire, and Borges 2018).

Dashboards are particularly common in business-style reports. They can be used to highlight brief and key summaries of a report. The layout of a dashboard is often grid-based, with components arranged in boxes of various sizes.

With the flexdashboard package, you can

• Use R Markdown to publish a group of related data visualizations as a dashboard.

• Embed a wide variety of components including HTML widgets, R graphics, tabular data, gauges, value boxes, and text annotations.

• Specify row or column-based layouts (components are intelligently re-sized to fill the browser and adapted for display on mobile devices).

• Create story boards for presenting sequences of visualizations and related commentary.

• Optionally use Shiny to drive visualizations dynamically.

To author a dashboard, you create an R Markdown document with the flexdashboard::flex_dashboard output format. You can also create a document from within RStudio using the File -> New File -> R Markdown dialog, and choosing a “Flex Dashboard” template.

If you are not using RStudio, you can create a new flexdashboard R Markdown file from the R console:

rmarkdown::draft(
"dashboard.Rmd", template = "flex_dashboard",
package = "flexdashboard"
)

The full documentation of flexdashboard is at https://rmarkdown.rstudio.com/flexdashboard/. We will only cover some basic features and usage in this chapter. Dashboards have many features in common with HTML documents (Section 3.1), such as figure options (Section 3.1.5), appearance and style (Section 3.1.4), MathJax equations (Section 3.1.8), header and before/after body inclusions (Section 3.1.10.2), and Pandoc arguments (Section 3.1.10.5), and so on. We also recommend that you take a look at the R help page ?flexdashboard::flex_dashboard to learn about more features and options.

References

Iannone, Richard, JJ Allaire, and Barbara Borges. 2018. Flexdashboard: R Markdown Format for Flexible Dashboards. https://CRAN.R-project.org/package=flexdashboard.


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### Home > APCALC > Chapter 12 > Lesson 12.1.1 > Problem12-13

12-13.

Multiple Choice: The interval of convergence of the power series $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } x ^ { n } } { n ^ { 2 } }$ is:

1. $[–3, 3]$

1. $(–3, 3]$

1. $[–1, 1]$

1. $[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } ]$

1. $( - \frac { 1 } { 3 } , \frac { 1 } { 3 } ]$

$\lim\limits_{n\to\infty}\Bigg|\frac{\frac{3^{n+1}x^{n+1}}{(n+1)^2}}{\frac{3^nx^n}{n^2}}\Bigg|$

$=\lim\limits_{n\to\infty}3|x|<1$


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Project deliverable Open Access

# D3.2 Maritime Use Case: Initial Evaluation and Validation Report

Project consortium members

### JSON-LD (schema.org) Export

{
"description": "
This report summarises the results of the preliminary evaluation of the components developed for the Maritime Use
\n\n
Case that will be integrated and tested in a final experimentation at sea toward the end of the Project INFORE.
\n\n
The Maritime Use Case consists in developing and deploying a hybrid sensor network for ship target detection,
\n\n
localization and classification. Sensor data for the global scale and the local scale surveillance scenario, resulting in
\n\n
an extreme scale data stream, will be integrated in a maritime situation awareness platform to perform high level
\n\n
fusion tasks and target behavioural analysis in order to detect behavioural anomalies of vessels with respect to the
\n\n
normal traffic in the area of interest. The developed platform will be integrated/using the INFORE architecture,
\n\n
utilizing its capabilities to analyse extreme scale streamed data.
\n\n
The evaluation is performed using a set of key performance indices (KPIs) defined for the single components of the
\n\n
system and for the system as a whole, including human factors. The data used in the evaluation process are historical
\n\n
data available at the beginning of the Project INFORE or generated by simulating surveillance scenarios.
\n\n
The tested components include i) the CMRE coordinated sea surface robotic sensor network for target detection and
\n\n
localization by acoustic data, the target detection and classification systems for satellite data and for thermal and RGB
\n\n
video camera data, and the MarineTraffic situation awareness platform algorithms for complex event detection.
\n\n
The results of the evaluation are in general in line with the expectations. The coordinated robotic sensor network
\n\n
outperforms the base network without coordination, the classification from satellite data and camera data performs
\n\n
with classification accuracy around or greater than 90% and around 80%, respectively. The complex event classifier
\n\n
system has analogous performance with accuracy achieving, in some cases, values greater than 90%.
\n\n
The report provides a complete description of each component, including theoretical details, the evaluation of the
\n\n
components and finally a preliminary plan of the final experiment at sea to evaluate the Maritime Use Case as a whole.
\n\n
Future work before the final experiment includes implementation of the control and cooperation software on board
\n\n
the surface robot vehicles and further test of the classification algorithms for satellite and camera data and for complex
\n\n
events. Furthermore, the detailed plan, including logistic aspects, of the final experiment at sea will be started soon
\n\n
this year and finalized next year.
",
"creator": [
{
"@type": "Person",
"name": "Project consortium members"
}
],
"url": "https://zenodo.org/record/4034088",
"datePublished": "2020-06-29",
"@context": "https://schema.org/",
"identifier": "https://doi.org/10.5281/zenodo.4034088",
"@id": "https://doi.org/10.5281/zenodo.4034088",
"@type": "CreativeWork",
"name": "D3.2 Maritime Use Case: Initial Evaluation and Validation Report"
}
31
37
views


text:
# Writing Samples

Rationale for Screening
Colorectal Cancer (CRC) is a good candidate for systematic screening in the general population. It is the second most prevalent cancer in the United States, with about 150,000 new cases and nearly 50,000 deaths per year,1 and its incidence has been rising, both overall and within specific population groups (e.g. blacks).2 Furthermore, early CRC usually has an indolent course and several screening tests—physical, laboratory, endoscopic, and radiologic—can detect the disease before it becomes incurable. Endoscopic techniques such as polypectomy and mucosectomy can even remove premalignant lesions and early cancers on an outpatient basis, further reducing morbidity and mortality.4 A fundamental aspect of CRC screening is risk stratification. This entails the assessment of each patient’s family history, including the diagnosis of hereditary cancer syndromes that predispose to CRC, such as the polyposis syndromes and hereditary non-polyposis colon cancer (HNPCC). Although rare, these pathologies account for a significant proportion (5-10%)2 of colorectal cancer cases, and their detection affords the possibility of diagnosing several early, potentially curable neoplasms in a single family.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.


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# fastest method to solve a quadratic equation modulo p^N

What is the fastest method to solve a quadratic equation

$ax^2 + bx + c = 0 (mod p^N)$

where p is prime?

## 1 Answer

I'd start by multiplying by the inverse of $a$, to make your lead coefficient $1$: $x^2+Bx+C\equiv 0$. Then you can complete the square, replacing your coefficient $B$ with $B-p^N$ if that helps you see how to understand $\frac{B}{2}$.

This will tell you what number has to be a quadratic residue, and you can work on finding its square root. For that, you probably want to find the square root mod $p$ first, and then use Hensel's lifting lemma until you get to $p^N$.

Does that answer your question, or do you need more details on any of those steps?

• Please tell me what is the fastest method of finding square root mod p. I will also appreciate if you guide me to an novice's explanation for Hensel's lifting lemma. – Talib Dec 25 '17 at 23:20
• First, you need to know whether your number even is a square - Legendre symbols are good for that. If it is, you can use a primitive root to find a square root for it. Whether this is really quicker than trial-and-error will vary from prime to prime. For the Hensel stuff, let me see what I can either find, or write up quickly... – G Tony Jacobs Dec 25 '17 at 23:39
• This is a pretty good resource on Hensel: brilliant.org/wiki/hensels-lemma – G Tony Jacobs Dec 25 '17 at 23:43
• Of course, it's also possible that $a$ is divisible by some smaller power of $p$, in which case you first solve a linear equation mod that power of $p$... – Robert Israel Dec 26 '17 at 1:21
• @RobertIsrael, because when someone isn't particularly used to this, they might find completing the square easier that way. Quick, what's 15/2 mod 19? Did you add or subtract 19 in your head to come up with the answer? – G Tony Jacobs Dec 26 '17 at 1:27


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# glGetActiveAtomicCounterBufferiv

## Name

glGetActiveAtomicCounterBufferiv — retrieve information about the set of active atomic counter buffers for a program

## C Specification

 void glGetActiveAtomicCounterBufferiv( Gluint program, GLuint bufferIndex, GLenum pname, GLint *params);

## Parameters

program

The name of a program object from which to retrieve information.

bufferIndex

Specifies index of an active atomic counter buffer.

pname

Specifies which parameter of the atomic counter buffer to retrieve.

params

Specifies the address of a variable into which to write the retrieved information.

## Description

glGetActiveAtomicCounterBufferiv retrieves information about the set of active atomic counter buffers for a program object. program is the name of a program object for which the command glLinkProgram has been issued in the past. It is not necessary for program to have been linked successfully. The link may have failed because the number of active atomic counters exceeded the limits.

bufferIndex specifies the index of an active atomic counter buffer and must be in the range zero to the value of GL_ACTIVE_ATOMIC_COUNTER_BUFFERS minus one. The value of GL_ACTIVE_ATOMIC_COUNTER_BUFFERS for program indicates the number of active atomic counter buffer and can be queried with glGetProgram.

If no error occurs, the parameter(s) specified by pname are returned in params. If an error is generated, the contents of params are not modified.

If pname is GL_ATOMIC_COUNTER_BUFFER_BINDING, then the index of the counter buffer binding point associated with the active atomic counter buffer bufferIndex for program is returned.

If pname is GL_ATOMIC_COUNTER_BUFFER_DATA_SIZE, then the implementation-dependent minimum total buffer object size, in baseic machine units, required to hold all active atomic counters in the atomic counter binding point identified by bufferIndex is returned.

If pname is GL_ATOMIC_COUNTER_BUFFER_ACTIVE_ATOMIC_COUNTERS, then the number of active atomic counters for the atomic counter buffer identified by bufferIndex is returned.

If pname is GL_ATOMIC_COUNTER_BUFFER_ACTIVE_ATOMIC_COUNTER_INDICES, then a list of the active atomic counter indices for the atomic counter buffer identified by bufferIndex is returned. The number of elements that will be written into params is the value of GL_ATOMIC_COUNTER_BUFFER_ACTIVE_ATOMIC_COUNTERS for bufferIndex.

If pname is GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_VERTEX_SHADER, GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_TESS_CONTROL_SHADER, GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_TESS_EVALUATION_SHADER, GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_GEOMETRY_SHADER, GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_FRAGMENT_SHADER, GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_COMPUTE_SHADER then a boolean value indicating whether the atomic counter buffer identified by bufferIndex is referenced by the vertex, tessellation control, tessellation evaluation, geometry, fragment or compute processing stages of program, respectively, is returned.

## Notes

glGetActiveAtomicCounterBufferiv is available only if the GL version is 4.2 or higher.

GL_ATOMIC_COUNTER_BUFFER_REFERENCED_BY_COMPUTE_SHADER is available only of the GL version is 4.3 or higher.

## Errors

GL_INVALID_VALUE is generated if program is not the name of a program object for which glLinkProgram has been called in the past.

GL_INVALID_VALUE is generated if bufferIndex is greater than or equal to the value of GL_ACTIVE_ATOMIC_COUNTER_BUFFERS for program.

GL_INVALID_ENUM is generated if pname is not one of the accepted tokens.

## Version Support

OpenGL Version
Function / Feature Name 2.0 2.1 3.0 3.1 3.2 3.3 4.0 4.1 4.2 4.3 4.4 4.5
glGetActiveAtomicCounterBufferiv - - - - - - - -


text:
[ CogSci Summaries home | UP | email ]

W. Woods, What's in a link: Foundations for Semantic Networks. Representation and Understanding, D. Bobrow and A. Collins (eds.), Academic Press, 1975.

Author of the summary: J. William Murdock, 1997, murdock@cc.gatech.edu

Cite this paper for:

• Representing the meaning of language requires that we represent not just what is true but also what is meant.

Keywords: Semantic Network, Natural Language, Representation

Systems: None

Summary: Discusses perspectives on the nature of semantics.  Takes the
view that semantics is the {\em relationship} between form and meaning
rather than either the form or the meaning itself.  Rejects the notion
of a canonical form for meanings.  Focuses heavily on the need for
"intensional" representation for representing entities and
distinctions between entities that are hypothetical, believed,
implied, etc.  Talks about some particular grammatical structures in
terms of how they should be represented.

Summary author's notes:

• This summary came from a file which had the following disclaimer: "The following summaries are the completely unedited and often hastily composed interpretations of a single individual without any sort of systematic or considered review. As such it is very likely that at least some of the following text is incomplete, inadequate, misleading, or simply wrong. One might view this as a very preliminary draft of a survey paper that will probably never be completed. The author disclaims all responsibility for the accuracy or use of this document; this is not an official publication of the Georgia Institute of Technology or the College of Computing thereof, and the opinions expressed here may not even fully match the fully considered opinions of the author much less the general opinions of the aformentioned organizations."

Back to the Cognitive Science Summaries homepage
Cognitive Science Summaries Webmaster:
JimDavies ( jim@jimdavies.org )


text:
# American Institute of Mathematical Sciences

April  2016, 12(2): 471-486. doi: 10.3934/jimo.2016.12.471

## $p$th Moment absolute exponential stability of stochastic control system with Markovian switching

 1 Department of Mathematics, College of Science, China University of Petroleum, Beijing 102249, China, China, China, China

Received  September 2014 Revised  February 2015 Published  June 2015

In this paper we discuss the $p$th moment absolute exponential stability of stochastic control system with Markovian switching. We first give a new concept of $p$th moment absolute exponential stability, then we establish some theorems under different hypotheses to guarantee the system $p$th moment absolutely exponentially stable. These sufficient conditions in our theorems are algebraic criteria in terms of matrix inequalities, and we introduce an $M$-method with MATLAB to compute them. Finally, some examples are given to illustrate our results.
Citation: Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471
##### References:
 [1] G. K. Basak, A. Bisi and M. K. Ghosh, Stability of a random diffusion with linear drift, Journal of Mathematical Analysis and Applications, 202 (1996), 604-622. doi: 10.1006/jmaa.1996.0336. [2] V. A. Brusin and V. A. Ugrinovskii, Stochastic stability of a class of nonlinear differential equations of Ito type, Siberian Mathematical Journal, 28 (1987), 381-393. [3] C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609. doi: 10.3934/jimo.2012.8.591. [4] R. E. Kalman, Lyapunov functions for the problem of Lur'e in automatic control, Proceedings of the National Academy of Sciences of the United States of America, 49 (1963), 201. doi: 10.1073/pnas.49.2.201. [5] D. G. Korenevskii, Algebraic criteria for absolute (relative to nonlinearity) stability of stochastic automatic control systems with nonlinear feedback, Ukrainian Mathematical Journal, 40 (1988), 616-621. doi: 10.1007/BF01057179. [6] H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering, Academic Press, New York, 1967. [7] X. Liao, L. Q. Wang and P. Yu, Stability of Dynamical Systems, Elsevier, 2007. doi: 10.1016/S1574-6917(07)05001-5. [8] X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, 2nd edition, Springer, 2008. doi: 10.1007/978-1-4020-8482-9. [9] D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8. [10] M. R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control, 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043. [11] A. I. Lurie and V. N. Postnikov, On the theory of stability of control systems, Applied Mathematics and Mechanics, 8 (1944), 246-248. [12] A. K. Mahalanabis and S. Purkayastha, Frequency-domain criteria for stability of a class of nonlinear stochastic systems, Automatic Control, IEEE Transactions on, 18 (1973), 266-270. [13] L. Li, Y. Gao and H. Wang, Second order sufficient optimality conditions for hybrid control problems with state jump, Journal of Industrial and Management Optimization, 11 (2015), 329-343. doi: 10.3934/jimo.2015.11.329. [14] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Processes and Their Applications, 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. [15] X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, WSEAS Trans. Circuits, 1 (2002), 68-73. [16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [17] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. doi: 10.1533/9780857099402. [18] P. V. Pakshin and V. A. Ugrinovskii, Stochastic problems of absolute stability, Automation and Remote Control, 67 (2006), 1811-1846. doi: 10.1134/S0005117906110051. [19] V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1962), 857-875. [20] Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Springer, 2011. doi: 10.1007/978-0-85729-256-8. [21] A. J. Van Der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer, 2000. doi: 10.1007/BFb0109998. [22] H. Xie, Theory and Application of Absolute Stability, Science Press, Beijing, 1986. [23] H. Xu and K. L. Teo, Exponential stability with-gain condition of nonlinear impulsive switched systems, Automatic Control, IEEE Transactions on, 55 (2010), 2429-2433. doi: 10.1109/TAC.2010.2060173. [24] H. Xu, K. L. Teo and W. Gui, Necessary and sufficient conditions for stability of impulsive switched linear systems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1185-1195. doi: 10.3934/dcdsb.2011.16.1185. [25] X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysis, 2014 (2014), Hindawi Publishing Corporation. doi: 10.1155/2014/126836. [26] V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl, 3 (1962), 620-623. [27] Y. Zhang, M. Wang, H. Xu and K. L. Teo, Global stabilization of switched control systems with time delay, Nonlinear Analysis: Hybrid Systems, 14 (2014), 86-98. doi: 10.1016/j.nahs.2014.05.004. [28] Y. Zhang, Y. Zhao, H. Xu, H. Shi and K. L. Teo, On boundedness and attractiveness of nonlinear switched delay systems, In Abstract and Applied Analysis, 2013 (2013), Hindawi Publishing Corporation.

show all references

##### References:
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text:
OpenStack Swift eventual consistency analysis & bottlenecks

Swift is the software behind the OpenStack Object Storage service.

This service provides a simple storage service for applications using RESTful interfaces, providing maximum data availability and storage capacity. I explain here how some parts of the storage and replication in Swift works, and show some of its current limitations. If you don't know Swift and want to read a more "shallow" overview first, you can read John Dickinson's Swift Tech Overview.

How Swift storage works

If we refer to the CAP theorem, Swift chose availability and partition tolerance and dropped consistency. That means that you'll always get your data, they will be dispersed on many places, but you could get an old version of them (or no data at all) in some odd cases (like some server overload or failure). This compromise is made to allow maximum availability and scalability of the storage platform.

But there are mechanisms built into Swift to minimize the potential data inconsistency window: they are responsible for data replication and consistency.

The official Swift documentation explains the internal storage in a certain way, but I'm going to write my own explanation here about this.

Consistent hashing

Swift uses the principle of consistent hashing. It builds what it calls a ring. A ring represents the space of all possible computed hash values divided in equivalent parts. Each part of this space is called a partition.

The following schema (stolen from the Riak project) shows the principle nicely:

In a simple world, if you wanted to store some objects and distribute them on 4 nodes, you would split your hash space in 4. You would have 4 partitions, and computing hash(object) modulo 4 would tell you where to store your object: on node 0, 1, 2 or 3.

But since you want to be able to extend your storage cluster to more nodes without breaking the whole hash mapping and moving everything around, you need to build a lot more partitions. Let's say we're going to build 210 partitions. Since we have 4 nodes, each node will have 210 ÷ 4 = 256 partitions. If we ever want to add a 5th node, it's easy: we just have to re-balance the partitions and move 1⁄4 of the partitions from each node to this 5th node. That means all our nodes will end up with 210 ÷ 5 ≈ 204 partitions. We can also define a weight for each node, in order for some nodes to get more partitions than others.

With 210 partitions, we can have up to 210 nodes in our cluster. Yeepee.

For reference, Gregory Holt, one of the Swift authors, also wrote an explanation post about the ring.

Concretely, when building one Swift ring, you'll have to say how much partitions you want, and this is what this value is really about.

Data duplication

Now, to assure availability and partitioning (as seen in the CAP theorem) we also want to store replicas of our objects. By default, Swift stores 3 copies of every objects, but that's configurable.

In that case, we need to store each partition defined above not only on 1 node, but on 2 others. So Swift adds another concept: zones. A zone is an isolated space that does not depends on other zone, so in case of an outage on a zone, the other zones are still available. Concretely, a zone is likely to be a disk, a server, or a whole cabinet, depending on the size of your cluster. It's up to you to chose anyway.

Consequently, each partitions has not to be mapped to 1 host only anymore, but to N hosts. Each node will therefore store this number of partitions:

number of partition stored on one node = number of replicas × total number of partitions ÷ number of node

Examples:

We split the ring in 210 = 1024 partitions. We have 3 nodes. We want 3 replicas of data.
→ Each node will store a copy of the full partition space: 3 × 210 ÷ 3 = 210 = 1024 partitions.
We split the ring in 211 = 2048 partitions. We have 5 nodes. We want 3 replicas of data.
→ Each node will store 211 × 3 ÷ 5 ≈ 1129 partitions.
We split the ring in 211 = 2048 partitions. We have 6 nodes. We want 3 replicas of data.
→ Each node will store 211 × 3 ÷ 6 = 1024 partitions.

Three rings to rule them all

In Swift, there is 3 categories of thing to store: accountcontainer and objects.

An account is what you'd expect it to be, a user account. An account contains containers (the equivalent of Amazon S3's buckets). Each container can contains user-defined key and values (just like a hash table or a dictionary): values are what Swift call objects.

Swift wants you to build 3 different and independent rings to store its 3 kind of things (accountscontainersand objects).

Internally, the two first categories are stored as SQLite databases, whereas the last one is stored using regular files.

Note that this 3 rings can be stored and managed on 3 completely different set of servers.

Data replication

Now that we have our storage theory in place (accounts, containers and objects distributed into partitions, themselves stored into multiple zones), let's go the replication practice.

When you put something in one of the 3 rings (being an account, a container or an object) it is uploaded into all the zones responsible for the ring partition the object belongs to. This upload into the different zones is the responsibility of the swift-proxy daemon.

But if one of the zone is failing, you can't upload all your copies in all zones at the upload time. So you need a mechanism to be sure the failing zone will catch up to a correct state at some point.

That's the role of the swift-{container,account,object}-replicator processes. This processes are running on each node part of a zone and replicates their contents to nodes of the other zones.

When they run, they walk through all the contents from all the partitions on the whole file system and for each partition, issue a special REPLICATE HTTP request to all the other zones responsible for that same partition. The other zone responds with information about the local state of the partition. That allows the replicator process to decide if the remote zone has an up-to-date version of the partition.

In case of account and containers, it doesn't check at the partition level, but check each account/container contained inside each partition.

If something is not up-to-date, it will be pushed using rsync by the replicator process. This is why you'll read that the replication updates are "push based" in Swift documentation.

# Pseudo code describing replication process for accounts
# The principle is exactly the same for containers
for account in accounts:
# Determine the partition used to store this account
partition = hash(account) % number_of_partitions
# The number of zone is the number of replicas configured
for zone in partition.get_zones_storing_this_partition():
# Send a HTTP REPLICATE command to the remote swift-account-server process
version_of_account = zone.send_HTTP_REPLICATE_for(account):
if version_of_account < account.version()
account.sync_to(zone)

This replication process is O(number of account × number of replicas). The more your number of account will increase and the more you will want replicas for your data, the more the replication time for your accounts will grow. The same rule applies for containers.

# Pseudo code describing replication process for objects
for partition in partitions_storing_objects:
# The number of zone is the number of replicas configured
for zone in partition.get_zones_storing_this_partition():
# Send a HTTP REPLICATE command to the remote swift-object-server process
verion_of_partition = zone.send_HTTP_REPLICATE_for(partition):
if version_of_partition < partition.version()
# Use rsync to synchronize the whole partition
# and all its objects
partition.rsync_to(zone)

This replication process is O(number of objects partitions × number of replicas). The more your number of objects partitions will increase, and the more you will want replicas for your data, the more the replication time for your objects will grow.

I think this is something important to know when deciding how to build your Swift architecture. Choose the right number the number of replicas, partitions and nodes.

Replication process bottlenecks

File accesses

The problem, as you might have guessed, is that to replicate, it walks through every damn things, things being accounts, containers, or object's partition hash files. This means it need to open and read (part of) a every file your node stores to check that data need or not to be replicated!

For accounts & containers replication, this is done every 30 seconds by default, but it will likely take more than 30 seconds as soon as you hit around 12 000 containers on a node (see measurements below). Therefore you'll end up checking consistency of accounts & containers on each all node all the time, using obviously a lot of CPU time.

For reference, Alex Yang also did an analysis of that same problem.

TCP connections

Worst, the HTTP connections used to send the REPLICATE commands are not pooled: a new TCP connection is established each time something has to be checked against the same thing stored on a remote zone.

This is why you'll see in the Swift's Deployment Guide this lines listed under "general system tuning":

# disable TIME_WAIT.. wait..
net.ipv4.tcp_tw_recycle=1
net.ipv4.tcp_tw_reuse=1

# double amount of allowed conntrack
net.ipv4.netfilter.ip_conntrack_max = 262144

In my humble opinion, this is more an ugly hack than a tuning. If you don't activate this and if you have a lot of containers on your node, you'll end up soon with thousands of connections in TIME_WAIT state, and you indeed risk to overload the IP conntrack module.

Container deletion

We also should talk about container deletion. When a user deletes a container from its account, the container is marked as deleted. And that's it. It's not deleted. Therefore the SQLite database file representing the container will continue to be checked for synchronization, over and over.

The only way to have a container permanently deleted is to mark an account as deleted. This way the swift-account-reaper will delete all its containers and, finally, the account.

Measurement

On a pretty big server, I measured the replications to be done at a speed of around 350 {account,container,object-partitions}/second, which can be a real problem if you chose to build a lots of partition and you have a low number_of_node ⁄ number_of_replicas ratio.

For example, the default parameters runs the container replication every 30 seconds. To check replication status of 12 000 containers stored on one node at the speed of 350 containers/seconds, you'll need around 34 seconds to do so. In the end, you'll never stop checking replication of your containers, and the more you'll have containers, the more your inconsistency window will increase.

Conclusion

Until some of the code is fixed (the HTTP connection pooling probably being the "easiest" one), I warmly recommend to chose correctly the different Swift parameters for your setup. The replication process optimization consists in having the minimum amount of partitions per node, which can be done by:

• decreasing the number of partitions
• decreasing the number of replicas
• increasing the number of node

For very large setups, some code to speed up accounts and containers synchronization, and remove deleted containers will be required, but this does not exist yet, as far as I know.

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OpenStack Swift

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OpenStack Swift 云存储技术详解

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Openstack Swift 原理、架构与API介绍

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搭建openstack swift环境的学习心得

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从零开始安装openstack swift

2015-08-26 17:55:24


text:
Kattis

# Prosjek

You are given an array of $N$ integers. Find a consecutive subsequence of numbers of the length at least $K$ that has the maximal possible average.

## Input

The first line of input contains two integers $N$ ($1 \leq N \leq 3 \cdot 10^5$) and $K$ ($1 \leq K \leq N$). The second line of input contains $N$ integers $a_ i$ ($1 \leq a_ i \leq 10^6$).

## Output

The first and only line of output must contain the maximal possible average. An absolute deviation of $\pm 0.001$ from the official solution is permitted.

Sample Input 1 Sample Output 1
4 1
1 2 3 4

4.000000

Sample Input 2 Sample Output 2
4 2
2 4 3 4

3.666666

Sample Input 3 Sample Output 3
6 3
7 1 2 1 3 6

3.333333


text:
# A rectangle is cut in half to create two squares that each has an area of 25. What is the perimeter of the original rectangle?

30

Explanation

The formula to find the area of a square is: Side × Side. If each square has an area of 25, then each side of the square must be 5 (because $$5 \times 5=25$$). So the the dimensions of the rectangle are $$5 \times 10$$. The perimeter is the sum of all the sides, and is therefore 5 + 10 + 5 + 10 = 30.


text:
Inverse tangent calculator.Enter the tangent value, select degrees (°) or radians (rad) and press the = button. Leibniz defined it as the line through a pair of infinitely close points on the curve. Tangent Line Calculator The calculator will find the tangent line to the explicit, polar, parametric and implicit curve at the given point, with steps shown. Once you have the slope of the tangent line, which will be a function of x, you can find the exact slope at specific points along the graph. Tangent, written as tan(θ), is one of the six fundamental trigonometric functions.. Tangent definitions. A tangent never crosses a circle, means it cannot pass through the circle. Advanced Math Solutions – Integral Calculator, the basics. Tangent also known as tan relates the angles and sides of a triangle. In the figure above with tangent line and secant line, (1) (Jurgensen et al. A vertical tangent touches the curve at a point where the gradient (slope) of the curve is infinite and undefined. 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This graph approximates the tangent and normal equations at any point for any function. Tangent. Tangents to Polar Curves. There also is a general formula to calculate the tangent line. However, as above, when $\theta=\pi$, the numerator is also $0$, so we cannot conclude that the tangent line is vertical. This calculator will find the equation of a line (in the slope-intercept, point-slope and general forms) given two points or the slope and one point, with steps shown. General Steps to find the vertical tangent in calculus and the gradient of a curve: Their Tangent Line calculator is bare bones and straightforward, making it easy to use for those who are familiar with Tangent Line calculators. image/svg+xml. In order to find the tangent line we need either a second point or the slope of the tangent line. Second calculator finds the line equation in parametric form, that is, . Show Instructions. Line Calculator. This video explains how to determine the equation of a tangent line to a curve defined by a vector valued function.http://mathispower4u.wordpress.com/ The radius of the circle OP is perpendicular to the tangent line RS. This is all that we know about the tangent line. x. y. The ‘h’ calculation does that. Simply write your equation below (set equal to f(x)) and set p to the value you want to find the slope for. Substitute x value from 0 to given x*2 in the tangent line equation. Free online tangent calculator. A tangent of a curve is a line that touches the curve at one point.It has the same slope as the curve at that point. Finding the Tangent Line. Tangent lines to one circle. ; Check the box Normal line to plot the normal line to the graph of at the point , and to show its equation. Graph. The tangent line equation calculator is used to calculate the equation of tangent line to a curve at a given abscissa point with stages calculation. Circle Tangent Line. Now we reach the problem. First Point. Tangent and Normal Line Calculator. Outer 2nd tangent line equation: Inner 1st tangent line equation: Inner 2nd tangent line equation: Tangent lines between two circles Equation of the two circles given by: (x − a) 2 + (y − b) 2 = r 0 2 (x − c) 2 + (y − d) 2 = r 1 2. y = sin(3x) sin2 (3x) given the point (0,0) How to Find the Vertical Tangent. On a graph, it runs parallel to the y-axis. 1963, p. 346). A tangent line t to a circle C intersects the circle at a single point T.For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. The length of two tangents from a … \[ … If the angle is unknown, but the lengths of the opposite and adjacent side in a right-angled triangle are known, then the tangent can be … But you can’t calculate that slope with the algebra slope formula because no matter what other point on the parabola you use with (7, 0) to plug into the formula, you’ll get a slope that’s steeper or less steep than the precise slope of 3 at (7, 9). Parent topic: Straight Lines. The tangent line equation calculator is used to calculate the equation of tangent line to a curve at a given abscissa point with stages calculation. The slope of the line tangent in the point P 1 will be the arithmetic mean of the slopes of the two secant lines.This method of calculation is possible because we have chosen the x 0 and x 2 points at equal distance from x 1. This applet illustrates the computation of the normal line and the tangent plane to a surface at a point .. 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## Probabilistic Causation in Hungary

20 12 2007

Budapest is a very nice city, and this sounds like an interesting program – I’m just not yet sure whether I can plan anything for that time period, or else I would certainly apply.

Course Dates: JULY 21 – AUGUST 1, 2008
Location: Central European University (CEU), Budapest, Hungary,
Detailed course description: http://www.sun.ceu.hu/causality

Faculty: Miklos Redei, Department of Philosophy, Logic and Scientific Method, London School of Economics, UK; Nancy Cartwright, London School of Economics and Political Science, UK; Damian Fennell, London School of Economics and Political Science, UK; Gabor Hofer-Szabo, King Sigismund College, Budapest, Hungary; Ferenc Huoranszki, Central European University, Budapest, Hungary
Laszlo E. Szabo, Eötvös University, Budapest, Hungary; Richard E. Neapolitan, Northeastern Illinois University

Target group: advanced graduate students, postdoctoral fellows, junior faculty and researchers in philosophy, physics, economics and computer science
Language of instruction: English
Tuition fee: EUR 500, financial aid is available.
The application deadline: February 14, 2008 (for scholarship places), April 30, 2008 (for fee-paying applications)
Online application: http://www.sun.ceu.hu/apply (attachments to be sent by email to causality@ceu.hu).

For further information queries can be directed to the SUN office by email (summeru@ceu.hu), via skype (ceu-sun) or telephone (00-36-1-327-3811).

## APA Interview Scheduling, 2007

13 12 2007

I realized from looking at the Philosophy Job Market Wiki that I had been waiting for interviews to be announced and worrying about them not being announced substantially earlier than I should have been. So I figured it would be useful to compile statistics on the time-line of when interviews were scheduled. I’ll try to keep updating this as the season goes on. I was considering compiling something like this after the APA was done, but some comments in this thread on the Philosophy Job Market Blog suggested that there would be demand for it now. I think it’ll be more useful for people in future years than for this year (since by definition, the data isn’t available until after the interviews are announced).

Also, for those paying attention in future years note that the distribution may well vary from year to year, both because weekends will fall at different points in the semester and in relation to the date of the APA, and also because there will be a different number and selection of departments hiring, and disciplinary trends will be changing. (For instance, in future years, more or fewer departments may decide to skip the APA interviews, or it might become more standard to e-mail candidates rather than call them or whatever.) Also, in each case, I’m just counting lines on the Philosophy Job Market Wiki – I know that sometimes a single line represents two or more potential hires, or that a single department may have multiple lines on the wiki that don’t all end up representing different hires, but this is the easiest way to count. “Leiter Ranked” means Leiter 2006-2008 top 54 in the US, 15 in UK, 4 in Canada, and 4 in Australasia – I know this unfairly excludes universities in Ireland and continental Europe, and is a fairly arbitrary cutoff in each region that is included, but at least it’s publicly available.

TOTAL TENURE-TRACK JOBS – 278
TOTAL LEITER RANKED – 49
TOTAL NON-US – 30

Number of jobs that first scheduled an interview (phone, APA, or otherwise) or campus visit on a given date (rows are grouped by week):

 DATE: by 12/2 12/3 12/4 12/5 12/6 12/7 TOTAL TT 20 1 7 10 7 12 LEITER RANKED 5 0 1 0 0 1 NON-US 4 0 0 0 0 0 DATE: 12/8-9 12/10 12/11 12/12 12/13 12/14 TOTAL TT 4 20 13 25 25 23 LEITER RANKED 0 7 2 6 7 5 NON-US 1 3 0 1 0 0 DATE: 12/15-16 12/17 12/18 12/19 12/20 12/21 12/22-23 TOTAL TT 5 16 13 12 8 2 1 LEITER RANKED 1 2 4 3 1 0 0 NON-US 0 0 2 0 1 0 0 Not entered in Wiki as of 12/24 TOTAL TT 75 LEITER RANKED 14 NON-US 22

If there are other categories that would be easy to track that people would like tracked, mention them in the comments. I might want to track liberal arts colleges, but I won’t necessarily recognize which names to track. I may well have missed a couple international jobs too, because the name wasn’t obviously Canadian.

## Probability and Bayesian Epistemology

10 12 2007

From the last Carnival of Philosophy, I found a post by another Kenny about the relation between Bayesian epistemology and probability! He puts forward three views of what this relation might be:

Here are brief definitions of each view, and how each one relates subjective degrees of rational confidence to probabilities (I will explain in more depth later).

* (P) takes subjective degrees of rational confidence as primitive. There is no state space for degrees of rational confidence, because they aren’t probabilities.
* (KPW) takes subjective degrees of rational confidence to be actual probabilities over the state space of all epistemically possible worlds, where the epistemically possible worlds are formal constructions that may or may not be objectively possible.
* (LPW) takes subjective degrees of rational confidence to be actual probabilities over the state space of the subset of the really possible worlds which are epistemically accessible.

However, he seems to be focused on a very particular understanding of the word “probability” that might not quite be what I would mean by it. The very fact that he talks about a relation between rational degrees of confidence and probabilities suggests that he’s understanding the word differently from how I am.

My understanding of the word is that “probability” refers to any function from a Boolean algebra to the real numbers satisfying the following three properties: (1) it is never negative; (2) the tautology is assigned value 1; (3) finite additivity (that is, given two elements whose conjunction is the contradiction, the probability of their disjunction is the sum of their probabilities). I’d also be willing to apply the term “probability” in cases where instead of a Boolean algebra in the strict mathematical sense, one uses any structure where the terms “tautology”, “conjunction”, “contradiction”, and “disjunction” have a natural interpretation.

It seems that Kenny Pearce, by contrast, understands the term to require that the algebra be an algebra of sets over some state space, and that there be some objective fact about the probability values. If this interpretation is right, then I don’t think I’d quite take any of the positions he mentions. At any rate, I think I support something more like (KPW) than the others, where “actual probabilities” isn’t taken in any objective sense. In explaining this position, I think I can give answers to three questions he raises:

1. Why should we suppose that we can use the math of probability theory in dealing with degrees of rational confidence?
2. The math of probability theory is generally interpreted in terms of sets called state spaces, but, ex hypothesi, degrees of rational confidence, not being probabilities, have no state spaces. What, then, does the math mean?
3. Why should we suppose that when an occurrence has a well defined objective probability, our subjective degree of rational confidence should be assigned a value equal to its probability?

In response to the first question, the standard answer would be to refer to something like a Dutch book argument – degrees of rational confidence can be described by the mathematics of probability theory because if degrees of confidence couldn’t, then the agent would be subject to a certain loss from a set of bets they would be willing to take, and therefore would be irrational. (There’s some slipperiness here with generating the bets from the confidences, and concluding irrationality based on a collection of bets the agent may take individually, but I think this can be cleaned up.) There’s also a host of other arguments for something like this same conclusion (though Alan Hájek raises issues for them in his (forthcoming?) “Arguments For – Or Against? – Probabilism”). As Kenny Pearce notes, nothing about these arguments requires there to be a state space, so they don’t end up being probabilities in his sense (due to Kolmogorov), but they do seem to be probabilities in the sense I use (and Popper, and Borel, and others).

As for the second question, I think that there actually is a state space that is relevant for degrees of rational confidence, which is why I lean more towards something like what Kenny Pearce calls (KPW) rather than (P). The state space here would be the set of epistemic possibilities (whatever those are – I don’t really have a good theory of them, do you?). Despite my lack of an account of them, I think they do need to play a role. I think we can’t make very good sense of the notion of a degree of confidence in p, supposing q, without a set of possibilities that we can restrict to the q-possibilities. Also, these epistemic possibilities seem to play an important role in other aspects of epistemic modality, not just degree of belief. And most importantly, I think there’s a rational difference between having an rational confidence of 0 in p and actually being certain that p will not happen. When measuring the speed of light, there’s a difference between my attitude towards it being exactly 2.9980000000000001 x 108 m/s, and my attitude towards it between 3 m/s – I consider the former possible given what I know, and the latter not. However, since there is some interval around 2.998 x 108 m/s that I can’t rule out, and there are infinitely many such values that I am indifferent between, I can’t give any of them a positive value without either violating additivity or assigning values larger than 1 to certain disjunctions. So I propose that the state space contains infinitely many epistemic possibilities, and that my degree of confidence in certain sets of these possibilities is 0, even though the set is non-empty. (Of course, for the empty set, I trivially have confidence 0 in that set of possibilities.) So I think this aspect of the math actually applies quite well to degrees of confidence, though I’m willing to concede that many people will want to challenge this point, and I don’t think it’s as important as the point that degrees of confidence must be probabilities in something like the general sense I outlined earlier.

However, I don’t think such a state space comes with objectively correct probabilities to assign – after all, it’s infinite, and Bertrand’s Paradox shows how all sorts of troubles arise when we think that symmetries of an infinite space constrain probability assignments.

As for the third question, I’m not sure I agree with its premise. I’m not totally convinced that when there is a well-defined objective probability, we should match it with our degrees of confidence. Consider a fair coin that has just been tossed. There is some sense in which it had an objective probability of 1/2 of coming up heads, so this principle would suggest having degree of belief 1/2 in heads. But if I also know that this coin was one of 10 fair coins flipped at that point, 9 of which happened to come up heads, then (in the frequency sense, as opposed to the chance sense) there is also an objective probability of 9/10 of that coin being heads up, so this principle would suggest the contradictory degree of belief of .9. Maybe in this situation one of the two principles wins out (my guess would be the latter), but I don’t really know under what circumstances something like this should be the case. Of course, I also don’t really know what sorts of objective things count as “well-defined objective probabilities” – is it chances, frequencies, or something else? There are many well-defined objective things that obey the mathematics of probability, but it’s an interesting question which (if any) should be tracked by our degrees of confidence.

Kenny Pearce suggests that on the (KPW) theory of degrees of confidence, it’s the fact that “the worlds … divide more or less evenly” that makes us assign 1/6 to each of the propositions about the way the die might land up. I don’t think there’s such thing as an objective measure over this infinite state space, so we can’t even make sense of the worlds dividing more or less evenly. Thus, if there is some objective reason for the degrees of belief we assign, I don’t know what it is yet, but I don’t think it could be anything like what Kenny Pearce suggests in either (KPW) or (LPW). (Also, I don’t think (LPW) is even a viable candidate, because this is supposed to be a theory of degree of rational belief, and actual possibilities have almost nothing to do with rational epistemic possibilities – one could try to make a modified 2-dimensionalist version of this strategy, as Frank Jackson does, but I’m not convinced that this will work.)

I think that these degrees of confidence exist, and are actually often much more precise than we realize (there’s no reason we should have transparent access to exactly what our degrees of belief are), but they’re not constitutively tied to any sort of objective probability in the sense that Kenny Pearce was expecting for a relation between Bayesian epistemology and probability. These degrees of belief are themselves probabilities, just in a different interpretation than Kenny Pearce was specifically considering.

## Foundations of Category Theory

1 12 2007

Yesterday, Solomon Feferman from Stanford was up at Berkeley to give the logic colloquium, and he had a very interesting discussion of foundations for category theory. I figured I’d write up the outlines of the basic theories he discussed, because mathematicians might well be interested in how this sort of thing could work. I’d also be interested in hearing what people who actually use category theory think of all this. The issues that came up are the reason why set theory suffices as a foundation for all mathematics before category theory, but has troubles here. The basic idea is that he wants some formal framework in which one can work with categories, in a way that the following four criteria are satisfied to a sufficient degree:

## The Criteria

1. There should be a category of all groups (with homomorphisms), a category of all sets (with functions), a category of all topological spaces (with continuous maps), and a category of all categories (with functors).

2. If A and B are two categories, then there should be a category consisting of all functors from A to B, with morphisms given by natural transformations.

3. Standard set-theoretic operations used throughout mathematics should be possible – things like taking unions and intersections of a set of sets, constructing powersets and ordered pairs and n-tuples and the like.

4. The framework should be provably consistent from some standard set-theoretic background. (This criterion is obviously going to seem unnecessary for someone like Lawvere, who wants to use category theory as an alternate foundation for mathematics. For someone like that, the really interesting criterion will probably be #3, because they’ll have to translate away all talk that mathematicians normally have of sets in other terms. Maybe this has been done, but I’ve had a lot of conceptual difficulty trying to read Lawvere’s textbook treatment of this stuff. At any rate, I don’t think most mathematicians will find this any more congenial than working in ZFC directly. As a result, this sort of thing wasn’t what Feferman was talking about, but he did a good job clarifying the interplay of category theory and foundational work in the more usual frameworks.)

Most of the talk was spent describing several systems that allow for some degree of satisfaction of all these criteria, but it looks like no one yet has come up with a framework that allows for full satisfaction of all of them.

## The Frameworks

### Mac Lane’s small and large categories

This is probably the most familiar way of talking about these issues for mathematicians. In the background, we use a standard theory to talk about sets and proper classes (Bernays-Gödel in particular). This theory gives us the full power of ZFC when talking about sets, but also allows us to talk about proper classes whose elements are sets. On this picture a category is a class of objects, together with a class of morphisms between them, satisfying the standard axioms. If both of these classes are sets, then the category is said to be small, while if either one is a proper class, then the category is said to be large.

We can see how well it fits all the criteria:

1. There are categories of all groups, of all sets, and all topological spaces, but each of these categories is itself large. There is a category of all small categories, but this category obviously doesn’t have any of the previous categories (or itself) as an object. Thus, it makes sense to talk about the fundamental group operation as a functor from the category of topological spaces to the category of groups, but this functor can’t itself be seen as a morphism in the category of categories.

(Note that these categories might be seen to leave out some members. Once we can talk about proper classes, there are some “large” groups, sets, and topological spaces, just as there are large categories. An example of a large group is Conway’s surreal numbers, which form a field, but are a proper class, so they are a large group, and large ring, as well as a large field. So we might also think there’s a problem with the category of groups since it doesn’t tell us about morphisms from arbitrary groups into the surreal numbers.)

2. If A is a small category, then functors from A to B can be represented by sets of ordered pairs (in the usual way that functions are) so that there is in fact a category containing all these functors as objects. If B is small, then this category is small, and if B is large, then this category is large as well. However, if A is a large category, then functors would themselves be proper classes, and therefore can’t be members of classes, and therefore there is no category containing these functors. This restriction may or may not be problematic. (I don’t know enough about what mathematicians actually do with categories, so I don’t know if it’s important to have this functor category between large categories.)

3. Fortunately, since the background theory is built from a standard set theory, all standard set-theoretic constructions work just fine.

4. Bernays-Gödel set theory was constructed specifically to be a conservative extension of ZF. Therefore, it is consistent iff ZF is. And just as ZFC is conservative iff ZF is, Bernays-Gödel is consistent with the Axiom of Choice iff it is consistent by itself.

### Grothendieck’s “axiom of universes”

This is another background I’ve heard of before. This framework basically extends Mac Lane’s hierarchy so that instead of just small and large categories, there are infinitely many levels of categories, each of which is “small enough” to be an object in a larger category.

The official technical details are as follows. We assume standard ZF set theory, and then add the axiom that every set is contained in a universe. A universe is just a transitive set (meaning that it contains everything contained in any of its members), which also contains the powerset of any of its members, and also contains the range of any definable function applied to any set inside the universe. Basically, this just means that a universe is a standard model of ZF. In this framework, a category is just a set of objects together with a set of morphisms, following the standard axioms (none of the proper class stuff that we get with Mac Lane).

1. Now, since the collection of all groups forms a proper class, there is no category of all groups. However, for any universe U, since this universe is a set, the collection of all groups contained in U is itself a set, so we get a category of all U-groups. Similarly, for any universe U there is a category of all U-sets, all U-topological spaces, and (importantly) all U-categories.

By the Axiom of Universes, we see that every set is contained in some universe. Since every category has some underlying set, this category is itself contained in some universe U, and thus is itself an object in the category of all U-categories. None of these categories contains itself, but no matter how big the categories are you’re talking about, there is some category of categories that contains everything that big. This lets you do lots of work that couldn’t be done in the Mac Lane framework, because on his framework, large categories aren’t objects in any category, while here we can always just go up one level. (However, Mac Lane does have the advantage that there is a single category containing all groups, while here no single category does.)

2. Again, if A and B are categories in some universe U (there must be some universe containing the set {A,B}, and this universe must contain both A and B) then every function from one to the other is itself in that universe U, and thus the category of functors from A to B is itself in U. This again is an improvement over the Mac Lane situation.

3. Since we’re using standard ZF set theory, this is straightforwardly satisfied.

4. There is a slight loss over Mac Lane’s framework here. The existence of a universe implies the consistency of ZF, because the universe is itself a model of ZF. Therefore, if the consistency of ZF implied the consistency of the Axiom of Universes, then the Axiom of Universes would prove its own consistency, which is impossible by Gödel’s theorem. Since this Axiom of Universes framework was implicitly used by Wiles in his proof of Fermat’s Last Theorem, we therefore don’t yet know that ZFC is strong enough to prove the result.

However, the situation is not so bad. The existence of a universe containing a set S is just the same thing as the existence of an inaccessible cardinal with higher cardinality than S. Therefore, the Axiom of Universes is equivalent to the existence of unboundedly large inaccessible cardinals. This is provably consistent if we assume the existence of any single large cardinal that is even larger (such as a Mahlo cardinal, or a measurable cardinal), so set theorists are basically just as certain that this theory is consistent as they are of ZFC. You might have further worries about whether these cardinals actually exist (so that the system has no false premises, and therefore leads only to true conclusions), but those are the sorts of worries ordinary mathematicians like to ignore.

### Feferman’s 1969 system

It turns out that Grothendieck’s system can be brought down in strength. Instead of using the Axiom of Universes, we just add a bunch of terms U1, U2, and so on (through all the ordinals, if necessary), and add some axioms. For each proposition P expressible in the language of set theory, add the axiom:
PU_i iff P
where PU_i is the same as P, with all quantifiers explicitly restricted to range only over Ui. This has the effect of making each set Ui “just like” the entire universe. Thus, we can just formulate Grothendieck’s system here. There are some slight differences: Grothendieck’s smallest universe satisfies the claim “there are no universes”, while other universes don’t – in this system, any two of these universes satisfy exactly the same sentences. But in general, those differences make this system better, because each universe represents the whole universe better than Grothendieck’s did.

However, there’s a nice advantage – this system is apparently equiconsistent with ZFC, rather than requiring the existence of additional inaccessibles. Thus, proving a theorem in this framework uses less strength than Grothendieck’s system does. I don’t know if the two systems are quite similar enough to translate the proof of FLT into this one, but that would bring down the strength needed quite a bit. So in some sense, mathematicians probably should use this system, rather than Grothendieck’s. There are still the limitations that Grothendieck has on the first two conditions, but the third and fourth are much better satisfied.

### Feferman’s 1974 system using Quine’s “New Foundations”

This was the system Feferman spent the most time discussing. This one is also the most confusing, because it uses Quine’s system NF instead of a more standard set theory. The basic way that this system works is that instead of the complicated set of axioms of ZFC, we just have two very intuitive axioms. One is the axiom of extensionality, which says that there are no two distinct sets with exactly the same members. The other is a restriced axiom of comprehension. The basic axiom of comprehension just says that for every property, there is a set consisting of all things that satisfy that property. But as Russell pointed out to Frege in 1902, this full axiom is inconsistent, because it leads to Russell’s paradox of the set of all sets that don’t contain themselves. In ZFC this paradox is avoided by using several axioms to prove the existence of sets defined by various types of properties. Quine decided instead to avoid this (and related) paradoxes by restricting the types of properties that can be used to define sets. The only properties he allowed were ones that could be written correctly in a theory of types. In particular, each variable in the formula could be assigned a number, such that in $x=y$, x and y get the same number, while in $x\in y$ the number assigned to y is exactly 1 greater than the number assigned to x. This prevents Russell’s paradox, because the formula $\lnot(x\in x)$ can’t be assigned numbers in this way.

The nice thing about NF is that it allows sets to contain themselves, and it allows there to be a set of all sets (this set is just defined as the set of all things satisfying $x=x$). Of course, there are some very messy things that go on with the theory of cardinalities, because the powerset of the universe has to be smaller than the universe itself. There are also messy things because the standard definitions of ordered pairs and powersets and sets of functions from one set to another need to use formulas that aren’t “stratified” in the appropriate way. Feferman discussed some ways of fixing this so it could all work. A further difficulty arises in that so far no one yet knows what sort of system would be strong enough to prove that NF is consistent. However, if we allow for many objects that aren’t sets (this requires weakening the axiom of extensionality so that it only applies to things with at least one element) it turns out that this system is actually provably weaker than ZFC. (In particular, it can be proven to be equiconsistent with Zermelo set theory.) We can then define categories to be given by a set of objects and a set of morphisms.

So now we can see how this theory stands up to the criteria.

1. Since there is a set of all sets, and sets can contain themselves, it turns out that in this theory we really do get a category of all groups, a category of all sets, a category of all topological spaces, and a category of all categories that really contains itself as an object! This is the only theory we’ve listed on which this really works, and we really get all categories into a single category.

2. Again, as long as A and B are categories, we really do get a category of all functors from A to B, with all the natural transformations as morphisms (I think).

3. Here is where we really run into trouble. We’ve patched things up so that we can take unions and intersections and products and powersets and the like. However, the Axiom of Choice is provably false in this theory. Additionally, sets of functions sometimes have bad problems. Feferman said in particular that the Yoneda Lemma can’t be proven in this system, as well as some other standard things that mathematicians want to use. I don’t really know what the problems are, but these seem bad. The problem is that no one yet really understands NF well enough to know how to fix these things. Perhaps for the sake of category theory, this would be worth doing. But, Dana Scott pointed out after the talk that NF has a history of destroying careers wasted a lot of people’s time, and it’s the only foundational-type system whose full consistency strength still hasn’t been decided, and it’s been around for 70 years. [Correction: I had mischaracterized what Scott said and updated the post to reflect this. He also points out that Jensen showed that NFU, allowing urelemente, is consistent, though I’m not sure what consistency strength it has.]

4. As mentioned above, allowing basic elements that aren’t sets makes this theory equiconsistent with Zermelo set theory, but this consistency proof is also part of the weakness of the theory, since it can’t do as much with functions as standard set theories. If we managed to fix that part, this would presumably blow up the consistency strength much higher than Grothendieck’s system.

## Conclusion

Thus, there’s a bunch of interesting ways out there to get something like what mathematicians want from category theory. However, none of these systems really gets everything that we want. The ones that we know are consistent, and we know have enough set-theoretic power for everyday mathematics, we also know can’t really talk about the category of all categories or the category of all groups. There are various replacement categories around (like the category of all U-categories, or the category of all small groups), and for many purposes these are enough. But they don’t quite get the conceptual ideas right. (One could level the same criticism directly at the set theories though – none of them really gets a set of all sets, or a class of all classes, except for the poorly understood New Foundations.)

This might motivate some further work by category theorists on foundations, and in particular something like Lawvere’s program, though his program requires a radical shift in understanding of the notions of set, function, collection, individual, and the like, so it doesn’t seem like it would be much more congenial to most mathematicians for fully formal work.

Of course, most mathematicians don’t need to do this stuff fully formally, so the situation is generally comfortable. But it’s nice to see where foundational work in set theory really could impact the lives of ordinary mathematicians.