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# Error Messages to Fear
The oldest and strongest emotion of mankind is fear, and the oldest and strongest kind of fear is fear of the unknown.
Supernatural Horror in Literature, HP Lovecraft, 1927.
Security error messages appear to take pride in providing limited information. In particular, they are usually some generic IOException wrapping a generic security exception. There is some text in the message, but it is often Failure unspecified at GSS-API level, which means "something went wrong".
Generally a stack trace with UGI in it is a security problem, though it can be a network problem surfacing in the security code.
The underlying causes of problems are usually the standard ones of distributed systems: networking and configuration.
Some of the OS-level messages are covered in Oracle's Troubleshooting Kerberos docs.
Here are some of the common ones seen in Hadoop stack traces and what we think are possible causes
That is: on one or more occasions, the listed cause was the one which, when corrected, made the stack trace go away.
## GSS initiate failed —no further details provided
WARN ipc.Client (Client.java:run(676)) - Couldn't setup connection for rm@EXAMPLE.COM to /172.22.97.127:8020
at org.apache.hadoop.ipc.Client$Connection.setupSaslConnection(Client.java:558) This is widely agreed to be one of the most useless of error messages you can see. The only ones that are worse than this are those which disguise a Kerberos problem, such as when ZK closes the connection rather than saying "it couldn't authenticate". If you see this connection, work out which service it was trying to talk to —and look in its logs instead. Maybe, just maybe, there will be more information there. ## Server not found in Kerberos database (7) or service ticket not found in the subject • DNS is a mess and your machine does not know its own name. • Your machine has a hostname, but the service principal is a /_HOST wildcard and the hostname is not one there's an entry in the keytab for. We've seen this in the stdout of a NN TGS_REQ { ... }UNKNOWN_SERVER: authtime 0, hdfs@EXAMPLE.COM for krbtgt/NOVALOCAL@EXAMPLE.COM, Server not found in Kerberos database ## No valid credentials provided (Mechanism level: Illegal key size)] Your JVM doesn't have the extended cryptography package and can't talk to the KDC. Switch to openjdk or go to your JVM supplier (Oracle, IBM) and download the JCE extension package, and install it in the hosts where you want Kerberos to work. ## Encryption type AES256 CTS mode with HMAC SHA1-96 is not supported/enabled [javax.security.sasl.SaslException: GSS initiate failed [Caused by GSSException: Failure unspecified at GSS-API level (Mechanism level: Encryption type AES256 CTS mode with HMAC SHA1-96 is not supported/enabled)]] This has surfaced in the distant past. Assume it means the same as above: the JVM doesn't have the JCE JAR installed. ## No valid credentials provided (Mechanism level: Failed to find any Kerberos tgt This may appear in a stack trace starting with something like: javax.security.sasl.SaslException: GSS initiate failed [Caused by GSSException: No valid credentials provided (Mechanism level: Failed to find any Kerberos tgt)] It's very common, and essentially means "you weren't authenticated" Possible causes: 1. You aren't logged in via kinit. 2. You have logged in with kinit, but the tickets you were issued with have expired. 3. Your process was issued with a ticket, which has now expired. 4. You did specify a keytab but it isn't there or is somehow otherwise invalid 5. You don't have the Java Cryptography Extensions installed. 6. The principal isn't in the same realm as the service, so a matching TGT cannot be found. That is: you have a TGT, it's just for the wrong realm. 7. Your Active Directory tree has the same principal in more than one place in the tree. 8. Your cached ticket list has been contaminated with a realmless-ticket, and the JVM is now unhappy. (See "The Principal With No Realm") 9. The program you are running may be trying to log in with a keytab, but it's got a Hadoop FileSystem instance before that login took place. Even if the service is not logged in. that FS instance is unauthenticated. ## Failure unspecified at GSS-API level (Mechanism level: Checksum failed) One of the classics 1. The password is wrong. A kinit command doesn't send the password to the KDC —it sends some hashed things to prove to the KDC that the caller has the password. If the password is wrong, so is the hash, hence an error about checksums. 2. There was a keytab, but it didn't work: the JVM has fallen back to trying to log in as the user. 3. Your keytab contains an old version of the keytab credentials, and cannot parse the information coming from the KDC, as it lacks the up to date credentials. 4. SPENGO/REST: Kerberos is very strict about hostnames and DNS; this can somehow trigger the problem. http://stackoverflow.com/questions/12229658/java-spnego-unwanted-spn-canonicalization; 5. SPENGO/REST: Java 8 behaves differently from Java 6 and 7 which can cause problems HADOOP-11628. ## javax.security.auth.login.LoginException: No password provided When this surfaces in a server log, it means the server couldn't log in as the user. That is, there isn't an entry in the supplied keytab for that user and the system (obviously) doesn't want to fall back to user-prompted password entry. Some of the possible causes • The wrong keytab was specified. • The configuration key names used for specifying keytab or principal were wrong. • There isn't an entry in the keytab for the user. • The spelling of the principal is wrong. • The hostname of the machine doesn't match that of a user in the keytab, so a match of service/host fails. Ideally, services list the keytab and username at fault here. In a less than ideal world —that is the one we live in— things are sometimes less helpful Here, for example, is a Zookeeper trace, saying it is the user null that is at fault. 2015-12-15 17:16:23,517 - WARN [main:SaslServerCallbackHandler@105] - No password found for user: null 2015-12-15 17:16:23,536 - ERROR [main:ZooKeeperServerMain@63] - Unexpected exception, exiting abnormally java.io.IOException: Could not configure server because SASL configuration did not allow the ZooKeeper server to authenticate itself properly: javax.security.auth.login.LoginException: No password provided at org.apache.zookeeper.server.ServerCnxnFactory.configureSaslLogin(ServerCnxnFactory.java:207) at org.apache.zookeeper.server.NIOServerCnxnFactory.configure(NIOServerCnxnFactory.java:87) at org.apache.zookeeper.server.ZooKeeperServerMain.runFromConfig(ZooKeeperServerMain.java:111) at org.apache.zookeeper.server.ZooKeeperServerMain.initializeAndRun(ZooKeeperServerMain.java:86) at org.apache.zookeeper.server.ZooKeeperServerMain.main(ZooKeeperServerMain.java:52) at org.apache.zookeeper.server.quorum.QuorumPeerMain.initializeAndRun(QuorumPeerMain.java:116) at org.apache.zookeeper.server.quorum.QuorumPeerMain.main(QuorumPeerMain.java:78) ## javax.security.auth.login.LoginException: Unable to obtain password from user Believed to be the same as the No password provided Exception in thread "main" java.io.IOException: Login failure for alice@REALM from keytab /etc/security/keytabs/spark.headless.keytab: javax.security.auth.login.LoginException: Unable to obtain password from user at org.apache.hadoop.security.UserGroupInformation.loginUserFromKeytab(UserGroupInformation.java:962) at org.apache.spark.deploy.SparkSubmit$.prepareSubmitEnvironment(SparkSubmit.scala:564)
at org.apache.spark.deploy.SparkSubmit$.submit(SparkSubmit.scala:154) at org.apache.spark.deploy.SparkSubmit$.main(SparkSubmit.scala:121)
at org.apache.spark.deploy.SparkSubmit.main(SparkSubmit.scala)
The JVM Kerberos code needs to have the password for the user to login to kerberos with, but Hadoop has told it "don't ask for a password'", so the JVM raises an exception.
Root causes should be the same as for the other message.
## failure to login using ticket cache file
You aren't logged via kinit, the application isn't configured to use a keytab. So: no ticket, no authentication, no access to cluster services.
you can use klist -v to show your current ticket cache
fix: log in with kinit
## Clock skew too great
GSSException: No valid credentials provided
(Mechanism level: Attempt to obtain new INITIATE credentials failed! (null))
GSSException: No valid credentials provided (Mechanism level: Clock skew too great (37) - PROCESS_TGS
kinit: krb5_get_init_creds: time skew (343) larger than max (300)
This comes from the clocks on the machines being too far out of sync.
This can surface if you are doing Hadoop work on some VMs and have been suspending and resuming them; they've lost track of when they are. Reboot them.
If it's a physical cluster, make sure that your NTP daemons are pointing at the same NTP server, one that is actually reachable from the Hadoop cluster. And that the timezone settings of all the hosts are consistent.
## KDC has no support for encryption type
This crops up on the MiniKDC if you are trying to be clever about encryption types. It doesn't support many.
## GSSException: No valid credentials provided (Mechanism level: Fail to create credential. (63) - No service creds)
Rarely seen. Switching kerberos to use TCP rather than UDP makes it go away
In /etc/krb5.conf:
[libdefaults]
udp_preference_limit = 1
Note also UDP is a lot slower to time out.
## Receive timed out
Usually in a stack trace like
Caused by: java.net.SocketTimeoutException: Receive timed out
at sun.security.krb5.KdcComm$KdcCommunication.run(KdcComm.java:390) at sun.security.krb5.KdcComm$KdcCommunication.run(KdcComm.java:343)
at java.security.AccessController.doPrivileged(Native Method)
at sun.security.krb5.KdcComm.send(KdcComm.java:327)
at sun.security.krb5.KdcComm.send(KdcComm.java:219)
at sun.security.krb5.KdcComm.send(KdcComm.java:191)
at sun.security.krb5.KrbAsReqBuilder.send(KrbAsReqBuilder.java:319)
at sun.security.krb5.KrbAsReqBuilder.action(KrbAsReqBuilder.java:364)
This means the UDP socket awaiting a response from KDC eventually gave up.
• The hostname of the KDC is wrong
• The IP address of the KDC is wrong
• There's nothing at the far end listening for requests.
• A firewall on either client or server is blocking UDP packets
Kerberos waits ~90 seconds before timing out, which is a long time to notice there's a problem.
Switch to TCP —at the very least, it will fail faster.
## javax.security.auth.login.LoginException: connect timed out
Happens when the system is set up to use TCP as an authentication channel, and the far end KDC didn't respond in time.
• The hostname of the KDC is wrong
• The IP address of the KDC is wrong
• There's nothing at the far end listening for requests.
• A firewall somewhere is blocking TCP connections
## GSSException: No valid credentials provided (Mechanism level: Connection reset)
We've seen this triggered in Hadoop tests after the MiniKDC through an exception; its thread exited and hence the Kerberos client got a connection error.
When you see this assume network connectivity problems, or something up at the KDC itself.
## Principal not found
The hostname is wrong (or there is more than one hostname listed with different IP addresses) and so a principal of the form user/_HOST@REALM is coming back with the wrong host, and the KDC doesn't find it.
## Defective token detected (Mechanism level: GSSHeader did not find the right tag)
Seen during SPNEGO Authentication: the token supplied by the client is not accepted by the server.
This apparently surfaces in Java 8 version 8u40; if Kerberos server doesn't support the first authentication mechanism which the client offers, then the client fails. Workaround: don't use those versions of Java.
This is now acknowledged by Oracle and has been fixed in 8u60.
## Specified version of key is not available (44)
Client failed to SASL authenticate:
javax.security.sasl.SaslException:
GSS initiate failed [Caused by GSSException: Failure unspecified at GSS-API level
(Mechanism level: Specified version of key is not available (44))]
The meaning of this message —or how to fix it— is a mystery to all.
One possibility is that the keys in your keytab have expired. Did you know that can happen? It does. One day your cluster works happily. The next your client requests are failing, with this message surfacing in the logs.
klist -kt zk.service.keytab
Keytab name: FILE:zk.service.keytab
KVNO Timestamp Principal
---- ----------------- --------------------------------------------------------
5 12/16/14 11:46:05 zookeeper/devix.cotham.uk@COTHAM
5 12/16/14 11:46:05 zookeeper/devix.cotham.uk@COTHAM
5 12/16/14 11:46:05 zookeeper/devix.cotham.uk@COTHAM
5 12/16/14 11:46:05 zookeeper/devix.cotham.uk@COTHAM
One thing to see there is the version number in the KVNO table.
Oracle describe the JRE's handling of version numbers in their bug database.
From an account logged in to the system, you can look at the client's version number
## Kerberos credential has expired
Seen on an IBM JVM in HADOOP-9969
javax.security.sasl.SaslException:
Failure to initialize security context [Caused by org.ietf.jgss.GSSException, major code: 8, minor code: 0
major string: Credential expired
minor string: Kerberos credential has expired]
The kerberos ticket has expired and not been renewed.
Possible causes
• The renewer did start, but didn't try to renew in time.
• A JVM/Hadoop code incompatibility stopped renewing from working.
• Renewal failed for some other reason.
• The client was kinited in and the token expired.
• Your VM clock has jumped forward and the ticket now out of date without any renewal taking place.
## SASL No common protection layer between client and server
Not Kerberos, SASL itself
16/01/22 09:44:17 WARN Client: Exception encountered while connecting to the server :
javax.security.sasl.SaslException: DIGEST-MD5: No common protection layer between client and server
at com.sun.security.sasl.digest.DigestMD5Client.checkQopSupport(DigestMD5Client.java:418)
at com.sun.security.sasl.digest.DigestMD5Client.evaluateChallenge(DigestMD5Client.java:221)
at org.apache.hadoop.ipc.Client$Connection.setupSaslConnection(Client.java:558) at org.apache.hadoop.ipc.Client$Connection.access$1800(Client.java:373) at org.apache.hadoop.ipc.Client$Connection$2.run(Client.java:727) at org.apache.hadoop.ipc.Client$Connection$2.run(Client.java:723) at java.security.AccessController.doPrivileged(Native Method) at javax.security.auth.Subject.doAs(Subject.java:422) at org.apache.hadoop.security.UserGroupInformation.doAs(UserGroupInformation.java:1657) at org.apache.hadoop.ipc.Client$Connection.setupIOstreams(Client.java:722)
at org.apache.hadoop.ipc.Client$Connection.access$2800(Client.java:373)
at org.apache.hadoop.ipc.ProtobufRpcEngine$Invoker.invoke(ProtobufRpcEngine.java:229) at com.sun.proxy.$Proxy23.renewLease(Unknown Source)
at sun.reflect.GeneratedMethodAccessor9.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.lang.reflect.Method.invoke(Method.java:497)
## On windows: No authority could be contacted for authentication
Reported on windows clients, especially related to the Hive ODBC client. This is kerberos, just someone else's library.
1. Make sure that your system is happy in the AD realm, etc. Do this first.
2. Make sure you've configured the ODBC driver according to the instructions.
## During service startup java.lang.RuntimeException: Could not resolve Kerberos principal name: + unknown error
This something which can arise in the logs of a service. Here, for example, is a datanode failure.
Could not resolve Kerberos principal name: java.net.UnknownHostException: xubunty: xubunty: unknown error
This is not a kerberos problem. It is a network problem being misinterpreted as a Kerberos problem, purely because it surfaces in security code which assumes that all failures must be Kerberos related.
2016-04-06 11:00:35,796 ERROR org.apache.hadoop.hdfs.server.datanode.DataNode: Exception in secureMain java.io.IOException: java.lang.RuntimeException: Could not resolve Kerberos principal name: java.net.UnknownHostException: xubunty: xubunty: unknown error
at org.apache.hadoop.http.HttpServer2$Builder.build(HttpServer2.java:290) at org.apache.hadoop.hdfs.server.datanode.web.DatanodeHttpServer.<init>(DatanodeHttpServer.java:108) at org.apache.hadoop.hdfs.server.datanode.DataNode.startInfoServer(DataNode.java:781) at org.apache.hadoop.hdfs.server.datanode.DataNode.startDataNode(DataNode.java:1138) at org.apache.hadoop.hdfs.server.datanode.DataNode.<init>(DataNode.java:432) at org.apache.hadoop.hdfs.server.datanode.DataNode.makeInstance(DataNode.java:2423) at org.apache.hadoop.hdfs.server.datanode.DataNode.instantiateDataNode(DataNode.java:2310) at org.apache.hadoop.hdfs.server.datanode.DataNode.createDataNode(DataNode.java:2357) at org.apache.hadoop.hdfs.server.datanode.DataNode.secureMain(DataNode.java:2538) at org.apache.hadoop.hdfs.server.datanode.DataNode.main(DataNode.java:2562) Caused by: java.lang.RuntimeException: Could not resolve Kerberos principal name: java.net.UnknownHostException: xubunty: xubunty: unknown error at org.apache.hadoop.security.AuthenticationFilterInitializer.getFilterConfigMap(AuthenticationFilterInitializer.java:90) at org.apache.hadoop.http.HttpServer2.getFilterProperties(HttpServer2.java:455) at org.apache.hadoop.http.HttpServer2.constructSecretProvider(HttpServer2.java:445) at org.apache.hadoop.http.HttpServer2.<init>(HttpServer2.java:340) ... 11 more Caused by: java.net.UnknownHostException: xubunty: xubunty: unknown error at java.net.InetAddress.getLocalHost(InetAddress.java:1505) at org.apache.hadoop.security.SecurityUtil.getLocalHostName(SecurityUtil.java:224) at org.apache.hadoop.security.SecurityUtil.replacePattern(SecurityUtil.java:192) at org.apache.hadoop.security.SecurityUtil.getServerPrincipal(SecurityUtil.java:147) at org.apache.hadoop.security.AuthenticationFilterInitializer.getFilterConfigMap(AuthenticationFilterInitializer.java:87) ... 14 more Caused by: java.net.UnknownHostException: xubunty: unknown error at java.net.Inet4AddressImpl.lookupAllHostAddr(Native Method) at java.net.InetAddress$2.lookupAllHostAddr(InetAddress.java:928)
... 18 more2016-04-06 11:00:35,799 INFO org.apache.hadoop.util.ExitUtil: Exiting with status 12016-04-06 11:00:35,806 INFO org.apache.hadoop.hdfs.server.datanode.DataNode: SHUTDOWN_MSG:
{code}
The root cause here was that the host xubunty which a service was configured to start on did not have an entry in /etc/hosts, nor DNS support. The attempt to look up the IP address of the local host failed.
The fix: add the short name of the host to /etc/hosts.
This example shows why errors reported as Kerberos problems, be they from the Hadoop stack or in the OS/Java code underneath, are not always Kerberos problems. Kerberos is fussy about networking; the Hadoop services have to initialize Kerberos before doing any other work. As a result, networking problems often surface first in stack traces belonging to the security classes, wrapped with exception messages implying a Kerberos problem. Always follow down to the innermost exception in a trace as the immediate symptom of a problem, the layers above attempts to interpret that, attempts which may or may not be correct.
## Against Active Directory: Realm not local to KDC while getting initial credentials
Nobody quite knows.
It's believed to be related to Active Directory cross-realm/forest stuff, but there are hints that it can also be raised when the kerberos client is trying to auth with a KDC, but supplying a hostname rather than the realm.
This may be because you have intentionally or unintentionally created A Disjoint Namespace.aspx))
If you read that article, you will get the distinct impression that even the Microsoft Active Directory team are scared of Disjoint Namespaces, and so are going to a lot of effort to convince you not to go there. It may seem poignant that even the developers of AD are scared of this, but consider that these are probably inheritors of the codebase, not the original authors, and the final support line for when things don't work. Their very position in the company means that they get the worst-of-the-worst Kerberos-related problems. If they say "Don't go there", it'll be based on experience of fielding those support calls and from having seen the Active Directory source code.
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## Tensors #2
Welcome back to the beginning of my journey into the world of multilinear algebra. In this post, I want to discuss the definition I previously gave of the tensor product of two vector spaces. There are many ways to define it, and the definition I gave was through a universal property. One thing I have found particularly enthralling is the way we can prove things about the tensor product by simply making use of this universal property. I will show an example of this shortly.
The main point of this post is to discuss in detail two different equivalent formulations of the universal property. I realize I still have not proved existence or uniqueness of the tensor product, but don’t worry. I will do this soon.
§3. Discussion of the universal property of the tensor product
The universal property of the tensor product. (Note that the labels used here are not consistent with those in my post).
I’ll omit the preamble and assume that from the last post, you know the general setup that we’ve put into place. Recall that we say that a pair $(\mathsf{T}, \otimes)$ consisting of a vector space $\mathsf{T}$ and a bilinear map $\otimes : \mathsf{V} \times \mathsf{W} \to \mathsf{T}$ has the universal property if
• (T) If $\mathsf{Z}$ is some other vector space and $f : \mathsf{V} \times \mathsf{W} \to \mathsf{Z}$ is bilinear, then there exists a unique linear map $\varphi : \mathsf{T} \to \mathsf{Z}$ such that $f = \varphi \circ \otimes$.
Condition (T) above turns out to be equivalent to the following two conditions holding:
• (T1) The span of $\mathrm{Img}(\otimes) := \{ v \otimes w : v \in V, w \in W \}$ is all of $\mathsf{T}$.
• (T2) If $\mathsf{Z}$ is some other vector space and $f : \mathsf{V} \times \mathsf{W} \to \mathsf{Z}$ is bilinear, then there exists a linear map $\varphi : \mathsf{T} \to \mathsf{Z}$ such that $f = \varphi \circ \otimes$.
At first glance you may have thought (T) and (T2) are the same. However, there is a subtle difference. (T) tells us that the map $\varphi$ for some fixed $f$ is unique. (T2) tells us that such a map $\varphi$ exists, but says nothing about uniqueness.
Let’s get right into it, then. Why are these equivalent? Well, let’s take care of the easy direction first: suppose (T1) and (T2) hold. Let an arbitrary $f$ (as described above) be given, and suppose that $\varphi_1, \varphi_2 : \mathsf{T} \to \mathsf{Z}$ are such that $f = \varphi_1 \circ \otimes = \varphi_2 \circ \otimes$. Our goal is to prove $\varphi_1 = \varphi_2$. However, this is quite easy. Note that by the above, we see that $f(v,w) = \varphi_1( v \otimes w ) = \varphi_2( v \otimes w ).$ However, by (T1), these vectors $v \otimes w$ generate all of $\mathsf{T}$! So therefore $\varphi_1$ and $\varphi_2$ agree on a generating set for $\mathsf{T}$, whereby immediately we obtain $\varphi_1 = \varphi_2$. Wonderful.
Now for the slightly more difficult direction. Suppose (T) holds. It is clear that (T2) holds, but (T1) is not so obvious. When I was thinking about this, I realized I didn’t really understand the proof of this fact given in the book I’m reading (Greub), because it made use of some odd notation I’m not familiar with. So here’s my own proof, which I hope will be clearer. How is uniqueness of the map related to whether or not these vectors $v \otimes w$ generate $\mathsf{T}$? Well, here’s a trick. Suppose to the contrary that the vectors $v \otimes w$ do not generate $\mathsf{T}$. Let $\mathsf{Z} = \mathsf{T}$ and pick our map $f$ to be nothing more than $\otimes$. Then (T) tells us there should be a unique $\varphi$ such that $\otimes = \varphi \circ \otimes$. You’re probably thinking already “OK, so we can just choose $\varphi$ to be the identity map”. That is correct, but it’s not the only choice. What we do know is that $\varphi$ restricted to the span of $\mathrm{Img}(\otimes)$ must be the identity map (why?). However, the key idea is that if $\mathrm{Img}(\otimes)$ does not generate all of $\mathsf{T}$, then there will be some “arbitrary choice” to be made in coming up with the map $\varphi$. As I previously stated, it is valid to choose $\varphi$ to be the identity, but we can also define $\varphi$ to be the identity on the subspace generated by $\mathrm{Img}(\otimes)$, and then map everything outside the subspace into the subspace (clearly, a map so obtained will not be the identity map on $\mathsf{T})$.
You can turn this into a formal proof if you want, but that’s the idea. In the next post, I will man up and prove existence and uniqueness. Cheers.
just another guy trying to make the diagrams commute.
This entry was posted in abstract algebra, linear algebra and tagged , , , . Bookmark the permalink.
### 3 Responses to Tensors #2
1. Saifuddin Syed says:
For the first direction, I don’t understand why T1 is needed to show \varphi_1 = \varphi_2 .
2. dx7hymcxnpq says:
The invocation of (T1) is crucial. Just because $\varphi_1( v \otimes w) = \varphi_2( v \otimes w)$ for all $v \in \mathsf{V}, w \in \mathsf{W}$ does not mean they are the same map on $\mathsf{T}$. All it means is that their restrictions to the subspace generated by the vectors $v \otimes w$ must be the same. If this subspace turns out to be contained properly within $\mathsf{T}$ then you can’t conclude $\varphi_1 = \varphi_2$ because you don’t know anything about how each map behaves outside of this subspace.
3. dx7hymcxnpq says:
Thing about the tensor product: it’s a space which is large enough to properly capture the possibilities for multilinear (well, in this case bilinear) maps, but it’s also small/tight enough to provide you with a unique means of decomposition (I’m referring to the maps $\varphi$ here). In that sense it’s very special.
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# Pluses and Minuses
Level pending
Let us write the numbers 1,2,3,...,2014 in order in a row. In how many ways can you put + and - symbols between the terms so that the sum of the 2014 numbers is 0?
$$\textbf{Details and Assumptions}$$
There must be an operator in all of the 2013 gaps between terms
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zurück
Abstract:
The method of convex integration originally comes from differential geometry. It was first used in 1954 in Nash's solution to the isometric embedding problem and was later brought to fluid dynamics. In this talk we will first visit some of the well-known results achieved by convex integration and then turn to an application of a concrete convex integration scheme, namely the failure of the chainrule for the divergence of vector fields. For given scalar functions $\beta$ and $f$, the aim is to construct via convex integration a Sobolev vector field $v$ and a scalar $L^p$-function $\rho$ such that the product is divergence-free, but $\div(\beta(\rho)v) = f.$ Finally, we discuss some possible generalisations and problems that may occur.
Wann?
02. Dezember 2021, 14:00-15:30
Wo?
TU Darmstadt FB Mathematik AG Analysis
S2/15 Raum 51
Schlossgartenstr. 7
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# The network complexity and the Turing machine complexity of finite functions
@article{Schnorr1976TheNC,
title={The network complexity and the Turing machine complexity of finite functions},
author={Claus-Peter Schnorr},
journal={Acta Informatica},
year={1976},
volume={7},
pages={95-107}
}
Let L(f) be the network complexity of a Boolean function L(f). For any n-ary Boolean function L(f) let $$TC(f) = min\{ T_p^{\bar A} (n){\text{ (}}\parallel p\parallel + 1gS_p^{\bar A} {\text{(}}n{\text{):}}res_p^{\bar A} {\text{(}}n{\text{) = }}f\}$$ . Hereby p ranges over all relative Turing programs and Ā ranges over all oracles such that given the oracle Ā, the restriction of p to inputs of length n is a program for L(f). ∥p∥ is the number of instructions of p. T p Ā (n) is the time bound… CONTINUE READING
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## An improved overlap argument for on - line multiplication . In : Complexity of Computation
• M. S. Paterson, M. J. Fischer, A. R. Meyer
• SIAM AMS Proceedings
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# How series resistors influence clearance-creepage requirements?
simulate this circuit – Schematic created using CircuitLab
I'm designing a circuit which should leak some amount of current from AC line to PE. The question is regarding clearance-creepage calculation for this specific case.
The part (triac optocoupler) is rated for 600V peak on TRIAC terminals, but the pin spacing is below minimum required by safety standard (2mm instead of 3mm required from line to PE).
The question is if I can count clearance-creepage added by each resistor leads spacing?
For example: If I have 4 resistors of 1206 size it will give me ~4x1.5mm=6mm donated by resistors + 2mm pin spacing of triac optocoupler.
The most intuitive answer will be yes, but in case triac is not conducting we will have line voltage at its terminals and it can probably arc the leads.
• The intuitive answer to me is no, but I'm not familiar with the details of the standards. I would suggest getting a different optocoupler. – Hearth Nov 6 '18 at 14:53
• Functionally, it makes very little difference unless we are talking about Gohm range. If you are trying to fulfill IEC 60950 or similar, then no, you can not use the resistors pin spacing. – winny Nov 6 '18 at 14:59
• Felthty, Winny, thanks for quick response. The problem there is only one manufacturer which has more than 3mm pin spacing. This is wierd that in components like TRIAC that are used for AC mains do not have sufficient clearence-creepage... So I asume that I misunderstood something and it is legal to use them somehow. – ILYA Shu. Nov 6 '18 at 15:01
• You have several ways to cheat here. Ask manufacurer which material group or CTI the plastic is. If CTI is 600, your creepage requirements are relaxed and you can make a cutout in your PCB. If not, you can coat it with conformal coating to get down to pollution degree 1 locally across the optocoupler. – winny Nov 6 '18 at 15:24
• @ILYAShu. You might have more success searching for it if you look for solid-state relays. TRIAC-type solid-state relays are basically the same thing as TRIAC-type optocouplers, just designed with more power in mind. I can easily find a few on digikey. – Hearth Nov 6 '18 at 15:29
As already stated by Felthry and winny, the spacing within the resistor terminals on the high voltage side of the circuit cannot be counted as a supplementary clearance/creepage distance across the isolation barrier offered by the optotriac $$\\mathrm{TRI}1\$$. The reason was already guessed by you: resistors do not offer galvanic insulation between their terminals, but only a functional one, meaning that they are properly spaced in order to work with an applied voltage sufficient to make them reach their maximum power dissipation without being destroyed. As long as no current flows through them, the highest voltage on the controlled load will appear at the optotriac anodes, leaving it as the only barrier for safety/galvanic isolation. Therefore you have two choices for solving your creepage/clearance problem:
1. If you need a clearance distance (i.e. across the air, "at sight") larger than 2mm, you should change the optotriac model you are using. No matter how you design your control and load circuit, there will be always a condition where the maximum voltage to be "blocked" will be present across the optotriac, and it will cause an arcing discharge through air to reach your input control circuit, usually with unwanted effects, to say the least.
2. If instead the clearance required can be relaxed, and you instead need absolutely to have a creepage distance (i.e. along a path laying on the board/component surfaces) larger than 2mm, then you can also decide to make a cut of sufficient thickness on the board just below the body of the optotriac. The requirement will be fulfilled (it depends also on how crowded is your board), with a modest increase of board costs.
• Daniele, thank you for quick response. Regarding 1: if my understanding of how arcs work right - arc is equivalent to a short-circuit. If this is true then it will case current flow in the resistors which will emidiately decrease voltage to 0V on TRIAC terminals, which will cause arc to stop. So arc can't even start, because to start it must have "insuficient" amount of energy (and in our case it's limited by resistors). Am I missing something? – ILYA Shu. Nov 6 '18 at 16:50
• @ILYAShu. You're right: the term "arcing" is not exact since the current cannot rise up to that levels (several amperes) needed to start a proper arc discharge. It is better to say that it will cause a black/glow discharge (you'll see the characteristic light spike or nothing at all) and few (perhaps tens of) milliamperes will pass from the load circuit part to the control circuit part, in spite of the attempt to create a safety barrier. – Daniele Tampieri Nov 6 '18 at 20:24
• didn't now about black/glow discharges, thanks for pointing out. But the isolation input-output is ok, the problem is distance between terminals of triac on load side. I'm afraid of possible arc from AC Line to PE across triac. – ILYA Shu. Nov 7 '18 at 7:45
I figured out that one can use parts which are rated for 600V in 230AC even if the pin spacing is below required with the following restriction:
• If the part is removed(not assembled) you'll need to keep the required clearance-creepage. This can be achieved in my case by disassembling one or more resistors when oproisolator is not present.
The answer is weird but I recieved it from verified source :) so I tend to believe this is true...
• +1. I have been busy so I had not the time to search a more accurate answer. Your verified source is right since, if your circuit is designed in order to leak some current from the AC power to the PE, you implicitly cannot keep the safety crepage/clearance distance from AC to PE. Therefore, when the optotriac is mounted and provides its "leakage" function, you have to respect only the functional isolation distances, lower than the safety ones: when it is not mounted, you have to remove also the resistors in order to reach the safety levels. – Daniele Tampieri Nov 11 '18 at 10:02
• Thanks for confirmation and deeper explonation, Daniele! – ILYA Shu. Nov 12 '18 at 13:52
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# 9.5.4 Gaussians On Surface Tesserae Simulate HYdrostatic Pressure (GOSTSHYP)
(July 14, 2022)
The Gaussians On Surface Tesserae Simulate HYdrostatic Pressure (GOSTSHYP) method, which was introduced by Scheurer and co-workers, 1058 Scheurer M. et al.
J. Chem. Theory Comput.
(2021), 17, pp. 583.
overcomes the problems associated with the mechanochemical models of pressure, i.e. HCFF and X-HCFF. GOSTSHYP uses a uniform field of Gaussian potentials that is placed on the tessellated molecular surface and that compresses the electron density. Each Gaussian potential G${}_{j}$ has the form
$G_{j}=p_{j}\cdot\exp\left(-w_{j}(\textbf{r}-\textbf{r}_{0})^{2}\right)$ (9.43)
During the GOSTSHYP routine, the parameters of the Gaussian potentials, $p_{j}$ and $w_{j}$, are adjusted such that a user-defined pressure is applied. Atoms and molecules can be treated, and the pressure-induced increase in the electronic energy is physically sound. During the SCF, the energy expression takes the form
$\displaystyle E_{\text{GOSTSHYP}}$ $\displaystyle=\sum_{j}E_{j}=\sum_{j}\int G_{j}(\textbf{r})\rho(\textbf{r})d% \textbf{r}$ $\displaystyle=\sum_{j}\sum_{\mu,\nu}\sum_{a}\left<\chi_{\mu}|G_{j}|\chi_{\nu}% \right>c_{\mu a}^{*}c_{\nu a}$ (9.44)
Due to the availability of nuclear gradients, geometry optimizations under pressure using the GOSTSHYP model are possible. At present, GOSTSHYP is implemented at the SCF level, allowing calculations with Hartree-Fock and Density Functional Theory (DFT).
Example 9.15 Geometry optimization of cyclopentadiene and ethylene under a pressure of 40 GPa using the GOSTSHYP model
$molecule 0 1 C 1.1148422354 -0.6418674001 0.7279292386 C 1.1148422354 -0.6418674001 -0.7279292386 C 0.5936432126 0.5363396649 1.1772168767 C -2.0464511598 -0.6129291257 0.6711240568 C -2.0464511598 -0.6129291257 -0.6711240568 C 0.5936432126 0.5363396649 -1.1772168767 C 0.2915208637 1.4128825196 0.0000000000 H 0.9756522868 2.2894492537 0.0000000000 H -0.7374232239 1.8214336422 0.0000000000 H 1.4681344173 -1.4690333337 -1.3527755131 H 1.4681344173 -1.4690333337 1.3527755131 H -2.3879086093 0.2541525765 1.2531118994 H -1.7231567891 -1.4887031107 1.2461940178 H -1.7231567891 -1.4887031107 -1.2461940178 H -2.3879086093 0.2541525765 -1.2531118994 H 0.4773764265 0.8454441265 2.2200767812 H 0.4773764265 0.8454441265 -2.2200767812$end
$rem JOBTYPE opt METHOD pbe BASIS cc-pvdz GEOM_OPT_MAX_CYCLES 150 SCF_ALGORITHM diis_gdm MAX_SCF_CYCLES 150 USE_LIBQINTS 1 GEN_SCFMAN 1 DISTORT 1$end
$distort model gostshyp pressure 40000 npoints_heavy 302 npoints_hydrogen 302 scaling 1.8$end
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# 2012 AMC 12B Problems/Problem 17
## Problem
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8$
## Solutions
$[asy] size(7cm); pair A=(0,0),B=(1,1.5),D=B*dir(-90),C=B+D-A; draw((-4,-2)--(8,-2), Arrows); draw(A--B--C--D--cycle); pair AB = extension(A,B,(0,-2),(1,-2)); pair BC = extension(B,C,(0,-2),(1,-2)); pair CD = extension(C,D,(0,-2),(1,-2)); pair DA = extension(D,A,(0,-2),(1,-2)); draw(A--AB--B--BC--C--CD--D--DA--A, dotted); dot(AB^^BC^^CD^^DA);[/asy]$
(diagram by [i] MSTang [/i])
### Solution 1
$[asy] size(14cm); pair A=(3,0),B=(5,0),C=(7,0),D=(13,0),EE=(4,0),F=(10,0),P=(3.4,1.2),Q=(5.2,0.6),R=(5.8,2.4),SS=(4,3),M=(4.6,1.8),G=(3.2,0.6),H=(7.6,1.8); dot(A^^B^^C^^D^^EE^^F^^P^^Q^^R^^SS^^M^^G^^H); draw(A--SS--D--cycle); draw(P--Q--R^^B--Q--C); draw(EE--M--F^^G--B^^C--H,dotted); label("A",A,SW); label("B",B,S); label("C",C,S); label("D",D,SE); label("E",EE,S); label("F",F,S); label("P",P,W); label("Q",Q,NW); label("R",R,NE); label("S",SS,N); label("M",M,S); label("G",G,W); label("H",H,NE);[/asy]$
Construct the midpoints $E=(4,0)$ and $F=(10,0)$ and triangle $\triangle EMF$ as in the diagram, where $M$ is the center of square $PQRS$. Also construct points $G$ and $H$ as in the diagram so that $BG\parallel PQ$ and $CH\parallel QR$.
Observe that $\triangle AGB\sim\triangle CHD$ while $PQRS$ being a square implies that $GB=CH$. Furthermore, $CD=6=3\cdot AB$, so $\triangle CHD$ is 3 times bigger than $\triangle AGB$. Therefore, $HD=3\cdot GB=3HC$. In other words, the longer leg is 3 times the shorter leg in any triangle similar to $\triangle AGB$.
Let $K$ be the foot of the perpendicular from $M$ to $EF$, and let $x=EK$. Triangles $\triangle EKM$ and $\triangle MKF$ also have legs in a 1:3 ratio, therefore, $MK=3x$ and $KF=9x$, so $10x=EF=6$. It follows that $EK=0.6$ and $MK=1.8$, so the coordinates of $M$ are $(4+0.6,1.8)=(4.6,1.8)$ and so our answer is $\boxed{\mathbf{(C)}\ 6.4}$.
### Solution 2
$[asy] size(7cm); pair A=(0,0),B=(1,1.5),D=B*dir(-90),C=B+D-A; draw((-4,-2)--(8,-2), Arrows); draw(A--B--C--D--cycle); pair AB = extension(A,B,(0,-2),(1,-2)); pair BC = extension(B,C,(0,-2),(1,-2)); pair CD = extension(C,D,(0,-2),(1,-2)); pair DA = extension(D,A,(0,-2),(1,-2)); draw(A--AB--B--BC--C--CD--D--DA--A, dotted); dot(AB^^BC^^CD^^DA);[/asy]$
Let the four points be labeled $P_1$, $P_2$, $P_3$, and $P_4$, respectively. Let the lines that go through each point be labeled $L_1$, $L_2$, $L_3$, and $L_4$, respectively. Since $L_1$ and $L_2$ go through $SP$ and $RQ$, respectively, and $SP$ and $RQ$ are opposite sides of the square, we can say that $L_1$ and $L_2$ are parallel with slope $m$. Similarly, $L_3$ and $L_4$ have slope $-\frac{1}{m}$. Also, note that since square $PQRS$ lies in the first quadrant, $L_1$ and $L_2$ must have a positive slope. Using the point-slope form, we can now find the equations of all four lines: $L_1: y = m(x-3)$, $L_2: y = m(x-5)$, $L_3: y = -\frac{1}{m}(x-7)$, $L_4: y = -\frac{1}{m}(x-13)$.
Since $PQRS$ is a square, it follows that $\Delta x$ between points $P$ and $Q$ is equal to $\Delta y$ between points $Q$ and $R$. Our approach will be to find $\Delta x$ and $\Delta y$ in terms of $m$ and equate the two to solve for $m$. $L_1$ and $L_3$ intersect at point $P$. Setting the equations for $L_1$ and $L_3$ equal to each other and solving for $x$, we find that they intersect at $x = \frac{3m^2 + 7}{m^2 + 1}$. $L_2$ and $L_3$ intersect at point $Q$. Intersecting the two equations, the $x$-coordinate of point $Q$ is found to be $x = \frac{5m^2 + 7}{m^2 + 1}$. Subtracting the two, we get $\Delta x = \frac{2m^2}{m^2 + 1}$. Substituting the $x$-coordinate for point $Q$ found above into the equation for $L_2$, we find that the $y$-coordinate of point $Q$ is $y = \frac{2m}{m^2+1}$. $L_2$ and $L_4$ intersect at point $R$. Intersecting the two equations, the $y$-coordinate of point $R$ is found to be $y = \frac{8m}{m^2 + 1}$. Subtracting the two, we get $\Delta y = \frac{6m}{m^2 + 1}$. Equating $\Delta x$ and $\Delta y$, we get $2m^2 = 6m$ which gives us $m = 3$. Finally, note that the line which goes though the midpoint of $P_1$ and $P_2$ with slope $3$ and the line which goes through the midpoint of $P_3$ and $P_4$ with slope $-\frac{1}{3}$ must intersect at at the center of the square. The equation of the line going through $(4,0)$ is given by $y = 3(x-4)$ and the equation of the line going through $(10,0)$ is $y = -\frac{1}{3}(x-10)$. Equating the two, we find that they intersect at $(4.6, 1.8)$. Adding the $x$ and $y$-coordinates, we get $6.4$. Thus, answer choice $\boxed{\textbf{(C)}}$ is correct.
### Solution 3
Note that the center of the square lies along a line that has an $x-$intercept of $\frac{3+5}{2}=4$, and also along another line with $x-$intercept $\frac{7+13}{2}=10$. Since these 2 lines are parallel to the sides of the square, they are perpendicular (since the sides of a square are). Let $m$ be the slope of the first line. Then $-\frac{1}{m}$ is the slope of the second line. We may use the point-slope form for the equation of a line to write $l_1:y=m(x-4)$ and $l_2:y=-\frac{1}{m}(x-10)$. We easily calculate the intersection of these lines using substitution or elimination to obtain $\left(\frac{4m^2+10}{m^2+1},\frac{6m}{m^2+1}\right)$ as the center or the square. Let $\theta$ denote the (acute) angle formed by $l_1$ and the $x-$axis. Note that $\tan\theta=m$. Let $s$ denote the side length of the square. Then $\sin\theta=s/2$. On the other hand the acute angle formed by $l_2$ and the $x-$axis is $90-\theta$ so that $\cos\theta=\sin(90-\theta)=s/6$. Then $m=\tan\theta=3$. Substituting into $\left(\frac{4m^2+10}{m^2+1},\frac{6m}{m^2+1}\right)$ we obtain $\left(\frac{23}{5},\frac{9}{5}\right)$ so that the sum of the coordinates is $\frac{32}{5}=6.4$. Hence the answer is $\framebox{C}$.
### Solution 4 (Fast)
Suppose
$$SP: y=m(x-3)$$ $$RQ: y=m(x-5)$$ $$PQ: -my=x-7$$ $$SR: -my=x-13$$
where $m >0$.
Recall that the distance between two parallel lines $Ax+By+C=0$ and $Ax+By+C_1=0$ is $|C-C_1|/\sqrt{A^2+B^2}$, we have distance between $SP$ and $RQ$ equals to $2m/\sqrt{1+m^2}$, and the distance between $PQ$ and $SR$ equals to $6/\sqrt{1+m^2}$. Equating them, we get $m=3$.
Then, the center of the square is just the intersection between the following two "mid" lines:
$$L_1: y=3(x-4)$$ $$L_2: -3y = x-10$$
The solution is $(4.6,1.8)$, so we get the answer $4.6+1.8=6.4$. $\framebox{C}$.
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1. ## Help
I am trying to simplify the end of this half angle problem
How do I get from $\sqrt{(9-\sqrt{17})/(18)}$
to:
$(1/3)\sqrt{(9-\sqrt{17})/(2)}$
2. ## Re: Help
Working backwards:
Since the two is on the denominator it is essentially multiplying the (9-(17)^(1/2)) by 1/2. So to get 1/3 into the denominator you would square it, which becomes 1/9. then you would multiply it by everything else which will get the denominator to be 18.
Working Forwards:
$\sqrt{(9-\sqrt{17})/(18)}$ =
$\sqrt{(9-\sqrt{17})(1/18)}$ =
$\sqrt{(9-\sqrt{17})(1/2)(1/9)}$ =
$(1/3)\sqrt{(9-\sqrt{17})(1/2)}$ =
$(1/3)\sqrt{(9-\sqrt{17})/(2)}$
Thank you
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# American Institute of Mathematical Sciences
August 2016, 9(4): 1025-1038. doi: 10.3934/dcdss.2016040
## Multiple homoclinic solutions for a one-dimensional Schrödinger equation
1 Department of Mathematics - University of Torino, Via Carlo Alberto, 10 - 10123 Torino 2 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, via delle Scienze 206, 33100 Udine, Italy
Received July 2015 Revised January 2016 Published August 2016
In this paper we study the problem of the existence of homoclinic solutions to a Schrödinger equation of the form $x''-V(t)x+x^3=0,$ where is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Ważewski's method.
Citation: Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040
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2020 Impact Factor: 2.425
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According to Artificial Intelligence: A Modern Approach 4th edition in IDA* the cutoff is the $f$-cost($g+h$); at each iteration, the cutoff value is the smallest $f$-cost of any node that exceeded the cutoff on the previous iteration. In other words, each iteration exhaustively searches an $f$-contour, finds a node just beyond that contour, and uses that ...
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# What percent of 1 is 3? make sure the answer is fully reduced.
What percent of 1 is 3? make sure the answer is fully reduced.
## This Post Has 3 Comments
1. korirosekc says:
0.3
step-by-step explanation:
2. Expert says:
Teri ma ki chhut
3. Expert says:
d
step-by-step explanation:
d
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## September 10, 2009
### Towards a Computer-Aided System for Real Mathematics
#### Posted by John Baez
I’ve known Arnold Neumaier for quite a while, thanks to many discussions on the newsgroup sci.physics.research. Recently he sent me a proposal for a system called FMATHL (Formal Mathematical Language), designed to be:
a formal framework which will allow — when fully implemented in some programming language — the convenient use of and communication of arbitrary mathematics on the computer, in a way close to the actual practice of mathematics, with emphasis on matching this practice closely.
Here’s a slightly edited version of what Arnold Neumaier sent me:
I am currently working on the creation of an automatic mathematical research system that can support mathematicians in their daily work, providing services for abstract mathematics as easily as Latex provides typesetting service, the arXiv provides access to preprints, Google provides web services, Matlab provides numerical services, or Mathematica provides symbolic services.
The mathematical framework (at present just a formal system – a kind of metacategory of all categories) is designed to be a formal framework for mathematics that will allow (some time in the future) the convenient use and communication of arbitrary mathematics on a computer, in a way close to the actual practice of mathematics.
I would like to make the system useful and easy to use for a wide range of scientists, and hence began to ask various people from different backgrounds for feedback.
At the present point where a computer implementation is not yet available (this will take at least two more years, and how useful it will be may well depend on your input), I’d most value:
• your constructive feedback on how my plans and the part of the work already done should be extended or modified in order to find widespread approval,
• your present views on what an automatic mathematical research system should be able to do to be most useful.
Here are two pdf files:
You can find more background work on my web page.
Posted at September 10, 2009 9:28 PM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2056
### Re: Towards a Computer-Aided System for Real Mathematics
Incredibly ambitious! I dare say even ridiculously ambitious…
Reading this reminded me of a much more modest idea. I’d love a plugin for my web browser and/or PDF previewer that can automagically follow citations like [Theorem 6.5, 17]. This would perhaps work by deciphering out what [17] is (by reading the bibliography of the same paper), extracting or locating online a copy of the paper, and searching through that for Theorem 6.5. I’d imagine the user interaction is just right-clicking on a citation, and having a pop-up box showing me the theorem statement, that I could then click on to get the statement in the original paper.
At least for papers on the arXiv, which are generated using hypertex, it shouldn’t be too hard to algorithmically locate “Theorem 6.5”.
Posted by: Scott Morrison on September 11, 2009 12:02 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Incredibly ambitious! I dare say even ridiculously ambitious…
Hear hear. But very exciting, I think. It seems to me that aspects of this project, at least, could be implemented in a finite time frame by dedicated people.
I’m particularly excited by the idea of formalizing mathematics starting in the middle. I’ve recently been playing around with formal proof assistants like HOL, Isabelle, Mizar, etc., and I have definitely been struck by their insistance on building everything from the ground up. The authors of such systems seem fond of invoking Bertrand Russell’s quip:
The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil.
I’m not sure exactly how Russell intended this (no doubt someone here can set me straight), but it seems to me that the way we really do mathematics is to start by setting up some theory, i.e. “postulating” some structure and axioms, and then we prove things based on that structure and axioms. Isabelle/Isar seems to have a bit of support for this with its “locales” (no relation to pointless topology, but by and large the attitude is different. But formalizing mathematics starting in the middle, not insisting that everything be completely machine-checkably justified at first, with a system that provides other benefits to mathematicians so that we’ll actually use it, seems like it has a good chance of actually building to critical mass.
I also like the idea of a “web of trust,” with theorems “signed” by sources of varying degrees of believability (from “verified with Isabelle” to “found in textbook X” to “I saw Karp in the elevator and he said it was probably true”). Of course it reminds me of the discussions we’ve been having about how to “certify” different research pages on the nLab. And the “semantic wiki” idea is one that I’ve mentioned before in those discussions.
I’ll need to think a bit more before I can come up with constructive feedback and suggestions, however.
Posted by: Mike Shulman on September 11, 2009 1:46 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
To see the context of the Russell quotation see here.
Posted by: David Corfield on September 11, 2009 8:28 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Could you please summarize the relevant context of the Russell quotation, for the benefit of the European readers who (because of unclarified copyright issues) cannot see any Google book pages?
Posted by: Arnold Neumaier on September 11, 2009 5:56 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I’ve recently been playing around with formal proof assistants like HOL, Isabelle, Mizar, etc., and I have definitely been struck by their insistance on building everything from the ground up. […] the way we really do mathematics is to start by setting up some theory, i.e. “postulating” some structure and axioms, and then we prove things based on that structure and axioms.
You certainly can do this sort of thing in formal proof assistants (Coq is the one that I know best). Of course you know that you can since they are Turing-complete, but Coq (at least) has support for doing this naturally, with commands like Assume and Hypothesis.
On the other hand, Coq comes with its own foundations of mathematics (the Calculus of Inductive and Coinductive Constructions), and if you want to use one that doesn't match theirs, then you not only have to write it yourself (not too hard) but choose terminology that doesn't conflict with what Coq already thinks a ‘Set’ is. If FMathL is designed with more flexibility in mind, then that would be a Good Thing.
Posted by: Toby Bartels on September 11, 2009 5:56 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Toby Bartels wrote:
Coq comes with its own foundations of mathematics (the Calculus of Inductive and Coinductive Constructions), and if you want to use one that doesn’t match theirs, then you not only have to write it yourself (not too hard) but choose terminology that doesn’t conflict with what Coq already thinks a Set’ is. If FMathL is designed with more flexibility in mind, then that would be a Good Thing.
FMathL has a concept of nested contexts in which reasoning happens; these contexts can be defined, opened, modified, and closed, according to established informal practice, just slightly formalized.
The intention is to take care of all common practices of mathematicians. Knowing that mathematicians freely use the same names for different concepts, depending on what they do (variable name conventions may even vary within the same document), FMathL will allow one to redefine everything (if necessary) by creating appropriate contexts.
The outermost context is always the standard FMathL context with its axioms, but in nested contexts one can override surface conventions (language constructs denoting concepts, relations, etc.) valid in an outer context in a similar way as, in programming, variable names in a subroutine can be chosen independent of variables in the calling routine. Internally, however, all this is disambiguated, and concepts are uniquely named.
One must also be able to define one’s own syntax for things, just by saying somewhere in the mathematical text things like “We say that (x,y) sign z word w if formula involving x,y,z,w”, overriding old, conflicting uses of sign and/or word valid in an outer contexts. (This creates high demands on the parser; we currently study how to meet this challenge.)
Such overloading of meanings or syntax is not really recommended, though, in the interest of transparency. Standard mathematics, i.e., what undergraduates should be able to understand without confusion and with limited effort, will be handled with a minimum of such artifices. (There are a number of ambiguities in traditional terminology and notation, which we hope to handle in some “natural” way, though we haven’t yet decided exactly how.)
Posted by: Arnold Neumaier on September 11, 2009 8:31 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
FMathL has a concept of nested contexts in which reasoning happens; these contexts can be defined, opened, modified, and closed, according to established informal practice, just slightly formalized.
OK, Coq has these too (called ‘Sections’); I think that Isabel's ‘locales’ are the same idea.
The outermost context is always the standard FMathL context with its axioms,
How strong are these? I would find it very nice if even the basic foundations were highly modular.
but in nested contexts one can override surface conventions (language constructs denoting concepts, relations, etc.) valid in an outer context.
Coq does not allow one to rename or redefine things within a Section, although it does allow one to rename things within a Module, which is basically like a Section except that it's stored in a different file. (The idea is that one introduces a Section for temporary convenience within a single document, but a Module should be reasonably self contained and is intended to be used by many different people.) It might be nice if FMathL is a little more forgiving than Coq about this.
One must also be able to define one’s own syntax for things, just by saying somewhere in the mathematical text things like “We say that (x,y) sign z word w if formula involving x,y,z,w”, overriding old, conflicting uses of sign and/or word valid in an outer contexts. (This creates high demands on the parser; we currently study how to meet this challenge.)
Coq allows this too (I mean symbols, since I've already dealt with redefining the things themselves), but people don't like to use it, since specifying the right order of operations is an annoying technicality. If you can program the parser to figure that out for us, that would be nice … if it's even possible.
Sorry to say a lot of ‘Yeah, the program that I know can already do all of this.’. The thing is, there are a lot of ways that people have developed to formalise rigorous mathematics on a computer (such as all of the ones that Mike mentioned), but none of them have caught on with practising mathematicians, so if you can create something with a better design, then that's good! There's nothing that anyone can point to and say ‘Instead of developing FMathL, just use that; it's good enough.’, because nothing is good enough yet.
Posted by: Toby Bartels on September 11, 2009 9:10 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
TB: There’s nothing that anyone can point to and say “Instead of developing FMathL, just use that; it’s good enough.”, because nothing is good enough yet.
Look at the links in FMathL - Formal Mathematical Language to see what I had already looked at before realizing the need (and a realistic possibility) to do it all in our Vienna group. Nothing that exists is easy to use, nothing looks like ordinary math, nothing attracts typical mathematicians.
I’d have preferred not to have to develop such a system myself. But it will not come without mathematicians playing a leading role in its development. They do not even do small, easy things such as How to write a nice, fully formalized proof, that would make things more readable without much effort.
Posted by: Arnold Neumaier on September 11, 2009 10:21 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
One must also be able to define one’s own syntax for things, just by saying somewhere in the mathematical text things like “We say that (x,y) sign z word w if formula involving x,y,z,w”,
Coq allows this too (I mean symbols, since I’ve already dealt with redefining the things themselves)
And Isabelle has something it calls “mixfix,” which seems to be along the same lines.
Posted by: Mike Shulman on September 12, 2009 5:08 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
AN: The outermost context is always the standard FMathL context with its axioms
TB: How strong are these? I would find it very nice if even the basic foundations were highly modular.
As Bourbaki, FMathL assumes classical logic and the global axiom of choice, but, weaker than Bourbaki, only the ability ot form the set of all subsets of the continuum, since this is sufficient to be able to reflect the whole FMathL conception inside itself and prove some natural properties. See Logic in context. More can be added in the standard way by making assumptions.
Since one must be able to use FMathL as a comfortable metalevel, intuitionistic logic is inadequate. Even treatments of intuitionistic logic usually use on the metalevel classical reasoning. (I’d like to know of a book that doesn’t, if such a book exists!)
The FMathL axioms are assumed on the specification level. But one can decide to work in a reflection level (one layer below the specification level), where one can define one’s own axioms and inference rules in a completely free way. Since FMathL will be reflected itself, in a number of partially nested, partially independent contexts, one can just take the part of the FMathL specifications one is happy with and augment it in one’s own way.
TB: specifying the right order of operations is an annoying technicality. If you can program the parser to figure that out for us, that would be nice … if it’s even possible.
There are no intrinsic difficulties; it is just a matter of getting the parser to do it correctly. This means that one needs to automatically generate the new grammar and update the parser accordingly. Thus we will provide a way easy for the user, but since we haven’t fixed the structure of our grammar yet (we need a grammar that works well in an incremental mode and can handle ambiguities and attributes), it will take some time before we can consider in detail, how.
Posted by: Arnold Neumaier on September 14, 2009 2:56 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I certainly can’t give constructive feedback on this amount of work, but while reading the (27) axioms I was thinking to myself:
are these axioms fixed and transcendent like in a true axiomatic framework
or
can these axioms be changed within FMathL (once it is working), a flexibility one might want to have when doing mathematics? Is FMathL its own metatheory?
Posted by: Uwe Stroinski on September 11, 2009 8:16 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
or can these axioms be changed within FMathL (once it is working), a flexibility one might want to have when doing mathematics? Is FMathL its own metatheory?”
—————————————-
“The mathematical framework (at present just a formal system – a kind of metacategory of all categories) is designed to be a formal framework for mathematics that will allow…”
————————————-
Neumaier wrote in “A Semantic Turing Machine.pdf
“Besides serving as a theoretical basis of all programming languages (λ-calculus being another), Turing machines have many interesting applications, reaching from problems in logic, e.g., the halting problem c.f. Odifreddi [11], to formal languages (see Cohen [2]).”
———————————-
SH: The halting problem (HP) applies to a formal language such as Lisp for instance. It seems to me that Neumaier should have explained why the halting problem could have no adverse impact on the USTM he proposes and my first thought was that the USTM would either be incomplete or inconsistent. Since Neumaier provides references regarding the HP, perhaps he has considered this. Another statement which troubled me was this,
…”or in other words, not every STM program, regarded as a function on the context, is Turing computable. For example, external processors might have access to the system clock etc..”
SH: This is a difference in architectures, but I learned that every computable process which runs on a PC is Turing computable and that has nothing to do with a PC having a system clock. A TM and a PC can compute exactly the same range of calculations (power) except because the TM is an ideal machine, it could compute more digits of Pi and similar cases for instance, because there are no physical constraints (time and memory). PCs have the physical computability limitations but both the TM and PC compute the same kind of effective procedures called Turing computability.
Posted by: Stephen Harris on September 11, 2009 11:35 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Steven Harris wrote:
It seems to me that Neumaier should have explained why the halting problem could have no adverse impact on the USTM he proposes.
It poses the same problems as for an ordinary computer. To deserve its name, the USTM (universal semantic Turing machine) halts iff the program it simulates halts. More is not needed.
The halting problem for a semantic Turing machine poses no problems for mathematics done in FMathL, except that some searches for a proof (or other searches) may never terminate. But mathematicians also sometimes search for a proof without getting a result. Of course, FMathL will have, like mathematicians, an option to quit a search early if it seems hopeless.
Stephen Harris:
I learned that every computable process which runs on a PC is Turing computable and that has nothing to do with a PC having a system clock.
This holds only if there is no external input. A clock, or a human being who types in a reply to a query, are not computable, at least not in any well-documented sense. But the result of the program depends on that input and hence is generally not computable either.
Posted by: Arnold Neumaier on September 11, 2009 1:25 PM | Permalink | Reply to this
### How many axioms may a foundation have?
US: …while reading the (27) axioms…
This is an advance over ZF, which needs infinitely many to express the existence of sets defined by properties. (NBG is finitely axiomatized, though.)
Posted by: Arnold Neumaier on September 14, 2009 12:19 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Uwe Stroinski wrote:
while reading the (27) axioms I was thinking to myself: are these axioms fixed and transcendent like in a true axiomatic framework, or can these axioms be changed within FMathL (once it is working), a flexibility one might want to have when doing mathematics? Is FMathL its own metatheory?
The framework has fixed axioms, since it defined the common part of different subject levels. Within the framework, one can do arbitrary mathematics, using the terminology of the framework as a metalevel.
Thus, if you like and it matters to you, you’ll be able to define your own (e.g., intuitionistic) logic, your own version of sets (say Bishop-style), functions (say, terminating algorithms), and real numbers (say, oracles defining one decimal digit after the other), and then reason in the resulting system.
You’ll be able to arrange with a presentation style file that the printed version of your theory does not show a trace of your assumption but regards them as well-known background knowledge, or that is outlined, or that it is explained in detail.
For other users of FMathL, this will just look like a particular context that you created, one that those who want to build upon your work can include into a context of their own. You can create as many such contexts as you like, and include in your current context any other context that you want - but you are responsible for maintaining consistency.
But the FMathL axioms were selected in such a way that, for most mathematics outside set theory and mathematical logic, one does not need these private contexts but can just work in the standard context which satisfies the FMathL axioms. Then one adds definitions and results from the desired fields until one has enough to do one’s own work.
At least the usual linear algebra, real and complex analysis, and elmentary algebra will be in standard contexts; if you can read German, you may look at
http://www.mat.univie.ac.at/~neum/FMathL/ALA.pdf
to see the possible content of such a standard context.
Posted by: Arnold Neumaier on September 11, 2009 1:07 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I use and develop mathematical formulae and relationships in computer vision (which obviously has a different flavour from both more “full theory” and “proof based” areas), but I’d say one important element of your new system should be: heed the lesson of internet search (google, etc) and design things in such a way that simple, brute force search is possible. To expand on that, they staggeringly amazing thing about even sophisticated search engines is that what the algorithms they use are SO elementary relative to what an actual human would do (although they’re sophisticated in their own way) they generally manage to produce search results which quite often help you on your task (even if only to help refine the vocabulary you use to express your goal). Likewise, one can do reasonably well at various tasks just using relatively dumb program scripts that churn away on some big database (eg, I remember something in the paper Scientific American about formalising biosciences paper results just enough to be able to do “brute force” connections between multiple paper’s results to suggest new things to try; unfortunately it appears SA’s paywall stops me finding a reference).
I know your primary focus is unambiguously communicating well-formed results with some level-of-trust certificate in a human centred way, but don’t design things in such a way that programs can’t get in at the contents in non-standard ways. (In particular, definitely make it possible to access individual statements in a “document” in the database directly if they fit some “pattern” (some mathematics oriented variant of a regular expression) without having to go through everything in the docuement, or having to commit to an axiom scheme, etc.) In one way it’s galling that brute force (rather than careful thought) can actually achieve so much, in another it’s exhilarating that new relationships and hints at entities may be uncovered by such means (in an analogous way to the monstrousness of the discovery of the Monstrous moonshine relationship).
Posted by: bane on September 11, 2009 1:44 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
bane: definitely make it possible to access individual statements in a “document” in the database directly if they fit some “pattern” (some mathematics oriented variant of a regular expression)
What kind of patterns would you like to search for in a math text? How would you like to pose such a query? How insensitive should the search be to details in formulas? Please give some telling examples.
Some sort of search will certainly be possible. But trying Wolfram|Alpha shows that even simple searches for mathematical patterns are difficult for today’s technology.
FMathL will have to rely for search, automatic proof, and other well-studied techniques on what others can do. We need to concentrate our efforts in order to have real impact. Therefore, FMathL is intended to be innovative mainly in the things that reside in so far neglected areas of relevance to mathematics for the computer, and otherwise just interfaces to known systems.
Thus I don’t know how far we’ll be able to proceed in the direction of structural search.
Posted by: Arnold Neumaier on September 11, 2009 8:51 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Thanks for your earlier clarification of my questions. Peter Schodl provided this rough summary at http://www.mat.univie.ac.at/~schodl/
“Roughly speaking, we want to teach a computer to understand a LaTeX-file well enough to communicate its essential content to other software. Firstly, we concentrate on mathematical text specifying optimization problems.”
————————————–
SH: I think the issue of not all blogging software being latex cross-compatible has come up on this forum. Does FMathL solve this problem or would the blogging software still need to be changed?
Fair use excerpt from my copy of “Introduction to Mathematical Philosophy”
By Bertrand Russell
“From the habit of being influenced by spatial imagination, people have
supposed that series _must_ have limits in cases where it seems odd if
they do not. Thus, perceiving that there was no _rational_ limit to the
ratios whose square is less than 2, they allow themselves to “postulate”
and _irrational_ limit, which was to fill the Dedekind gap. Dedekind in
the above-mentioned work, set up the axiom that the gap must always be
filled, i.e. that every section must have a boundary. It is for this
reason that series where his axiom is verified are called “Dedekindian.”
But there are an infinite number of series for which it is not verified.
The method of “postulating” what we want has many advantages; they are
the same as the advantages of theft over honest toil. Let us leave them
to others and proceed with our honest toil.”
Posted by: Stephen Harris on September 12, 2009 12:10 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I expressed things badly: what I meant was “definitely don’t make any design decisions that will make it impossible (for others to write software)to search…”, not that your project should actively provide search technology, particularly if that’s not an area of direct interest. I say this because of two “facts of life”:
(1) when designing GUI based software for humans it’s easy to choose programming constructs and data representations so that they are effectively only usable with the GUI.
(2) If something becomes popular it will require a significant functionality increase to displace it, and even if it does get displaced it’s very rare for old “documents” to get properly converted. Eg, the TeX language was “set in stone” in 1983 (AIUI), and most of the core LaTeX “language” by about 1988 (ie, new user level commands not “implementation”). Various people, including Wolfram Research, have tried to displace it and none have gathered significant marketshare. I don’t expect TeX/LaTeX to be displaced until someone figures out inferring pen-based mathematical writing. Yet LaTeX doesn’t provide ways of indicating even simple semantic information, such as distinguishing eqnarrays with multiple = signs based upon whether they represent cases in a definition or steps of simplification, etc. (I know it could, but it doesn’t and I doubt such a feature could be “made” to be used by everyone at this stage.)
So basically all I’m saying is: imagine both that your project is technically successful and that it becomes very widespread and the world will have 20+ years of “documents” in this format, are there any design decisions that you’d come to regret.
I’ve got to go out now, but I’ll try and think of some concrete search examples and post later.
Posted by: bane on September 12, 2009 3:19 PM | Permalink | Reply to this
### design decisions that you’d come to regret.
bane: imagine both that your project is technically successful and that it becomes very widespread and the world will have 20+ years of “documents” in this format, are there any design decisions that you’d come to regret.
If I’d know it today, I’d certainly try to avoid it. In any case, the advantage of a fully semantic description of a subject matter in a self-reflected environment such as FMathL is designed to be makes it a fully automatic task to upgrade the whole database to a new representation.
Writing a program for doing so would be much easier than writing a program that upgrades LaTeX (which has an ill-defined semantics) to a different environment.
Posted by: Arnold Neumaier on September 14, 2009 4:09 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I’m just starting to get into the description of FMathL; here are some initial thoughts.
I think it is misleading, as is done in the introduction to the description of FMathL, to conflate CCAF and ETCS. CCAF (the Category of Categories As a Foundation for mathematics) is, in my opinion, a convoluted setup, since in order to do any mathematics, you need a notion of set, so before you can get anywhere you first have to define sets in terms of categories. ETCS (the Elementary Theory of the Category of Sets), on the other hand, is a set theory, not a “version” of CCAF. The axioms of CCAF are about things called “categories” and “functors,” while the axioms of ETCS are about things called “sets” and “functions” (among those axioms being that sets and functions are the objects and morphisms of a category, the titular category of sets).
I like to state the different between ETCS and ZFC by saying that ETCS is a “structural” set theory while ZFC is a “material” set theory. In my (biased) opinion, structural set theories solve the interpretation/implementation problems of material set theories, and are also more in line with mathematical practice (for instance, they do not permit nonsensical questions such as whether $1\in\sqrt{2}$ or whether $\pi$ is equal to the Cayley graph of $F_2$). Also, structural set theory is closely related to type theory, which is used by some existing proof assistants like HOL and Isabelle. So my question is, what is the advantage of FMathL over structural set theory or type theory as a foundation?
(By the way, while it is true that Lawvere’s original description of ETCS used the single-sorted definition of a category so that “every set is regarded as a mapping,” this (mis)feature is easily discarded (and usually is, in practice). By contrast, the feature of ZFC by which a function is a particular type of set really seems essential to the development of the theory. So I don’t think it is fair to speak of the two in the same breath as reasons that existing foundations are inadequate.)
Posted by: Mike Shulman on September 12, 2009 6:44 AM | Permalink | Reply to this
### CCAF, ETCS and type theories
MS: I think it is misleading, as is done in the introduction to the description of FMathL, to conflate CCAF and ETCS. CCAF (the Category of Categories As a Foundation for mathematics) is, in my opinion, a convoluted setup, since in order to do any mathematics, you need a notion of set, so before you can get anywhere you first have to define sets in terms of categories. ETCS (the Elementary Theory of the Category of Sets), on the other hand, is a set theory, not a “version” of CCAF.
I hadt’t called ETCS a version of CCAF. It is a part of CCAF, and I discussed it as such.
But the situation is not as simple as you describe it.
Any metatheory needs a concept of sets or collections, in order to speak about the objects, properties, and actions it is going to define on the formal level. And a metatheory that may serve as a foundation of mathematics must be able to model itself by reflection.
Thus you cannot have ETCS first without having categories even “firster”. There are four possible scenarios for foundations involving categories:
(I) Start with informal sets, create a formal definition of sets, and from it a formal definition of categories.
(II) start with informal sets, create a formal definition of categories, and from it a formal definition of set.
(III) start with informal categories, create a formal definition of categories, and from it a formal definition of set.
(IV) start with informal categories, create a formal definition of sets, and from it a formal definition of categories. ZFC + inaccessible cardinal + conventional category theory realizes (I). conventional category theory + ETCS realizes (II). CCAF (with ETCS built in) realizes (III). ETCS + conventional category theory realizes (IV).
The spirit of categorial foundations requires (III), I think, which needs both CCAF and ETCS.
MS: ETCS is a “structural” set theory while ZFC is a “material” set theory. In my (biased) opinion, structural set theories solve the interpretation/implementation problems of material set theories, and are also more in line with mathematical practice (for instance, they do not permit nonsensical questions such as whether $1\in\sqrt{2}$
They only replace one interpretation/implementation problem by another. Sets of sets cannot be formed in ETCS, but are very common mathematical practice. Thus categories are too “immaterial” to be useful as background theory.
MS: So my question is, what is the advantage of FMathL over structural set theory or type theory as a foundation?
Mathematics is type-free; so is FMathL. Mathematics does not have a type element and a type set (as in ETCS). At best there is a very weak typing that interprets membership in an arbitrary set as a type. (The development of type theory goes in this direction, too, e.g., with dependent types; but as it does so, types become less and less distinguishable from sets.)
FMathL models the actual practice, and does not compromise to gain formal conciseness. The elegance of common mathematical language lies in its power to be expressive, short, and yet fairly easily intelligible, which is in stark contrast to the many different type theories I have seen.
MS: “every set is regarded as a mapping,” this (mis)feature is easily discarded (and usually is, in practice).
True, but the existing versions of CCAF still differ a lot, so that the general conclusion about the categorial foundations is justified. The paper was not meant to give a fair discussion of the relative merits of the existing foundations, but just to point out that neither is adequate for the practice of mathematics.
MS: By contrast, the feature of ZFC by which a function is a particular type of set really seems essential to the development of the theory.
In the usual expositions, yes. But not intrinsically. ZFC is essentially equivalent with NBG, which was first formulated by Von Neumann (the N in NBG) as a theory in which everything was a function, and sets were special functions. It is easy to create equivalent versions of NBG and ZFC in which functions and sets are both fundamental concepts related by axioms that turn characteristic functions into sets and graphs of functions into functions.
The possibility of these apparent differences that do not lead to essential differences in the power of the theory shows that these are matters of implementation, and not of essence. FMathL intends to capture the latter.
Posted by: Arnold Neumaier on September 14, 2009 4:01 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
I hadn’t called ETCS a version of CCAF.
Sorry, I misinterpreted one of your sentences.
It is a part of CCAF
I disagree. One may regard it as a part of a general universe of “categorial foundations of mathematics,” but I think CCAF has a very specific meaning which is distinct from ETCS (e.g. Lawevere, “The category of categories as a foundation for mathematics”). ETCS does not require (or, usually, have anything to do with) CCAF, while the development of math from CCAF does not necessarily go through ETCS.
Any metatheory needs a concept of sets or collections, in order to speak about the objects, properties, and actions it is going to define on the formal level.
Well, any metatheory at least needs a logic. Most theories such as ZFC, ETCS, CCAF, etc. are formulated in first-order logic. If one then wants to talk about models for that logic, then one needs a “place” in which to consider such models, which will generally involve a notion of set/collection.
The spirit of categorial foundations requires (III), I think, which needs both CCAF and ETCS.
Well, maybe. But if that’s so, then I would not argue in favor of “categorial foundations” (a phrase I generally do not use). What I’m proposing as “structural set theory” is, I think, your (IV). Note, though, that ETCS does not require a prior (formal or informal) definition of category; it can be stated in pure first-order logic.
Mathematics is type-free
When I look at mathematics, I see types everywhere. What is the objection to $1\in\sqrt{2}$ if not a type mismatch?
By the way, I observe that $1\in\sqrt{2}$ is in fact a legal statement in FMathL. Isn’t that exactly the sort of “extraneous information” that’s problematic about ZFC? I don’t actually see how FMathL solves any of the problems of material set theory.
Here’s what I see when I look at mathematics as it is done by mathematicians:
• Mathematical structures (groups, rings, topological spaces, manifolds, vector spaces) are built out of sets and functions/relations between these sets. From the point of view of the general theory of any such structure, the elements of such sets have no internal structure; they are featureless. In particular, the general theory of a type of structure is invariant under isomorphism.
• Elements of sets can have “internal meaning” relative to elements of the same set or other sets, as specified by functions and relations. For instance, the elements of a cartesian product $A\times B$ are interpreted as pairs $(a,b)$ relative to the sets $A$ and $B$, with the relationship specified via the projection functions $A\times B\to A$ and $A\times B\to B$. But once we start thinking about $A\times B$ as an object in its own right without its relationship to $A$ and $B$, its elements lose their “internal” properties and become featureless, like the elements of any other set.
For instance, when we construct $\mathbb{Q}$ as a set of ordered pairs of integers, we consider a subset of $\mathbb{Z}\times\mathbb{Z}$ and use the interpretation of its elements as pairs to construct operations and properties of it. However, once the construction is finished, we generally forget about it and treat each rational number as an independent entity.
I think this property (that structure on a set comes only from the outside) is precisely what allows us to capture “the essence of mathematical concepts;” otherwise we will always be carrying around baggage about the internal properties of the elements of our sets.
Sets of sets cannot be formed in ETCS, but are very common mathematical practice.
I would also argue that the above interpretation of cartesian products also applies to “sets of sets.” The elements of a power set $P A$ are given meaning as subsets of $A$ only via the “is an element of” relation from $A$ to $P A$, just as the elements of $A\times B$ are given meaning as ordered pairs only via the projections to $A$ and $B$. Once we forget that $P A$ was constructed in relation to $A$, the meaning of its elements as subsets of $A$ vanishes. This is the case, for instance, when we construct the real numbers via Dedekind cuts: $\mathbb{R}$ starts out as a subset of $P \mathbb{Q}$, just as $\mathbb{Q}$ started out as a subset of $\mathbb{Z}\times\mathbb{Z}$, but once we have proven enough about it we generally forget about the identification of a real number with a set of rationals and treat it as an independent entity.
It is true that mathematicians usually think of the elements of $P A$ as “being” subsets of $A$, rather than being merely “associated” to them. But this is perfectly in line with the structural philosophy that anything and everything can be transported along an isomorphism: every subset of $A$ corresponds to a unique element of $P A$, so we might as well consider elements of $P A$ to “be” subsets of $A$ (as long as we continue to keep the “is an element of” relation in mind). But the structural point of view also allows us to discard the “is an element of” relation, which we can’t do in any system where the elements of $P A$ really are subsets of $A$.
By the way, this transport along a bijection is, I think, easily hidden from the user of any computer system. After all, what does one do with subsets of $A$? The basic thing is to talk about which elements of $A$ are elements of them—and this is handled directly by the “is an element of” relation. All other operations and properties on elements of $P A$ are most naturally defined in terms of “is an element of,” so the user won’t ever be aware that the elements of $P A$ “aren’t really” subsets of $A$.
The elegance of common mathematical language lies in its power to be expressive, short, and yet fairly easily intelligible, which is in stark contrast to the many different type theories I have seen.
I think type theory (along with many other kinds of formal logic) should be viewed as like assembly language. Hardly anyone writes code in assembly language, but not because it has been replaced; rather because more expressive, concise, and intelligible languages have been built on top of it. The description of FMathL seems to me like trying to build a CPU that natively runs on Fortran, rather than writing a Fortran compiler. Logicians have put a lot of effort into developing a theory that works very smoothly at a low level and is very flexible; why not build on it instead of pulling it out and replacing it wholesale?
By the way, one of the things I like about Isabelle is that it is designed with a very weak metalogic, on top of which arbitrary other logics can be implemented. People usually use either HOL or ZFC as the “object logic” but in theory one could use any logic that can be expressed via natural deduction rules. I think there are good reasons to build in this sort of modularity at a low level, just as there are good reasons for hiding it from the end user.
the feature of ZFC by which a function is a particular type of set really seems essential to the development of the theory.
In the usual expositions, yes. But not intrinsically.
You’re right.
(Regarding “sets being mappings,” it sounds like your real objection is the absence of “sets of sets,” which I’ve addressed above.)
Posted by: Mike Shulman on September 14, 2009 5:18 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
AN: It [ETCS] is a part of CCAF
MS: I disagree. One may regard it as a part of a general universe of “categorial foundations of mathematics,” but I think CCAF has a very specific meaning which is distinct from ETCS.
Distinct, yes, but including it; I only claimed that. I referenced C. McLarty, Introduction to Categorical Foundations for Mathematics, version of August 14, 2008. On p.55, he states as part of the CCAF axioms: There is a category Set whose objects and arrows satisfy the ETCS axioms.
Without such an axiom, or some replacement (as in Makkai’s version also referenced) there are no sets, and hence no complete categorial foundations for mathematics.
AN: Mathematics is type-free
MS: When I look at mathematics, I see types everywhere.
When I look at mathematics, I see structures everywhere, not types. Trying to see types, I see that typing violations abound, being justified as abuse of notation.
MS: What is the objection to $1\in\sqrt{2}$ if not a type mismatch?
The objection is that different constructions of the reals (considered equivalent by mathematicians) answer the question differently.
MS: I observe that $1\in\sqrt{2}$ is in fact a legal statement in FMathL.
Yes, but it is undecidable. It should be, since people who implement mathematics in ZFC, say, may get different (subjective, implementation-dependent) anwsers. Thus the formal status of this statement should be the same as that of the the continuum hypothesis, say.
MS: Mathematical structures (groups, rings, topological spaces, manifolds, vector spaces) are built out of sets and functions/relations between these sets. From the point of view of the general theory of any such structure, the elements of such sets have no internal structure; they are featureless. In particular, the general theory of a type of structure is invariant under isomorphism.
Earlier in my life I had been working and publishing on finite symmetry groups ($E_8$, the Leech lattice, etc.). Everyone in this field was obviously regarding any permutation group as a group. Alt(5) acting on 5 points and PSL(2,5) acting on 6 points were nonisomorphic permutation groups, but isomorphic as groups. What was meant by ”the same” was context-dependent, taking into account more or less structure, as needed.
FMathL respects this context-dependence in a natural way; the meaning of an asserted equality depends on the context. This feature allows FMathL to remain much closer to actual practice than previous foundations.
In category theory (and in ZFC), permutation groups and groups are completely different objects, related by functors (as you describe). And much more of that, which must be handled in the traditional foundations by a pervasive misuse of notation and language. In my view, what is regarded by the purists as “misuse” is the true usage: mathematicians generally think in this falsely called misused language rather than in the clumsy purist way. The functors only appear when one forces standard mathematics into the straightjacket of category theory.
In categorial language, the harmless statement $\sqrt{2}\in \mathbb{R}$ (where $\mathbb{R}$ denotes the ordered field of real numbers) would as much be a type mismatch as the dubious statement $1\in\sqrt{2}$. Writing out all the functors needed to match types would make ordinary mathematical language as clumsy to use as proof assistants based on type theory.
MS: I think type theory (along with many other kinds of formal logic) should be viewed as like assembly language. Hardly anyone writes code in assembly language, but not because it has been replaced; rather because more expressive, concise, and intelligible languages have been built on top of it.
I fully agree. This is why FMathL distinguishes between subjective levels (where particular implementations sit, like different assembler programs for the same functionality, written perhaps even in different assembler languages), and the object level, which gives the essence.
MS: The description of FMathL seems to me like trying to build a CPU that natively runs on Fortran, rather than writing a Fortran compiler.
No. The current description of FMathL is that of a language, not of a CPU. (The CPU’s correspond to the subject levels in the FMathL paper.) In your picture, FMathL tries to be a formal, easy-to-use high-level programming language, while traditional foundations are low-level assembler languages.
MS: Logicians have put a lot of effort into developing a theory that works very smoothly at a low level and is very flexible; why not build on it instead of pulling it out and replacing it wholesale?
The full system MathResS (which uses FMathL as its high level semantics) will build upon it. There will be compilers from the high level to various low levels, i.e., interfaces between FMathL and Coq, Isabelle/Isar, or HOLlight, say. (This doesn’t show in the paper on the FMathL mathematical framework, but is amply reflected in the vision document.)
But past work on proof assistants etc. produced only the assembler level, not a high level comparable to Fortran that would make writing formal mathematics easy. The high level currently only exists as informal mathematical language.
Imagine there were no Fortran or C++, and all programs would have to be specified in ordinary language plus formulas, to be translated by specialists directly into assembler. Computer science would be as little accepted by the world at large as proof assistants are by mathematicians.
The purpose of FMathL is to create a Mat-tran for translating high level mathematics that is as easy to use as modern For-tran for translating high level algorithms.
Posted by: Arnold Neumaier on September 15, 2009 11:38 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Arnold Neumaier wrote about $1 \in \sqrt{2}$
Thus the formal status of this statement should be the same as that of the the continuum hypothesis, say.
I disagree with that. The generalised continuum hypothesis $GCH$ is a meaningful statement, at least as long as one is doing mathematics with enough power to recursively define families of sets indexed by the ordinals using the power set operation. Then you can prove useful things, such as $V = L \;\Rightarrow\; GCH$ and $GCH \;\Rightarrow\; AC$.
To make $GCH$ come out true or false, you need to make certain assumptions beyond the standard ones, and these should be regarded as a matter of convention; similarly, to make $1 \in \sqrt{2}$ come out true or false, you need to adopt some convention beyond the standard ones. But you need to adopt such a convention even to make $1 \in \sqrt{2}$ meaningful, which is not necessary for the continuum hypothesis, or to prove the theorems in the las paragraph.
If you only want to talk about true statements (and thus also false ones, through their negations), then you can treat these similarly; you will be unable to prove or refute either in the standard context, but you will be able to prove or refute it in some other contexts. But I would like a system to complain of a type mismatch the moment that I write down $1 \in \sqrt{2}$, until I've adopted conventions that make it meaningful; I don't want to have the statement accepted, just considered neither proved nor refuted yet.
All that said, I appreciate your point that mathematicians use a high-level language in which a host of standard type conversions are suppressed, and it will be very useful to have a formal system that already knows about all of this. Still, I would like a distinction between ‘meaningless’ and merely ‘undecidable’ statements in any given context.
Posted by: Toby Bartels on September 16, 2009 2:10 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TB: you need to adopt such a convention even to make $1\in\sqrt{2}$ meaningful, which is not necessary for the continuum hypothesis,
The statement is equally meaningful in ZF with the standard add-ons for numbers as is the continuum hypothesis.
TB: I would like a system to complain of a type mismatch the moment that I write down $1\in\sqrt{2}$ until I’ve adopted conventions that make it meaningful; I don’t want to have the statement accepted, just considered neither proved nor refuted yet.
You can make the system complain by adding to your standard context the requirement that $x\in y$ is nominal for numbers $x$ and $y$. Similarly, the user can enforce any desired typing rules by specifying them.
But to have a system in which one can use ZF naturally, one cannot make the typed behavior the default. The default must be as much agnosticism as possible, while there must be simple ways for users to make the system more restrictive, to have it conform to the amount of typedness they want.
Posted by: Arnold Neumaier on September 16, 2009 8:18 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
The statement is equally meaningful in ZF with the standard add-ons for numbers as is the continuum hypothesis.
Yes, but in other systems it is meaningless, whereas the continuum hypothesis is meaningful in any system in which you have the syntax to write it down.
The default must be as much agnosticism as possible
I agree whole-heartedly, but I do not agree that this is what you are doing.
Agnosticism, to me, includes considering syntax meaningless unless it has been given some meaning. Phrases like ‘$_\pi(=^2($’ and ‘$1 \in \sqrt{2}$’ are meaningless to me until somebody explains to me what they mean.
In order of increasing knowledge: meaningless, undecidable, true. Or rather, it should be: meaningless so far, undecided so far, known to be true. The ‘so far’s are because additional assumptions/conventions can push things farther along in the list; the change from ‘undecidable’ to ‘undecided’ is because I know that your system, however great it might be (^_^), can't calculate whether something is undecidable in an arbitrary context.
You also, by default, include other assumptions that should be optional. All this, I suppose, to make ‘a system in which one can use ZF naturally’ out of the box. But if I think that $\mathbf{ZF}$ is a kludge, then that's not a feature for me. By all means, include a $\mathbf{ZF}$ option as a standard module, but making it the default is not ‘as much agnosticism as possible’.
there must be simple ways for users to make the system more restrictive
That is, more restrictive in typing, or less restrictive in assumptions made and conventions adopted. You do this through reflection, right? I'm inclined so far to prefer Coq (or Isabel), which also allows reflection but makes fewer assumptions up front. On the other hand, if you make this more user-friendly, then that would be a Good Thing.
Posted by: Toby Bartels on September 16, 2009 7:07 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Arnold Neumaier wrote
Earlier in my life I had been working and publishing on finite symmetry groups ($E_8$, the Leech lattice, etc.). Everyone in this field was obviously regarding any permutation group as a group. $Alt(5)$ acting on 5 points and $PSL(2,5)$ acting on 6 points were nonisomorphic permutation groups, but isomorphic as groups. What was meant by ”the same” was context-dependent, taking into account more or less structure, as needed.
FMathL respects this context-dependence in a natural way; the meaning of an asserted equality depends on the context. This feature allows FMathL to remain much closer to actual practice than previous foundations.
In category theory (and in ZFC), permutation groups and groups are completely different objects, related by functors (as you describe). And much more of that, which must be handled in the traditional foundations by a pervasive misuse of notation and language. In my view, what is regarded by the purists as “misuse” is the true usage: mathematicians generally think in this falsely called misused language rather than in the clumsy purist way. The functors only appear when one forces standard mathematics into the straightjacket of category theory.
In categorial language, the harmless statement $2 \in \mathbb{R}$ (where $\mathbb{R}$ denotes the ordered field of real numbers) would as much be a type mismatch as the dubious statement $1 \in 2$. Writing out all the functors needed to match types would make ordinary mathematical language as clumsy to use as proof assistants based on type theory.
It seems to me that a structural set theory such as ETCS also upholds the principle of context-dependence as fundamental.
Let’s take for example “$\sqrt{2}$”. In a structural set theory, it’s quite true that there is no object (taken in isolation) called $\sqrt{2}$. Instead, it is part and parcel of such an approach that we declare the ambient context in which $\sqrt{2}$ is embedded: $\sqrt{2}$ as real number, $\sqrt{2}$ as complex number, etc., by writing down for example
$\sqrt{2}: 1 \to \mathbb{R}$
Having declared the context, it is then meaningful in such a theory to ask, given a subset such as $\mathbb{Q} \subseteq \mathbb{R}$, a question like: is $\sqrt{2} \in \mathbb{Q}$? The question is equivalent to asking whether the point $\sqrt{2}: 1 \to \mathbb{R}$ factors (evidently uniquely) through the given inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}$. Or, whether the pair
$(\sqrt{2}, [\mathbb{Q}]): 1 \to \mathbb{R} \times P(\mathbb{R})$
factors through the local membership relation
$\in_{R} \hookrightarrow \mathbb{R} \times P(\mathbb{R})$
Thus, once such contexts have been declared, the proposition $\sqrt{2} \in_{\mathbb{R}} \mathbb{Q}$ becomes completely meaningful in a structural set theory like ETCS, and reflective of how mathematicians posit questions in actual practice.
Thus, symbol $\in$ is not formalized in a structural set theory as a global relation on objects; it too is contextualized (or localized) by referring to a specified domain, as in the case $\in_{\mathbb{R}}$.
Continuing this train of thought: where a mathematician might traditionally write
$\forall_{x \in \mathbb{R}} \exists_{y \in \mathbb{R}} x + y \in \mathbb{Q}$
a “purist” working in a formal setting like ETCS would disambiguate the different senses of the symbol $\in$, where the instances appearing below the quantifiers are interpreted as declaring the types or contexts of the variables, and the one in the predicate being quantified refers to a local membership relation (such as $\in_{\mathbb{R}}$) pertaining to that type. But I hardly feel like this detail is cumbersome: writing the expression
$\forall_{x: \mathbb{R}} \exists_{y: \mathbb{R}} x + y \in_{\mathbb{R}} \mathbb{Q}$
is certainly no worse than how it would appear in a fully formal expression in ZFC, and IMO comes closer to expressing how people think (which we seem to agree is heavily context-dependent).
Regarding phrases such as “straitjacket of category theory” and the final sentence of the material quoted above: these strike me as bald assertions. What is the evidence behind them?
Posted by: Todd Trimble on September 16, 2009 1:51 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
$\forall_{x\in\mathbb{R}}\exists_{y\in\mathbb{R}}x+y\in\mathbb{Q}$
vs
$\forall_{x:\mathbb{R}}\exists_{y:\mathbb{R}}x+_\mathbb{R}y\in_{\mathbb{R}}\mathbb{Q}$
You forgot the subscript on ‘$+$’. (^_^)
But there is need to abolish the first form in favour of the second. If one adopts structural foundations, then it is possible to algorithmically decode the first as meaning the second, as long as it appears in a context where $\mathbb{R}$ has been declared as a set equipped with an operation $+$, $\mathbb{Q}$ has been declared as a subset of $\mathbb{R}$ (which includes declaring it as a set and equipping it with an injection to $\mathbb{R}$, as you would be likely to do if you had earlier defined $\mathbb{R}$ in terms of $\mathbb{Q}$), the quantifiers have their usual meanings, and $x$ and $y$ are free to be introduced as variables. I would argue that the default context in ordinary mathematics has these features, so the first expression is unambiguous.
Posted by: Toby Bartels on September 16, 2009 7:18 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
And I’d agree with your so arguing, particularly with the fact that translation from the traditional notation to the “purist” one is a routine algorithm. But not everyone has thought about these nuances, so it’s perhaps just as well to spell them out on occasion.
You’ve actually further strengthened the argument for the viability of structural foundations as not being at all cumbersome.
Posted by: Todd Trimble on September 16, 2009 7:41 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
But there is need to abolish the first form in favour of the second.
Whoops!, that should be ‘no need to abolish’.
Hopefully that's clear from what I wrote afterwards, but I still should watch out for missing negations.
Posted by: Toby Bartels on September 16, 2009 7:55 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Actually, this disambiguation algorithm is implemented in modern proof assistants.
E.g. Coq would use type classes for this.
Posted by: Bas Spitters on September 25, 2009 8:02 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
AN: The functors only appear when one forces standard mathematics into the straightjacket of category theory. […] Writing out all the functors needed to match types would make ordinary mathematical language as clumsy to use as proof assistants based on type theory.
TT: Regarding phrases such as “straitjacket of category theory” and the final sentence of the material quoted above: these strike me as bald assertions. What is the evidence behind them?
These assertions were to be taken with a grain of salt. The statement on proof assistants was referring to the overhead incurred when one writes a piece of math in Coq, say, vs. in LaTeX - roughly a factor of 10 in time (this factor is Freek Wiedijk’s estimate, not my exaggeration). Adding all the categorial annotations needed to be able to rigorously say things one likes to say, and making sure everything is correctly in place, is not quite as expensive but still a significant nuisance, a straightjacket that encumbers writing and reading math.
TT: translation from the traditional notation to the “purist” one is a routine algorithm
Routine, but tedious, not much different from converting a lemma and its proof from LaTeX to Coq. It makes the difference between being widely accepted and being used only by afficionados.
I’ll answer to the remainder of your post once I know how to reproduce the formulas without recomposing them myself….
Posted by: Arnold Neumaier on September 16, 2009 8:56 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Routine, but tedious, not much different from converting a lemma and its proof from LaTeX to Coq. It makes the difference between being widely accepted and being used only by afficionados.
I am fully in agreement with this. However, what I’m trying to say is that I think the best solution is not to discard typing information altogether, but rather to automate the process of inferring and adding type annotations. We should be able to omit this type information when writing mathematics, and a computer should be able to infer it just as a human reader infers it, but the information is nevertheless there and should be modeled by the formal language.
Automating this conversion/inference sounds like much the same thing that you’ve said, regarding FMathL as a “higher level language” which can be “compiled” to Coq, Isabelle/Isar, or HOL. But I don’t see how an untyped language can be compiled to a typed one in a meaningful way. When a FMathL user types $1\in \sqrt{2}$, what does that get compiled to in a “lower level” typed language?
Posted by: Mike Shulman on September 17, 2009 2:05 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
MS: Automating this conversion/inference sounds like much the same thing that you’ve said, regarding FMathL as a “higher level language” which can be “compiled” to Coq, Isabelle/Isar, or HOL. But I don’t see how an untyped language can be compiled to a typed one in a meaningful way.
In a similar way as one can pose to HOL any query formulated in the untyped first order language ZF. It is well-known that systems like HOLlight can prove much of ZF theory; so, where is the problem?
MS: When a FMathL user types $1\in\sqrt{2}$, what does that get compiled to in a “lower level” typed language?
It gets compiled into a precise specification format that adds all relevant contextual information, in this case, that the user seems to have intended an interpretation of numbers as a set.
Most of real math involves such guessing of intentions, which is maintained until strange conclusions are automatically derived. In interactive mode, the formula would be found suspicious, and a query window would pop up to ask the user to confirm the interpretation, to correct the formula, or to supply additional context.
From the specification format, a dedicated spec2HOL translator would map the part of the problem to be checked for correctness into an appropriate query in HOL.
Posted by: Arnold Neumaier on September 17, 2009 3:11 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
In interactive mode, the formula would be found suspicious, and a query window would pop up to ask the user to confirm the interpretation, to correct the formula, or to supply additional context.
Oh, that makes me happy! (^_^)
And I take it that if, after developing a theory of ordinals, I were to write down the generalised continuum hypothesis (that the $\aleph$ series equals the $\beth$ series), then no such window would pop up?
Posted by: Toby Bartels on September 17, 2009 10:52 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TB: if, after developing a theory of ordinals, I were to write down the generalised continuum hypothesis (…), then no such window would pop up?
Then FMathL would just take notice that you added an assumption to your context. It would have made a few checks to see that one can rewrite the formula in a useful way (which cannot be done with $1\in\sqrt{2}$ since there are no rules for manipulating $\in$ between numbers) and conclude that things make enough sense not to bother the user with a query.
FMathL might become suspicious, though, when you subsequently add that the simple coninuum hypothesis is violated, and would ask whether you intended to close the previous context with the GCH, since otherwise the context becomes trivially inconsistent.
In noninteractive mode, it would write all the queries into a logbook that can be inspected after compilation, and proceed on a tentative basis. (As mathematicians do when they are not sure about the meaning of a cryptic passage.)
Posted by: Arnold Neumaier on September 18, 2009 12:06 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
FMathL might become suspicious, though, when you subsequently add that the simple coninuum hypothesis is violated, and would ask whether you intended to close the previous context with the GCH, since otherwise the context becomes trivially inconsistent.
That's good.
(Of course, we can have no guarantee that FMathL will notice that I'm in an inconsistent context, especially if I write CH as a statement about subsets of $\mathbb{R}$ instead of as a statement about $\aleph$s and $\beth$s. But the more contradictions that it can find in a reasonable amount of time, the better.)
Posted by: Toby Bartels on September 18, 2009 9:55 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TB: Of course, we can have no guarantee that FMathL will notice that I’m in an inconsistent context, especially if I write CH as a statement about subsets of ℝ
Yes. In particular, we might all be working in an inconsistent context without knowing it, like Cantor was for 25 years.
If FmathL discovers this and raises a query, I’d rather think of a bug in FMathL than of one in standard math…
Posted by: Arnold Neumaier on September 18, 2009 10:58 PM | Permalink | Reply to this
### Cantor vs Frege
In particular, we might all be working in an inconsistent context without knowing it, like Cantor was for 25 years.
You mean Frege; Cantor knew what he was doing, even though he didn't formalise it.
Posted by: Toby Bartels on September 18, 2009 11:07 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Do you mean that you interpret an untyped theory like ZF in a typed theory by having just one type? To me that seems unfaithful to how typed systems are intended to be used.
Posted by: Mike Shulman on September 18, 2009 9:22 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
MS: Do you mean that you interpret an untyped theory like ZF in a typed theory by having just one type? To me that seems unfaithful to how typed systems are intended to be used.
Do you see a better way? How would you pose a ZF problem to HOLlight, say?
I don’t think that there is any other option. In the The CADE ATP System Competition - The World Championship for 1st Order Automated Theorem Proving, people pose each year lots of untyped problems that they want to see solved (or were already solved) by some theorem prover. Do you think HOLlight should not be allowed to solve these?
On the other hand, if you have a naturally typed problem, the types would form in FMathL a particular family of constructive sets, and a translator to HOLlight would be able to recognize this and then make better use of the typing in the proof assistant.
Posted by: Arnold Neumaier on September 18, 2009 10:09 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
McLarty in Learning from Questions on Categorical Foundations says (on p. 51):
“If the major theorems of category theory are proved in set theory, and then I want to axiomatize them, is that not a kind of dependence on set theory? Well in the first place these theorems are not exactly proved in set theory. Their usual naïve versions are incorrect in set theory. They quantify over collections too large to be ZF sets, and manipulate them too freely for Gödel-Bernays classes, and treat them too uniformly for Grothendieck universes. There are many well-known and sufficiently workable set-theoretic fixes for handling these theorems but they are all just that—fixes.”
This suggests to me that there is no known natural way to code category theory in set theory. Is this a reasonable point of view? If it is, what’s a solution to this problem?
Posted by: Eugene Lerman on September 16, 2009 10:32 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
I don't think that I agree with McLarty there, and I'd like to know what in category theory he thinks can't be formulated in $\mathbf{ZFC} + GU$, where $GU$ is the axiom that every set belongs to some Grothendieck universe, about as easily as anything in ordinary mathematics is axiomatised in $\mathbf{ZFC}$. Categorists' language even tells you when the Grothendieck universes are coming in: whenever they say ‘small’ (or something that may be defined in terms of smallness, like ‘accessible’ or ‘complete’).
I certainly agree with McLarty's second point (not quoted above), so it doesn't matter for his overall argument. I also believe that formulating mathematics in $\mathbf{ZFC}$ is generally to perpetrate an act of violence upon it; I just don't see how category theory is particularly special in that regard.
Posted by: Toby Bartels on September 17, 2009 12:59 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Well, I wrote a whole paper about ways to formalize category theory in (mostly material) set theory. (By which I mean, of course, ways to deal with the size distinctions in category theory; the theory of small categories never poses any problem.) Of course, “natural” is in the eye of the beholder, but I don’t think there’s a whole lot to object to, at least in the more effective versions. Any of them can be “structuralized” to eliminate any objections on that score.
I think it’s a bit misleading to say that the theorems of category theory “are not exactly proved in set theory.” One has to choose a set-theoretic formalization, but there are a number which suffice perfectly well. And when it comes to it, hardly any mathematics is actually “proved in set theory”–it’s proved in informal mathematical language which we all trust could be translated into set theory (if we’re the sort of people who believe that all mathematics should be founded on set theory). I don’t think category theory is much different there; the only thing is that it matters a bit what set-theoretic foundation you choose, but that isn’t unique to category theory either.
Of course, set-theoretic foundations are perhaps not philosophically in line with category theory, which may be more along the lines of what McLarty is getting at. For instance, there is the problem of evil: as long as your categories have sets of objects, you’ll be able to talk about equality of those objects. But I don’t see any way to solve that unless your foundational axioms are really about the 2-category of categories; even the 1-category of categories isn’t good enough.
Posted by: Mike Shulman on September 17, 2009 1:58 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Todd and Toby have said most of what I would say as well. I agree that a lot of what is written in everyday mathematics does not naively typecheck, and also that such “abuses” of notation are not “wrong” but are the real usage. However, I don’t think this is a reason to discard types, which are, I still believe, everywhere in mathematics and carry important information. Rather, we should improve the type system.
Consider your example regarding $x\in \mathbb{R}$ not type-checking because $\mathbb{R}$ is not a set but a field. I would argue that a better way to describe this is that the symbol $\in$ is overloaded, in the precise sense of computer science. Thus, what appears on the RHS of $\in$ does not always have to be a set, but can be anything for which an appropriate semantics is defined. One could additionally supply a default semantics: when the RHS is a structure of some sort having only one underlying set, then the meaning of $\in$ should default to referring to that underlying set. This is completely precise and maintains the important type information. Moreover, it is already possible in Isabelle:
record 'a magma =
elements :: "'a set"
times :: "'a => 'a => 'a" (infixl "$\star$\<index>" 70)
definition in_magma :: "'a => 'a magma => bool"
where "in_magma x M = (x $\in$ elements M)"
notation in_magma ("_ $\in$ _")
axioms closed: "[| x $\in$ M ; y $\in$ M |] ==> (x$\star_M$y) $\in$ M"
(Of course, this isn’t the way one would actually work in Isabelle, I’m just trying to illustrate overloading of $\in$ with code that typechecks.)
Posted by: Mike Shulman on September 16, 2009 8:56 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Hmm, I guess this is actually more or less the same thing that Toby said above.
Posted by: Mike Shulman on September 16, 2009 9:42 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
MS: Consider your example regarding $x\in \mathbb{R}$ not type-checking because $\mathbb{R}$ is not a set but a field. I would argue that a better way to describe this is that the symbol $\in$ is overloaded, in the precise sense of computer science.
Yes, this is more reasonable. But this doesn’t solve all type problems. For example, Todd Trimble’s suggestion that “2 as a real number” and 2 as a complex number” are different objects with different types is really worrying This gives a multitude of distinct objects that most mathematician would consider to be identical, e.g., “2 as an element of $(\mathbb{N},suc)$”, “2 as an element of $(\mathbb{N},+)$”, “2 as an element of $(\mathbb{N},+,*)$”, “2 as an element of $(\mathbb{N},\lt)$”, “2 as an element of $(\mathbb{N},+,\lt)$”, “2 as an element of $(\mathbb{Z},+)$”, “2 as an element of $(\mathbb{Q},+,*)$”, to name only a few.
One gets a lot of accidental relations when formalizing numbers in ZF, which are avoided in category theory, but instead one gets a lot of accidental duplication. This shows that category theory is like ZF a useful mode of viewing mathematics, but that it does not capture its full essence.
Compare this with the way categories are defined in FMathL (Section 2.13). This is not quite the same as the traditional definition, but it is the same from the point of view of essence. In FMathL, objects and morphisms can belong very naturally to several categories.
Posted by: Arnold Neumaier on September 17, 2009 2:29 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Permit me then to try to assuage your worries! :-)
In ETCS (to give an example of a categories-based set theory, but there are ways of doing structural set theory, including one that Mike Shulman is working out in the nLab, called SEAR), a [global] element is conceived as a map of the form $1 \to X$; the $X$ can be thought of as the ‘type’. The context $X$ is given with the element as its codomain; it is a datum of that element. Thus “elements” are not free-floating or given in vacuo; they come already attached to sets.
In just the same way, “objects” are not free-floating either – they come attached to categories in which they are conceived as being objects of. Thus we have $\mathbb{N}$ qua set, $\mathbb{N}$ qua structure with 0 and successor, qua ordered abelian group, qua ring, etc. – these are all objects in different categories. But each of those structures you mentioned allow one to define an element $2: 1 \to \mathbb{N}$ in the underlying set, and insofar as all those structures on $\mathbb{N}$ you mentioned are standardly defined by exploiting the “Peano postulates” [in which the set $\mathbb{N}$ comes equipped with a 0 and a successor, satisfying a principle of primitive recursion or universal property as natural numbers object that makes the recursive definitions of each of these structures possible], it’s provably the same element 2 (that is, same morphism $1 \to \mathbb{N}$ in $Set$) we’re referring to in each of those cases.
Even if we have several natural numbers objects, there is an invariant meaning of ‘2’ in each of these, insofar as the unique isomorphism from one natural numbers object to another (as natural numbers objects) takes the element ‘2’ in one to the element ‘2’ in the other. I’m puzzled why any of this should be considered a problem.
Posted by: Todd Trimble on September 17, 2009 9:36 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
I meant to say $\mathbb{N}$ as “ordered commutative monoid”, not “ordered abelian group”. :-P
Posted by: Todd Trimble on September 17, 2009 9:42 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
I’m puzzled why any of this should be considered a problem.
Well, it's very different from the way that most mathematicians think about these things.
(And to handle $2 \in \mathbb{Z}$, $2 \in \mathbb{C}$, etc, you also have to talk about some injections between the sets before you can prove that various guises of $2$ are the same underneath.)
So while I agree with you that mathematics is like this underneath, I also agree with Arnold that it would be nice to have a user-friendly system where it doesn't explicitly look like that. I would like a that does have all of that underneath it, at some level, but hides it from the user, at least until they ask for more details.
Posted by: Toby Bartels on September 17, 2009 10:57 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
And to handle $2 \in \mathbb{Z}$, $2 \in \mathbb{C}$, etc, you also have to talk about some injections between the sets before you can prove that various guises of $2$ are the same underneath.
That goes without saying. In a categories-based set theory, such injections often arise as universal arrows (e.g., $\mathbb{N}$ is the initial rng, affording a unique comparison map $\mathbb{N} \to \mathbb{Q}$ in $Rng$). As you know, of course.
Could you say a little more what you mean by “it’s very different from the way most mathematicians think about these things”? It seems to me that, now that categorical ideas have seeped into the general consciousness, many mathematicians do in effect “think structurally” – simply put, that context matters – for example that 2 in $\mathbb{Q}$ is different from 2 in $\mathbb{Z}$ since the first 2 is invertible and the second isn’t, and that $2$ in $\mathbb{R}$ is different because it has a square root.
Posted by: Todd Trimble on September 18, 2009 12:24 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Could you say a little more what you mean by “it's very different from the way most mathematicians think about these things”? It seems to me that, now that categorical ideas have seeped into the general consciousness, many mathematicians do in effect “think structurally” – simply put, that context matters – for example that 2 in $\mathbb{Q}$ is different from 2 in $\mathbb{Z}$ since the first 2 is invertible and the second isn’t, and that 2 in $\mathbb{R}$ is different because it has a square root.
I don't have any statistical evidence to back it up, but my feeling is that this is still a minority. It's one thing to say that $2$ behaves differently in $\mathbb{Q}$ from how it behaves in $\mathbb{Z}$, but another thing to say that $2$ in $\mathbb{Q}$ is a different object from $2$ in $\mathbb{Z}$. Like many philosphical differences, there is no practical distinction here, but I think that a lot of people would have difficulty even understanding what you were saying up above —especially the very young (undergraduates who have had little or no experience yet with abstract concepts like groups, metric spaces, and other kinds of structured sets) and the very old (and set in their ways; I can think of some examples from my days at UCR who I'm sure would not have understood what you were saying, although I'd rather not name them here). I think that most of the others would understand it but think it weird; it probably seems unnecessarily complicated.
Again, no scientific evidence; that's my feeling from talking with non-categorially-inclined mathematicians.
Posted by: Toby Bartels on September 18, 2009 9:26 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TT: it’s provably the same element 2 (that is, same morphism $1\to\mathbb{N}$ in Set) we’re referring to in each of those cases. […] I’m puzzled why any of this should be considered a problem.
I don’t understand; I am truly puzzled. How can you even speak of (let alone prove it) the same morphism $1\to\mathbb{N}$ in Set when the $\mathbb{N}$’s are different (being a set, a set with successor, etc., which are all different things)?
Posted by: Arnold Neumaier on September 18, 2009 12:22 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Easy: these objects ($\mathbb{N}$ as set, $\mathbb{N}$ as ordered commutative monoid, etc., etc.) are treated as belonging to different categories: $Set$, $OrdCMon$, etc. When one applies the appropriate “forgetful functor” [from each of these categories to $Set$, the functor that forgets or strips off structure] to each of these structured objects, in each case the output is the same: the underlying set $\mathbb{N}$. (Strictly speaking, I have just said something “evil”, but I am not going to worry about this on first pass.)
Are you worried about the fact that the naturals as ordered rig “is” a sextuple $(\mathbb{N}, 0, 1, +, \cdot, \lt)$, and hence a complicated type of “set”, a set different from $(\mathbb{N}, 0, succ)$ say? Yes, you can shoehorn any mathematical structure to make it a “set” (as one might do if ZF were one’s religion), but that’s not how mathematicians would normally think of it – they just bear in mind whatever structure (e.g., an ordered rig is a set equipped with certain operations and a binary relation) is relevant to the discussion at hand. And, in ever-increasing numbers, they think of the collection of structures of given species or signature as belonging to a category in its own right, with different types of structure belonging to different categories. So the set $\mathbb{N}$ can come equipped with different types of structure, giving rise to objects in different categories, but one can always forget the structure and come back to the same underlying set $\mathbb{N}$.
Posted by: Todd Trimble on September 18, 2009 1:46 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TT: Are you worried about the fact that the naturals as ordered rig “is” a sextuple, and hence a complicated type of “set”, a set different from ($\mathbb{N}$,0,succ) say?
For me, as I believe for most mathematicians, the naturals have the maximal structure, and therefore are simultaneously a set, a Peano structure, a free monoid, a semiring, etc.. This view is simple, harmless, consistent with actual practice, and minimizes the amount of trivial add-on needed to formalize it.
But to understand your position, I tried to walk in your shoees and treat them as different objects, Let’s call them $N$ and $N'$ for simplicity. (iTex has no macros…) I do not mind that $N$ and $N'$ are not sets; the problem lies somewhere else.
$1\to N$ and $1\to N'$ are both sets, but $1\to N$ and $1\to N'$ consist of arrows between different categories. Since an arrow, as I suppose is the purist view (maybe I am wrong here?) knows its origin and destination as part of its identity, it is impossible that the arrows $2\in(1\to N)$ and $2'\in(1\to N')$ are the same, as you said is provable. They can be identified only after applying an unspoken isomorphism.
But mathematicians generally differentiate between “same” and “isomorphic”; one really needs two words for these; only confusing them is truly evil (except perhaps when taking extreme case in how one formulates things).
This is what puzzles me. In your version of the foundation, there are too many copies of everything, and trivial functors that semi-identify them again. One gets a myriad of different objects for what is naturally the same object, and this propagates into everything constructed from these objects. One can pass the buck to a myriad of unspoken isomorphisms, but one cannot remove the myriad of things needed to make everything really precise.
But that is needed for a machine that efficiently handles all of undergraduate mathematics (say) in a single, consistent implementation.
Doesn’t this look like an ideal opportunity to apply Ockham’s razor?
Purist will still be able to discuss “2 as an element of $N$” and “2 as an element of $N'$”, but these would be regarded as objects different from the simple natural “2”.
That this is the mathematically natural way of regarding matters can already seen by the way we refer to them in the present discussion: To be understandable we must spell out the full form of the object. This is “2” for the natural number 2, but “2” does not denote an instance of the concept of “a natural number viewed in the context of a particular category”.
Posted by: Arnold Neumaier on September 18, 2009 2:37 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
But to understand your position, I tried to walk in your shoes and treat them as different objects, Let’s call them N and N′ for simplicity. (iTex has no macros…) I do not mind that N and N′ are not sets; the problem lies somewhere else.
$1 \to N$ and $1 \to N′$ are both sets, but $1 \to N$ and $1 \to N′$ consist of arrows between different categories.
There seems to be a lot of confusion just in this one sentence. For now I’ll try to carry on this discussion in good faith, but I may run out of steam soon – these are after all very elementary topics we’re spending time on, and frankly I’m beginning to lose patience.
No, those arrows are not sets.
“Consist of arrows between different categories” is not good mathematical English, but I think what you were trying to say is that those arrows are morphisms of different categories. But that’s not right either, and certainly not what I said. For one thing, there may not be any such arrows. The arrows, actually arrow I had been talking about, $2: 1 \to \mathbb{N}$, is in $Set$.
Let me give an example to help make this clearer. Let me call $C'$ the category of sets equipped $X$ with a point $x: 1 \to X$ and an endofunction function $s: X \to X$. The set $\mathbb{N}$ equipped with its standard Peano (or Lawvere natural number object) structure is an example. We could call this object $\mathbb{N}'$, if you like. A one-element set $1$ carries a unique such structure, where the point is $id: 1 \to 1$ and the endofunction is $id: 1 \to 1$, so that’s another example. Does there exist a morphism of the form $1 \to \mathbb{N}'$, in the category $C'$? NO!
However, if $U': C' \to Set$ is the evident forgetful functor, so that $U'(\mathbb{N}') = \mathbb{N}$, the set $\mathbb{N}$, then there is of course an arrow $2: 1 \to \mathbb{N}$. That’s in the category $Set$. Not in $C'$. In $Set$.
Each of those standard structures you mentioned a while back ($(\mathbb{N}, 0, succ)$, $(\mathbb{N}, 0, 1, +)$, and so on), sets equipped with prescribed operations (that are themselves arrows in $Set$), give us enough structure to pick out an element $2: 1 \to \mathbb{N}$, as an arrow in $Set$. In each case, it’s the same damned 2, provided that the structures we’re talking about are the standard ones. That’s what is provable (for Pete’s sake!). One can, and one often does, think of sets-with-structure as objects in a category in its own right, and I brought that up hoping it would help explain that yes, we can think of $\mathbb{N}$ as bearing many different types of structure (which need to be distinguished in order to have a coherent conversation), and thinking of these as objects in different categories may help us bear such distinctions in mind, but all those structures you mentioned, as sets equipped with operations of various sorts, can be used to define one and the same 2, as a morphism of the form $1 \to \mathbb{N}$ in $Set$. Is what I’m saying clear now?
For me, as I believe for most mathematicians, the naturals have the maximal structure, and therefore are simultaneously a set, a Peano structure, a free monoid, a semiring, etc.. This view is simple, harmless, consistent with actual practice, and minimizes the amount of trivial add-on needed to formalize it.
I can’t really speak for most mathematicians, and I’m not sure you can either, but I’m not sure what the heck is meant by “the maximal structure”. The maximal structure on $\mathbb{N}$ consists, I guess, of all possible functions $\mathbb{N}^n \to \mathbb{N}$ of any arbitrary arity $n$ (finite or infinite), and all possible relations on $\mathbb{N}$ (again of arbitrary arity). Is that how most mathematicians think of $\mathbb{N}$? I don’t think so. I think what mathematicians do is think of $\mathbb{N}$ however the hell they wish to think of it, referring to as much or as little structure as will suit whatever purpose they have in mind. Of course, a good mathematician will tell us what he has in mind. For example, if he is investigating decidability issues, he has to tell us whether he means $\mathbb{N}$ as monoid or rig or whatever. (I guess if the context is unstated, the usual default is to think of $\mathbb{N}$ as ordered rig, but there are lots of other things people do with $\mathbb{N}$, e.g., they can think of it as dynamical system or as a group representation in various extremely creative ways. Thus “the maximal structure” doesn’t carry a lot of coherence for me.)
Ultimately, we are on the same side: we would all like flexible, easy-to-use foundations. I entered this conversation hoping to clarify the role of context dependence in structuralist points of view on set theory, but as of this writing it’s not clear to me whether you are honestly confused by what I’m saying but still trying to understand, or just trying to pick faults in what you take my position to be.
Posted by: Todd Trimble on September 18, 2009 5:32 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TT: it’s not clear to me whether you are honestly confused by what I’m saying but still trying to understand, or just trying to pick faults in what you take my position to be.
I have no time to waste, and discuss here only to learn and to clarify.
I learnt category over 30 years ago as a student, mainly as a tools to see universal constructions from a unifying point of view.
I never needed to make any use of it in a long career in pure and applied mathematics, until this year when my vision of the foundations of mathematics was developed enough to merit a comparison with other foundations that are around. So I looked at the categorial approach, its benefits and its weaknesses. But unlike for you it is for me a mainly foreign language to which I was exposed as a youth but never had spoken it myself, apart from doing theexercises in the course where I leant it.
The FMathL mathematical framework uses categories on the foundational level just a little bit since it provides some nice features. So I had asked John Baez about his opinion, and he moved the discussion to here. I’ll keep posting to the discussion as long as I expect to learn something from it.
TT: all those structures you mentioned, as sets equipped with operations of various sorts, can be used to define one and the same 2, as a morphism of the form 1→ℕ in Set. Is what I’m saying clear now?
Yes. So the N here is always the set of natural numbers without structure, and N’ only enters as 1→U’(N’). My confusion came from the fact that before you had talked of “2 in ℚ is different from 2 in ℤ”, and I had thought that your new comment was a commentary on this, except with various forms of ℕ in place of ℚ and ℤ.
But you can see how difficult it is for someone with little practice in category theory to apply the right invisible functors at the right places.
It is definitely not something that belongs to the essence of mathematics, otherwise everyone would have to practice it before being allowed to forget it again.
And I have to find a way to teach the FMathL system to detect and overcome such misunderstandings (which can arise in reading any math in a field one is not fluent in).
TT: I’m not sure what the heck is meant by “the maximal structure”.
This was short for your 6-tuple plus succ, i.e., the union of all the stuff that is usually present in discussions about natural numbers. Depending on the context, a mathematician can add any conservative extension by defined operations to make this maximal structure rich enough. In the usual view one picks up from reading math papers, ℕ doesn’t suddenly stop to have a multiplication simply because someone doesn’t need it at the moment.
TT: Ultimately, we are on the same side: we would all like flexible, easy-to-use foundations. I entered this conversation hoping to clarify the role of context dependence in structuralist points of view on set theory,
Yes, and appreciate that. I learn from this discussion, though perhaps not at the speed you’d like to see. But this doesn’t prevent me form continuing to see all the trivial dead weight a pure categorial approach brings to fully formalized mathematics.
I want a formalization that is as free from artificialities as possible.
Posted by: Arnold Neumaier on September 18, 2009 6:32 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Arnold:
I have no time to waste, and discuss here only to learn and to clarify.
Excellent. Neither do I, and that’s also why I come here: to learn and help explain when I’m able.
I learnt category over 30 years ago as a student, mainly as a tools to see universal constructions from a unifying point of view.
Okay, thanks, this is good to know.
I never needed to make any use of it in a long career in pure and applied mathematics, until this year when my vision of the foundations of mathematics was developed enough to merit a comparison with other foundations that are around. So I looked at the categorial approach, its benefits and its weaknesses. But unlike for you it is for me a mainly foreign language to which I was exposed as a youth but never had spoken it myself, apart from doing theexercises in the course where I leant it.
The FMathL mathematical framework uses categories on the foundational level just a little bit since it provides some nice features. So I had asked John Baez about his opinion, and he moved the discussion to here. I’ll keep posting to the discussion as long as I expect to learn something from it.
It restores it some. But (and I’m not trying to browbeat you here), but I have to say in all honesty: based on what you’ve written here and elsewhere, it seems to me you have a pretty hazy understanding of category theory and what it has to say about foundations. I don’t have a problem with that, unless someone in that position then holds forth on categorical foundations, its strengths and weaknesses, what’s impossible and what’s provable, etc. I happen to know a bit about the subject myself. Not as much as some people, but “more than your average bear” as Yogi Bear used to say.
But you can see how difficult it is for someone with little practice in category theory to apply the right invisible functors at the right places.
Excuse me then. Although I did try to write clearly, it’s possible that I was talking to some degree as mathematicians often do, assuming some unstated context and background. I wasn’t aware of what your background in category theory was, perhaps, and explained things too rapidly (?).
It is definitely not something that belongs to the essence of mathematics, otherwise everyone would have to practice it before being allowed to forget it again.
Well, that’s another bald claim, but it’s not the first time I’ve heard that sort of thing.
Let’s forget ETCS and all that, then; the kernel of the complaint seems to be that the categorical approach is hard to get into – even very carefully written texts like Mac Lane and Moerdijk’s Sheaves in Geometry and Logic, a nice introduction to topos theory with a strongly logic-oriented point of view, take some effort to penetrate and master. I’ve made some forays myself into writing up some of the details of ETCS (as reproduced on the nLab), with an eye to eventually being able to explain it smoothly to undergraduates, but that series is still very unfinished and not nearly to my satisfaction. Long story short, you’re right, no one has succeeded yet in making categorical set theory look like a snap.
However, it looks like Mike Shulman has written down a very interesting program of study, different to the categories-based approach (which is extremely hands-on and bottom-up) but which is very smooth and top-down (invoking for example a very powerful comprehension principle) while still being faithful to the structuralist POV. He calls it SEAR (Sets, Elements, and Relations). That may be a more congenial place to start; in fact I strongly recommend it to gain a better appreciation of this POV (easier to read certainly than Lawvere).
If you’d like to learn more however about categorical set theory and “categorical foundations”, it might be a good idea to read some of the category texts written by people with significant formal philosophic training, e.g., Awodey, J.W. Bell, Goldblatt, and McLarty (and do the exercises!). They tend to write at a more accessible level than Johnstone say or Lawvere.
This was short for your 6-tuple plus succ, i.e., the union of all the stuff that is usually present in discussions about natural numbers. Depending on the context, a mathematician can add any conservative extension by defined operations to make this maximal structure rich enough. In the usual view one picks up from reading math papers, ℕ doesn’t suddenly stop to have a multiplication simply because someone doesn’t need it at the moment.
Obviously I agree with the spirit of the last sentence, and we obviously agree that a mathematician should have the freedom to “call” the multiplication whenever the need is felt or pass to a conservative extension rich enough for any desired purpose. What’s to disagree with. But I still don’t believe in maximal structures per se much.
For example, model theorists often consider what are called o-minimal structures, and it is widely believed (although unproven as far as I know) that there exists no maximal o-minimal structure. One can extend, world without end, amen.
But this doesn’t prevent me form continuing to see all the trivial dead weight a pure categorial approach brings to fully formalized mathematics.
It might be a good idea not to be too insistent about that before you develop a better understanding of that approach. There’s a lot going on there, and also a lot has happened in the past thirty years which you might not be fully aware of.
Posted by: Todd Trimble on September 18, 2009 8:55 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TT: it seems to me you have a pretty hazy understanding of category theory and what it has to say about foundations.
While this may be the case, I have looked at the less formula-intensive discussion of categorial foundations to get a view of what I can expect from the approach. This is what I have done in any field I entered, and if you look at my homepage you can see that I have successfully entered many fields. Time is bounded; so I need a way to see where to concentrate and to invest learning all the details. With category theory the goods to expect were never sufficient to motivate me to practice the formalism. Nevertheless, i can decipher any categorical statement with some effort, and have done so quite a number of time.
But it feels like very occasionally programming in C++ when all the time you program in an easy-to-use language like Matlab - One has to remind oneself each time what rules are applicable when, and where one needs care. (We do even the FMathL proptoyping in Matlab rather than in Haskell, say, since Matlab is much easier to use.)
I just lack the practice to express myself easily, and I’ll aquire this practice only if the benefits are high.
The main reason I cannot see why category theory might become a foundation for a system like FMathL (and this is my sole interest in category theory at present) is that a systematic, careful treatment already takes 100 or more pages of abstraction before one can discuss foundational issues formally, i.e., before they acquire the first bit of self-reflection capabilities.
In FMathL, the reflection cycle must be very short, otherwise an implementation of the system is impossible to verify by hand.
TT: I wasn’t aware of what your background in category theory was, perhaps, and explained things too rapidly (?).
Well, since this forum is read by people wil all sorts of background knowledge on categories, it pays to give a little attention in adding a bit of redundant information to remind those not doing categories every day about some context. Often a few words or an extra phrase is sufficient.
I remember when, on my very first conference, Peter Cameron, one of the big people in finite geometries, started his lecture with reminding the audience (all finite geometers) of the definition of a permutation group, a statement everyone must haver known (but not everyone used it on a daily basis). For me, it was an eye-opener for how to communicate well. (You may wish to look at my theoretical physics FAQ to get an idea on how it bore fruit.) AN: It is definitely not something that belongs to the essence of mathematics
TT: Well, that’s another bald claim, but it’s not the first time I’ve heard that sort of thing.
I wasn’t saying this of category theory, but of treating it as a foundation rather than as a tool, forcing the need for type-matchng everything by invisible functors.
TT: SEAR (Sets, Elements, and Relations).
TT: But I still don’t believe in maximal structures per se much.
Note that maximal structure is something very constructive: The union of the structure assembled about an object at a given time in a given finite context.
Of course, any context can be indefinitely extended, and as this is done, the meaning of maximal changes in a similar way as the meaning of the largest element in a finite set of numbers may change when augmenting the set.
But this doesn’t invalidate the meaning of the concept of the maximum of a finite set of numbers.
But at any time, it is well-defined. Context management is one of the basic tasks a system like MathML has to do (besides the ability to precisely specify concepts), and the way this is done decides on the feasibility of the whole projects.
AN: But this doesn’t prevent me form continuing to see all the trivial dead weight a pure categorial approach brings to fully formalized mathematics.
TT: It might be a good idea not to be too insistent about that before you develop a better understanding of that approach.
I can see the dead weight (100 pages overhead) already with the limited understanding I have now.
Show me a paper that outlines a reasonably short way to formally define all the stuff needed to be able to formally reflect in categorial language a definition that characterizes when an object is a subgroup of a group.
This is not difficult in categorial terms when you are allowed to use the standard informal mathematical metalanguage.
But the way I measure the potential merit of a conceptual framework is by how easy it is to say the same things rigorously without using the metalanguage, in particular, avoiding all the abuses that makes informal mathematics so powerful and short.
Posted by: Arnold Neumaier on September 18, 2009 11:43 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Show me a paper that outlines a reasonably short way to formally define all the stuff needed to be able to formally reflect in categorial language a definition that characterizes when an object is a subgroup of a group.
Show me that paper for $\mathbf{ZFC}$ (or any other foundation that isn't specifically geared towards group theory), and I'll show you the same for $\mathbf{ETCS}$.
Posted by: Toby Bartels on September 19, 2009 12:14 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
To make this a serious but fair challenge: Give me a paper whose source is available (say from the arXiv), formalised in ZFC (or whatever), and I'll rewrite it to be formalised in ETCS. (The paper can include its own specification of ZFC too, and mine will include its own specification of ETCS.)
Posted by: Toby Bartels on September 19, 2009 12:20 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Show me that paper for ZFC (or any other foundation that isn’t specifically geared towards group theory), and I’ll show you the same for ETCS.
I’m guessing that the response to this may be “well, ZFC is no better.”
However, I must be misunderstanding the statement, because I don’t see why it is at all difficult. A subgroup of a group is a subset which is closed under the group operations. In ETCS “subset” means “injective function”. But why is that at all hard to formalize?
Along the same lines, exactly what 100 pages are you referring to?
Posted by: Mike Shulman on September 19, 2009 3:06 AM | Permalink | PGP Sig | Reply to this
### Introduction to Categorical Logic
MS: exactly what 100 pages are you referring to?
I was thinking of the lecture notes Introduction to Categorical Logic by Awodey and Bauer. They take endless preparations (partly moved to the appendix) before they have reflected enough logic that would semantically adequately encode stuff like the metalanguage of ETCS (which was not encoded there). Maybe far less suffices, but the way the lecture notes are organized doesn’t make it easy to find out what can be deleted.
I believe that standard axiomatic set theory + first order logic as organized in standard textbooks is much shorter. One only needs the introductory part of axiomatic set theory (until one has functions and natural numbers) since not even the general notion of a cardinal is needed in first order logic.
But of course, these were just estimates from the literature. If you know of a shorter Introduction to Categorical Logic also starting from scratch, I’d appreciate a reference.
Posted by: Arnold Neumaier on September 19, 2009 6:48 PM | Permalink | Reply to this
### Re: Introduction to Categorical Logic
I think I’m still failing to understand exactly what you mean by “reflect”. Do you mean being able to give a formal definition, in some theory, of the syntax and semantics of first-order logic? If so, that is just as easy in type theory as in ZF.
All the work in those lecture notes is geared towards describing the categorical semantics for first-order logic. This is hard work no matter what theory you are working in, whereas a simple set-based semantics is easy. I think books on categorical logic usually focus on this difficult job, often assuming that their readers are familiar with the easy version.
Posted by: Mike Shulman on September 19, 2009 8:36 PM | Permalink | PGP Sig | Reply to this
### Re: Introduction to Categorical Logic
MS: I think I’m still failing to understand exactly what you mean by “reflect”. Do you mean being able to give a formal definition, in some theory, of the syntax and semantics of first-order logic?
I discussed what I mean in Sections 1.5 and 3.2 of the FMathL mathematical framework paper. It means to prepare enough formal conceptual and algorithmic ground that enables one to formally write down everything needed to explain the meaning of an ordinary mathematical text that defines the system in the usual informal way, and together with the meaning the algorithmic steps that are allowed to be performed.
In partuclar, for your vision of category theory, the system should be able to know formally what it means to treat the theory in a morally correct way.
MS: If so, that is just as easy in type theory as in ZF.
I think there is a type error in your statement. the left hand side is a way of structuring the logic, the right hand side is a way of adding mathematical functionality to it. How can they be compared for easiness?
Foundations consist of two sides - the logic and the axiom system. With respect to (classical) logic, the choice is between FOL and HOL (first and higher order logic) and between typed and type-free theories. With respect to the axiom system, the choice is between some version of set theory and some version of category theory.
I argued mainly against having as basic axiom system that of a category theory. This has nothing to do with types.
Type theories have proved their foundational value. The only complaint I have against type theoretic foundations coupled with some set theory - realized in many theorem provers - is that the cores of these provers are huge, and very hard to check by hand for correctness. But the untyped theorem provers are also far from satisfying in this respect.
Posted by: Arnold Neumaier on September 19, 2009 9:03 PM | Permalink | Reply to this
### Re: Introduction to Categorical Logic
When I said “type theory” I meant to refer, not to typed first-order logic in general, but to a specific type theory including type constructors for products, subsets, quotients, powersets, exponentials, etc – a theory such as is usually used for the internal language of a topos. Perhaps this is what you are calling “some version of category theory,” although it is not intrinsically categorial.
Posted by: Mike Shulman on September 20, 2009 6:16 AM | Permalink | PGP Sig | Reply to this
### Re: CCAF, ETCS and type theories
But it feels like very occationally programming in C++ when all the time you program in an easy-to-use language like Matlab - One has to remind oneself each time what rules are applicable when, and where one needs care.
Interesting that you should say that. I was having the same thought that this argument is much like the argument between statically typed and dynamically typed programming languages. Unsurprisingly, I prefer statically typed ones. I actually used to like dynamically typed ones like Perl and Python better, but over time I came to realize that there is a cleanness and elegance to a well-designed typed language (in which category I am hesitant to include C++), and that by forcing you to specify through the problem precisely, static typing eliminates many subtle errors and guides you to a conceptual solution. These days, if I could, I would do all my programming in Haskell. But, unfortunately, most of the world does not agree with me. (-:
Anyway, I feel that some of the same considerations may apply to mathematics. In particular, the role of mathematical foundations is not necessarily restricted to a descriptive one. One must, of course, resist the temptation to be overly prescriptive and thus alienate the majority of mathematicians, but I think that a knowledge of and appreciation for formal logic and foundations has the potential to change one’s practice of mathematics for the better, at least in small ways. I also don’t think it’s wrong for a system for computer formalization of mathematics to attempt small, incremental improvements in the way mathematicians write and reason.
Posted by: Mike Shulman on September 19, 2009 4:43 AM | Permalink | PGP Sig | Reply to this
### Re: CCAF, ETCS and type theories
MS: over time I came to realize that there is a cleanness and elegance to a well-designed typed language (in which category I am hesitant to include C++), and that by forcing you to specify through the problem precisely, static typing eliminates many subtle errors and guides you to a conceptual solution.
The formal specification part of FMathl will in fact have a kind of type system to make internally formal argueing and checking easy. But:
This type system does not look like math but like programming, and hence is not appropriate at the abstract level. An ordinary user should therefore never notice that it exists.
The FMathL concept of a type is quite different from that of a type theory. I’ll talk about it here once the design of the specification level is reasonably stable to merit discussion. (Maybe around Christmans?)
Posted by: Arnold Neumaier on September 19, 2009 7:15 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
I actually don’t see why that’s any different. If $\in$ can be overloaded, why can’t $2$ be overloaded? Numeric literals are polymorphic in Haskell.
Posted by: Mike Shulman on September 18, 2009 5:47 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
In FMathL, objects and morphisms can belong very naturally to several categories.
As a category theorist, I find that very worrying! (-:
I expect you are right that many mathematicians think of the real number $2$ as “the same” as the natural number $2$, but I think there are a fair number who realize that they are, strictly speaking, different. (They are different in material set theory too, of course: whether you define real numbers using Dedekind cuts or Cauchy sequences, in no case will $2$ be $\{\emptyset,\{\emptyset\}\}$ or whatever you used for your naturals.) But as I said in my previous post, I think that the usage of the former collection of mathematicians is adequately addressed by notational overloading. (In fact, $2$ has a meaning more general than that! It also means $1+1$ in the generality at least of any abelian group. And some people use it to mean a 2-element set, or the interval category. You just don’t have any hope of capturing mathematical usage without heaps of notational overloading, so as long as it’s there, why not make full use of it?)
However, as a category theorist, I feel perfectly within my rights to object that morally, nothing should ever be an object of two categories at the same time. If that were possible, then people could start talking about things like taking the “intersection” of two categories, and that would just be all wrong. The set $\mathbb{R}$ is an object of $Set$, the field $\mathbb{R}$ is an object of $Fld$, and so on, but these are different objects.
Posted by: Mike Shulman on September 18, 2009 5:58 AM | Permalink | Reply to this
### objects and morphisms can belong naturally to several categories
MS: It also seems to me that all the things about real numbers you are saying that FMathL does right, it only does right specifically for real (or complex) numbers, because you have built them into the system by fiat. If FMathL didn’t include axioms specifying that there was a particular set called $\mathbb{R}$ with particular properties
In fact, it doesn’t. The axioms only defined when a complex number is real. To define the set of real numbers, one needs to add
(1)$\mathbb{R}:=\{x\in\mathbb{C}\mid \overline x = x\},$
but to be allowed to do that needs later axioms.
AN: In FMathL, objects and morphisms can belong very naturally to several categories.
MS: As a category theorist, I find that very worrying! (-:
But every undergraduate student is very thankful for not having to distinguish between the many incarnations of 2 (and of compound objects that involve 2) in the many different structures it is in! It simplifies life dramatically without sacrificing the slightest bit of rigor!
MS: I think there are a fair number who realize that they are, strictly speaking, different. (They are different in material set theory too, of course: whether you define real numbers using Dedekind cuts or Cauchy sequences, in no case will 2 be $\{\emptyset,\{\emptyset\}\}$ or whatever you used for your naturals.)
This is why I regard both foundations as inadequate.
Even defining precisely the details of the process of overloading necessary to make mathematics work as usual is a nightmare (of the same magnitude as that for the needed abuse of notation in ZF-based foundations), and nobody has ever done it.
It is clear that this overloading is not intrinsic to mathematics but only to the attempt to give it a clategorial or set-theoretic foundation. After all, real numbers existed long before Dedekind invented the first straitjacket for it!
MS: You just don’t have any hope of capturing mathematical usage without heaps of notational overloading, so as long as it’s there, why not make full use of it?
When talking of “you”, I think you projected your own hopelessness onto me! I not only have hopes but have a good deal of evidence for its realizability within easily formalizable limits - otherwise I wouldn’t have embarked on the FMathL project.
MS: as a category theorist, I feel perfectly within my rights to object that morally, nothing should ever be an object of two categories at the same time.
In FmathL, you can exercise this right by defining your own version of categories, just as those who want to base their mathematics on ZF can define their own version of ZF-numbers.
MS: If that were possible, then people could start talking about things like taking the “intersection” of two categories,
In FMathL you can form it, but it would not automatically be a category, but an object whose only useful property would be the elements it contains.
MS: and that would just be all wrong.
Wrong at best in the current tradition, but traditions can change. Such a change of tradition might have many advantages, once systematically explored: For example, one can express with this naturally (and every mathematician immediately understands without the need for explanations) that ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation $R$ (suitably defined), and many other similar constructs.
But please justify your claim of wrongness by giving an example of a major categorial result that becomes wrong when one drops the requirement that different categories may not have common objects.
For I haven’t this seen stated as a formal requirement of the concepts of category theory. It seems to be a mere metarequirement of some category theorists only, and at best of a status like that of the implicit overloading generally employed without saying so explicitly.
For example, Wikipedia says that “Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category”
How does this square with you feeling that morally, nothing should ever be an object of two categories at the same time?
But perhaps “morally” is the qualifier that makes the difference to “in practice”. I am interested in describing mathematics as done in practice, since a good automatic research system must understand the practice, but not necessarily any subjective moral associated with it.
… back to the conversion issue…
It takes a lot of training for a mathematician not already immersed into category theory to believe (and feel happy with) the multitude of trivial conversions needed to state rigorously what you want to consider the moral state of affairs.
This is characteristic for any theory that thinks in constructive terms (like ZF or categories) rather than in specification terms (like FMathL).
Generations of students had to be forced into an unnatural ZF (or ZF-like) straitjacket since, for a long time, that was the only respectable foundation. Traditional categorial foundations only exchange this straitjacket for a different one.
FMathL shows that no such straitjacket is needed since actual mathematical practice can be fully rigorously formalized without any need for accidentals that are not actually used after a concept was defined into existence.
Posted by: Arnold Neumaier on September 18, 2009 11:50 AM | Permalink | Reply to this
If FMathL didn’t include axioms specifying that there was a particular set called $\mathbb{R}$ with particular properties
In fact, it doesn’t. The axioms only defined when a complex number is real.
Yes, but that is completely irrelevant to the point I was trying to make. The point is that the axioms supply, however they do it, a set $\mathbb{R}$ without specifying how it is constructed.
But every undergraduate student is very thankful for not having to distinguish between the many incarnations of 2
So is every Haskell programmer. But that doesn’t mean that the different incarnations of 2 aren’t different at a fundamental level.
It is clear that this overloading is not intrinsic to mathematics but only to the attempt to give it a clategorial or set-theoretic foundation.
It is clear to me that this overloading is fundamental to the way mathematics is spoken and written by mathematicians. I’m guessing that even you will not claim that the real number $2$ is the same as the element of $\mathbb{Z}/p\mathbb{Z}$ denoted by $2$ — especially when $p=2$ (the integer) because then $2=0$ in $\mathbb{Z}/p\mathbb{Z}$, which it assuredly does not in $\mathbb{R}$! So any system which can parse mathematics as it is actually written by mathematicians will have to allow overloading, regardless of how nightmarish it might be, and regardless of the foundations one chooses. (And I thought you already agreed that overloading $\in$ was reasonable! why is overloading $2$ different?)
I not only have hopes but have a good deal of evidence for [capturing mathematical usage without heaps of notational overloading]
I would like to see some of this evidence.
It takes a lot of training for a mathematician not already immersed into category theory to believe (and feel happy with) the multitude of trivial conversions needed to state rigorously what you want to consider the moral state of affairs.
But I thought the whole point of this project is so that the mathematician not already immersed in any sort of foundations doesn’t have to believe in or feel happy with those foundations; they can just write mathematics as they usually do and the system will interpret it correctly.
Posted by: Mike Shulman on September 18, 2009 6:21 PM | Permalink | Reply to this
MS: even you will not claim that the real number 2 is the same as the element of ℤ/pℤ denoted by 2 — especially when p=2 (the integer) because then 2=0 in ℤ/pℤ, which it assuredly does not in ℝ! So any system which can parse mathematics as it is actually written by mathematicians will have to allow overloading.
I call this context-dependent ambiguity. This has nothing to do with types.
Today I was grading a paper where c was defined as a certasin product, and a few lines later was a formula involving c in the denominator, explained to be the speed of light. And the student went on saying: Therefore c is very small and can be neglected. I was very puzzled in the first round of reasding until I noticed that there were two different meanings of the symbol c.
I conclude that the existence of context-dependent ambiguity must be accounted for. But it doesn’t give licence to generate more versions of 2 than are absolutely needed.
MS: I would like to see some of this evidence.
It is difficult to convey direct evidence before we have fixed the formal framework for representing mathematical context, but here is the idea:
I mentioned already elsewhere in this discussion that one can avoid the multiple meanings of $\mathbb{N}$ by interpreting it in each context as the richest structure that can be picked up from the context.
This principle works very generally, and is, I believe, consistent with the way mathematicians work (excepting perhaps category theorists).
As an automatical research system must have anyway the capacity to build up context, this accommodated the principle without problems, without the need of overloading. Each context has its own collection of interpretations.
in some sense, this uses the same idea as the categorial approach to foundations, but to turn each context into a category in order to be able to use the categorial version of this idea for general context changes in mathematics would create another straitjacket…
MS: I thought the whole point of this project is so that the mathematician not already immersed in any sort of foundations doesn’t have to believe in or feel happy with those foundations; they can just write mathematics as they usually do and the system will interpret it correctly.
This is the main point, but not the whole point. The whole point also involves making the system trustworthy in that everyone interested in checking out how the system arrives at its answers should be able to get as much detail as wanted, and also in the simplest possible form. In particular, someone extremely critical should be able to check for himself (if possible without any use of a machine) that the whole system works in a sound way. This means that the basics must be as transparent as possible.
Compare with Coq. I enter a conjecture; Coq works on it for a week and then says: “The conjecture is true, and here are 3749MB of proof text that Coq_V20.57 verified to be correct.
Nobody is going to check that, except perhaps another machine. Humans must take it on trust. But on which basis? At least you’d want to check the implementation of Coq to get assurance. But Coq is a huge package….
The core of FMathL will have to be a small package, with programs transparent enough to be checked by hand. This is possible only if things are kept as simple as possible.
Posted by: Arnold Neumaier on September 18, 2009 7:30 PM | Permalink | Reply to this
I’m confused; are you saying that $2$ should not be used to mean $1+1$ in $\mathbb{Z}/p\mathbb{Z}$? I think that is a perfectly justifiable notation and, I believe, so do many other mathematicians.
Posted by: Mike Shulman on September 18, 2009 8:45 PM | Permalink | PGP Sig | Reply to this
MS: are you saying that 2 should not be used to mean 1+1 in ℤ/pℤ?
No. I am saying that 2 has a context-dependent ambiguous meaning.
In a context where ℤ/pℤ (or another ring with 1) appears as a structure whose elements are discussed, 2 should mean 1+1 in this ring, wheres in the absence of such a context, 2 should be considered as the default: a complex number (and at the same time as a real, rational, integral, and natural number).
Thus I accept notation ambiguity - to be able to recognize ambiguity and resolve it from context is essential for any automatic math system.
But I do not regard notation ambiguity as something to be captured by a concept of overloading and subtyping, since not every notation ambiguity can be considered as an instance of the latter.
Therefore I accept that “2 in ℤ/pℤ” and “2 as natural number” are different objects, They are rarely used in the same context.
But I do not accept that “2 as natural number” and “2 as complex number” are different objects. Almost everyone will agree with me. (In algebra I was even taught how to achieve this uniqueness of 2 in ZF using a construction called identification.)
Posted by: Arnold Neumaier on September 18, 2009 9:00 PM | Permalink | Reply to this
But I do not regard notation ambiguity as something to be captured by a concept of overloading and subtyping, since not every notation ambiguity can be considered as an instance of the latter.
Can you give some examples of the sort of notation ambiguity you are thinking of which can’t be captured by overloading and subtyping?
Posted by: Mike Shulman on September 19, 2009 2:28 AM | Permalink | PGP Sig | Reply to this
MS: Can you give some examples of the sort of notation ambiguity you are thinking of which can’t be captured by overloading and subtyping?
For example, writing $x \circ y *z$ without having defined priorities of the operations (and where different priority rules exist in different traditions), and both ways to interpret them make sense. The dangling else is a famous instance of that.
Of course, one can write formal papers that avoid any sort of ambiguity. This is what is done in Mizar, but it accounts for the huge overhead that makes Mizar not a very practical tool for the ordinary mathematician.
FMathL must resolve these issues by reasoning, figuring out which version makes sense in the context. Typing is just the simplest way of doing such reasoning in cases where it applies.
Posted by: Arnold Neumaier on September 19, 2009 5:25 PM | Permalink | Reply to this
Thanks for the examples, now I know what you mean.
Typing is just the simplest way of doing such reasoning in cases where it applies.
Maybe this is the core of our disagreement. I do not see types as “just” a way of resolving notational ambiguity. Rather, types carry important semantic information in their own right.
It seems unlikely that either of us will convince the other, however, since versions of this argument have been raging for years. But I’m very glad to have had / be having the discussion; I think I’ve learned a lot.
Posted by: Mike Shulman on September 19, 2009 6:50 PM | Permalink | PGP Sig | Reply to this
MS: types carry important semantic information in their own right.
Yes, they do. But the common type systems are far too inflexible to capture all of the semantics. Since a system like MathML needs anyway a way to represent arbitrary semantics, the more general semantic system will automatically take over the function a typing system would have.
Once one has the semantic system, one can choose to represent things there in any way one likes. It will be easy to create in FMathL contexts that work exactly like HOL and allow you to do everything with overloading if you believe this is the way mathematics should be done on the base level. But the standard context for doing mathematics will most likely be different, since typing introduces a lot of unnecessary representation overhead.
On the other hand, since the semantics of everything is clearly structured, it will not be very difficult to create automatic translators from one sort of mathematical representation to another sort. Indeed, this will be one of the strengths of the FMathL system and its view of many subject levels with a common object level.
Every mathematician can view mathematics in his or her preferred way, asnd still be sure that everything translates correctly to every other mathermatician’s view.
MS: But I’m very glad to have had / be having the discussion; I think I’ve learned a lot.
The same hold for me. It is very interesting and helpful.
Posted by: Arnold Neumaier on September 19, 2009 8:07 PM | Permalink | Reply to this
But I do not accept that “2 as natural number” and “2 as complex number” are different objects. Almost everyone will agree with me.
I would be very interested to see a wide-ranging survey of professional mathematicians on this point, broken down by field and perhaps age. I’m not saying you’re wrong that “almost everyone” will agree with you, but I don’t really have data to judge one way or the other.
I find it markedly inconsistent and arbitrary to view $2\in\mathbb{Z}/p\mathbb{Z}$ and $2\in\mathbb{Z}$ as different, but $2\in\mathbb{Z}$ and $2\in\mathbb{C}$ as the same. What about $2\in\mathbb{Q}_p$? Or $2\in\overline{\mathbb{Q}}$? What’s the general rule?
Posted by: Mike Shulman on September 19, 2009 6:59 PM | Permalink | PGP Sig | Reply to this
MS: I find it markedly inconsistent and arbitrary to view $2\in\mathbb{Z}/p\mathbb{Z}$ and $2\in\mathbb{Z}$ as different, but $2 \in\mathbb{Z}$ and $2 \in\mathbb{C}$ as the same. What about $2\in \mathbb{Q}_p$? Or $2\in \overline\mathbb{Q}$? What’s the general rule?
I didn’t invent that. For Bourbaki (whom I am following in this), the rule is that an object $x\in A$ remains $x$ even when it is considered as an element of $B$ where $B$ contains $A$.
Bourbaki has a general construction called identification that replaces (for example) the pseudo-rationals in the Dedekind reals by the true rationals from which the Dedekind reals were constructed, and adapts the operations in such a way that the resulting field of reals contains the ordinary rationals.
Thus what (in my understanding of) category theory (maybe I am not using the correct term here) is an embedding functor is for Bourbaki the identity mapping.
Upon having described identification at the first occasion where it occurs, it suffices later (when discussing reals) to say that “by identification, we may take Q to be a subset of R.” Somewhere, an abuse of language is introduced to say that identification will be made silently if the embedding is canonical. This establishes the standard mathematical terminology in a completely rigorous way.
Thus the $2\in \mathbb{Q}_p$ and the $2\in \overline\mathbb{Q}$ are the same as the $2\in \mathbb{C}$, as all three sets contain $\mathbb{Z}$, which in turn contains 2.
Posted by: Arnold Neumaier on September 19, 2009 8:08 PM | Permalink | Reply to this
After a long conversation with a friend this evening, I feel like I have a better understanding of how and why many/most people may think of $2\in\mathbb{Z}$ and $2\in \mathbb{R}$ as identical. Perhaps I have unknowingly trained myself to think of them as different, because that is what the structural approach to mathematics requires (and I think of mathematics structurally because that is what I see when I look at mathematics—although I now accept that you see something different).
Honestly, it just feels really messy to me to think about the category $Ring$ (say) if some elements of some rings might be equal to some elements of other rings, depending on how one constructed them and whether an embedding of one into another is “canonical” or not. But aesthetics certainly differ! (-:
Posted by: Mike Shulman on September 20, 2009 6:26 AM | Permalink | PGP Sig | Reply to this
### Re: objects and morphisms can belong naturally to several categories
ordered monoids are the objects in the intersection of Order and Monoid
As far as I can tell, this is not true even in FMathL (and I don’t see how it could ever be true). An order is a set equipped with an order relation, and monoid is a set equipped with a multiplication and unit. An ordered monoid is a set equipped with both. What is true instead is that ordered monoids are the objects of the pullback of the two forgetful functors from $Order$ and $Monoid$ to $Set$.
But please justify your claim of wrongness by giving an example of a major categorial result that becomes wrong when one drops the requirement that different categories may not have common objects.
Mathematics is not just about results. Mathematics, and especially category theory, is also about definitions and concepts. The point is that one object being in two categories is foreign to the practice of mathematics and can only lead to confusion. You’ve supplied me with a case in point above: if it were possible for one object to be in two categories, then I can easily see some people thinking that ordered monoids are the intersection of $Order$ and $Monoid$, when in fact this is false.
Your example of opposite categories is also a good one. The opposite of the category of frames is the category of locales, but a frame is not the same as a locale. For instance, the terminal frame is very different from the terminal locale. Suppose I wanted to study “localic frames,” i.e. the point-free version of “topological frames,” which in turn would be frames whose frame operations are continuous. If someone has been told that categories can be intersected, and maybe is still laboring under the misapprehension that ordered monoids are the intersection of $Order$ and $Monoid$, they might immediately try to define localic frames as the interesction of $Locale$ and $Frame$. But if $Locale$ is defined as $Frame^{op}$ and opposite pairs of categories “have the same objects,” then $Locale \cap Frame$ would consist just of the frames/locales, a far cry from “localic frames.” Trying to intersect categories should be a type error.
Posted by: Mike Shulman on September 18, 2009 6:38 PM | Permalink | PGP Sig | Reply to this
### Re: objects and morphisms can belong naturally to several categories
AN: Such a change of tradition might have many advantages […] ordered monoids are the objects in the intersection of Order and Monoid that ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation R (suitably defined),
MS: As far as I can tell, this is not true even in FMathL (and I don’t see how it could ever be true). An order is a set equipped with an order relation, and monoid is a set equipped with a multiplication and unit.
I was speaking in subjunctive mode (and clearly not about the meaning of the terms in existing foundations) since this is not a reality but only a possibility that might (or might not) work out. (I thought the cafe is also about exloring possibilities, not only about presenting truths.) Since the current FMathL framework does not have a formal way to say what a monoid is, nobody knows yet whether this could be true in FMathL. This brings us to semantical questions that need to be resolved on the next layer of FMathL that we are designing at the moment, but it is not yet ready:
What does it formally mean to speak of “a set equipped with a relation”?
In ZF, it means to have a pair (S,R) where S is a set and R is a relation, and this is easily formalized in ZF itself.
In category theory, its formal meaning must be something different; I don’t know exactly what. How would this be formally expressed, using only terms intrinsic to the language of categories?
FMathl is neither ZF nor category theory, hence can give it again a different meaning. (What it will be, will be decided by the end of the year, I guess.)
Probably, but I haven’t checked the details, one can give the intersection of categories in FMathL a meaning that makes intersections of categories of two algebraic structures inherit the properties of both structures in such a way that it is consistent with Axiom A10 governing intersection.
MS: Your example of opposite categories is also a good one. The opposite of the category of frames is the category of locales, but a frame is not the same as a locale.
There must be something wrong either with your statement or with the definition of opposites in Wikipedia.
Wikipedia said explicitly that the two categories have the same objects. And this seems to be the canonical definition: Adámek et al, Abstract and Concrete Categories say something amounting to the same in Definition 3.5. So does Definition 1.3.2 of Asperti and Longo, Categories, Types and Structures. So says Section 1.4 in Barr and Wells, Toposes, Triples and Theories. So says Section 1.6.1 in Schalk and Simmons, An introduction to Category Theory in four easy movements.
Not being an expert in category theory, I need to rely on trustworthy sources for the accepted meaning of the basic concepts. There seems to be full agreement in the literature (at least that available online) that at least certain distinct categories have the same objects.
MS: But if Locale is defined as Frame$^{op}$ and opposite pairs of categories “have the same objects,” then Locale $\cap$ Frame would consist just of the frames/locales, a far cry from “localic frames.”
I agree. But backed by the orthodox definition of opposites, and with the usual moral of a mathematician, I’d draw the conclusion that your statement “The opposite of the category of frames is the category of locales, but a frame is not the same as a locale” is incompatible with the definition of opposites when taken literally. (FMathL would raise here a popup window and ask for support or correction or clarifying context.)
But perhaps there is something else wrong with my moral of reading category theory texts besides what Todd Trimble pointed out. (Just as I learn through such a discussion, FMathL would have an internal moral code that learns from the feedback from popup windows.)
Posted by: Arnold Neumaier on September 18, 2009 8:46 PM | Permalink | Reply to this
### can objects and morphisms belong to several categories?
I think that making ordered monoids the intersection of $Order$ and $Monoid$, in any formal system, would be a very bad idea. (I also have my doubts about whether it is possible in a consistent way.) Everywhere in mathematics that I have seen “intersection” used, it refers to adding properties, rather than structure as in this case. Furthermore, a category should only be considered as defined up to equivalence, and this sort of “intersection” seems unlikely to be invariant under equivalence.
What does it formally mean to speak of “a set equipped with a relation”?
In ETCS, a set equipped with a relation consists of an object $A$, an object $R$, and a monomorphism $R\to A\times A$.
There must be something wrong either with your statement or with the definition of opposites
Very interesting! I didn’t realize that the basic texts of category theory could be misinterpreted in this way by someone unfamiliar with our way of thinking.
I think that all category theorists, as well as most of the mathematicians they talk a lot to (algebraists, topologists, etc.), are so immersed in a structural way of thinking that an object of a category only has meaning as an object of that category. When you construct one category from another, you might use the “same” set of objects, but once you’ve constructed it, there is no relationship between the objects, because after all any category is only defined up to equivalence.
This phenomenon isn’t special to categories. For instance, any group has an opposite group with “the same elements” obtained by reversing the order of multiplication, but I don’t think any group theorist would then consider it meaningful to take the intersection of a group and its opposite group. Using “the same elements” is a construction of the opposite, with the same status as Dedekind cuts or Cauchy sequences—once the construction is performed, the fact that you used “the same” objects is discarded. And plenty of other constructions of the opposite are possible.
In fact, this same principle applies to basically all constructions I am familiar with in mathematics. The quotient of one group by another, the polynomial algebra of a ring, the Postnikov tower of a topological space—in each case you may give a specific construction in terms of the input (e.g. maybe an element of $R[x]$ “is” a function from $\{0,\dots,n\}$ to $R$, thought of as the coefficients of a polynomial), but in each case once the construction is performed, its details are forgotten.
I always assumed, without really thinking much about it, that all modern mathematicians thought in this way, except for maybe ZF-theorists (on some days of the week). But apparently not!
Posted by: Mike Shulman on September 18, 2009 9:18 PM | Permalink | PGP Sig | Reply to this
### Re: can objects and morphisms belong to several categories?
MS: In ETCS, a set equipped with a relation consists of an object A, an object R, and a monomorphism R→A×A.
This only goes halfway towards answering my question. You reduced it to another informal construct.
What does it formally mean that something consists of three typed things?
Posted by: Arnold Neumaier on September 18, 2009 10:02 PM | Permalink | Reply to this
### what are structures, structurally?
“A set equipped with a binary relation” is not a single object in the discourse of structural set theory the way a pair $(A,R)$ is a single object in the discourse of material set theory. But that doesn’t make it informal. If I want to make a statement about all sets equipped with binary relations, I can construct a formal sentence of the form
$\forall A. \forall R:P(A\times A). \dots$
Posted by: Mike Shulman on September 19, 2009 5:03 AM | Permalink | PGP Sig | Reply to this
### Re: can objects and morphisms belong to several categories?
MS: a category should only be considered as defined up to equivalence, and this sort of “intersection” seems unlikely to be invariant under equivalence.
This appears to be part of the unspoken moral code of categorist. But all categories they write down are defined as particular categories, and equivalence of categories is a concept that doesn’t appear on page 1, where it should figure if categories are really defined only up to equivalence.
Let’s not further discuss intersection; maybe this is still unbaked. But clarifying the moral code so that it is intelligible to an outsider who learns through self-study (ultimately I am thinking of the FMathL system to be that outsider) seems quite important!
AN: There must be something wrong either with your statement or with the definition of opposites
MS: Very interesting! I didn’t realize that the basic texts of category theory could be misinterpreted in this way by someone unfamiliar with our way of thinking.
I have aquired an eye for all these subtleties (where an automatic system would stumble upon) because I’ve been observing myself closely the last few years how I read math, in order to ba able to teach it to the FMathL system.
I can’t see how the definitions can be read in any other way by someone reared in the tradition of Bourbaki. If you take ZF as the metatheory then none of the usual rules for identifying abuses of terminology give you the slightest clue that something else could have been intended!
MS: I think that all category theorists, as well as most of the mathematicians they talk a lot to (algebraists, topologists, etc.), are so immersed in a structural way of thinking …
… that hey have lost contact with those mathematicians whose daily work is a bit less abstract?
I was taught structural thinking but in the way of Bourbaki, rather than in the categorial way. There one clearly distinguishes between equality and isomorphism, although one knows that in many cases only the isomorphism-invariant propoerties are relevant. But being able to distinguish the two modes has lots of advantages; in particular, there is no problem of evil.
One knows that Alt(5) and PSL(2,5) are isomorphic groups, but as individuals the two groups are naturally distinguishable by their construction. Onene distinguishes clearly between the group Alt(5) with its intrinsic action on 5 elements (though we allowed abuse of notation to label these elements arbitrarily, but idf pressed, we’d have undone that) and a group Alt(5), which is just an arbitrary group isomorphic to Alt(5). Thus one could say that “Alt(6) contains the groups Alt(5) and PSL(2,5), and therefore two conjugacy classes of alternating groups of 5 elements”, with a perfectly clear meaning.
I know that this can be reformulated in categorial terms, but if we want to maintain the same degree of precision, it becomes clumsy.
MS: Using “the same elements” is a construction of the opposite, with the same status as Dedekind cuts or Cauchy sequences - once the construction is performed, the fact that you used “the same” objects is discarded.
I find this a weakness of the constructive approach to math…
MS: And plenty of other constructions of the opposite are possible.
It is interesting, though, that all books use the same construction.
MS: I always assumed, without really thinking much about it, that all modern mathematicians thought in this way, except for maybe ZF-theorists (on some days of the week). But apparently not!
Maybe I am not a modern mathematician. I was still taught the old virtues of precise definitions, and the advantage of having two different words (such as “same” and “isomorphic”) for concepts whose confusion causes confusion.
Posted by: Arnold Neumaier on September 18, 2009 10:41 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
This reminds me of a question about forgetful functors that I’ve had for a while.
The definition of “forgetful function” is (in every account I’ve seen) specified in terms of removing structure from a set equipped with some structure.
This very much depends on the presentation of the relevant categories. How can we define “forgetful functor” in a way that’s invariant under equivalence?
Posted by: Tom on September 18, 2009 11:21 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
How can we define “forgetful functor” in a way that’s invariant under equivalence?
I'd say (and Mike did say, at [[forgetful functor]]) that any functor can be a forgetful functor, depending on your point of view; calling it that simply establishes a (perhaps temporary) point of view.
Posted by: Toby Bartels on September 19, 2009 12:14 AM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
I have aquired an eye for all these subtleties (where an automatic system would stumble upon)
I would humbly suggest that an automatic system need only stumble upon them if it had been designed by a human mathematician who was unconversant with structural ways of thinking.
Thus one could say that “Alt(6) contains the groups Alt(5) and PSL(2,5), and therefore two conjugacy classes of alternating groups of 5 elements”, with a perfectly clear meaning.
Not very clear to me; perhaps our linguistic conventions are equally mutually unintelligible. Do you mean that Alt(6), considered together with its canonical action on a 6-element set, contains the group Alt(5) with its canonical action on a 5-element set and the group PSL(2,5) with its canonical action on some other set? If so, I don’t see why it should become clumsy in typed language; we just overload Alt(5) so that it can mean either “the group Alt(5)” or “the group Alt(5) with its canonical action.”
But being able to distinguish the two modes has lots of advantages; in particular, there is no problem of evil.
I am puzzled by this statement; it seems to me that only if there is a notion of equality, in addition to a notion of isomorphism, can the problem of evil even be posed.
MS: Using “the same elements” is a construction of the opposite, with the same status as Dedekind cuts or Cauchy sequences - once the construction is performed, the fact that you used “the same” objects is discarded.
I find this a weakness of the constructive approach to math…
But this seems to me exactly what FMathL is doing, when you first construct something, then define the class of things isomorphic to it, and redefine that object to be something arbitrarily chosen from that class. You use a particular construction, then you discard its details.
And actually, I don’t know what you mean by the “constructive approach” here, nor what other approach you are contrasting it with.
MS: And plenty of other constructions of the opposite are possible.
It is interesting, though, that all books use the same construction.
Well, of course. It’s the simplest one. I expect that basically all books use the same construction of the rational numbers in terms of the integers.
Maybe I am not a modern mathematician. I was still taught the old virtues of precise definitions
I’m having trouble not being insulted by that. Everything we’ve said here is perfectly precise.
Posted by: Mike Shulman on September 19, 2009 3:02 AM | Permalink | PGP Sig | Reply to this
### Precise definitons
AN: Maybe I am not a modern mathematician. I was still taught the old virtues of precise definitions
MS: I’m having trouble not being insulted by that. Everything we’ve said here is perfectly precise.
The first was not intended; I just was explaining my trainig, not makin a comparison with yours. (Please do not take anything I say personal.) The second is simply false.
I was specifically referring to the fact that all authoritative definitions I could find on the web on the definition of opposite categories say explicitly that they have the same objects, while you said equally explicitly (and now even supposedly perfectly precisely) that “morally, nothing should ever be an object of two categories at the same time”.
Please give me the perfectly precise meaning of the term “morally” that you had in mind when writing “Everything we’ve said here is perfectly precise.” It is the only point of dispute here. For without the moral part, my interpretation is correct.
Not a single word in the definitions of Wikipedia (or any of the other sources quoted earlier) tells me that two different categories must have disjoint object classes. If this were really part of category theory, it would be trivial to state it as part of the definition of caategories, and surely some author would have cared enough about precision to do so.
But no author I know of has done it, and for good reasons. For it would make the standard definition contradictory, since it is easy to construct two different categories in the sense of the standard definition that share some objects.
Posted by: Arnold Neumaier on September 19, 2009 6:10 PM | Permalink | Reply to this
### Re: Precise definitons
How about this, Arnold: two categories may have an object in common, but you should never use that fact. The construction that shows that opposite categories exist (inside a model of set theory as foundation) uses the same objects as the original category (so we’re sure that there are “enough” objects around) but from that point we never actually use that fact.
Here’s an analogous situation not using categories: the von Neumann construction showing that a model of the Peano axioms exists within set theory defines 4 as the set {0,1,2,3}. Thus within this construction it happens that 3 is an element of 4, but from this point we never use this fact, since numbers being elements of other numbers is a property of the model, not of the structure. Technically you can make this statement within the model, but “morally” you shouldn’t.
Similarly, it’s easy to come up with an equivalent category which also behaves like the opposite category, but which shares no components at all with the original category. But it really doesn’t matter, since equality of components of two distinct categories is not part of the structure.
Posted by: John Armstrong on September 19, 2009 7:15 PM | Permalink | Reply to this
### Re: Precise definitons
Thanks, that’s about what I meant.
I wasn’t including statements prefixed by “morally” in my analysis of what is precise and what isn’t; perhaps I should have said something like “every mathematical definition we’ve given has been precise.”
Posted by: Mike Shulman on September 19, 2009 7:54 PM | Permalink | PGP Sig | Reply to this
### Re: Precise definitons
JA: two categories may have an object in common, but you should never use that fact. The construction that shows that opposite categories exist (inside a model of set theory as foundation) uses the same objects as the original category (so we’re sure that there are “enough” objects around) but from that point we never actually use that fact.
The construction that shows that opposite categories exist must use the fact that two categories may have an object in common, otherwise it loses its simplicity.
The same holds in many other instances in current category theory texts.
The point is that in constructions, one freely uses (and needs) the definition of categories in the Bourbaki sense (i.e., as ordinary algebraic structures, allowing categories to share objects).
But after one has constructed an instance of the desired category, one forms its isomorphy class and chooses an anonymous element from it (in the sense of Bourbaki’s - actually Hilbert’s - choice operator used also in FMathL) to get rid of the accidentals of the construction.
Thus one needs both interpretations of categories, the concrete one (an instance) and the generic one (an anonymous instance).
FMathL naturally provides both views, and does not need to impose on the user a moral (“you should not”) that mathematics never had, beyond sticking to what is set down in the axioms.
Posted by: Arnold Neumaier on September 19, 2009 8:35 PM | Permalink | Reply to this
### Re: Precise definitons
The construction that shows that opposite categories exist must use the fact that two categories may have an object in common.
No more than the von Neumann construction shows that the number 3 must be an element of the number four. The usual construction of the opposite category is one of many, but the nature of the opposite category is not defined by this construction or any artifacts of this construction.
And I even said that it’s possible to construct another version of the opposite category whose components are completely disjoint from those of the original category. Please to go back and read what I wrote.
On another note, when did mathematics not regard the syntactic validity of statements like “3 is an element of 4” as problematic at best? You can ask “is 3 an element of 4?”, or “is this object from one category ‘the same as’ that object from another category?”, or “does a baseball shortstop have red hair?”, but these questions are all meaningless because they’re not part of the relevant structures. That’s what “moral” means here. You can ask whether two categories share objects all you like, but the question is as completely beside the point as either of the other two.
Posted by: John Armstrong on September 19, 2009 11:08 PM | Permalink | Reply to this
### Re: Precise definitons
JA: two categories may have an object in common, but you should never use that fact.
AN: The construction that shows that opposite categories exist must use the fact that two categories may have an object in common.
JA: No more than the von Neumann construction shows that the number 3 must be an element of the number four.
If you define the number 3 as a von Neumann numeral, it must have this property. If you define the number 3 in a different way it doesn’t. It depends how you formulate your definition.
I was referring to the definiton as found in any of the standard sources quoted.
I think a “should” has no place in mathematics, apart from the requirement that one should take as true only what axioms, definitions, and proved theorems say.
Thus any should in mathematics must be formalized in such terms. This is the only way to keep the semantics of mathematics precise.
JA: “You can ask “is 3 an element of 4”, […] but these questions are all meaningless because they’re not part of the relevant structures. That’s what “moral” means here.
I am insisting on giving this a formal meaning since otherwise it is not possible to teach it to an automatic system. But I don’t see any way to make the moral you want to impose formally precise in any system at all without violating this morality somewhere.
Posted by: Arnold Neumaier on September 20, 2009 12:12 PM | Permalink | Reply to this
### Re: Precise definitons
I am insisting on giving this a formal meaning since otherwise it is not possible to teach it to an automatic system.
You have this precisely backwards. It’s perfectly simple to teach a formal system not to ask whether one number “is an element of” another number. Just don’t define “is an element of” to have any meaning for numbers.
Posted by: John Armstrong on September 20, 2009 3:36 PM | Permalink | Reply to this
### Re: Precise definitons
JA:two categories may have an object in common, but you should never use that fact.
JA: “You can ask “is 3 an element of 4”, […] but these questions are all meaningless because they’re not part of the relevant structures. That’s what “moral” means here.
AN: I am insisting on giving this a formal meaning since otherwise it is not possible to teach it to an automatic system.
JA: You have this precisely backwards. It’s perfectly simple to teach a formal system not to ask whether one number “is an element of” another number. Just don’t define “is an element of” to have any meaning for numbers.
My “this” referred to “the categorial moral”, not to this specific example.
In a typed system, the example already has the precise meaning of “undefined”. But the moral also contains your statement at the top of this message.
How do you make this precise enough that an automatic system does not feel entitled after having seen the definition of a category (Definition 1.1.1 in Asperti and Longo) to make Definition 1.3.1 (which uses $\subseteq$, which is defined only in terms of equality between objects)?
And why should one follow your injunction when the standard textbooks don’t follow it? (Definition 1.3.1 is completely standard.)
Posted by: Arnold Neumaier on September 21, 2009 2:46 PM | Permalink | Reply to this
### Re: Precise definitons
And why should one follow your injunction when the standard textbooks don’t follow it? (Definition 1.3.1 is completely standard.)
You’re completely (intentionally?) missing the distinction I drew between a construction demonstrating the existence of a model of a structure and the subsequent use of the properties of a structure. As I said before, “moral” (which was someone else’s term) refers to the latter segment, not the former.
Von Neumann numerals are also “completely standard”, and in their construction some numbers are elements of other numbers, but once we’ve constructed this model (to show that a NNO exists) we never again use that fact because it’s a property of the model and not of the structure.
I’m done here.
Posted by: John Armstrong on September 21, 2009 3:51 PM | Permalink | Reply to this
### Re: Precise definitons
I think a “should” has no place in mathematics, apart from the requirement that one should take as true only what axioms, definitions, and proved theorems say.
That’s very different from my philosophy, and from my experience doing mathematics and talking about it with other mathematicians. Some “should”s that I can think of off the top of my head: one should use enough generality but not too much, one should not use confusing variable names (even if they are formally correct), one should not use redundant or unnecessary axioms, one should choose the names of defined terms in a consistent way, and one should not invent and study concepts that have no motivation or relation to the rest of mathematics. Of course, as always, not everyone agrees on what one should and shouldn’t do, but I think mathematics is rife with normative judgements beyond truth and falsity.
Posted by: Mike Shulman on September 21, 2009 5:37 AM | Permalink | PGP Sig | Reply to this
### Re: Precise definitons
AN: I think a “should” has no place in mathematics, apart from the requirement that one should take as true only what axioms, definitions, and proved theorems say.
MS: That’s very different from my philosophy, and from my experience doing mathematics and talking about it with other mathematicians. Some “should”s that I can think of off the top of my head: one should use enough generality but not too much […]
I don’t think these are shoulds that make a difference in mathematical understanding but in the quality of the resulting mathematics. Of course good mathematics is governed by lots of shoulds.
But mathematics itself is not. Shoulds have no place in the interpretation of the meaning of a well-formed piece of (even irrelevant, too abstract, too special, or too longwinded) mathematical text, and on what a mathematician is allowed to do with it without leaving the realm of the theory.
In our present discussion you had wondered about how much misunderstanding is possible by not knowing the shoulds.
I still think that I interpreted the definitions in an impeccable way, and indeed in the way the definitions are used (at least at times) by category theorist. I even believe that they cannot be interpreted in any other way from a strictly formal perspective. The moral seems to lie only in the labeling of some of it as evil or unnatural.
Anyway, I see signs of slow convergence towards a consensus!
Posted by: Arnold Neumaier on September 21, 2009 3:20 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
MS: I would humbly suggest that an automatic system need only stumble upon them if it had been designed by a human mathematician who was unconversant with structural ways of thinking.
I have not the slightest idea how a system should be constructed that, without having any prior notion of category theory, being exposed to the Wikipedia article quoted would not be able to construct categories satisfying the axioms that have common objects.
MS: I don’t see why it should become clumsy in typed language; we just overload Alt(5) so that it can mean either “the group Alt(5)” or “the group Alt(5) with its canonical action.”
Yes, of course. I didn’t say that one can solve many of these problems by overloeading. I just explained the way such things were treated in the tradition I grew up with, and that it worked well.
MS: it seems to me that only if there is a notion of equality, in addition to a notion of isomorphism, can the problem of evil even be posed.
But there must be such a notion of equality intrinsic in any definition of category theory (as opposed to categories). One cannot purge this sort of evil.
One must know that if you take two objects A,B from a category C that any two mentions of A are identical objects (not only the same up to isomorphy), while a mention of A and a mention of B are possibly identical. Otherwise one cannot embed category theory into standard logic, and structural concepts such as automorphisms would not make sense.
One also needs it in order to be able to define opposite categories in the standard way, even if one subsequently forgets how they were constructed.
MS: this seems to me exactly what FMathL is doing, when you first construct something, then define the class of things isomorphic to it, and redefine that object to be something arbitrarily chosen from that class. You use a particular construction, then you discard its details.
Of course, this was frequently done in mathematics, already before categories were born. But thwe point is that FMathL can choose when to do it, and indicates it (as Bourbaki would have done), while in category theory, one is forced to do it, even when one does not want to do it.
For example, if one treats posets arising in practical programming as categories, one almost always needs them in their concretely defined form, and not only up to isomorphism. And it is clear that categories, ad defined everywhere, may have this concrete from.
Thus FMathL preserves an important freedom of mathematicians that an equality-free categorial approach tries to forbid for purist reasons. But it also accommodates a purist categorial approach as you propose it, just because of your observation.
MS: I don’t know what you mean by the “constructive approach” here, nor what other approach you are contrasting it with.
I had contrasted the constructive approach that defines quaternions via a construction and the specification approach that defines quaternions via a characterization. You had then mentioned that initially, a characterization might not be available, and I had replied why this does not make the FMathL approach unattractive.
MS: I expect that basically all books use the same construction of the rational numbers in terms of the integers.
Here is an exception: Arnold Neumaier, Analysis und lineare Algebra, Lecture Notes in German
Posted by: Arnold Neumaier on September 19, 2009 7:13 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
I have not the slightest idea how a system should be constructed that, without having any prior notion of category theory, being exposed to the Wikipedia article quoted would not be able to construct categories satisfying the axioms that have common objects.
How about a system which is structural, and therefore which regards the question of whether two different structures (such as two different categories) have common elements as a type error?
I didn’t say that one can[‘t] solve many of these problems by overloeading.
No, but what you did say was:
I know that this can be reformulated in categorial terms, but if we want to maintain the same degree of precision, it becomes clumsy.
I was pointing out that we can maintain the same degree of precision in a structural theory with overloading, without it becoming clumsy.
One must know that if you take two objects A,B from a category C that any two mentions of A are identical objects (not only the same up to isomorphy)
Actually, it’s good enough if any two mentions of A are connected by a specified isomorphism in a coherent way. But there is a distinction between naming a given object and asking whether two given objects are identical.
in category theory, one is forced to do it, even when one does not want to do it.
This is not true. You can always keep the extra structure around which you used to construct the gadget; you don’t have to forget about it. If you construct the reals as a subset of $P(\mathbb{Q})$ using Dedekind cuts, you don’t have to then forget about the $\in$ relation relating them to $\mathbb{Q}$.
Posted by: Mike Shulman on September 19, 2009 8:03 PM | Permalink | PGP Sig | Reply to this
### Re: can objects and morphisms belong to several categories?
MS: How about a system which is structural, and therefore which regards the question of whether two different structures (such as two different categories) have common elements as a type error?
It would report a type error in the standard definition of the opposite category, if written down formally.
MS: I was pointing out that we can maintain the same degree of precision in a structural theory with overloading, without it becoming clumsy.
Not on the surface, because of the overloading, but inside the system, which must track all that overloading. One needs to consider both aspects - efficiency for the user and efficiency for the system.
MS: Actually, it\u2019s good enough if any two mentions of A are connected by a specified isomorphism in a coherent way
So with $n$ mentions, you created an internal overhead of $O(n^2)$. In a lengthy proof, $n$ can be large. Thus at least an efficient implementation may not pretend that it adheres to the categorial moral.
But a solid foundation must also be able to describe what happens on the implementation level.
MS: But there is a distinction between naming a given object and asking whether two given objects are identical.
I don’t think this is good enough. For often one may want to derive results that hold for general A, B, and later one may want to use this result for the special case where A and B are the same.
MS: .If you construct the reals as a subset of $P*(\mathbb{Q})$ using Dedekind cuts, you don’t have to then forget about the $\in$ relation relating them to $mathbb{Q}$.
How do I then refer to the reals with $\in$ as opposed to the reals without $\in$? They are no longer the same objects, although I constructed the latter to be the former.
Such a schizophrenic state of affairs is avoided when using Bourbaki’s choice operator.
Posted by: Arnold Neumaier on September 19, 2009 8:38 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
It would report a type error in the standard definition of the opposite category, if written down formally.
It wouldn't, because we never ask (then or afterwards) if an object of $C$ is the same as an object of $C^op$.
In formalising mathematics without a fundamental global equality, it's important to distinguish the external judgement that two terms are syntactically identical from the internal proposition that two terms refer to the same object. If $A$ is an object of $C$, then you may interpret $A$ as an object of $C^op$, without even an abuse of language. (I'm not sure that the last clause is correct in Mike's favourite foundations, but it is in mine, which are more type-theoretically oriented.) But you can't introduce $A$ as an object of $C$ and $B$ as an object of $C^op$ and ask whether $A = B$, or even whether $A \cong B$; that's a type error.
[…] often one may want to derive results that hold for general A, B, and later one may want to use this result for the special case where A and B are the same.
I would handle this by substitution, just as I would if I wanted the result for the special case where $A$ is $x^2 + 2$ and $B$ is $x + y - \Sigma$ (in a context where those terms make sense —and have the right type). I know that people write ‘If $A = B$, then […]’, but I take this as abuse of language (or syntactic sugar) for ‘Setting $A$ to $B$, […]’ or ‘Setting $B$ to $A$, […]’ (depending on which symbol is used in the sequel).
Posted by: Toby Bartels on September 19, 2009 9:22 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
Thanks, Toby, that’s exactly what I meant.
If $A$ is an object of $C$, then you may interpret $A$ as an object of $C^op$, without even an abuse of language. (I’m not sure that the last clause is correct in Mike’s favourite foundations
It is. At least, insofar as I have a favorite foundation.
Posted by: Mike Shulman on September 20, 2009 5:42 AM | Permalink | PGP Sig | Reply to this
### Re: can objects and morphisms belong to several categories?
TB (in an earlier mail): I’d also like to see a formalisation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), as literally true! It seems doubtful to me; I know what they mean, but I need to translate it.
I’d like to see a formulation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), and rewrites it in a form that it can be taken as literally true. The moral of mathematics requires that a definition can be read as literally true in the sense that that any abuses of language are explained somewhere in sufficient detail that they can be undone.
Looking at other treatises, I can see nothing in Definitions 1.1.1 and 1.3.1 that is not standard. (At least two of the other sources I had quoted have identical requirements.)
So please provide a reading that has no unexplained abuses of language.
MS: How about a system which is structural, and therefore which regards the question of whether two different structures (such as two different categories) have common elements as a type error?
AN: It would report a type error in the standard definition of the opposite category, if written down formally.
TB: It wouldn’t, because we never ask (then or afterwards) if an object of C is the same as an object of C$^{op}$. […] If A is an object of C, then you may interpret A as an object of C$^{op}$, without even an abuse of language. But you can’t introduce A as an object of C and B as an object of C$^{op}$ and ask whether A=B, or even whether A$\cong$B; that’s a type error.
On the surface, this looks like a solution. Your suggestion amounts to having equality on the metalevel but forbidding it on the object level. But I think this does not hold water. I don’t think your suggestion can be implemented consistently in a fully formalized way without producing contradictions.
For to say that A is an object of C is formalized as $A\in Ob_C$, and to say that B is an object of $C^{op}$ is formalized as $B\in Ob_{C^{op}}$. Now Definition 1.3.2 in Asperti and Longo, Categories, Types and Structures implies, using Definition 1.1.1, that $Ob_C=Ob_{C^{op}}=:X$, say. Now X is a collection, and (at least when identifying collections with certain ETCS-sets or SEAR-sets, say) one can compare elements from X for equality.
But A and B are elements from X, so they can be compared for equality. A formal theorem explorer has no way to avoid this conclusion. Once it draws this conclusion it produces a type error, and exits, not being able to continue to explore the current context.
One cannot escape here to a metalevel since there is no way to feed a theorem explorer unformalized stuff.
The same problem appears with Definition 1.3.1 of a subcategory. Once you have $Ob_D\subseteq Ob_C$, nothing may forbid to compare an element of $Ob_D$ with an element of $Ob_C$ without an inconsistency.
Note that this book was written for readers not exposed to categories before. It is difficult for any such reader who takes these definitions seriously to arrive at any other conclusion.
Posted by: Arnold Neumaier on September 20, 2009 11:35 AM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
I'd also like to see a formalisation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), as literally true! It seems doubtful to me; I know what they mean, but I need to translate it.
I’d like to see a formulation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), and rewrites it in a form that it can be taken as literally true.
Yes, I see that this is your quest. If FMathL can do that with this passage, then that would show its power.
So please provide a reading that has no unexplained abuses of language.
Honestly, I think that (especially since this is an introductory text) the phrasing of the Exercise was probably a mistake. They might just try this:
$\mathbf{Set}$ is a subcategory of $\mathbf{Rel}$. Is it a full subcategory?
Of course, that's not the same exercise, and it should go immediately after Definition 1.3.1.
The real meaning of Exercise 1.3.2 is
$\mathbf{Set}^{\mathbf{op}}$ is obviously equivalent to a subcategory of $\mathbf{Rel}$. Is it a full subcategory?
But since they haven't defined equivalence of categories yet, this is an inappropriate exercise at that point. (Actually, the relation of $\mathbf{Set}^{\mathbf{op}}$ to the desired subcategory of $\mathbf{Rel}$ is stricter than equivalence, but still not anything that they've defined yet.)
Another way out is to interpret Definition 1.3.1, particularly the requirement that $\mathbf{D}[a,b] \subseteq \mathbf{C}[a,b]$, in a structural way to mean that $\mathbf{D}[a,b]$ is equipped with an injection to $\mathbf{C}[a,b]$. I don't think that this is how they intended it, since in an introductory book you ought to explain that sort of thing. But if the authors are deep into the structural framework, then they might have been thinking this without realising it.
I'm interested in how you interpret the claim that $\mathbf{Set}^{\mathbf{op}}$ is a subcategory of $\mathbf{Rel}$. Is it easy to understand what it means, and what exactly does it mean? Should FMathL accept it, and how should it interpret it?
If $A$ is an object of $C$, then you may interpret $A$ as an object of $C^{op}$, without even an abuse of language. But you can't introduce $A$ as an object of $C$ and $B$ as an object of $C^{op}$ and ask whether $A = B$, or even whether $A \cong B$; that's a type error.
On the surface, this looks like a solution. Your suggestion amounts to having equality on the metalevel but forbidding it on the object level. But I think this does not hold water. I don’t think your suggestion can be implemented consistently in a fully formalized way without producing contradictions.
You're right; what I've said here is contradictory. If (as I said) you may interpret an object $A$ of $C$ as an object of $C^{op}$, then you can compare it to the object $B$ of $C^{op}$, since any two objects of $C^{op}$ may be compared (for isomorphism at least). I wasn't thinking about carefully enough, and I apologise.
(I stand by what I said about distinguishing identity judgements from equality propositions, although this does not appear to be a place that it applies. In fact, I think that it must be irrelevant to what we're discussing, since it's a criticism of $\mathbf{ETCS}$ as much as of anything else. So never mind.)
Since we're worrying here about typing errors when comparing two objects of two categories, maybe I should go back to the beginning and say what I think about that, without suggesting that Mike or anybody else would agree with me. (I know that there are differences between Mike's and my philosophy, and we are getting close to some of them.)
Given two arbitrary types (where a type might be the type of elements of a set, or the type of objects of a category, or something else) $X$ and $Y$, and given $A$ of type $X$ and $B$ or type $Y$, it doesn't normally make sense to ask whether $A = B$. However, it is not good design for a mathematical formaliser to throw up an error whenever anybody writes $A = B$ in this context. First it should try to reduce the expressions for $X$ and $Y$ (especially if this is something that can always be efficiently strongly normalised) to see if they come out the same. Even if that fails (and especially if reduction is not confluent or was not completed), then it should give the user an opportunity to specify a type $Z$ (which might be either $X$ or $Y$) and operations to $Z$ from $X$ and $Y$ respectively. If this works, then $A$ and $B$ may now be interpreted as having the same type, and $A = B$ presumably makes sense.
In particular, if $G$ and $H$ have just been introduced as two groups in a context appropriate for group theory, with $X$ and $Y$ the types of elements of $G$ and $H$ respectively, then there is no way to avoid the type error. But if instead $Y$ is the type of elements of $G^{op}$, then it is easy to avoid the type error; probably the system can do it automatically (in fact, probably one has $Y$ defined directly as $X$).
I have used groups here instead of categories to avoid the evil of asking whether two objects in a single given category are equal, which is a different issue (related to that stuff about identity judgements). Of course, the elements of a group correspond to the morphisms of a category, but I don't think that this makes a difference here.
Posted by: Toby Bartels on September 20, 2009 7:42 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
TB: If FMathL can do that with this passage, then that would show its power.
To teach FMathL this power, I first need to understand what “should” be understood after having read Definition 1.1.1 and what after Definition 1.3.1.
You explained only how then the exercise should be understood, in terms of concepts not yet introduced.
How can an automatic system understand things at the very introduction of a theory when the intentions are formulated so poorly?
This is why I had asked. So let me rephrase my request:
I’d like to see a few pages of text that introduce in a formally precise way the full supposed content of what the trained category theorist understands that these two definitions and the exercise should have conveyed to the reader, including any moral an automatic system should follow in interpreting the remainder of category theory, and explaining any abuse of notation or language that is apparent from the presentation of these two definitions in the standard textbooks.
If I get such a description, and if it is logically consistent, I will guarantee that FMathL will be provided with a generic mechanism for interpreting the definitions as written by Asperti and Longo in the correct way, and for identifing a meaningful interpretation of the exercise (together with raising a flag for having discovered a sloppiness.)
But first I need to understand myself clearly enough what you read into the text morally although it is not written there formally.
Posted by: Arnold Neumaier on September 21, 2009 2:43 PM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
I first need to understand what “should” be understood after having read Definition 1.1.1 and what after Definition 1.3.1.
I think that what “should” be understood at this point is that the authors made a mistake in stating the exercise. Seasoned categorists can guess what the authors might have meant, but I would not expect an undergraduate without experience in category theory to be able to.
Posted by: Mike Shulman on September 21, 2009 6:03 PM | Permalink | PGP Sig | Reply to this
### Equality between objects of different types
TB: given A of type X and B or type Y, it doesn’t normally make sense to ask whether A=B. […] it should give the user an opportunity to specify a type Z (which might be either X or Y) and operations to Z from X and Y respectively. If this works, then A and B may now be interpreted as having the same type, and A=B presumably makes sense.
I think something like that is feasible in FMathL. I’ll keep it in mind in its design.
In this connection, would you consider each category to be a separate type? Each object of a category? Each Homset? (I am not sure whether all three simultaneously may be required consistently.)
Posted by: Arnold Neumaier on September 21, 2009 2:52 PM | Permalink | Reply to this
### Re: Equality between objects of different types
I would consider the class of objects of each category to be a separate type, and each homset in each category to be a separate type. In general, I don’t think of a single object of a category as a type (in general, there’s no way for it to have “elements”), although for some particular categories such as $Set$ they can be interpreted that way.
Posted by: Mike Shulman on September 21, 2009 6:13 PM | Permalink | PGP Sig | Reply to this
### Re: Equality between objects of different types
In general, I don’t think of a single object of a category as a type (in general, there’s no way for it to have “elements”), although for some particular categories such as $Set$ they can be interpreted that way.
For students and curious lurkers, see [[concrete category]].
Posted by: Eric Forgy on September 21, 2009 7:54 PM | Permalink | Reply to this
### Reflection
TB: I should have asked if there was an easy user-friendly way to import it.
What can be more user-friendly than typing “import file.con”, or dragging an icon for file.con into the current context window? If you accept everything the imported context is regulating, this is enough. Otherwise you need to edit file.con to suit your needs before importing it. Just deleting something is easy; other things depend on what you want to change, and how.
TB: I’m worried about an analogue of the mismatch between the internal and external languages of a topos that is not well-pointed.
I can’t tell exactly what you are aiming at, but there is always a kind of mismatch between a subject level (probably your external language) and the object level (probably your internal language).
Because of Goedel’s theorem, a subject level is always strictly stronger in proving power than the object level (unless both are inconsistent).
However, there is no analogue of Goedel’s theorem for the descriptive power of a system. A system with weak proving power can still have a descriptive power sufficient to represent all mathematics including their proofs. The reason is that while finding proofs is undecidable in general, checking proofs is constructively possible under quite modest assumptions about the logic.
Thus even though the current FMathL framework supports only a truncated set theory, having power sets only for sets of size up to the continuum, the reflected level (in which all formal reasoning happens) can check all proofs in axiomatic set theory, even those involving inaccessible cardinals. It is only required that you specify the latter in the FMathL specification language, which is based on the truncated set theory.
And it can check all proofs presented in Bishop-type constructive mathematics, if you specify the latter in the FMathL specification language, although the logic in which the specification language is defined is classical.
TB: By Excluded Middle, a set is either inhabited or not; an uninhabited set is (by definition) empty; hence a nonempty set is inhabited. So if there is a global choice function for inhabited sets, then there is one for uninhabited sets.
I see. So one would have to restrict the choice somehow, and change that part of the context. I am not very familiar with the various ways of defining restricted choice, though; maybe you can help me in saying how much choice you want to allow. In any case, I’ll give it thought.
To teach the system the moral of categories, it seems to me necessary and probably sufficient to have choice for elements of equivalence classes.
TB: Give me a paper whose source is available (say from the arXiv), formalised in ZFC (or whatever), and I’ll rewrite it to be formalised in ETCS.
AN: I do not have the patience to undo all the abuses of notation traditionally used in the context of ZFC.
TB: I don’t understand what you’re asking of categorial foundations, then, if no other foundation does it.
You were asking for something formalized in ZFC. I can’t supply that, since I regard abuses of notation as lack of formalization. I mentioned here what I consider a sufficient level of formality.
Thus I am satisfied if you can translate a typical introduction to axiomatic set theory (until natual numbers and functions are available) and an introduction to logic (until variable substitutions and ZFC are available) into a document starting from scratch with the axioms for a category, at a comparable level of rigor and only need a comparable number of pages.
TB: Or do you claim that FMathL does this? Can I take http://www.mat.univie.ac.at/~neum/ms/fmathl.pdf as my text?
The reflection part is only promised there and outlined, not realized. We are working on a prototype version, and hope to have one by the end of the year. At least, FMathL will not have abuses of notation since whatever remains of them will be valid notation rather than abuse. Thus the level or rigor will be higher than in a typical textbook.
TB: we still don’t need a full-fledged set theory (or other foundation of all mathematics), just a way to talk about recursively enumerable sets of natural numbers.
Yes, something like this is sufficient. But if you think that this saves a lot of work, you are mistaken. You can relax the axiom of power sets and delete the axiom of foundations, but you need equivalents of all the rest to be able to define recursively enumerable sets of natural numbers. And you need to build up quite a lot of conceptual machinery before you have all the concepts and properties one is using there without thinking.
The complexity of complete formal foundations do not become apparent before one hasn’t tried to find sone!
TB: In particular, the requirement of a countable set of variables can be replaced with a single variable x and the requirement that X’ is a variable whenever X is; there is no need to say Cantor’s word “countable”.
True. But then you must stick to that convention later on in whatever you do since you do not have anyhing else. This will make your foundation nearly incomprehensible to a human reader. For example, if you agree with the typing paradigm, it will make type checking extremely tedious. But foundations should be easily checkable by hand and teachable in the classroom!
Posted by: Arnold Neumaier on September 19, 2009 9:53 PM | Permalink | Reply to this
### Re: Reflection
So one would have to restrict the choice somehow, and change that part of the context. I am not very familiar with the various ways of defining restricted choice, though; maybe you can help me in saying how much choice you want to allow.
I really only want (as the default, until I choose something stronger) the axiom of unique choice: When $A$ is an inhabited set such that any two elements of $A$ are equal, then we have an element $Choice(A)$ of $A$. Of course, this isn't going to work for the uses to which you put the global choice operator.
Another possibility is to refuse to allow, from the hypothesis that $A = B$, the conclusion that $Choice(A) = Choice(B)$ (even given that $A$ and $B$ are inhabited). This should prevent the proof of Excluded Middle (in a framework with intuitionistic logic), while still allowing a definition like $Choice(Comp Ord Fld)$ for $\mathbb{R}$. I don't think that you should ever need to deduce, say, $Choice(Comp Ord Fld) = Choice(Comp Arch Fld)$ from $Comp Ord Fld = Comp Arch Fld$ (if you ever even want to say the latter).
Posted by: Toby Bartels on September 19, 2009 11:04 PM | Permalink | Reply to this
### Re: Reflection
TB: I really only want (as the default, until I choose something stronger) the axiom of unique choice: Of course, this isn’t going to work for the uses to which you put the global choice operator.
Yes, this is too weak for FMathL purposes. But note that there is no default on the reflection level. The user decides which parts of the FMathL framework reflection should be used.
The FMathL default defined in the framework paper only controls the meaning of the specification language in which everything formal is then represented.
TB: Another possibility is to refuse to allow, from the hypothesis that A=B, the conclusion that Choice(A)=Choice(B)
This is against the basic principle of FMathL that substitution of equals is unrestricted.
This means that someone would have to write a UniqueChoice context and create workarounds of all the uses of standard Choice in the FMathL reflection.
Once this is done, working formally in your setting is as easy as working in the standard context.
But the library of theorems in that contexts would have to be recreated from the standard library by checking which proofs still hold in the new context, and by trying to repair the others.
However this is not more difficult than what anyone has to do who changes something in established foundations. In FMathL it is probably even easier since the precise semantics will help in automatic translation and checking.
Posted by: Arnold Neumaier on September 20, 2009 11:55 AM | Permalink | Reply to this
### Re: Reflection
Thus I am satisfied if you can translate a typical introduction to axiomatic set theory (until natual numbers and functions are available) and an introduction to logic (until variable substitutions and ZFC are available) into a document starting from scratch with the axioms for a category, at a comparable level of rigor and only need a comparable number of pages.
As I feel that I’ve said numerous times, a structural/type-theoretic foundation does not need to start with the axioms for a category.
I feel sure that I could do this, but unfortunately I don’t have anywhere near the time it would take at the moment. (-: Sorry if that sounds like a cop-out, but actually I don’t really even have the time to be engaging in this discussion…
Posted by: Mike Shulman on September 20, 2009 6:26 AM | Permalink | PGP Sig | Reply to this
### Re: Reflection
MS: a structural/type-theoretic foundation does not need to start with the axioms for a category.
Yes, and then there are fewer problems. I’ll look at SEAR from this point of view, once I have more time to study it more deeply.
MS: I feel sure that I could do this, but unfortunately I don’t have anywhere near the time it would take at the moment. (-: Sorry if that sounds like a cop-out, but actually I don’t really even have the time to be engaging in this discussion…
I understand. This rhetorical request was thought to be an explication of what would revise my current judgment on the overhead of categorial foundations rather than as a request that you or TB should actually do this; especially since I know that you prefer structural non-categorial foundations.
Posted by: Arnold Neumaier on September 20, 2009 12:22 PM | Permalink | Reply to this
### Re: Reflection
I am satisfied if you can translate a typical introduction to axiomatic set theory (until natual numbers and functions are available) and an introduction to logic (until variable substitutions and ZFC are available) into a document starting from scratch with the axioms for a category, at a comparable level of rigor and only need a comparable number of pages.
For the first one, how about Sections 1.3 and 2.1–5 of http://www.math.uchicago.edu/~mileti/teaching/math278/settheory.pdf?, which I found by Googling "axiomatic set theory". For the second one, I'm not sure what kind of work you're thinking of, so an example would help.
I reserve the right for lack of time not to complete these, but I'll try to at least indicate how they would be done.
Posted by: Toby Bartels on September 20, 2009 7:44 PM | Permalink | Reply to this
### Re: Reflection
AN: a typical introduction to axiomatic set theory (until natual numbers and functions are available) and an introduction to logic (until variable substitutions and ZFC are available)
TB: For the first one, how about Sections 1.3 and 2.1–5 of http://www.math.uchicago.edu/~mileti/teaching/math278/settheory.pdf?
I think to be able to reflect the notion of an expression one also needs finte products and recursive definitions; thus you’d add Sections 2.7 and 2.8. Section 2.9.1 is also used in the standard descriptions of logic.
Thus 17 pages comprising elementary axiomatic set theory are sufficient to reflect logic.
TB: For the second one, I’m not sure what kind of work you’re thinking of, so an example would help.
I have no good online reference (although there might well be some - didn’t search thoroughly). But Section 1-13 of Chapter 5 of the book “The mathematics of metamathematics” by Rasiowa and Sikorski is a suitable template. It presents in about 50 verbose pages (one may skip Section 13A-D) intuition and formality about reflecting mathematical theories and in particular reflects ZF in Section 13E, using on the informal level informal notions of sets, functions, sequences (i.e., what axiomatic set theory tells us can be encoded as such).
Thus 50 pages of elementary logic are sufficient to reflect axiomatic set theory.
Therefore, self-reflective (and hence fully self-explaining) foundations of traditional set-based mathematics can be developed in about 70 pages of mathematical text, at the usual level of mathematical rigor and in a language easily understood by anyone who passed the undergraduate math stage.
Posted by: Arnold Neumaier on September 21, 2009 11:01 AM | Permalink | Reply to this
### Re: Reflection
All right, I'll do Section 1.3 and start on Chapter 2 of Mileti, and hopefully the point at which it is clear that I could continue indefinitely will come before the point at which it is tiresome to continue. (^_^)
I should be able to look at Rasiowa & Sikorski later this week. (It's been recommended to me before, in other contexts, so it's about time that I did!) Hopefully, that will just be a matter of checking that the informal notions used have already been successfully formalised in the translation of Mileti to categorial foundations.
Posted by: Toby Bartels on September 21, 2009 11:00 PM | Permalink | Reply to this
### Re: Reflection
Temporarily at http://tobybartels.name/settheory.ps does this through Section 2.5. The first parts are completely redone, but afterwards it becomes almost a matter of cutting and pasting. The rest (except Section 2.6, not in the assignment) would be even more like that, although I could do it if there are still questions as to how it would go.
I tried to make the development as tight as possible, although I couldn't help but point out a few things, like the universal property of the Cartesian product and which proofs require classical logic. As Mike did with SEAR, I took axioms that focus on elements rather than arbitrary functions. However, since the assignment was to start with the elementary axioms of a category, I defined an element to be a certain sort of function, as in ETCS.
Personally, I would start with something more like SEAR (or SEPS) and take a leisurely pace, pointing out all of the sights along the way. But that would not meet the assignment.
Posted by: Toby Bartels on October 15, 2009 11:45 AM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
One needs to consider both aspects - efficiency for the user and efficiency for the system.
That’s a fair point, although I feel that types are important enough that if I were designing a system, I would want to put in whatever extra effort is necessary to include them.
BTW, I didn’t mean my mention of the possibility of connecting occurrences of $A$ by isomorphisms as a serious suggestion that one implement; my preferred solution is what Toby said in response.
How do I then refer to the reals with ∈ as opposed to the reals without ∈? They are no longer the same objects
They are the same objects. $\in$ is extra structure on the same set of elements.
Posted by: Mike Shulman on September 20, 2009 6:13 AM | Permalink | PGP Sig | Reply to this
### Re: can objects and morphisms belong to several categories?
AN: How do I then refer to the reals with ∈ as opposed to the reals without ∈? They are no longer the same objects
MS: They are the same objects. ∈ is extra structure on the same set of elements.
This makes sense only if you agree that two categories based on the same set of objects with different extra structure on it have the same objects. But I thought this is something you wanted to avoid!
The problem is that an automatic system must translate these sorts of statements into a well-defined fully formalized statement that can be fed to a theorem prover for checking proofs involving them.
I still don’t see how this can be done, given the standard axioms for category theorem and its derived conceptual basis as given in standard works.
Posted by: Arnold Neumaier on September 20, 2009 8:51 AM | Permalink | Reply to this
### Re: can objects and morphisms belong to several categories?
They are the same objects. $\in$ is extra structure on the same set of elements.
This sounds so strange to me that I'm sure that there must be a misunderstanding here, either of Mike or by Mike.
We have the complete ordered field $\mathbb{R}$ of real numbers, and we have $\mathbb{R}$ equipped with a binary relation $\in$ from $\mathbb{Q}$. (Presumably, $\in$ is either $\lt$, $\gt$, $\leq$, or $\geq$, but we haven't specified.) These are different things, not in the sense that we would put $\ne$ between them (which would be a typing error), but in the sense that we would not accept $=$ between them (which would also be a typing error).
However, there is also an obvious forgetful operation (I won't say ‘functor’ since I haven't specified categories, although it would be easy to do that) from the latter to the former. A user-friendly system for mathematics should even be able to detect this and put it in automatically wherever it's needed.
Posted by: Toby Bartels on September 20, 2009 6:18 PM | Permalink | Reply to this
### Re: objects and morphisms can belong naturally to several categories
Definition 1.3.2 of Asperti and Longo, Categories, Types and Structures
I'd also like to see a formalisation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), as literally true! It seems doubtful to me; I know what they mean, but I need to translate it.
Posted by: Toby Bartels on September 18, 2009 10:32 PM | Permalink | Reply to this
### Re: objects and morphisms can belong naturally to several categories
I’d also like to see a formalisation that takes both their definition, and their statement that begins the Exercise in that section (using the previous Definition 1.3.1), as literally true!
This is a really good example of the sort of mistakes you can end up making when you do anything with categories non-structurally. Category theory wants to be purely structural with all its heart ♥. (-:
Posted by: Mike Shulman on September 19, 2009 2:33 AM | Permalink | PGP Sig | Reply to this
### Re: objects and morphisms can belong naturally to several categories
ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation $R$ (suitably defined)
I would like to see you formalise this! Or the (related but perhaps simpler) statement that $Group$ is a subcategory of $Set$. I think that a lot of mathematicians do think that way, but it is difficult to formalise; the difficulties have to do with the difference between structure and properties. I usually see it as a hallmark of sophisticated understanding to drop these ideas, but sometimes I also wonder how far they can be maintained.
Posted by: Toby Bartels on September 18, 2009 9:55 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
It also seems to me that all the things about real numbers you are saying that FMathL does right, it only does right specifically for real (or complex) numbers, because you have built them into the system by fiat. If FMathL didn’t include axioms specifying that there was a particular set called $\mathbb{R}$ with particular properties, then you’d have to construct it just like in any other set theory, and in particular you’d have to choose an implementation (thereby making $1\in\sqrt{2}$ either true or false), and it would also no longer be true that $2\in\mathbb{N}$ and $2\in\mathbb{R}$ were the same thing.
But $\mathbb{R}$ and $\mathbb{C}$ are by no means the only mathematical object that has such problems! For example, suppose I want to study the quaternions $\mathbb{H}$ in FMathL. I could define them as ordered pairs of complex numbers, or as $2\times 2$ complex matrices, or as $4\times 4$ real matrices, and in each case different “accidental” things would be true about them just as in ZF, and in no case would $\mathbb{C}$ actually be a subset of $\mathbb{H}$. And so on.
Posted by: Mike Shulman on September 18, 2009 9:24 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
MS: you’d have to choose an implementation (thereby making $1\in\sqrt{2}$ either true or false)
No. I can define the domain $COF$ of all complete ordered fields, show by some construction that $COF$ is not empty, and then specify $\mathbb{R}$ as $Choice(COF)$. This selects a unique, implementation-dependent copy of the real numbers without any accidental properties. (Maybe this will be the form of the actual later implementation. These things will be decided only after we have done enough experiments.)
MS: But $\mathbb{R}$ and $\mathbb{C}$ are by no means the only mathematical object that has such problems! For example, suppose I want to study the quaternions $\mathbb{H}$ in FMathL. I could define them as ordered pairs of complex numbers, or as 2×2 complex matrices, or as 4×4 real matrices, and in each case different “accidental” things would be true about them just as in ZF, and in no case would $\mathbb{C}$ actually be a subset of $\mathbb{H}$.
Indeed, if you define them in this way, this is what happens. But I would call this not definitions but constructions, and constructions usually have accidental properties.
To get things right without any artifacts, one needs to think more categorially, and define $\mathbb{H}$ as $Choice(X)$, where $X$ denotes the domain of all skew fields that contain $\mathbb{C}$ as a subfield of index 2. Any of the constructions you gave shows that $X$ is nonempty, so that this recipe defines a unique existing object whose only decidable properties are those that one can derive from the assumptions made. (It has in addition lots of undecidable, implementation-dependent properties, though.)
The use of Bourbaki’s global choice operator is essential for this golden road to the essence of mathematics.
Posted by: Arnold Neumaier on September 18, 2009 1:10 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
It sounds like you’re saying that in order to construct anything in FMathL without extraneous details I need to find a way to describe it in terms that characterize it uniquely. That seems to me like a mighty tight straightjacket!! Suppose I’m Hamilton inventing the quaternions, and maybe I’m ahead of my time and I’ve realized that they could be constructed from the complex numbers in several different ways, but I don’t yet have any idea how to characterize them uniquely. I didn’t set out to study skew fields containing $\mathbb{C}$ as a subfield of index 2, I set out to study a particular thing, which could be constructed in several ways, and only later discovered that it was the unique skew field containing $\mathbb{C}$ with index 2. It seems to me that uniqueness theorems such as these generally come later, after a new object has been studied in its own right for a while and its essential properties isolated.
Here’s a more modern example: what about the stable homotopy category? It has lots of different constructions; you can start with lots of different kinds of point-set-level spectra. But although all these constructions give the same result, off the top of my head I’m not at all sure how to characterize the result uniquely without referring to the specific constructions of it.
Posted by: Mike Shulman on September 18, 2009 6:17 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
MS: It sounds like you’re saying that in order to construct anything in FMathL without extraneous details I need to find a way to describe it in terms that characterize it uniquely.
No. Of course one can construct in FMathL a skewfield via complex 2 x 2 matrices, and call it the quaternions. Then one can construct another skew field vial real 4x4 matrices and call it the tetranions. Then one discovers the theorem that tetranions are isomorphic to quaternions. At this point, it makes sense to revise notation (a cvommon process in mathematics), consider the domain $X$ of all skewfields isomorphic to these particular skew fields, and to define the quaternions as Choice($X$). When later the characterization theorem is discovered, it just gives a simpler description for $X$, but the definition is already stable once you know that you want to abstract from the accidentals of the construction. One can do this even if only a single construction exists (e.g., for the Monster simple group).
Posted by: Arnold Neumaier on September 18, 2009 6:57 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
one can avoid the multiple meanings of $\mathbb{N}$ by interpreting it in each context as the richest structure that can be picked up from the context.
This isn’t “avoiding” the multiple meanings of $\mathbb{N}$, it is merely inferring which meaning is meant, which I am all in favor of. Category theorists do this all the time just like other mathematicians. But the fact that one has to do an interpretation means that the multiple meanings still exist.
The fact that $\mathbb{N}$ can indicate different levels of structure depending on the context is precisely what I mean by “overloading.”
this uses the same idea as the categorial approach to foundations, but to turn each context into a category
I don’t think I ever advocated turning every context into a category. And, as I’ve said, I think that calling this the “categorial” approach to foundations is misleading; it doesn’t necessarily have anything to do with categories. The point I’m trying to make is that mathematics is typed, and that’s just as true in type theory and non-categorial structural set theory as it is in ETCS or CCAF.
Posted by: Mike Shulman on September 18, 2009 8:52 PM | Permalink | PGP Sig | Reply to this
### Re: CCAF, ETCS and type theories
MS: The point I’m trying to make is that mathematics is typed, and that’s just as true in type theory and non-categorial structural set theory as it is in ETCS or CCAF.
I made some first comments on your SEAR page, but need to look at the concepts more thoroughly.
The point I’m trying to make is that typing does not solve many of the disambiguation problerms that an automatic math system must be able to handle. it solves only some of them. Since a more flexible disambiguation system is needed anyway, it can as well replace the typing. By doing so in an FMathL like fashion, it automatically eliminates the multiplicity of typed instances of the same thing.
Posted by: Arnold Neumaier on September 18, 2009 10:43 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
The use of Bourbaki’s global choice operator is essential for this golden road to the essence of mathematics.
That's too bad, because I would like to do mathematics in which the axiom of choice is optional, without having to code it all myself. I think that I can work well in a system where $\mathbb{N}$ has a maximal structure (at least in an ever-growing, potential infinity sort of way), even though that's not exactly how I would normally think of $\mathbb{N}$, but I really won't find it useful if choice is essential.
I hope that it isn't really essential for what you're doing here. After all, you don't care which particular object $Choice(X)$ is, and the user won't even have access to those details. So I hope that there is a way to implement this that avoids anything that proves the axiom of choice.
Posted by: Toby Bartels on September 18, 2009 10:07 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
AN: The use of Bourbaki’s global choice operator is essential for this golden road to the essence of mathematics.
TB: That’s too bad, because I would like to do mathematics in which the axiom of choice is optional, without having to code it all myself.
The way FMathL will be implemented is fully reflective. The current paper essentially serves to define a common metalevel, within which one can objectively (i.e., with the same meaning in all subjective implementations) talk about a constructive description of the FMathL implementation. The latter is what is actiually carried out. Thus, one must trust the axiom of choice to trust the system, but what is proved inside the system is fully configurable, since you can simple build your context without importing all the modules needed for a full reflection of FMathL. Thus you’d only have to create your own constructive restricted version of Choice (if this hasn’t been done already by someone else), most likely that Choice is defined only for inhabited sets rather than for all nonempty sets, and things will work as before. You need some construction to know that you can choose, but you have that in a constructive approach anyway.
Actually, after we have successfully reflected the whole FMathL framework, we’ll take stock to see what is really needed for a minimal part that can already reflect the whole framework. Then we redefine this as the core, and there is a possibility that this will be constructive. Only the core need to be trusted and checked for correctness, since the remainder will be definable in terms of the core.
Of course, consistency of the core does not imply consistency of the theories built later with the help pof the core and user specifications. Thus, even in a weak, but fully reflective core it will be possible to specify ZFC with all sorts of inaccessible cardinals, say, but their consistency is left to the user.
AN: ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation R (suitably defined)
TB: I would like to see you formalise this! Or the (related but perhaps simpler) statement that Group is a subcategory of Set.
I’ll try to do that sooner or later, but in my experience this may take days or weeks to crystallize into something practical (if it is at all possible). So don’t expect a quick reply.
Posted by: Arnold Neumaier on September 18, 2009 11:22 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
Thus you'd only have to create your own constructive restricted version of Choice […] and things will work as before.
And I can import everything that's already been defined your way? That might work.
most likely that Choice is defined only for inhabited sets rather than for all nonempty sets,
For the record, that won't work. (At least, if it did, then you'd have Excluded Middle implies Choice, which I wouldn't want either!)
Actually, after we have successfully reflected the whole FMathL framework, we’ll take stock to see what is really needed for a minimal part that can already reflect the whole framework. Then we redefine this as the core, and there is a possibility that this will be constructive.
That sounds good!
I would like to see you formalise this! Or the (related but perhaps simpler) statement that Group is a subcategory of Set.
I’ll try to do that sooner or later, but in my experience this may take days or weeks to crystallize into something practical (if it is at all possible). So don’t expect a quick reply.
I understand. But I would be very interested to see it.
Posted by: Toby Bartels on September 19, 2009 12:15 AM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
TB: And I can import everything that’s already been defined your way?
Yes, since that is what reflection is all about. A self-reflective foundation (and only that is a real foundation) can explain formally all the stuff its talking about informally. To explain it formally means to have a module that defines its syntax and semantics. Of course, in a well-designed package, you have access to that and can use, combine, and modify it in any way you like. (In the latter case, of course, the trust certificates will be reset to trusted by you only.)
AN: most likely that Choice is defined only for inhabited sets rather than for all nonempty sets,
TB: For the record, that won’t work.
Actually, I just noticed that Axiom A19, is alredy formulated as an axiom of global constructive choice.
TB: At least, if it did, then you’d have Excluded Middle implies Choice, which I wouldn’t want either!
I don’t understand; please indicate the argument.
However, it is well-known that Choice implies Excluded Middle, and this remains true in FMathL; see Section 3.1. If you don’t like this, you’d have to relax the axioms for sets (Section 2.14) in a way that makes this proof invalid.
TB: Give me a paper whose source is available (say from the arXiv), formalised in ZFC (or whatever), and I’ll rewrite it to be formalised in ETCS. (The paper can include its own specification of ZFC too, and mine will include its own specification of ETCS.)
I will not be able to meet that challenge since I do not have the patience to undo all the abuses of notation traditionally used in the context of ZFC. I argue that precisely this should not be demanded from a really good foundation of mathematics.
However, Bourbaki’s Elements of Mathematiks at least clarify each form of abuse of notation on first use, so that an automatic system going through it in linear order will pick up all the language updates along the way. So I believe that, in principle, Bourbaki meets your request, though not in a minimally short way since they aimed at completeness, not at minimal reflection.
TB: Are you telling me that even ZFC needs a set theory to serve as its foundation? I can’t think of any way to interpret this to make it true.
Yes, of course. This is done in books on logic. They start with an informal set theory, then introduce the machinery to formally talk about first order logic, and then produce at some stage the definition of ZFC.
To interpret it without circularity, logicians distinguish between the metalevel (the informal set theory) and the object level (the formal theory), and then build hierarchies for reflection. They build them upwards to metametalevels etc..
FMathL rather reflects downwards to objectobject levels, etc., which is more appropriate from an implementation point of view.
TB: You have to say something like this to reflect first-order logic in some mathematical foundation. But not to do first-order logic.
To do it, you just need a mind in which the logic is already implemented.
But to communicate objectively what you do, you need to reflect it: You need to understand what it means that you can choose any indexed letter as a variable, and what it means to have a substitution algorithm, etc.. Once you try to communicate this to a novice who hasn’t already an equivalent implementation, you find yourself teaching him informal concepts of sets and functions with their properties.
A foundation is just supposed to have that done rigorously.
Posted by: Arnold Neumaier on September 19, 2009 5:23 PM | Permalink | Reply to this
### Re: CCAF, ETCS and type theories
That's fine.
Yes, since that is what reflection is all about. A self-reflective foundation (and only that is a real foundation) can explain formally all the stuff its talking about informally. To explain it formally means to have a module that defines its syntax and semantics. Of course, in a well-designed package, you have access to that and can use, combine, and modify it in any way you like. (In the latter case, of course, the trust certificates will be reset to trusted by you only.)
I should have asked if there was an easy user-friendly way to import it. Also I should learn more about the practical problems of reflection, because I have another question, which I don't know how to ask; but I'm worried about an analogue of the mismatch between the internal and external languages of a topos that is not well-pointed. And I'm concerned about the trust certificates; since I'm reflecting in order to use a weaker foundation (that is, fewer assumptions, stricter requirements), I would like FMathL to verify for me those results that go through.
most likely that Choice is defined only for inhabited sets rather than for all nonempty sets,
For the record, that won't work. At least, if it did, then you'd have Excluded Middle implies Choice, which I wouldn't want either!
I don’t understand; please indicate the argument.
By Excluded Middle, a set is either inhabited or not; an uninhabited set is (by definition) empty; hence a nonempty set is inhabited. So if there is a global choice function for inhabited sets, then there is one for uninhabited sets. Conversely, the argument that Choice implies Excluded Middle needs only the version of choice for inhabited sets (in fact, only for quotient sets of $\{0,1\}$). I would like this to be optional.
Give me a paper whose source is available (say from the arXiv), formalised in ZFC (or whatever), and I’ll rewrite it to be formalised in ETCS.
I will not be able to meet that challenge since I do not have the patience to undo all the abuses of notation traditionally used in the context of ZFC. I argue that precisely this should not be demanded from a really good foundation of mathematics.
I don't understand what you're asking of categorial foundations, then, if no other foundation does it. Or do you claim that FMathL does this? Can I take http://www.mat.univie.ac.at/~neum/ms/fmathl.pdf as my text?
Are you telling me that even ZFC needs a set theory to serve as its foundation? I can't think of any way to interpret this to make it true.
Yes, of course. This is done in books on logic. They start with an informal set theory, then introduce the machinery to formally talk about first order logic, and then produce at some stage the definition of ZFC.
I would like to see a book on logic (syntactic logic, not model theory) that does this. To describe the logic, we need a metalanguage (as you say), but we still don't need a full-fledged set theory (or other foundation of all mathematics), just a way to talk about recursively enumerable sets of natural numbers. (But I think that it's more common to talk about finite lists from a fixed finite language than natural numbers.)
In particular, the requirement of a countable set of variables can be replaced with a single variable $x$ and the requirement that $X'$ is a variable whenever $X$ is; there is no need to say Cantor's word ‘countable’.
Posted by: Toby Bartels on September 19, 2009 8:58 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
I (perhaps wrongly) assumed that “CCAF” meant the same thing as Lawvere originally meant by it, and I don’t think this included ETCS. But I can’t find my copy of Lawvere’s original paper at the moment, so I could be wrong.
Posted by: Mike Shulman on September 16, 2009 9:01 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
http://138.73.27.39/tac/reprints/articles/11/tr11abs.html Longer version of the 1964 paper
“Philosophers and logicians to this day often contrast “categorical” foundations for mathematics with “set-theoretic” foundations as if the two were opposites. Yet the second categorical foundation ever worked out, and the first in print, was a set theory—Lawvere’s
axioms for the category of sets, called ETCS, (Lawvere 1964). These axioms were written soon after Lawvere’s dissertation sketched the category of categories as a foundation, CCAF, (Lawvere 1963). They appeared in the PNAS two years before axioms for CCAF were published (Lawvere 1966). The present longer version was available since April 1965 in the Lecture Notes Series of the University of Chicago Department of Mathematics.1 It gives the same definitions and theorems, with the same numbering as the 5 page PNAS version, but with fuller proofs and explications.”
Posted by: Stephen Harris on September 17, 2009 1:01 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
The open question was: Does his original version of CCAF have an axiom saying that there is a category of sets satisfying ETCS?
Posted by: Arnold Neumaier on September 17, 2009 4:04 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
Posted by: Arnold Neumaier
“The open question was: Does his original version of CCAF have an axiom saying that there is a category of sets satisfying ETCS?”
————————————
SH: Mike said “original paper” which I wasn’t sure meant Lawvere’s thesis which was also published 40 years later. Precisely speaking, Lawvere doesn’t present formalized axioms but more informally.
http://138.73.27.39/tac/reprints/index.html [tr5]
From the Author’s Comments on his (Lawvere) 1963 PhD. thesis Describing January, 1960
————————
“My dream, that direct axiomatization of the category of categories would help in overcoming alleged set-theoretic difficulties, was naturally met with skepticism by Professor Eilenberg when I arrived (and also by Professor Mac Lane when he visited Columbia).”
————————–
From the thesis Introduction
“One so inclined could of course view all mathematical assertions of Chapter I as axioms.” …
“Since all these notions turn out to have first-order characterizations (i.e. char acterizations solely in terms of the domain, codomain, and composition predicates and the logical constants =, ∀, ∃, ⇒, ∧, ∨, ¬ ), it becomes possible to adjoin these characterizations as new axioms together with certain other axioms, such as the axiom of choice, to the usual first-order theory of categories (i.e. the one whose only axioms are associativity, etc.) to obtain the first-order theory of the category of categories. Apparently a great deal of mathematics (for example this paper) can be derived within the latter theory. We content ourselves here with an intuitively adequate description of the basic operations and special objects in the category of categories, *leaving the full formal axioms to a later paper*. We assert that all that we do can be interpreted in the theory ZF3, and hence is consistent if ZF3 is consistent. By ZF3 we mean the theory obtained by adjoining to ordinary Zermelo-Fraenkel set theory…”
—————————
SH: I think the formal axioms were presented later in
Lawvere, F. William (1966)*, The category of categories as a foundation for mathematics, in S.Eilenberg et al., eds, ‘Proceedings of the Conference on Categorical Algebra, La Jolla, 1965’, Springer-Verlag, pp. 1–21.
Colin McLarty said [tr11], “Lawvere’s axioms for the category of sets, called ETCS, (Lawvere 1964). These axioms were written soon *after Lawvere’s dissertation sketched the category of categories as a foundation, CCAF, (Lawvere 1963). They appeared in the PNAS two years before axioms for CCAF were published (Lawvere 1966)*. [cited above]
SH: So in my inexpert opinion, the original version of CCAF would be Lawvere’s dissertation (1963) and there are no formal axioms of ETCS so perhaps they could be called assertions, although that is not how Lawvere thought about them.
For other non-experts: I’m including some of my notes from the thesis which include ideas tantamount to informal axioms.
Seven ideas introduced in the 1963 thesis
(1) The category of categories is an accurate and useful framework for algebra, geometry, analysis, and logic, therefore its key features need to be made explicit.
(2) The construction of the category whose objects are maps from a value of one given functor to a value of another given functor makes possible an elementary treatment of adjointness free of smallness concerns and also helps to make explicit both the existence theorem for adjoints and the calculation of the specific class of adjoints known as Kan extensions.
(3)* Algebras (and other structures, models, etc.) are actually functors
to a background category from a category which abstractly concentrates the essence of a certain general concept of algebra, and indeed homomorphisms are nothing but natural transformations between such functors. Categories of algebras are very special, and explicit *axiomatic characterizations of them can be found, thus providing a general guide to the special features of construction in algebra.
(4) The Kan extensions themselves are the key ingredient in the unification of a large class of universal constructions in algebra (as in [Chevalley, 1956]).
(5) The dialectical contrast between presentations of abstract concepts and the abstract concepts themselves, as also the contrast between word problems and groups, polynomial calculations and rings, etc. can be expressed as an explicit construction of a new adjoint functor out of any given adjoint functor. Since in practice many abstract concepts (and algebras) arise by means other than presentations, it is more accurate to apply the term “theory”, not to the presentations as had become traditional in formalist logic, but rather to the more invariant abstract concepts themselves which serve a pivotal role, both in their connection with the syntax of presentations, as well as with the semantics of representations.
(6) The leap from particular phenomenon to general concept, as in the leap from cohomology functors on spaces to the concept of cohomology operations, can be analyzed as a procedure meaningful in a great variety of contexts and involving functorality and naturality, a procedure actually determined as the adjoint to semantics and called extraction of “structure” (in the general rather than the particular sense of the word).
(7) The tools implicit in (1)–(6) constitute a “universal algebra” which should not only be polished for its own sake but more importantly should be applied both to constructing more pedagogically effective unifications of ongoing developments of classical algebra, and to guiding of future mathematical research.
In 1968 the idea summarized in (7) was elaborated in a list of solved and unsolved problems, which is also being reproduced here.”
——————————-
“My stay in Berkeley tempered the naive presumption that an important
preparation for work in the foundations of continuum mechanics would
be to join the community whose stated goal was the foundations of
mathematics.”
——————————-
I read the Preface to Yves Bertot’s book on Coq and it took about 20 years to develop which makes me think that your time frame of 5 years isn’t very long.
Posted by: Stephen Harris on September 18, 2009 1:37 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
SH: I read the Preface to Yves Bertot’s book on Coq and it took about 20 years to develop which makes me think that your time frame of 5 years isn’t very long.
Fortunately, I can build upon all this previous work rather than having to develop it all again. I have it easier with lessns that Coq had to learn the hard way.
Nevertheless, the 5 years are based on the assumption that 10 people work full-time on it for 5 years. I am trying to get financial support to achieve this, but it isn’t easy. At the moment I have only 2 people for the next two years, and one more for 1/2 a year.
SH: It seems Mike is right about ETCS and CCAF being quite distinct.
This is undisputed. But my claim was that CCAF contains ETCS, whereas his claim was that CCAF is completely independent of ETCS. (If it were so, how could CCAF be a foundation of mathematics, where sets are necessary?)
Posted by: Arnold Neumaier on September 18, 2009 9:48 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
his claim was that CCAF is completely independent of ETCS. (If it were so, how could CCAF be a foundation of mathematics, where sets are necessary?)
I don't understand this. How can ETCS be a foundation of mathematics when it's independent of ZFC?
Posted by: Toby Bartels on September 18, 2009 9:39 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
AN: his claim was that CCAF is completely independent of ETCS. (If it were so, how could CCAF be a foundation of mathematics, where sets are necessary?)
TB: I don’t understand this. How can ETCS be a foundation of mathematics when it’s independent of ZFC?
I don’t understand how your question relates to my remark, which had no reference to ZFC.
ETCS is a set theory based on category theory, which is based on a set theory. Thus one can claim that ETCS is a different set-theoretic foundation.
CCAF is a category theory based on category theory. It needs some way to create sets, otherwise it cannot serve as its own metatheory. (For eexample, tt needs to be able to formalize things as “there is a countable set of variables” before it can talk about predicate logic. McLarty’s version of CCAF does this by explicitly requiring that CCAF contains a copy of ETCS.
I see now way to avoid something like this, but Mike Schulman had proposed from memory that Lawvere’s CCAF had no ETCS inside it. I have no access to Lawvere’s CCAF paper, so I can’t check.
Posted by: Arnold Neumaier on September 18, 2009 10:54 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
I don’t understand how your question relates to my remark, which had no reference to ZFC.
Sorry, I didn't express myself very well. I meant that the question that you asked me makes no more sense in that context (to me) than the question that I asked you.
ETCS is a set theory based on category theory, which is based on a set theory.
And now I don't understand the second clause of this sentence! Or rather, I think that I understand, but if so, then it's wrong. The category theory that ETCS is based is not based on a set theory; it's elementary.
Posted by: Toby Bartels on September 18, 2009 11:02 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
TB: The category theory that ETCS is based is not based on a set theory; it’s elementary.
Even elementary stuff expressed in first order logic needs a set theory to be able to serve as foundation. For example, it needs to be able to formalize statements such as “there is a countable set of variables” before it can talk about predicate logic.
TB: I should clarify further that CCAF certainly has a set theory in it, in the same way that any set theory has a category theory in it. That is, you can define sets in terms of categories, and go on from there.
This is precisely what the ETCS inside McLarty’s CCAF does. And I’d be surprised if Lawvere had taken a different road.
Posted by: Arnold Neumaier on September 18, 2009 11:33 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
Even elementary stuff expressed in first order logic needs a set theory to be able to serve as foundation.
What??? Are you telling me that even ZFC needs a set theory to serve as its foundation? I can't think of any way to interpret this to make it true.
“there is a countable set of variables”
You have to say something like this to reflect first-order logic in some mathematical foundation. But not to do first-order logic.
Posted by: Toby Bartels on September 19, 2009 12:26 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
Even elementary stuff expressed in first order logic needs a set theory to be able to serve as foundation.
Everything needs to be based on set theory, therefore no matter what alternate foundations you consider, they must be based on set theory, therefore the only foundation for mathematics is set theory.
Wait, what?
Posted by: John Armstrong on September 19, 2009 2:32 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
It seems that, again, the notion needed is some kind of reflection (or maybe simulation?); since Set Theory untidily encapsulates “all” of mathematics, any competing foundational system ought to allow a reasonable reflection or simulation of Set Theory.
Maybe we need a category of foundational systems, simulations/reflections as morphisms between them, (higher-dimensional stuff?) … ?
Posted by: some guy on the street on September 19, 2009 5:38 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
TB: I should clarify further that CCAF certainly has a set theory in it, in the same way that any set theory has a category theory in it. That is, you can define sets in terms of categories, and go on from there.
AN: This is precisely what the ETCS inside McLarty’s CCAF does. And I’d be surprised if Lawvere had taken a different road.
—————————————
Category theory used to have a set-theoretic background. Lawvere provided an alternative. The Category = Set is fundamental. But I don’t think that ETCS is fundamental to CCAF because ETCS is just one formulation of Cat = Set.
http://plato.stanford.edu/entries/category-theory/
“An alternative approach, that of Lawvere (1963, 1966), begins by characterizing the category of categories, and then stipulates that a category is an object of that universe.
Identity, morphisms, and composition satisfy two axioms:
Associativity xxx…
Identity xxx…
This is the definition one finds in most textbooks of category theory. As such it explicitly relies on a set theoretical background and language.
An alterative, suggested by Lawvere in the early sixties, is to develop
an adequate language and background framework for a category of categories.”
SH: The Cat named Set is just one category in the category of categories. I don’t think that Set, has to be ETCS and so ETCS is not foundational to CCAF. I don’t think the Cat = Set is rigid in the axioms that it allows, so is not limited to ETCS. After all ETCS, was just invented for the benefit on a two semester course, and later the axioms in it were simplified.
Posted by: Stephen Harris on September 19, 2009 9:13 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
I should clarify further that CCAF certainly has a set theory in it, in the same way that any set theory has a category theory in it. That is, you can define sets in terms of categories, and go on from there. Depending on exactly how CCAF works, you would define a set as a discrete category, or perhaps a discrete skeletal category, or something like that.
Posted by: Toby Bartels on September 18, 2009 11:12 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
AN wrote: “I see now way to avoid something like this, but Mike Schulman
had proposed from memory that Lawvere’s CCAF had no ETCS inside it. I have no access to Lawvere’s CCAF paper, so I can’t check.” ———————
SH: From the quote below you will see that Lawvere’s version of Category Theory does not require a set-theoretical background. I think this is assumed on this forum. But the Category of Categories, CCAF, can have the category of Set, as one of its objects. That doesn’t make set theory foundational to CT in Lawvere’s approach. The axioms in ETCS might well qualify as axioms in the Cat = Set. Remember that Lawvere invented ETCS for a simplification of set theory for a one year class. Later, he simplified the axioms which ETCS contained. I don’t think ETCS is intrinsic to CCAF, but it is (I think) one of the possible axiomatic formulations for the Category of Set which is intrinsic to CCAF in the same sense that all possible categories in the category of categories are in some sense intrinsic, at least after being identified. Even if Lawvere had used ETCS as his example for the Cat = Set, in his 1963 thesis or 1966 paper, I think other set axiomatic systems could have replaced it; I don’t think cat = Set has a rigid definition in terms of axioms.
If I’m wrong about this last part, Toby be sure to tell me!;-) So I think you, AN, have overvalued ETCS because you thought it was foundational to a category theory which must have a set-theoretic foundation. But the Cafe mostly uses Lawevere’s not-set-theoretic foundation AFAIK.
http://plato.stanford.edu/entries/category-theory/
“An alternative approach, that of Lawvere (1963, 1966), begins by characterizing the category of categories, and then stipulates that a category is an object of that universe.
Identity, morphisms, and composition satisfy two axioms:
Associativity xxx…
Identity xxx…
“This is the definition one finds in most textbooks of category theory. As such it explicitly relies on a set theoretical background and language.
An alterative, suggested by Lawvere in the early sixties, is to develop an adequate language and background framework for a category of categories.”
Posted by: Stephen Harris on September 19, 2009 8:57 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
SH: Even if Lawvere had used ETCS as his example for the Cat = Set, in his 1963 thesis or 1966 paper, I think other set axiomatic systems could have replaced it
I agree. But some notion of set is needed in any categorial foundations of mathematics deserving its name. I took ETCS as the standard notion, as I took ZF as the standard notion of set although people successfully used alternative set theories as foundation, too.
Posted by: Arnold Neumaier on September 19, 2009 7:33 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
AN wrote: “This is undisputed. But my claim was that CCAF contains ETCS, whereas his claim was that CCAF is completely independent of ETCS. (If it were so, how could CCAF be a foundation of mathematics, where sets are necessary?)” ——————-
SH: I haven’t noticed a rapid meeting of the minds yet, so I thought I would look into some outside opinion. I too thought that maybe Category Theory might no be such a good basis for FMathL. I now think my opinion is wrong. FMathL is an algorithm and this idea seems to be a key point: “This means that we can view category theory as a collection of algorithms.” FMathL will be an algorithm. This post is my research and notes.
TT wrote:
“However, it looks like Mike Shulman has written down a very interesting program
of study, different to the categories-based approach (which is extremely hands-on and bottom-up) but which is very smooth and top-down (invoking for example a very powerful comprehension principle) while still being faithful to the structuralist POV. He calls it SEAR (Sets, Elements, and Relations).”
AN wrote:
“The main reason I cannot see why category theory might become a foundation for a system like FMathL (and this is my sole interest in category theory at present) is that a systematic, careful treatment already takes 100 or more pages of abstraction before one can discuss foundational issues formally, i.e., before they acquire the first bit of self-reflection capabilities. …
Show me a paper that outlines a reasonably short way to formally define
all the stuff needed to be able to formally reflect in categorial language
a definition that characterizes when an object is a subgroup of a group.”
SH: This paper seems to fit the bill?
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.9846
“Herein we formalize a segment of category theory using the implementation of Calculus of Inductive Construction in Coq. Adopting the axiomatization proposed by Huet and Saibi we start by presenting basic concepts, examples and results of category theory in Coq. Next we define adjunction and cocartesian lifting and establish some results using the Coq proof assistant.”
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.9846
“This paper aims to bring the benefits of the use of Category Theory to the field of Semantic Web, where the coexistence of intrinsically different models of local knowledge makes difficult the exchanging of information. The paper uses categorical limit and colimit to define operations of breaking and composing ontologies, formalizing usual concepts in ontologies (alignment, merge, integration, matching) and proposing a new operation (the hide operation). The presented set of operations form a useful framework that makes easier the manipulation and reuse of ontologies.”
“Computational Category Theory”
“Another reason why computer scientists might be interested in category theory is that it is largely constructive. Theorems asserting the existence of objects are proven by explicit construction. *This means that we can view category theory as a collection of algorithms*. These algorithms have a generality beyond that normally encountered in programming in that they are parameterized over an arbitrary category and so can be specialized to different data structures.
One writes expressions to denote mathematical entities rather than defining the transitions of an abstract machine. ML also provides types which make a program much more intelligible and prevent some programming mistakes. ML has polymorphic types which allow us to express in programs something of the generality of category theory.
However, the type system of ML is not sufficiently sophisticated to prevent the illegal composition of two arrows whose respective source and target do not match.”
Posted by: Stephen Harris on September 19, 2009 5:45 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
AN: Show me a paper that outlines a reasonably short way to formally define all the stuff needed to be able to formally reflect in categorial language
It fits half of the bill. The missing half is to give Coq a categorial foundation. Only then is the reflection complete.
SH: FMathL is an algorithm and this idea seems to be a key point: “This means that we can view category theory as a collection of algorithms.”
I don’t see how a theory can be an algorithm. A theory consists of concepts, their semantic interpretation, and algorithms for manipulating the concepts consistent with that interpretation. All three parts are needed.
If the world consists of algorithms only, they perform meaningless tasks.
Posted by: Arnold Neumaier on September 19, 2009 7:31 PM | Permalink | Reply to this
### Re: CCAF vs ETCS
SH: FMathL is an algorithm and this idea seems to be a key point:
“This means that we can view category theory as a collection of algorithms.”
———————————-
AN replied
I don’t see how a theory can be an algorithm. A theory consists of concepts, their semantic interpretation, and algorithms for manipulating the concepts consistent with that interpretation. All three parts are needed.
If the world consists of algorithms only, they perform meaningless tasks.”
SH: I provided the full context of the basis for my quote below. I think the analogy is very precise. As far as meaning goes, the difference is between human “original intentionality” and a program which is considered to have “derived intentionality”. That means that the programmer provides the meaning that the program carries for other observers or users. Likewise, the mathematician defines or constructs some category and the purpose or meaning of that category that is communicated to other mathematicians. It flows from the mind of the mathematician into an abstract symbolic language which means something to the reader. This is pretty much the same definition of natural language which operates by a shared, agreed upon meaning.
“An alternative approach, that of Lawvere (1963, 1966), begins by characterizing the category of categories, and then stipulates that a category is an object of that universe.”
“The point is that the category of categories is not just a category, but what is known as a 2-category; that is, its arrows are functors, but two functors between the same two categories in turn form a category, the arrows being natural transformations of functors. Thus there are 1-arrows (functors) between objects (categories), but there are also 2-arrows (natural transformations) between 1-arrows.”
Encyclopedia of Computer Science and Technology By Allen Kent, James G. Williams
Computational Aspects
“In this section we indicate something of the computational nature of category theory that has attracted the interest of computer scientists and led to applications which we describe in the Applications section.
Observation one is that category theory, like logic, operates on the same level of generality as computer programming. It is *not tied to specific structures like sets or numbers* but as in programming, where we may define types to represent a wide range of structures, so in category theory, objects may range over many kinds of structures. This generality is exploited in describing the semantics of computation and can also be used to write highly generic codes.
Being based upon arrows and their compositions, category theory is an abstract theory of typed functions, with objects corresponding to types and arrows to functions. Notice that composition rather than application is the primitive operation. Through this identification, features of functional programming find categorical analogues. Type constructors
correspond to maps between categories (called functors), higher order functions are described in categories with additional structure (cartesian closed categories) and polymorphic types similarly. This correspondence between programming constructs and category theory formalizes structural properties of programs in an elementary equational language. Models of programming languages are then categories with suitable internal structure.
There is something more going on here. This correspondence translates functional language (like typed lambda-calculus) into an arrow-theoretic language: that is, translates a language with variables, where we can substitute values for names, into a variable-free combinator language.
Languages with variables seem more appropriate for programming and other
descriptive activities, whereas combinator languages are more suited to
algebraic manipulation and possibly more efficient evaluation. These ideas have been used to build abstract machines for implementing functional languages based upon categorical combinators.
Somewhat belying the abstraction of category theory is the fact that it is largely constructive-theorems asserting existence are proven by explicit construction. These constructions provide algorithms which can be coded as computer programs-programs with an unusual degree of generarality. They are abstracted over categories and so apply to a range of different data types, the same program performing analogous operations on types such as sets, graphs, and automata. In a sense, the core of category theory is
just a *collection of algorithms*.
Posted by: Stephen Harris on September 21, 2009 5:40 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
I haven’t read the following paper so I’m not certain what it includes about axioms.
Lawvere, F. William (1966), The category of categories as a foundation for mathematics, in S.Eilenberg et al., eds, ‘Proceedings of the Conference on Categorical Algebra, La Jolla, 1965’, Springer-Verlag, pp. 1–21.
—————————-
SH: I don’t think the formal axioms were presented in Lawvere’s 1963 thesis. I think they can be found in this 1964 paper
www.pubmedcentral.nih.gov/articlerender.fcgi?artid=300477
“We adjoin eight first-order axioms to the usual first-order theory of an abstract Eilenberg-Mac Lane category’ to obtain an elementary theory with the following properties:
(a) There is essentially only one category which satisfies these eight axioms together with the additional (non-elementary) axiom of completeness, namely, the category of sets and mappings. Thus our theory distinguishes 8 structurally from other complete categories, such as those of topological spaces, groups, rings, partially ordered sets, etc.”
This paper is also online with extended commentaries (long version) added later.
http://138.73.27.39/tac/reprints/articles/11/tr11abs.html
Posted by: Stephen Harris on September 18, 2009 2:37 AM | Permalink | Reply to this
### Re: CCAF vs ETCS
It seems Mike is right about ETCS and CCAF being quite distinct. However, Lawvere doesn’t appear to see the merits to be quite as disconnected as apparently Mike does.
———————-
Colin McLarty wrote:
“Yet the second categorical foundation ever worked out, and the first in print,
was a set theory —- Lawvere’s axioms for the category of sets, called ETCS,
(Lawvere 1964).”
Lawvere, ETCS paper, page 34, wrote:
“However, it is the author’s feeling that when one wishes to go substantially
beyond what can be done in the theory [ETCS] presented here, a much more
satisfactory foundation for practical purposes will be provided by a theory
of the category of categories.”
“Part of the summer of 1963 was devoted to designing a course based on the
concluded that the category of categories is the best setting for “advanced”
mathematics).”
“This elementary theory of the category of sets arose from a purely practical
educational need. …
But I soon realized that even an entire semester would not be adequate for
explaining all the (for a beginner bizarre) membership-theoretic definitions
and results, then translating them into operations usable in algebra and
analysis, then using that framework to construct a basis for the material I
planned to present in the second semester on metric spaces.
However I found a way out of the ZF impasse and the able Reed students could
indeed be led to take advantage of the second semester that I had planned.
The way was to present in a couple of months an explicit axiomatic theory of
the mathematical operations and concepts (composition, functionals, etc.) as
actually needed in the development of the mathematics.”
Posted by: Stephen Harris on September 18, 2009 7:14 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
When we write maths on a blackboard what do we actually do? This has to be analysed satisfactorily if we are ever to have decent graphical mathematical editors - a necessity if we want the communication of maths by computer to approximate the ease of a blackboard and coloured chalk.
What we do has two components, one visible and the other invisible. The visible component is a tree-structured graphic. The nodes of the tree represent the meaningful subexpressions. The editor must be able to ‘explode’ this tree, for example by a perspective representation, or by using a separate window to show different levels, so that the user can select subtrees for editing, dragging and dropping into other windows, etc.
The invisible component is the semantic content of the tree, rather than its visual display. This is the formalization that is in the back of the user’s mind. It may not be a complete formalization, so this is the tricky part. What formalization is appropriate for the computer? I suspect that there may be many answers to this.
The visible component needs software that may possibly already exist, but I do not know of it. The !Draw application for RISC OS is a structured vector-graphics editor, that has been around for a quarter of a century, and it has some of the features of what is required.
Posted by: Gavin Wraith on September 12, 2009 7:56 AM | Permalink | Reply to this
### Ideocosm puzzles; Re: Towards a Computer-Aided System for Real Mathematics
“The invisible component is the semantic content of the tree, rather than its visual display. This is the formalization that is in the back of the user’s mind.”
Yes, but that’s where it gets infinitely tricky.
As I’ve said before, on other threads, we don’t know much at all about the Topology of the Ideocosm – the space of all possible ideas.
Here we admit that we don’t know how to formalize, illustrate, or automate the subspace of all possible “Mathematical ideas.”
I’m not sure we can even define it, given that Mathematics is at any time partly instantiated by what is in the heads of all Mathematicians, a society that changes over time both locally and globally.
I am not clear on how we might even define a hyperplane that separates the “Mathematical” ideas from the “nonmathematical” ideas within the Ideocosm.
Nor, for that matter, how we might even define a hyperplane that separates the “Physical” ideas from the “Nonphysical” ideas within the subspace of the Ideocosm of Theories of Mathematical Physics.
Posted by: Jonathan Vos Post on September 12, 2009 4:23 PM | Permalink | Reply to this
### Re: Ideocosm puzzles
GW: The invisible component is the semantic content of the tree, rather than its visual display. This is the formalization that is in the back of the user’s mind.
JVP: Yes, but that’s where it gets infinitely tricky. As I’ve said before, on other threads, we don’t know much at all about the Topology of the Ideocosm - the space of all possible ideas.
It is just what can be represented on the semantic web. What is not known is only the part of it that is potentially useful. But the subspace of well-defined mathematical statements can be delineated up to semantic equivalence. It will just be what can be processed by FMathL, since the latter is designed to be able to process all mathematics. (Already Coq and Isabelle/Isar can do that, though not really conveniently.)
Posted by: Arnold Neumaier on September 14, 2009 4:10 PM | Permalink | Reply to this
### Godel-numbering to “game” the metasystem; Re: Ideocosm puzzles
“the subspace of well-defined mathematical statements can be delineated up to semantic equivalence.”
I understand. But in a dynamic metasystem, where “new” mathematical ideas can be introduced coherently, as Category Theory historically came after Set Theory, it is not obvious to me that the semantics and pragmatics (“potentially useful”) are guaranteed always to be well-defined, once gadgets such as Godel-numbering are used to “game” the metasystem. But I’m eager to know more.
Posted by: Jonathan Vos Post on September 14, 2009 8:09 PM | Permalink | Reply to this
### Re: Godel-numbering to “game” the metasystem; Re: Ideocosm puzzles
JVP: in a dynamic metasystem, where “new” mathematical ideas can be introduced coherently, as Category Theory historically came after Set Theory, it is not obvious to me that the semantics and pragmatics (“potentially useful”) are guaranteed always to be well-defined
Web ontology languages like RDF/OWL have a very general representation for semantics that can handle the way arbitrary concepts were looked at from antiquity till today, and hence probably far into the future. Semantical content is represented as a collection of triples of names.
For FMathL, it turned out to be more convenient to consider only collections of triples where the first two entries determine the third, leading to a partial binary dot operation that associates to certain pairs of objects a third one:
f . is_continuous = true
customer_1147 .f irst_name = Otto
etc.. As is easi to see, this still can hold arbitrary semantical relations between objects. The operation table of the dot operation is an infinite matrix that we call a semantic matrix.
The collective knowledge about mathematics can be considered as a huge and growing semantic matrix, of which the FMathL system is to capture the most important part.
JVP: … well-defined, once gadgets such as Godel-numbering are used to “game” the metasystem.
Consistency depends on the context, and is maintained in the usual way. Goedel’s results put limits on what is achievable constructively, but do not threaten well-definedness.
Posted by: Arnold Neumaier on September 15, 2009 12:23 PM | Permalink | Reply to this
### What mathematicians carry around in their heads; Re: Godel-numbering to “game” the metasystem; Re: Ideocosm puzzles
Excellent answer. This suggests to me how carefully you’ve thought through your system, metasystem, metametasystem…
“The collective knowledge about mathematics can be considered as a huge and growing semantic matrix…”
Cf.:
“Most mathematicians can go through their entire careers without learning anything about proof theory and intuitionistic logic, and I think the reason is that both undermine the naive model of mathematical foundations that most mathematicians carry around in their heads. Mathematicians hate thinking about foundations.”
Posted by: Jonathan Vos Post on September 16, 2009 6:25 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
GW: What we do has two components, one visible and the other invisible.
This not only holds for the blackboard case (which I leave to the OCR people, who do not too badly in turning formulas into latex) but also to the typed case.
Thus the visible component gives the expression tree (syntwx, e.g., presentation MathML), while the invisible component gives the intentions (e.g., Content MathML). The latter is the more difficult, poorly understood on the formal level thing.
Of course, when we write math on the blackboard, there are also our voice and gestures that convey helpful information for the semantical interpretation, e.g., a broad smile while saying “Let $\epsilon\lt 0$”.
On the other hand, for the syntactical side - do you really think that a visual view of the syntax tree of a formula such as $a(b+c)=ab+ac$ is more helpful than the equation itself? The longer the formula the less intelligible is the tree. (Consider a sequence of equalities $A = B = C = D + E = F$ with long expressions $A,\dots,F$ that are the same apart from successive substitutions.)
Posted by: Arnold Neumaier on September 20, 2009 12:52 PM | Permalink | Reply to this
### Herding cats
I’m someone who spends a fair amount of time trying to tease as much semantics as possible out of the LaTeX that people type.
“Presentational” MathML contains far more semantics than one typically finds in the LaTeX people actually generate, “in the wild.”
As a consequence, the output of itex2MML generally falls short of what would be possible in hand-crafted Presentational MathML (let alone Content MathML or the fancy-schmancy system discussed above).
Take as an example, the expression
f(a+b)^2
Of course, the exponent is not applied to the right parenthesis – even though that is literally how the expression is written.
Rather, what the author probably means is either
{ f(a+b) }^2
or, perhaps,
f { (a+b) }^2
Which it is, depends on what the expression
f(a+b)
is supposed to mean. In MathML, there are entities, ⁢ and ⁡, which would normally be placed between the <mi>f</mi> and the <mo>(</mo> to indicate whether we mean “the variable f times (a+b)” or “the function f applied to a+b.”
But, of course, there’s nothing like that in LaTeX.
What an ordinary LaTeX user can do is use brace brackets, as above, to — at least — indicate the desired grouping.
When I use itex (in the comments here on golem, or in Instiki), I try to always use braces to indicate grouping. That gets translated into the placement of <mrow> elements, which produces the right semantics (and not merely the right visual appearance) in the resulting MathML.
As far as I can tell, I am the only person who does that.
The problem of getting people to insert semantic information, into the stuff they write, was eloquently shown to be hopeless, long ago, in a famous essay by Cory Doctorow.
Posted by: Jacques Distler on September 14, 2009 4:25 AM | Permalink | PGP Sig | Reply to this
### Re: Herding cats
Just a short technical followup, for those unfamiliar with LaTeX and/or MathML.
In LaTeX, grouping is indicated by brace brackets and the matching of left- and right-braces is strictly enforced. Parentheses do not indicate grouping (and the matching of left- and right-parentheses is not enforced).
In Presentational MathML, grouping is indicated by the <mrow> element. itex2MML translates matched pairs of braces into <mrow>s.
Posted by: Jacques Distler on September 14, 2009 5:27 AM | Permalink | PGP Sig | Reply to this
### Re: Herding cats
Here you hit a problem in “worldwide inference”: one feature of brace pairs is that they turn TeX mathop’s into normal characters (without the wider mathop spacing), and years of having to deal with two-column proceedings styles have trained me to stick braces around any plus signs, etc, in all displayed equations to increase the likelihood of it fitting on one line (due to tight page limits). (Stylists will say I shouldn’t do this, but I’ve never received a referee report that indicates they’ve ever noticed, let alone object.) I don’t know what the itex parser would infer about the equations from this habit :-)
I’ve read a lot of the papers that Knuth wrote about the design and implementation of TeX and, whilst what he writes makes it clear he cares deeply about “exact” reproducibility of typesetting both in different installations and years later, I can’t recall any statements about the direct electronic use of TeX documents, particularly by algorithms. So it’s a reasonable hypothesis that he just didn’t think this would be relevant to documents produced in the lifetime of the system. But the TeX family has clearly lived far longer than expected and issues in it’s design are starting to affect new uses for the documents.
Posted by: bane on September 14, 2009 9:56 AM | Permalink | Reply to this
### Re: Use of braces
Wouldn’t that effect be better achieved with
\everydisplay={\mathcode\+="002B}
(forcing $+$ to be a mathord instead of a mathbin in displayed equations)?
Posted by: Mike Shulman on September 14, 2009 8:04 PM | Permalink | Reply to this
### Re: Use of braces
That’s probably a higher level way to do it, although it obviously needs extending to all the other mathethatical operations and relations I tend to use in displayed equations. (In case anyone’s wondering, there’s a greater tendency to have “word” variable names and subscripts in CS, which combined with 2 column means most displayed equations take just over a line in the “natural” spacing.)
My reason for doing it this way is only that of my knowledge came from the TeXbook with a bit of LaTeX knowledge bolted on.
The bigger point was that curly braces don’t always have no effect on appearance, and hence using them to denote structure will run in to some corner cases.
Posted by: bane on September 15, 2009 10:09 AM | Permalink | Reply to this
### Re: Herding cats
itex2MML translates matched pairs of braces into <mrow>s.
I didn’t know that; where is it documented? Now that I know it, maybe I’ll make an effort.
However, the way I read the proposal, the idea was for a system that would be able to infer this sort of missing information from the context. It seems to me that in many cases, such as your example, this is just a matter of type-checking. If $a$, $b$, and $f$ are all variables representing numbers, then $f(a+b)$ can only mean $f$ times $(a+b)$, whereas if $a$ and $b$ are numbers and $f$ is a function, then $f(a+b)$ probably means $f$ applied to $(a+b)$ (unless you’re multiplying functions by numbers pointwise–but that’s often written only with the number on the left). In cases where more than one interpretation type-checks, it seems plausible to me that a computer could still sometimes infer the probable intent from the context, just as a human does. For example, if later on one encounters the statement $g(f(a+b)) = (g\circ f)(a+b)$, it is a good bet that $f(a+b)$ meant function application and not pointwise multiplication.
Posted by: Mike Shulman on September 14, 2009 6:39 AM | Permalink | Reply to this
### Re: Herding cats
itex2MML translates matched pairs of braces into <mrow>s. I didn’t know that; where is it documented?
Alas, there isn’t any technical documentation on how itex2MML is implemented.
You might guess that this is how it works, based the description of how the \color command works. But, really, that would only occur to you if you knew what an <mrow> element was, in the first place. And that would put you in a very small minority indeed.
Now that I know it, maybe I’ll make an effort.
Great! Welcome to a very elite club.
Next thing you know, you’ll be using the \tensor{}{} and \multiscripts{}{}{} commands.
(Jason Blevins and I worked quite hard to write the LaTeX macros to implement those commands. You can see them in the TeX export in Instiki.)
However, the way I read the proposal, the idea was for a system that would be able to infer this sort of missing information from the context.
If you have sufficient context, you may be able to guess (humans, after all, manage to). Otherwise, you have to rely on people entering (correct!) metadata about what all the symbols mean (e.g., whether $f$ is a function or a variable).
That’s when you run into (some of) the problems mentioned in Cory Doctorow’s article.
Posted by: Jacques Distler on September 14, 2009 7:23 AM | Permalink | PGP Sig | Reply to this
### Re: Herding cats
MS: However, the way I read the proposal, the idea was for a system that would be able to infer this sort of missing information from the context.
JD: If you have sufficient context, you may be able to guess (humans, after all, manage to). Otherwise, you have to rely on people entering (correct!) metadata about what all the symbols mean (e.g., whether f is a function or a variable).
A typical mathematical document together with the background of a trained reader contains everything needed to understand the paper. FMathL is therefore supposed to guess the interpretation from the context and from past experience, just as any mathematician does.
If this is not enough, a mathematician decides that the formula (or sentence, or article, or book) is too poorly written to merit understanding, and skips to the next formula (or sentence, or article, or book), perhaps coming back later, when the context has become richer. FMathL will be taught to do the same.
But since you seem to know MathML well, I wonder what you say to our study Limitations in Content MathML.
Posted by: Arnold Neumaier on September 14, 2009 4:25 PM | Permalink | Reply to this
### Re: Herding cats
A typical mathematical document together with the background of a trained reader contains everything needed to understand the paper. FMathL is therefore supposed to guess the interpretation from the context and from past experience, just as any mathematician does.
Sounds like you’re trying (among other things) to develop a knowledge representation for mathematics.
Good luck with that!
But since you seem to know MathML well, I wonder what you say to our study Limitations in Content MathML.
I’ll take a look. But, off the top of my head, I’d say that one’s view of the (in)adequacy of CMML, depends on what you think its purpose is.
I see the primary use of CMML as a common data-interchange format between symbolic manipulation programs. For that purpose, I think it works passably well.
If you have some fancier use-case in mind, your answer may be different …
Posted by: Jacques Distler on September 14, 2009 5:18 PM | Permalink | PGP Sig | Reply to this
### Re: Herding cats
JD: But, off the top of my head, I’d say that one’s view of the (in)adequacy of Content MathML depends on what you think its purpose is.
We were looking for what we could use to support FMathL activities (in particular, the representation of common formulas in mathematics, including block matrices in linear algebra and index notation for tensors) and simply recorded our failure to find it in Content MathML.
The Conten MathML document MathML2 of course takes a much more modest view on what it wants to achieve:
“It would be an enormous job to systematically codify most of mathematics - a task that can never be complete. Instead, MathML makes explicit a relatively small number of commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it provides a mechanism for associating semantics with new notational constructs. In this way, mathematical concepts that are not in the base collection of elements can still be encoded”
Unfortunately, the mechanism provided turned out to be almost useless.
Fortunately, the outlook is not as pessimistic as this disclaimer lets one guess, and we are close to a good solution (but not using MathML).
JD: I see the primary use of CMML as a common data-interchange format between symbolic manipulation programs. For that purpose, I think it works passably well.
I never tried to use automatic symbolic manipulation involving the definition of a covariant derivative in index notation. But if there is a differential geometry package that can do that, it will not be able to use Content MathML.
Posted by: Arnold Neumaier on September 15, 2009 12:04 PM | Permalink | Reply to this
### Re: Herding cats
{ f(a+b) }^2 f { (a+b) }^2
From a strictly presentational point of view (which is, in this case, the point of view of LaTeX), these are wrong, since they put the superscript on the wrong element (the group instead of simply the right parenthesis.
The difference in these cases is tiny, but it exists; replace a with \sum_{i=1}^n a_i in a displayed equation to see better how it works. Of course, in that case, you probably want to use larger parentheses, so go ahead and use \left and \right; the spacing works differently. (But now the effect will be tiny again and in fact too subtle for iTeX2MathML.)
The problem is that grouping has meaning for TeX that may or may not match the semantic meaning that you intend to convey to MathML. It would be better (at least theoretically) to have a grouping command that is ignored by TeX but interpreted in MathML.
{(a+b)}^2_2
(a+b)^2_2
{(\sum_{i=1}^n a_i+b)}^2_2
(\sum_{i=1}^n a_i+b)^2_2
{\left(\sum_{i=1}^n a_i+b\right)}^2_2
\left(\sum_{i=1}^n a_i+b\right)^2_2
${(a+b)}^2_2$
$(a+b)^2_2$
${(\sum_{i=1}^n a_i+b)}^2_2$
$(\sum_{i=1}^n a_i+b)^2_2$
${\left(\sum_{i=1}^n a_i+b\right)}^2_2$
$\left(\sum_{i=1}^n a_i+b\right)^2_2$
Posted by: Toby Bartels on September 14, 2009 10:23 AM | Permalink | Reply to this
### Re: Herding cats
Here’s a screenshot of the same examples in TeX.
To me, the first and second and the fifth and sixth examples look nearly identical.
The third and fourth, of course, look radically different. But I’d say that’s because the parentheses are too small, and neither one looks “right” to me.
In each case, at least in my browser, the MathML, generated by itex2MML, looks pretty darned close to the TeX output.
Posted by: Jacques Distler on September 14, 2009 3:26 PM | Permalink | PGP Sig | Reply to this
### Re: Herding cats
To me, the first and second and the fifth and sixth examples look nearly identical.
To tell the difference between the fifth and the sixth, I have to stack two of one on top of one of the other and look carefully at the vertical positions where the subscript of one line comes near the superscript of the next line. I can tell the difference between the first and second simply by looking at the gap between the multiscripts, but I still agree that they look nearly identical.
If you buy TeX's philosophy about how mathematical typesetting should be built out of boxes, then one is technically right and the other technically wrong, despite the small size of the practical effect. But if you think that CMML or something like it is the wave of the future, then this shouldn't matter to you, and putting grouping in iTeX is a good idea, even it produces technically incorrect TeX. Someday we should be able to print MathML just as nicely as we can now print TeX (and it's already close enough for the screen, at least when MathML supports everything), and then there will never be a need to use the TeX export.
Now here's a more practical consideration. I don't like the size of the delimeters produced by \left and \right; I've written macros that replace them with something slightly smaller. To keep things simple (even though it produces something slightly smaller yet than what I would like), let's do it with \bigg:
{\bigg(\sum_{i=1}^n a_i+b\bigg)}^2_2
\bigg(\sum_{i=1}^n a_i+b\bigg)^2_2
${\bigg(\sum_{i=1}^n a_i+b\bigg)}^2_2$
$\bigg(\sum_{i=1}^n a_i+b\bigg)^2_2$
Actually, the difference in the MathML here is still pretty subtle, at least on my browser. I can see it better in actual TeX (with displayed equtions).
If MathML is the future, then fudging the heights of the delimiters like this would be a job for a stylesheet. I have no idea to do such a thing, however.
Posted by: Toby Bartels on September 14, 2009 9:45 PM | Permalink | Reply to this
### Re: Cat On A Hot Tin Roof
Category Theory began — I’m talking about Aristotle here — with the observation that signs, symbols, syntax, and so on … are inherently equivocal, which means that we must refer them to the right categories of interpretation if we want to resolve their ambiguities.
In other words — to borrow a word that Peirce borrowed from Aristotle — there is an inescapably abductive element to the task of interpretation.
See, for example, Interpretation as Action : The Risk of Inquiry
Posted by: Jon Awbrey on September 14, 2009 6:30 PM | Permalink | Reply to this
### Re: Cat On A Hot Tin Roof
Category Theory began — I’m talking about Aristotle here — […]
Although our term ‘category’ does come from Aristotle (via Kant), I would say that what Aristotle discusses is more type theory than category theory. He's still correct, however!
Posted by: Toby Bartels on September 14, 2009 9:44 PM | Permalink | Reply to this
### Re: Cat On A Hot Tin Roof
Although our term ‘category’ does come from Aristotle (via Kant), I would say that what Aristotle discusses is more type theory than category theory.
It might be nice to have a paragraph with discussion of this terminology issue at category theory.
Posted by: Urs Schreiber on September 15, 2009 7:36 AM | Permalink | Reply to this
### Re: Cat On A Hot Tin Roof
We’ve already aired the Kant connection.
Posted by: David Corfield on September 15, 2009 8:51 AM | Permalink | Reply to this
### Philosophical Excavations
I cited my favorite locus classicus from Aristotle here:
Peirce’s first cut — it’s the deepest — is here:
Posted by: Jon Awbrey on September 15, 2009 2:12 PM | Permalink | Reply to this
### Re: Cat On A Hot Tin Roof
I wrote:
It might be nice to have a paragraph with discussion of this terminology issue [the two meanings of “category”] at category theory.
David reacted:
We’ve already aired the Kant connection.
I don’t see any of this at $n$Lab:category theory
At least the MacLane-quote that Jon points to should be copied there.
I’ll take care of that now. But I won’t try to talk about Aristotle et al. That’s not my job.
Posted by: Urs Schreiber on September 15, 2009 5:52 PM | Permalink | Reply to this
### Technical questions on the input interface
I have some technical questions on the blogging software. Maybe some things can be improved and/or explained.
1. When I use the text filter “itex to MathML with parbreaks”, how do I quote a piece of text from a previous mail? Trying to copy it with the mouse produces unintelligible output.
2. The interface provides the possibility to “Remember personal info”, but why can’t it remember the text filter used last time? I regularly forget to set it in the first preview to what I want.
3. It would be nice if the switch between “view chronologically” and “view threaded”, at present at the bottom of the whole page, would appear at the bottom of each message.
4. Why are the “Previous Comments and Trackbacks” repeated in each response window? It only makes navigating in the window more difficult (tiny motions have large consequences for long discussions like this one). I’d prefer to have a larger comment window.
5. The options are visible only before the first preview, which I found a bit of a nuisance. Also, it would be nice if they (nd the name information) appeared after the command window rather than before it, since this saves scrolling in the first round.
Posted by: Arnold Neumaier on September 16, 2009 8:34 PM | Permalink | Reply to this
### Re: Technical questions on the input interface
When I use the text filter “itex to MathML with parbreaks”, how do I quote a piece of text from a previous mail? Trying to copy it with the mouse produces unintelligible output.
Copy it with the mouse and put a > character in front of it. This doesn’t work for math symbols, however. Several of us have been complaining about this for a while, but no one has fixed it yet.
The interface provides the possibility to “Remember personal info”, but why can’t it remember the text filter used last time?
Posted by: Mike Shulman on September 16, 2009 8:45 PM | Permalink | Reply to this
### Re: Technical questions on the input interface
When I use the text filter “itex to MathML with parbreaks”, how do I quote a piece of text from a previous mail? Trying to copy it with the mouse produces unintelligible output.
Copy it with the mouse and put a > character in front of it.
No, that doesn't work with that filter; instead see the stuff about ‘blockquote’ at this FAQ.
Or try using the ‘Markdown with itex to MathML’ filter; it's a lot more powerful and includes this feature.
Posted by: Toby Bartels on September 16, 2009 9:52 PM | Permalink | Reply to this
### Re: Technical questions on the input interface
Posted by: Toby Bartels on September 16, 2009 9:47 PM | Permalink | Reply to this
### Re: Technical questions on the input interface
There's a thread for this stuff …
I've copied it there.
Posted by: Toby Bartels on September 16, 2009 10:16 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Just a note from the sidelines…
As a layman, I’m thoroughly enjoying this conversation. I’m starting to see an image of language itself being explained by category theory, which is mind boggling but makes perfect sense.
I always half-jokingly described mathematics as a “foreign language” similar to the way some organizations recognize fluency in a programming language as a “foreign language”.
Mathematics (maybe at the undergraduate level?) then seems like a perfect progression in the attempt to understand language and communication “arrow theoretically”.
I obviously don’t know what I’m talking about, but that is the hazy picture beginning to form in my mind.
By the way, since the goal seems to be to formalize the language of mathematics and eventually implement a computer system, then it also seems like it would make sense to develop the most fundamental mathematics concepts using the most fundamental computational concepts. Think about how computers encode information. Bits. Packets.
Now I’m just thinking out loud…
Posted by: Eric Forgy on September 18, 2009 3:49 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
After posting this, my mind started racing and I remembered that probably the most primordial object in category theory is the (-2)-category “True”. There are two (-1)-categories “True” and “False”.
Is it a coincidence that the most fundamental concept in a computer is the bit?
It would be fun to trace the development information content via bits on a computer with the development of information content via category theory beginning with the (-2)-category True.
Posted by: Eric Forgy on September 18, 2009 4:42 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Eric wrote “Is it a coincidence that the most fundamental concept in a computer is the bit?”
Just to note that modern digital computers are not the only kinds of computational devices. Eg, there were the old-time analogue computers, there are “multi-layer perceptrons”, there’s cellular automata (which, whilst having discrete states, binary states don’t seem “specially nice”). So, whilst there’s a very good case to be made that binary digital circuitry is the fundamental idea of computation, it’s not completely obvious that this is the case.
(And that’s without discussing if quantum mechanics changes things.)
Posted by: bane on September 18, 2009 5:19 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Let me change my question to:
Is it a coincidence that the most fundamental concept in a digital computer is the bit?
I find it somehow compelling that the building block of the periodic table is also the building block of the digital computer.
Maybe I am giddy because it is Friday, but that somehow seems profound to me.
Posted by: Eric Forgy on September 18, 2009 5:42 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I wasn’t trying to dismiss the concept of a connection; indeed given that digital computers are created by human beings who fond of very of True and False there’s almost certainly a deep connection. I was just pointing out that it’s currently unclear whether the “fundamental human approach to computation” is close to the “fundamental approach to computation”.
In terms of other connections, there’s obviously relations of information theory to: the most basic questions you can ask are ones with True or False as answers. But that’s another difficult to formalise connection.
Posted by: bane on September 18, 2009 6:21 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Eric, I think you would enjoy this paper which packs a lot of thinking into 15 pages. I like the idea of category theory describing structures, and then structures within those structures as an organization of rather than a foundation of mathematics.
philosophy.ucdavis.edu/landry/2CategoryTheoryTheLanguage.pdf
CATEGORY THEORY: THE LANGUAGE OF MATHEMATICS by Elaine Landry
“Rather, ‘like’ in the sense that just as mathematics, in virtue of its ability to classify empirical and/or scientific objects according to their structure, presents us with those generalized structures which can be variously interpreted. Likewise, then, a specific category, in virtue of its ability to classify mathematical concepts and their relations according to their structure, presents us with those frameworks which can be variously interpreted.7 It is in this sense that specific categories act as “linguistic frameworks” for concepts: they allow us to organize our talk of the content of various theories in terms of structure, because “[i]n this description of a category, one can regard “object,” “morphism,” “domain,” “codomain,” and “composites” as undefined terms or predicates” (Mac Lane 1968, 287).
In like manner, a general category, in virtue of its ability to classify mathematical theories and their relations according to their shared structure, presents us with those frameworks which can be variously interpreted.8 It is in this sense that general categories act as “linguistic frameworks” for theories: they allow us to organize our talk of the common structure of various theories in terms of structure, because in this description of a category once can regard “object,” “functor,” etc., as undefined terms or predicates. That is, general categories allow us to organize our talk of the structure of various theories in the same manner in which the various theories of mathematics are used to talk about the structure of their objects, viz., as “positions in structures.”
We say that category theory is the language of mathematical concepts and relations because it allows us to talk about their specific structure in various interpretations, that is, independently of any particular interpretation. Likewise, our talk of the relationship between mathematical theories and their relations is represented by general categories. We say that category theory is the language of mathematical theories and their relations because it allows us to talk about their general structure in terms of “objects” and “functors,” wherein such terms are likewise taken as ‘syntactic assemblages waiting for an interpretation of the appropriate sort to give their formulas meaning’.
Posted by: Stephen Harris on September 22, 2009 2:26 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Thanks for the reference Stephen! I’ve been traveling lately, hence the slow response, but managed to read this on a plane.
You certainly don’t owe me a coffee, but I’ll take it as an invitation and if you ever find yourself near LA with some free time, let me know as well :)
The thing that I find fascinating about this whole conversation is the glimmers of “arrow theoretic” formulation of communication itself. For example, what is “w” arrow theoretically?
Then a close cousin (or ancestor) of communication is information. How does category theory encode information? Can that be quantified?
In principle, it would seem to be possible to formulate a complete “arrow theoretic” means of communication.
Posted by: Eric Forgy on October 1, 2009 7:09 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
I’m starting a new thread because this is a reply to several different comments and because the nesting level in threaded view is getting ridiculous.
I think there is one fundamental problem that is cropping up repeatedly in several guises in this discussion, which I mentioned up here. It appears to have led me astray in a few places as well (as it’s done before in the past; argh!). Namely, in unaugmented structural set theory (including ETCS and SEAR and type theory), a structured set is not a single object in the domain of discourse. For instance, in none of these cases does the formal language allow one to talk about a group. A group in SEAR (to be specific) consists of a set $G$ and an element $e\in G$ and a function $m:G\times G \to G$, such that certain axioms are satisfied. By contrast, in ZF, a group can be defined to consist of an ordered triple $(G,e,m)$ such that $e\in G$, $m$ is a function $G\times G\to G$, and certain other axioms are satisfied.
This is what I meant when I said here that the relation $\in$ between rationals and (Dedekind) reals is extra structure on the same set of reals. The ordered field of reals consists of the set $\mathbb{R}$, some elements $0,1\in \mathbb{R}$, some functions $+,\cdot:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$, a relation $({\le}): \mathbb{R} \looparrowright \mathbb{R}$, etc. The relation $({\in}): \mathbb{Q} \looparrowright \mathbb{R}$ is one more piece of structure on the same set $\mathbb{R}$, which you can use or not use as you please.
Likewise, this is what I think is going on with opposites. If I’m working with a group as above, then I can construct its opposite group to consist of the same set $G$, the same element $e$, and a reversed function $m$. Now of course it makes sense to ask whether an element of $G$ and an element of $G^{op}$ are equal, since they are elements of the same set. On the other hand, if I am just given two groups $G$ and $H$, it makes no sense to ask whether an element of $G$ is equal to an element of $H$. So what I said here is not quite right, and I apologize: what I should have said is that it would be a type error to compare elements of (sets underlying) two different structures unless those structures are built on the same underlying set(s).
(If you’re going to object to the notion of sets being “the same,” I think the answer was provided by Toby: we mean the external judgment that two terms are syntactically equal, rather than a (disallowed) internal proposition that two terms refer to the same object.)
So I think it is misleading to speak about two different groups “having elements in common”—either we are talking about two different group structures on the same set, in which case the two have exactly the same elements by definition, or we are talking about group structures on different sets, in which case asking whether an element of one is equal to an element of the other is a type error. Therefore, I was also not quite right when I said that a structural system would not be able to construct categories having common objects: it can construct pairs of categories (such as a category and its opposite) that have the same collection of objects, but no more.
Now going back to the original subject of intersections, in a structural theory the operation of “intersection” does not apply to arbitrary pairs of sets, but rather to pairs of subsets of the same fixed ambient set. (One might argue that the intersection of a set with itself should be defined (and equal to itself), but I think this probably derives from a misconception that distinct sets in structural set theory are “disjoint,” rather than it just not being meaningful to ask whether their elements are equal.) In particular, it is not meaningful to speak of the intersection of the sets of objects of two categories.
It never really occurred to me before to consider this peculiarity (that structured sets are not single things) as a problem, since “for any group, …” can always be interpreted as shorthand for “for any set $G$, element $e\in G$, and function $m:G\times G\to G$ such that <blah>, …”, and similar sorts of interpretations happen in other foundations like ZF. But I guess that this sort of implicit interpretation is part of the “compilation from high-level language to assembly language,” and what you want is a formalization of the higher-level language that (among other things) includes “a group” as a fundamental object of study.
I admit that this is definitely something I have found frustrating about existing proof assistants: they do not seem to really understand that “a group” is a thing. But somehow I never really isolated the source of my frustration before.
One way to deal with this (which I believe is adopted by many type theorists?) is to introduce one set called a “universe” $U$ whose elements are (interpreted as) sets in some way. Then a triple of sets can be modeled by an element of $U\times U\times U$, and so on. But a really structural approach would insist that sets are not the elements of any set, but rather the objects of a category—so what we really need is structural category theory, which doesn’t quite exist yet. Possibly this is what Lawvere was thinking of when he said that “when one wishes to go substantially beyond [ETCS], a much more satisfactory foundation… will be provided by a theory of the category of categories”—although I’ve always felt that one should really be thinking about the 2-category of categories.
Regardless, it does seem that existing structural frameworks fall short here. Now I feel all fired up to improve them! But that’s a whole nother kettle of worms.
I’m also feeling bad about hijacking this discussion with a long branching argument about the merits of structural/categorial set theory. If people have the energy, let’s also go back to FMathL and your larger proposal and see what other constructive things we can discuss about it. As I said way back at the beginning, I am really excited about the overall idea—which is perhaps what is driving me to be overly critical of FMathL, since I would like such a system to “get it right” (and for better or for worse, I usually seem to think I know what’s “right”). But as I’ve said, I feel like I understand a little better now where you are coming from and what problems you are trying to solve.
Posted by: Mike Shulman on September 21, 2009 7:26 AM | Permalink | PGP Sig | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
Namely, in unaugmented structural set theory (including ETCS and SEAR and type theory), a structured set is not a single object in the domain of discourse. For instance, in none of these cases does the formal language allow one to talk about a group.
One reason why material ($\mathbf{ZFC}$-like) foundations want to package things up into a tuple is that one might want to make these tuples elements of some other set. In structural foundations, you couldn't do that anyway; if you want a family of groups, then you need that to be parametrised by some index set $I$, and you have a group $G_k$ for every element $k$ of $I$. You could do that in material foundations too, of course, but if instead you want to allow a family to ‘parametrise itself’, then you need each of the objects in the family (groups, in this case) to formally be single objects that can be elements of a set.
I admit that this is definitely something I have found frustrating about existing proof assistants: they do not seem to really understand that “a group” is a thing.
In Coq, you can do this with Records. Formally, this is based on having a Type of Sets and all that (Records are just user-friendly sugar), but you never need to use anything about that type (such as the equality predicate on its elements, which would be evil).
But a really structural approach would insist that sets are not the elements of any set, but rather the objects of a category—so what we really need is structural category theory, which doesn’t quite exist yet.
I'd like to see structural $\infty$-groupoid theory; I think that I could do a lot with that, possibly everything that I want. Of course, there are already $\infty$-groupoids hidden in ordinary intensional type theory, but it's not clear that there are enough; I would really want types that are explicit $\infty$-groupoids, and nothing more.
Posted by: Toby Bartels on September 21, 2009 7:59 AM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
MS: it would be a type error to compare elements of (sets underlying) two different structures unless those structures are built on the same underlying set(s).
You need to make more exceptions to account for subcategories. There may be a category with object set C and two subsets A and B such that there are subcategories with object sets A and B. In this case, one must be able to compare their elements, too, because of the definition of a subcategory. With ZF or NBG as metalanguage (as usual), this already implies that one can compare objects from any two categories.
With SEAR as metatheory it might be different since I have not yet a good intuition about SEAR. (But I added a number of comments on the SEAR page that make me doubt that it is a mature enough theory.)
MS: let’s also go back to FMathL and your larger proposal and see what other constructive things we can discuss about it. As I said way back at the beginning, I am really excited about the overall idea
Yes, I’d appreciate that.
Posted by: Arnold Neumaier on September 21, 2009 3:25 PM | Permalink | Reply to this
### Re: Towards a Computer-Aided System for Real Mathematics
You need to make more exceptions to account for subcategories. There may be a category with object set C and two subsets A and B such that there are subcategories with object sets A and B. In this case, one must be able to compare their elements, too, because of the definition of a subcategory.
Structurally, a “subcategory” of $C$ means a category equipped with an injective functor to $C$, just like a “subset” means a set equipped with an injective function (except in SEAR, where subsets are technically distinguished from their tabulations—but even there, it is only the tabulation which is itself a “set” and therefore can be the set of objects of a category). Therefore, if $A$ is a subcategory of $C$, then objects of $A$ cannot be compared directly to objects of $C$, but only after applying the inclusion functor.
Posted by: Mike Shulman on September 21, 2009 5:56 PM | Permalink | PGP Sig | Reply to this
### still on objects common to different categories
MS: if A is a subcategory of C, then objects of A cannot be compared directly to objects of C, but only after applying the inclusion functor.
Then please tell me why, formally, the following reasoning is faulty.
I am using Definitions 1.1.1 (category) and 1.1.3 (subcategory) taken from Asperti and Longi, modified to take account of your statement above. If you think these are faulty, please give me a reliable replacement that I may take as authoritative.
But I do not accept any moral injunction unless it is presented as a formal restriction to what a theorem prover would be allowed to do.
Let $C_{abcd}$ be the category whose objects are the symbols $a,b,c,d$, with exactly one morphism between any two objects, composing in the only consistent way. Let the categories $C_{abc}$ and $C_{abd}$ be defined similarly. Clearly, these are both subcategories of $C_{abcd}$, with the identity as the inclusion functor. But I can compare their objects for equality.
[related snippets of other mails]
MS: I would consider the class of objects of each category to be a separate type,
Would this work consistently in the above example?
AN: I first need to understand what “should” be understood after having read Definition 1.1.1 and what after Definition 1.3.1.
MS: I think that what “should” be understood at this point is that the authors made a mistake in stating the exercise.
This requires having also read the exercise. But my example above seems to indicate that something nontrivial and unstated should already be understood at the point these two definitions have been read.
What we do in this whole discussion is in fact the typical process of how mathematicians growing up in different traditions learn to align their language so that they may speak with each other without generating (permanent) misunderstandings. As a result, the language and awareness of all participating parties is sharpened, and then applicable to a wider share of mathematical documents.
Posted by: Arnold Neumaier on September 22, 2009 2:39 PM | Permalink | Reply to this
### Re: still on objects common to different categories
if A is a subcategory of C, then objects of A cannot be compared directly to objects of C, but only after applying the inclusion functor.
I am not a mathematician and I think this has nothing to do with category theory and even with mathematics:
You assume that you can talk about objects (in the lay meaning of the word, not categorically) from A and C without taking care of defining the proper “universe of discourse” in which the denotations for objects in A and objects in C can validly appears in the same sentence.
This is a platonist stance, the objects “exist” independently of the discourse about them.
I would say, not so, Platonism has been poisoning mathematics and philosophy for more than two millenia.
Posted by: J-L Delatre on September 22, 2009 8:49 PM | Permalink | Reply to this
### Re: still on objects common to different categories
Let $C_abcd$ be the category whose objects are the symbols $a,b,c,d$, with exactly one morphism between any two objects, composing in the only consistent way. Let the categories $C_abc$ and $C_abd$ be defined similarly. Clearly, these are both subcategories of $C_abcd$, with the identity as the inclusion functor. But I can compare their objects for equality.
Yes, this is like the example of opposite categories. You started with the four objects $a,b,c,d$ and constructed things out of them. It should be possible to force you to use only copies of these four objects when you construct new things out of them, so that equality between them would not make literal sense, but I don't see the point in doing so. The constructions will still come equipped with operations to the type $\{a,b,c,d\}$ from the types of objects of the various categories, and we could compare them for equality along those operations. So why not formalise those operations as identity? That's how I would do it.
I don't know if that's how Mike would do it. One could not do that in $\mathbf{SEAR}$ (although one could for the example of opposite categories).
I think that it was John Armstrong who first wrote
But it really doesn’t matter, since equality of components of two distinct categories is not part of the structure.
I would not put it quite that way. I would say that equality of objects (or morphisms, etc) of two arbitrary categories is not meaningful; that is, in a context where all that is said about two categories $C$ and $D$ is that they are categories, then equality of their objects is not meaningful. However, if $C$ and $D$ are given to us in some more complicated way (such as $D := C^{op}$, or even $D := C$ for that matter), then that might give some meaning to equality of their objects.
There is nothing particularly special about categories in this respect; the same goes for elements of groups, for example.
Posted by: Toby Bartels on September 22, 2009 8:59 PM | Permalink | Reply to this
### Re: still on objects common to different categories
TB: I would say that equality of objects (or morphisms, etc) of two arbitrary categories is not meaningful; that is, in a context where all that is said about two categories C and D is that they are categories, then equality of their objects is not meaningful. However, if C and D are given to us in some more complicated way, then that might give some meaning to equality of their objects.
In the standard interpretation of mathematical language that everyone learns as an undergraduate, the standard definition of a category implies the following:
Equality between two objects of two arbitrary categories is undecidable, while that of two categories given by some explicit construction may be decidable. (Something analogous holds for elements of groups, etc. in place of objects of categories.)
With this modification, I agree with you. Indeed, this is what you have in FMathL. But undecidable and meaningless are different notions - the first says that you cannot assign any definite truth value to it, the second says that it is not well-formed. With your formulation (using meaningless), one does not get something consistent.
Posted by: Arnold Neumaier on September 23, 2009 10:20 AM | Permalink | Reply to this
### Re: still on objects common to different categories
In the standard interpretation of mathematical language that everyone learns as an undergraduate,
And which I unlearnt as a graduate (^_^)
But undecidable and meaningless are different notions - the first says that you cannot assign any definite truth value to it, the second says that it is not well-formed.
Right.
With your formulation (using meaningless), one does not get something consistent.
I can't imagine what you mean by this. What is inconsistent?
Posted by: Toby Bartels on September 25, 2009 6:50 PM | Permalink | Reply to this
### Re: still on objects common to different categories
From another point of view, the construction $\{a,b,c,...\} \mapsto C_{a,b,c,...}$ is a functor from the Category of Sets to the Category of Categories. The Category of Categories doesn’t itself have a privileged functor from $C_{a,b,c}$ to $C_{a,b,c,d}$. The one that looks natural comes from the natural-looking map from $\{a,b,c\}$ to $\{a,b,c,d\}$ — which is precisely the absent thing in SEAR or (i.i.r.c.) ETCS.
But it gets worse (or better!): we often want to consider the 2-category of categories, functors, and natural transformations (or of groupoids, functors and natural (automatically) isomorphisms). And in this setting, there isn’t much reason to privilege the natural-looking functor from $C_{a,b,c}$ to $C_{a,b,c,d}$ over the functor that sends each object in $\{a,b,c\}$ to $d$! It’s a naturally isomorphic functor, and in this case all of them are equivalences, anyways.
But you asked about what’s wrong “formally”. Since I haven’t read all of what you mean by “formal”, I similarly don’t know if this addresses that question at all.
Posted by: some guy on the street on September 23, 2009 6:43 AM | Permalink | Reply to this
### Re: still on objects common to different categories
sg: But you asked about what’s wrong “formally”. Since I haven’t read all of what you mean by “formal”, I similarly don’t know if this addresses that question at all.
Formally = in a way that it is clear how to teach it to an automatic system like FMathL or Coq.
How do you prevent such a system from drawing the conclusions I draw when the only context given are a definition of a category and of a subcategory, but the system already knows how to handle the language of naive set theory (with {x in A | property} but not {x | property}) and of elementary algebra as taught in undergraduate courses?
None of the three answers given so far resolves this. My conclusions are perfectly allowed according to the usual conventions of reading mathematics.
Thus in order to unambiguously defining the intended meaning, one needs to specify (without using the concept of a category since this is not yet born) either a different way of interpreting the same wording, or one needs to give a different wording to the standard definitions.
Posted by: Arnold Neumaier on September 23, 2009 9:42 AM | Permalink | Reply to this
### Re: still on objects common to different categories
How do you prevent such a system from drawing the conclusions I draw when the only context given are a definition of a category and of a subcategory, but the system already knows how to handle the language of naive set theory (with {x in A | property} but not {x | property}) and of elementary algebra as taught in undergraduate courses?
A strongly typed system would not conclude that there exists an $x$ such that $x$ is an object of $C_{a,b,c}$ and $x$ is an object of $C_{a,b,c,d}$, because none of that can be expressed in the language. Quantification requires a domain (which we can fix here), and being an object of some category is not a predicate (which we can't fix without changing what it says a bit).
I imagine that a less strongly typed system might be able to conclude something like that, while still rejecting $1 \in \sqrt 2$ as meaningless (at least in the default context). You would probably do this through subtyping. But I don't have much experience with subtyping.
A strongly typed system can still handle {x in A | property}. Assuming that property uses the variable x only where an element of A makes sense, then the system should accept this as specifying a subset of A (however the notion of subset is formalised).
So if you start with $C_{a,b,c,d}$, then you can construct the set of objects of $C_{a,b,c}$ as [the underlying set of] the subset {x in A | property}, where A is the set of objects of $C_{a,b,c,d}$ and property states that an element of $A$ equals $a$, $b$, or $c$. Then you can continue to get the entire category $C_{a,b,c}$. You can also give $C_{a,b,c}$ the structure of a subcategory of $C_{a,b,c,d}$. A system that knows about category theory could helpfully construct all of this for us as soon as we write down property and ask it to construct the corresponding full subcategory.
Now, any system (if it's any good for category theory) should be able to conclude this: There exists an object $x$ of $C_{a,b,c,d}$ that belongs to the subcategory $C_{a,b,c}$. (There is a formal distinction between $C_{a,b,c}$ as a subcategory of $C_{a,b,c,d}$ and $C_{a,b,c}$ as a category in its own right, which we normally ignore by abuse of language.) You already know about the distinction between being an element of a set —a typing declaration— and belonging to a subset —a relation between elements and subsets of a given set—; the same holds for being an object of a category and belonging to a subcategory.
Posted by: Toby Bartels on September 25, 2009 8:04 PM | Permalink | Reply to this
### Re: still on objects common to different categories
From another point of view, the construction $\{a,b,c, \ldots\} \mapsto C_{a,b,c, \ldots}$ is a functor from the Category of Sets to the Category of Categories. The Category of Categories doesn’t itself have a privileged functor from $C_{a,b,c}$ to $C_{a,b,c,d}$. The one that looks natural comes from the natural-looking map from $\{a,b,c\}$ to $\{a,b,c,d\}$ — which is precisely the absent thing in SEAR or (i.i.r.c.) ETCS.
I’m not sure what this is supposed to mean. The desired inclusion $\{a, b, c\} \hookrightarrow \{a, b, c, d\}$ in $Set$ is constructed in ETCS by interpreting these sets as a 3-fold and 4-fold coproduct of copies of a terminal set $1$ and invoking universal properties of coproducts. So I have no idea what is meant by saying it’s “absent” from ETCS (or SEAR for that matter).
There’s a meta-theorem that ETCS and Bounded Zermelo set theory with Choice are bi-interpretable in one another, so that anything you can express in one is expressible in the other. This might be helpful in realizing what can and cannot be said in ETCS. Mike has also written down bi-interpretability statements in this vein in the SEAR article.
Posted by: Todd Trimble on September 23, 2009 1:35 PM | Permalink | Reply to this
### Re: still on objects common to different categories
OK, I stand corrected.
Posted by: some guy on the street on September 23, 2009 4:09 PM | Permalink | Reply to this
### Re: still on objects common to different categories
To be fair, though, I didn’t say quite what I should have. Better would have been: in ETCS, given a 4-fold coproduct of copies of $1$, whose four coproduct inclusions $1 \to 1 + 1 + 1 + 1$ are given names $a$, $b$, $c$, $d$, we can think of those inclusions as providing subsets, and then construct the union of the subsets $a, b, c$. If we give the 4-element set the name “$\{a, b, c, d\}$”, this union gives a subset which interprets what is standardly meant by the inclusion $\{a, b, c\} \hookrightarrow \{a, b, c, d\}$ in naive set-theoretic language.
Continuing the bridge between naive language and more formal language (and keeping in mind that in structural set theory, elements of $S$ are defined to be morphisms $1 \to S$), we go on to define a membership relation between elements of a set like $S = \{a, b, c, d\}$ and subsets of $S$, like the one we just named $\{a, b, c\} \hookrightarrow \{a, b, c, d\}$: we say an element $x$ of $S$ is a member of a subset $T \hookrightarrow S$ if $x: 1 \to S$ factors through the subset inclusion. Then the members of the subset $\{a, b, c\} \hookrightarrow \{a, b, c, d\}$ are indeed the elements of $S$ we called $a, b, c$, and everything is as it should be. But as you can see, some slight care is needed to give the naive language rigorous meaning in ETCS.
Posted by: Todd Trimble on September 23, 2009 5:05 PM | Permalink | Reply to this
### Re: still on objects common to different categories
in structural set theory, elements of $S$ are defined to be morphisms $1\to S$
This is true in ETCS, but not in SEAR.
Posted by: Mike Shulman on September 23, 2009 5:18 PM | Permalink | PGP Sig | Reply to this
### Re: still on objects common to different categories
Thanks. That’s what I meant.
Posted by: Todd Trimble on September 23, 2009 7:20 PM | Permalink | Reply to this
### Re: still on objects common to different categories
Let $C_{abcd}$ be the category whose objects are the symbols $a,b,c,d$
I’m going to assume we’re talking about small categories, so that we can make formal sense of them in any set theory. As we’ve said repeatedly, there is nothing special about categories here and the presence of evil can muddy the waters, but since you’re insisting on talking about categories instead of, say, groups, let’s go on that way.
In structural set theory you can’t just pull things out of the air and make them into a set. They have to be given to you as elements of some other set. So where are those symbols coming from? I’m guessing that you have in mind some infinite set of symbols, which could be represented by $\mathbb{N}$, so that you can construct its subset $\{a,b,c,d\}$ (using, for example, an encoding $a=0,b=1,c=2,d=3$), which is (or, in SEAR, its tabulation is) another set equipped with a specified injection into $\mathbb{N}$. Now you can of course construct a further subset $\{a,b,c\}$ with a further injection into $\{a,b,c,d\}$. Those injections are then how you then compare their elements.
Posted by: Mike Shulman on September 23, 2009 3:41 PM | Permalink | PGP Sig | Reply to this
### What is a structured object?
MS: either we are talking about two different group structures on the same set, in which case the two have exactly the same elements by definition, or we are talking about group structures on different sets, in which case asking whether an element of one is equal to an element of the other is a type error.
Are we allowed in SEAR to talk about group structures on different subsets of the same set? Then we can again ask these questions.
Or is this question meaningless? You haven’t defined how to create in SEAR objects such as groups or group structures. Are the latter sets, elements, or relations?
Or are they a new type of formal objects that were not present before? Then how did they come into existence? (If you want to have objects of each category to be of a different type, you better first allow for a countable set of types in SEAR.)
Or are they only metaconcepts without a formal version, just a way of talking? (But on the metalevel, one seems to be able to compare arbitrary semantical constructs. Or do you want to impose restrictions on what qualifies for valid metastatements?)
MS: I guess that this sort of implicit interpretation is part of the “compilation from high-level language to assembly language,” and what you want is a formalization of the higher-level language that (among other things) includes “a group” as a fundamental object of study.
This seems necessary in order that a theorem prover can understand all conventions.
But it seems that in category theory one has “groups” as formal objects (of the category of groups), while “group structure on a set” is a meta-object only, consisting of a set $S$ and a group $G$ with $set(G)=S$, where $set$ is the forgetful functor that removes the operations.
So part of the problem appears to lie in that you switch between different points of view (formal object or only a way of speaking tha can be formalized only by eliminating the concept) about what a group is.
Asking the system to rewrite all occurences of groups (and other structural concepts that in ZF would be tuples) by elimination of these concept on the formal level probably may create a huge overhead in view of the nested object constructions we often have in mathematics.
I commented on a related issue at the pure set entry of the nLab (under Membership trees). [I just see that the double opening apostrophe lead to an unsuspected result there. Unfortunately, the nLab editing has no preview facitly that would allow one to see things before posting.]
Too much is fuzzy for me to see what you really want to have.
Posted by: Arnold Neumaier on September 22, 2009 3:42 PM | Permalink | Reply to this
### Re: What is a structured object?
You haven’t defined how to create in SEAR objects such as groups or group structures. Are the latter sets, elements, or relations?
Given a set $G$, a group structure on $G$ is a function from $G \times G \to G$ such that …. Functions and products have already been discussed, and the condition … can be stated in the language of $\mathbf{SEAR}$. I claim, as a partisan of structural set theory, that all definitions and proofs in ordinary mathematics are like this, modulo abuses of language (such as suppressing the inclusion function $X \to Y$ when $X$ is defined as a subset of $Y$) that are no worse than the abuses used with $\mathbf{ZFC}$.
(If you want to have objects of each category to be of a different type, you better first allow for a countable set of types in SEAR.)
This is already present; $\mathbf{SEAR}$ is a theory in first-order logic, so it already has a countable set of variables. It is a dependent type theory, and each pair of variables for a set gives a type of relations; the only other type in it is the type of sets. As $1 + 2 \aleph_0 = \aleph_0$, that is the number of types. (Of course, this is all on the metalevel.)
Asking the system to rewrite all occurences of groups (and other structural concepts that in ZF would be tuples) by elimination of these concept on the formal level probably may create a huge overhead in view of the nested object constructions we often have in mathematics.
On the contrary, a series of definitions like ‘A group is a set equipped with a function ….’, ‘A ring is a group equipped with a function ….’, and ‘An ordered field is a ring equipped with a relation ….’ leads in the ‹A structure is a tuple.› view to an ordered field being a pair with a pair with a pair $(((S,+),\cdot),\lt)$; quite a mess! While in the ‹A structure consists of several objects.› view, it simply leads to a set, two functions, and a relation.
On the other hand, one can take the structure as tuple view in a structural foundation, using something like Coq's Records, if one wants to. But this requires a richer ground type theory than $\mathbf{SEAR}$ has.
[I just see that the double opening apostrophe lead to an unsuspected result there. Unfortunately, the nLab editing has no preview facitly that would allow one to see things before posting.]
[Yes, many others have asked for a Preview; the downside is that one of the biggest complaints about MediaWiki is that it's too easy to lose your edit since you forgot that the Preview was not a Save! The philophy in Instiki is that your Sumbit is a preview; if you don't like what you see, then you edit again, and it counts as only one edit in the history if your Submits are all within 30 minutes of each other no other editor slips in between. See discussion here.]
Posted by: Toby Bartels on September 22, 2009 9:45 PM | Permalink | Reply to this
### Re: What is a structured object?
AN: You haven’t defined how to create in SEAR objects such as groups or group structures. Are the latter sets, elements, or relations?
TB: Given a set G, a group structure on G is a function from G×G to G such that ….
OK. Thus a group structure on a set G is an object of type relation(GxG,G).
Now what is a group? Does one have to eliminate the concept of group in favor of that of a group structure when going from informal SEAR to formal SEAR as a first order logic with dependent types?
AN: Asking the system to rewrite all occurences of groups (and other structural concepts that in ZF would be tuples) by elimination of these concept on the formal level probably may create a huge overhead in view of the nested object constructions we often have in mathematics.
TB: On the contrary, a series of definitions like “A group is a set equipped with a function …” leads in the “A structure is a tuple” view to an ordered field being a pair with a pair with a pair (((S,+),$\cdot$),$\lt$); quite a mess!
At present, every formalization of a piece of mathematics is a mess; this was not the point.
What I was referring to was the overhead in the length of the formalization. With ZF, you can formalize a concept once as a tuple, and then always use the concept on a formal level.
But with a composite thing that exist only as a way of speaking, the formalization must replace this thing in each occurrence by the defining way of speaking. If this happens recursively (and much of mathematics is deep in the sense of data structures), the size of the formal expression may explode to the point of making the automatic verification of simple high-level statements a very complex task.
TB: On the other hand, one can take the structure as tuple view in a structural foundation, using something like Coq’s Records, if one wants to. But this requires a richer ground type theory than SEAR has.
This is what I was aiming at. For reflection purposes, one cannot work in pure SEAR, while one can do that in pure ZF.
TB: The philophy in Instiki is that your Sumbit is a preview; if you don’t like what you see, then you edit again, and it counts as only one edit in the history if your Submits are all within 30 minutes of each other no other editor slips in between.
good to know.
Posted by: Arnold Neumaier on September 23, 2009 1:58 PM | Permalink | Reply to this
### Re: What is a structured object?
Now what is a group? Does one have to eliminate the concept of group in favor of that of a group structure when going from informal SEAR to formal SEAR as a first order logic with dependent types?
I already answered this up here: “A group in SEAR consists of a set $G$ and an element $e\in G$ and a function $m:G\times G \to G$, such that certain axioms are satisfied.”
A group in SEAR is not a single thing in the universe of discourse. This is not a problem for formalization at a low level, but it may be undesirable when trying to formalize at a higher level, for all the reasons that you’ve given. But it doesn’t prevent SEAR from reflecting on itself formally.
In Isabelle, at least, “a group” is really defined as follows: given a type 'a, one constructs the type 'a group of group structures on 'a, and then defines a group (of type 'a) to be an element of 'a group. I presume this is what Coq’s Records are like as well. I don’t see why this couldn’t be done in SEAR just as well, though: given a set $A$ we can define the set $grp(A)$ of group structures on $A$ as a subset of $A^{A\times A}$. Of course just knowing an element of $grp(A)$ doesn’t give you anything unless you remember that $grp(A)$ was constructed from $A$ in a particular way.
Posted by: Mike Shulman on September 23, 2009 3:33 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
Arnold Neumaier wrote:
OK. Thus a group structure on a set $G$ is an object of type $relation(GxG,G)$.
Now what is a group? Does one have to eliminate the concept of group in favor of that of a group structure when going from informal SEAR to formal SEAR as a first order logic with dependent types?
In SEAR, yes.
I would not found FMathL directly on SEAR, if I were you. Besides any efficiency problems, it's simply more user-friendly to treat a group as a single object. I would probably give any computer system for abstract mathematics a simple dependent type theory with support for dependent sums (and probably only depedent sums, unless I really want to found the whole thing on type theory) and implement the interface similarly to Coq's Records.
For reflection purposes, one cannot work in pure SEAR, while one can do that in pure ZF.
I don't see what reflection has to do with it. It's a matter of convenience and (if you say so) efficiency. Writing SEAR in a dependent type theory with direct sums doesn't add any strength to it (as long as you don't add quantification over types), since you can eliminate them (down to the base types).
Mike Shulman wrote in response:
In Isabelle, at least, “a group” is really defined as follows: given a type 'a, one constructs the type 'a group of group structures on 'a, and then defines a group (of type 'a) to be an element of 'a group. I presume this is what Coq’s Records are like as well. I don’t see why this couldn’t be done in SEAR just as well, though: given a set $A$ we can define the set $grp(A)$ of group structures on $A$ as a subset of $A^{A \times A}$. Of course just knowing an element of $grp(A)$ doesn’t give you anything unless you remember that $grp(A)$ was constructed from $A$ in a particular way.
What seems to be missing is the concept of just having a group simple, rather than a group of type 'a or an element of a set constructed from $A$ in a particular way.
Here is how you would define the type of groups in Coq:
Record Group: Type := {uGroup: Set; sGroup: GroupStructure uGroup}.
That is, a group consists of a set and a group structure on that set. If you have G of type Group, then uGroup G is of type Set and sGroup G is of type GroupStructure uGroup G. I'm assuming that we've already defined GroupStructure, since that is not a problem even in SEAR, but it might be more user-friendly not to do this but instead to put everything into the Record:
Record Group: Type := {uGroup: Set; mGroup: uGroup -> uGroup -> uGroup; aGroup: forall (x y z: uGroup), mGroup (mGroup x y) z = mGroup x (mGroup y z);
and so on.
Posted by: Toby Bartels on September 25, 2009 8:11 PM | Permalink | Reply to this
### Re: What is a structured object?
Let me say more explicitly the following: I am not asserting (any more) that structural set theory is sufficient for what you want to do. I can’t speak for anyone else, but I accept that structural set theory, whether ETCS or SEAR or whatever, has flaws preventing it from being used for the high-level computerized mathematical tool you want to create. They are different flaws from the flaws of ZF, and they are different from the flaws that it sounded to me like you were ascribing to it in your introduction to FMathL, but they are flaws nonetheless.
I do assert that structural set theory is a sufficient low-level foundation for mathematics on a par with ZF, and I believe that it is closer to the way mathematicians treat sets in everyday practice. (Although the language they use, in particular words like “subset,” tends to evoke material set theory. This can certainly be a barrier to understanding structural set theory; I blame it on the accident of history and the current ascendancy of ZF as a foundation.) Thus, I would like it if a higher-level language could be based on, or at least more in line with the ideas of, structural set theory. You clearly disagree with these assertions as well, and I’m more than happy to continue discussing them, but let’s not confuse them with the difficulty of implementing things on a computer.
Posted by: Mike Shulman on September 23, 2009 4:06 PM | Permalink | PGP Sig | Reply to this
### Material vs. structural foundations of mathematics
MS: I do assert that structural set theory is a sufficient low-level foundation for mathematics on a par with ZF, and I believe that it is closer to the way mathematicians treat sets in everyday practice.
I had a few days off to do things neglected while occupied with this time-consuming discussion. Instead of replying individually to the single contributions, let me summarize how things look form my perspective.
My main conclusion from the present discussion and from reading the nLab pages on SEAR and pure sets is the following:
In a material theory, structural objects are constructed as anonymous objects chosen from the equivalence classes of mathematical structures of some kind with respect to isomorphism. Then one can do all structural mathematics inside suitable such collections of equivalence classes. However, to do so for nontrivial mathematics requires numerous abuses of language and notation; otherwise the description becomes tedious and virtually incomprehensible.
In a structural theory, material objects are constructed as the rigid objects in some category, with being isomorphic as equality. Then one can do all material mathematics inside suitable such collections of rigid objects. However, to do so for nontrivial mathematics requires numerous (but different) abuses of language and notation; otherwise the description becomes tedious and virtually incomprehensible.
Thus from a descriptive point of view, the material interpretation and the structural interpretation form equivalent approaches, both not describing the essence of mathematics but only straightjackets into which this essence can be forced in a Procrustean way, and in which one feels better is a matter of taste. My taste is that neither of these should be used. I want to be free of straitjackets. Thus I favor a declarative theory similar to FMathL, which accounts for the actual mathematical language and needs no abuses of language.
From a logical point of view, there is the additional question of proof power of the two views. I’d find it surprising if there were a structural theory with proof strength equivalent to that of ZFC, but I’d find it plausible that there are hierarchies of structural theories and hierarchies of material theories such that each of the former has a proof strength inferior to one of the latter, and conversely. Thus form the logical point of view, the structural and the material approach are still equivalent but in a weaker sense.
So what ultimately counts is the practical point of view. Here the advantage of the material point of view is very clear. After all, we already need a material free monoid to communicate mathematics. Then, the material point of view is nearly obvious to any newcomer, making for a simple entrance and plenty of very elementary exercises that lead to mathematical insight, while the structural point of view emerges only after having digested enough of more elementary material mathematics.
On the other hand, for many problems, both the material and the structural perspective offer insights. Therefore a good foundation of mathematics should offer both views.
Thus for me the priorities are clear: Describe mathematics in a declarative way that allows naturally both material and structural constructs, but support it constructively with a material mathematical universe in which the structural realm is constructible in a transparent way that can be easily used.
Posted by: Arnold Neumaier on September 30, 2009 12:34 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I’d find it surprising if there were a structural theory with proof strength equivalent to that of ZFC
As has already been mentioned several times, there are real theorems here. The book by Mac Lane and Moerdijk proves that ETCS is equiconsistent with a fragment of ZFC (Bounded Zermelo with Choice), but as shown by Colin McLarty, one can easily strengthen ETCS with structural axioms so that ETCS+ is equiconsistent (bi-interpretable) with full ZFC. Mike and others (Toby, David Roberts, and there may be others too) have written down details of similar equiconsistency statements involving SEAR.
In each case, the idea is the same; see Mac Lane-Moerdijk for details. To reflect ZFC in a structural set theory like ETCS+, one reflects material sets using well-founded rooted trees; “elements” of such a tree with root $r$ are subtrees rooted at the children of $r$. (There’s also the example of algebraic set theory – see the book by Joyal-Moerdijk, where models for ZFC are constructed as certain types of initial algebras.)
Certainly in the case of ETCS and algebraic set theory, this material has been well worked over, so I’m curious as to why you express doubt about its validity.
Posted by: Todd Trimble on September 30, 2009 3:37 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I believe the strengthening of ETCS to a theory equivalent with ZFC actually predates McLarty by quite some time (although McLarty’s axiom is a bit different). One reference (which I wanted to mention earlier but failed to remember) is
• Osius, Gerhard, “Categorical set theory: a characterization of the category of sets”, JPAA 1974
Posted by: Mike Shulman on September 30, 2009 5:14 PM | Permalink | PGP Sig | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: I’d find it surprising if there were a structural theory with proof strength equivalent to that of ZFC
TT: As has already been mentioned several times, there are real theorems here. The book by Mac Lane and Moerdijk proves that ETCS is equiconsistent with a fragment of ZFC (Bounded Zermelo with Choice), but as shown by Colin McLarty, one can easily strengthen ETCS with structural axioms so that ETCS+ is equiconsistent (bi-interpretable) with full ZFC.
I didn’t express doubts but said I’d find it surprising, meaning that it would reveal something to me that would extend my intoition. Clearly, my intuition about structural foundations is more limited than yours, so I can be easier surprised than you.
Maybe there is a theorem there, but the SEAR material here on the web doesn’t give complete proofs, and ETCS is, as you say, weaker than ZFC.
So I’d like to see Colin McLarty’s stronger version in order to understand what is missing in my intuition. Can you give me a reference to his work?
Posted by: Arnold Neumaier on September 30, 2009 5:33 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Osius’ paper I cited above is a good one to look at. I think the paper of McLarty’s that we are referring to is “Exploring categorical structuralism” in Philos. Math.
I am also planning to include a more detailed proof, dealing with the non-well-founded case as well as the well-founded one, in my forthcoming paper “Unbounded quantifiers and strong axioms in topos theory,” which I will post about when it is in a state to be read by others.
Posted by: Mike Shulman on September 30, 2009 5:42 PM | Permalink | PGP Sig | Reply to this
### Re: Material vs. structural foundations of mathematics
Yes, that was the paper by McLarty that I had in mind. Unfortunately I am not familiar with the paper of Osius, although I’m aware of it.
Posted by: Todd Trimble on September 30, 2009 6:07 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
MS: Osius’ paper I cited above is a good one to look at. I think the paper of McLarty’s that we are referring to is “Exploring categorical structuralism” in Philos. Math.
I got the latter from the web; it refers to Osius for the crucial part. The latter is not free online; so it will take a while for me to get it and read it.
Posted by: Arnold Neumaier on September 30, 2009 6:15 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I didn’t express doubts but said I’d find it surprising, meaning that it would reveal something to me that would extend my intoition. Clearly, my intuition about structural foundations is more limited than yours, so I can be easier surprised than you.
Hmm. Here is what you wrote:
I’d find it surprising if there were a structural theory with proof strength equivalent to that of ZFC, but I’d find it plausible that there are hierarchies of structural theories and hierarchies of material theories such that each of the former has a proof strength inferior to one of the latter, and conversely. Thus from the logical point of view, the structural and the material approach are still equivalent but in a weaker sense.
The last sentence, a straightforward declaration, sure reads like a rejection of the stronger sense. If you didn’t mean that way, it is seriously misleading.
Posted by: Todd Trimble on September 30, 2009 6:29 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: Thus from the logical point of view, the structural and the material approach are still equivalent but in a weaker sense.
TT: The last sentence, a straightforward declaration, sure reads like a rejection of the stronger sense. If you didn’t mean that way, it is seriously misleading.
I should have written: … but (to my present understanding) in a weaker sense.
But I thought that everything anyone says is to be considered subject to the restriction “according to the writer’s present understanding”.
Posted by: Arnold Neumaier on September 30, 2009 7:08 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
But I thought that everything anyone says is to be considered subject to the restriction “according to the writer’s present understanding”.
Yes of course, but that still doesn’t erase the fact that it’s a declaration of belief. My question was why you believe(d) it.
Posted by: Todd Trimble on September 30, 2009 7:33 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
TT: but that still doesn’t erase the fact that it’s a declaration of belief. My question was why you believe(d) it.
My present intuition tells me that equivalence is unlikely to hold. You tell me otherwise, and evidence of a proof (which I hope to gather by reading the paper by Osius - McLarty doesn’t have the details) may well change my belief.
Posted by: Arnold Neumaier on September 30, 2009 7:59 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
My present intuition tells me that equivalence is unlikely to hold.
Okay, thank you, that’s a good honest positive declaration. But you still haven’t said WHY. Why does your intuition tell you that?
Posted by: Todd Trimble on September 30, 2009 8:18 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
In a structural theory, material objects are constructed as the rigid objects in some category, with being isomorphic as equality. Then one can do all material mathematics inside suitable such collections of rigid objects.
I am forced to conclude that you have not understood anything that we have been saying.
One can construct a model of material set theory inside structural set theory by using rigid trees or other models. This may be interesting to do if one doubts that they are equally strong. However, it is irrelevant for matheamtical practice, because this is not, not, not how one does mathematics in a structural set theory! A group in structural set theory is a set with a multiplication operation and an identity satisfying the axioms—there is no need to equip this set with the superfluous extra structure of a rigid tree.
After all, we already need a material free monoid to communicate mathematics.
I have no idea what that means.
the material point of view is nearly obvious to any newcomer, making for a simple entrance and plenty of very elementary exercises that lead to mathematical insight, while the structural point of view emerges only after having digested enough of more elementary material mathematics.
My experience in teaching newcomers to mathematics is that even material set theory is fraught with conceptual hurdles. At present, one tends to appreciate the structural point of view only after digesting some abstract mathematics (or, perhaps, never), but there’s no evidence that it has to be that way. I would argue that that’s an artifact of the fact that almost everyone is taught material set theory first, and hardly anyone is ever taught structural set theory.
Posted by: Mike Shulman on September 30, 2009 5:21 PM | Permalink | PGP Sig | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: In a structural theory, material objects are constructed as the rigid objects in some category, with being isomorphic as equality. Then one can do all material mathematics inside suitable such collections of rigid objects.
MS: I am forced to conclude that you have not understood anything that we have been saying.
Maybe, but then the communication barrier is deeper than we both think.
MS: A group in structural set theory is a set with a multiplication operation and an identity satisfying the axioms—there is no need to equip this set with the superfluous extra structure of a rigid tree.
But as far as this goes there is no difference at all to the material point of view. A group in material set theory is also a set with a multiplication operation and an identity satisfying the axioms.
ZF adds superfluous extra stuff in terms of tuples that are sets of sets of sets, while SEAR adds superfluous extra structure in terms of lots of trivial conversion and embeddign functors.
None of this stuff is relevant for doing mathematics as it is done in practice.
But some of it is needed (in different ways) if one wants to force mathematics into either a purely material or a purely structural straitjacket. This is why I like neither of these constructive foundations. I want to avoid both extremes. (But find the material straitjacket still preferable to the structural one.)
AN: After all, we already need a material free monoid to communicate mathematics.
MS: I have no idea what that means.
The text displayed on the screen where you are reading this is composed of material elements of such a monoid.
Without language no communication of mathematics. But language needs a material monoid.
MS: strengthening of ETCS to a theory equivalent with ZFC actually predates McLarty
Thanks for the reference. I’ll try to get it…
Posted by: Arnold Neumaier on September 30, 2009 5:59 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
The text displayed on the screen where you are reading this is composed of material elements of such a monoid.
Yes, I think we knew you were referring to words, but how should we interpret what you mean by ‘material’?
For example, ‘word’ is a 4-tuple. In material set theory, there are various ways of representing 4-tuples, e.g.,
$\{w, \{\{w\}, o\}, \{\{\{w\}, o\}, r\}, \{\{\{\{w\}, o\}, r\}, d\}\}$
Is that what you meant by a “material element” of the free monoid? If so, why are you convinced that we need such constructions? If not, then what did you mean?
Posted by: Todd Trimble on September 30, 2009 7:04 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
TT: For example, ‘word’ is a 4-tuple. In material set theory, there are various ways of representing 4-tuples,
ZF and its relatives are not the only material theories.
And ‘word’ is not a 4-tuple; Kuratowski tuples form a monoid only under a very unnatural operation. Instead, it is an element of a free monoid generated by 4 material characters w, o, r, and d.
FMathL takes this into account.
Posted by: Arnold Neumaier on September 30, 2009 7:20 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
What, sir, do you mean by “material element of a free monoid”? For I take it you were saying that words (not characters, words) are “material elements of free monoids”.
What about “material” is necessary here?
Posted by: Todd Trimble on September 30, 2009 7:47 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
TT: What, sir, do you mean by “material element of a free monoid”? For I take it you were saying that words (not characters, words) are “material elements of free monoids”.
What about “material” is necessary here?
If w and o are material characters then their product (well-defined in any monoid) is material, too.
At least this is the understanding I gained from the use of material and structural in your community.
But even if this is not what you understand by these terms, it is the meaning I want to give the term (and is how I used it in all my mails), since this is the way it works in FMathL.
Posted by: Arnold Neumaier on September 30, 2009 8:14 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
a free monoid generated by 4 material characters w, o, r, and d
I (at least) have no idea what you mean when you call these characters ‘material’. Surely you don't mean that they are themselves sets, with their own elements? Perhaps you mean that they can be compared for equality with any other mathematical object, but I fail to see how this is needed for anything.
To discuss words, we need set a $A$, called the alphabet and whose elements are called letters; then a word is an element of the free monoid on $A$. We only need to test letters for equality with other letters, which is provided by the set $A$. We need to test words for equality with other words, which the free monoid construction also provides; it even provides a test for equality of composites of tuples of words. This is all perfectly structural. Indeed, the concept of ‘free monoid’ is inherently categorial.
Posted by: Toby Bartels on September 30, 2009 7:49 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
TB: I (at least) have no idea what you mean when you call these characters ‘material’. Surely you don’t mean that they are themselves sets, with their own elements? Perhaps you mean that they can be compared for equality with any other mathematical object
No, I mean that they have an identity such that w can be recognized as the letter w’, and not only as an anonymous element from some set. One needs not only know that w is different from o but also that w is in fact w’!
Structurally, there is no difference between any two 4-letter words with distinct letters.
But to know what a word means you need to know the identity of each letter. This is what makes things material in the sense I find most natural to give to this word, not that it is written as a set or that one can compare for equality.
TB: This is all perfectly structural. Indeed, the concept of ‘free monoid’ is inherently categorial.
I don’t think that material and structural are always in opposition.
If it were, it were not possible to translate reasonably smoothly from a traditionally material view of mathematics to a structural view.
Posted by: Arnold Neumaier on September 30, 2009 8:15 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
But to know what a word means you need to know the identity of each letter.
And so you do. If formalised in $\mathbf{ETCS}$ (for example), each letter is a function from $1$ to $A$, and these have their own identities.
This is what makes things material in the sense I find most natural to give to this word
I no longer remember who suggested to Mike that set theory in the style of $\mathbf{ZFC}$ be called ‘material’, and I don't think that I ever knew the reason. But unless you're claiming that this requires a foundation in the style of $\mathbf{ZFC}$ (in particular, with global equality and a global membership predicate), then I have no disagreement with you … but I don't see the relevance, either.
Posted by: Toby Bartels on September 30, 2009 8:29 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: But to know what a word means you need to know the identity of each letter.
TB: And so you do. If formalised in ETCS (for example), each letter is a function from 1 to A, and these have their own identities.
Then please tell me which function from 1 to A is the letter w.
Posted by: Arnold Neumaier on September 30, 2009 9:10 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
It’s whatever function $1 \to A$ has been named $w$.
It’s no different in principle from telling numbers apart. You could for example specify the subset $A$ of $\mathbb{N}$ consisting of the first 26 elements (with respect to say the standard ordering of $\mathbb{N}$), and decide to name the 23rd element $w$. From that point on, you know which function $1 \to A$ is meant by “$w$”.
I think I can understand what’s behind the question. For example, the complex numbers $i$ and $-i$ behave exactly alike. But of course they are not the same. To deal with that, you can decide to represent the complex numbers as the quotient field $\mathbb{R}[x]/(x^2 + 1)$ and then say “I’ve decided to name the residue class of $x$$i$’.” You could have named it $-i$ of course, but once you settle on the name, you stick with that.
Posted by: Todd Trimble on September 30, 2009 9:56 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: please tell me which function from 1 to A is the letter w.
TT: It’s whatever function 1→A has been named w. […] You could for example specify the subset A of ℕ consisting of the first 26 elements (with respect to say the standard ordering of ℕ), and decide to name the 23rd element w. From that point on, you know which function 1→A is meant by “w”.
Thus you don’t need just a set A, called the alphabet, but you need a particular well-ordering of the set A before your prescription makes sense. In my view, giving a well-ordering to A is materializing the set A.
Strictly speaking you also need to name the letters, which is to give a mapping from A to the set of names for the letters, which must be material. Othewrwise you cannot tell someone else formally which element represents which symbol.
But I think I understand your point of view, without agreeing with that it is any improvement over a more naive material point of view.
Posted by: Arnold Neumaier on September 30, 2009 11:28 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
In my view, giving a well-ordering to A is materializing the set A.
Well, that has very little (or nothing) to do with how I have been using the word “material.” Should we take “giving a well-ordering” as your definition of “materializing”? So that in particular, when you said “we already need a material free monoid to communicate mathematics,” we should have interpreted that as meaning “we need the free monoid on a well-ordered set?” I don’t think I would disagree with that latter assertion, but I don’t think it has anything to do with the material/structural divide in the way we have been using the words.
Posted by: Mike Shulman on October 1, 2009 5:03 AM | Permalink | PGP Sig | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: In my view, giving a well-ordering to A is materializing the set A.
MS: Well, that has very little (or nothing) to do with how I have been using the word “material.”
Then please give your definition of how the material and the structural point of view should be recognized.
MS: Should we take “giving a well-ordering” as your definition of “materializing”?
No. A set is materialized if it is given extra structure which makes its elements uniquely identifiable by giving a formal expression identifying it.
In particular, a well-ordering of a finite set materializes it since you can point to each particular element by a formal expression identifying it: ”the first element”, ”the second element”, etc.
If this is not the meaning of material then I have no clue why you can refer to ZF set theory as a material theory.
Posted by: Arnold Neumaier on October 1, 2009 10:33 AM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
No. A set is materialized if it is given extra structure which makes its elements uniquely identifiable by giving a formal expression identifying it.
I’m glad this has finally come out, although we’ve probably been doing an awful lot of talking past each other because we misunderstood how we each intended the word ‘material’.
It reminds me of a Buddhist story I once read. There was a man who worshipped Amitabha, who in traditional iconography is bright red, but the man had misunderstood or mistranslated and thought the color was gray, like ash from the fire. So whenever he meditated on and envisioned Amitabha, it was always a gray Amitabha. Finally the guy is on his deathbed, and just to be sure, asks his teacher what color Amitabha is, and on finding out bursts into laughing, saying, “Well, I used to think him the color of ash, and now you tell me he is red,” and died laughing.
‘Material’ as in “material set theory” is something I’d only heard in the last few months at latest. I just assumed it meant we were talking about a form of set theory founded on a global membership relation, like ZF, Bernays-Gödel, Morse-Kelly, etc. The “material” signified to me that elements had “substance” (I used the phrase ‘internal ontology’ before): could have elements which themselves could have elements, and so on.
Posted by: Todd Trimble on October 1, 2009 12:44 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I have tried to clarify what I intended to mean by “material set theory” and “structural set theory” at the set theory page on the nlab. This is pretty close to what Todd said. In particular, “material” is a property of a theory, not of a set. When you start talk about giving a set extra structure, that can of course be done structurally just as naturally (as the word suggests).
Posted by: Mike Shulman on October 1, 2009 2:36 PM | Permalink | PGP Sig | Reply to this
### Re: Material vs. structural foundations of mathematics
MS: I have tried to clarify what I intended to mean by “material set theory” and “structural set theory” at the set theory page on the nlab. […] In particular, “material” is a property of a theory, not of a set.
I added some remarks there. It doesn’t seem to define when an arbitrary theory is material, and hence does not define a property of a theory. It only defines the compund concept of a “material set theory”, and does this in terms too vague that one could decide questions such as whether FMathL is or isn’t a material set theory.
I very much prefer the concept of material vs. structural that I presented in a previous mail and extracted from your usage in the present discussion. (There you also used the terms “material foundations”, Todd Trimble and Toby Bartels used “material sets”, TB also used “material framework”; so the term clearly wants to be generalized…)
Posted by: Arnold Neumaier on October 1, 2009 4:13 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Todd Trimble and Toby Bartels used “material sets”, TB also used “material framework”
For the record, I'll specify what I mean by these.
I use ‘material’ as short for ‘membership-based’, which itself really means ‘featuring a global membership predicate’, which means ‘featuring a binary predicate which, given any two terms for a set, returns a proposition whose intended meaning is that the first set is a member of the other’. This is not a purely syntactic concept; it depends on the intended meaning.
In front of ‘set theory’, ‘foundations’, or ‘framework’, this is exactly what ‘material’ means; but ‘material sets’ really means ‘sets in a material set theory’, which in turn might literally mean ‘terms for sets in a material set theory’ or ‘the intended meaning of terms for sets in a material set theory’.
Posted by: Toby Bartels on October 1, 2009 9:17 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: A set is materialized if it is given extra structure which makes its elements uniquely identifiable by giving a formal expression identifying it.
TT: we misunderstood how we each intended the word ‘material’. […] ‘Material’ as in “material set theory” is something I’d only heard in the last few months at latest. […] The “material” signified to me that elements had “substance” (I used the phrase ‘internal ontology’ before): could have elements which themselves could have elements, and so on.
I hadn’t heard at all the term “material” in this context. Judging from a Google search, the term was coined in the n-Lab.
I guessed at the likely meaning from the examples of usage given by those discussing here. Being clearly a contrast to “structural” I was trying to see what sort of meaning I could give it that made sense in my general view of mathematics.
The only natural pair of informal contrasts I could find that matched reasonably were
structural = defined only up to isomorphism, independent of any particular construction
material = given in terms of concrete building blocks.
After having seen how material set theory was constructed within SEAR, I was able to make the second more specific to
material = being able to identify the elements uniquely by giving a formal expression identifying it.
This seemed to match, giving both a precise meaning to the terms and showing that the two concepts are not in complete opposition but having a common intersection that explains why both points of views can be taken as foundations and still be some sort of equivalent.
I am still in doubt about the precise nature of this equivalence. You had asked why? about my intuition, but I can’t pinpoint it at the moment. Perhaps reading Osius will help me understand my and his intuition.
But I think these terms, with the above meanign, are useful general notions, the endpoints of a continuum of ways of thinking about mathematics.
FMathL is trying to plough here middle ground.
Posted by: Arnold Neumaier on October 1, 2009 2:59 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
The only natural pair of informal contrasts I could find that matched reasonably were
structural = defined only up to isomorphism, independent of any particular construction
material = given in terms of concrete building blocks.
After having seen how material set theory was constructed within SEAR, I was able to make the second more specific to
material = being able to identify the elements uniquely by giving a formal expression identifying it.
There’s a missing ingredient in your (informal) characterization of “structural” which I think is crucial to the discussion, and which actually is very close in spirit to the characterization of “material” quoted at the very end. Properly understood, there is no clash whatsoever between “material mathematics” as I understand your use of the term now, and structural mathematics.
The missing ingredient is that in general, structures defined by means of “universal elements” are defined up to canonical (uniquely determined) isomorphism.
The bit I recently wrote about what we mean precisely in describing $\mathbb{R}[x]$ as ‘the’ “free $\mathbb{R}$-algebra on one generator” should suffice to illustrate what I mean. There can be many such structures (many realizations of such structure), but given any two of them, say $(A, a: 1 \to U(A))$ and $(B, b: 1 \to U(B))$, there is exactly one homomorphism $f: A \to B$ such that $U(f)(a) = b$. By a famous argument, this homomorphism must be an isomorphism. It is the (unique) canonical isomorphism between these two universal structures.
In particular, the only structure-preserving automorphism from $(A, a: 1 \to U(A))$ to itself is the identity, and once this structure is given, we can uniquely specify elements therein by means of formal expressions. For instance, we are given a specified (explicitly named) formal generator $a$, and other elements are uniquely and formally specified by applying algebra operations in recursive fashion, starting with that $a$.
Of course, this is just standard practice of mathematicians; structural mathematicians shouldn’t be seen as doing anything different.
Posted by: Todd Trimble on October 1, 2009 6:05 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
‘Material’ as in “material set theory” is something I’d only heard in the last few months at latest. I just assumed it meant we were talking about a form of set theory founded on a global membership relation, like ZF, Bernays-Gödel, Morse-Kelly, etc.
Mike introduced the term to the discussion here. That is exactly what it means.
Posted by: Toby Bartels on October 1, 2009 8:11 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Arnold wrote:
Thus you don’t need just a set A, called the alphabet, but you need a particular well-ordering of the set A before your prescription makes sense. In my view, giving a well-ordering to A is materializing the set A.
Well, that’s just one prescription, just something simple off the top of my head. It doesn’t have to be a well-ordering, but yes, the naming is definitely an additional structure, just as in the parable of $i$ and $-i$.
For example, in ETCS, if $[n]$ represents the coproduct of $n$ copies of a chosen terminal object 1, then there are exactly $n$ elements $1 \to [n]$; they are all coproduct inclusions, and certainly they are all distinct. It may help to think of $[26]$ as a blob of twenty six distinct points. The points are clearly distinct, but they look exactly alike, are clones if you will.
Then, you may assign them names however you please, writing next to them (or on their identical red shirts), ‘A’, ‘B’, …, ‘Z’ say. If you choose to close your eyes and they take off their shirts (in other words, if you forget the naming) and they permute among themselves, you obviously can’t retrieve the original naming. But, as along as the names are firmly attached, as long as you bear in mind the naming structure, you are free to use it, knowing for example where Mr. P went to under some specified mapping $f: [26] \to \Delta$.
This sort of thing happens at the formal level too. For example, part of the structure of $[2]$ as so-called “subobject classifier” is a given element $1 \to [2]$ which is traditionally called “true”. Such an element is considered part of the structure of the subobject classifier as such. With that structure firmly attached, you are then in the position to set up a well-defined bijective correspondence between functions $f: X \to [2]$ and subsets of $X$, by considering $f^{-1}(true) \subseteq X$. You could have chosen the other element of $[2]$ of course as your “true”, but whichever element you chose, you stick to it and remember it for future reference.
Posted by: Todd Trimble on October 1, 2009 2:13 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
TT: there are exactly n elements 1→[n]; they are all coproduct inclusions, and certainly they are all distinct. It may help to think of [26] as a blob of twenty six distinct points.
Yes, this is a different way of creating materially a set of 26 nameable elements, and again, it is not a pure set but a set with additional structure. Mathematicians very rarely use pure sets!
This reminds me of the $C_{abcd}$ problem, which still puzzles me. I’d like to know your answer to my query:
Let $C_{abcd}$ be the category whose objects are the symbols a,b,c,d, with exactly one morphism between any two objects, composing in the only consistent way. Let the categories $C_{abc}$ and $C_{abd}$ be defined similarly. Clearly, these are both subcategories of $C_{abcd}$, with the identity as the inclusion functor. But I can compare their objects for equality.
Do you agree that from the material point of view (e.g., with categories modelled inside ZF, as in Lang’s book), this reasoning is correct?
If not, what is contrary to the axioms?
And if my reasoning is right from the material point of view, which extra axioms (in addition to what is in Wikipedia, or Lang, or Asperti and Longi) characterize the permitted ways of structural reasoning?
Posted by: Arnold Neumaier on October 1, 2009 3:20 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Todd’s description of the complex numbers reminds me of debates I heard seven or so years ago about the structuralism then popular in the philosophy of mathematics which said that mathematical entities are patterns, and that all that mattered about elements of the pattern are their properties invariant under isomorphism. The idea here was to explain how $2$ is merely a place in a pattern however it is realised set theoretically.
Someone pointed out that this would entail identifying $i$ and $-i$ in the complex numbers since nothing distinguishes them according to their place within the structure of the complex numbers. After discussion with John, I realised that we are often not careful saying what we mean by $\mathbb{C}$. There’s a difference between the field $\mathbb{R}[x]/(x^2 + 1)$ and the same field with the extra structure of a choice of a residue class to be designated $i$. They belong to different categories.
In the first case, there are two automorphisms on the object; in the second case, only one, but there’s another object with the same image under the functor which forgets the structure.
Posted by: David Corfield on October 1, 2009 11:06 AM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
DC: There’s a difference between the field ℝ[x]/($x^2+1$) and the same field with the extra structure of a choice of a residue class to be designated i. They belong to different categories.
Does choosing a notation really change the category an object belongs to? This would make the conversion headache in the structural approach even worse.
Does the monoid $\mathbb{N}$ of natural numbers under addition no longer belong to the category monoids if I add the conservative definition 2:=1+1?
Similarly, why can’t I put $i:=x mod x^2+1$ to define the imaginary unit in ℝ[x]/($x^2+1$) without changing the category the latter object belongs to?
This does not affect the existence of the automorphism induced by $i\to -i$.
Or do you hold that each definition changes the type of an algebraic structure? This would make things extremely unworkable formally!
Posted by: Arnold Neumaier on October 1, 2009 3:33 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Does the monoid $\mathbb{N}$ of natural numbers under addition no longer belong to the category monoids if I add the conservative definition 2:=1+1?
Similarly, why can’t I put $i:= x mod x^2+1$ to define the imaginary unit in $\mathbb{R}[x]/(x^2+1)$ without changing the category the latter object belongs to?
This does not affect the existence of the automorphism induced by $i \mapsto -i$.
I think David said it right, but it’s slightly subtle. The way to reconcile it with the point you’re making is by recognizing that, considering $\mathbb{R}[x]$ as an abstract $\mathbb{R}$-algebra, it’s not clear which element is $x$ until you say so. Thus, there’s an automorphism on $\mathbb{R}[x]$ which sends $x$ to $-x$, and either (or indeed any $a x + b$ with $a \neq 0$) could be considered a distinguished generator of the polynomial algebra. Giving a generator $1 \to U(\mathbb{R}[x])$ (here $U$ denotes the appropriate underlying-set functor) is thus adding some extra structure to the algebra.
A typical categorical response to all this is to define $\mathbb{R}[x]$ to be the free $\mathbb{R}$-algebra on one generator, which has a materializing or concretizing effect. More explicitly, this involves a universal property: when we say “free algebra on one generator”, we mean (to be precise) that there is given a function $i: 1 \to U(\mathbb{R}[x])$, traditionally called ‘$x$’, such that for every function $f: 1 \to U(A)$ into the underlying set of an $\mathbb{R}$-algebra $A$, there exists a unique $\mathbb{R}$-algebra homomorphism $\phi: \mathbb{R}[x] \to A$ such that $f = U(\phi) \circ i$. And there: this formulation involving the universal function $i$ gives you a distinguished element which people usually call $x$.
Also note that $\mathbb{R}[x]$ equipped with this distinguished element $i: 1 \to U(\mathbb{R}[x])$ has no non-trivial automorphisms. This is just an instance of a feature holding true for general universal properties.
(People often also say “free” to refer to a property: there exists a distinguished element such that… rather than giving the element at the outset as extra structure. Caveat lector.)
Posted by: Todd Trimble on October 1, 2009 5:05 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: why can’t I put $i:=x mod x^2+1$ to define the imaginary unit in ℝ$[x]/(x^2+1)$ without changing the category the latter object belongs to?
TT: indeed any ax+b with a≠0 could be considered a distinguished generator of the polynomial algebra.
I don’t understand:
This should not matter in a purely structural view. If you change the generator, you also change the ideal and hence the resulting field, but in any case, the $i$ so defined will be the distinguished square root of -1 of this field. Since structrally everything is defined anyway only up to isomorphism, this gives exactly the right result, with a canonical $i$ that changes with the field considered.
TT: ℝ[x] equipped with this distinguished element i:1→U(ℝ[x]) has no non-trivial automorphisms.
This is true if you require that $i$ is preserved, but this is another reason why I find a purely structural point of view awkward.
I find it unacceptable that the concept of an automorphism changes simply by labeling an element. The world of pure structure is a strange world, not the world of the average mathematician.
The complex numbers as mathematicians generally use them have complex conjugation as an automorphism, although $i$ is distinguished but not preserved by this automorphism.
TT: The missing ingredient is that in general, structures defined by means of “universal elements” are defined up to canonical (uniquely determined) isomorphism.
Again I do not understand:
Two instances of the field of complex numbers (without a distinguished imaginary unit) are not structurally the same since there is no canonical isomorphism? This would be very strange indeed.
Posted by: Arnold Neumaier on October 1, 2009 6:59 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Two instances of the field of complex numbers (without a distinguished imaginary unit) are not structurally the same since there is no canonical isomorphism? This would be very strange indeed.
I wouldn't say that they are not, in some sense, the same just because there is more than one isomorphism. (There is always a sense in which they are not the same, if they are represented differently syntactically. But that is not itself a question for mathematics.) I would say this: It is not only important whether things are isomorphic, but also in how many ways they are isomorphic; after all, $Iso(A,B)$ is not just a truth value, but a set (a meta-set, although usually also realisable internally as a set). In higher category theory, we even have $Equiv(A,B)$ as (in general) an $\infty$-groupoid!
Of course, there is more to say than just the cardinality of $Iso(A,B)$, such as the action on it by the monoid $Hom(B,B)$ and so on. But when $Iso(A,B)$ is a singleton, then things become much simpler, to the point that simply writing $A = B$ is an abuse of language that is easy to handle. If $Iso(A,B)$ is inhabited but (possibly) not a singleton, then writing $A = B$ is a little more dangerous; the danger only really comes to fruition, however, when you get loops $A = B = C = A$ since the composite isomorphism $A \to B \to C \to A$ might not be the identity.
Posted by: Toby Bartels on October 1, 2009 9:48 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I am afraid, Arnold, that you did not attend carefully to what I wrote. I hope at least it was clear that I was trying to build a bridge of understanding between what you wrote and what David wrote. But, as I said, the mathematical point involved was slightly subtle, so I ask you to read again, with care.
Let me try again.
AN: why can’t I put $i := x mod x^2+1$ to define the imaginary unit in $\mathbb{R}[x]/(x^2+1)$ without changing the category the latter object belongs to?
TT: indeed any $a x + b$ with $a \neq 0$ could be considered a distinguished generator of the polynomial algebra.
I don’t understand:
This should not matter in a purely structural view. If you change the generator, you also change the ideal and hence the resulting field, but in any case, the $i$ so defined will be the distinguished square root of -1 of this field. Since structrally everything is defined anyway only up to isomorphism, this gives exactly the right result, with a canonical $i$ that changes with the field considered.
First of all, the part of mine that you quoted was lifted from between a pair of parentheses, where it was indeed a parenthetical aside. I now regret that aside, because it seems to have distracted you from the point I was trying to make.
Second, please note that the ideal generated by $x^2 + 1$ does not change if you replace $x$ by $-x$. That’s the point! There are two candidates in the polynomial algebra whose residue class modulo this ideal yields a square root of -1, but these residue classes are different square roots of -1. It follows that if you haven’t chosen a candidate to work with, you haven’t uniquely specified which so-called canonical square root of -1 in this model you intended to label $i$! (And if you’ll recall, unique specification was what this discussion was originally about.)
You may think, “well, clearly I meant to choose $x$”, but knowledge of which element that is is not encoded within the polynomial algebra structure, hence it is an extra piece of information in addition to the algebra structure.
TT: $\mathbb{R}[x]$ equipped with this distinguished element $i: 1 \to U(\mathbb{R}[x])$ has no non-trivial automorphisms.
This is true if you require that $i$ is preserved, but this is another reason why I find a purely structural point of view awkward.
I find it unacceptable that the concept of an automorphism changes simply by labeling an element. The world of pure structure is a strange world, not the world of the average mathematician.
The complex numbers as mathematicians generally use them have complex conjugation as an automorphism, although $i$ is distinguished but not preserved by this automorphism.
We are not “simply labeling an element”, we are also choosing an element to label. This is important for the purpose of making unique specifications, which are important for ‘material’ constructions according to the sense “material = being able to identify the elements uniquely by giving a formal expression identifying it.”
Since we are not simply assigning a label but choosing an element to label, and since this choice is an extra datum or structure, it is logical for this discussion (which was to elucidate a point David made, not to discuss the behavior of “average mathematicians”) that we consider automorphisms which “remember” (respect) this extra structure.
What categories average mathematicians choose to work in is their business. It’s fine if they want their morphisms to ignore preservation of the chosen “$i$”. Me: I’m flexible – I’ll work in whatever category is best suited to the discussion I’m having.
(With the little polemical dig “strange world”, I can’t resist adding my own: category theory in fact teaches one great flexibility in thinking. But this point is perhaps lost on someone who often whines about categorical straitjackets, on rather thin and not terribly well-informed evidence.)
I’ll also add, for what it’s worth, that this category, the one whose objects are pairs $(A, a: 1 \to U(A))$ consisting of algebras and elements in their underlying sets, and whose morphisms are algebra homomorphisms that preserve elements thus distingished, is an example of what we category theorists call a comma category, a very important tool. Comma categories are extremely relevant to discussions in which adjoint pairs of functors crop up (just about everywhere, in case you didn’t know), including in particular free functors which are adjoint to forgetful functors, and more particularly the polynomial algebra functor which is left adjoint to the forgetful functor from algebras to sets, which I touched upon over here.
TT: The missing ingredient is that in general, structures defined by means of “universal elements” are defined up to canonical (uniquely determined) isomorphism.
Again I do not understand:
Two instances of the field of complex numbers (without a distinguished imaginary unit) are not structurally the same since there is no canonical isomorphism? This would be very strange indeed.
Your quotation is taken from another comment, here. But please attend closely to what I said: I said structures defined by means of universal elements. The main example from that comment was the polynomial algebra $\mathbb{R}[x]$ equipped with a universal element $i: 1 \to U(\mathbb{R}[x])$. Did I speak of the complex numbers there? No, I did not. But could I speak of canonical isomorphisms if $\mathbb{C}$ is considered as also coming equipped with a an element $i: 1 \to \mathbb{C}$ which is universal among $\mathbb{R}$-algebras equipped with a chosen square root of -1? Yes, I could.
Please observe as well that I added that missing ingredient because I thought the little sound-bite you gave for “structural” was a bit thin and needed more. That ingredient I consider particularly relevant for building a bridge between ‘structural’ and your sense of ‘material’. But please also note that I said “a missing ingredient” – I wasn’t pretending to exhaust the meaning of ‘structural’.
As to your question, though, Toby has given an informed reply. There is rather more to be said than can be encapsulated within a brief aphorism.
Posted by: Todd Trimble on October 2, 2009 5:44 AM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Then please tell me which function from 1 to A is the letter w.
The letter ‘w’, of course.
Are you suggesting that you have another way to answer the question, which letter is the letter w?
Posted by: Toby Bartels on October 1, 2009 9:10 AM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
AN: Then please tell me which function from 1 to A is the letter w.
TB: The letter ‘w’, of course.
This is like answering ‘the expression $A_5$’ in response to ‘Which group is $A_5$?’. It doesn’t explain anything. You are simply pushing things you don’t like to the metalevel, as if this would solve the problem.
TB: Are you suggesting that you have another way to answer the question, which letter is the letter w?
I didn’t ask which letter is the letter w but which function form 1 to A is the letter w.
In a material set theory with urelements, you have A={a,…,w,x,y,z}, and w is a well-defined urelement.
The point is that there must be a way to tell a computer what is meant by w, and this can only be done on a formal level involving material objects.
Posted by: Arnold Neumaier on October 1, 2009 10:25 AM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
I didn’t ask which letter is the letter w but which function form 1 to A is the letter w.
Yes, and I defined a letter to be (following the framework of ETCS) a function from 1 to A.
In a material set theory with urelements, you have A={a,…,w,x,y,z}, and w is a well-defined urelement.
But which urelement is w?
Posted by: Toby Bartels on October 1, 2009 7:29 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
Arnold writes:
So what ultimately counts is the practical point of view. Here the advantage of the material point of view is very clear. After all, we already need a material free monoid to communicate mathematics. Then, the material point of view is nearly obvious to any newcomer, making for a simple entrance and plenty of very elementary exercises that lead to mathematical insight, while the structural point of view emerges only after having digested enough of more elementary material mathematics.
I don’t understand what you mean by “we already need a material free monoid to communicate mathematics.” Please explain.
The advantage of material set theory from a practical point of view will not be at all clear to some of us here; quite the contrary. In fact, I argued here that there are strong practical advantages of structural set theory – “practical” in the sense of being faithful to working practice of contemporary mathematics. In particular, I argued that a categories-based set theory, by focusing on the relevancy of universal properties, is at a formal level very directly concerned with mathematical essence – getting at the heart of what contemporary mathematicians need sets for and what they do with them – while at the same time eliminating extraneous and irrelevant features which manifest themselves in material set theory.
The basic argument you seem to be making is that structural set theory is harder to learn than material set theory. I think Mike, with his SEAR, makes a good case that that need not be true. Thus, I reject
the structural point of view emerges only after having digested enough of more elementary material mathematics
as mere assertion. Clearly the real test of the pedagogical viability of structural set theory is in the classroom. I’m happy to say that I’ve incorporated structural ways of thinking into undergraduate courses I’ve taught, and Toby says the same for himself. So these are not completely idle claims.
Posted by: Todd Trimble on September 30, 2009 5:55 PM | Permalink | Reply to this
### Re: Material vs. structural foundations of mathematics
There is an asymmetry that is still missing.
In a material theory, structural objects are constructed as […]. Then one can do all structural mathematics inside suitable such […]. However, to do so for nontrivial mathematics requires numerous abuses of language and notation; otherwise the description becomes tedious and virtually incomprehensible.
In a structural theory, material objects are constructed as […]. Then one can do all material mathematics inside suitable such […]. However, to do so for nontrivial mathematics requires numerous (but different) abuses of language and notation; otherwise the description becomes tedious and virtually incomprehensible.
The asymmetry is this: While one can construct material sets as you described, still for the purposes of normal mathematics there is no reason whatsoever to do so. When we structuralists hear an ordinary mathematician describe something about, say, Lie algebras, we immediately turn it into our own language (where, as Goethe would say, it might mean something completely different) and think about it that way. We definitely do not construct material pure sets and think of a Lie algebra as a Kuratowski pair. And we find that we have no difficulty in communicating with the Lie algebraist this way; they can't even tell that we are doing this.
Even when we hear set theorists talk about large cardinals, we still don't bother to construct material sets; if we only care about sets up to cardinality, then we're still talking about objects of the category $Set$ of structural sets. (Now sometimes the set theorists can tell if we're using categorial model theory, but that's perfectly valid in a material framework too.) Only if the set theorists bring up the von Neumann hierarchy do we need to construct material sets.
To be fair, the material set theorist doesn't really have to construct structural objects either, at least not in the formal way that you describe. But they still have to deal with certain categories and ignore the membership structure of the objects and morphisms in these categories; they know intuitively what to ignore (which is why anything that they say can be translated so readily into our language), but for us it is automatic.
Thus I favor a declarative theory similar to FMathL, which accounts for the actual mathematical language and needs no abuses of language.
I still want to see how you will interpret ‘An ordered monoid is a set that is both ordered and a monoid, such that ….’ with no abuse of language. Good for you if you can do it! But if abuses of language are unavoidable, and one must work to formalise their meaning rather than to define everything in such a way that they are already literally valid, then I'm just as happy to add one more for ‘a function on $A$’ when $A$ was declared to be a subset.
From a logical point of view, there is the additional question of proof power of the two views. I’d find it surprising if there were a structural theory with proof strength equivalent to that of ZFC
Notice that the only reason that anyone ever linked to pure set was to indicate (very roughly, of course) how such an equivalence would be proved. (If you don't accept that we've put in enough details to establish that, very well; even I am relying more on my intuition and Mike's judgement than a careful check of Mike's argument about Collection.) But this is not necessary to understand how ordinary mathematics may be formalised in structural set theory (especially since ordinary mathematics doesn't even need high-powered set-theoretic axioms like Collection).
After all, we already need a material free monoid to communicate mathematics.
I don't understand what you mean by this. Aren't the elements of this free monoid simply words? (properly, strings of characters). Why do words need to have material elements??? (Of course, they need to have letters, but those are different.)
Then, the material point of view is nearly obvious to any newcomer,
I dispute this too.
The newcomer will think that they know what a set is, until you tell them that everything is a set, which they will find odd. You can avoid this, at least at first, with Urelemente, but eventually you'll do something like construct the set of real numbers, and then they will learn that a real number is a set, which is odd. Meanwhile, the structural set theorist, whose sets all have anonymous elements, has all along said that a set is merely a way to encode or describe certain things; we have never pretended that the elements of the set are those things, and so it is no surprise when a real number may be encoded as or described by, say, a set of rational numbers.
And at some point you must tell them that they are not allowed to take the set of all sets (or if they are, that they are at any rate not allowed to take the set of all sets that do not belong to themselves), which is no worse than telling them that they are not allowed to compare elements of two sets without some explicit way (such as a bijection between the sets) of comparing them. At least I know how to motivate the latter (but to be fair, the former also has to be explained, although perhaps later, by our group).
On the other hand, for many problems, both the material and the structural perspective offer insights. Therefore a good foundation of mathematics should offer both views.
I agree with this (well, at least the second sentence). But you've already agreed that either perspective allows one to formalise the other.
Posted by: Toby Bartels on September 30, 2009 6:42 PM | Permalink | Reply to this
### Re: What is a structured object?
So part of the problem appears to lie in that you switch between different points of view (formal object or only a way of speaking tha can be formalized only by eliminating the concept) about what a group is.
This is a fair criticism; I think we’ve been a bit sloppy about this in the foregoing discussion. The problem is that category theory which deals with large categories is hard to formalize in any kind of set theory. Neither ZF nor SEAR nor ETCS has an intrinsic object called “a large category.” In ZF, one “defines” a “proper class” to be specified by a first-order formula, and then a “large category” to be a “meta-category” whose objects and arrows are proper classes.
In structural set theory, one way to “define” a “large category” is to give a finite graph $D_C$ together with a couple of first-order formulas $obj_C$ and $arr_C$ with free variables labeled by the vertices and edges of $D_C$. “An object” of this category is then a diagram of shape $D_C$ in $Set$ (hence, a collection of sets and functions) such that $obj_C$ holds with the appropriate variables substituted, and likewise for “a morphism”.
Neither of these situations is really completely satisfactory. In ZF one can extend the theory to NBG or MK or add universes, and redefine “large category” to mean “category whose set of objects is not necessarily an element of the universe.” One structural counterpart of this is algebraic set theory in which classes, rather than sets, are the objects of the basic category under consideration, and there is a notion of “smallness” such that “sets” are the small classes. I feel that a more structural version of this considers a 2-category of large categories, rather than a category of classes, since in practice one rarely cares about the objects of a proper class up to more than isomorphism; I have some axioms for such a 2-category written down but haven’t put them up anywhere yet.
So, although when talking informally about category theory, I tend to think “2-structurally,” I’m not sure whether there yet exists a formal system which really captures what I mean by this. Thus there are really two questions here: the suitability of structural set theory for “small mathematics,” and its potential extensions to a “structural category theory” or “structural class theory” adequate for dealing with large categories (and which could hopefully be extended to treat extra-large 2-categories, XXL 3-categories, etc.).
Posted by: Mike Shulman on September 23, 2009 4:16 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
BTW, if we want to keep talking about the structural viewpoint on large categories and we want a formal setting in which to do it, universes in structural set theory should be perfectly adequate. Their main flaw is that they permit evil, but this shouldn’t be essential for understanding the issues in a structural viewpoint. David Roberts and I have been working on the axioms for universes in SEAR.
I don’t have time right now to explain how one goes about constructing a category of small sets, and thence a category of small groups, from a universe, but maybe someone else can.
Posted by: Mike Shulman on September 24, 2009 5:41 AM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
MS: A group in SEAR consists of a set G and an element e∈G and a function m:G×G→G, such that certain axioms are satisfied. A group in SEAR is not a single thing in the universe of discourse.
AN: So part of the problem appears to lie in that you switch between different points of view (formal object or only a way of speaking that can be formalized only by eliminating the concept) about what a group is.
MS: This is a fair criticism; I think we’ve been a bit sloppy about this in the foregoing discussion. The problem is that category theory which deals with large categories is hard to formalize in any kind of set theory.
This has nothing at all to do with large categories. Consider the category of finite groups. Its objects are finite groups, not group structures on a finite set. Thus you need to have the concept of finite group as an object rather than as a metaobject that cannot be formalized except by eliminating it from the formal representation.
MS: This is not a problem for formalization at a low level, but it may be undesirable when trying to formalize at a higher level, for all the reasons that you’ve given. But it doesn’t prevent SEAR from reflecting on itself formally.
It does. Reflection means being able to define a copy of SEAR inside SEAR, including all the language used to define this copy. (This is independent on any computer implementation. The latter, of course, must in addition care about efficiency, which causes some additional problems for theories where important concepts like that of a group are not a single thing in the universe of discourse.)
This means you need to start by calling the elements of a certain SEAR sets characters, then creating a SEAR model of text, then creating a SEAR model of context-free languages to express formulas and phrases, then create a model of what are variables, type declarations, axioms, definitions, assertions, proofs, and then state in this language the SEAR axiom system together with the definitions and assertions needed to explain the terminology used in the axioms.
Then, and only then, you can speak of having SEAR as a foundation.
MS: I do assert that structural set theory is a sufficient low-level foundation for mathematics on a par with ZF, and I believe that it is closer to the way mathematicians treat sets in everyday practice.
With ZF in place of SEAR, all of the above has be done at various levels of detail, and one can find for each step literature expanding on it in fairly detailed ways.
But I do not see how you can do this consistently with SEAR.
Apparently, you cannot even define formally the concept of a category without avoiding the problems with the category $C_{1234}$ I had mentioned. (For definiteness, here I specialize abcd to elements from the natural numbers inside SEAR.)
And there are ZF-based texts like Bourbaki and Lang who introduce each permitted abuse of language before using it. SEAR abuses the language without any excuse, and without saying how to undo the abuses if one wants to be more careful.
I know that it is a time-consuming task to repeat this for new foundations, and I neither expect that you do this quickly or that a single person should be expected to do this.
But I’d expect that you don’t assert something you are so far from having achieved.
Posted by: Arnold Neumaier on September 24, 2009 9:14 AM | Permalink | Reply to this
### Re: What is a structured object?
AN: Apparently, you cannot even define formally the concept of a category without avoiding the problems with the category $C_{1234}$ I had mentioned.
Just to clarify: I didn’t mean obstacles related to large cardinals. The problem arises even for categories all of whose objects are finite sets equipped with extra structure.
Posted by: Arnold Neumaier on September 24, 2009 11:04 AM | Permalink | Reply to this
### Re: What is a structured object?
I’ve enjoyed reading this vigorous exchange.——————–
In “Introduction to higher order categorical logic” by Joachim Lambek, P. J. Scott
I noticed that they recommended type theory as a foundation to mathematics rather than either category theory or set theory.
Bertot and Casteran, Interactive Theorem Proving (Coq)
“Amokrane Saibi showed that a notion of subtype with inheritance and implicit coercions could be used to develop modular proofs in universal algebra, and most notably, to express elegantly the main notions in category theory.”
http://pauillac.inria.fr/~saibi/Cat.ps by Amokrane Saibi (Coq)
“We then construct the Functor Category, with the natural definition of natural transformations. We then show the Interchange Law, which exhibits the 2categorical structure of the Functor Category. We end this paper by giving a corollary to Yoneda’s lemma.
This incursion in Constructive Category Theory shows that Type Theory is adequate to represent faithfully categorical reasoning. Three ingredients are essential: \Sigma types, to represents structures, dependent types, so that arrows are indexed with their domains and codomains, and a hierarchy of universes, in order to escape the foundational difficulties. Some amount of type reconstruction is necessary, in order to write equations between arrows without having to indicate their type other than at their binder, and notational abbreviations, allowing e.g. infix notation, are necessary to offer the formal mathematician a language close to the ordinary informal categorical notation.”
SH: Perhaps this is interesting.
Posted by: Stephen Harris on September 24, 2009 2:29 PM | Permalink | Reply to this
### Re: What is a structured object?
Type theory is, indeed, a very nice foundation for mathematics, which is very closely related to structural set theory. In fact, Bounded SEAR is nearly indistinguishable from type theory, and ETCS is also basically equivalent to it. However, my opinion (and this is only my opinion) is that type theory is harder for mathematicians without training in logic to understand, whereas they are quite used to thinking in terms of sets, relations, and functions. Perhaps this is only a relic of the ascendancy of material set theory as a foundation for so many years. Perhaps it is an artifact of the viewpoint taken by most textbooks on type theory.
Posted by: Mike Shulman on September 24, 2009 6:02 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
Perhaps it is an artifact of the viewpoint taken by most textbooks on type theory.
I blame this. Most books on ‘type theory’ are about logic; most books on ‘set theory’ (even if structural) are about mathematics. But I see ‘type’ and ‘set’ as nearly interchangeable, although ‘type’ can also be used in a broader context (for which there are other words if I want to be more specific, such as ‘preset’, ‘class’, or even —conjecturally for me— ‘$\infty$-groupoid’).
Posted by: Toby Bartels on September 25, 2009 8:28 PM | Permalink | Reply to this
### Re: What is a structured object?
But I see ‘type’ and ‘set’ as nearly interchangeable
Here are some differences in the way I think of them:
• The elements of a set are always equipped with a notion of equality, while the elements of a type need not be.
• In type theory, one cannot quantify over all types (although one can fake it with universes), whereas in set theory one (potentially) can.
• The previous point is perhaps a consequence of a “level” distinction. Constructions on sets are either specified by operations or by axioms which are part of the theory. But I think type constructors are usually viewed as syntactic judgements external to any theory. (Probably I’m not using the buzzwords correctly here, but hopefully you get my meaning.)
• Type theory can be more flexible, e.g. it can be interpreted in fibered preorders rather than in categories. I’m not sure how to do that with set theory.
Admittedly, these are all subtle distinctions.
Posted by: Mike Shulman on September 25, 2009 10:29 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
I wrote:
‘type’ can also be used in a broader context
and Mike wrote:
Type theory can be more flexible
I would say that not every type theory is a set theory, far from it; but every set theory is a type theory. Types can (and usually do, in my experience) have equality predicates but (as you note) need not; in Martin-Löf's original ‘impredicative’ Intuitionistic Type Theory (the one that turned out inconsistent by Burali-Forti), you can quantify over all types, so the term ‘set’ doesn't have a monopoly on that idea. I don't see type constructors as external to type theory; I don't know what you're trying to say there.
And I wouldn't be too averse to somebody's using the term ‘set’ more flexibly either. It's not that different from our use of ‘set’ to mean, basically, a structured set with all of the extra structure removed, which AN correctly objects is inconsistent with its use, by Cantor and the material set theorists who followed him, to mean a part of some universe (originally the real line, eventually the von Neumann hierarchy). We can respond to AN that there is now substantial literature that uses the term in this way (and a vast literature in which it is easily interpreted in this way), which this hypothetical more flexible person may not have; but if we discover some group of mathematicians that does use ‘set’ for, say, something without an equality predicate, then I wouldn't have any standing to complain (even though I would rather call that particular sort of thing ‘preset’ myself).
Posted by: Toby Bartels on September 26, 2009 12:53 AM | Permalink | Reply to this
### Re: What is a structured object?
in Martin-Löf’s original ‘impredicative’ Intuitionistic Type Theory (the one that turned out inconsistent by Burali-Forti), you can quantify over all types, so the term ‘set’ doesn’t have a monopoly on that idea.
Is there a consistent type theory in which you can quantify over all types? The way I think of type theory, quantifiers are tied to quantifying over elements of some type.
I don’t see type constructors as external to type theory; I don’t know what you’re trying to say there.
I think what I mean is that where type theory has type constructors, which are operations on types, set theory often has existence axioms about sets. Admittedly the distinction is not always possible to see.
Posted by: Mike Shulman on September 26, 2009 4:57 AM | Permalink | PGP Sig | Reply to this
### Sets vs types
Is there a consistent type theory in which you can quantify over all types?
Sure, $\mathbf{SEAR}$ for example.
I know, you call $\mathbf{SEAR}$ a ‘set theory’ instead of a ‘type theory’, but if that's only because it allows quantification over all types, then the argument is circular. Meanwhile, we've got Arnold Neumaier objecting that $\mathbf{SEAR}$ is not a set theory because it's not material; membership in sets should be a predicate, and making it a typing declaration is a give-away that you've really got a type theory (although AN said ‘copies of cardinal numbers’, and later ‘universes’, instead of ‘types’). There is a historical basis for either distinction.
I don't see type constructors as external to type theory; I don't know what you're trying to say there.
I think what I mean is that where type theory has type constructors, which are operations on types, set theory often has existence axioms about sets. Admittedly the distinction is not always possible to see.
No wonder you didn't want me to introduce Cartesian products as an operation in $\mathbf{SEPS}$; you were trying to build a set theory rather than a type theory! Of course, if one's type theory sticks to propositions-as-types, then it really can't tell the difference between these. On the other hand, even material set theory can be written down using operations; I can't think of a reference now, but 10 years ago I was working out how to eliminate existential quantifiers from the $\mathbf{ZF}$ axioms entirely.
Posted by: Toby Bartels on September 27, 2009 1:37 AM | Permalink | Reply to this
### Re: Sets vs types
You seem to have a more expansive notion of what “type theory” means than I’ve encountered anywhere else. In part D of the Elephant, or in Jacobs’ Categorical Logic and Type Theory, type theory is given a specific meaning: there are types, function symbols, terms, type constructors (such as products and sums, possibly dependent), and so on. If we allow a logic on top of the the type theory (or fake it with propositions-as-types), then there are relation symbols and formula constructors as well, such as $\wedge$, $\vee$, $\Rightarrow$, $\exists$, etc., with inference judgements such as “if $\phi$ is a formula containing a free variable $x$ of type $A$, then $\exists x:A.\phi$ is a formula without such a free variable.” Type theory together with logic might also be called “typed first-order logic.”
By contrast, SEAR is formulated in a typed first-order logic, but the types involved are “set”, “relation”, and “element.” Just like ZF is formulated in a single-sorted first-order logic, where the elements of the single sort are called “sets”. SEAR looks kind of like type theory because when $A$ is has the type “set,” the dependent type “element of $A$” looks a lot like calling $A$ itself a type. But in type theory as I have learned it from the references above, one cannot write something like “for all types $A$”, since every variable must have a type and there is no type of all types (at least, not if you want to avoid paradoxes). But perhaps I have learned too narrow a meaning of “type theory;” can you point me to any references that use it more expansively?
Posted by: Mike Shulman on September 27, 2009 9:22 PM | Permalink | PGP Sig | Reply to this
### Re: Sets vs types
By contrast, SEAR is formulated in a typed first-order logic, but the types involved are “set”, “relation”, and “element.”
Yes, but type theory itself is also formulated in a typed first-order logic, where the types involved are ‘type’, ‘term’, ‘proposition’, and the like. There is, in my opinion, a significant difference between a type theory such as that which underlies $\mathbf{SEAR}$, in which all of the types are listed up front once and for all, and a type theory such as Martin-Löf's, in which enough generic type constructors are given that one can formalise all of ordinary mathematics. In fact, I would say this difference is greater than that between the second kind of type theory and structural set theory, and the difference between material and structural set theory is not really smaller.
But in type theory as I have learned it from the references above, one cannot write something like “for all types $A$”, since every variable must have a type and there is no type of all types (at least, not if you want to avoid paradoxes). But perhaps I have learned too narrow a meaning of “type theory;” can you point me to any references that use it more expansively?
I think that the problem is that type theorists never invented a word analogous to ‘class’ in set theory; if they had, then nobody would say that every variable must have a ‘type’, since they would use this new word instead. But suppose that material set theory had developed differently, never inventing the word ‘class’, but instead always using ‘set’ for the general notion and ‘small set’ for the more restrictive case. Then the axioms of separation and collection (to keep their meaning the same as they have now in $\mathbf{ZFC}$) would only apply to formulas whose variables are all bounded by some set, and while one can write down other formulas, one cannot actually do anything with them; all that we have done is to develop $\mathbf{NBG}$ in a different language.
As I said, Martin-Löf wrote down a theory in which one can say ‘for all types $A$’, but it was inconsistent. One can make a consistent version as follows: replace the word ‘type’ everywhere by ‘small type’, except in the phrase ‘type of all types’, where only the second ‘type’ is replaced; this would be perfectly analogous to the use of ‘small set’ above. I would now like to cite that Martin-Löf did just this, but he did not; instead, he developed a stronger theory with a hierarchy of universes, in each of which all type constructors may be used. But it seems to me that if type theory without universes is ‘type theory’ and type theory with a hierarchy of universes is ‘type theory’, then type theory with a single fixed universe of small types, in between these two, is also ‘type theory’.
Some people (I think Beeson, and since I'm already going to look up something else in that for you, I'll try to check this too) distinguish ‘set’ and ‘type’ by whether the theory is material or structural; to them, $\mathbf{SEAR}$ is, like $\mathbf{ETCS}$, already a ‘type’ theory. (For what it's worth, that's how I used the words before you convinced me that there was no reason to do this.) You seem to distinguish them by whether one can quantify over all of them when defining one of them, which is also reasonable but not the only way to do things (and then $\mathbf{ETCS}$ is still a ‘type’ theory). Another way to distinguish them is to say that a ‘set’ has an arbitrary equality relation, while a ‘type’ has none (or has only syntactic identity); that is done here for example (although using ‘preset’ is probably a more precise way to do this). There are many distinctions that can be made in one's style of foundations, but I don't see any of them as an essential or universal distinction between these two words, nor do I see the need for such a distinction.
Posted by: Toby Bartels on September 28, 2009 1:57 AM | Permalink | Reply to this
### Re: Sets vs types
I fully agree that there is a continuum of theories, and it is by no means a priori clear where to draw the line between “type theory” and “set theory.” But we have to have words that mean something, or we’ll never know what we’re talking about!
I had a lengthy email exchange with Thomas Streicher several months ago about more or less this question. We did a lot of not understanding what each other was saying, and we got especially confused because we were also talking about interpretability of theories internal to a non-well-pointed topos. The metric of quantifiers over all sets/types to distinguish “set theory” from “type theory,” which I’ve been adhering to here, is what came out of that discussion as a convention we could both agree on. (BTW, I don’t agree that ETCS is a type theory by that metric—the question is not whether quantifiers over sets are allowed in the separation axiom, but whether they exist at all in the language.)
It is certainly true that for many people, “set theory” means “material set theory,” so perhaps we structural-set-theorists should have just stuck with “type” instead of “set.” (Thomas also mentioned that when he was first learning topos theory, the use of “set theory” for the internal logic of a topos confused him because it was clear that set theory was stronger than type theory—another possible axis along which one could distinguish.) I do of course feel that there is something important to be gained by calling structural set theory “set theory” rather than “type theory”; in particular, it points out that this (and not material set theory) is really how sets are used by mathematicians (although apparently this can be harder to convince people of than I realized, pace AN!).
And I still think there is a difference between structural set theory and type theory.
By contrast, SEAR is formulated in a typed first-order logic, but the types involved are “set”, “relation”, and “element.”
Yes, but type theory itself is also formulated in a typed first-order logic, where the types involved are “type”, “term”, “proposition”, and the like.
I agree that type theory can be formulated in such a way, but it can also stand alone as such a theory itself. To borrow the metaphor of programming languages, type theory is a part of logic, which is the machine language of mathematics. You can write an interpreter for machine language in machine language (and you might want to, in order to run it on some other architecture), but you can also run it directly on the machine it was written for. But SEAR must be compiled/interpreted into type theory/logic; it is not the machine language of any machine.
Posted by: Mike Shulman on September 28, 2009 4:20 AM | Permalink | PGP Sig | Reply to this
### Re: Sets vs types
Here is a contentful and important mathematical consequence of that difference. Type theories (in the sense that I am using the word) have a term model. That is, you can construct a topos (or a category with less structure, if your theory doesn’t require as much) which is the free topos containing an internal model of that theory. In particular, applying this to “IHOL” (the type theory corresponding to an ordinary topos) there is a free topos.
This is not true (at least, not as far as I can tell) for SEAR and other “structural set theories” which allow quantifiers over sets in their axioms. (You might have seen a draft of my UQ&SA paper in which I claimed that it was, but now I believe that is incorrect.)
In both cases you can also interpret the logic as happening “one level up,” as you suggested, and now in both cases there is a free model. But this sort of free model looks very different: now instead of a category whose individual objects represent the individual types/sets, we have a category containing a single “object of types” and a single “object of elements.”
What we get in this latter case can be thought of as a “free category of classes.” The category of small objects in a category of classes is a topos—but even if the category of classes satisfies its version of the stronger axioms like unbounded separation and collection, it does not in general follow that its category of small objects satisfies its version of them. All we can say is that the internal category of small objects satisfies these axioms in the internal logic of the category of classes.
Posted by: Mike Shulman on September 29, 2009 3:26 PM | Permalink | PGP Sig | Reply to this
### Re: Sets vs types
I wrote:
Some people (I think Beeson, and since I’m already going to look up something else in that for you, I'll try to check this too) distinguish ‘set’ and ‘type’ by whether the theory is material or structural
Nothing so clear cut as that. Actually, Beeson seems to be confused; in Chapter II (Informal Foundations of Constructive Mathematics), he claims (Section II.3) to use Bishop's concept of set (which is definitely structural) and even notes that $x = y$ is not globally meaningful. But then (Section II.9) he defines $x \in Y$ whenever $x \in X$ and $X \subseteq Y$, calling this a ‘difference in use of language’ from Bishop. And so it is, but it's not clearly explained.
All of the formal ‘set theories’ in Beeson are both material and based on first-order logic, while the only ‘type theories’ are those of Martin-Löf, so that doesn't help. The same is true in other references that I've just checked.
Posted by: Toby Bartels on October 3, 2009 12:58 AM | Permalink | Reply to this
### Re: What is a structured object?
This has nothing at all to do with large categories. Consider the category of finite groups.
Ah, okay, I misunderstood your complaint.
The way to deal with this is the same as the way to deal with any sort of family of objects in structural set theory. A small category in structural set theory consists of a set $C_0$ of objects, a set $C_1$ of morphisms, functions $s,t:C_1\to C_0$, $i:C_0\to C_1$, and $c:C_1\times_{C_0}C_1\to C_1$ with axioms as defined for instance here. If you want to consider the objects of such a category as “being” sets with structure, then you simply consider a $C_0$-indexed family of sets with structure and a $C_1$-indexed family of morphisms between them.
(A small equivalent of) the category of finite groups, for instance, would be a category as above equipped with a $C_0$-indexed family of finite groups $G$ and a $C_1$-indexed family of morphisms $H$ between them, such that any morphism between groups in $G$ occurs exactly once in $H$, and such that any finite group is isomorphic to one in $G$.
Unfortunately I don’t have time to explain in more detail right now exactly what is meant by “family” in all these cases, but it is not hard.
This is not a problem for formalization at a low level, but it may be undesirable when trying to formalize at a higher level, for all the reasons that you’ve given. But it doesn’t prevent SEAR from reflecting on itself formally.
It does. Reflection means being able to define a copy of SEAR inside SEAR, including all the language used to define this copy.
That is in fact what reflection means, but you haven’t explained why not having “a group” as a single object in the domain of discourse prevents it.
Then, and only then, you can speak of having SEAR as a foundation.
I don’t understand why reflection should be the defining test of a foundation. To me, saying that something is a foundation for mathematics means that it can be used to formalize all (or a substantial part) of mathematics. Logic is, indeed, an important part of mathematics, but only a part. Being able to compile its own compiler is an important test of a (compiled) programming language, but it is not the defining feature that enables us to call something a “programming language.” My impression is that generally by the time that a language is able to compile its own compiler, it is fairly well-accepted that it is, in fact, a programming language.
Regardless, if formalizing logic is what you want, I claim that logic, just like most of the rest of mathematics, is already written in an essentially structural way. For example, suppose one chooses to code logical sentences as natural numbers. This never depends on the specific definition of natural numbers as finite von Neumann ordinals or what-have-you; it only depends on the fact that they satisfy the induction property. Well, so do the natural numbers in SEAR or ETCS. Consider for simplicity a one-sorted theory with $n$ binary function symbols, which we code by the natural numbers $0,1,\dots,(n-1)$, and $m$ binary relation symbols, coded similarly. We can then use the separation property to define a subset $F$ of $\mathbb{N}$ consisting of those natural numbers that code well-formed formulas in this language. A logical theory then consists of a subset of $F$, the axioms. A structure for this language is a set $M$, together with a function $\{0,1,\dots,(n-1)\}\times M\times M\to M$ coding the function operations and a subset of $\{0,1,\dots,(m-1)\}\times M\times M$ coding the relation symbols. (Here, of course, $\{0,1,\dots, (n-1)\}$ denotes an $n$-element set equipped with a specified injection into $\mathbb{N}$ that gives its elements meaning as natural numbers.) The inductive property of $\mathbb{N}$ enables us to define the truth value of any formula on such a structure, so we can define a model of a theory to be a structure in which all the axioms are true.
In other words, all the work of reflection is already done. All that remains for structural set theory to do is point out that existing mathematics is already structural.
Posted by: Mike Shulman on September 24, 2009 5:58 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
This has been a very interesting discussion, and I hope Mike won’t mind (since he says he’s busy) if I touch upon some of what he was saying above, and outline a construction of an internal category of finite groups within a structural set theory.
As a warmup, let’s construct an internal category $Fin$ equivalent to the category of finite sets. We take the set of objects $Fin_0$ to be $\mathbb{N}$, the set of natural numbers, with one element $n \geq 0$ for each finite cardinality.
As Mike was saying, in order to construe objects $n \in \mathbb{N}$ as giving actual finite sets, we construct a “family” $\phi: F \to \mathbb{N}$ where each fiber $F_n$ is a set of cardinality $n$. For example, consider the function
$\phi: \mathbb{N} \times \mathbb{N}\to \mathbb{N}: (m, n) \mapsto m + n + 1$
Then, for each $n \geq 0$, the fiber $\phi^{-1}(n)$ is a set of cardinality $n$. This fiber will also be denoted $[n]$.
Next, using the existence of dependent products in a structural set theory like ETCS, one may construct the family of morphisms between finite sets,
$\psi: Fin_1 \to \mathbb{N} \times \mathbb{N},$
where the fiber over $(m, n) \in \mathbb{N} \times \mathbb{N}$ is $[n]^{[m]}$, the set of functions from $[m]$ to $[n]$. In other words, an element $f$ of $Fin_1$ “is” a function between finite sets. Let us write $dom(f)$ for the first component of $\psi(f)$ and $cod(f)$ for the second component, so that $\psi(f) = \langle dom(f), cod(f) \rangle$. This gives us functions
$dom, cod: Fin_1 \overset{\to}{\to} Fin_0$
which are part of the structure of an internal category $Fin$; the rest of the structure consists of identity and composition functions
$id: Fin_0 \to Fin_1 \qquad c: Fin_1 \times_{Fin_0} Fin_1 \to Fin_1,$
which are not hard to construct. In the end, the internal category constructed is equivalent to the category of finite sets.
Now let us continue by sketching the internal category of finite groups. To construct a set $G_0$ whose elements represent all isomorphism classes of finite groups, we construct a family
$card: G_0 \to \mathbb{N}$
where each fiber $card^{-1}(n)$ is the set of all group structures on the set $[n]$: the subset of
$[n]^{[n] \times [n]} \times [n] \times [n]^{[n]}$
whose members $(m, e, i)$ are those triples which obey the equational axioms (appropriate to the theory of groups) for multiplication $m$, identity $e$, and inversion $i$. We may construe elements $g$ of $G_0$ as “finite groups”. In particular, the “underlying set” of a finite group $g \in G_0$ is
$U(g) = \phi^{-1}(card(g))$
Finally, we construct the set $G_1$ of finite group homomorphisms. This is the set of those triples
$(g, f, h) \in G_0 \times Fin_1 \times G_0$
such that $dom(f) = card(g)$, $cod(f) = card(h)$, and the function $f$ satisfies the equations necessary to make it a homomorphism from the group structure $g$ to the group structure $h$.
This completes the sketch of an internal category equivalent to the category of finite groups. While it’s just a sketch, all the formal details can be filled in within the framework of a structural set theory such as ETCS or SEAR.
Which brings me to a question. Sometime earlier Arnold wrote:
At present, every formalization of a piece of mathematics is a mess; this was not the point.
What I was referring to was the overhead in the length of the formalization. With ZF, you can formalize a concept once as a tuple, and then always use the concept on a formal level.
and then
This is what I was aiming at. For reflection purposes, one cannot work in pure SEAR, while one can do that in pure ZF.
As a matter of fact there are bi-interpretability theorems which show that any construction in Zermelo set theory (Bounded Zermelo theory with Choice to be more precise) can be expressed in the structural theory ETCS, and vice-versa, and certainly one can augment ETCS with additional axioms to recover the full power of ZF. Similarly, if I recall correctly, Mike has basically said in his article that SEAR is bi-interpretable with (has the same expressive power as) ZF. So it is not clear to me why Arnold believes that for reflection purposes, one can work with ZF but not with SEAR. For example, what was sketched above indicates that one can reflect finite groups within (say) ETCS at a formal level. Mike said a little more about reflection in his later comment here.
Posted by: Todd Trimble on September 25, 2009 6:59 AM | Permalink | Reply to this
### Re: What is a structured object?
I hope Mike won’t mind (since he says he’s busy)
I should hope I wouldn’t mind either, no matter how busy I am! (-: I hope I haven’t given the impression that I own structural set theory or something. As many people have been saying, all of this stuff (except perhaps some details of SEAR) is decades old.
Posted by: Mike Shulman on September 25, 2009 8:00 AM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
MS: As many people have been saying, all of this stuff (except perhaps some details of SEAR) is decades old.
If this is true, it should be easy to point to a paper or book that contains in terms of ETCS the definition of the basic concepts of category theory, including the examples of a few concrete categories (comparable in richness of structure to the category of finite groups).
Thus I’d appreciate getting such a decades old reference that backs up your claim.
Posted by: Arnold Neumaier on September 25, 2009 10:09 AM | Permalink | Reply to this
### Re: What is a structured object?
it should be easy to point to a paper or book that contains in terms of ETCS the definition of the basic concepts of category theory, including the examples of a few concrete categories (comparable in richness of structure to the category of finite groups).
As I’ve been saying over and over again, I don’t think anyone has felt the need to do this sort of thing, because once the basic structure of ETCS (say) is developed sufficiently it becomes “obvious” to people who think like we do that the rest of mathematics can follow, and everyone would rather spend their time pushing the boundaries. Rewriting Bourbaki by changing a word here and there isn’t a really fun way to spend one’s time, nor likely to be counted as a significant contribution to mathematics when one is applying for jobs. That isn’t to say that I don’t wish that someone had, so that I could point you to it! Mathematics is full of things that are “understood” by people who work in a given field for a long time before being carefully written down with enough details to make sense to others.
You will find this perspective running implicitly through many books on topos theory, and they are actually doing something more general: considering how mathematics can be developed on the basis of any elementary topos. But again, they probably don’t supply enough details about how to do this to satisfy you.
Posted by: Mike Shulman on September 25, 2009 3:49 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
I’ll second what Mike said: for those people who have absorbed the methods that are explained in a book like Moerdijk and Mac Lane’s text, the sort of explicit detail of the sort I laid out is more along the lines of an exercise whose solution would be well-understood by many. It’s probable that it would be carried out in more explicit detail only when an outsider comes along and begins asking a different set of questions like you are doing here, so what you are looking for exactly might be hard to track down in the literature.
Posted by: Todd Trimble on September 25, 2009 4:20 PM | Permalink | Reply to this
### Re: What is a structured object?
If this is true, it should be easy to point to a paper or book that contains in terms of ETCS the definition of the basic concepts of category theory, including the examples of a few concrete categories (comparable in richness of structure to the category of finite groups).
This may not exist, because any basic textbook on category theory has to mention foundations to deal with size issues, and this discussion is unlikely to be independent of material vs structural foundations.
However, any modern algebra book, if it doesn't talk about either set theory or category too much, will do this. For example, take Lang, remove (or rewrite) only the two pages on Logical Prerequisites, and the rest (including the Appendix on more advanced set theory!) is fine as it is. (I haven't checked every page, but I did skim through Chapter I and Appendix 2.)
There is a constant abuse of language (which should probably be remarked upon if one rewrites the Logical Prerequisites) where a subset $S$ of a set $X$ is conflated with the underlying set of $S$ (and also an element $a$ of $X$ that belongs to $S$ is conflated with the unique corresponding element of the underlying set of $S$), but this is no worse than the abuse (not remarked upon!) that begins Section V.1 in my (1993) edition:
Let $F$ be a field. If $F$ is a subfield of a field $E$, […]
Literally, a subfield of $E$ is (as Lang defined it) a subset of $E$, not a field in its own right. (In $\mathbf{ZFC}$, a subset of $E$ might happen to equal the ordered triple that is a field, but if so then that is not what Lang wants here!) Structural set theory uses the same abuse of language, although now also for unstructured sets just as much as for structured sets such as fields.
Lang also discusses category theory, but he doesn't indicate how to formalise it, so that text doesn't need any changing either. (What is a ‘collection’? Lang doesn't say. The unwary reader may assume that it's the same as a ‘set’ and be led to a paradox on the next page!)
Posted by: Toby Bartels on September 25, 2009 9:38 PM | Permalink | Reply to this
### Re: What is a structured object?
Mike: you didn’t give me that impression (or even that you were pretending to such ownership (-: ). In fact, I salute both you and Toby for all your hard work in providing all those many thoughtful responses. I think all of us have been learning a lot from the exchange.
Posted by: Todd Trimble on September 25, 2009 12:30 PM | Permalink | Reply to this
### Re: What is a structured object?
TT: This completes the sketch of an internal category equivalent to the category of finite groups.
OK, I get the idea of how to reflect things. Once one has the group as a single object (and in contrast to Mike Shulman, you modelled it that way), the basic obstacle to full reflection is gone.
One builds some machinery that mimicks the material structure of ZF, for example by providing triples that encode the group. Then one uses this structure to do what one is used to do in the standard reduction of mathematics to ZF.
I agree that one can probably fill in all details, and that this gives a way to define formally what the category FG of finite groups is, and hence what a finite group is, namely an element of Ob(FG).
Thus I now grant that (and understand how) ETCS - and maybe SEAR in a similar way - may be viewed as being a possible foundation of all of mathematics (when enhanced with enough large cardinals to handle large categories).
What I no longer understand now, however, is the claim that this way of organizing mathematics is superior to that of basing it on ZF since it is structural rather than material.
For I find the meaning of a finite group implied by the construction you gave not any more natural than the meaning of a natural number implied by its ZF construction by von Neumann.
It is ugly, and no mathematician thinks of this as being the essence of finite groups.
Moreover, for a (general) group, one has a similar messy construction, and a finite group is no longer a group but only ”becomes” a group under the application of a suitably define functor.
This flies in the face of the ordinary understanding of every algebraist of the notions of group and finite group.
In the attempts (in this discussion) to capture the essence of mathematics the proponents introduce so much artificial stuff in the form of trivial but needed functors that the result no longer resembles the essence to be captured.
Thus the structural, ETCS-based approach is no better in capturing the essence of mathematics as the material, ZF-based approach.
Both create lots of structure accidental to the construction, structure that is not in the nature of the mathematics described but in the nature of forcing mathematics into a ETCS-theoretic or ZF-theoretic straitjacket.
Posted by: Arnold Neumaier on September 25, 2009 10:36 AM | Permalink | Reply to this
### Re: What is a structured object?
For I find the meaning of a finite group implied by the construction you gave not any more natural than the meaning of a natural number implied by its ZF construction by von Neumann.
I think you are misunderstanding the point of the construction. The meaning of a finite group is still “a finite set $G$ equipped with a multiplication $m:G\times G\to G$ and a unit $e\in G$ such that …”. Just like the meaning of a Cauchy sequence of rationals is “a function $\mathbb{N}\to \mathbb{Q}$ such that …”. It’s only when you want to consider “the category of finite groups” or “the set of Cauchy sequences” as an abstract object that you need to construct a set whose elements code for finite groups or Cauchy sequences.
Posted by: Mike Shulman on September 25, 2009 3:31 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
Arnold wrote:
What I now longer understand now, however, is the claim that this way of organizing mathematics is superior to that of basing it on ZF since it is structural rather than material.
For I find the meaning of a finite group implied by the construction you gave not any more natural than the meaning of a natural number implied by its ZF construction by von Neumann.
It is ugly, and no mathematician thinks of this as being the essence of finite groups.
Moreover, for a (general) group, one has a similar messy construction, and a finite group is no longer a group but only ”becomes” a group under the application of a suitably define functor.
This flies in the face of the ordinary understanding of every algebraist of the notions of group and finite group.
In the attempts (in this discussion) to capture the essence of mathematics the proponents introduce so much artificial stuff in the form of trivial but needed functors that the result no longer resembles the essence to be captured.
Thus the structural, ETCS-based approach is no better in capturing the essence of mathematics as the material, ZF-based approach.
Okay, a lot of opinions are being expressed here. Let me first say that the charge of “ugliness” is an aesthetic judgment, not part of formalized mathematics. Given the strictures I placed myself under (showing that a group could be expressed as a single element), to satisfy your demands, the notion was bound to look harder than the ordinary understanding of the algebraist, whose “essence” [as you like to say] is simply, as we have been saying over and over,
• A group is a set equipped with a group structure
which I maintain is structural in essence: there is no reference in that definition to the fact that elements may themselves have elements. The word “structural” means that it is abstract structure that is paramount, not the internal ontology of elements which is necessarily uninvariant under isomorphism – internal ontology of elements is a consideration which is alien to the practice of working mathematicians (unless they are investigating ZF perhaps, from a platonist point of view).
Presumably, if FMathL is well-developed, the human user can work in the customary style of sets+structure, and it is the job of the computer to then translate (or shoehorn) that into a single object or element. I don’t think the computer would care or have an opinion whether that’s done in ZF or SEAR or whatever, although obviously consideration must be given to what is the most efficient way to do the shoehorning.
Rather than say the structuralist view is “better” (it may certainly be better for certain purposes), and bring in aesthetic disagreements which may well be irreconcilable, I would say that at least in some respects, the structural view is closer to the way mathematics has traditionally been practiced. For example, the idea that a point on the real line may have elements which themselves have elements is, I think you will admit, peculiar to twentieth-century mathematics (and maybe to some extent now), and is an idea that is utterly irrelevant to working practice. And yet this abnormality is an undeniable consequence if one takes ZF and particularly a global membership relation as one’s foundations. I believe there’s some merit in rejecting those consequences as abnormal and irrelevant to mathematics.
On the other hand, a different twentieth-century development which has proven itself extremely relevant to current practice is category theory, which emphasizes universal properties and invariance of structure with respect to isomorphism. A structural development like ETCS takes those precepts very seriously indeed and embeds them as part of the formal development, whereas those precepts for a committed ZF-er would have to remain at the level of “morality” and are not part of the formal set-up.
Don’t get me wrong – as an abstract structure, the cumulative hierarchy is a recursively rich, powerful, and interesting mathematical structure. But as foundations, it’s not particularly pertinent to how mathematicians think about $L^2$ and such things. Those of us committed to category theory have come a bit closer to the essence, I believe, by focusing on things like universal properties as far more relevant to practice.
Posted by: Todd Trimble on September 25, 2009 4:04 PM | Permalink | Reply to this
### Re: What is a structured object?
On the other hand, a different twentieth-century development which has proven itself extremely relevant to current practice is category theory, which emphasizes universal properties and invariance of structure with respect to isomorphism.
I’d like to add my 5 cents worth to this discussion by agreeing with Todd. I am not a category theorist and never will be — category theory hurts my head. On the other hand I find it very useful to try to think like a category theorist. Even (especially!) when I am working on something that appears quite far from category theory, like dynamical systems or symplectic toric geometry.
Posted by: Eugene Lerman on September 25, 2009 9:49 PM | Permalink | Reply to this
### Re: What is a structured object?
Eugene wrote:
I am not a category theorist and never will be — category theory hurts my head.
The only thing stopping you is that you still think it’s bad for your head to feel that way. It’s actually good — it’s the feeling of new neurons growing.
It’s sort of like the aches and pains you get after lifting more weights than you’re used to. Good weightlifters still feel those aches; they just learn to like them.
Posted by: John Baez on September 26, 2009 3:46 AM | Permalink | Reply to this
### its the feeling of new neurons growing; Re: What is a structured object?
No pain, no gain, in the visceral brain, or the complex plane.
Posted by: Jonathan Vos Post on September 26, 2009 4:55 PM | Permalink | Reply to this
### Re: What is a structured object?
The worrying thing about your weight-lifter analogy is that body builders tear their muscles to promote growth.
Posted by: David Corfield on September 26, 2009 6:23 PM | Permalink | Reply to this
### Re: What is a structured object?
David wrote:
The worrying thing about your weight-lifter analogy is that body builders tear their muscles to promote growth.
And what’s worrying about that? I bet the ‘aching head’ feeling I get when struggling to learn new concepts is somehow analogous to the ‘torn muscle’ feeling I get whenever I up the amount of weight I lift at the gym. I bet there’s some real ‘damage’ to ones conceptual/neurological structure whenever one struggles really hard to master difficult new ideas: comfortable old connections are getting torn apart. But then new improved connections grow to take their place!
I think the people who do well at learning new things are the ones who learn to enjoy the ache. In the case of the ‘torn muscle’ feeling, the pleasure comes from 1) knowing that one is getting stronger, 2) the endorphin high, 3) a learned association between the two. Maybe something similar happens in the intellectual realm.
Posted by: John Baez on September 26, 2009 10:13 PM | Permalink | Reply to this
### Re: What is a structured object?
I will say: as someone who has begun a strength-training regime fairly recently, and whose aching arms feel like useless appendages right now, this mini-thread is helping a little bit. Thanks!
Posted by: Todd Trimble on September 27, 2009 4:20 PM | Permalink | Reply to this
### No fiber bundle pain, no gain; Re: What is a structured object?
Ironically, the pain from body building comes from fiber bundles. Or, actually, tearing the membranes surrounding bundles of fibers.
Skeletal muscle is made up of bundles of individual muscle fibers called myocytes. Each myocyte contains many myofibrils, which are strands of proteins (actin and myosin) that can grab on to each other and pull. This shortens the muscle and causes muscle contraction.
It is generally accepted that muscle fiber types can be broken down into two main types: slow twitch (Type I) muscle fibers and fast twitch (Type II) muscle fibers. Fast twitch fibers can be further categorized into Type IIa and Type IIb fibers.
These distinctions seem to influence how muscles respond to training and physical activity, and each fiber type is unique in its ability to contract in a certain way. Human muscles contain a genetically determined mixture of both slow and fast fiber types. On average, we have about 50 percent slow twitch and 50 percent fast twitch fibe
Andersen, J.L.; Schjerling, P; Saltin, B. Scientific American. “Muscle, Genes and Athletic Performance” 9/2000. Page 49
McArdle, W.D., Katch, F.I., and Katch, V.L. (1996). Exercise physiology : Energy, nutrition and human performance
Lieber, R.L. (1992). Skeletal muscle structure and function : Implications for rehabilitation and sports medicine. Baltimore : Williams and Wilkins.
Andersen, J.L.; Schjerling, P; Saltin, B. Muscle, Genes and Athletic Performance. Scientific American. Sep 2000
Thayer R., Collins J., Noble E.G., Taylor A.W. A decade of aerobic endurance training: histological evidence for fibre type transformation. Journal of Sports Medicine and Phys Fitness. 2000 Dec; 40(4).
Posted by: Jonathan Vos Post on September 28, 2009 7:09 AM | Permalink | Reply to this
### Clues To Reversing Aging Of Human Muscle Discovered; Re: No fiber bundle pain, no gain; Re: What is a structured object?
DOING Math (what Erdos called “being alive”) also helps reverse the effects of aging on the Brain. I don’t much like the common analogy: “The brain is a muscle; use it or lose it” because, you know, the brain is NOT a muscle. Yet regular and vigorous use IS beneficial, and to an extent that surprises many people.
Clues To Reversing Aging Of Human Muscle Discovered
… “Our study shows that the ability of old human muscle to be maintained and repaired by muscle stem cells can be restored to youthful vigor given the right mix of biochemical signals,” said Professor Irina Conboy, a faculty member in the graduate bioengineering program that is run jointly by UC Berkeley and UC San Francisco, and head of the research team conducting the study. “This provides promising new targets for forestalling the debilitating muscle atrophy that accompanies aging, and perhaps other tissue degenerative disorders as well.”…
Morgan E. Carlson, Charlotte Suetta, Michael J. Conboy, Per Aagaard, Abigail Mackey, Michael Kjaer, Irina Conboy. Molecular aging and rejuvenation of human muscle stem cells. EMBO Molecular Medicine, 2009; DOI: 10.1002/emmm.200900045
Posted by: Jonathan Vos Post on September 30, 2009 9:01 PM | Permalink | Reply to this
### Re: What is a structured object?
If you say this …
I agree that one can probably fill in all details, and that this gives a way to define formally what the category FG of finite groups is, and hence what a finite group is, namely an element of Ob(FG).
then naturally you will say this …
What I no longer understand now, however, is the claim that this way of organizing mathematics is superior to that of basing it on ZF since it is structural rather than material.
A finite group ‘is’ a set equipped with a group structure. If it vital to encode this formally as a single object, then supplement SEAR or ETCS with a dependent type theory with dependent sums. But it is not essential to mathematical practice to do so.
If you want to have a collection of finite groups (or whatever), then any foundations requires some reasoning to show that your collection is valid. (After all, a collection of literally ‘all’ finite groups is impossible in ZFC, as is a collection of all groups whatsoever in either ZFC or ETCS.) Although other methods may be available in some cases, the uniform way to do this is by using the Axiom of Collection: you find some way to index your objects by a set, and the axiom gives you your collection.
In material set theory, you can set things up so that each object is literally an element of the collection, which is convenient; this wouldn't make sense in structural set theory, so you instead introduce an abuse of language in which the ‘elements’ of the collection are actually the fibres over the elements of the index set (together with the structures defined on those fibres).
I said that material set theory is convenient, but in fact it is not convenient enough! Even in ZFC, there is no small category FG such that a finite group is literally the same as an object of FG. Instead, if you insist on recovering the notion of finite group from the category FG, then you can define a finite group to be a set $U$ together with an object $S$ of FG and a bijection between $U$ and the underlying set of $S$. In ZFC, presumably the ‘underlying set’ of $S$ is the first entry in a tuple $(S,m)$; in ETCS, the ‘underlying set’ of $S$ is as defined in Todd's comment. (In both cases, it takes another step to recover the group in the usual sense, as a set together with a group operation.) Once again, structural set theory prevents a potential mistake (thinking that $G$ is not a finite group because it is not literally an object of FG) by throwing up a typing error.
Moreover, for a (general) group, one has a similar messy construction, and a finite group is no longer a group but only “becomes” a group under the application of a suitably define functor.
Hopefully you see now that this is not true in ETCS, but even so … this is no worse than the fact that a Riemannian manifold only “becomes” a manifold under the application of (in the structured-sets-as-tuples formalisation) projection onto the first entry (or possibly even something a bit more complicated).
Posted by: Toby Bartels on September 25, 2009 9:39 PM | Permalink | Reply to this
### Re: What is a structured object?
AN: Moreover, for a (general) group, one has a similar messy construction, and a finite group is no longer a group but only “becomes” a group under the application of a suitably define functor.
TB: this is no worse than the fact that a Riemannian manifold only “becomes” a manifold under the application of (in the structured-sets-as-tuples formalisation) projection onto the first entry (or possibly even something a bit more complicated).
I think the standard mathematical language teaches something different that gets lost both by encoding it into ZF and by encoding it in ETCS or SEAR, though in different ways.
In mathematical practice, to say that an object is a group or a manifold says that it has certain properties. To say that it is a finite group or a Riemannian manifold adds properties but of course preserves all previous properties.
Similarly, to say that a subset H of a group G is a subgroup if it is closed under products and inversion is not an abuse of notation (as was claimed in the discussion on SEAR), since the subset H is not only a set and a subset of G but inherits from the group a product mapping from H x H to G (and even one from H x G to G, etc), and if the subgroup condition holds, this is a mapping from H x H to H and hence the binary operation alluded to in calling it a subgroup.
Similarly, to say that $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^3)$ are separable Hilbert spaces does not strip them of any distinguishing property these spaces have by construction, although the category of separable Hilbert spaces contains only one object up to isomorphism.
Thus almost all objects mathematicians talk about are almost always equipped with lots of stuff through their context, but neither the formalization in ZF nor that in SEAR or ETCS (or Coq, etc.) takes account of that.
That one doesn’t use all these extra structure all the time is not to be handled by deleting entries from the tuple (in ZF) or by applying a forgetful functor (in the structural approach) but by the same common sense that logicians use when they list in some formal natural deduction only the stuff they actually used.
That the categorial way alone cannot capture this essence of mathematics is quite obvious from simple examples:
If $G\in Ob(Grp)$ any sane mathematician infers that $G$ is equipped with a set structure with which the assumption $x,y,z\in G$ makes sense, and infers that there is a product operation for which $xy\in G$ and $(xy)z=x(yz)$.
But this only holds if $Grp$ is the category materially constructed by the definition of $Grp$, and not (as claimed in this discussion - I don’t remember by whom) if one forgets this construction once the category is formed, and only retains the isomporphism class of the category.
Thus the “structural” point of view actually loses structure!
Sometimes the loss of structure is dramatic: The category $CLOF$ of closed linearly ordered fields and the category $E7$ of undirected graphs isomorphic to the $E_7$ Dynkin diagram are isomorphic, but objects from these two categories have very different properties. There is not even a canonical isomorphism betwee the two categories. Here the essence is completely lost.
Posted by: Arnold Neumaier on September 30, 2009 12:27 PM | Permalink | Reply to this
### Re: What is a structured object?
I assume that by “properties” you mean “properties or structure or stuff” (around here we use a precise meaning of property according to which a finite group is a group with extra properties, but a Riemannian manifold is not a manifold with extra properties (but rather extra structure)).
I agree that both ZF and ETCS/SEAR handle this issue clumsily, albeit clumsily in different ways. However, I think this argument:
If G∈Ob(Grp) any sane mathematician infers that G is equipped with a set structure with which the assumption x,y,z∈G makes sense, and infers that there is a product operation for which xy∈G and (xy)z=x(yz).
But this only holds if Grp is the category materially constructed by the definition of Grp, and not… if one forgets this construction once the category is formed, and only retains the isomporphism class of the category.
misses the point. If one wants to treat Grp as an abstract category, then one forgets how its objects were constructed (which has nothing to do with materiality), just as if one wants to treat $A_5$ as an abstract group, one forgets that its elements have a natural action on some 5-element set. However, nothing forces us to do that forgetting as soon as the object is formed, and quite often we don’t.
But, as I said, I agree that both ZF and ETCS/SEAR are clumsy about moving between different levels of properties or structure. This would be something that would be great for a higher-level formalization to improve on.
Actually, it strikes me right now that this issue is very similar to class inheritance in object-oriented programming. When we say that a Riemannian manifold is a manifold, the “is a” really has the same meaning as in OOP: a Riemannian-manifold object can be used anywhere that a manifold is expected, but it doesn’t thereby lose its Riemannianness (although if we access it only through a manifold ptr then we can’t use any of its Riemannianness). From this point of view, the clumsiness of existing foundations amounts to requiring all upcasts to be explicit.
Posted by: Mike Shulman on September 30, 2009 5:53 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
MS: I assume that by “properties” you mean “properties or structure or stuff”
Yes. For me extra properties and extra structure are synonymous. It is just something more that can be profitably exploited for reasoning.
MS: However, nothing forces us to do that forgetting as soon as the object is formed, and quite often we don’t.
This piece of moral sounds quite different and much more agreeable than the many times repeated one I had to put up with before:
“However, once the construction is finished, we generally forget about it” “but in each case once the construction is performed, its details are forgotten. I always assumed, without really thinking much about it, that all modern mathematicians thought in this way” “once the construction is performed, the fact that you used “the same” objects is discarded.” “When you construct one category from another, you might use the “same” set of objects, but once you’ve constructed it, there is no relationship between the objects, because after all any category is only defined up to equivalence.” (quotes from your earlier mails)
“ two categories may have an object in common, but you should never use that fact.” “You’re completely (intentionally?) missing the distinction I drew between a construction demonstrating the existence of a model of a structure and the subsequent use of the properties of a structure. As I said before, “moral” (which was someone else’s term) refers to the latter segment, not the former. […] (John Armstrong)
You now seem to say that all the categories can be considered as concrete categories or as abstract categories depending on the purpose the mathematician wants to achieve. This is fine with me. Indeed, the standard (material) definition of a category is precisely that of a concrete category. And in concrete categories I am allowed to do all the stuff you wanted to forbid: compare objects of different categories, check whether the objects of one form a subclass of those of the others, create intersections of the class of objects of two different categories, etc.. once this is allowed, I have no problems at all with the categorial language (except for lack of fluency in expressing myself in it). One has all this structure around unless one deliberately forgets it. There is no moral that tells one that one should forget it, except if one wants to forget it.
It was only the strange moral that was imposed on it without having any formal justification that bothered me.
MS: it strikes me right now that this issue is very similar to class inheritance in object-oriented programming. […] From this point of view, the clumsiness of existing foundations amounts to requiring all upcasts to be explicit.
Yes. This is why FMathL will have on the specification level a much more flexible type-like system that borrows much more from the theory of formal languages than from the theory of types.
Posted by: Arnold Neumaier on September 30, 2009 6:49 PM | Permalink | Reply to this
### Re: What is a structured object?
You now seem to say that all the categories can be considered as concrete categories or as abstract categories depending on the purpose the mathematician wants to achieve.
Yes, of course. The comments you quoted were in a different context, explaining that (for example) the particular construction of the real numbers as Dedekind cuts is usually forgotten once we have the real numbers, so that it is better if you can forget it rather than actually have the real numbers be Dedekind cuts as in material set theory.
I made this same point here.
Indeed, the standard (material) definition of a category is precisely that of a concrete category.
No, I don’t think so. Some people have a precise definition of a “concrete category,” but here I’m thinking of it in a more vague way like “a category together with some information preserved from its construction.” I don’t see what this has to do with materiality.
And in concrete categories I am allowed to do all the stuff you wanted to forbid: compare objects of different categories, check whether the objects of one form a subclass of those of the others, create intersections of the class of objects of two different categories, etc.
No. If two concrete categories $C$ and $D$ both have a forgetful functor to $Set$ (being part of the information you remembered from their constructions), then you can ask whether the underlying sets of an object $x\in C$ and $y\in D$ are isomorphic, or whether every set that underlies an object of $C$ also underlies an object of $D$, or consider the collection of all sets that underlie both an object of $C$ and an object of $D$, but these are quite different things from the forbidden ones.
Posted by: Mike Shulman on September 30, 2009 8:25 PM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
AN: Indeed, the standard (material) definition of a category is precisely that of a concrete category.
MS: No, I don’t think so. Some people have a precise definition of a “concrete category,”
You had at least the additional qualifier of an abstract category, which seems to be something different from the category as defined in the textbooks.
MS: but here I’m thinking of it in a more vague way like “a category together with some information preserved from its construction.”
I am referring to the standard definition of a (small) category C found everywhere, with the standard interpretation of Ob(C) as class (or set) in the traditional sense (not SEAR, not ETCS, which,in most textbooks, do not figure early if at all).
There is nothing vague in this definition beyond what is vague in any mathematical discourse.
This definition does not ask you to forget anything about the category you constructed.
Indeed, the definition does not even provide a formal mechanism for forgetting. The reason is presumably either that such an automatic mechanism was never intended by those who invented and traded the definition, or that it is difficult to formalize rigorously at this stage.
On the contrary, to forget something you need to do something to the category, and no such doing is formally sepcified in any introduction to category theory.
It is an additional moral that you want to impose without specifying it axiomatically.
But what is not in the axioms can be ignored by anyone working with them, without harming in the least the correctness of what is done, and without affecting any consistent interpretation of the axioms.
AN: And in concrete categories I am allowed to do all the stuff you wanted to forbid: compare objects of different categories, check whether the objects of one form a subclass of those of the others, create intersections of the class of objects of two different categories, etc.
MS: No.
My example of the categories $C_{abcd}$ etc. is still there; nobody has shown me any conflict with the standard definitions of a category and a subcategory (interpreted with Ob(C) as class in the traditional sense).
If you want to consistently uphold your No, you’d prove my assertions there wrong!
Posted by: Arnold Neumaier on September 30, 2009 9:08 PM | Permalink | Reply to this
### Re: What is a structured object?
We have already been over this same territory several times. In my view we have given adequate responses to all of these issues, including your category $C_{a b c d}$. I don’t have time to repeat the same arguments again, especially since I have no reason to believe the communication would be any more successful the second or third time around. So I guess we’re at an impasse here.
Posted by: Mike Shulman on October 1, 2009 5:00 AM | Permalink | PGP Sig | Reply to this
### Re: What is a structured object?
I am referring to the standard definition of a (small) category C found everywhere
So are we. You're focussing on the question of whether you can take two arbitrary categories C and D and ask whether C is a subcategory of D, but really you should (to avoid confusion with other issues around categories) start with the question of whether you can take two arbitrary sets C and D and ask whether C is a subset of D. (Or perhaps use groups instead of sets.) Certainly you can do the former if you can do the latter, which is obvious enough looking at the standard definition. But doing the latter is already objectionable (at the very least, an abuse of language) from a structuralist perspective.
with the standard interpretation of Ob(C) as class (or set) in the traditional sense (not SEAR, not ETCS, […]).
How can you tell?
Saunders Mac Lane, one of the two people who first defined categories, is on record as preferring ETCS as a foundation of mathematics. (This is quoted in that McLarty paper that's been linked here.) He considers his concept of category perfectly well formalised by structural set theory.
There is an additional complication, which goes beyond merely having structural foundations, that even within a single arbitrary small category, one should not be able to compare objects for equality (only for isomorphism); this is the problem of evil. ETCS and SEAR do allow this, while I would prefer a foundation of category theory that does not. I know some ways to approach this, but I don't think that it's a solved problem yet.
Posted by: Toby Bartels on October 1, 2009 9:07 AM | Permalink | Reply to this
### Re: What is a structured object?
TB: really you should (to avoid confusion with other issues around categories) start with the question of whether you can take two arbitrary sets C and D and ask whether C is a subset of D.
According to the first paragraph of the Prerequisites in Serge Lang, Algebra, second printing 1970 (who treats categories in Chapter I.7), I am allowed to do this. I take this to be the standard point of view.
Lang’s context allows me to do everything I did with $C_{abcd}$ etc., although your moral forbids it.
TB: Saunders Mac Lane, one of the two people who first defined categories, is on record as preferring ETCS as a foundation of mathematics.
So he allows only bounded comprehension in mathematics?
If your view is right, it depends on the foundations of mathematics whether one is allowed to do the things I do. This would mean that the foundations are not equivalent.
But this would conflict with the result by Osius (which I still need to check) that ETCS+R is equivalent to ZFC.
I think you cannot consistently claim both.
Posted by: Arnold Neumaier on October 1, 2009 10:47 AM | Permalink | Reply to this
### Re: What is a structured object?
TB: Saunders Mac Lane, one of the two people who first defined categories, is on record as preferring ETCS as a foundation of mathematics.
So he allows only bounded comprehension in mathematics?
Toby didn’t say that; he said “preferred foundations”. Saunders would have been very happy to allow you to speak if you were giving him an instance of unbounded separation, and was well familiar with ZFC and its cousins.
Saunders’ position was that just about all core mathematics (what goes on in basic courses on functional analysis, algebraic topology, and so on) can be developed on the basis of ETCS. Not all developments – he was well aware that some set-theoretic constructions required going beyond ETCS. I think he chose not to be too exercised by that, but he may have had some occasional doubts. (I got to know Saunders rather well during my Chicago years, so I think I can say this.) He was also much concerned with making ETCS more accessible to people; I think this worried him more than any limitations of ETCS.
Posted by: Todd Trimble on October 1, 2009 2:40 PM | Permalink | Reply to this
### Re: What is a structured object?
According to the first paragraph of the Prerequisites in Serge Lang, Algebra, second printing 1970 (who treats categories in Chapter I.7), I am allowed to do this. I take this to be the standard point of view.
And so it is. And yet, nowhere does Lang actually use the idea that one can take two arbitrary sets and ask whether one is contained in the other; he never needs to. He may take a set $U$ and then consider an arbitrary subset of $U$; what this means can be defined (or even taken as axiomatic) in structural foundations. And he may take two arbitrary subsets of some set $U$ and ask whether one is contained in the other, which can also be defined structurally. But there is no need in ordinary mathematics to take two arbitrary sets and ask whether one is contained in the other; even if one thinks it meaningful, it never matters.
Lang’s context allows me to do everything I did with $C_{abcd}$ etc., although your moral forbids it.
I'm not sure why you keep saying this. Is there anything that you did with $C_{abcd}$ etc that we have not yet formalised structurally?
If your view is right, it depends on the foundations of mathematics whether one is allowed to do the things I do. This would mean that the foundations are not equivalent.
ETCS is equivalent to BZC (which is ZFC without replacement and with only bounded separation). ETCS+R is equivalent to ZFC (since replacement and bounded separation together imply full separation).
Posted by: Toby Bartels on October 1, 2009 7:58 PM | Permalink | Reply to this
### Re: What is a structured object?
In mathematical practice, to say that an object is a group or a manifold says that it has certain properties.
This connects with the idea earlier that ‘ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation’. As I said then, I would be very interested to see a formalism in which this can be taken literally!
But it would be tricky. We should be able to say, for example, that a ring (that is, an associative unital ring) is an object that is both an abelian group and a monoid, satsfying a compatibility relation. But since every abelian group is already a monoid, surely $AbGrp \cap Mon = AbGrp$, so now it has only one structure, which the compatibility condition forces to be trivial! (For an even worse example, try a commutative rig, where now both structures are commutative monoid structures.)
One thing that you could do is to say that a ring is an object that is both an additive abelian group and a multiplicative monoid, satisfying a compatibility condition. Then you seem to have to define monoids twice, and you're forbidden to say that $(]0,\infty[, \cdot, (a,b \mapsto a^{\log b})$ is a ring, even though we have found it useful to say so. Of course, there may be ways around that, but I don't know them.
The way that I do know to formalise the idea that a ring may be defined as somehow both an abelian group and a monoid is to start with $AbGrp \times Mon$ and then carve out $Ring$ with a compatibility condition which includes having the same underlying set. (On the face of it, this is evil, but I know ways around that. And in any case, there's no point worrying about evil if one doesn't even have structuralism.) This only makes sense, as far as I can tell, if a group is a set equipped with some structure rather than simply a set satisfying some property.
Once one has grown out of the idea that a group is literally simply a certain kind of set, then it's not so hard that an abelian group might not be literally simply a certain kind of group, even when that can still be done.
Sometimes the loss of structure is dramatic: The category $CLOF$ of closed linearly ordered fields and the category $E7$ of undirected graphs isomorphic to the $E_7$ Dynkin diagram are isomorphic, but objects from these two categories have very different properties. There is not even a canonical isomorphism betwee the two categories. Here the essence is completely lost.
Again, you can always put that structure back if you want it. Then you have categories equipped with some structure rather than just categories. (In particular, you might equip $CLOF$ with the structure of its inclusion into the category of fields, and you might equip $E7$ with its inclusion into the discrete category of Dynkin diagrams, which itself has functors to various categories, such as $Lie Alg$.) Incidentally, there is a canonical relationship between these categories; there is an adjoint equivalence between them that is unique up to unique isomorphism, which is enough. But of course, the structures that we like to put on them are very different (even though we could put each structure on the other category if we wished).
Posted by: Toby Bartels on September 30, 2009 6:32 PM | Permalink | Reply to this
### Re: What is a structured object?
AN: In mathematical practice, to say that an object is a group or a manifold says that it has certain properties.
TB: This connects with the idea earlier that ‘ordered monoids are the objects in the intersection of Order and Monoid satisfying the compatibility relation’. As I said then, I would be very interested to see a formalism in which this can be taken literally!
I am working on that and will soon show you how it can be done in FMathL.
TB: Once one has grown out of the idea that a group is literally simply a certain kind of set, then it’s not so hard that an abelian group might not be literally simply a certain kind of group, even when that can still be done.
I never had this idea, hence could not outgrow it.
I always had the idea that although a group G is different from the set making up the elements of G, G contains precisely these elements. Thus I always doubted the semantic legitimacy of the extensivity property of sets, since mathematical practice does not support it.
For exactly the same reasons I oppose the idea that an abelian group should not be literally a group.
It is like claiming that a person is not literally a man or a woman, because you add structure in the form of a gender. This is completely foreign to my understandign of language.
mathematical language shares this additive property of natural language, and good foundations should preserve this important feature.
TB: you can always put that structure back if you want it.
In ordinary language, in informal mathematical langualge, and in FMathL you never lose it, unless you want to lose it.
Posted by: Arnold Neumaier on September 30, 2009 8:35 PM | Permalink |
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# Why does the pressure change on uniformly mixing two liquids?
The two cylinders are connected the upper cylinder has a cross-section of A and the lower one has a cross-section of 2A (I've taken the cross-sections to be A and 2A as they are easier to work with than the radii r and 2r).
Initially the liquid present in the upper cylinder has a density of $$\rho$$ and the lower one has a density of $$2\rho$$. The pressure at the bottom in this scenario can be calculated as:
$$P_1=\rho\cdot g\cdot 2h+2\rho\cdot g \cdot h=4\rho\cdot g\cdot h$$
Now, if we were to mix the two liquids present in the cylinders to create a new solution with a uniform density of $$\frac32\rho$$ (The volume of both the cylinders is the same so we can simply take the mean of the two densities).
The pressure in this case can be calculated as:
$$P_2=\frac32\rho\cdot g \cdot 3h= \frac92\cdot g \cdot h$$
This was one approach to calculate the pressures another method is by simply calculating $$\frac{F}{A}$$
Since the entire liquid system is in equilibrium and the only external force balancing the gravitational force is the normal applied at the bottom surface the pressure will be:
$$P=\frac{Mg}{A}$$
$$P=\frac{\rho\cdot 2h \cdot A\cdot g + 2\rho\cdot 2A \cdot h \cdot g}{2\cdot A }= 3\rho \cdot g \cdot h$$
Which is matching with neither of the cases but i can't see why this is wrong.
First consider the case of a cylinder of uniform cross section $$A$$, in which liquids of density $$\rho_1,\rho_2,$$ with corresponding volumes $$V_1,V_2,$$ are present. Using the hydrostatic equation, the pressure at the bottom is $$P_1=\rho_1gh_1+\rho_2gh_2=\rho_1g(V_1/A)+\rho_2g(V_2/A)=(\rho_1V_1+\rho_2V_2)g/A$$. If after mixing the total volume doesn't change then mass conservation says that the final density should be $$\rho_f=(\rho_1V_1+\rho_2V_2)/(V_1+V_2)$$. The corresponding pressure at the bottom will be: $$\rho_fgh_f=\rho_fg(h_1+h_2)=\rho_fg(V_1+V_2)/A=(\rho_1V_1+\rho_2V_2)g/A,$$ the same as before. Therefore, when the cylinder has uniform cross-section, mixing doesn't change the pressure at the bottom, which is logical since the same weight of fluid is supported by the bottom in both cases.
In your problem the cross-section varies. Let $$\rho_1,V_1,A_1$$ and $$\rho_2,V_2,A_2$$ be the density, volume and cross-section of the two fluids. The interface between the two fluids lies exactly where the cross-section changes. Using the hydrostatic equation, the pressure is: $$P_1=\rho_1gh_1+\rho_2gh_2=\rho_1g(V_1/A_1)+\rho_2g(V_2/A_2)$$, which can't be simplified further. After mixing the density becomes $$\rho_f=(\rho_1V_1+\rho_2V_2)/(V_1+V_2)$$ as before. But now the pressure at the bottom changes to $$\rho_fgh_f\neq P_1,$$ as you can verify by substitution. Since the same weight of fluid is being supported before and after mixing, is the change in pressure a contradiction? No, it isn't.
Above result derived using hydrostatic equation is correct. If you want to derive the pressure on the bottom using force balance then you must recognise that in the case where cross-section of the container varies, weight of the fluid is not the only force acting on the bottom. There is also a downward reaction force due to the horizontal wall where the cross-section changes (denoted by black arrows in the figure below). This reaction force is equal to the product of hydrostatic pressure at the horizontal wall (equal to $$\rho_1gh_1$$ before mixing and $$\rho_fgh_1$$ after mixing) and its area. When this is accounted for you get the same answer as above.
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### Why so many significant phase III results in clinical trials?. (arXiv:1907.00185v2 [econ.GN] UPDATED)
Planning and execution of clinical research and publication of results should conform to the highest ethical standards, given that human lives are at stake. However, economic incentives can generate conflicts of interest for investigators, who may be inclined to withhold unfavorable results or even tamper with the data. Analyzing p-values reported to the ClinicalTrials.gov registry…
### Subexponential LPs Approximate Max-Cut. (arXiv:1911.10304v1 [cs.DS])
We show that for every $\varepsilon > 0$, the degree-$n^\varepsilon$ Sherali-Adams linear program (with $\exp(\tilde{O}(n^\varepsilon))$ variables and constraints) approximates the maximum cut problem within a factor of $(\frac{1}{2}+\varepsilon’)$, for some $\varepsilon'(\varepsilon) > 0$. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension…
### Communication: Curing basis set overcompleteness with pivoted Cholesky decompositions. (arXiv:1911.10372v1 [physics.chem-ph])
The description of weakly bound electronic states is especially difficult with atomic orbital basis sets. The diffuse atomic basis functions that are necessary to describe the extended electronic state generate significant linear dependencies in the molecular basis set, which may make the electronic structure calculations ill-convergent. We propose a method where the over-complete molecular basis…
### Arbitrary access temporal pulse cloaking and restoring in periodically poled lithium niobate. (arXiv:1911.10378v1 [physics.optics])
Temporal cloaks have inspired the innovation of research on security and efficiency of quantum and fiber communications for concealing temporal events. The existing temporal cloaking approaches possessing ps ~ns cloaking windows employed the third-order nonlinearity mostly. Here we explore a temporal cloak for perpetually concealing pulse events using high efficiency second-order nonlinearity. A temporal pulse…
### The Liquid Argon In A Testbeam (LArIAT) Experiment. (arXiv:1911.10379v1 [physics.ins-det])
The LArIAT liquid argon time projection chamber, placed in a tertiary beam of charged particles at the Fermilab Test Beam Facility, has collected large samples of pions, muons, electrons, protons, and kaons in the momentum range 300-1400 MeV/c. This paper describes the main aspects of the detector and beamline, and also reports on calibrations performed…
### Synthesis and Properties of Non-Curing Graphene Thermal Interface Materials. (arXiv:1911.10383v1 [physics.app-ph])
Development of the next generation thermal interface materials with high thermal conductivity is important for thermal management and packaging of electronic devices. We report on the synthesis and thermal conductivity measurements of non-curing thermal paste, i.e. grease, based on mineral oil with the mixture of graphene and few-layer graphene flakes as the fillers. It was…
### Standardless EDXRF technique using bremsstrahlung radiation from a transmission type x-ray generator. (arXiv:1911.10396v1 [physics.atom-ph])
We demonstrate the use of bremsstrahlung radiation in energy dispersive x-ray fluorescence technique as a tool to perform elemental analysis of solid samples employed in inter-disciplinary science research. The bremsstrahlung radiation can be taken from a small, portable, transmission type x-ray generator. Theoretically generated bremsstrahlung spectra are found to be in good agreement with the…
### Modified deformation behaviour of self-ion irradiated tungsten: A combined nano-indentation, HR-EBSD and crystal plasticity study. (arXiv:1911.10397v1 [physics.comp-ph])
Predicting the dramatic changes in material properties caused by irradiation damage is key for the design of future nuclear fission and fusion reactors. Self-ion implantation is an attractive tool for mimicking the effects of neutron irradiation. However, the damaged layer of implanted samples is only few microns thick, making it difficult to estimate macroscopic properties.…
### Ground Truth Simulation for Deep Learning Classification of Mid-Resolution Venus Images Via Unmixing of High-Resolution Hyperspectral Fenix Data. (arXiv:1911.10442v1 [eess.IV])
Training a deep neural network for classification constitutes a major problem in remote sensing due to the lack of adequate field data. Acquiring high-resolution ground truth (GT) by human interpretation is both cost-ineffective and inconsistent. We propose, instead, to utilize high-resolution, hyperspectral images for solving this problem, by unmixing these images to obtain reliable GT…
### Regularized and Smooth Double Core Tensor Factorization for Heterogeneous Data. (arXiv:1911.10454v1 [stat.ML])
We introduce a general tensor model suitable for data analytic tasks for heterogeneous data sets, wherein there are joint low-rank structures within groups of observations, but also discriminative structures across different groups. To capture such complex structures, a double core tensor (DCOT) factorization model is introduced together with a family of smoothing loss functions. By…
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# Spurious comma by use of \glsaddallunused
I am compiling the following document with glossaries v4.11:
\documentclass[a4paper]{report}
\usepackage{glossaries}
\makeglossaries
\newglossaryentry{foo}{name={foo},description={Foo}}
\newglossaryentry{bar}{name={bar},description={Bar}}
\newglossaryentry{baz}{name={baz},description={Baz}}
\begin{document}
\printglossaries
\end{document}
The entries "foo" and "baz" appear as intended, but there is a spurious comma in the "bar" entry that was added using \glsadd:
Removing \glsaddallunused fixes the problem, but then the "baz" entry does not appear any more.
The entry bar is added twice without using it by \glsadd{bar} and \glsaddallunused. In the description of the latter, the documentation says:
If you want to use \glsaddallunused, it's best to place the command at the end of the document to ensure that all the commands you intend to use have already been used. Otherwise you could end up with a spurious comma or dash in the location list.
Apparently, this also applies, when an entry is added twice.
Possible workaround: Using bar by putting it in an unused box: \sbox0{\gls{bar}} or \glsunset{bar} to mark it used:
\documentclass[a4paper]{report}
\usepackage{glossaries}
\makeglossaries
\newglossaryentry{foo}{name={foo},description={Foo}}
\newglossaryentry{bar}{name={bar},description={Bar}}
\newglossaryentry{baz}{name={baz},description={Baz}}
\begin{document}
• @SörenSchulze \sbox0{\gls{bar}} is intended as addition to \glsadd{bar}. The command for this purpose is \glsunset if I have understood the documentation correctly. Answer updated. Jul 2 '15 at 11:33
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# Synopsis: Getting closer to the Bohr model
An algebraic version of Bohr’s collective model is shown to be an effective tool for the analysis of rotational and vibrational spectra in nuclei.
In the algebraic collective model, the five variables define the quadrupole moments of a nucleus. A group theoretical approach is used to separate the variables into a “radial” coordinate ($\beta$) and four angular variables. The radial wave functions can be chosen corresponding to a specific mean deformation in $\beta$ while calculations involving the angular coordinates are made simple by group theoretical techniques. This leads to huge computational savings over calculations in a spherical basis ($\beta =0$), used in some previous models, for which a very much larger set of basis functions is required. The Bohr Hamiltonian can now be solved for virtually any assumed potential.
In a paper appearing in Physical Review C, David Rowe and Trevor Welsh of the University of Toronto in Canada and Mark Caprio of the University of Notre Dame in the US demonstrate the practical utility of the algebraic collective model when applied to various well-known solvable limits of the Bohr model. In several cases, they find a substantial amount of centrifugal stretching (elongation of the nucleus with increasing angular momentum), which is neglected in adiabatic approximations to the Bohr model.
They argue that, as in the case of the interacting boson model, the ease of carrying out collective model calculations for a wide range of Hamiltonians can be used to quickly characterize a large body of nuclear phenomena and test for limitations of the Bohr model. – John Millener
### Announcements
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### Viewpoint: Cyclotron Radiation from One Electron
An electron’s energy can be determined with high accuracy by detecting the radiation it emits when moving in a magnetic field. Read More »
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# Inequality without factor table
Gold Member
I am trying to solve this inequality without using a factor table.
The problem
$$\frac{x+4}{x-1} > 0$$
The attempt at a solution
As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two cases where x-1 is bigger and smaller than 0 because the sign gets inverted when multiplying (or dividing) by a negative number.
Case 1:
## x-1 > 0 \\ x >1 ## gives
$$\frac{x+4}{x-1} > 0 \\ \\ x+4 > 0 \\ x > -4$$
Case 2:
## x-1 < 0 \\ x < 1 ## gives
$$\frac{x+4}{x-1} > 0 \\ \\ x+4 < 0 \\ x < -4$$
This is the place where I get stuck. I am not sure how to take all this information and produce an answer.
Related Precalculus Mathematics Homework Help News on Phys.org
member 587159
I am trying to solve this inequality without using a factor table.
The problem
$$\frac{x+4}{x-1} > 0$$
The attempt at a solution
As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two cases where x-1 is bigger and smaller than 0 because the sign gets inverted when multiplying (or dividing) by a negative number.
Case 1:
## x-1 > 0 \\ x >1 ## gives
$$\frac{x+4}{x-1} > 0 \\ \\ x+4 > 0 \\ x > -4$$
Case 2:
## x-1 < 0 \\ x < 1 ## gives
$$\frac{x+4}{x-1} > 0 \\ \\ x+4 < 0 \\ x < -4$$
This is the place where I get stuck. I am not sure how to take all this information and produce an answer.
Well, you can't just multiply both sides with x - 1 since it contains information to solve the inequality. From what you have now, x can be any real number except 4 or 1. What if x = 0? I don't know whether you know a sign table. You can use this to solve this exercise.
PeroK
Homework Helper
Gold Member
2020 Award
This is the place where I get stuck. I am not sure how to take all this information and produce an answer.
Let's take case 1. ##x > 1## What can you say about the expression ##\frac{x+4}{x-1}## in this case?
Last edited by a moderator:
Gold Member
Let's take case 1. ##x > 1## What can you say about the expression ##\frac{x+4}{x-1}## in this case?
It is always positive for x > 1.
Merlin3189
PeroK
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It is always negative for x > 1.
Hmm! Really? What about ##x = 2##?
Gold Member
Hmm! Really? What about ##x = 2##?
You were fast! Edited the answer just before you commented :D. It is positive. I cannot tell that from my expression that I get in case 1. Not sure how to interpret x > -4
PeroK
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2020 Award
You were fast! Edited the answer just before you commented :D. It is positive.
So, that's half your answer: If ##x > 1## the expression is always positive.
What about when ##x < 1##?
Gold Member
So, that's half your answer: If ##x > 1## the expression is always positive.
What about when ##x < 1##?
Well, that is the tricky part! :D x = 0 gives a negative number. x = - 5 gives a positive number. Not sure how to use my case studies to come to an answer though.
PeroK
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Well, that is the tricky part! :D x = 0 gives a negative number. x = - 5 gives a positive number.
Is there another critical value for ##x##? Perhaps something you've already calculated?
Gold Member
Is there another critical value for ##x##? Perhaps something you've already calculated?
I have a feeling that it is the interval that I calculated in case 2 but I am not sure why.
PeroK
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I have a feeling that it is the interval that I calculated in case 2 but I am not sure why.
What you did in the OP was correct, but it created a logically slightly difficult solution that you weren't able to interpret. I suggest you solve the problem and I can try to explain what happened in your OP, if you like.
You actually should have used 3 cases here. The next case is ##-4 \le x < 1##. Let's call this case 2.
Rectifier
Gold Member
What you did in the OP was correct, but it created a logically slightly difficult solution that you weren't able to interpret. I suggest you solve the problem and I can try to explain what happened in your OP, if you like.
You actually should have used 3 cases here. The next case is ##-4 \le x < 1##. Let's call this case 2.
In the third case gives a negative outcome.
I feel like its much easier to solve these with a sign table :D
PeroK
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In the third case gives a negative outcome.
Not quite. When ##x = -4## the expression is zero. But, there are no solutions for ##-4 \le x < 1##.
Rectifier
Gold Member
Gives a positive outcome.
PeroK
Homework Helper
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Gives a positive outcome.
So, what's the overall solution?
Gold Member
So, what's the overall solution?
x>1 and x<-4
PeroK
Homework Helper
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What happened in your original post?
You started by assuming that ##x > 1## and looked for solutions. You found solutions when ##x > -4##. This looked strange but it was actually quite logical. What you had was:
##x > 1## and ##x > -4##
Which, logically, is simply ##x > 1##.
There's something else that might have happened in a different problem. You might have got something like:
##x > 1## and ##x < -2##
Again, that is not a problem, but would tell you that there are no solutions for ##x > 1##.
Rectifier
Gold Member
What happened in your original post?
You started by assuming that ##x > 1## and looked for solutions. You found solutions when ##x > -4##. This looked strange but it was actually quite logical. What you had was:
##x > 1## and ##x > -4##
Which, logically, is simply ##x > 1##.
There's something else that might have happened in a different problem. You might have got something like:
##x > 1## and ##x < -2##
Again, that is not a problem, but would tell you that there are no solutions for ##x > 1##.
Oh okay, thank you for the explanation! But why do I choose x>1 and not x > -4? I suspect that the answer is trivial.
PeroK
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Oh okay, thank you for the explanation! But why do I choose x>1 and not x > -4? I suspect that the answer is trivial.
Which numbers satisfy both equations?
Rectifier
Gold Member
Which numbers satisfy both equations?
x > 1. So I should basically take the one with the smallest common solution set?
PeroK
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x > 1. So I should basically take the one with the smallest common solution set?
Yes, but best to understand the logic. Here's the full logic of your original post:
##(x > 1 \ AND \ x> -4) \ OR \ (x < 1 \ AND \ x < -4)##
##(x > 1) \ OR \ (x < -4)##
So, now I'm going to slightly correct your overall solution:
x>1 and x<-4
Perhaps ##x > 1## or ##x < -4## is better!
If you ever do some computer programming, this sort of logical thinking can be quite useful!
Rectifier
Gold Member
Thank you for your help. I will take try to understand that approach a bit later today. Thank you for your help once more.
Mark44
Mentor
x>1 and x<-4
PeroK said:
Perhaps ##x > 1## or ##x < -4## is better!
If you ever do some computer programming, this sort of logical thinking can be quite useful!
The problem with x > 1 AND x < -4 is that x can't simultaneously be larger than 1 and smaller than -4. There are no numbers that satisfy both inequalities. That's why PeroK's version is better.
Rectifier
Gold Member
The problem with x > 1 AND x < -4 is that x can't simultaneously be larger than 1 and smaller than -4. There are no numbers that satisfy both inequalities. That's why PeroK's version is better.
Thank you for the clarification :)
SammyS
Staff Emeritus
Homework Helper
Gold Member
x > 1. So I should basically take the one with the smallest common solution set?
Maybe that's not the best way to look at it.
Case 1 holds only for x > 1. No matter what the result you get for this case, in this problem you got x > -4, that result is only good for x > 1.
Case 2 holds only for x < 1. No matter what the result you get for this case, in this problem you got x < -4, that result is only good for x < 1.
In the end, it's more like the intersection. Well, I suppose that is the smallest common set for each case.
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pylops.signalprocessing.Patch3D¶
pylops.signalprocessing.Patch3D(Op, dims, dimsd, nwin, nover, nop, tapertype='hanning', scalings=None, name='P')[source]
3D Patch transform operator.
Apply a transform operator Op repeatedly to patches of the model vector in forward mode and patches of the data vector in adjoint mode. More specifically, in forward mode the model vector is divided into patches, each patch is transformed, and patches are then recombined together. Both model and data are internally reshaped and interpreted as 3-dimensional arrays: each patch contains a portion of the array in every axis.
This operator can be used to perform local, overlapping transforms (e.g., pylops.signalprocessing.FFTND or pylops.signalprocessing.Radon3D) on 3-dimensional arrays.
Note
The shape of the model has to be consistent with the number of windows for this operator not to return an error. As the number of windows depends directly on the choice of nwin and nover, it is recommended to first run patch3d_design to obtain the corresponding dims and number of windows.
Warning
Depending on the choice of nwin and nover as well as the size of the data, sliding windows may not cover the entire data. The start and end indices of each window will be displayed and returned with running patch3d_design.
Parameters: Op : pylops.LinearOperator Transform operator dims : tuple Shape of 3-dimensional model. Note that dims[0], dims[1] and dims[2] should be multiple of the model size of the transform in their respective dimensions dimsd : tuple Shape of 3-dimensional data nwin : tuple Number of samples of window nover : tuple Number of samples of overlapping part of window nop : tuple Size of model in the transformed domain tapertype : str, optional Type of taper (hanning, cosine, cosinesquare or None) scalings : tuple or list, optional Set of scalings to apply to each patch. If None, no scale will be applied name : str, optional Name of operator (to be used by pylops.utils.describe.describe) Sop : pylops.LinearOperator Sliding operator ValueError Identified number of windows is not consistent with provided model shape (dims).
Sliding1D
Sliding2D
Sliding3D
Patch2D
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11.8 Cohesion and adhesion in liquids: surface tension and capillary
Page 1 / 11
• Understand cohesive and adhesive forces.
• Define surface tension.
• Understand capillary action.
Children blow soap bubbles and play in the spray of a sprinkler on a hot summer day. (See [link] .) An underwater spider keeps his air supply in a shiny bubble he carries wrapped around him. A technician draws blood into a small-diameter tube just by touching it to a drop on a pricked finger. A premature infant struggles to inflate her lungs. What is the common thread? All these activities are dominated by the attractive forces between atoms and molecules in liquids—both within a liquid and between the liquid and its surroundings.
Attractive forces between molecules of the same type are called cohesive forces . Liquids can, for example, be held in open containers because cohesive forces hold the molecules together. Attractive forces between molecules of different types are called adhesive forces . Such forces cause liquid drops to cling to window panes, for example. In this section we examine effects directly attributable to cohesive and adhesive forces in liquids.
Cohesive forces
Attractive forces between molecules of the same type are called cohesive forces.
Attractive forces between molecules of different types are called adhesive forces.
Surface tension
Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called surface tension . Molecules on the surface are pulled inward by cohesive forces, reducing the surface area. Molecules inside the liquid experience zero net force, since they have neighbors on all sides.
Surface tension
Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called surface tension.
Making connections: surface tension
Forces between atoms and molecules underlie the macroscopic effect called surface tension. These attractive forces pull the molecules closer together and tend to minimize the surface area. This is another example of a submicroscopic explanation for a macroscopic phenomenon.
The model of a liquid surface acting like a stretched elastic sheet can effectively explain surface tension effects. For example, some insects can walk on water (as opposed to floating in it) as we would walk on a trampoline—they dent the surface as shown in [link] (a). [link] (b) shows another example, where a needle rests on a water surface. The iron needle cannot, and does not, float, because its density is greater than that of water. Rather, its weight is supported by forces in the stretched surface that try to make the surface smaller or flatter. If the needle were placed point down on the surface, its weight acting on a smaller area would break the surface, and it would sink.
the meaning of phrase in physics
is the meaning of phrase in physics
Chovwe
write an expression for a plane progressive wave moving from left to right along x axis and having amplitude 0.02m, frequency of 650Hz and speed if 680ms-¹
how does a model differ from a theory
what is vector quantity
Vector quality have both direction and magnitude, such as Force, displacement, acceleration and etc.
Besmellah
Is the force attractive or repulsive between the hot and neutral lines hung from power poles? Why?
what's electromagnetic induction
electromagnetic induction is a process in which conductor is put in a particular position and magnetic field keeps varying.
Lukman
wow great
Salaudeen
what is mutual induction?
je
mutual induction can be define as the current flowing in one coil that induces a voltage in an adjacent coil.
Johnson
how to undergo polarization
show that a particle moving under the influence of an attractive force mu/y³ towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v²k² and distance uk²/√u-vk as origin
show that a particle moving under the influence of an attractive force mu/y^3 towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v^2k^2 and distance uk^2/√u-k as origin
No idea.... Are you even sure this question exist?
Mavis
I can't even understand the question
yes it was an assignment question "^"represent raise to power pls
Gabriel
Gabriel
An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10kg . Pendulum 2 has a bob with a mass of 100 kg . Describe how the motion of the pendula will differ if the bobs are both displaced by 12º .
no ideas
Augstine
if u at an angle of 12 degrees their period will be same so as their velocity, that means they both move simultaneously since both both hovers at same length meaning they have the same length
Modern cars are made of materials that make them collapsible upon collision. Explain using physics concept (Force and impulse), how these car designs help with the safety of passengers.
calculate the force due to surface tension required to support a column liquid in a capillary tube 5mm. If the capillary tube is dipped into a beaker of water
find the time required for a train Half a Kilometre long to cross a bridge almost kilometre long racing at 100km/h
method of polarization
Ajayi
What is atomic number?
The number of protons in the nucleus of an atom
Deborah
type of thermodynamics
oxygen gas contained in a ccylinder of volume has a temp of 300k and pressure 2.5×10Nm
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# zbMATH — the first resource for mathematics
The topology of free topological groups. (Russian. English summary) Zbl 1068.22002
Fundam. Prikl. Mat. 9, No. 2, 99-204 (2003); translation in J. Math. Sci., New York 131, No. 4, 5765-5838 (2005).
This paper deals with the notion of the free topological group $$F(X)$$ in the sense of Markov and with the topological Abelian group $$A(X)$$ which is generated by a Tikhonov space $$X$$. A variety of results is presented to demonstrate how useful explicit descriptions of the topologies on these groups are. Moreover, a plenty of interesting examples is given to illustrate the use of such descriptions in many topological problems. In addition, several approaches to describe topologies on free topological groups are presented. This paper is worth to be read by any mathematician working with the notion of free topological group.
##### MSC:
22A05 Structure of general topological groups 54H11 Topological groups (topological aspects)
##### Keywords:
topological groups; topology of groups; Suslin numbers
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## Content description
Describe translations, reflections in an axis, and rotations of multiples of 90 ° on the Cartesian plane using coordinates. Identify line and rotational symmetries (ACMMG181)
Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)
There is a separate resource for line and rotational symmetry
### Rotations of multiples of 90° about a point
In the figure above, $$\triangle ABC$$ is rotated 90 ° in an clockwise direction about point $$D$$ to the image $$\triangle A^\prime B^\prime C^\prime$$.
The point $$A$$ in the diagram above is rotated 90 ° in a clockwise direction about point $$D$$. Note that $$AD = A^\prime D$$ and $$\angle AD A^\prime = 90 ^\circ$$.
Using coordinates we can describe the rotation of the vertices:
$$A(4, 4) \rightarrow A^\prime (9, 3); B(3, 6) \rightarrow B^\prime (7, 2); C(5, 7) \rightarrow C^\prime (6, 4)$$.
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# Car collision
1. Sep 7, 2009
### Bryon
1. The problem statement, all variables and given/known data
I am trying to find the time a collision occurs of car 1 that is travelling 31m/s and can accelerate at -1.8m/s and car 2 that is traveling at a constant velocity of 6m/s.
2. Relevant equations
v(final)^2=v(initial)+ 2a(x(final) - x(initial))
v(final) = v(initial) + at
x(final) = x(initial) + v(initial)t + .2at^2
3. The attempt at a solution
I found change in velocity of car 1 over the 30 meter distance.
v(final)^2 = 31^2 - 2(-1.8)(-30) = 28,837
28.837 = 31 + (-1.8)t ............t = 0.996
the distance car 2 traveled over the 0.996s is 5.976m
so adding the distance car 2 traveled plus the distance car 1 is initially from car 2...
v(final)^2 = 31^2 - 2(-1.8)(-35.976) = 28.835
28.835 = 31 + (-1.8)t..................t =1.204s
Which 1.204 seconds turned out to be the wrong answer. Would I have to find the relative velocity between the cars over the 30 meters? Would the relative velocity be the average over the 30m? I am not sure what else to look at.
Thanks for the help!
2. Sep 7, 2009
### kuruman
Can you state the problem exactly as it is given? Specifically, how far apart are the cars initially?
3. Sep 7, 2009
### Bryon
Here is the problem: A certain automobile can decelerate at 1.8 m/s^2. Traveling at a constant car 1 = 31m/s, this car comes up behind a car traveling at a constant car 2 = 6m/s. The driver of car 1 applies the brakes until it is just 30m behind the slower car. Call the instant which the brakes are applied t = 0. At what time does the inevitable collision occur?
4. Sep 7, 2009
### Jebus_Chris
A collision is when their positions are the same. So create two equations, 1 for each car, that model each cars position.
$$x=x_o+v_ot-\frac{1}{2}gt^2$$
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Recent questions and answers in States of Matter: Liquids and Gases
A gas occupies one litre under atmosphere pressure. What will be the volume of the same amount of gas under 750mm of Hg at the same temperature?
To see more, click for all the questions in this category.
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# Constructing a List from a Decimal Number
I am trying to get from a number such as $12.345$ to a list $\{1,2,3,4,5\}$. My best attempt so far has been:
First[RealDigits[12.345]]
however this of course gives $\{1, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ since it assumes the number to have infinite precision.
Any help is greatly appreciated
• First[RealDigits[Rationalize[12.345]]]? – J. M.'s technical difficulties May 21 '17 at 18:44
• What output do you expect for 100.00 ? – Shadowray May 21 '17 at 19:00
• Related: (110596) – Mr.Wizard Jul 21 '18 at 3:13
12.345 does not have infinite precision, rather it has machine precision. If you specify the precision correctly your formulation will work as written:
First @ RealDigits[12.3455]
{1, 2, 3, 4, 5}
I believe a modified input form like this is necessary to remove ambiguity; see:
J. M. proposed Rationalize in a comment but this cannot be relied upon, e.g.:
First @ RealDigits @ Rationalize[12.345678]
{1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0}
First @ RealDigits[12.345678] /. {a___, 0 ...} :> {a}
{1, 2, 3, 4, 5, 6, 7, 8}
• Hi Mr.W. Could not respond to your comment re my deleted answer. I deleted it because it cannot handle 100.00 (which, I presumed, should be transformed to {1, 0, 0, 0, 0}). – kglr May 21 '17 at 19:11
• @kglr I see. But I think that is a problem with the OP's input format, i.e. it is an unreasonable expectation. We could perhaps drop only zeros to the right of the decimal point, but there is no way to differentiate 100.00 from 100.00000` etc.; the precision must be included in the input I believe. – Mr.Wizard May 21 '17 at 19:14
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Winning spelling algorithm
New Book Reviews!
Winning spelling algorithm
Written by Lucy Black
Sunday, 24 July 2011
There were over 300 participants in the Microsoft Research/Bing Speller Challenge. The winners were selected automatically and were presented with prizes at a workshop last month.
The problem that this Microsoft challenge set out to tackle was getting a better set of results from search queries that could include mistyped or misspelled words - the subtlety, as explained in last December's blog post announcing the competition being that:
"One person's spelling error could be another's perfect query".
The contest was open to researchers and students worldwide and the blog post reporting the winners notes that there were entries from every continent except Antarctica.
For the purposes of the contest participants were given access to real-world data at web scale by using the Microsoft Research Web N-gram Services and were able to improve their algorithms and see how it compared to other spelling correction systems by using an evaluation service provided by Microsoft Research.
Given that the task was to devise an automatic algorithm, the winning entries were also selected automatically based on performance in figuring out the best spelling alternatives, for example, "Britney Spears" for "briteny spears".
Prizes were awarded as follows:
• First place (US$10,000): Gord Lueck – Canada • Second place (US$8,000): Yanen Li, Huizhong Duan, and ChengXiang Zhai – United States
• Third place (US$6,000): Yasser Ganjisaffar, Andrea Zilio, Sara Javanmardi, Inci Cetindil, Manik Sikka, Sandeep P. Katumalla, Narges Khatib, and Chen Li – United States • Fourth place (US$4,000): Dan Ştefănescu, Radu Ion, and Tiberiu Boroş – Romania
• Fifth place (US$2,000): Yoh Okuno – Japan In the video below Harry Shum, corporate vice president of Bing, explains the idea behind the contest and introduces the overall winner Gord Lueck who tells us how he set about the challenge. Related articles:$30,000 for a better spell checker
Microsoft Web N-gram Services go public
Mozilla Firefox Replacing Gecko With Servo30/01/2017The time has finally come for Firefox to upgrade the aging Gecko rendering engine that served it for 20 or some years.The upgrade will enable Firefox to take full advantage of modern CPU's and GPU's, [ ... ] + Full Story //No Comment - Turmits are Turing-universal, The Whale Swarm Algorithm & Rules That Govern Fish21/02/2017• Nontrivial Turmites are Turing-universal • Whale swarm algorithm for function optimization • What Is the Rule that Gives Rise to Coordinated Swimming in Fish? + Full Story More News
Last Updated ( Sunday, 24 July 2011 )
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# Simulating Exploding Dice
Your task is to make a program that takes in an integer n > 1, and outputs the roll of a single n-sided die. However, this dice follows the rules for exploding dice.
When you roll the die, check what value you rolled. If you got the maximum for that kind of die (on a standard d4 that would be 4, or 6 on a d6, etc.), roll again and add the new roll to that total. Each roll continues adding to the total, until you don't roll the max number anymore. That final number is still added though.
Your program should take in a single integer n, and roll the exploding n-sided die. Here's an example distribution to show what it should look like for n=4. Note that you should never output any multiples of n, since they will always explode.
You can assume the stack size for any recursion you do is infinite, and your random function must meet our standards for randomness (built-in random generator or time/date). Your random function should also be as uniform as possible, vs. something like a geometric distribution, since these are dice we're talking about.
• does the program have to be perfect? Like can its distribution be off by some extremely low amount? Apr 12 '19 at 14:38
• To: Riker; RE: @Maltysen's comment above; or extremely high amount? Apr 12 '19 at 15:09
• @ArtemisFowl See our standards for randomness. Also, here. Apr 12 '19 at 22:46
# Excel VBA, 46 bytes
Thanks to @TaylorScott
Do:v=-Int(-[A1]*Rnd):t=t+v:Loop While[A1]=v:?t
Executed in the command window.
As a user-defined function.
# Excel VBA, 108 67 bytes
Function z(i)
Do
v=Int((i*Rnd)+1)
z=z+v
Loop While v=i
End Function
• You can get this down quite a bit by using a do..loop while loop, and converting the function into an immediate window function. - Do:v=-Int(-[A1]*Rnd):t=t+v:Loop While[A1]=v:?t - 46 Bytes Apr 15 '19 at 19:03
• @TaylorScott Thanks, I forgot that Do x While y existed in Excel VBA. Apr 15 '19 at 20:23
# Excel (pure), 43 bytes
=A1*INT(-LOG(RAND(),A1))-INT(RAND()*(1-A1))
Takes input from A1, outputs in the cell you put this formula.
# Explanation
This uses the observation from Robin Ryder's R solution that the requested distribution is the sum of n times a Geom(1-1/n) and an independent Uniform(1..n-1) distribution. The clever bit is that
INT(-LOG(RAND(),A1))
uses a logarithm to scale a Uniform(0,1) distribution exactly to the desired geometric distribution: for instance, with A1=6, the LOG maps the interval (1/6,1) to the interval (-1,0), the interval (1/36,1/6) to the interval (-2, -1), and so on. I also save a +1 on the uniform distribution by scaling a random number "backwards" to (1-A1, 0), then subtracting the floor of this negative number.
import System.Random
f n=do i<-randomRIO(1,n);last$((i+)<$>f n):[return i|i<n]
Try it online!
Original
Quite similar to @dfeuer's, but using do notation.
• -13 bytes by removing whitespace, thanks to @dfeuer
• -3 bytes thanks to this
• 81 bytes Apr 13 '19 at 2:13
• Wuups, fixed the typo! And thanks for the suggestion :)
– bugs
Apr 13 '19 at 10:16
# Octave/MATLAB with Statistics Package/Toolbox, 30 bytes
@(n)geornd(1-1/n)*n+randi(n-1)
Try it online!
### How it works
This is an anonymous function which takes n as input and produces a number obtained as follows. The function generates a geometric random variable with parameter 1-1/n (this models the number of rolls that produce n), multiplies by n, and adds a random variable uniformly distributed from 1 to n-1 (this models the last roll).
# Vyxal, 25 10 bytes
λ?:ɾ℅~=[x+
Try it Online!
-15 thanks to Aaron Miller. This is a fully functional program as lambdas are automatically called at the end.
• recursion would probably make this simpler. May 8 at 13:46
• @Razetime Yes probably, I'll try May 8 at 20:21
• 11 bytes with recursion Jun 22 at 20:28
• @AaronMiller 10 as you can close the if in the footer. Jun 22 at 20:41
• Ok, I wasn't sure. I haven't really done many submissions that were just functions. Thanks for saying so! Jun 22 at 20:43
# ><>, 90 bytes
0& v
v:::< <
v/1>{1-:?!v}
>x0^v10~ <
^
5=?v>:@}}*{+{2*l
&n;>,*:1%-1+:&+&=?^
Try it online!
The whitespace on the second line is bugging me. I'll work on golfing that out.
><> doesn't have a nice method for producing uniform random integers. This approach generates, for input $$\N\$$, a random number produced by generating $$\N\$$ random bits, then taking the resulting binary integer and dividing it by $$\2^N\$$. This process is repeated until $$\N\$$ is not generated by this process.
• – Jo King
Apr 13 '19 at 0:07
37 bytes: ${NestWhile[{_+Random[1,x]},{x|_},0]} # Perl 6, 26 bytes {[+] {^$_ .roll+1}...$_>*} Try it online! # Python 3, 80 bytes This is pretty much just a tail-call recursive version of @Artemis Fowl's answer, but I liked doing it without unrolling into a while loop. Uses an accumulator parameter to return the total rolled value once the exploding stops. from random import* def r(n,a=0):v=randint(1,n);a+=v;return r(n,a)if v==n else a • I don’t think python does tail-call optimization, does it? Regardless, good answer anyways! Apr 15 '19 at 17:21 • Unfortunately not. I've heard that Guido doesn't want python to turn into a functional language 😜 Apr 15 '19 at 17:23 # Charcoal, 17 bytes ＮθＷ⁼Ｌυ№υθ⊞υ⊕‽θＩΣυ Try it online! Link is to verbose version of code. Explanation: Ｎθ Input n. Ｗ⁼Ｌυ№υθ Repeat while the predefined empty list only contains ns (i.e. its length equals its count of ns)... ⊞υ⊕‽θ ... push a random integer from 1 to n to the list. ＩΣυ Print the total. # C# (Visual C# Interactive Compiler), 60 bytes int f(int n,int k=1)=>(k=new Random().Next(n)+1)<n?k:k+f(n); Saved 4 bytes thanks to @ExpiredData! Try it online! • 60 bytes Apr 12 '19 at 19:48 • @ExpiredData Using new Random() gives it the same seed if you call it again, which doesn't make it truly random Apr 12 '19 at 20:11 • Nah it's a new seed, run my tio a few times. But even if it was same seed random() isn't true random anyway it's a prng.... Apr 12 '19 at 20:40 • I've noticed when running on TIO, you either get values less than n or way, way, higher. I think creating a new Random in a tight loop causes the same seed to be used? In any case, this might save a byte? int f(int n,int k=1)=>(k+=new Random().Next(n))<n?k:k+f(n); – dana Apr 13 '19 at 5:38 • This is longer, but seems to give better results? var r=new Random();int k;int f(int n)=>(k=r.Next(n)+1)<n?k:k+f(n); – dana Apr 13 '19 at 5:42 # Perl 6, 25 bytes {sum {.rand+|0+1}...$_>*}
Try it online!
## F#, 83 bytes
let r=System.Random()
let e d=
let mutable t=0
while t%d=0 do t<-t+r.Next(d)+1
t
Try it online! The first argument in the TIO program is the number of sides on the die, the second is the amount of tests to make. The output is printed to the console and shows each total and the number of times it has been rolled.
The random number generator r is initialised outside of the function. Initialising it within the function, combined with calling this function many times in succession, will cause it to return the same random numbers over and over again (try it for yourself!)
For the life of me I could not figure out how to write this without using mutable, especially since the last roll must be counted (Seq.takeWhile would not include the terminating element). I thought about using Seq.mapFold but it seems like it evaluates the input sequence first, which was a no-go for me.
• You're implementing so many extra things. They don't appear to be in the code you're counting, though, so...okay. Apr 17 '19 at 8:18
• What do you mean by extra things? Apr 17 '19 at 20:04
• "shows ...the number of times it has been rolled", not a requirement. Apr 17 '19 at 21:14
• ..You need to roll one exploding die. Not 10,000. I'm not sure this is valid...? Apr 17 '19 at 21:22
• The submission does only roll one die. The TIO program demonstrates the function being run 10000 times to show that its results are distributed similar to the example distribution. It's common to have a TIO to show the function being used and that its output is valid. And that is not included in the byte count. Apr 18 '19 at 21:34
# C (gcc), 36 32 bytes
-4 bytes thanks to Ben Voigt
f(n,x){x=rand()%n;x=x?x:n+f(n);}
Try it online!
Here a Test with 100k d4
• Shouldn't srand(time(0)) be included in the byte count? AFAIK rand() will always return the same value if it was never seeded Apr 12 '19 at 16:42
• I'm not entirely sure, but I remember the meta consensus being that using an unseed PRNG is acceptable @Tau Apr 12 '19 at 16:50
• This is not even close to being C code. Your recursive call has the wrong number of parameters, and you are missing a return. Apr 13 '19 at 18:46
• Even on gcc it returns 0 all the time (if compiled with -O3) or total garbage (if compiled with -O1). If you have optimization-unstable code that depends on particular compile settings, that should be mentioned. Apr 15 '19 at 14:32
• @BenVoigt unless mentioned in the post, it is assumed compilers use default options. The default for gcc is -O0, which works perfectly fine for this code. The purpose of codegolfing is to write code in as few bytes as possible, not write functional, production-ready code. If it works, it works. Apr 15 '19 at 14:55
# Ruby, 28 bytes
f=->n{(a=rand n)>0?a:n+f[n]}
Try it online!
# Java (JDK), 61 bytes
int f(int n){int r=1;r+=Math.random()*n;return r<n?r:r+f(n);}
Try it online!
# Funky, 38 bytes
f=n=>ifn==x=math.random(n)f(n)+x elsex
Try it online!
# C# (.NET Core), 155 153 145 144 bytes
using C=System.Console;class A{static void Main(){int i=int.Parse(C.ReadLine()),j=0;while((j+=new System.Random().Next(i)+1)%i<1){}C.Write(j);}}
Try it online!
This code instantiates a new RNG every time, but that uses time as a seed anyway so it should still satisfy randomness requirements.
I wonder if console input is cheaper in the long run.
• Saved 2 bytes by using [0,i)+1 rather than [1,i+1) Apr 17 '19 at 19:35
• Saved 8 bytes by forgoing principles about RNG and just making a new one in the loop. Apr 17 '19 at 21:19
• Saved one byte w/ <1 instead of ==0 Apr 20 '19 at 7:35
# Clojure, 63 bytes
#(loop[n 0](let[r(+ n(rand-int %)1)](if(<(- r n)%)r(recur r))))
A naive solution to the problem.
Expanded:
(defn exploding-die [sides]
(loop [total 0]
(let [new-total (+ total (rand-int sides) 1)]
(if (< (- new-total total) sides) new-total
(recur new-total)))))
## VTL-2, 5453 51 bytes
1 A=?
2 C=C+B
3 B='/A*0+%+1
4 #=B=A*2
5 ?=C+B
Line 1 takes input into variable A, this will be our die's sidedness. Line 2 does what it appears to, though it's important to note that in VTL-2, referencing a variable that hasn't been initialized assumes 0, so for the first pass, this is C=0+0. Line 3 divides a random number by our dice sides (' is the system variable for a random number) and then turns this into a roll - % is the remainder of the last division operation; add 1 and put this in B. Line 4 is a little cryptic: B=A is evaluated first. It evaluates to 1 if the two are equal (if our die roll is the same as its number of sides), otherwise it evaluates to 0. This result is multiplied by two (*2), and then the final value here is handed to #=, which is equivalent to a GOTO. If this is given a zero, it ignores it; otherwise we GOTO 2, adding the roll to the total and rolling again. Line 6 prints the total of C+B.
Had an epiphany right after posting this, and golfed off one byte. I was doing 3 B='/A and 4 B=%+1 because % is a system variable with the remainder of the last division operation; you need to do a division operation to get that value. But it occurred to me that I could do the division and then multiply that by zero since I don't need it. But since I've done it, the remainder is now in % and so I can add that to the zero I just made, and add 1 to get the die roll. This is long, but still shorter than two lines - line numbers always take two bytes in VTL-2, plus a space to separate the line number, plus a CR.
Second edit, did my math wildly wrong on byte count both times. Eesh. Third, golfed off two leftover parens.
• Is this the language from the Altair systems? Google seems to imply so. But this is really cool! May 4 '19 at 20:55
• @Rɪᴋᴇʀ Yup! Stumbled across it recently while looking for some totally unrelated 680 info. I'm running the interpreter in an 8800 emulator. It's a very interesting language, a lot of fun design decisions were made to fit it in ~1KB (VTL-1 was closer to ~700B!). May 4 '19 at 21:29
• Huh, that's pretty cool. I'm glad you decided to answer my challenge in that language, of all things. May 4 '19 at 23:38
# MMIX, 40 bytes (10 instrs)
Assumes TRAP 82,78,71¹ gets a randomly-generated uint64_t, by some method the OS knows about, but we don't have to.
(jxd -T)
00000000: e3010000 f7010000 00524e47 1effff00 ẉ¢¡¡ẋ¢¡¡¡RNGœ””¡
00000010: feff0006 220101ff 22010100 43fffffb “”¡©"¢¢”"¢¢¡C””»
00000020: 26000100 f8010000 &¡¢¡ẏ¢¡¡
Disassembled:
expl SETL $1,0 PUT rD,0 0H TRAP 82,78,71 // loop: tmp = sysrand() DIVU$255,$255,0 GET$255,rR // tmp = tmp % n
ADDU $1,$1,$255 ADDU$1,$1,$0 // t += tmp + n
BZ $255,0B // if(!tmp) goto loop SUBU$0,$1,$0
POP 1,0 // return t - n
The main hack here is to treat every roll as being in the range $$\[n,2n-1]\$$, then subtracting $$\n\$$ once after everything is over.
¹ Yes, those numbers were carefully picked.
# Ly, 16 bytes
00ns[p+1l?l=]p+u
Try it online!
This one is pretty much a direct translations of the rules of the challenge.
00n # Push "0", "0" and the input number on the stack
s # Save the input number in the backup cell
[ ] # Loop, quits when the random number doesn't match the die size
p+ # Delete loop iterator check, add the remain two entries
# (The stack holds the accumulated value and the last roll)
1l? # Generate a random number between 1 and die size, inclusive
l= # Compare the random roll to the saved die size
p+u # Add the last role and print as a number
`
|
{}
|
# Packages
The packages you will need for this workshop are as follows. I am also including the hex codes for a colorblind-friendly palette, which I use for my plots.
library(tidyverse)
library(ordinal)
library(MASS)
library(ggeffects)
library(effects)
cbPalette <- c("#999999", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")
# What is ordinal logistic regression?
Ordinal logistic regression (henceforth, OLS) is used to determine the relationship between a set of predictors and an ordered factor dependent variable. This is especially useful when you have rating data, such as on a Likert scale. OLS is more appropriate to use than linear mixed effects models in this case because although a Likert scale might include numeric values to choose from, these values are inherently categorical. For example, it is unacceptable to choose 2.743 on a Likert scale ranging from 1 to 5. The most common form of an ordinal logistic regression is the “proportional odds model”.
Note that an assumption of ordinal logistic regression is the distances between two points on the scale are approximately equal. (That is, if on a scale of 1 to 5, the distance between 1 and 2 is similar to the distance between 4 and 5.) This is a difficult assumption to test, as it involves knowledge about the rating system of language users. We will therefore assume for this workshop that this is the case.
There are two packages that currently run ordinal logistic regression. The polr() function in the MASS package works, as do the clm() and clmm() functions in the ordinal package. Here, I will show you how to use the ordinal package. Note that the difference between the clm() and clmm() functions is the second m, standing for mixed. This package allows the inclusion of mixed effects. The results from the two packages are comparable.
The coefficients for OLS are given in ordered logits, or ordered log odds ratios. It is therefore important to remember what a log odds ratio is.
## Probabilities, odds, and odds ratios
Recall the difference between odds and probabilities. Probabilities are considered proportions or percentages, defined by dividing the occurrences of an event divided by the total number of observations. Odds are the ratio of the probability of one event to the probability of another event, which can be simplified as the ratio of the frequency of X to the frequency of Y.
For probabilities, if the chances of two events are equal, the probability of either outcome is 0.5, or 50%. Probability ranges from 0 to 1 (0% to 100%).
If the odds equal 1, the probabilities of the outcomes are equal. If the odds are lower than 1, the probability of the second event is greater than the first (aka, if m/p < 1, then P is more likely than M). If odds is higher than 1, the probability of the first event is greater than the second event (if m/p > 1, then M more likely than P).
Log odds are logarithmically transformed odds. Log odds are also called logits. Note that logarithmically transformed here means the natural log, not base-10 log.
An odds ratio is the ratio of two odds. It tells you if the odds for a particular event is more or less likely in a particular scenario over another.
A log odds ratio is the log of the odds ratio. If a log odds ratio is positive, the specified level boosts the chances of a selected outcome. If a log odds ratio is negative, the specified level decreases the chances of a selected outcome. Log odds are centered around 0 (because ln(1) = 0, so when odds are equal, ln(odds) = 0.
In order to convert from log odds ratios to odds ratios, use exp(X). To convert from log odds ratios to probabilities, use the following formula: probability = exp(X)/(1 + exp(X)). You can also use the plogis() function to do this conversion.
Probability Odds Logit
0.001 0.001 -6.91
0.01 0.01 -4.6
0.05 0.05 -2.94
0.01 0.11 -2.2
0.25 0.33 -1.1
0.5 1 0
0.75 3 1.1
0.9 9 2.2
0.95 19 2.94
0.99 99 4.6
0.999 999 6.91
## Set-up of the model
The format of the OLS proportional odds model is as follows. Note that this will become important when we calculate log odds ratios, and by extension, probabilities, of events getting a certain rating (or below).
$$logit[P(Y \leq j)] = \alpha_j - \beta x, j = 1 ... J-1$$
We can read this as such: The log odds of the probability of getting a rating less than or equal to J is equal to the equation $$\alpha_j - \beta x$$, where $$\alpha_j$$ is the threshold coefficient corresponding to the particular rating, $$\beta$$ is the variable coefficient corresponding to a change in a predictor variable, and $$x$$ is the value of the predictor variable. Note, $$\beta$$ is the value given to each coefficient corresponding to a variable, which is similar to a coefficient in a linear model. However, while we have in a linear model the coefficient for a variable in the original units of the response variable, this model gives us the change in log odds. The $$\alpha$$ value can be considered an intercept of sorts (if comparing to a linear model) - it is the intercept for getting a rating of J or below, again in log odds.
Since we have defined the relationship between probability and log odds as P(X) = exp(X)/(1 + exp(X)) (where X is the log odds ratio), we can extend our definition of the proportional odds model to be:
$$P(Y \leq j) = \frac{exp(\alpha_j - \beta x)}{1+exp(\alpha_j - \beta x)}, j = 1 ... J-1$$.
# Running an ordinal logistic regression in R
## Data
The data I will be using was kindly provided by Prof. Ionin. The data is comprised of ratings of acceptability of Brazilian Portuguese noun phrases in a variety of positions. The fixed effects of interest are as follows:
• NP type (bare singular vs. bare plural)
• position (subject vs. object)
• NP number (single-NP vs. list-NP)
In addition, because these are categorical variables, I have simulated a fourth fixed effect, called FreqSim, which is a numeric value between 1 and 10. While data analysis should not include the simulation of random variables, I have done this in order to show you the effect of continuous variables on a model.
## ID survey list label position NP rating item FreqSim
## 1 100 listNP 1 3.1barepl object barepl 4 3.1xx 2.595569
## 2 100 listNP 1 3.2barepl object barepl 4 3.2xx 2.507130
## 3 100 listNP 1 3.3barepl object barepl 4 3.3xx 1.782511
## 4 100 listNP 1 3.4barepl object barepl 4 3.4xx 1.683113
## 5 101 listNP 1 3.1barepl object barepl 3 3.1xx 1.722588
## 6 101 listNP 1 3.2barepl object barepl 4 3.2xx 1.396689
## 'data.frame': 1152 obs. of 9 variables:
## $ID : int 100 100 100 100 101 101 101 101 102 102 ... ##$ survey : Factor w/ 2 levels "listNP","singleNP": 1 1 1 1 1 1 1 1 1 1 ...
## $list : int 1 1 1 1 1 1 1 1 1 1 ... ##$ label : chr "3.1barepl" "3.2barepl" "3.3barepl" "3.4barepl" ...
## $position: Factor w/ 2 levels "object","subject": 1 1 1 1 1 1 1 1 1 1 ... ##$ NP : Factor w/ 2 levels "barepl ","baresg": 1 1 1 1 1 1 1 1 1 1 ...
## $rating : Ord.factor w/ 4 levels "1"<"2"<"3"<"4": 4 4 4 4 3 4 4 4 4 4 ... ##$ item : chr "3.1xx" "3.2xx" "3.3xx" "3.4xx" ...
## $FreqSim : num 2.6 2.51 1.78 1.68 1.72 ... head(BrP) ## ID survey list label position NP rating item FreqSim ## 1 100 listNP 1 3.1barepl object barepl 4 3.1xx 2.595569 ## 2 100 listNP 1 3.2barepl object barepl 4 3.2xx 2.507130 ## 3 100 listNP 1 3.3barepl object barepl 4 3.3xx 1.782511 ## 4 100 listNP 1 3.4barepl object barepl 4 3.4xx 1.683113 ## 5 101 listNP 1 3.1barepl object barepl 3 3.1xx 1.722588 ## 6 101 listNP 1 3.2barepl object barepl 4 3.2xx 1.396689 str(BrP) ## 'data.frame': 1152 obs. of 9 variables: ##$ ID : int 100 100 100 100 101 101 101 101 102 102 ...
## $survey : Factor w/ 2 levels "listNP","singleNP": 1 1 1 1 1 1 1 1 1 1 ... ##$ list : int 1 1 1 1 1 1 1 1 1 1 ...
## $label : chr "3.1barepl" "3.2barepl" "3.3barepl" "3.4barepl" ... ##$ position: Factor w/ 2 levels "object","subject": 1 1 1 1 1 1 1 1 1 1 ...
## $NP : Factor w/ 2 levels "barepl ","baresg": 1 1 1 1 1 1 1 1 1 1 ... ##$ rating : Ord.factor w/ 4 levels "1"<"2"<"3"<"4": 4 4 4 4 3 4 4 4 4 4 ...
## $item : chr "3.1xx" "3.2xx" "3.3xx" "3.4xx" ... ##$ FreqSim : num 2.6 2.51 1.78 1.68 1.72 ...
Before we get started, we should plot the data and see if we see any patterns. To do so, I will use ggplot2 syntax.
BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating)))) %>% ggplot( aes(x = position, fill = rating)) + geom_bar(position = "fill") + facet_grid(survey~NP) + scale_fill_manual(values = cbPalette) + theme_minimal()
## Initial model
The syntax for an OLS is similar to that of a linear mixed effects model using lme4. I will first plot the data based on the variables we are considering, and then run the model.
BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating))), position = as.factor(position)) %>% ggplot( aes(x = position, fill = rating)) + geom_bar(position = "fill") + scale_fill_manual(values = cbPalette) + theme_minimal()
ols1 = clm(rating~position,data = BrP, link = "logit")
summary(ols1)
## formula: rating ~ position
## data: BrP
##
## logit flexible 1152 -1419.08 2846.17 4(0) 4.77e-07 2.3e+01
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## positionsubject -1.095 0.113 -9.693 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Threshold coefficients:
## Estimate Std. Error z value
## 1|2 -2.41897 0.11131 -21.733
## 2|3 -1.39129 0.09292 -14.973
## 3|4 -0.37825 0.08329 -4.541
What does this mean? We have one variable coefficient, corresponding to the difference between the two positions (similar to the slope estimate in linear models), and three threshold coefficients.
The coefficient, here just for position = subject, takes the $$\beta$$ value in our model specification given above.
We can consider the coefficient similarly to coefficients in linear models. Compared to position = object, the variable position = subject has a log-odds value of -1.095. That means when we calculate the log odds ratios, whenever we are looking at an observation with position = subject, we include the $$\beta$$ as -1.095, and $$x$$ = 1. When we are looking at an observation with position = object, we include the $$\beta$$ as -1.095, and $$x$$ = 0.
In terms of its actual meaning in relationship to the variables, we would say that for a one unit increase in position (i.e., going from 0 to 1, or object to subject), we expect a -1.095 increase (or a 1.095 decrease) in the expected value of rating on the log odds scale, given all of the other variables in the model are held constant. In other words, when going from object to subject, the likelihood of a 4 versus a 1-3 on the rating scale decreases by 1.095 on the log odds scale, the likelihood of a 3 versus a 1-2 on the rating scale decreases by 1.095 on the log odds scale, and the likelihood of a 2 versus a 1 on the rating scale decreases by 1.095 on the log odds scale.
If we want to look at this on the odds scale, we can exponentiate using exp() to look at odds.
exp(coef(ols1))
## 1|2 2|3 3|4 positionsubject
## 0.08901285 0.24875433 0.68505888 0.33439899
Now, when position = subject, the likelihood of a 4 versus a 1-3 on the rating scale is multiplied by 0.2488 (compared to the likelihood of position = object), the likelihood of a 3 versus a 1-2 on the rating scale is multiplied by 0.2488, and the likelihood of a 2 versus a 1 on the rating scale is multiplied by 0.2488.
Now, what about the threshold coefficients? These are the coefficients, again in log odds, for receiving a rating of below J. They can be considered the “cut points” or thresholds between the two variables. So the coefficient reading 1|2 is the likelihood of receiving a 1 rating as opposed to a 2, 3, or 4 rating, the coefficient 2|3 is the likelihood of receiving a 1 or 2 as opposed to a 3 or 4, and the 3|4 coefficient is the likelihood of receiving a 1, 2, or 3, as opposed to a 4.
Why is this relevant? Because we can use the formula given above to calculate the log odds, and therefore the probability by extension, of receiving a certain score or below for each value of the predictor variable. We can also extend this to get the log odds ratio of receiving an exact rating.
First, let’s look at the log odds of receiving a 2 or below for both subject and object positions. The calculation looks like this:
Subject:
$$logit[P(Y \leq 2)] = \alpha_{2|3} - \beta (subject) = -1.39129 - (-1.095 \times 1) = -0.29629$$
Object: $$logit[P(Y \leq 2)] = \alpha_{2|3} - \beta (subject) = -1.39129 - (-1.095 \times 0) = -1.39129$$
If we want to get the probabilities for each of these, we can use the formula given above:
Subject:
$$P(Y \leq 2) = \frac{exp(\alpha_{2|3} - \beta (subject))}{1+exp(\alpha_{2|3} - \beta (subject)} = \frac{exp(-0.29629)}{1+exp(-0.29629)} = \frac{0.2487542}{1+0.2487542} = 0.4264647$$
Object: $$P(Y \leq 2) = \frac{exp(\alpha_{2|3} - \beta (subject))}{1+exp(\alpha_{2|3} - \beta (subject)} = \frac{exp(-1.39129)}{1+exp(-1.39129)} = \frac{1.097374}{2.097374} = 0.1992019$$
Note, you can also do this calculation using the plogis function:
#subject
plogis(-0.29629)
## [1] 0.4264647
#object
plogis(-1.39129)
## [1] 0.1992019
What about the probability of getting a rating of exactly 2? We can calculate this as $$P(Y = 2) = P(Y \leq 2) - P(Y \leq 1)$$.
For subject position, $$P(Y \leq 2) = 0.4264647$$. Using the calculations above, we can see that $$P(Y \leq 1) = plogis(-2.41897 - (-1.095 * 1)) = 0.2101585$$. Therefore, $P(Y = 2) = 0.4264647 - 0.2101585 = 0.2163062 For object position, $$P(Y \leq 2) = 0.1992019$$. Using the calculations above, we can see that $$P(Y \leq 1) = plogis(-2.41897 - (-1.095 * 0)) = 0.08173753$$. Therefore,$P(Y = 2) = 0.1992019 - 0.08173753 = 0.1174644
Finally, what about the probabilities/logits for the rating level 4? Since probability needs to add up to 1, we can take the probabilitiies for 1, 2, and 3, and subtract them from 1. If we would like to convert to logits, we can use the inverse of the equation above… or the qlogis() function.
Note that these results look similar to the proportions seen in the graph (though are not identical, due to the modeling parameters.)
Note that we can calculate all of these probabilties using the ggpredict() function from the ggeffects package, or using predict() in base R. I will continue using the ggpredict() version because it automatically gives confidence intervals.
newdat= data.frame(position = c("subject", "object")) %>% mutate(position = as.factor(position))
ols1predict =cbind(newdat, predict(ols1, newdat, type = "prob")$fit) ols1predict ## position 1 2 3 4 ## 1 subject 0.21022759 0.2163400 0.2454159 0.3280165 ## 2 object 0.08173719 0.1174648 0.2073469 0.5934511 ggpredictions_ols1 = ggpredict(ols1, terms = c("position")) ggpredictions_ols1 ## ## # Predicted probabilities of rating ## # x = position ## ## # Response Level = 1 ## ## x | Predicted | 95% CI ## ---------------------------------- ## object | 0.08 | [0.06, 0.10] ## subject | 0.21 | [0.18, 0.25] ## ## # Response Level = 2 ## ## x | Predicted | 95% CI ## ---------------------------------- ## object | 0.12 | [0.10, 0.14] ## subject | 0.22 | [0.19, 0.25] ## ## # Response Level = 3 ## ## x | Predicted | 95% CI ## ---------------------------------- ## object | 0.21 | [0.18, 0.24] ## subject | 0.25 | [0.22, 0.28] ## ## # Response Level = 4 ## ## x | Predicted | 95% CI ## ---------------------------------- ## object | 0.59 | [0.55, 0.64] ## subject | 0.33 | [0.29, 0.37] #Note that ggpredicts doesn't give the original labels for position - you need to give it the names of the factor labels, which will be in the order of the original model. ggpredictions_ols1$x = factor(ggpredictions_ols1$x) levels(ggpredictions_ols1$x) = c("object", "subject")
colnames(ggpredictions_ols1)[c(1, 5)] = c("Position", "Rating")
ggpredictions_ols1
##
## # Predicted probabilities of rating
## # x = position
##
## Position | Predicted | Rating | 95% CI
## --------------------------------------------
## object | 0.08 | 1 | [0.06, 0.10]
## object | 0.12 | 2 | [0.10, 0.14]
## object | 0.21 | 3 | [0.18, 0.24]
## object | 0.59 | 4 | [0.55, 0.64]
## subject | 0.21 | 1 | [0.18, 0.25]
## subject | 0.22 | 2 | [0.19, 0.25]
## subject | 0.25 | 3 | [0.22, 0.28]
## subject | 0.33 | 4 | [0.29, 0.37]
ggplot(ggpredictions_ols1, aes(x = Rating, y = predicted)) + geom_point(aes(color = Position), position =position_dodge(width = 0.5)) + geom_errorbar(aes(ymin = conf.low, ymax = conf.high, color = Position), position = position_dodge(width = 0.5), width = 0.3) + theme_minimal() + scale_color_manual(values = cbPalette[2:3])
Now, let’s run another model, here with two terms. Can you figure out how to interpret these results?
BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating)))) %>% ggplot( aes(x = position, fill = rating)) + geom_bar(position = "fill") + scale_fill_manual(values = cbPalette) + theme_minimal()+ facet_wrap(~NP)
ols2 = clm(rating~position + NP,data = BrP, link = "logit")
summary(ols2)
## formula: rating ~ position + NP
## data: BrP
##
## logit flexible 1152 -1327.81 2665.63 4(0) 6.61e-07 3.4e+01
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## positionsubject -1.2492 0.1178 -10.60 <2e-16 ***
## NPbaresg -1.5581 0.1195 -13.04 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Threshold coefficients:
## Estimate Std. Error z value
## 1|2 -3.4937 0.1487 -23.49
## 2|3 -2.3564 0.1273 -18.51
## 3|4 -1.2032 0.1115 -10.80
Here, we can extend our model in a similar method we would use for a linear model:
$$logit[P(Y \leq j)] = \alpha_j - \beta_1 x_1 - \beta_2 x_2, j = 1 ... J-1$$
So what would the logit be for position = subject and NP = barepl, for a rating of 3 or less?
$$logit[P(Y \leq 3)] = \alpha_{3|4} - \beta_{subject} x_1 - \beta_{baresg} x_2 = -1.2032 - (-1.2492 \times 1) - (-1.5581 \times 0) = 0.046$$
To get probability, we use plogis():
plogis(0.046)
## [1] 0.511498
What about for position = object and NP = baresg, with rating == 3?
$$logit[P(Y \leq 3)] = \alpha_{3|4} - \beta_{subject} x_1 - \beta_{baresg} x_2 = -1.2032 - (-1.2492 \times 0) - (-1.5581 \times 1) = 0.3549$$
$$logit[P(Y \leq 2)] = \alpha_{2|3} - \beta_{subject} x_1 - \beta_{baresg} x_2 = -2.3564 - (-1.2492 \times 0) - (-1.5581 \times 1) = 0.3549$$
plogis(0.3549) - plogis(-0.7983)
## [1] 0.277416
Once again, we can use the ggpredict() function to get all probabilities:
ggpredictions_ols2 = data.frame(ggpredict(ols2, terms = c("position", "NP")))
ggpredictions_ols2
## x predicted conf.low conf.high response.level group
## 1 object 0.02949259 0.02130969 0.04068710 1 barepl
## 2 object 0.05706886 0.04364658 0.07429811 2 barepl
## 3 object 0.14433638 0.11885852 0.17419586 3 barepl
## 4 object 0.76910217 0.72180702 0.81046804 4 barepl
## 5 object 0.12613119 0.09970982 0.15832255 1 baresg
## 6 object 0.18426038 0.15364252 0.21939808 2 baresg
## 7 object 0.27739525 0.24472316 0.31262378 3 baresg
## 8 object 0.41221318 0.35853232 0.46806668 4 baresg
## 9 subject 0.09582361 0.07464488 0.12221766 1 barepl
## 10 subject 0.15257022 0.12539854 0.18438826 2 barepl
## 11 subject 0.26308316 0.23064212 0.29831791 3 barepl
## 12 subject 0.48852301 0.43302368 0.54430675 4 barepl
## 13 subject 0.33482478 0.28420004 0.38955957 1 baresg
## 14 subject 0.27602253 0.23849848 0.31699280 2 baresg
## 15 subject 0.22172775 0.19099760 0.25583859 3 baresg
## 16 subject 0.16742494 0.13554617 0.20502302 4 baresg
ggpredictions_ols2$x = factor(ggpredictions_ols2$x)
levels(ggpredictions_ols2$x) = c("object", "subject") colnames(ggpredictions_ols2)[c(1, 5,6)] =c("Position", "Rating", "NP") ggpredictions_ols2 ## Position predicted conf.low conf.high Rating NP ## 1 object 0.02949259 0.02130969 0.04068710 1 barepl ## 2 object 0.05706886 0.04364658 0.07429811 2 barepl ## 3 object 0.14433638 0.11885852 0.17419586 3 barepl ## 4 object 0.76910217 0.72180702 0.81046804 4 barepl ## 5 object 0.12613119 0.09970982 0.15832255 1 baresg ## 6 object 0.18426038 0.15364252 0.21939808 2 baresg ## 7 object 0.27739525 0.24472316 0.31262378 3 baresg ## 8 object 0.41221318 0.35853232 0.46806668 4 baresg ## 9 subject 0.09582361 0.07464488 0.12221766 1 barepl ## 10 subject 0.15257022 0.12539854 0.18438826 2 barepl ## 11 subject 0.26308316 0.23064212 0.29831791 3 barepl ## 12 subject 0.48852301 0.43302368 0.54430675 4 barepl ## 13 subject 0.33482478 0.28420004 0.38955957 1 baresg ## 14 subject 0.27602253 0.23849848 0.31699280 2 baresg ## 15 subject 0.22172775 0.19099760 0.25583859 3 baresg ## 16 subject 0.16742494 0.13554617 0.20502302 4 baresg ggplot(ggpredictions_ols2, aes(x = Rating, y = predicted)) + geom_point(aes(color = Position), position =position_dodge(width = 0.5)) + geom_errorbar(aes(ymin = conf.low, ymax = conf.high, color = Position), position = position_dodge(width = 0.5), width = 0.3) + theme_minimal() +facet_wrap(~NP) + scale_color_manual(values = cbPalette[2:3]) ## Including a continuous predictor It is also possible to include a continuous predictor in a model. Here, I will include the FreqSim variable I simulated. BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating)))) %>% ggplot( aes(x = FreqSim, fill = rating)) + geom_histogram(binwidth = 0.1) + scale_fill_manual(values = cbPalette) + theme_minimal() BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating)))) %>% ggplot( aes(x = cut(FreqSim, 5), fill = rating)) + geom_bar(position = "fill") + scale_fill_manual(values = cbPalette) + theme_minimal() Once again you can include the continuous predictor in the model with the same syntax as for a linear model. ols3 = clm(rating~position + FreqSim, data = BrP) summary(ols3) ## formula: rating ~ position + FreqSim ## data: BrP ## ## link threshold nobs logLik AIC niter max.grad cond.H ## logit flexible 1152 -1419.02 2848.03 4(0) 4.77e-07 2.3e+02 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## positionsubject -1.09443 0.11304 -9.682 <2e-16 *** ## FreqSim -0.03531 0.09566 -0.369 0.712 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Threshold coefficients: ## Estimate Std. Error z value ## 1|2 -2.4896 0.2215 -11.240 ## 2|3 -1.4617 0.2123 -6.884 ## 3|4 -0.4485 0.2078 -2.158 Here, our result for FreqSim is not significant, which is not surprising since this is simulated data. However, we can still use this to show the how to interpret the results of the model. In this case, we say that for a one unit increase in FreqSim, we expect a -0.03531 increase (or a 0.03531 decrease) in the expected value of rating on the log odds scale, given all of the other variables in the model are held constant. In other words, the likelihood of a 4 versus a 1-3 on the rating scale decreases by 0.03531 on the log odds scale, the likelihood of a 3 versus a 1-2 on the rating scale decreases by 0.03531 on the log odds scale, and the likelihood of a 2 versus a 1 on the rating scale decreases by 0.03531 on the log odds scale. If we wanted to calculate the probability of getting a rating value of 1 for a word with FreqSim = 2 and position = subject, we use our formula: $$logit[P(Y \leq 1)] = \alpha_{1|2} - \beta_{subject} x_1 - \beta_{FreqSim} x_2 = -2.4896 - (-1.09443 \times 1) - (-0.03531 \times 3) = -1.28924$$ plogis(-1.28924) ## [1] 0.2159815 Once again, the ggpredict() function can be used to predict values for all combinations of the predictor variable. ggpredictions_ols3 = data.frame(ggpredict(ols3, terms = c("position", "FreqSim [1, 1.5, 2, 2.5, 3]"))) ggpredictions_ols3 ## x predicted conf.low conf.high response.level group ## 1 object 0.07912849 0.05822380 0.1066885 1 1 ## 2 object 0.11453662 0.08946980 0.1455041 2 1 ## 3 object 0.20448986 0.17380786 0.2390212 3 1 ## 4 object 0.60184503 0.53256920 0.6672683 4 1 ## 5 object 0.08042470 0.06248520 0.1029490 1 1.5 ## 6 object 0.11601261 0.09451516 0.1416348 2 1.5 ## 7 object 0.20595625 0.17860685 0.2362888 3 1.5 ## 8 object 0.59760644 0.54519364 0.6478802 4 1.5 ## 9 object 0.08174025 0.06486185 0.1025292 1 2 ## 10 object 0.11749914 0.09724485 0.1413117 2 2 ## 11 object 0.20740739 0.18112912 0.2363975 3 2 ## 12 object 0.59335322 0.54763988 0.6375048 4 2 ## 13 object 0.08307538 0.06460484 0.1062269 1 2.5 ## 14 object 0.11899600 0.09696862 0.1452224 2 2.5 ## 15 object 0.20884266 0.18119998 0.2394690 3 2.5 ## 16 object 0.58908596 0.53589732 0.6402707 4 2.5 ## 17 object 0.08443031 0.06224939 0.1135577 1 3 ## 18 object 0.12050302 0.09432392 0.1527228 2 3 ## 19 object 0.21026142 0.17931894 0.2449496 3 3 ## 20 object 0.58480525 0.51415248 0.6521350 4 3 ## 21 subject 0.20426926 0.15921706 0.2581553 1 1 ## 22 subject 0.21349866 0.17933561 0.2521700 2 1 ## 23 subject 0.24625350 0.21706275 0.2779761 3 1 ## 24 subject 0.33597858 0.27531230 0.4025866 4 1 ## 25 subject 0.20715426 0.17072961 0.2490165 1 1.5 ## 26 subject 0.21491462 0.18353715 0.2500137 2 1.5 ## 27 subject 0.24588026 0.21692545 0.2773312 3 1.5 ## 28 subject 0.33205086 0.28588557 0.3816865 4 1.5 ## 29 subject 0.21006924 0.17737136 0.2469851 1 2 ## 30 subject 0.21631244 0.18584104 0.2502444 2 2 ## 31 subject 0.24547195 0.21664845 0.2767753 3 2 ## 32 subject 0.32814637 0.28850006 0.3704046 4 2 ## 33 subject 0.21301423 0.17658792 0.2546309 1 2.5 ## 34 subject 0.21769149 0.18619078 0.2528657 2 2.5 ## 35 subject 0.24502879 0.21613852 0.2764189 3 2.5 ## 36 subject 0.32426550 0.27944156 0.3725619 4 2.5 ## 37 subject 0.21598921 0.17011998 0.2701993 1 3 ## 38 subject 0.21905113 0.18492454 0.2574860 2 3 ## 39 subject 0.24455102 0.21529237 0.2763855 3 3 ## 40 subject 0.32040864 0.26247998 0.3844575 4 3 ggpredictions_ols3$x = factor(ggpredictions_ols3$x) levels(ggpredictions_ols3$x) = c("object", "subject")
colnames(ggpredictions_ols3)[c(1, 5,6)] =c("Position", "Rating", "FreqSim")
ggpredictions_ols3$Rating = as.factor(ggpredictions_ols3$Rating)
ggplot(ggpredictions_ols3, aes(x = FreqSim, y = predicted)) + geom_smooth(aes(color = Rating, group = Rating), se = FALSE)+ theme_minimal() +facet_wrap(~Position) + scale_color_manual(values = cbPalette)
## geom_smooth() using method = 'loess' and formula 'y ~ x'
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : span too small. fewer data values than degrees of freedom.
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.98
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.02
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0804
As we can see here, FreqSim really doesn’t add much. This isn’t totally surprising. A dataset with a meaningful research question behind it will possibly show more significant results.
## Interactions
Including an interaction, like with a linear model, will require a theoretical motivation. I will include here a two-way interaction in order to show how the model can be interpreted with an interaction. The syntax is the same as with a linear model.
ols4 = clm(rating~position * NP,data = BrP, link = "logit")
summary(ols4)
## formula: rating ~ position * NP
## data: BrP
##
## logit flexible 1152 -1325.63 2663.27 4(0) 3.91e-07 8.9e+01
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## positionsubject -1.5325 0.1826 -8.392 <2e-16 ***
## NPbaresg -1.8399 0.1833 -10.038 <2e-16 ***
## positionsubject:NPbaresg 0.4917 0.2367 2.077 0.0378 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Threshold coefficients:
## Estimate Std. Error z value
## 1|2 -3.6605 0.1725 -21.218
## 2|3 -2.5350 0.1580 -16.049
## 3|4 -1.3834 0.1461 -9.469
Now, we have an extra term in our model coefficients, being the interaction term. This term is significant, indicating that there is a modulating effect between the position and NP terms.
Now, our model looks like this:
$$logit[P(Y \leq j)] = \alpha_j - \beta_1 x_1 - \beta_2 x_2 - \beta_3 x_1x_2, j = 1 ... J-1$$
I will do the calculation for a rating of exactly 2, for a subject/baresg word:
$$logit[P(Y \leq 2)] = \alpha_{2|3} - \beta_{subject} x_1 - \beta_{baresg} x_2 - \beta_{subject:baresg} x_1x_2 = -2.5350 - (-1.5325 \times 1) - (-1.8399 \times 1) - (0.4917 \times 1 \times 1) = 0.3457$$
$$logit[P(Y \leq 1)] = \alpha_{1|2} - \beta_{subject} x_1 - \beta_{baresg} x_2 - \beta_{subject:baresg} x_1x_2 = -3.6605 - (-1.5325 \times 1) - (-1.8399 \times 1) - (0.4917 \times 1 \times 1) = 0.3457$$
plogis(0.3457)
## [1] 0.5855745
plogis(-0.7798)
## [1] 0.314363
plogis(0.3457) - plogis(-0.7798)
## [1] 0.2712115
Once again, we can plot the results:
ggpredictions_ols4 = data.frame(ggpredict(ols4, terms = c("position", "NP")))
ggpredictions_ols4
## x predicted conf.low conf.high response.level group
## 1 object 0.02507574 0.01717219 0.03648189 1 barepl
## 2 object 0.04836596 0.03455575 0.06731063 2 barepl
## 3 object 0.12702267 0.09819175 0.16279077 3 barepl
## 4 object 0.79953562 0.74191333 0.84694677 4 barepl
## 5 object 0.13936432 0.10809666 0.17787264 1 baresg
## 6 object 0.19353390 0.16103018 0.23079362 2 baresg
## 7 object 0.27928169 0.24672221 0.31434492 3 baresg
## 8 object 0.38782009 0.33007465 0.44889857 4 baresg
## 9 subject 0.10640447 0.08125202 0.13817215 1 barepl
## 10 subject 0.16203947 0.13256674 0.19657975 2 barepl
## 11 subject 0.26875191 0.23601925 0.30421648 3 barepl
## 12 subject 0.46280415 0.40195917 0.52477742 4 barepl
## 13 subject 0.31434659 0.26162291 0.37233754 1 baresg
## 14 subject 0.27120307 0.23388011 0.31205606 2 baresg
## 15 subject 0.23160509 0.19919887 0.26752207 3 baresg
## 16 subject 0.18284525 0.14560262 0.22708207 4 baresg
ggpredictions_ols4$x = factor(ggpredictions_ols4$x)
levels(ggpredictions_ols4$x) = c("object", "subject") colnames(ggpredictions_ols4)[c(1, 5,6)] =c("Position", "Rating", "NP") ggpredictions_ols4 ## Position predicted conf.low conf.high Rating NP ## 1 object 0.02507574 0.01717219 0.03648189 1 barepl ## 2 object 0.04836596 0.03455575 0.06731063 2 barepl ## 3 object 0.12702267 0.09819175 0.16279077 3 barepl ## 4 object 0.79953562 0.74191333 0.84694677 4 barepl ## 5 object 0.13936432 0.10809666 0.17787264 1 baresg ## 6 object 0.19353390 0.16103018 0.23079362 2 baresg ## 7 object 0.27928169 0.24672221 0.31434492 3 baresg ## 8 object 0.38782009 0.33007465 0.44889857 4 baresg ## 9 subject 0.10640447 0.08125202 0.13817215 1 barepl ## 10 subject 0.16203947 0.13256674 0.19657975 2 barepl ## 11 subject 0.26875191 0.23601925 0.30421648 3 barepl ## 12 subject 0.46280415 0.40195917 0.52477742 4 barepl ## 13 subject 0.31434659 0.26162291 0.37233754 1 baresg ## 14 subject 0.27120307 0.23388011 0.31205606 2 baresg ## 15 subject 0.23160509 0.19919887 0.26752207 3 baresg ## 16 subject 0.18284525 0.14560262 0.22708207 4 baresg ggplot(ggpredictions_ols4, aes(x = Rating, y = predicted)) + geom_point(aes(color = Position), position =position_dodge(width = 0.5)) + geom_errorbar(aes(ymin = conf.low, ymax = conf.high, color = Position), position = position_dodge(width = 0.5), width = 0.3) + theme_minimal() +facet_wrap(~NP) + scale_color_manual(values = cbPalette[2:3]) ## Random effects As I mentioned before, the ordinal package has an advantage over MASS, in that it has the ability to include random effects. The syntax is similar to that of lme4’s lmer() function. Note that the function being used here is clmm() - the extra m stands for mixed. ols5 = clmm(rating~position*NP + (1|ID), data = BrP) summary(ols5) ## Cumulative Link Mixed Model fitted with the Laplace approximation ## ## formula: rating ~ position * NP + (1 | ID) ## data: BrP ## ## link threshold nobs logLik AIC niter max.grad cond.H ## logit flexible 1152 -1243.45 2500.90 442(1697) 1.50e-03 8.6e+01 ## ## Random effects: ## Groups Name Variance Std.Dev. ## ID (Intercept) 1.265 1.125 ## Number of groups: ID 72 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## positionsubject -1.8757 0.1996 -9.397 <2e-16 *** ## NPbaresg -2.2848 0.2022 -11.300 <2e-16 *** ## positionsubject:NPbaresg 0.6422 0.2510 2.559 0.0105 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Threshold coefficients: ## Estimate Std. Error z value ## 1|2 -4.4650 0.2457 -18.170 ## 2|3 -3.0915 0.2258 -13.694 ## 3|4 -1.6945 0.2097 -8.081 Now we have the same information as before, plus a little more. At the top of the summary, there is the information regarding the random effects - specifically the ID of the participant. The coefficients have changed slightly , though we have the same significance levels. Note that we have an extra column in our prediction matrix, with the standard error. ggpredictions_ols5 = data.frame(ggpredict(ols5, terms = c("position", "NP"), type = "fe")) ## Loading required namespace: emmeans ggpredictions_ols5 ## x predicted std.error conf.low conf.high response.level group ## 1 object 0.01137409 0.002763256 0.005958204 0.01678997 1 barepl ## 2 object 0.03208654 0.006904418 0.018554125 0.04561894 2 barepl ## 3 object 0.11172434 0.019023724 0.074438530 0.14901016 3 barepl ## 4 object 0.84481504 0.027491683 0.790932327 0.89869775 4 barepl ## 5 object 0.10154649 0.017748501 0.066760071 0.13633292 1 baresg ## 6 object 0.20705908 0.024420592 0.159195599 0.25492256 2 baresg ## 7 object 0.33483305 0.018034067 0.299486926 0.37017917 3 baresg ## 8 object 0.35656138 0.041163010 0.275883361 0.43723940 4 baresg ## 9 subject 0.06983095 0.012987834 0.044375260 0.09528663 1 barepl ## 10 subject 0.15884947 0.021620967 0.116473149 0.20122578 2 barepl ## 11 subject 0.31649341 0.020664737 0.275991269 0.35699555 3 barepl ## 12 subject 0.45482618 0.044261182 0.368075856 0.54157650 4 barepl ## 13 subject 0.27956441 0.036329195 0.208360493 0.35076832 1 baresg ## 14 subject 0.32556600 0.021210648 0.283993896 0.36713811 2 baresg ## 15 subject 0.25589766 0.024266581 0.208336037 0.30345929 3 baresg ## 16 subject 0.13897193 0.022345994 0.095174586 0.18276927 4 baresg ggpredictions_ols5$x = factor(ggpredictions_ols5$x) levels(ggpredictions_ols5$x) = c("object", "subject")
colnames(ggpredictions_ols5)[c(1, 6,7)] =c("Position", "Rating", "NP")
ggpredictions_ols5
## Position predicted std.error conf.low conf.high Rating NP
## 1 object 0.01137409 0.002763256 0.005958204 0.01678997 1 barepl
## 2 object 0.03208654 0.006904418 0.018554125 0.04561894 2 barepl
## 3 object 0.11172434 0.019023724 0.074438530 0.14901016 3 barepl
## 4 object 0.84481504 0.027491683 0.790932327 0.89869775 4 barepl
## 5 object 0.10154649 0.017748501 0.066760071 0.13633292 1 baresg
## 6 object 0.20705908 0.024420592 0.159195599 0.25492256 2 baresg
## 7 object 0.33483305 0.018034067 0.299486926 0.37017917 3 baresg
## 8 object 0.35656138 0.041163010 0.275883361 0.43723940 4 baresg
## 9 subject 0.06983095 0.012987834 0.044375260 0.09528663 1 barepl
## 10 subject 0.15884947 0.021620967 0.116473149 0.20122578 2 barepl
## 11 subject 0.31649341 0.020664737 0.275991269 0.35699555 3 barepl
## 12 subject 0.45482618 0.044261182 0.368075856 0.54157650 4 barepl
## 13 subject 0.27956441 0.036329195 0.208360493 0.35076832 1 baresg
## 14 subject 0.32556600 0.021210648 0.283993896 0.36713811 2 baresg
## 15 subject 0.25589766 0.024266581 0.208336037 0.30345929 3 baresg
## 16 subject 0.13897193 0.022345994 0.095174586 0.18276927 4 baresg
ggplot(ggpredictions_ols5, aes(x = Rating, y = predicted)) + geom_point(aes(color = Position), position =position_dodge(width = 0.5)) + geom_errorbar(aes(ymin = conf.low, ymax = conf.high, color = Position), position = position_dodge(width = 0.5), width = 0.3) + theme_minimal() +facet_wrap(~NP) + scale_color_manual(values = cbPalette[2:3])
## The full model
Following Dr. Ionin’s paper from which this data comes, let’s take a look at a full model. Note that this took me about 15 seconds to run on my computer.
#raw data
BrP %>% mutate(rating = ordered(rating, levels=rev(levels(rating)))) %>% ggplot( aes(x = position, fill = rating)) + geom_bar(position = "fill") + facet_grid(survey~NP) + scale_fill_manual(values = cbPalette) + theme_minimal() + ggtitle("Raw Data")
ols6 = clmm(rating~position*NP*survey + (1|ID) + (1|item), data = BrP)
summary(ols6)
## Cumulative Link Mixed Model fitted with the Laplace approximation
##
## formula: rating ~ position * NP * survey + (1 | ID) + (1 | item)
## data: BrP
##
## logit flexible 1152 -1181.20 2386.40 979(3593) 4.39e-04 3.6e+02
##
## Random effects:
## Groups Name Variance Std.Dev.
## ID (Intercept) 1.0032 1.0016
## item (Intercept) 0.5767 0.7594
## Number of groups: ID 72, item 24
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## positionsubject -2.6630 0.4504 -5.912 3.38e-09
## NPbaresg -2.9280 0.3255 -8.996 < 2e-16
## surveysingleNP -1.9855 0.4270 -4.650 3.32e-06
## positionsubject:NPbaresg 1.6000 0.3812 4.197 2.70e-05
## positionsubject:surveysingleNP 1.0033 0.4183 2.399 0.016462
## NPbaresg:surveysingleNP 0.6596 0.4172 1.581 0.113873
## positionsubject:NPbaresg:surveysingleNP -1.8573 0.5332 -3.483 0.000495
##
## positionsubject ***
## NPbaresg ***
## surveysingleNP ***
## positionsubject:NPbaresg ***
## positionsubject:surveysingleNP *
## NPbaresg:surveysingleNP
## positionsubject:NPbaresg:surveysingleNP ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Threshold coefficients:
## Estimate Std. Error z value
## 1|2 -5.9204 0.4285 -13.815
## 2|3 -4.3697 0.4105 -10.645
## 3|4 -2.8239 0.3945 -7.158
ggpredictions_ols6 = data.frame(ggpredict(ols6, terms = c("position", "NP", "survey"), type = "fe"))
ggpredictions_ols6
## x predicted std.error conf.low conf.high response.level
## 1 object 0.002676824 0.001144066 0.0004344957 0.004919153 1
## 2 object 0.009819713 0.003966276 0.0020459550 0.017593472 2
## 3 object 0.043548519 0.015966933 0.0122539060 0.074843133 3
## 4 object 0.943954943 0.020871473 0.9030476070 0.984862279 4
## 5 object 0.019172061 0.007138515 0.0051808296 0.033163293 1
## 6 object 0.065211294 0.021399504 0.0232690372 0.107153552 2
## 7 object 0.217483944 0.048145526 0.1231204464 0.311847442 3
## 8 object 0.698132700 0.074938724 0.5512554992 0.845009901 4
## 9 object 0.047769286 0.014953506 0.0184609530 0.077077618 1
## 10 object 0.143511540 0.034894762 0.0751190622 0.211904018 2
## 11 object 0.334719905 0.033750949 0.2685692596 0.400870550 3
## 12 object 0.473999270 0.077505563 0.3220911576 0.625907382 4
## 13 object 0.158884489 0.045998360 0.0687293595 0.249039618 1
## 14 object 0.312188810 0.042784764 0.2283322140 0.396045405 2
## 15 object 0.335823305 0.035862221 0.2655346435 0.406111966 3
## 16 object 0.193103397 0.053080916 0.0890667138 0.297140080 4
## 17 subject 0.037061496 0.011898905 0.0137400702 0.060382921 1
## 18 subject 0.116531310 0.030321648 0.0571019717 0.175960647 2
## 19 subject 0.306271318 0.039653973 0.2285509581 0.383991677 3
## 20 subject 0.540135877 0.077606340 0.3880302456 0.692241509 4
## 21 subject 0.093196213 0.029333044 0.0357045026 0.150687924 1
## 22 subject 0.233205815 0.047014085 0.1410599022 0.325351729 2
## 23 subject 0.368110683 0.019593769 0.3297076023 0.406513764 3
## 24 subject 0.305487288 0.071595621 0.1651624504 0.445812126 4
## 25 subject 0.126822969 0.034951905 0.0583184949 0.195327443 1
## 26 subject 0.279634215 0.042680589 0.1959817979 0.363286631 2
## 27 subject 0.356176848 0.025948068 0.3053195688 0.407034127 3
## 28 subject 0.237365968 0.056150724 0.1273125719 0.347419365 4
## 29 subject 0.562301081 0.084786699 0.3961222049 0.728479958 1
## 30 subject 0.295996488 0.045751098 0.2063259838 0.385666992 2
## 31 subject 0.107709792 0.031408243 0.0461507667 0.169268818 3
## 32 subject 0.033992639 0.011888492 0.0106916222 0.057293655 4
## group facet
## 1 barepl listNP
## 2 barepl listNP
## 3 barepl listNP
## 4 barepl listNP
## 5 barepl singleNP
## 6 barepl singleNP
## 7 barepl singleNP
## 8 barepl singleNP
## 9 baresg listNP
## 10 baresg listNP
## 11 baresg listNP
## 12 baresg listNP
## 13 baresg singleNP
## 14 baresg singleNP
## 15 baresg singleNP
## 16 baresg singleNP
## 17 barepl listNP
## 18 barepl listNP
## 19 barepl listNP
## 20 barepl listNP
## 21 barepl singleNP
## 22 barepl singleNP
## 23 barepl singleNP
## 24 barepl singleNP
## 25 baresg listNP
## 26 baresg listNP
## 27 baresg listNP
## 28 baresg listNP
## 29 baresg singleNP
## 30 baresg singleNP
## 31 baresg singleNP
## 32 baresg singleNP
ggpredictions_ols6$x = factor(ggpredictions_ols6$x)
levels(ggpredictions_ols6\$x) = c("object", "subject")
colnames(ggpredictions_ols6)[c(1, 6,7, 8)] =c("Position", "Rating", "NP", "survey")
ggpredictions_ols6
## Position predicted std.error conf.low conf.high Rating NP
## 1 object 0.002676824 0.001144066 0.0004344957 0.004919153 1 barepl
## 2 object 0.009819713 0.003966276 0.0020459550 0.017593472 2 barepl
## 3 object 0.043548519 0.015966933 0.0122539060 0.074843133 3 barepl
## 4 object 0.943954943 0.020871473 0.9030476070 0.984862279 4 barepl
## 5 object 0.019172061 0.007138515 0.0051808296 0.033163293 1 barepl
## 6 object 0.065211294 0.021399504 0.0232690372 0.107153552 2 barepl
## 7 object 0.217483944 0.048145526 0.1231204464 0.311847442 3 barepl
## 8 object 0.698132700 0.074938724 0.5512554992 0.845009901 4 barepl
## 9 object 0.047769286 0.014953506 0.0184609530 0.077077618 1 baresg
## 10 object 0.143511540 0.034894762 0.0751190622 0.211904018 2 baresg
## 11 object 0.334719905 0.033750949 0.2685692596 0.400870550 3 baresg
## 12 object 0.473999270 0.077505563 0.3220911576 0.625907382 4 baresg
## 13 object 0.158884489 0.045998360 0.0687293595 0.249039618 1 baresg
## 14 object 0.312188810 0.042784764 0.2283322140 0.396045405 2 baresg
## 15 object 0.335823305 0.035862221 0.2655346435 0.406111966 3 baresg
## 16 object 0.193103397 0.053080916 0.0890667138 0.297140080 4 baresg
## 17 subject 0.037061496 0.011898905 0.0137400702 0.060382921 1 barepl
## 18 subject 0.116531310 0.030321648 0.0571019717 0.175960647 2 barepl
## 19 subject 0.306271318 0.039653973 0.2285509581 0.383991677 3 barepl
## 20 subject 0.540135877 0.077606340 0.3880302456 0.692241509 4 barepl
## 21 subject 0.093196213 0.029333044 0.0357045026 0.150687924 1 barepl
## 22 subject 0.233205815 0.047014085 0.1410599022 0.325351729 2 barepl
## 23 subject 0.368110683 0.019593769 0.3297076023 0.406513764 3 barepl
## 24 subject 0.305487288 0.071595621 0.1651624504 0.445812126 4 barepl
## 25 subject 0.126822969 0.034951905 0.0583184949 0.195327443 1 baresg
## 26 subject 0.279634215 0.042680589 0.1959817979 0.363286631 2 baresg
## 27 subject 0.356176848 0.025948068 0.3053195688 0.407034127 3 baresg
## 28 subject 0.237365968 0.056150724 0.1273125719 0.347419365 4 baresg
## 29 subject 0.562301081 0.084786699 0.3961222049 0.728479958 1 baresg
## 30 subject 0.295996488 0.045751098 0.2063259838 0.385666992 2 baresg
## 31 subject 0.107709792 0.031408243 0.0461507667 0.169268818 3 baresg
## 32 subject 0.033992639 0.011888492 0.0106916222 0.057293655 4 baresg
## survey
## 1 listNP
## 2 listNP
## 3 listNP
## 4 listNP
## 5 singleNP
## 6 singleNP
## 7 singleNP
## 8 singleNP
## 9 listNP
## 10 listNP
## 11 listNP
## 12 listNP
## 13 singleNP
## 14 singleNP
## 15 singleNP
## 16 singleNP
## 17 listNP
## 18 listNP
## 19 listNP
## 20 listNP
## 21 singleNP
## 22 singleNP
## 23 singleNP
## 24 singleNP
## 25 listNP
## 26 listNP
## 27 listNP
## 28 listNP
## 29 singleNP
## 30 singleNP
## 31 singleNP
## 32 singleNP
ggplot(ggpredictions_ols6, aes(x = Rating, y = predicted)) + geom_point(aes(color = Position), position =position_dodge(width = 0.5)) + geom_errorbar(aes(ymin = conf.low, ymax = conf.high, color = Position), position = position_dodge(width = 0.5), width = 0.3) + theme_minimal() +facet_grid(survey~NP) + scale_color_manual(values = cbPalette[2:3])+ ggtitle("Probabilities of responses, full model")
ggplot(ggpredictions_ols6, aes(x = Position, y = predicted, fill = Rating)) + geom_bar(position = "dodge", stat = "identity") + facet_grid(survey~NP) + scale_fill_manual(values = cbPalette) + theme_minimal()+ ggtitle("Probabilities of responses, full model")
ggpredictions_ols6 %>% mutate(Rating = ordered(Rating, levels=rev(levels(Rating)))) %>% ggplot( aes(x = Position, y = predicted, fill = Rating)) + geom_bar(position = "fill", stat = "identity") + facet_grid(survey~NP) + scale_fill_manual(values = cbPalette) + theme_minimal() + ggtitle("Probabilities of responses, full model (reversed color scheme)")
ggplot(ggpredictions_ols6, aes(x = Position, y = predicted, fill = Rating)) + geom_bar(position = "fill", stat = "identity") + facet_grid(survey~NP) + scale_fill_manual(values = cbPalette) + theme_minimal() + ggtitle("Probabilities of responses, full model")
Questions?
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# heavytailed
## November 17, 2014
### The Libor Market Model
Filed under: Uncategorized — Tags: , , , — heavytailed @ 9:15 pm
Among statisticians (particularly biostatisticians), hearing ‘LMM’ immediately triggers “Linear Mixed Model” and scary thoughts about random effects. Among quant circles the association is completely different: LMM means the Libor Market Model, the most commonly used model of interest rates among the squalor of the fixed-income desks.
## What is LIBOR?
Does it help if I tell you the acronym stands for London Inter-Bank Offer Rate? No? Simply put, a selection of 18 banks tell the Intercontinental Exchange what they think a fair price is if they were to borrow money.
# The Model
So after all that time describing LIBOR, it’s time to reveal that The LIBOR Market Model is a complete misnomer. The model is, itself, a simple model of forward contracts. In fact, it should really be called the NIRMM: Nominal Interest Rate Market Model, as any effective interest rate can be modeled so long as forwards contracts can be observed.
## Forward Rates
A forward contract is an up-front agreement for a future loan. No bank would do this in real life, but pretend you know you’re going to buy a house in three years. You walk into Local Municipal Bank and say “Three years from now, I’m going to get a €300,000, 5-year loan from you, let’s lock in a rate.” The bank says “OK we’ll give you 2.5% at that time.” Boom. Forward rate is 0.025 [1] In the actual market, there are many different forward rates, corresponding to different loan durations (“tenors”) and ending dates (“maturities”). Each of these contracts, naturally, has its own rate with which it is associated.
Forward rates don’t have a simple floating value based merely on supply-and-demand (like, say, onions): they face a significant no-arbitrage constraint: If I borrow €300,000 for 3 years, and also hash out a €300,000 forward contract which begins in 3 years and lasts for 5 years, then the total rate I pay ought to be the same as if I’d simply borrowed €300,000 for 8 years. Mathematically if $P(a,b)$ is a bond from time a to time b, and $F(t, a, b)$ is a forward contract at time t from time a to time b: $P(t, T_1 + T_2) = P(t, T_1)F(t, T_1, T_2)$.
If this equality fails to hold, and $P(t, T_1 + T_2) > P(t, T_1)F(t, T_1, T_2)$ then I can make money by borrowing money using the right-side structure (borrow from now to $T_1$, and engage a future contract from $T_1$ to $T_2$), and lending money using the left-side structure. Similarly, if $P(t, T_1 + T_2) < P(t, T_1)F(t, T_1, T_2)$ then I can make money by borrowing money using the left-side structure, and lending using the right-side structure.
If the equality above holds, then the forward contract rate has an easy definition in terms of zero-coupon bonds $P: \;\; F(t, T_1, T_2) = \frac{P(t, T_1+T_2)}{P(t, T_1)}$. The simply compounded interest rate $L(t, T_1, T_2)$ is defined by
$\frac{1}{1 + (T_2 - T_1)L(t, T_1, T_2)} = F(t, T_1, T_2)$
This is a bit of a silly definition as written; let’s parse this out:
$\frac{1}{1 + \delta L_i} = \frac{P_{i+1}}{P_i} \Rightarrow 1 + \delta L_i = \frac{P_i}{P_{i+1}} \Rightarrow L_i = \frac{P_{i} - P_{i+1}}{\delta P_{i+1}}$
This gives the nifty equation
$P_{i+1}(1 + \delta L_i) = P_i$
So we can calculate the market Libors from prices, spots, or forwards (e.g. using the market prices for each, without needing to convert). The Libor Market Model is a model for $L$, which models the way in which forward prices responds to stochastic noise. It has the form:
$dL = \mathbf{\Xi}Ldt + L \odot \Omega \; dW$
where $L$ is a vector, $\mathbf{\Xi}$, $\Omega$ are matrices; all of these are functions of time. While $\Omega(t)$ is an exogenous model of volatility, the no-arbitrage constraint will enable us to define the matrix $\; \mathbf{\Xi}$.
Consider a simple case where $\; \mathbf{\Xi} = 0$ and $\Omega = I$. Then each component of $L$ evolves as independent geometric brownian motion, and so
$L_i(t) = L_i(0)\exp\left(-\frac{1}{2}t + W_t^i\right)$ and $\mathbb{E}[L_i(t)] = L_i(0)e^{-\frac{1}{2}t}\mathbb{E}[\exp(W_t^i)] = L_i(0)e^{-\frac{1}{2}t + \frac{1}{2}t} = L_i(0)$
There’s a problem here: no-arbitrage is not enforced. If $L_i(0)$ is not in an arbitrage-free state, or if $dW$ ever knocks the system out of an arbitrage-free position (which happens almost surely), then the relationship between forwards and spots (e.g. the $L_i$ and $L_j$) evolves with no regard to market forces.
So how do we enforce our sample paths to be approximately zero-arbitrage? It turns out that simply baking this assumption in suffices to do so; that is: the assumption of geometric brownian motion under the risk-neutral measure is more powerful than one might initially think.
It’s important to note that the “risk-neutral measure” is one of those terms of art that has only a practical meaning. It almost always means whatever measure makes my thing a martingale. Consider that the definition of $L_i(t) = L(t, T_i, T_{i+1})$ gives
$L_i(t)P_{i+1}(t) = \frac{1}{T_{i+1}-T_i}(P_i(t) - P_{i+1}(t)) \Rightarrow L_i(t) = \frac{1}{\delta}\left(\frac{P_i(t)}{P_{i+1}(t)} - 1\right)$
Now we want to differentiate. Using the derivative from deterministic calculus will get us into trouble; so we need to use a stochastic derivative, and in particular the stochastic division rule.
Ito’s Lemma and the Stochastic Quotient
Stochastic calculus anywhere outside of engineering is overburdened with theory, acronyms, and poor notation. Most approaches to teaching it get far too technical far too quickly. Seriously, go look at Wikipedia and try not to get completely lost. Google around for some notes and see if you fare any better.
In my mind, the entirety of brownian stochastic calculus is summed up as follows:
$dW^2 \sim dt$
This is intuitive in the following sense: if $X(0) = 0$ and $dX = \sigma dW$ we have that $X(t) \sim N(0, \sigma t)$. Of course then $\frac{X(t)}{\sqrt{\sigma dt}}^2 \sim \chi^2(1)$. In an infinitesimal step $X = 0 + dW = dW$ and so $\sigma dt \chi^2(1) \sim X^2 \approx dW^2$. Then we make the approximation $dt \chi^2(1) \approx \delta(dt)$ since $\mathbb{P}[dt \chi > \epsilon] \rightarrow 0$ for any $\epsilon$ not depending on $dt$. Ergo $dW^2 \sim dt$.
This is very handwavy because we bypass all concern about the competing notions of convergence. We have a random variable converging to a distribution, and a function (that distribution) converging to an infinitesimal; and the infinitesimal converging to 0. Nevertheless, the above is exactly correct when used in a Taylor expansion (the only place you’d want to use it anyway), and can be used to derive many things, including the stochastic quotient.
Consider two geometric brownian motions
$dA = A\alpha dt + A\sigma dW_a$
$dB = B\beta dt + B\gamma dW_b$
with $\mathbb{E}[W_aW_b] = \rho$. Let $f(a,b) = \frac{a}{b}$ and then
$df = \frac{\partial f}{\partial a} da + \frac{\partial f}{\partial b}db + \frac{\partial^2 f}{\partial a \partial b}dadb + \frac{1}{2}\frac{\partial^2 f}{\partial a^2}da^2 + \frac{1}{2}\frac{\partial^2 f}{\partial b^2}db^2 + \dots$
$= \frac{1}{b}da - \frac{a}{b^2}db - \frac{1}{b^2}dadb + \frac{a}{b^3}db^2 +\dots$
Which means that
$d(A/B) = \frac{1}{B}(A \alpha dt + A \sigma dW_a) - \frac{A}{B} \frac{1}{B}(B \beta dt + B \gamma dW_b) - \dots$
$... - \frac{1}{B}(A\alpha dt + A\sigma dW_a)(\beta dt + \gamma dW_b) + \frac{A}{B^3}(B\beta dt + B\gamma dW_b)^2$
We observe a lot of A/B terms in here. Writing C=A/B:
$dC = C(\alpha-\beta) dt + C(\sigma - \gamma)dW - 2C\sigma\gamma dW_adW_b + C\gamma^2dW_b^2 + O(dt^2) + o(dtdW)$
Dropping terms smaller than $O(dt)$ and setting $dW^2 = dt$ then we have
$dC = C(\alpha - \beta + \gamma^2 - \sigma\gamma\rho)dt + C(\sigma - \gamma)dW$
The spot rates $P_i$ follow (independent) geometric brownian motion (this is, again, the no-arbitrage assumption)
$dP_i = P_i \mu_i(t)dt + P_i\sigma_i(t)dW$
Using the quotient rule, and defining $R_{i} = \frac{P_i}{P_{i+1}}$ we have
$\frac{dR_{i}}{R_{i}} = (\mu_i - \mu_{i+1} + \sigma^2_{i+1} - \sigma_i\sigma_{i+1})dt + (\sigma_i - \sigma_{i+1})dW$
This gives then
$dL_i = \frac{1}{\delta}R_i(\mu_i - \mu_{i+1} + \sigma^2_{i+1} - \sigma_i\sigma_{i+1})dt + (\sigma_i - \sigma_{i+1})dW$
The crux of the Libor Market Model is that $L_i$ should follow brownian motion. This implies two things, first, that the volatility difference is very special: $\sigma_i - \sigma_{i+1} = \frac{\delta}{R_i} L_i \xi_i(t)$ and also that the drift terms should be linked to the volatilities as $\mu_i(t) = \sigma_i(t) r(t)$. These then give
$dL_i = \frac{1}{\delta} R_i\left[((\sigma_i - \sigma_{i+1})r + \sigma_{i+1}(\sigma_i - \sigma_{i+1}))dt + (\sigma_i - \sigma_{i+1})dW\right]$
$dL_i = L_i\left[(r + \sigma_{i+1}) \xi_i dt + \xi_i dW\right]$
And now we have brownian motion. It’s worth parsing out what the spot rates now have to look like based on the central assumptions we made in order to shove $L_i$ into a brownian motion form. Because we have $\mu_i(t) = \sigma_i(t)r(t)$, we therefore must have
$dP_i = P_i\sigma_ir dt + P_i \sigma_i dW \Rightarrow \log \frac{P_t}{P_0} = (r\sigma_i - \frac{\sigma_i^2}{2})t + \sigma dW = \sigma_i\left[(r - \frac{\sigma_i}{2})t + dW\right]$
with the right-hand equation assuming constant volatility. In addition, the required relationship
$\sigma_{i+1} = \sigma_i - \frac{\delta}{R_i} L_i \xi$
telescopes. One thing to keep in mind is that $\sigma_i$ is the volatility of a bond that exercises at time $T_i$, so $\sigma_i(t) = 0$ for $t > T_i$. Letting $j(t)$ be the first index for which $t < t_i$ the above telescopes to
$\sigma_{i+1} = \omega(t) - \sum_{j(t)}^i \frac{\delta}{R_i}L_i$
And since $L = \frac{1}{\delta}(R - 1) \Rightarrow R = \delta L + 1 \Rightarrow \frac{\delta}{R_i}L_i = \frac{\delta L_i}{1 + \delta L_i}$
which is the term one is most often used to seeing.
If you recall, above I claimed that the derivation of this model only required certain no-arbitrage assumptions. There are, however, seemingly unrelated-to-arbitrage assumptions on volatility differences, and a linkage between volatility and drift. However, if $L_i$ were not martingales (e.g. if we couldn’t take a reference frame where $dL_i = L_i \xi_i dW$) then because $P_{i+1}(1 + \delta L_i) = P_i$, it would imply that there is an arbitrage in the $P_i$. The hand-wavy way to demonstrate this is to note that, by the stochastic quotient above, that if $A$ is a martingale, and $B$ is a martingale on the same space, then $A/B$ must also be a martingale. Taking the contrapositive: if $A/B$ is not a martingale at least one of: A is not a martingale, B is not a martingale, or A and B are not defined on the same space must be true. Since $L_i$ is a ratio, then the failure of $L_i$ to be a martingale implies that there must be an arbitrage in the bond prices. Therefore the “strange” conditions on the drift and volatility difference terms are actually just no-arbitrage constraints. The mathematics of this is worked out in Heath, Jarrow, & Morton, in the section on ‘Existence of Market Prices for Risk.’ One of the drawbacks of HJM and BGM is that the notation used is that of stochastic integrals rather than stochastic differentials, which is different from the notation presented here.
## Measures, Calibration, and Pricing
You may have noticed in the previous discussion my borrowing the term ‘reference frame’ to refer to dropping the drift term from the differential equation. This is typically referred to as a ‘change of measure’ or even more confusingly, ‘change of numeraire.’ There are appeals to Girsanov’s theorem and the Radon-Nikodym derivative, which involves developing the theory of filtrations. It’s all rather complicated for a very simple piece of intuition: you’ve got a particle moving with certain dynamics. Classically, you can write those dynamics in any number of suitable reference frames, including one in which the particle is not moving at all. The stochastic equivalent is to adopt a reference frame where the particle is moving only stochastically, and this amounts to adopting a reference frame that moves deterministically (and instantaneously) with the drift alone; that is if
$\frac{dX}{X} = \alpha(t)dt + \sigma(t)dW$
then we can adopt the reference frame
$\frac{dF}{X} = \alpha(t)dt$
and note the presence of $X$ in the denominator. Sure this is now a system of linked equations, which is precisely the point. Establishing that this is a valid transformation probabilistically is a bit involved, but the actual transformation is intuitively simple. Girsanov’s theorem basically says “You can pick a reference frame.”
That rant aside, how many degrees of freedom do we have in this model? The $\sigma_i(t)$ are fixed (possibly up to a single baseline function $\omega(t)$) by the preceding $L_j$. That just leaves the $\xi_i$; so there are basically n+1 degrees of freedom. The n of course can be increased arbitrarily by choosing $\xi_i(t)$ to be of higher order than a constant term. Given that $L_j$ is a martingale with $L_j =_{\mathbb{Q}_j} \xi_j(t)dW_j$ the $\xi_j$ are typically referred to (and can be interpreted as) Libor volatilities. Again, a reminder, that the $\xi_i$ are not volatilities for the actual LIBOR which is reported daily: these are instead forward rates. Of course, application of Ito’s rule brings them into line:
$F_i = f(L_i) = (1 + \delta L_i)^{-1}$
$df = \frac{\partial f}{\partial x}dx + \frac{1}{2}\frac{\partial^2f}{\partial x^2}dx^2 + \dots$
$dF_i = -(1 + \delta L_i)^{-2}\delta L_i \xi_i dW + (1 + \delta L_i)^{-3}\delta^2\xi_i^2 L_i^2 dW^2$
Recalling that $\frac{\delta L_i}{1 + \delta L_i} = \frac{\sigma_i - \sigma_{i+1}}{\xi}$ we get out (well, what did you expect??)
$dF_i =_{\mathbb{Q}_i} (\sigma_{i+1}-\sigma_i)^2F_idt + (\sigma_{i+1}-\sigma_i)F_idW$
Despite the fact that the $\sigma_i$ can be computed from the $\xi_i$ (given $L_i$), and visa versa, which means that one could choose to model the forward, spot, or Libor volatilities, the philosophy of the LMM is that the “truth” lies with the Libor volatilities, $\xi_i$, and that spot and forward volatilities are derived quantities. The Libor volatilities can be modeled in much the same way forward volatilities would be (for instance Ho-Lee or HJM like models). For instance, we could let
$\xi(t) = (a + bt)e^{-ct} + d$
$\xi_i(t) = k_i \xi(t)$
which gives us $n+4$ free parameters. The Libor values $L(t, T_i, T_{i+1})$ are directly observable, as defined above, and thus their volatilities can be fit directly.
There are alternatives to direct fitting as well. In much the same way that one can use option prices to back out parameters of the underlying asset, one can back out parameters of the forward rates from caps and swaps. This method of calibration requires a slight digression on pricing these instruments.
Derivatives pricing under LMM
The standard interest rate derivatives are caps and swaps. There are of course others, including a plethora of exotics, but caps and swaps are the two which are germane to LMM calibration. We start with a swap agreement.
Interest Rate Swaps
An interest rate swap agreement references two dates: a “reset date” $T_i$ and a “settlement date” $T_{i+1}$. The difference $\delta_i = T_{i+1}-T_i$ is the time interval for the contractual fixed rate $\kappa$. The contract is as follows:
Party A is looking to hedge against volatility in forward interest rates, and wants to fix a set interest rate $\kappa$ over the time period $(T_i, T_{i+1})$. Therefore at time $T_{i+1}$ Party A disburses $\delta_i \kappa$ to Party B.
Party B is looking to speculate on the volatility on the forward interest rates, and specifically wants to capitalize on the difference between $\kappa$ and the short rate $r(t=T_i, T_{i+1})$. In exchange for receiving $\delta_i \kappa$ from Party A, Party B disburses whatever the short rate was over that period, $r(t=T_i, T_{i+1})$.
The short rate $r(t, T)$ can be calculated from zero-coupon bond prices as $r(t, T) = \left(\frac{\mathrm{Face\;Value}}{P(t, T)}\right)^{1/\mathrm{num \; compounds}} - 1$. We will take the face value to be $\texttt{f}$, and we’ll assume for simplicity that there is only the single application of the rate at $T_{i+1}$. Therefore the contract stipulates
$\mathrm{Receive}_A = \frac{\texttt{f}}{P(t=T_i, T_{i+1})} - 1$
$\mathrm{Receive}_B = \delta_i \kappa$
The present value of these are
$\mathrm{PV}_A = \frac{P(t, T_{i+1})}{\texttt{f}}\left(\frac{\texttt{f}}{P(t=T_i, T_{i+1})} - 1\right)$
$\mathrm{PV}_B = \frac{P(t, T_{i+1}) \delta_i \kappa }{\texttt{f}}$
There’s some simplification for $\mathrm{PV}_A$, since $\frac{P(t, T_{i+1})}{\mathtt{f}} = \frac{P(t, T_i)P(t=T_i, T_{i+1})}{\mathtt{f}^2}$ (in expectation) then we can rewrite
$\mathrm{PV}_A = \frac{P(t, T_i)}{\mathtt{f}} - \frac{P(t, T_{i+1})}{\texttt{f}}$
The fair value of $\kappa$ can be identified by setting $\mathrm{PV}_A = \mathrm{PV}_B$ so that
$\frac{P(t, T_i)}{\mathtt{f}} - \frac{P(t, T_{i+1})}{\texttt{f}} = \frac{P(t, T_{i+1})}{\texttt{f}} \cdot \delta_i \kappa$
Yielding
$\kappa = \frac{P(t, T_i) - P(t, T_{i+1})}{\delta_i P(t, T_{i+1})} = \frac{1}{\delta_i}\left(\frac{P_i}{P_{i+1}}-1\right) = L_i$
This implies that the Libor parameters can be calibrated to the interest rate swap market. The details were worked out by Jamshidian, and extended to a model similar to the LMM, but using forward swap rates. A direct comparison can be found here. For us, we continue to Caps.
Interest Rate Caps
An interest rate cap is effectively a call option on an interest rate. For those already familiar with the lingo, I’m subsuming what’s typically called a “caplet” into this definition.
Party A is looking to hedge against an increase in forward interest rates, and has in mind a maximum rate payment $\zeta$ over the period $(T_i, T_{i+1})$. Party A seeks a payment from Party B if the interest rates exceed $\zeta$ during this period, using the start-of-period interest rate $r(t=T_{i}, T_{i+1})$ as a proxy for the interest rate of the whole period. Thus at time $T_{i+1}$, Party A receives $\delta_i \mathrm{max}(r(T_i, T_{i+1}) - \zeta, 0)$.
Party B is looking to bet that interest rates will not exceed a particular value, also $\zeta$, over the same period. Thus at time $t$, Party B receives a payment of $V_B$ from Party A in anticipation of possible future disbursement, should the interest rate at $t=T_{i+1}$ exceed $\zeta$.
This means that $V_B$ is the present value of the expected payoff of the cap, e.g. [I’m taking face-value to be 1 for bonds]
$V_B = \frac{1}{P(t, T_{i+1})}\delta_i \mathbb{E}[\mathrm{max}(r(T_i, T_{i+1}) - \zeta, 0)]$
We can rewrite the interest rate as above:
$V_B =\frac{\delta_i}{P(t, T_{i+1})}\mathbb{E}\left[\left(\frac{1}{P(t=T_i, T_{i+1})} - 1 - \zeta\right)^+\right]$
$V_B =\frac{\delta_i}{P(t, T_{i+1})}\mathbb{E}\left[\left(\frac{P(t, T_i)}{P(t, T_{i+1})}-1 - \zeta\right)^+\right]$
$V_B = \frac{\delta_i}{P(t, T_{i+1})}\mathbb{E}[(\delta_iL_i - \zeta)^+]$
Admittedly, this looks very strange. The units just don’t work out; the only way this is consistent is if $\delta_i$ is not measured in time, but is a dimensionless scalar. In fact, $\delta_i$ here is measured as fraction of a compound time, so if $T_{i+1}-T_i = 6 \mathrm{mo}$ for yearly compounds, then $\delta_i = 0.5$. In fact, if you look back over the derivation, you’ll notice that this has to be the case, otherwise the libor rate has units 1/t, which would give a forward rate units of time, which is nonsensical for a rate.
Usual cap agreements are contracts that reference a large number of the above caps; for instance a 5-year monthly cap agreement would reference $5 \times 12 = 60$ caps. Because of this, the values of a cap for period $T_i$ has to be deconvolved from publicly-traded cap contracts. In addition, the optimization problem, while convex, is nonlinear, making it somewhat more difficult to back out from caps than swaps. See here for more details (fair warning: poor notation rears its ugly head).
Originally I was planning on doing some model calibration myself as an example; perhaps in the future. The post, as it stands, is long enough as it is. There’s one obvious question remaining, though:
Why don’t we calibrate LMM to trasury rates?
Recall that the Libor rate was defined in terms of the spot and forward rate
$L_i = \frac{P_i - P_{i+1}}{\delta P_{i+1}}$
The answer is that there is a divergence between the ZCP rates calculated from (for instance) U.S. Treasuries, and the ZCP implied by calibrating the LMM to interest rate caps. The point is that government rates are different from interbank rates, whereas the risk-free zero-coupon bond is a theoretical, unobservable quantity. While coupon theory can be applied to things like treasury bonds, and the resulting forward and spot rates analyzed, there is no guarantee that government bonds are, really and truly, a risk-free ZCP. (Indeed, we expect things like exchange rate risk to differentially effect interbank rates and government rates). So calibrating the interbank rates to government rates will make you a very sad panda.
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# Convert to Regular Notation -4.12*10^-3
-4.12×10-3
Since the exponent of the scientific notation is negative, move the decimal point 3 places to the left.
-0.00412
Convert to Regular Notation -4.12*10^-3
Scroll to top
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# Logistic Growth Models (interpreting r value)
1. Jan 28, 2014
### thelema418
I originally posted this on the Biology message boards. But I have not received any responses.
In models of exponential growth, we have an intrinsic growth rate (r) that is calculated as the difference of birth rates to death rates.
With the logistic growth model, we also have an intrinsic growth rate (r). How then do birth rates and death rates relate to the intrinsic growth rate in the context of this model? Specifically, if you have a model where you have been given values for r and K, does the birth rate and death rate associated with r occur at a particular time? I'm wondering if this specifically relates to P(t) = K/2 since this is where the maximum growth occurs.
Thanks.
2. Jan 28, 2014
### pasmith
Logistic growth of a population $P(t)$ is governed by the ODE
$$\dot P = aP - bP^2$$
where $a$ is (birth rate - death rate), which is assumed to be constant, and $b \geq 0$, which is assumed to be constant, is a parameter representing the effects of competition for resources. In effect $bP$ is the death rate from competition, which is not constant but is proportional to the size of the population, whereas $a$ is birth rate less death rate from all other causes. When $b = 0$ we recover exponential growth and there are no competition-related deaths.
For $b \neq 0$ the ODE can also be written in the form
$$\dot P = rP(K - P)$$
where $r = b$ and $K = a/b$, or in the form
$$\dot P = sP\left(1 - \frac{P}{K}\right)$$
where $s = a$ and again $K = a/b$.
3. Jan 29, 2014
### thelema418
Yes, those are the models I'm speaking about.
But my question concerns, I guess, "practical guidance" of the model. Consider a model where the population reaches capacity at t = 500. If a researcher measures the birth rate and death rate after t = 500, the number of births would be the same as the number of deaths.
This is again why I'm wondering if the inflection point is significant to the concept of r. If I have a birth rate and death rate relative to a specific time and I know the model is logistic, is this enough information to find r for the logistic equation?
4. Jan 29, 2014
### pasmith
I don't think so. To determine $r$ and $K$ you need to know $P$ and $dP/dt$ at two different times. Knowledge of $P^{-1} dP/dt$, which is really all that the birth and death rates give you, at just a single time is not sufficient.
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How to operate with function body without evaluation
I have several functions (let a[x] and b[x]), defined as expressions with integrals and differentiations. I want to perform some manipulations with these expressions - for example, ask Mathematica to simplify a[x] + b[x] without evaluations of integrals. Hold doesn't works (as I tried) as I used to expand functions a[x] and b[x] first. I think, that I could replace Integrate and D with some meaningless heads integrate and d, and then expand functions with another meaningless symbol x, but I can't achieve this, because Mathematica tries first to perform integrations and fails.
Example:
a[t_] := Integrate[t, {x, -∞, ∞}]
b[t_] := Integrate[t^2, {x, -∞, ∞}]
a[t]+b[t] (* this doesn't works, as it "evaluates" integrals;
I want to see simple sum *)
ADDED: I want to see something like myIntegrate[t, {x, -∞, ∞}]+myIntegrate[t^2, {x, -∞, ∞}]. Really, a and b are more complicated, and they contain more than one integrals. I want Mathematica to perform expression simplification keeping integrals unevaluated. I could manually modify definition for a and b, but they are rather complicated, and don't want to keep two copies (one with Integrate, and one with some myIntegrate).
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What's the result you'd like to achieve for your example? – Leonid Shifrin Apr 2 '14 at 9:19
@LeonidShifrin Edited question, added an output example that I want to achieve. – Yury Apr 2 '14 at 13:46
@Yury As it looks after added explanation it seems me that you want to make unnecessary complication of the problem. In such a case as a first step I simply define the functions under the integrals (in your case t and t^2) and then operate with them, transforming and adding a Jacobian, if needed, but not integrating them. I integrate the result in a second step as soon as I am satisfied with the results of the first step.. My solution offered below is only useful, if you want to suppress the integration for the purpose of a demonstration, such as giving a talk. – Alexei Boulbitch Apr 2 '14 at 13:53
One way of doing this would be using Defer or HoldForm. For example, let us define the functions a and b as follows:
a[t_] := Defer[Integrate[t, {x, -∞, ∞}]]
b[t_] := Defer[Integrate[t^2, {x, -∞, ∞}]]
and
a1[t_] := HoldForm[Integrate[Exp[-t], {x, 0, ∞}]]
b1[t_] := HoldForm[Integrate[Exp[-t^2], {x, 0, ∞}]]
They both return unevaluated function. Then the function:
mySum[expr1_, expr2_] := ReplacePart[expr1, {1, 1} -> expr1[[1, 1]] + expr2[[1, 1]]]
does the job.
mySum[a1[t], b1[t]]
The result looks as follows:
The same will take place with the functions a[t] and b[t]
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# State Space in TI-Basic
Just for fun, running a state-space simulation in TI-Basic
January 2, 2017 -
TI-Basic holds a special place in my heart - It's a terrible language on a terrible device, but I have fond memories of wasting time in my Algebra II/Trig class programming 2048, so that I could wast time in class doing something more fun than programming in TI-Basic! Recently, I remembered that my TI-84 supports matrices, and figured that it should be possible to get a state-space simulation up and running on it pretty quickly.
(As an aside, that graph is the exact same thing as the matlab code from part four)
I figured that it would be easy, but I didn't expect it to be nearly as easy as it was! Within 8 lines of TI-Basic, I'd put together a program to simulate a closed-loop state-space system:
[F] is the state, and [E] is the K-matrix (because TI-Basic doesn't let you use letters higher than J for matrices ).
Once the program is done running, the timestep and $$x_{1,1}$$ values are in L1 and L2, and can be graphed via statplot.
This isn't very useful for a few reasons:
• It's really really really slow. It can take around 30 seconds to simulate a simple system.
• Lists can only contain up to 999 values, so, for example if dt=0.01, you can only simulate for 9.99 seconds!
• It's way less convenient than tools like python or matlab/octave
Despite all that, it was fun to make, and I was surprised that it worked as well as it did! This really shows how generic state space is - it's likely possible to implement a lot more that I did, since most of the techniques used in state-space are just simple matrix operations.
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# glmmrBase
(Version 0.2.2) R package to support the specification of generalised linear mixed models using the R6 object-orientated class system.
## Generalised linear mixed models
A generalised linear mixed model (GLMM) has a mean function for observation $$i$$ is
$\mu_i = \mathbf{x}_i\beta + \mathbf{z}_i \mathbf{u}$
where $$\mathbf{x}_i$$ is the $$i$$th row of matrix $$X$$, which is a $$n \times P$$ matrix of covariates, $$\beta$$ is a vector of parameters, $$\mathbf{z}_i$$ is the $$i$$ th row of matrix $$Z$$, which is the $$n \times Q$$ “design matrix” for the random effects, $$\mathbf{u} \sim N(0,D)$$, where $$D$$ is the $$Q \times Q$$ covariance matrix of the random effects terms that depends on parameters $$\theta$$, and $$\mathbf{\mu}$$ is the $$n$$-length vector of mean values. The assumed data generating process for the study is
$y_i \sim G(h(\mu_i);\phi)$
where $$\mathbf{y}$$ is a $$n$$-length vector of outcomes $$y_i$$, $$G$$ is a distribution, $$h(.)$$ is the link function, and $$\phi$$ additional scale parameters to complete the specification.
## Generating data
The package includes the function nelder(), which we use to generate data for the examples below. Nelder (1965) suggested a simple notation that could express a large variety of different blocked designs. The notation was proposed in the context of split-plot experiments for agricultural research, where researchers often split areas of land into blocks, sub-blocks, and other smaller divisions, and apply different combinations of treatments. However, the notation is useful for expressing a large variety of experimental designs with correlation and clustering including cluster trials, cohort studies, and spatial and temporal prevalence surveys. We have included the function that generates a data frame of a design using the notation.
There are two operations: * > (or $$\to$$ in Nelder’s notation) indicates “clustered in”. * * (or $$\times$$ in Nelder’s notation) indicates a crossing that generates all combinations of two factors.
The function takes a formula input indicating the name of the variable and a number for the number of levels, such as abc(12). So for example ~cl(4) > ind(5) means in each of five levels of cl there are five levels of ind, and the individuals are different between clusters. The formula ~cl(4) * t(3) indicates that each of the four levels of cl are observed for each of the three levels of t. Brackets are used to indicate the order of evaluation.
## Specifying covariance
The specification of a covariance object requires three inputs: a formula, data, and parameters. A new instance of each class can be generated with the $new() function, for example Covariance$new(...).
A covariance function is specified as an additive formula made up of components with structure (1|f(j)). The left side of the vertical bar specifies the covariates in the model that have a random effects structure. The right side of the vertical bar specify the covariance function f for that term using variable named in the data j. Covariance functions on the right side of the vertical bar are multiplied together, i.e., (1|f(j)*g(t)). The table below shows the currently implemented covariance functions
Function $$Cov(x_i,x_{i'})$$ $$\theta$$ Code
Identity/ Group membership $$\theta_1^2 \mathbf{1}(x_i = x_{i'})$$ $$\theta_1 > 0$$ gr(x)
Exponential $$\theta_1 \text{exp}(- \vert x_i - x_{i'}\vert / \theta_2 )$$ $$\theta_1,\theta_2 > 0$$ fexp(x)
$$\text{exp}(- \vert x_i - x_{i'}\vert /\theta_1)$$ $$\theta_1 > 0$$ fexp0(x)
Squared Exponential $$\theta_1 \text{exp}(- (\vert x_i - x_{i'}\vert / \theta_2)^2)$$ $$\theta_1,\theta_2 > 0$$ sqexp(x)
$$\text{exp}(-( \vert x_i - x_{i'}\vert/\theta_1)^2 )$$ $$\theta_1 > 0$$ sqexp0(x)
Autoregressive order 1 $$\theta_1^{\vert x_i - x_{i'} \vert}$$ $$0 < \theta_1 < 1$$ ar1(x)
Bessel $$K_{\theta_1}(x)$$ $$\theta_1$$ > 0 bessel(x)
Matern $$\frac{2^{1-\theta_1}}{\Gamma(\theta_1)}\left( \sqrt{2\theta_1}\frac{x}{\theta_2} \right)^{\theta_1} K_{\theta_1}\left(\sqrt{2\theta_1}\frac{x}{\theta_2})\right)$$ $$\theta_1,\theta_2 > 0$$ matern(x)
Compactly supported*
Wendland 0 $$\theta_1(1-y)^{\theta_2}, 0 \leq y \leq 1; 0, y \geq 1$$ $$\theta_1>0, \theta_2 \geq (d+1)/2$$ wend0(x)
Wendland 1 $$\theta_1(1+(\theta_2+1)y)(1-y)^{\theta_2+1}, 0 \leq y \leq 1; 0, y \geq 1$$ $$\theta_1>0, \theta_2 \geq (d+3)/2$$ wend1(x)
Wendland 2 $_1(1+(_2+2)y + ((_2+2)^2 - 1)y2)(1-y){_2+2}, 0 y$ $$\theta_1>0,\theta_2 \geq (d+5)/2$$ wend1(x)
$$0, y \geq 1$$
Whittle-Matern $$\times$$ Wendland** $$\theta_1\frac{2^{1-\theta_2}}{\Gamma(\theta_2)}y^{\theta_2}K_{\theta_2}(y)(1+\frac{11}{2}y + \frac{117}{12}y^2)(1-y), 0 \leq y \leq 1; 0, y \geq 1$$ $$\theta_1,\theta_2 > 0$$ prodwm(x)
Cauchy $$\times$$ Bohman*** $$\theta_1(1-y^{\theta_2})^{-3}\left( (1-y)\text{cos}(\pi y)+\frac{1}{\pi}\text{sin}(\pi y) \right), 0 \leq y \leq 1; 0, y \geq 1$$ $$\theta_1>0, 0 \leq \theta_2 \leq 2$$ prodcb(x)
Exponential $$\times$$ Kantar**** $$\theta_1\exp{(-y^{\theta_2})}\left( (1-y)\frac{\sin{(2\pi y)}}{2\pi y} + \frac{1}{\pi}\frac{1-\cos{(2\pi y)}}{2\pi y} \right), 0 \leq y \leq 1$$ $$\theta_1,\theta_2 > 0$$ prodek(x)
$$0, y \geq 1$$
$$\vert . \vert$$ is the Euclidean distance. $$K_a$$ is the modified Bessel function of the second kind. Variable $$y$$ is defined as $$x/r$$ where $$r$$ is the effective range. For the compactly supported functions $$d$$ is the number of dimensions in x. Permissible in one or two dimensions. Only permissible in one dimension. ****Permissible in up to three dimensions.
One combines functions to provide the desired covariance function. For example, for a stepped-wedge cluster trial we could consider the standard specification with an exchangeable random effect for the cluster level, and a separate exchangeable random effect for the cluster-period, which would be ~(1|gr(j))+(1|gr(j,t)) or ~(1|gr(j))+(1|gr(j)*gr(t)). Alternatively, we could consider an autoregressive cluster-level random effect that decays exponentially over time so that, for persons $$i$$ in cluster $$j$$ at time $$t$$, $$Cov(y_{ijt},y_{i'jt}) = \theta_1$$, for $$i\neq i'$$, $$Cov(y_{ijt},y_{i'jt'}) = \theta_1 \theta_2^{\vert t-t' \vert}$$ for $$t \neq t'$$, and $$Cov(y_{ijt},y_{i'j't}) = 0$$ for $$j \neq j'$$. This function would be specified as ~(1|gr(j)*ar1(t)).
Parameters are provided to the covariance function as a vector. The covariance functions described in the Table have different parameters $$\theta$$, and a value is required to be provided to generate the matrix $$D$$ and related objects for analyses and which serve as starting values for model fitting. The elements of the vector correspond to each of the functions in the covariance formula in the order they are written.
A full call to create a new covariance object is:
R> df <- nelder(~ (j(10)* t(5)) > ind(10))
R> cov <- Covariance$new(formula = ~(1|gr(j)*ar1(t)), R> parameters = c(0.05,0.8), R> data= df) in this call, the parameters are optional, and if provided as a list of arguments to a Model object (see below), then the data argument is also optional. A compactly supported function is used, then the effective range parameters should be provided in the order the function appears in the formula. R> cov <- Covariance$new(formula = ~(1|prodwm(x,y)),
R> parameters = c(0.25,0.5),
R> eff_range = 0.5,
R> data= df)
## Mean function specification
Specification of the mean function follows standard model formulae in R. A vector of values of the mean function parameters is required to complete the model specification along with the distribution as a standard R family object. A complete specification is thus:
R> mf <- MeanFunction$new(formula = ~ factor(t)+ int - 1, R> data = df, R> parameters = rep(0,6), R> family = gaussian()) As before, the parameters, data, and family are optional and can instead be provided directly to the Model call below. Note that factor in this function does not drop one level, unlike standard R formulae, so removing the intercept is required to prevent a collinearity problem. ## Model specification A model can be created by specifying a Covariance and MeanFunction object: R> model <- Model$new(covariance = cov,
R> mean = mf,
R> var_par = 1)
Alternatively, we can provide a list of arguments to the covariance and mean arguments:
R> model <- Model$new(covariance = list(formula = ~(1|gr(j)*ar1(t))), R> mean = list(formula = ~ factor(t)+ int - 1), R> data = df, R> family = gaussian(), R> var_par = 1) where, as required, parameters can be supplied to covariance and mean function argument lists. For Gaussian models, and other distributions requiring an additional scale parameter $$\phi$$, one must also specify the option var_par which is the conditional variance $$\phi = \sigma$$ at the individual level. The default value is 1. Alternatively, one can specify a design by providing the list of arguments directly to covariance and mean.function instead of model objects. ## Supported Families The package and associated packages (glmmrMCML and glmmrOptim) currently support the following families and link functions | Family | Link functions | |——–|————————-| | Gaussian | Identity, log | | Binomial | Logit, log, identity | | Poisson | Log, Identity | | Gamma | Log, Inverse, Identity| | Beta | Logit | The Beta family is provided by the package function Beta(), which generates a barebones list specifying the family and link. We use a mean-variance parameterisation of the Beta family. The likelihood is: $f(y_i | \mu_i, \phi) = \frac{y_i^{\mu_i\phi - 1}(1-y_i)^{(1-\mu_i)\phi - 1}}{B(\mu_i\phi, (1-\mu_i)\phi)}$ where $$B()$$ is the Beta function, and we use logit link $\log\left( \frac{\mu_i}{1-\mu_i} \right) = \mathbf{x}_i\beta + \mathbf{z}_i \mathbf{u}$ We similarly use a mean-variance parameterisation for the Gamma regression function: $f(y_i | \mu_i, \nu) = \frac{1}{\Gamma(\nu)}\left( \frac{\nu y_i}{\mu_i} \right)^\nu \exp \left( -\frac{\nu y_i}{\mu_i} \right) \frac{1}{y}$ where we also provide logit, inverse, and identity link functions for the specification of $$\mu_i$$. ## Accessing computed elements Each class holds associated matrices and has member functions to compute basic summaries and analyses. The Matrix package is used for matrix operations in R. For example, a Covariance object holds the matrix $$D$$ R> cov$D[1:10,1:10]
10 x 10 sparse Matrix of class "dsCMatrix"
[1,] 0.002500 0.00200 0.0016 0.00128 0.001024 . . . . .
[2,] 0.002000 0.00250 0.0020 0.00160 0.001280 . . . . .
[3,] 0.001600 0.00200 0.0025 0.00200 0.001600 . . . . .
[4,] 0.001280 0.00160 0.0020 0.00250 0.002000 . . . . .
[5,] 0.001024 0.00128 0.0016 0.00200 0.002500 . . . . .
[6,] . . . . . 0.002500 0.00200 0.0016 0.00128 0.001024
[7,] . . . . . 0.002000 0.00250 0.0020 0.00160 0.001280
[8,] . . . . . 0.001600 0.00200 0.0025 0.00200 0.001600
[9,] . . . . . 0.001280 0.00160 0.0020 0.00250 0.002000
[10,] . . . . . 0.001024 0.00128 0.0016 0.00200 0.002500
## Use of glmmrBase in other packages
This package provides a foundation for other packages providing analysis or estimation of generalised linear mixed models. For example, we have the glmmrMCML package, which provides Markov Chain Monte Carlo Maximum Likelihood estimations for these models, and glmmrOptim, which finds approximate optimal designs based on these models. New classes can be defined that inherit from the classes included in this package. glmmrMCML defines the modelMCML class that adds the member function MCML. Then the new functions can access the model elements, such as covariance matrices, and benefit from automatic updating of these elements when specifications or parameters change. As an example we can define a new class that has a member function that returns the determinant of the matrix $$D$$:
R> CovDet <- R6::R6Class("CovDet",
R> inherit = Covariance,
R> public = list(
R> det = function(){
R> return(Matrix::determinant(self$D)) R> })) R> cov <- CovDet$new(formula = ~(1|gr(j)*ar1(t)),
R> parameters = c(0.05,0.8),
R> data= df)
R> cov$det()$modulus
[1] -340.4393
attr(,"logarithm")
[1] TRUE
\$sign
[1] 1
attr(,"class")
[1] "det"
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# Simple understanding of convex co-compactness
I was looking for the definition of the term "convex co-compact" in simple cases. But most references I find are looking into a little bit sophisticated notions such as mapping class group, Schottky subgroup and higher dimensional hyperbolic spaces. I would like to understand the definition just in the simple case of a discrete subgroup of $SL_2(\mathbb R)$ acting on the Poincaré half-plane.
What I could find was that the action is convex co-compact if the action is co-compact on the convex hull of the limit set $L$ . What I am doubting is the term "convex hull". Does this mean the collection of all geodesic segments connecting each pair of points in the set $L$ ?
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Yes, exactly. ${}$ – t.b. May 9 '11 at 13:36
Well slightly more than the union of all geodesics connecting points in L. You need to take the convex hull of this union. Think of a group G, a Scottish group generated by two hyperbolic isometrics. Start with four very small (euclidean) geodesics (1,2,3,4) in the Poincare unit disc model. Choose two hyperbolic isometrics: one identifying 1 with 2 and the other identifying 3 and 4. Remember that L will lie 'under' these small geodesics. If you only take the union of this example then you get 'holes', you get something that is clearly not a convex set!
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# Timing A Function in C++
March 31, 2020 - John Law - 4 mins read
Recently, I encountered the following interview question:
Write a function that returns true if it has been called 3 times in 3 seconds.
To solve this, I have used a traditional C-style approach (in C++):
#include<iostream>
#include<ctime>
#include<unistd.h>
using namespace std;
time_t start;
int count;
bool being_called(time_t call_time){
count++;
if(call_time - 3 <= start){
return !(count < 3);
}else{
start = call_time;
count = 1;
return false;
}
}
int main(){
start = time(NULL);
cout << being_called(time(NULL)) << endl;
cout << being_called(time(NULL)) << endl;
cout << "=====" << endl;
sleep(4);
cout << being_called(time(NULL)) << endl;
cout << being_called(time(NULL)) << endl;
cout << being_called(time(NULL)) << endl;
cout << being_called(time(NULL)) << endl;
return 0;
}
This works perfectly well and it actually solves the problem. However, there might be other alternatives.
### Precision
time_t time (time_t* timer) returns a Unix time, which is simply a integer value. Since it is an integer, $t=1.0001$ would be recorded the same as $t=1.9888$. This is not very good as if we call the function at $t_1=0.011, t_2=3.511, t_3=3.611,$ then the function returns true and the flaw is obvious. This is verified in the last section.
This can be solved with a int gettimeofday(struct timeval *tv, struct timezone *tz) in sys/time.h. The struct timeval *tv contains seconds and microseconds of the current time. The precision is slightly improved, despite the fact that there will be still some errors when dealing with doubles.
### A "C++" Approach
Although the previous ways work in C++, they are not 100% C++-ish. I could have used chrono. It has an extremely rich functionality comparing to ctime.
#include<iostream>
#include<chrono>
using namespace std;
using namespace std::chrono;
int count;
high_resolution_clock::time_point start;
bool being_called(high_resolution_clock::time_point call_time){
count++;
if(duration_cast<seconds>(call_time - start).count() <= 3){
return !(count < 3);
}else{
start = call_time;
count = 1;
return false;
}
}
int main(){
using namespace literals::chrono_literals;
high_resolution_clock c;
start = c.now();
cout << being_called(c.now()) << endl;
cout << being_called(c.now()) << endl;
cout << "=====" << endl;
cout << being_called(c.now()) << endl;
cout << being_called(c.now()) << endl;
cout << being_called(c.now()) << endl;
cout << being_called(c.now()) << endl;
return 0;
}
One should notice that C++14 is required to access std::literals::chrono_literals. There is a difference between a duration cast and plain duration. duration<double, milli>, for example, requires a milli ratio type to process the time. On the other hand, duration cast is useful for getting the whole integral number of seconds of time, like the above case. For a better precision, we can do the following.
bool being_called(high_resolution_clock::time_point call_time){
count++;
if(duration<double, milli>(call_time - start).count() <= 3000.0){
return !(count < 3);
}else{
start = call_time;
count = 1;
return false;
}
}
### Verifying the Guess
Previously, we made the following guess:
This is not very good as if we call the function at $t_1=0.011, t_2=3.511, t_3=3.611,$ then the function returns true and the flaw is obvious.
#include<iostream>
#include<chrono>
using namespace std;
using namespace std::chrono;
int count;
high_resolution_clock::time_point start;
bool being_called(high_resolution_clock::time_point call_time){
count++;
if(duration_cast<seconds>(call_time - start).count() <= 3){
return !(count < 3);
}else{
start = call_time;
count = 1;
return false;
}
}
int main(){
using namespace literals::chrono_literals;
high_resolution_clock c;
start = c.now();
cout << "=====" << endl;
high_resolution_clock::time_point t = c.now();
cout << being_called(c.now()) << endl;
cout << "=====" << endl;
cout << being_called(c.now()) << endl;
cout << "=====" << endl;
cout << being_called(c.now()) << endl;
cout << "=====" << endl;
cout << "Time elapsed: " <<
duration<double, milli>(c.now() - t).count() << "ms" << endl;
return 0;
}
=====
0
=====
0
=====
1
=====
Time elapsed: 3801.72ms
This can be extremely sophisticated if we expand our solution, but it looks good to me for now!
This is John Law, signing off. You read 282 words.
|
{}
|
Computationaly hard detokenization algorithm for credit card numbers
I am designing a vault that tokenizes credit card numbers (a plaintext that consists of 16 decimal digits), with the following requirements:
1. Given some plaintext, the vault returns an index. If the plaintext does not exist it returns a new index. If it does exist it returns the same index it had returned before for that plaintext.
2. The vault allows detokenization (given index return the plaintext), however this is quite rare (once a month) and involves authentication procedure.
3. The vault stores the plaintext encrypted using an algorithm so that detokenization of one index is computationally possible but hard. Let the costs of detokenization be X per entry.
4. An attacker that got access to the vault should find it too computationally expensive to detokenize the entire vault.
5. A variation: the attacker may have access to a limited list of (index, plaintext) pairs that they injected into the vault before they compromised it. Let the length of this list be Y.
What is the encryption algorithm that should be used for this use case? Can it be designed so that X is configurable (given Y=0)? Can detokenizing the entire vault be computationally hard regardless of Y, or must we assume Y = 0?
Edit:
I've modified the original question. Instead of tokenizing a 6 decimal digit plaintext, the question is now about tokenizing a 16 decimal digit credit card number.
Both a legitimate user and a very skilled attacker has access to partial credit card numbers. A partial credit card number consists of the first 6 plaintext digits, the last 4 plaintext digits, and the index received from the vault. In order to find the remaining middle 6 digits the legitimate user would authenticate and detokenize, while the skilled attacker would have three options:
• Perform a limited number of tokenizing operations without compromizing the vault.
• Compromise the vault and perform 50,000 tokenizing operations per plaintext.
• Compromise the vault and perform one detokenization per plaintext.
• Many credit cards share the same first 6 digits.
• Werther the plaintext is the complete 16 digits number or only the middle 6 digit number is part of the design.
• The index length is part of the design
• Once the vault is compromised the attacker has access to the vault storage/memory which contains everything except one-time keys that were properly erased. The same attacker has also compromised the detokenizer's private key stored elsewhere if there are such keys.
Edit 2:
• Removed the requirement that the index is random. It just need not to be some hash of the plaintext to avoid brute forcing the index without compromizing the vault.
• Changed brute force tokenization per plaintext from 100,000 to 50,000
• Obviously the plaintext could be encrypted with a public key, and assume that only the authorized detokenizer has the private key. Alas, if a skilled attacker has enough resources to compromise the vault (by hacking into the vault machine) , the attacker has enough resources to compromise the private key (by hacking into the machine containing the private key). Assuming the hack is discovered in timely fashion, the cards would be canceled before the hacker sold the Cards in the black market. For that purpose, the detokenization of the entire vault must be so hard that it would delay the attacker for a long time, or would require vast amounts of compute resource to make it non-economical in the first place.
• It is expected to make the tokenization computationally hard - so 50,000 tokenization operations per credit card number would be non-economical or would take too long (configurable)
• It is expected to make the detokenization computationally harder than 1 and less than 50,000 tokenization operations (configurable).
-
Given a procedure that did the above, wouldn't one attack be to go through all 1,000,000 possible plaintexts and insert them into the vault. By rule (1), if they're already in the vault, that'll return the same index. Won't this efectively detokenize the entire vault without that much expense (hence violating requirement (4)? – poncho Feb 6 '14 at 5:52
The vault will most likely store only the CCN (16 decimal digit PAN) without the expiry date or service code. I suppose this may change in the future, but it is a reasonable assumption for now. I am updating the question from 10^6 to 10^5. – itaifrenkel Feb 7 '14 at 14:13
Why does the new index need to be random? $\;$ – Ricky Demer Feb 8 '14 at 2:54
It does not have to be random, it just need not contain information from the plaintext itself. I'll update the question – itaifrenkel Feb 8 '14 at 6:25
As fgrieu suggests, itaifrenkel should probably think again and see if he can change the situation. Placing the tokenizer/detokenizer inside a smart card may be feasible and may provide real security. – K.G. Feb 10 '14 at 20:44
I am assuming that the vault shall store arbitrary-length messages and associate with each message a token consisting of six decimal digits. Otherwise, as has been noted (see below), the problem is probably either impossible or trivial.
I interpret your requirements to mean that the detokenization algorithm is also available to an attacker that has gotten access to the vault, which means that the cost of detokenizing the entire vault is at most $nX$, where $n$ is the number of items stored in the vault and $X$ is the cost of detokenizing one index.
This means that the best you can do is to balance your $X$ such that $X$ is feasible, but $nX$ is economically infeasible. (Note that $n$ is very small, so it seems unlikely that $X$ cost can be feasible for you, but $nX$ cost infeasible for an attacker.)
(Note that if the vault's detokenization algorithm is not available to the attacker, your problem has a very simple solution using deterministic public key encryption: Encrypt the message using the detokenizer's public key. If you already have the ciphertext, you're done. Otherwise, pick a fresh index and store the index-ciphertext pair. The detokenizer simply decrypts the ciphertext. This is the best you can do unless the vault's tokenization algorithm also isn't available to the attacker, in which case everything is trivial.)
The system consists of an interactive algorithm $V$ and an algorithm $D$. The correctness requirement can then be stated as follows:
• We can send $m$ to $V$, in which case $V$ responds with an integer $i$. As long as $V$ has been sent less than $10^6$ distinct messages, the following requirement is satisfied: For any $m'$ previously sent to $V$ to which $V$ replied with $i'$, then $i=i'$ if and only if $m=m'$.
• We can send $i$ to $V$, in which case $V$ responds with $z$ or $\bot$. The response satisfies the following requirement: If $V$ previously replied with $i$ when sent message $m$, then $D$ will output $m$ on input of $z$, using expected time $X$.
• We can send compromise to $V$, in which case $V$ responds with any secret keys it may have and a list $(i_1, z_1), (i_2, z_2), \dots, (i_n, z_n)$. If $V$ was sent the messages $m_1, m_2, \dots, m_k$ with replies $j_1,j_2, \dots, j_k$, then $k=n$, $i_1=j_1$, ..., $i_n=j_n$, and $D(z_i) = m_1$, $D(z_2) = m_2$, ..., $D(z_n) = m_n$.
The security requirement is the following:
• If $V$ has been sent $n$ distinct messages, of which $k$ have been sent by the attacker (or can be predicted with reasonable effort) and the remaining $n-k$ are unpredictable to the attacker, then no adversary sending compromise to $V$ can recover the remaining $n-k$ messages with expected cost significantly smaller than $(n-k)X$.
The following scheme seems to satisfy these requirements. We need
• a hash function $H$ (modeled as a random oracle),
• a deterministic one-time symmetric encryption scheme $(\mathcal{E}, \mathcal{D})$, and
• a large family of groups $\mathcal{G}$ such that we can efficiently sample a group $G$, a generator $g$ and a verifiably random element $y$ in the subgroup generated by $g$, such that $(G,g)$ are suitable for ElGamal encryption, and the best algorithm for computing logarithms smaller than $T$ in $G$ requires expected cost $\sqrt{T}$. (All reasonable assumptions.)
The interactive algorithm $V$ stores a list of tuples $(i, c, G, g, y, x, w)$:
• On input of $m$ to $V$, let $K = H(m)$ and $c = \mathcal{E}(K,m)$. If $(i,c,-,-,-,-,-)$ is stored, respond with $i$ and stop. Otherwise sample fresh $i$. Sample $G$ from $\mathcal{G}$ as well as $g,y$, then choose random $r < X^2$. Compute $x = g^r$, $w=y^r K$. Store $(i, c, G, g, y, x, w)$.
• On input of $i$ to $V$, if $(i, c, x, w)$ is stored, respond with $z = (c, G, g, y, x, w)$.
• On input of compromise, respond with the entire list of tuples.
The algorithm $D$ on input of $(c, G, g, y, x, w)$ does:
1. Compute the discrete logarithm of $x$ to the base $g$. (Expected cost $X$.)
2. Compute $K = w y^{-r}$.
3. Compute $m = \mathcal{D}(K, c)$.
4. (Optional) Verify that $H(m) = K$.
This scheme is correct, and a five-minute analysis suggests that it also satisfies the security requirement. The only minor subtlety in the scheme is that every entry must have a fresh group, otherwise algorithms for computing many discrete logarithms in one group could allow the adversary to decrypt with expected time smaller than $(n-k)X$.
I would not actually use this scheme without a proper security analysis.
A few minor notes: The families of groups can be either based on finite fields or elliptic curves, where the latter would probably involve point counting and therefore be moderately expensive. In either case, naïve ElGamal is probably not the right scheme to use, some variant of DHIES or ECIES or something would be better.
You can make tokenization cost $T$ work as follows: Let $K' = H(m)$. Use $K'$ to generate an elliptic curve of prime order $\approx T^2$ and two verifiably random points $P$ and $Q$ on the curve. Compute the discrete logarithm $U = \log_P Q$. Then let $K = H(m,U)$. Another (simpler) option (as was pointed out in another answer) is to simply use an expensive hash function such as PBKDF2 or scrypt with appropriate parameters.
-
Would it be possible to make the tokenization also computationaly hard (configurable?) to mitigate the fact that the plaintext is not arbitrary length. – itaifrenkel Feb 7 '14 at 20:54
I think so. I've added a paragraph at the end. – K.G. Feb 8 '14 at 15:48
"We can send compromise to V, in which case V responds with a list (i1,z1)...". When compromised V would also respond with any "secret" keys it needs to perform the deterministic symmetric encryption. How would that change the security analysis of this solution? – itaifrenkel Feb 8 '14 at 23:13
Yes, $V$ should respond with any secrets. I've fixed it. There are no long-term secrets in the solution, so it doesn't matter. – K.G. Feb 9 '14 at 12:52
Given the nature (Credit Card Numbers) of the 16-digit decimal numbers, they include one Luhn check digit, and it is trivial to reconstruct any unknown digit from the 15 others. With their 6 first digits and 4 last digits assumed known, the remaining 6 digits have at most as much entropy as 5 decimal digits, that is $b=5\cdot\log_2(10)\approx16.6$ bit. The following generalizes to a short plaintext (say at most 100 bytes) with an unknown portion taking $2^b$ equally likely values trivially determinable from the rest of the plaintext, and would be adaptable to $b$ bit of entropy and a publicly known distribution, like a bias towards small values.
An unavoidable limitation is that, with access to the vault's internals (even without the detokenization credentials), partial information on a plaintext in the vault, and matching index, an attacker can repeatedly query candidate plaintexts (say sequentially from a random starting point) into a simulation of the vault, and check if the simulation is returning the index. Expected cost is that of $2^{b-1}$ tokenizations. Expected time is $2^{b-1}/n$ queries with $n$ simulations of the vault running in parallel at the same speed as the vault, for $n\ll2^b$. Thus one query to the vault must require significant work $W_q$ on average. If we are willing to wait 1 second per query to the vault, and have the vault consume 10 kW (about the design power of the NEMA 14-50 plug sometime used for charging an electric car) during that (bringing the cost of electricity alone to $\$0.0004$at my home's rate), an attacker using the same hardware and rate could recover a plaintext every 5.8 days (at a cost of$\$200$ in electricity) per plaintext; and we should fear the adversary is significantly more efficient that we are. That is not satisfactorily safe by normal cryptographic standards, but better than nothing.
Another unavoidable limitation is that with access to the vault's internals, the detokenization credentials, and an index, an attacker can recover the plaintext for that index by the method used by the vault for detokenization. Thus one detokenization must require significant work $W_d$ on average. Say, if we're willing to spend 100 seconds per detokenization, and have the vault consume 10 kW during that, an attacker using the same hardware and rate can do so and recover each plaintext at that cost; and again we should fear the adversary is significantly more efficient than we are. That is not safe by any stretch of imagination, and I thus question the rationality of considering an adversary with simultaneously detokenization credentials, valid indexes, and partial information on plaintexts, as in the question right now.
Thus I first describe a system that disregards an adversary with detokenization credentials, but I think matches all the requirements as currently worded (including not holding a detokenization private key outside the vault, if that still allows a passphrase unknown to the adversary as detokenization credentials). I'll then sketch how to modify that to add any feeble resistance we can have against an adversary with detokenization credentials.
• At initialization:
• The vault is given a passphrase $P$, which subsequently will be required only for detokenization.
• The vault stretches $P$ into a 256-bit key $K$ using scrypt and constant salt $S$ unique to the vault; the parameters determining the amount of work (iterations, memory, number of threads/cores) are set for $W_d$.
• The vault deterministically generates an RSA key $(N,e,d)$, using as the necessary source of random bits a CSPRNG seeded with $K$.
• The vault stores the public key $(N,e)$ and zeroizes $P,K,d$ and any other intermediary result.
• The vault initializes its internal variable $I=0$, and an internal database to empty (that will hold one cryptogram per plaintext stored).
• At query:
• The vault deterministically and slowly turns the plaintext it receives into a cryptogram as follows:
• The vault applies scrypt with the plaintext as password, and some constant salt $S'$ unique to the vault, yielding a 256-bit result $R$; work parameters are set for $W_q$.
• The vault enciphers the plaintext into the cryptogram using RSAES-OAEP of PKCS#1v2, using as the necessary source of random bits a CSPRNG seeded with $R$.
• That cryptogram is searched in the database:
• If absent, it is stored in the database at index $I$; $I$ is incremented; and the former $I$ is returned as the index for the plaintext just stored.
• If present, its index is returned.
• At detokenization:
• The vault accepts the index to detokenize, and alleged passphrase $P$.
• The vault stretches the alleged $P$ into alleged $K$ as during initialisation.
• The vault deterministically generates the alleged RSA key $(N,e,d)$ from $K$ as during initialization.
• If the alleged $(N,e)$ matches the stored $(N,e)$, then
• if the index to detokenize is less than current $I$, then
• the vault fetches the cryptogram at the index, deciphers it using $(N,d)$, and outputs the plaintext.
• The vault zeroizes $P,K,d$ and any other intermediary result.
As pointed in comment, it is enough to use a moderate $N$ when $b$ is small, since the system can't be very safe anyway. Given use of RSAES-OAEP, $e=3$ is safe and allows to spend more effort in scrypt (but security authorities frown at $e=3$, thus we might bow and use $e=2^{16}+1$).
The system is such that it is twice safer for any extra unknown bit of information in the partial plaintext, which is a nice-to-have. I do not see that Y known plaintexts as in 5 of the question helps more than by allowing to weed out records corresponding to these Y plaintexts.
If we really want to present some symbolic resistance to an adversary with detokenization credentials, there are options. I'll assume $W_d/W_q\ll2^b$ (in any system, the contrary would be useless against any adversary also holding indexes and partial information on plaintexts, as assumed in the question). Sketch of one possibility:
• We modify query by enciphering, rather than the full plaintext, the plaintext excluding the secret portion $M$ (here of 6 decimal digits), which we replace with $M\bmod\lceil2^{b+1}\cdot W_q/W_d\rceil$ or other suitable hint giving $b+1-\log_2(W_d/W_q)$ bit of information about $M$.
• We modify detokenisation to recover the full plaintext by trying the about $2\cdot W_d/W_q$ candidates that remain (in random order or at least starting from a random point to avoid timing attacks), thus with expected work about $W_d$.
• We modify initialization and detokenisation to stretch $P$ into $K$ with work only a fraction of $W_d$.
Many improvements seem feasible, but I lack the energy to do more than list some:
• random-like indexes as in the question initially;
• reducing the memory used in the vault, e.g. by using a public-key cryptosystem with shorter cryptograms than RSA with small plaintext, or perhaps radically by creeping ciphertext in the indexes;
• allowing plaintext of arbitrarily large size, e.g. by using hybrid encryption;
• storing ciphertext outside the vault without compromising security;
• improved security against an adversary with detokenisation credentials but without the random-like indexes, or/and partial plaintext information.
Further, if we turned around the problem and removed the assumption that the vault is insecure, replacing that with say a security-evaluated Smart Card IC with redundant CPUs, or perhaps just an off-the-shelf Java Card or programmable HSM, we could have much enhanced security without drawing kilowatts during operation or requiring too much of a huge investment. In the simplest embodiment
• Initialization chooses an AES key at random; and initializes an 8-digit PIN.
• Query accepts the plaintext as 16 bytes in ASCII; waits as long as bearable; enciphers the plaintext using AES; outputs the 16-byte ciphertext as index (encodable as a 22-characters base-64 string);
• Detokenization checks the PIN code as familiar in bank Smart Cards and SIM cards, with an error counter, zeroizing the device after three consecutive failed attempts; accepts the index; waits as much as bearable; deciphers the index; and outputs the result.
Note: plaintext can be verified to be valid on Query and Detokenization; that can only help, by limiting the information that an adversary can get.
-
The PKE should be fast so that more effort can be spent on scrypt. $\hspace{2.55 in}$ (I would suggest $\:\operatorname{length}(N\hspace{.02 in}) = 1280\:$ and $\:e=3\;$.) $\;\;\;$ – Ricky Demer Feb 7 '14 at 21:44
I updated the question to clarify that the private key should be assumed to be compromised too. – itaifrenkel Feb 8 '14 at 6:50
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{}
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# Probabilistic data structures. Quotient filter
In this article, we continue our acquaintance with implementations of probabilistic sets and consider a modern successor of the Bloom filter that is called Quotient filter. Such data structures can effectively work in situations when we need to handle billions of elements and have optimized memory access.
We continue to discover a problem of approximate membership and if you miss the Bloom filter article, you might like to read it first.
One of the main limitations of Bloom filters that we mentioned early was significantly bad performance when your data doesn't pass into main memory (filter is too big). Of course, one can use a disk to store some parts of the filter. But remember, to add or test elements in Bloom filter we need to perform as many random accesses as we have hash function, which is fast enough in the memory, but not on your HDD (on SSD it's a bit faster, but still not so good). So, if you need to be able to store a lot of elements, prepare enough memory ...
But that if we can change the storage algorithm to store all data related to one element really close with the need to have random access only once. Sounds good, the approach is to use only one hash function, but then we need to deal with high collision probability. Quite smart solution is known as Quotient filter.
## Quotient filter
Quotient filter was introduced by Michael Bender et al. in 2011. It is a space-efficient probabilistic data structure that implements a probabilistic set with 4 operations:
• add element into the set
• delete element from the set
• test whether an element is a member of the set
• test whether an element is not a member of the set
Quotient filter can be described by 2 parameters:
• p - size (in bits) for fingerprints
• 1 hash function that generates such fingerprints
Quotient filter doesn't store the element itself, an only p-bit fingerprint is stored.
Quotient filter is represented as a compact open hash table with m = 2^q buckets. The hash table employs quotienting, a technique suggested by D. Knuth:
• the fingerprint f is partitioned into:
• the r least significant bits (f_r = f mod 2^r, the remainder)
• the q = p - r most significant bits (f_q = ⌊\frac{f}{2^r}⌋, the quotient)
The remainder is stored in a bucket indexed by the quotient. Each bucket contains 3 bits, all 0 at the beginning: is_occupied, is_continuation, is_shifted.
If two fingerprints f and f′ have the same quotient (f_q = f′_q) - it is a soft collision. All remainders of fingerprints with the same quotient are stored contiguously in a run.
If necessary, a remainder is shifted forward from its original location and stored in a subsequent bucket, wrapping around at the end of the array.
• is_occupied is set when the bucket j is the canonical bucket (f_q = j) for some fingerprint f, stored (somewhere) in the filter
• is_continuation is set when the bucket is occupied but not by the first remainder in a run
• is_shifted is set when the remainder in the bucket is not in its canonical bucket
## Algorithm
### To test an element
• Apply the hash function the to element and calculate fingerprint f.
• Calculate quotient f_q and remainder f_r for the fingerprint f.
• If bucket f_q is not occupied, then the element definitely not in the filter.
• If bucket f_q is occupied:
• starting with bucket f_q, scan left to locate bucket without set is_shifted bit.
• scan right with running count (is_occupied: +1, is_continuation: -1) until the running count reaches 0 - when it's the quotient's run.
• compare the stored remainder in each bucket in the quotient's run with f_r
• if found, than element is (probably) in the filter, else - it is definitely not in the filter.
• Apply the hash function the to element and calculate fingerprint f.
• Calculate quotient f_q and remainder f_r for the fingerprint f.
• Follow a path similar to test procedure until certain that the fingerprint is definitely not in the filter
• Choose bucket in the current run by keeping the sorted order and insert remainder f_r (set is_occupied bit)
• Shift forward all remainders at or after the chosen bucket and update the buckets' bits.
## Example
Consider Quotient filter with quotient size q = 3 and 32-bit signed MurmurHash3 as h.
• f_q("amsterdam") = 1, f_r("amsterdam") = 164894540
• f_q("berlin") = 4, f_r("berlin") = -89622902
• f_q("london") = 7, f_r("london") = 232552816
Insertion at this stage is easy since all canonical slots are not occupied. We just store our reminder in their canonical slots.
Add element madrid: f_q("madrid") = 1, f_r("madrid") = 249059682.
The canonical slot 1 is already occupied. The shifted and continuation bits are not set, so we are at the beginning of the cluster which is also the run's start.
The reminder f_r("madrid") is strongly bigger than the existing reminder, so it should be shifted right into the next available slot 2 and shifted bit and continuation bit should be set (but not the occupied bit, because it pertain to the slot, not the contained reminder).
Add element ankara: f_q("ankara") = 2, f_r("ankara") = 62147742.
The canonical slot 2 is not occupied, but already in use. So, the f_r("ankara") should be shifted right into the nearest available slot 3 and its shifted bit should be set. In addition, we need to flag the canonical slot 2 as occupied by setting the occupied bit.
Add element abu dhabi: f_q("abu dhabi") = 1, f_r("abu dhabi") = -265307463.
The canonical slot 1 is already occupied. The shifted and continuation bits are not set, so we are at the beginning of the cluster which is also the run's start.
The reminder f_r("abu dhabi") is strongly smaller than the existing reminder, so all reminders in slot 1 should be shifted right and flagged as continuation and shifted.
If shifting affects reminders from other runs/clusters, we also shift them right and set shifted bits (and mirror the continuation bits if they are set there).
Test element ferret: f_q("ferret") = 1, f_r("ferret") = 122150710.
The canonical slot 1 is already occupied and it's the start of the run. Iterate through the run and compare f_r("ferret") with existing reminders until we found the match, found a reminder that is strongly bigger, or hit the run's end.
We start from slot 1 which reminder is smaller, so we continue to slot 2. Reminder in the slot 2 is already bigger than f_r("ferret"), so we conclude that ferret is definitely not in the filter.
Test element berlin: f_q("berlin") = 4, f_r("berlin") = -89622902.
The canonical slot 4 is already occupied, but shifted bit is set, so the run for which it is canonical slot exists, but is shifted right.
First, we need to find a run corresponding to the canonical slot 4 in the current cluster., so we scan left and count occupied slots. There are 2 occupied slots found (indices 1 and 2), therefore our run is the 3rd in the cluster and we can scan right until we found it (count slots with not set continuation bit).
Our run starts in the slot 5 and we start comparing f_r("berlin") with existing values and found exact match, so we can conclude that berlin is probably in the filter.
## Properties
• False positives are possible. The situations when an element is not a member, but filter returns like it is a member. Fortunately, it's not too probable situation and we can even estimate it's probability:
P(e \in S| e \notin S) \leq \frac{1}{2^r}
• It's possible to tune probability of false positives responses. As we can see from the formula above, such probability depends on the size of the fingerprint's reminder.
• False negatives are not possible. Filter returns that an element isn't a member only if it's definitely not a member:
P(e \notin S| e \in S) = 0
• Hash function should generate uniformly distributed fingerprints.
• The length of most runs is O(1) and it is highly likely that all runs have length O(log m)
• Quotient filter efficient for large number of elements (~1B for 64-bit hash function)
## Quotient filter vs. Bloom filter
• Quotient filters are about 20% bigger than Bloom filters, but faster because each access requires evaluating only a single hash function.
• Results of comparison of in-RAM performance (M. Bender et al.):
• inserts. BF: 690 000 inserts per second, QF: 2 400 000 insert per second
• lookups. BF: 1 900 000 lookups per second, QF: 2 000 000 lookups per second
• Lookups in Quotient filters incur a single cache miss, as opposed to at least two in expectation for a Bloom filter.
• Two Quotient Filters can be efficiently merged without affecting their false positive rates. This is not possible with Bloom filters.
• Quotient filters support deletion.
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# [NRG] Reminder: NRG Meeting: "Modern DNS cache network" (Nicaise Eric Cho... @ Mon Sep 23, 2013 11am - 12pm (NRG Calendar)
Sun Sep 22 11:00:07 EDT 2013
This is a reminder for:
Title: NRG Meeting: "Modern DNS cache network" (Nicaise Eric Choungmo
Fofack)
Title: Modern DNS cache network
Presenter: Nicaise Eric Choungmo Fofack
PhD student, U. Nice Sophia Antipolis / INRIA (France)
http://www-sop.inria.fr/members/Nicaise.Choungmo_Fofack
Abstract: Caching is undoubtedly one of the most popular solution that
easily scales up with a world-wide deployment of resources. The Domain Name
System (DNS) is a well-known example of such cache deployment. Recent
experiments on DNS hierarchy reveal that most of DNS caches over Internet
have deviated from their traditional behavior by violating the Time-To-Live
(TTL) rule specified in the RFC 6195. Theses DNS caches are called {\em
modern DNS} in the literature. In this paper, we provide an analytic tool
to assess the performance of the modern DNS hierarchy. We first introduce a
class of expiration-based (or TTL-based) cache systems which enables us to
describe the observed characteristics of modern DNS caches. We then compute
the metrics of interest and establish several properties from the
perspective of the modern caching hierarchy and end users, respectively
using simple arguments of renewal theory. We evaluate our model on a single
cache using real DNS traces and at network level through Event-driven and
Monte-Carlo simulations, jointly used with Fourier Amplitude Sensitivity
Test to explore the space of the input parameters. We observe that our
analytic model predicts remarkably well all performance metrics at all
caches of the network with relative errors smaller than $1\%$. Finally, we
use our theoretic findings to characterize the optimal TTL configuration,
explain some observed phenomenons and share the lessons learned on this
modern DNS hierarchy.
Bio: Graduated in June 2009, he got his Engineering degree in
Telecommunications Systems and Networks form National School of Applied
Science (ENSA) of Tanger. The year after in August 2010, he received the
MSc degree in Ubiquitous Networking and Computing at University of Nice
Sophia Antipolis (UNS). He was granted by The 2010 French Goverment MESR of
the University of Nice Sophia Antipolis for a PhD. This PhD thesis is
hosted in the team-project MAESTRO at INRIA Sophia Antipolis for the 3rd
and last year. He received the Best student Paper Award of the 6th
International Conference on Performance Evaluation Methodologies and Tools
(ValueTools) 2012 for the paper "Analysis of TTL-based Cache Networks",
joined work with Philippe Nain (his PhD advisor), Giovanni Neglia and Don
Towsley (who he is currently visiting).
When: Mon Sep 23, 2013 11am – 12pm Eastern Time
Where: MCS-148, 111 Cummington Mall, Boston, MA 02215
Calendar: NRG Calendar
Who:
* Larissa Spinelli - creator
Event details:
You are receiving this email at the account nrg-l at cs.bu.edu because you are
subscribed for reminders on calendar NRG Calendar.
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# I Chain rule in a multi-variable function
Tags:
1. May 7, 2016
### Ananthan9470
Suppose you have a parameterized muli-varied function of the from $F[x(t),y(t),\dot{x}(t),\dot{y}(t)]$ and asked to find $\frac{dF}{dt}$, is this the correct expression according to chain rule? I am confused because of the derivative terms involved.
$\frac{dF}{dt}=\frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt}$
Or similar terms containing $\dot{x}(t)$ etc should also be included or it is something else altogether?
2. May 7, 2016
### Staff: Mentor
If the function had parameters x, y, z, and w, the total derivative would have four terms, with the last two being $\frac{\partial F}{\partial z} \frac{dz}{dt} + \frac{\partial F}{\partial w} \frac{dw}{dt}$. I believe that the derivative you're trying to find needs similar terms, with the partials being with respect to $\dot{x}$ and $\dot{y}$.
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# Digital signal processing
## z-Transformation
The infinite impulse response is a property of a filter of digital signal processing. The filter response of the impulse signal does not end infinitely. The filter can be represented in either the time domain or frequency domain. Time-domain filter modifies time-domain input signal to time-domain output signal. The frequency domain lets us understand or to design the effect of the filter. The discrete Fourier transformation $$\Sigma^{N-1}_{n=0} x(n)e^{-i2\pi nm/N}$$ transforms time domain to frequency domain in the finite impulse response.
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# Team:Aachen/Interlab Study/Hardware
(Difference between revisions)
Revision as of 22:13, 22 August 2014 (view source)Mjoppich (Talk | contribs) (→Open-Source DIY Hardware)← Older edit Revision as of 22:24, 22 August 2014 (view source)Mjoppich (Talk | contribs) (→Open-Source DIY Hardware)Newer edit → Line 6: Line 6: {{Team:Aachen/BlockSeparator}} {{Team:Aachen/BlockSeparator}} - = Open-Source DIY Hardware = + = Open Source DIY Hardware = + + Being in the measurement track and having a team of highly motivated engineering and computer science students, we tackled the challenge to build, document and evaluate our open source hardware approach. + + For our daily tasks in the lab, two key devices were detected: fluorometer and OD-meter. As we use GFP most of the time, the fluorometer is designed to work best with GFP. For modularity reasons, and re-usability, it is designed such that a change to another fluorescence protein is easy. + + Besides the mandatory $\mu$-controller architecture, we worked together with the [https://hci.rwth-aachen.de/fablab Fablab Aachen] to construct the device. There we have the chance to use laser cutters and 3D printers. + + The core component for detecting the light intensity is the cuvette holder. Please find the 3D model we printed below: + + +
+ +
+ + + This cuvette holder can be used for both devices: the whole in the bottom is used for fluorescence measurement, the two opposite wholes are used for the light sensor and the LED for optical density measurement respectively. == Fluorescence == == Fluorescence ==
# Open Source DIY Hardware
Being in the measurement track and having a team of highly motivated engineering and computer science students, we tackled the challenge to build, document and evaluate our open source hardware approach.
For our daily tasks in the lab, two key devices were detected: fluorometer and OD-meter. As we use GFP most of the time, the fluorometer is designed to work best with GFP. For modularity reasons, and re-usability, it is designed such that a change to another fluorescence protein is easy.
Besides the mandatory $\mu$-controller architecture, we worked together with the Fablab Aachen to construct the device. There we have the chance to use laser cutters and 3D printers.
The core component for detecting the light intensity is the cuvette holder. Please find the 3D model we printed below:
This cuvette holder can be used for both devices: the whole in the bottom is used for fluorescence measurement, the two opposite wholes are used for the light sensor and the LED for optical density measurement respectively.
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Black holes: A first pass
What is a black hole?
The central idea behind black holes is the notion of trapping. Can there be a region of space where one can go in, but from which one cannot come out? Such a region is called a black hole.
Why should we think such regions might exist? The reason is gravity. Gravity is a universal force that makes everything attract everything else. If we put a lot of mass in some region, then any object coming near this region will be strongly pulled in by the gravitational attraction created by this mass. The question then is: Can this gravitational force trap objects that fall in, so that they can never come out?
Quite remarkably, the answer to this question has changed each time we have incorporated a new principle of physics in our thinking. It is likely that the best way to arrive at a final theory unifying all principles is to look for a consistent description of the black hole. The theory of black holes is therefore the cutting edge of research into the ultimate laws of nature.
The story of black holes has gone through 4 iterations. In each new iteration, we incorporate a new principle of physics, and find a change in what can and cannot be trapped.
A black hole is a region of space into which one can fall in but from which one cannot come out.
The term 'black hole' was coined by John Wheeler in 1967
First iteration
We assume that:
• Gravity is described by Newton's law of gravitation, where the attraction between two bodies falls off as the square of their separation.
• The motion of particles is described by Newton's laws of motion, where the acceleration of a particle is proportional to the force acting on it.
Using these principles, John Michell conjectured in 1783 that sufficiently massive stars may be invisible: the gravitational pull of the body may be so strong that even light cannot escape from their surface and reach our eyes. He called such objects dark stars.
Today we have learnt that nothing can travel faster than the speed of light. If we add this assumption to our Newtonian thinking, then it would seem that nothing can fly out of a dark star. Are such dark stars therefore black holes?
It turns out the answer is no. Even though light and other objects may not be able to fly out on their own from such dark stars, they can still be extracted out by applying a suitable force. Consider a person standing on such a star. He cannot jump up and escape the gravitational pull of the star. But he can sit in a rocket, and the thrust provided by the rocket can lift him out to empty space, away from the star.
Thus we see that with Newtonian physics, nothing is really 'trapped' by gravity, and we cannot get a black hole.
The situation changed with the work of Einstein.
John Michell ,
1724-1793.
A normal star emits light.
Michell argued that light will not be able to escape the gravity of a sufficiently massive object, so we will get a 'dark star'.
But the dark star is not a black hole, since an object (the black dot) can still be extracted away from the star using the force supplied by a rocket.
Second iteration
We assume that:
• Gravity is described by Einstein's theory of general relativity. In this theory the effects of gravity are incorporated by making space and time curved rather than flat.
• The behavior of matter in general relativity is given by a natural extension to curved spacetime of the behavior one had in flat spacetime. A key feature is that no form of matter - particles or waves - can move faster than the speed of light.
In this theory the curvature of spacetime can be made such that we get a trapping region: i.e. we get a black hole. There is a boundary surface called the horizon: things can fall in through the horizon, but they cannot come out. In fact more is true: once things fall through the horizon, they must keep moving towards the center of the hole, where they eventually get crushed at a location called the singularity.
When stars run out of the fuel they can burn, they start to compress under their own gravitational attraction. Computations show that if a star is more than 3 times heavier than the sun, then it would compress in a runaway fashion. That is, it would get more and more dense as it compresses, till it becomes an infinite density point which forms the singularity. This process is called the gravitational collapse of the star, and it results in the formation of a black hole.
The existence of such black holes marked a very unusual step in the development of physics. In Newton's world (first iteration) anything which could be done could be undone: for example we could collect many atoms together to make a planet, but we could then also prise these atoms apart to break the planet back to individual atoms. By contrast, in general relativity (second iteration) we can make a black hole by throwing a lot of matter together into a region. But we can never 'unmake' the hole, since nothing that went in the hole can ever come out.
While this irreversibility of black hole formation is a strange phenomenon, it is not in contradiction with anything. Thus general relativity is a consistent theory on its own, and this theory predicts the existence of black holes which trap anything that falls into them.
We will find a serious problem, however, at the next iteration.
In 1915 Albert Einstein argued that Newton's law of gravitation should be replaced by a more accurate theory: General Relativity
A typical star (the dark disc) creates a gentle curvature of spacetime
Allowing more curvature eventually leads to a black hole. The exterior of the hole, depicted above, ends in a horizon (the green circle); anything falling past the horizon cannot come back out.
A schematic picture of the black hole interior. Even a rocket trying its best to fly outwards will get dragged back into the singularity and be crushed.
Third iteration
We assume that:
• Gravity is still described by Einstein's theory of general relativity.
• Matter is described by quantum theory, in which all matter is described by waves. In the context of curved spacetime, the appropriate formulation is called quantum field theory in curved spacetime.
With this setup, Hawking made two remarkable discoveries, in 1974 and 1975.
In 1974 Hawking found that due to quantum mechanical effects, the black hole slowly leaked out energy in the form of long wavelength radiation. This radiation is called Hawking radiation, and it leads to a slow evaporation of the hole. We can think of mass and energy as equivalent notions, due to Einstein's relation $$E=mc^2$$. Thus the leakage of energy in the form of Hawking radiation leads to a slow decrease in the mass of the hole, and Hawking concluded that the hole would eventually disappear. So in this third iteration, we cannot trap mass (or equivalently, energy)
The radiation emerging from the black hole has a strange property: it does not carry the information of the matter that made the black hole in the first place. The reason for this is simple: the radiation is produced at the horizon, while the matter which made the hole has been sucked away to a different location - the central singularity.
In 1975 Hawking showed that this situation leads to a very serious problem, as follows:
1. Start with two stars, each having the same mass, but having different compositions; for example one can be made of Hydrogen atoms, and one of Helium atoms.
2. Let each star collapse to a black hole. Since the two black holes have the same mass, they produce the same curvature of spacetime.
3. Hawking radiation is produced by the curvature of spacetime; thus the radiation produced is identical in the two cases.
4. When the two holes evaporate, we are left with radiation in each case. But since the radiation is exactly the same in the two cases, we cannot look at the radiation and tell whether it came from the star made of Hydrogen or from the star made of Helium. Thus the information about the detailed composition of the star has been lost; the radiation only knows about the total mass of the initial star.
This problem is called the Black hole information paradox. The reason it is a paradox is that in all formulations of physics considered before, information was not lost. For example in Newtonian physics, we may start with particles in one configuration and after some time end with a different configuration; but we can always look at the final configuration and figure out exactly which initial configuration it came from. Similarly, in quantum theory we start with an initial wavefunction, and this evolves to a final wavefunction; but we can look at the final wavefunction and know which initial wavefunction it came from. Thus information is never lost in normal physical processes; it just gets pushed into different forms.
But in the Hawking process we have seen that we cannot look at the final wavefunction (describing radiation) and say which initial wavefunction (i.e. which star) it came from. Thus, Hawking argued, the process of black hole formation and evaporation cannot be described by the usual rules of quantum theory; i.e., black holes violate quantum mechanics.
Most people were unhappy with this conclusion, since they did not want to give up quantum theory. But they could not find anything wrong with Hawking's computation of radiation or the fact that this radiation carried no information about the hole. Many people therefore accepted the simplest possible way out: the existence of remnants; defined as follows.
Remnants: Suppose we accept Hawking's assumptions and computations when the black hole is large. But when the black hole is about to vanish, it will be vey small, and for such small black holes the gravity theory should also be modified to include quantum effects; i.e., we should replace Einstein's general relativity by a theory of quantum gravity. It may then happen that some (heitherto unknown) effect in this quantum gravity theory stops the further evaporation of the hole, so that we are left with a tiny remnant. In that case the information that went into the hole (e.g. the composition of a collapsing star) would not be lost, but would instead stay trapped forever in this tiny remnant. We will therefore avoid a conflict with the foundations of quantum theory.
To summarize, in this third iteration we find that mass (or equivalently, energy) is not trapped, while information is either lost or trapped, depending on our assumptions about the endpoint of black hole evaporation.
Surprisingly, the picture of the black hole was to change once again.
Stephen Hawking argued in 1975 that the dynamics of black holes violates quantum theory.
The Hawking evaporation process. Quantum effects create a particle-antiparticle pair near the horizon. One member of the pair escapes as radiation, while the other falls into the hole. Since the radiation carries away energy, the hole becomes smaller.
The information paradox: we cannot look at the final state of radiation and say which star we started with; thus information has been 'lost'.
The remnant idea: the evaporation leaves a tiny remnant which stores the needed information.
Fourth iteration
We assume that:
• Gravity is described by a quantum theory of spacetime; thus Einstein's general relativity needs to be extended to incorporate the principles of quantum mechanics. The most successful theory giving such a unification is string theory.
• Matter is described by quantum theory as well. In string theory, gravity and matter appear together in a unique, unified way. .
Computations with string theory indicate that we get a remarkable change in the dynamics of the hole once again: the hole takes the form of a fuzzball, which traps neither energy nor information. A fuzzball is in principle just like a planet, which also does not trap energy or information.
In a planet, the region inside its surface is filled with a complicated structure of atoms, which carry the 'information' about the planet. A fuzzball also has a complicated structure inside its boundary, and this structure carries the information of the fuzzball. The idea that all black holes are replaced by fuzzballs in quantum gravity is called the fuzzball paradigm.
Since we had defined a black hole as something which creates 'trapping', strictly speaking we should say that there are no black holes in string theory. But interestingly, fuzzballs retain several properties of the traditional black hole (i.e. the black hole that we had in the third iteration). The radius of the typical fuzzball is approximately the same as the horizon radius of the black hole with the same mass. The gravity felt by a particle outside the fuzzball boundary is approximately the same as the gravity outside the horizon of the black hole. Further, the fuzzball emits radiation at the same temperature that Hawking found for his radiation from the traditional black hole. For these reasons we sometimes refer to fuzzballs as being the 'actual states of black holes in string theory'.
Resolution of the information paradox: The crucial fact is that with fuzzballs, there is no black hole information paradox. Consider again the gravitational collapse of a star made of Hydrogen and a star made of Helium. Even though these two stars may have the same total mass, the detailed internal structure of the fuzzballs they create will be different. This difference gets imprinted on the radiation produced by the fuzzballs. Thus when the fuzzballs evaporate away, we can look at the final radiation and know all the information about the star we had started with.
By contrast, in the third iteration the black hole was was 'empty' except for a central singularity. Thus the hole was described only by its spacetime curvature, which did not have any information about the initial star apart from its total mass. Since the radiation was produced by this curvature, it could not carry any information, and we had a paradox.
Non-triviality of the fuzzball paradigm: Since fuzzballs do not create an information problem, one might wonder why people did not seek a fuzzball-like structure right away, instead of worrying about the paradox created by the traditional black hole. The reason is that with ordinary matter, it is impossible to make a horizon sized object that does not collapse under the pull of its own gravity. It is the unique features of string theory: extra dimensions, extended objects like strings and branes etc. that allow for the back hole to get replaced by a fuzzball and resolve the information problem.
Alternative proposals: A very satisfying aspect of the fuzzball paradigm is that it resolves the information paradox without violating any of the fundamental principles of relativity or quantum theory. Alternative resolutions of the paradox have also been proposed, where the traditional structure of the hole is left essentially unchanged; i.e., there is a horizon, and the all the matter inside the horizon has been collapsed to a highly dense ball at the center. In these alternative proposals one typically solves the problem by violating one of the familiar tenets of physics: causality (the principle that nothing can travel faster than light), locality (the principle that objects at one point can directly influence only the things in their immediate vicinity) or quantum unitarity (the evolution rule in quantum theory which says that states do not appear or disappear during time evolution).
A fuzzball is a horizon sized structure that does not collapse because of special features of string theory. The red circles depict 'holes' in space and the green bubbles are spheres that join them. Such configurations result from novel topologies which arise because of the extra dimensions in string theory.
How fuzzballs resolve the information paradox: different stars give different fuzzball structures, and thus different radiation. We can then look at the final radiation and know which star we started with.
in 1959 Buchdahl proved that a spherical ball of normal matter will collapse to a black hole if its radius is shrunk below 1.25 times the horizon radius. Fuzzballs bypass this theorem and resist collapse because string theory allows novel structures not possible with ordinary matter.
One alternative to fuzzballs is the wormhole proposal. A wormhole connects the emitted Hawking radiation particles back to the black hole. Since the radiation goes very far from the hole, this is a kind of long distance nonlocality.
The lesson from the information paradox
Traditionally, any theory of quantum gravity was expected to be relevant only over very tiny distances -- distances shorter than the planck length $$l_p$$ which is about $$\sim 10^{-33} \, cm$$. How then can string theory change the entire interior of a black hole, which is typically several Km across?
The answer is that is that a black hole is characterized by a very large number $$N$$ : the number of elementary constituents that make the hole. For a typical stellar hole, we have $$N \sim 10^{40}$$. We then need to ask: if a large number $$N$$ of elementary constituents are involved in a physical process, then is the length scale relevant for quantum gravity still $$l_p$$, or could it change to something like $$N$$ times $$l_p$$?
The fuzzball paradigm teaches us that in string theory, the distance over which quantum gravity is relevant indeed grows with $$N$$, in such a way that the range of these effects is always comparable to the horizon radius. Thus quantum gravity emerges from the microscopic to the macroscopic domain.
This lesson appeared so radical that at first many people did not accept it, and kept looking for other ways to resolve the information paradox. One suggestion was that the black hole remained as in the third iteration to a good approximation, but tiny quantum gravity effects encoded the information in subtle ways among the large number of particles emitted in Hawking radiation. In fact Hawking himself surrendered a bet to John Preskill in 2004, saying in essence that such tiny corrections might have a cumulative effect that would invalidate his 1975 claim that black holes posed a puzzle. But most people did not agree with Hawking's 2004 reasoning, and Hawking's co-signer on the bet - Kip Thorne - refused to surrender his position that black holes posed a puzzle.
In 2009, the small corrections theorem was proved, which used results from quantum information theory to show that tiny corrections could not encode the information in the emitted radiation. This theorem provided a natural closure for the fuzzball paradigm: the theorem showed that large modifications were needed at the horizon of the black hole used in the third iteration, and the fuzzball structure provided exactly such large modifications. Meanwhile more and more configurations for black holes were constructed, and in each case the structure was found to be that of a fuzzball, rather than of the traditional hole with horizon.
Nevertheless some people continued with efforts to preserve the traditional structure of the black hole while evading the information paradox. These efforts typically involve altering some fundamental principle of quantum theory or gravity. In particular, some of these efforts try to evade the information problem by saying that particles that float far away as Hawking radiation are not distinct from entities that live inside the black hole; this identification of 'data bits' is represented by the wormholes mentioned above.
In our full exposition of the information paradox, we will explain these ideas in detail.
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# Selina solutions for Concise Physics Class 10 ICSE chapter 1 - Force [Latest edition]
Course
Textbook page
## Chapter 1: Force
Exercise -1 (A)Exercise - 1 (A)OthersExercise -1(A)Exercise - 1(A)Exercise - 1 (B)Exercise - 1 (C)
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise -1 (A), Exercise - 1 (A), Exercise -1(A), Exercise - 1(A) [Pages 9 - 10]
Exercise -1 (A) | Q 1.1 | Page 9
State the condition when on applying a force, the body has:
the translational motion
Exercise - 1 (A) | Q 1.2 | Page 9
State the condition when on applying a force, the body has:
the rotational motion.
Q 2 | Page 9
Define moment of force and state its S.I unit.
Exercise -1 (A) | Q 3 | Page 9
State whether the moment of force is a scalar or vector quantity?
Exercise - 1 (A) | Q 4 | Page 9
State two factors affecting the turning effect of a force.
Exercise -1 (A) | Q 5 | Page 10
When does a body rotate? State one way to change the direction of rotation of a body. Given a suitable example to explain your answer.
Exercise -1 (A) | Q 6 | Page 10
Write the expression for the moment of force about a given axis.
Exercise -1(A) | Q 7 | Page 10
What do you understand by the clockwise and anticlockwise moment of force? When is it taken positive?
Exercise - 1 (A) | Q 8 | Page 10
State one way to reduce the movement of given force about a given axis of rotation.
Exercise -1 (A) | Q 9 | Page 10
State one way to obtain greater moment of a given force about a given axis of rotation.
Exercise - 1 (A) | Q 10 | Page 10
Why is it easier to open a door by applying the force at the free end of it?
Exercise - 1(A) | Q 11 | Page 10
The stone of hand flour grinder is provided with a handle near its rim. Give reason.
Exercise - 1 (A) | Q 12 | Page 10
It is easier to turn the steering wheel of a large diameter than that of a small diameter. Given reason.
Exercise - 1 (A) | Q 13 | Page 10
A spanner (or wrench) has a long handle. Why?
Exercise - 1 (A) | Q 14 | Page 10
A jack screw is provided with a long arm. Explain why?
Exercise - 1 (A) | Q 15 | Page 10
A, B and C are the three forces each of magnitude 4 n acting in the plane of the paper as shown in Figure. The point O lies in the same plane.
1) Which force has the least moment about O? Give a reason.
2) which force has the greatest moment about O? Give a reason.
3) Name the forces producing (a) Clockwise (b) anticlockwise moments.
4) what is the resultant torque about the point O?
Exercise - 1 (A) | Q 16 | Page 10
The adjacent diagram shows a heavy roller, with its axle at O. which its axle at O. which is to be raised on a pavement XY by applying a minimum possible force. show by an arrow on the diagram the point of application and the direction in which the force should be applied.
Exercise - 1 (A) | Q 17.1 | Page 10
A body is acted upon by two force of magnitude F but in opposite direction. State the effect of the force if
both forces act at the same point of the body.
Exercise - 1 (A) | Q 17.2 | Page 10
A body is acted upon by two force of magnitude F, but in opposite directions. State the effect of the force if
the two forces act at two different points of the body at a separation r.
Exercise - 1 (A) | Q 18 | Page 10
Draw a neat labelled diagram to show the direction of two forces acting on a body to produce rotation in it. Also, mark the point about which rotation takes place by the letter O.
Exercise - 1 (A) | Q 19.1 | Page 10
What do you understand by the term couple?
Exercise - 1 (A) | Q 19.2 | Page 10
State the couple effect. Give two example of couple action in our daily life.
Exercise - 1 (A) | Q 20 | Page 10
Define moment of couple. Write its S.I unit
Exercise - 1 (A) | Q 21 | Page 10
Prove that Moment of couple = Force x couple arm.
Exercise - 1 (A) | Q 22 | Page 10
What do you mean by an equilibrium of a body?
Exercise - 1 (A) | Q 23.1 | Page 10
State the condition when a body is in static equilibrium. Give one example of static equilibrium.
Exercise - 1 (A) | Q 23.2 | Page 10
State the condition when a body is in dynamic equilibrium. Give one example of dynamic equilibrium.
Exercise - 1 (A) | Q 24 | Page 10
State two condition for a body acted upon by several forces to be in equilibrium.
Exercise - 1 (A) | Q 25 | Page 10
State the principle of moments. Give one device as an application of it
Exercise - 1 (A) | Q 26 | Page 10
Describe a simple experiment to verify the principle of moments, if you are supplied with a metre rule, a fulcrum and two springs with slotted weights.
Exercise - 1 (A) | Q 27.1 | Page 10
Complete the following sentence :
The S.I. unit of moment of force is _________.
Exercise - 1 (A) | Q 27.2 | Page 10
Complete the following sentence :
In equilibrium algebraic sum of moments of all forces about the point of rotation is ______________.
Exercise - 1 (A) | Q 27.3 | Page 10
Complete the following sentence :
In a beam balance when the beam is balanced in a horizontal position, it is in ____________equilibrium.
Exercise - 1 (A) | Q 27.4 | Page 10
Complete the following sentence :
The moon revolving around the earth is in ____________ equilibrium.
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (A) [Pages 10 - 11]
Exercise - 1 (A) | Q 1 | Page 10
Multiple choice Type:
The moment of a force about axis depends:
• Only on the magnitude of force
• Only on the perpendicular distance of force from the axis
• Neither on the force nor on the perpendicular distance of force from the axis
• Both on the force and its perpendicular distance from the axis.
Exercise - 1 (A) | Q 2 | Page 11
Multiple choice Type:
A body is acted upon by two unequal forces in opposite directions, but not in the same line. The effect is that:
• The body will have only the rotational motion
• The body will have only the translational motion
• The body will have neither the rotational motion nor the translational motion
• The body will have rotational as well as translational motion.
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (A) [Pages 11 - 12]
Exercise - 1 (A) | Q 1 | Page 11
The moment of a force of 10 N about a fixed point O is 5 N m. Calculate the distance of the point O from the line of action of the force.
Exercise - 1 (A) | Q 2 | Page 11
A nut is opened by a wrench of length 10 cm. if the least force required is 5.0 N. find the moment of force needed to turn the nut.
Exercise - 1 (A) | Q 3 | Page 11
A wheel of diameter2 m is shown in the figure with the axle at O. A force F = 2 N is applied at B in the direction shown in the figure. Calculate the moment of force about Centre O and point A.
Exercise - 1 (A) | Q 4 | Page 11
The diagram in the figure shows two forces F1 = 5 N and F2 = 3N acting at point A and B of a rod pivoted at a point O, such that OA = 2m and OB = 4m
Calculate:
1) Moment of force F1 about O
2) Moment of force F2 about O
3) Total moment of the two forces about O.
Exercise - 1 (A) | Q 5 | Page 11
Two forces each of magnitude 10 N act vertically upwards and downwards respectively at the two ends of a uniform road of length 4m which is pivoted at its midpoint as shown in the figure. Determine the magnitude of the resultant moment of forces about the pivot O.
Exercise - 1 (A) | Q 6 | Page 11
The figure shows two forces each of magnitude 10 N acting at the point A and B at a separation of 50 cm, in opposite directions. Calculate the resultant moment of two forces about the point A, B and O, situated exactly at the middle of the two forces.
Exercise - 1 (A) | Q 7 | Page 11
A steering wheel of diameter 0.5 m is rotated anticlockwise by applying two forces each of magnitude 5 N. Draw a diagram to show the application of forces and calculate the moment of couple applied.
Exercise - 1 (A) | Q 8 | Page 11
A uniform metre rule is pivoted at its mid-point. A weight of 50 gf is suspended at one end of it. where should a weight of 100 gf be suspended to keep the rule horizontal?
Exercise - 1 (A) | Q 9.1 | Page 11
A uniform metre rule balance horizontally on a knife-edge placed at the 58 cm mark when a weight of 20 gf is suspended from one end.
Draw a diagram of the arrangement.
Exercise - 1 (A) | Q 9.2 | Page 11
A uniform metre rule balances horizontally on a knife edge placed at the 58 cm mark when a weight of 20gf is suspended from one end.
What is the weight of the rule?
Exercise - 1 (A) | Q 10 | Page 11
The diagram below Shows a uniform bar supported at the middle point O. A weight of 40 gf is placed at a distance of 40 cm to the left of the point O. How can you balance the bar with a weight of 80 gf?
Exercise - 1 (A) | Q 11.1 | Page 11
The figure shows a uniform metre rule placed on a function at its mid-point O and having a weight 40 gf at the 10 cm mark and a weight of 20 gf at the 9.0 cm mark.
Is the metre rule in equilibrium? If not how will the rule turn?
Exercise - 1 (A) | Q 11.2 | Page 11
Figure shows a uniform metre rule placed on a fulcrum at its mid-point O and having a weight 40gf at the 10 cm mark and a weight of 20gf at the 90 cm mark.
How can the rule be brought in equilibrium by using an additional weight of 40 gf ?
Exercise - 1 (A) | Q 12 | Page 12
When a boy weighing 20 kgf sits at one end of a 4 m long see saw, it gets depressed at this end. How can it be brought to the horizontal position by a man weighing 40 kgf
Exercise - 1 (A) | Q 13 | Page 12
A physical balance has its arms of length 60 cm and 40 cm. What weight kept on pan of the longer arm will balance an object of weight 100 gf kept on other pan?
Exercise - 1 (A) | Q 14.1 | Page 12
The diagram in the figure shows a uniform metre rule weighing 100 gf, pivoted as its centre O. two weight 150 gf and 250 gf hang from the point A and B respectively of the metre rule such that OA = 40 cm and OB = 20 cm . Calculate
the total anticlockwise moment about o.
Exercise - 1 (A) | Q 14.2 | Page 12
The diagram shows a uniform metre rule weighing 100gf, pivoted at its centre O. Two weights 150gf and 250gf hang from the point A and B respectively of the metre rule such that OA = 40 cm and OB = 20 cm. Calculate :
the total clockwise moment about O
Exercise - 1 (A) | Q 14.3 | Page 12
The diagram shows a uniform metre rule weighing 100gf, pivoted at its centre O. Two weights 150gf and 250gf hang from the point A and B respectively of the metre rule such that OA = 40 cm and OB = 20 cm. Calculate :
the difference of anticlockwise and clockwise moment
Exercise - 1 (A) | Q 14.4 | Page 12
The diagram shows a uniform metre rule weighing 100gf, pivoted at its centre O. Two weights 150gf and 250gf hang from the point A and B respectively of the metre rule such that OA = 40 cm and OB = 20 cm. Calculate :
the distance from O where a 100gf weight should be placed to balance the metre rule.
Exercise - 1 (A) | Q 15.1 | Page 12
A uniform metre rule of weight 10gf is pivoted at its 0 mark.
What moment of force depresses the rule?
Exercise - 1 (A) | Q 15.2 | Page 12
A uniform metre rule of weight 10gf is pivoted at its 0 mark.
How can it be made horizontal by applying a least force ?
Exercise - 1 (A) | Q 16 | Page 12
A uniform half metre rule can be balanced at the 29.0 cm mark when a mass 20g is hung from its one end.
1) Draw a diagram of the arrangement.
2) Find the mass of the half metre rule.
Exercise - 1 (A) | Q 17.1 | Page 12
A uniform metre rule of mass 100 g is balanced on the fulcrum at mark 10 cm by suspending an unknown mass M at the mark 20 cm.
Find the value of M.
Exercise - 1 (A) | Q 17.2 | Page 12
A uniform metre rule of mass 100g is balanced on a fulcrum at mark 40cm by suspending an unknown mass m at the mark 20cm.
To which side the rule will tilt if the mass m is moved to the mark 10cm ?
Exercise - 1 (A) | Q 17.3 | Page 12
A uniform metre rule of mass 100g is balanced on a fulcrum at mark 40cm by suspending an unknown mass m at the mark 20cm.
What is the resultant moment now ?
Exercise - 1 (A) | Q 17.4 | Page 12
A uniform metre rule of mass 100g is balanced on a fulcrum at mark 40cm by suspending an unknown mass m at the mark 20cm.
How can it be balanced by another mass 50 g ?
Exercise - 1 (A) | Q 18 | Page 12
In following figure , a uniform bar of length l m is supported at its ends and loaded by a weight W kgf at its middle. In equilibrium, find the reactions R1 and R2 at the ends.
["Hint" : "In equilibrium" "R"_1 + "R"_2 = "W" "and" "R"_1 xx 1/2 = "R"_2 xx 1/2]
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (B) [Pages 14 - 15]
Exercise - 1 (B) | Q 1 | Page 14
Define the term centre of gravity of a body.
Exercise - 1 (B) | Q 2 | Page 14
Can the centre of gravity be situated outside the material of the body? Give an example
Exercise - 1 (B) | Q 3 | Page 14
On what factor does the position of the centre of gravity of a body depend? Explain your answer with an example.
Exercise - 1 (B) | Q 4.1 | Page 14
What is the position of the centre of gravity of a rectangular lamina?
Exercise - 1 (B) | Q 4.2 | Page 14
What is the position of the centre of gravity of a cylinder?
Exercise - 1 (B) | Q 5.1 | Page 14
At which point is the centre of gravity situated in a triangular lamina.
Exercise - 1 (B) | Q 5.2 | Page 14
At which point is the centre of gravity situated in a circular lamina?
Exercise - 1 (B) | Q 6 | Page 14
Where is the centre of gravity of a uniform ring situated?
Exercise - 1 (B) | Q 7 | Page 14
A square cardboard is suspended by passing a pin through a narrow hole at its one corner. Draw a diagram to show its rest position. In the diagram, mark the point of suspension by the letter S and the centre of gravity by the letter G.
Exercise - 1 (B) | Q 8 | Page 15
Explain how you will determine the position of the centre of gravity experimentally for a triangular lamina (or a triangular piece of cardboard)
Exercise - 1 (B) | Q 9.1 | Page 15
State True or False
The position of the centre of gravity of a body remains unchanged even when the body is deformed.
• True
• False
Exercise - 1 (B) | Q 9.2 | Page 15
State True or False
Centre of gravity of a freely suspended body always lies vertically below the point of suspension.
Exercise - 1 (B) | Q 10 | Page 15
A uniform flat circular rim is balanced on a sharp vertical nai by supporting it at point A, as shown in the figure. Mark the position of the centre of gravity of the rim in the diagram by the letter G.
Exercise - 1 (B) | Q 11 | Page 15
The figure shows three pieces of cardboard of uniform thickness cut into three different shapes. On each diagram draw two lines to indicate the position of the centre of gravity G.
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (B) [Page 15]
Exercise - 1 (B) | Q 1 | Page 15
The centre of gravity of a uniform ball is
a) at its geometrical centre
b) at its bottom
c) at its topmost point
d) at any point on its surface
• at its geometrical centre
• at its bottom
• at its topmost point
• at any point on its surface
Exercise - 1 (B) | Q 2 | Page 15
The centre of gravity of a hollow cone of height h is at distance x from its vertex where the value of x is:
• "h"/3
• "h"/4
• "2h"/3
• "3h"/4
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (C) [Page 18]
Exercise - 1 (C) | Q 1 | Page 18
Explain the meaning of uniform circular motion. Give one example of such motion.
Exercise - 1 (C) | Q 2 | Page 18
Draw a neat labelled diagram for a particle moving in a circular path with a constant speed. In you diagram show the direction of velocity at any instant.
Exercise - 1 (C) | Q 3 | Page 18
Is it possible to have an accelerated motion with a constant speed? Name such type of motion.
Exercise - 1 (C) | Q 4 | Page 18
Give an example of motion in which speed remains uniform, but the velocity changes.
Exercise - 1 (C) | Q 5 | Page 18
A uniform circular motion is an accelerated motion explain it. State whether the acceleration is uniform or variable? Name the force responsible to cause this acceleration. What is the direction of force at any instant? Draw diagram in support of your answer.
Exercise - 1 (C) | Q 6 | Page 18
Differentiate between Uniform linear motion and Uniform circular motion.
Exercise - 1 (C) | Q 7 | Page 18
Name the force required for circular motion. State its direction.
Exercise - 1 (C) | Q 8 | Page 18
What is a centripetal force?
Exercise - 1 (C) | Q 9 | Page 18
Explain the motion of a planet around the sun in a circular path.
Exercise - 1 (C) | Q 10.1 | Page 18
With reference to the direction of action, how does a centripetal force differ from centrifugal force?
Exercise - 1 (C) | Q 10.2 | Page 18
Is centrifugal force the force of rection of centripetal force?
Exercise - 1 (C) | Q 10.3 | Page 18
Compare the magnitudes of centripetal and centrifugal force.
Exercise - 1 (C) | Q 11 | Page 18
Is centrifugal force a real force?
Exercise - 1 (C) | Q 12 | Page 18
A small pebble is placed near the periphery of a circular disc which is rotating about an axis passing through it centre.
(a) What will be your observation when you are standing outside the disc? Explain it
(b) What will be your observation when you are standing at the centre of the disc. Explain it
Exercise - 1 (C) | Q 13 | Page 18
A piece of stone tied at the end of a thread is whirled in a horizontal circle. Name the force which provides the centripetal force.
1. Is the velocity of stone uniform or variable?
2. Is the acceleration of stone uniform or variable?
3. What is the direction of acceleration of stone at any instant?
4. What force does provide the centripetal force required for circular motion?
5. Name the force and its direction which acts on the hand.
Exercise - 1 (C) | Q 14 | Page 18
State two differences between the centripetal and centrifugal force.
Exercise - 1 (C) | Q 15.1 | Page 18
State True or False
The earth moves around the sun with a uniform.
• True
• False
Exercise - 1 (C) | Q 15.2 | Page 18
State True or False
The motion of the moon around the earth in a circular path is an accelerated motion.
• True
• False
Exercise - 1 (C) | Q 15.3 | Page 18
State True or False:
A uniform linear motion is unaccelerated , while a uniform circular motion is an accelerated motion.
• True
• False
Exercise - 1 (C) | Q 15.4 | Page 18
State True or False
In a uniform circular motion, the speed continuously changes because of the direction of motion changes.
• True
• False
Exercise - 1 (C) | Q 15.5 | Page 18
State whether the following statement are true or false by writing T/F against them.
A Boy experiences a centrifugal force on his hand when he rotates a piece of stone tied at one end of a string, holding the other end in the hand.
• True
• False
#### Selina solutions for Concise Physics Class 10 ICSE Chapter 1 Force Exercise Exercise - 1 (C) [Pages 0 - 18]
Exercise - 1 (C) | Q 1 | Page 18
Which of the following quantity remains constant in uniform circular motion:
• Velocity
• Speed
• Acceleration
• Both velocity and speed
Exercise - 1 (C) | Q 2
The centrifugal force is:
• a real force
• the force of reaction of centripetal force
• a fictitious force
• directed towards the centre of circular path
## Chapter 1: Force
Exercise -1 (A)Exercise - 1 (A)OthersExercise -1(A)Exercise - 1(A)Exercise - 1 (B)Exercise - 1 (C)
## Selina solutions for Concise Physics Class 10 ICSE chapter 1 - Force
Selina solutions for Concise Physics Class 10 ICSE chapter 1 (Force) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Concise Physics Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster.
Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Selina textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.
Concepts covered in Concise Physics Class 10 ICSE chapter 1 Force are Turning Forces Concept, Moment of a Force, Forces in Equilibrium, Centre of Gravity, Force - Uniform Circular Motion, Concept of Force, Elementary Introduction of Translational and Rotational Motions, Turning Effect of Force (Moment of Force).
Using Selina Class 10 solutions Force exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Selina Solutions are important questions that can be asked in the final exam. Maximum students of CISCE Class 10 prefer Selina Textbook Solutions to score more in exam.
Get the free view of chapter 1 Force Class 10 extra questions for Concise Physics Class 10 ICSE and can use Shaalaa.com to keep it handy for your exam preparation
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# By you
Geometry Level pending
Let $$M$$ be the midpoint of side $$BC$$ of triangle $$ABC$$. Let the median $$AM$$ intersect the incircle of $$ABC$$ at $$K$$ and $$L, K$$ being nearer to $$A$$ than $$L$$, where $$AK=KL=LM$$. If the ratio of the sides of triangle $$ABC$$ are in ratio $$x:y:z$$. Find $$x+y+z$$.
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# Journal of Operator Theory
Volume 73, Issue 1, Winter 2015 pp. 71-90.
On fluctuations of traces of large matrices over a non-commutative algebra
Authors: Yong Jiao (1) and Mihai Popa (2)
Author institution:(1) Institute of Probability and Statistics, Central South University, Changsha 410075, China
(2) University of Texas at San Antonio, Department of Mathematics, One UTSA Circle, San Antonio, TX 78249, U.S.A. and Institute of Mathematics Simion Stoilow'' of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-014700, Romania
Summary: The paper investigates the asymptotic behavior of (non-normalized) traces of certain classes of matrices with non-commutative random variables as entries. We show that, unlike in the commutative framework, the asymptotic behavior of matrices with free circular, respectively with Bernoulli distributed Boolean independent entries is described in terms of free, respectively Boolean cumulants. We also present an example of relation of monotone independence arising from the study of Boolean independence.
DOI: http://dx.doi.org/10.7900/jot.2013sep12.1997
Keywords: non-commutative random variables, random matrices, fluctuation moments, free independence, Boolean independence, monotone independence
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(Redirected from Angular mil)
Not to be confused with Minute of arc.
The PSO-1 reticle in a Dragunov sniper rifle has 10 horizontal lines with 1-mil spacing, which can be used to compensate for wind drift or for range estimation.
Unit system SI derived unit
Unit of Angle
Unit conversions
1 mil in ... ... is equal to ...
turns 1/2000π turn
degrees ≈ 0.057296°
gons ≈ 0.063662g
An angle of 1 radian (marked green, approximately 57.3°) corresponds to an angle where the length of the arc (blue) is equal to the radius of the circle (red), while a milliradian therefore corresponds to an angle equal to a thousandth of that value.
A milliradian, often called a mil or mrad (sometimes capitalized MRAD), is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Mils are widely used in adjustment of firearm sights, where an adjustment of 0.1 mil equals 1 cm at 100 meters.
Scopes with mil-dots or marks in the reticle can be used for range estimation if the target size is known (or vice versa to determine a target size if the distance is known). In such applications, the metric units millimeters for target size and meters for target distance is useful, because they coincide with the definition of the milliradian where arc length is defined as 11000 of the radius.
Just like a circle can be divided into 360 degrees or 2π radians, a circle can instead be divided into 2000π ≈ 6283 milliradians. One milliradian approximately equals ≈ 0.057296° or ≈ 3.4377′ (minutes of arc). While the true definition of a mil (2000π ≈ 6283.185… milliradians in a circle) is used in scope adjustment knobs ("turrets") and optical reticles, there are other rounded definitions used for land mapping and artillery which are easier to divide into smaller parts. For instance there are compasses with 6400 NATO mils, 6000 Warsaw Pact mils or 6300 Swedish "streck's" per circle instead of 360°, achieving higher resolution than a 360° compass while also being easier to divide into parts than if true milliradians were used. The term angular mil is used in artillery.[citation needed]
## History
The milliradian (circle/6283.185…) was first used in the mid nineteenth Century by Charles-Marc Dapples (1837–1920), an engineer and professor at the University of Lausanne.[1] Degrees and minutes were the usual units of angular measurement but others were being proposed, with "grads" (circle/400) under various names having considerable popularity in much of northern Europe. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles (60°, circle/6) and hence 600 units to a circle.
Around the time of the start of World War I, France was experimenting with the use of milliemes (circle/6400) for use with artillery sights instead of decigrades (circle/4000). The United Kingdom was also trialing them to replace degrees and minutes. They were adopted by France although decigrades also remained in use throughout World War I. Other nations also used decigrades. The United States, which copied many French artillery practices, adopted mils (circle/6400). Before 2007 the Swedish defence forces used "streck" (circle/6300, streck meaning lines or marks) (together with degrees for some navigation) which is closer to the milliradian but then changed to NATO mils. After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced "units of base" with meters) the Red Army expanded the 600 unit circle into a 6000 mil one. Hence the Russian mil has nothing to do with milliradians as its origin.
In the 1950s, NATO adopted metric units of measurement for land and general use. Mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.
## Mathematical principle
Use of the milliradian is practical because it is concerned with small angles, and at small angles ${\displaystyle \sin \theta \simeq \theta }$. This allows a user to dispense with trigonometry and use simple ratios to determine size and distance with high accuracy for rifle and short distance artillery calculations. More in detail, because ${\displaystyle {\text{subtension}}\simeq {\text{arc length}}}$, instead of finding the angular distance denoted by θ by using the tangent function
${\displaystyle \theta =\arctan {\frac {\text{subtension}}{\text{range}}}}$,
one can instead make a good approximation by using the definition of a radian and the simplified formula:
${\displaystyle \theta ={\frac {\text{subtension}}{\text{range}}}}$
## Firearm sights
The angular mil is commonly used both in the military and civilian shooting sports as a unit for clicks on scope adjustments knobs (turrets), and in optical reticles allowing rough range estimation and precise shot correction. The mils relationship to the trigonometric radian gives rise to the handy property of subtension: One mil approximately subtends one metre at a distance of one thousand metres. More formally the small angle approximation for skinny triangles shows that the angle in radians approximates to the sine of the angle.
Since mil is an angular measurement, the length of the area covered by the angle increase with distance. Mils are very easy to use with metric units. A common scope adjustment increment in European scopes is 0.1 mil, which are sometimes called "centimeter clicks" since 0.1 mil equals 1 cm at 100 meters. Similarly, scopes with 0.2 mil adjustment clicks can be referred to as having "two centimeters clicks", etc.
Angle @ 100 m @ 200 m @ 300 m @ 400 m @ 500 m @ 600 m @ 700 m @ 800 m @ 900 m @ 1000 m
0.1 mil 1 cm 2 cm 3 cm 4 cm 5 cm 6 cm 7 cm 8 cm 9 cm 10 cm
0.2 mil 2 cm 4 cm 6 cm 8 cm 10 cm 12 cm 14 cm 16 cm 18 cm 20 cm
1 mil 10 cm 20 cm 30 cm 40 cm 50 cm 60 cm 70 cm 80 cm 90 cm 100 cm
5 mil 50 cm 100 cm 150 cm 200 cm 250 cm 300 cm 350 cm 400 cm 450 cm 500 cm
Mil adjustment is commonly used in the mechanic adjustment of iron and scope sights in shooting sports, where sight adjustment using mils is particularly useful together with metric units when shooting at regular distances such as 100 m or 300 m, because for instance one click of a sight adjustment of 0.1 mil will move the point of impact exactly 1 cm at 100 m and 3 cm at 300 m respectively. This is not the case when using minutes of arc with imperial units, where one often simplifies 1′ being equal to 1 inch at 100 yards while in reality 1′ at that distance equals 1.047 inches, producing a small error that will increase the more the sight is adjusted or the longer the shooting distance. Therefore, in particular a spotter in long range shooting (i.e. 1000 m and above) theoretically can provide more precise shot corrections using a mil reticle.
### Mil reticles
"FinDot" reticle as used by Finnish Defence Forces snipers (a regular Mil-dot reticle with the addition of 400 m – 1200 m holdover (stadiametric) rangefinding brackets for 1 meter high or 0.5 meter wide targets at 400, 600, 800, 1000 and 1200 m).
Many telescopic sights used on rifles have reticles that are marked in angular mils. This can either be accomplished with lines or dots, and the latter is generally called mil-dots. The mil reticle serves two purposes, range estimation and trajectory correction.
With a mil reticle-equipped scope the distance to an object can be estimated with a fair degree of accuracy by a trained user by determining how many angular mils an object of known size subtends. Once the distance is known, the drop of the bullet at that range (see external ballistics), converted back into angular mils, can be used to adjust the aiming point. Generally mil-reticle scopes have both horizontal and vertical crosshairs marked; the horizontal and vertical marks are used for range estimation and the vertical marks for bullet drop compensation. Trained users, however, can also use the horizontal dots to compensate for bullet drift due to wind. Mil-reticle-equipped scopes are well suited for long shots under uncertain conditions, such as those encountered by military and law enforcement snipers, varmint hunters and other field shooters. These riflemen must be able to aim at varying targets at unknown (sometimes long) distances, so accurate compensation for bullet drop is required.
### Mixing mil and minutes of arc
It is possible to purchase rifle scopes with a mix of for instance a mil reticle and minute-of-arc turrets (or vice versa), but it is general consensus that such mixing should be avoided. It is preferred to either have both a mil reticle and mil adjustment (mil/ mil), or a minute-of-arc reticle and minute-of-arc adjustment to utilize the strength of each system. Then the shooter can know exactly how many clicks to correct based on what he sees in the reticle.
### Mil and minutes of arc conversion table
In the table below conversions from mil to metric values are exact (e.g. 0.1 mil equals exactly 1 cm at 100 meters), while conversions of minutes of arc to both metric and imperial values are approximate.
Minute of arc equivalent
(decimal)
Mil equivalent mm @ 100 m cm @ 100 m in @ 100 m in @ 100 y
1/8′ 0.125′ 0.036 mil 3.64 mm 0.36 cm 0.14 in 0.13 in
0.05 mil 0.172′ 0.05 mil 5 mm 0.5 cm 0.197 in 0.18 in
1/4′ 0.25′ 0.073 mil 7.27 mm 0.73 cm 0.29 in 0.26 in
0.1 mil 0.344′ 0.1 mil 10 mm 1 cm 0.39 in 0.36 in
1/2′ 0.5′ 0.145 mil 14.54 mm 1.45 cm 0.57 in 0.52 in
0.15 mil 0.516′ 0.15 mil 15 mm 1.5 cm 0.59 in 0.54 in
0.2 mil 0.688′ 0.2 mil 20 mm 2 cm 0.79 in 0.72 in
1′ 1.0′ 0.291 mil 29.1 mm 2.91 cm 1.15 in 1.047 in
• 0.1 mil equals exactly 1 cm at 100 m
• 1 mil ≈ 3.44′
• 1′ ≈ 0.291 mil (or 2.91 cm ≈ 3 cm at 100 m)
### Adjustment range and base tilt
The horizontal and vertical adjustment range of a firearm sight is often advertised by the manufacturer using mils. For instance a rifle scope may be advertised as having a vertical adjustment range of 20 mils, which means that by turning the turret the bullet impact can be moved a total of 2 meters at 100 meters (or 4 m at 200 m, 6 m at 300 m etc.). The horizontal and vertical adjustment ranges can be different depending on the particular sight. With a neutral mount, roughly half of the elevation is then usable:
${\displaystyle {\text{usable elevation in neutral mount}}={\frac {\text{scope's total elevation}}{2}}}$
In most regular sport and hunting rifles (except for in long range shooting), rifle scopes are usually mounted without tilt which means that the mounted scope points reasonably parallell to the barrel when it is dialed to the middle of its horizontal adjustment range. This is done because the optical quality of the scope is best in the middle of its adjustment range. While not normally a problem at short and medium range shooting, using a "0 mil" or non-tilted mount means that only about half of the vertical adjustment can be used to compensate for bullet drop. For example, on a scope with a 20 mil vertical elevation range mounted in a level mount, only about 10 mil of the vertical adjustment can be used to compensate for bullet drop at longer ranges.
In long range shooting, tilted scope mounts are often used since it is important to have enough vertical adjustment to compensate for the bullet drop for the given caliber at the given distance. For this purpose scope mounts are sold with varying degrees of tilt, but common values are 3, 6 or 9 mil (10.3′, 20.6′ or 31′ respectitvely) which corresponds to 0.3 m, 0.6 m and 0.9 m at 100 m. If the same 20 mil scope in the example above is mounted with a 9 mil tilt, the scope adjustment has to be bottomed out for short range shooting, but in return the setup will have about 19 mils of vertical adjustment that can be used for bullet drop compensation at long range as opposed to about 10 mils with a neutral mount. Then the maximum scope elevation can be found by:
${\displaystyle {\text{maximum elevation with tilted mount}}={\frac {\text{scope's total elevation}}{2}}+{\text{base tilt}}}$
Total elevation differ between models, but about 10–11 mils are common in hunting scopes, while scopes made for long range shooting usually can have an adjustment range of 30–50 mils.[citation needed] The adjustment range needed to shoot at a certain distance vary with firearm, caliber and load. For example, with a certain .308 load and firearm combination, the bullet may drop 13 mils at 1000 meters (13 meters). To be able to reach out, one could either:
• Use a scope with 26 mils of adjustment in a neutral mount, to get a usable adjustment of 26 mils/2 = 13 mils
• Use a scope with 14 mils of adjustment and a 6 mil tilted mount to achieve a maximum adjustment of 14 mils/2 + 6 = 13 mils
## Range estimation
Estimating mils with hands
Mildot chart as used by snipers
Angle can be used for either calculating target size or range if one of them are known. Where the range is known the angle will give the size, where the size is known then the range is given. When out in the field angle can be measured approximately by using calibrated optics or roughly using one's fingers and hands. With an outstretched arm one finger is approximately 30 mils wide, a fist 150 mils and a spread hand 300 mils.
Mil reticles in optics can easily be used for range estimation because of the precise mathematical simplification that can be made with such small angular measurements, exploiting the attribute of radians that a small angle is a good approximation to its sine, that is, for small angles sin θ ≈ θ. Mil reticles often have dots or marks with a spacing of one mil in between, but graduations can also be finer and coarser (i.e. 0.8 or 1.2 mil).
While a radian is defined as an angle on the unit circle where the arc and radius have equal length, a milliradian is defined as the angle where the arc length is one thousandth of the radius. Therefore, when using milliradians for range estimation, the unit used for target distance needs to be thousand times as large as the unit used for target size. Metric units are particularly useful in conjunction with a mil reticle because the mental arithmetic is much simpler with decimal units, thereby requiring less mental calculation in the field. Using the range estimation formula with the units meters for range and millimeters for target size it is just a matter of moving decimals and do the division, without the need of multiplication with additional constants, thus producing fewer rounding errors.
${\displaystyle {\text{distance in meters}}={\frac {\text{target in millimeters}}{\text{angle in mils}}}}$
The same holds true for calculating target distance in kilometers using target size in meters.
${\displaystyle {\text{distance in kilometers}}={\frac {\text{target in meters}}{\text{angle in mils}}}}$
If using the imperial units yards for distance and inches for target size, one has to multiply by a factor of 100036 ≈ 27.78, since there are 36 inches in one yard.
${\displaystyle {\text{distance in yards}}={\frac {\text{target in inches}}{\text{angle in mils}}}\times 27.78}$
Also, in general the same unit can be used for subtension and range if multiplied with a factor of thousand, i.e.
${\displaystyle {\text{distance in meters}}={\frac {\text{target in meters}}{\text{angle in mils}}}\times 1,000}$
### Examples
Land Rovers are about 3 to 4 m long, "smaller tank" or APC/MICV at about 6 m (e.g. T-34 or BMP) and about 10 m for a "big tank." From the front a Land Rover is about 1.5 m, most tanks around 3 - 3.5 m. So a SWB Land Rover from the side are one finger wide at about 100 m. A modern tank would have to be at a bit over 300 m.
If for instance a target known to be 1.5 m wide (1500 mm) is measured to 2.8 mils in the reticle, the range can be estimated to:
${\displaystyle {\text{distance in meters}}={\frac {1500~{\text{mm}}}{2.8~{\text{mils}}}}=535.7~{\text{m}}}$
So if the above-mentioned 6 m long BMP (6000 mm) is viewed at 6 mils its distance is 1000 m, and if the angle of view is twice as large (12 mils) the distance is half as much, 500 m.
When used with some riflescopes of variable objective magnification and fixed reticle magnification (where the reticle is in the second focal plane), the formula can be modified to:
${\displaystyle {\text{distance in meters}}={\frac {\text{size in mm}}{\text{angle in mils}}}\times {\frac {\text{mag}}{10}}}$
Where mag is scope magnification. However, a user should verify this with their individual scope since some are not calibrated at 10×. As above target distance and target size can be given in any two units of length with a ratio of 1000:1.
## Definitions for maps and artillery
Map measure M/70 of the NATO member Denmark with the full circle divided into 6400 NATO mils
In the Swiss Army, 6400 "artillery per milles" ("Artilleriepromille") are used to indicate an absolute indication of direction by using the notation that 0 A ‰ (corresponding to 6400 A ‰) points to the north, instead of using NATO mils where direction is always relative to the target (0 or 6400 NATO mils is always towards target).
There are 2000π milliradians (≈ 6283.185 mil) in a circle; thus a milliradian is just under 16283 of a circle, or ≈ 3.438 minutes of arc. Each of the definitions of the angular mil are similar to that value but are easier to divide into many parts.
• 16283 The "real" trigonometric unit of angular measurement of a circle in use by telescopic sight manufacturers using stadiametric rangefinding in reticles.
• 16400 of a circle in NATO countries.
• 16000 of a circle in the former Soviet Union and Finland (Finland phasing out the standard in favour of the NATO standard).
• 16300 of a circle in Sweden. The Swedish term for this is streck, literally "line". Sweden (and Finland) have not been part of NATO nor the Warsaw Pact. Note however that Sweden has changed its map grid systems and angular measurement to those used by NATO, so the "streck" measurement is obsolete.
### Conversion table
Conversions between units
Milliradian NATO mil Warsaw Pact Mil Swedish streck Degrees Minute of arc
1 milliradian = 1 1.018592 0.954930 1.002677 0.057296 3.437747
1 NATO mil = 0.981719 1 0.9375 0.984375 0.05625 3.375
1 Warsaw Pact mil = 1.047167 1.066667 1 1.05 0.06 3.6
1 Swedish streck = 0.997302 1.015873 0.952381 1 0.057143 3.428572
1 degree = 17.452778 17.777778 16.666667 17.5 1 60
1 minute of arc = 0.290880 0.296297 0.277778 0.291667 0.016667 1
(Values in bold face are exact.)
• 1 trigonometric milliradian (mil) ≈ 3.43774677078493′
• 1 NATO mil = 3.375′ exactly
• 1 Warsaw Pact mil = 3.6′ exactly
### Use in artillery sights
Artillery uses angular measurement in gun laying, the azimuth between the gun and its target many kilometres away and the elevation angle of the barrel. This means that artillery uses mils to graduate indirect fire azimuth sights (called dial sights or panoramic telescopes), their associated instruments (directors or aiming circles), their elevation sights (clinometers or quadrants), together with their manual plotting devices, firing tables and fire control computers.
Artillery spotters typically use their calibrated binoculars to walk fire onto a target. Here they know the approximate range to the target and so can read off the angle (+ quick calculation) to give the left/right corrections in metres.
## References
1. ^ Renaud, Hugues (2002-05-31). Dictionnaire historique de la Suisse. Fonds, AV Laussane. Dapples: ... Charles-Marc (1837-1920), ingénieur, professeur à l'université de Lausanne, municipal à Lausanne, est l'inventeur de l'unité appelée "millième" pour mesurer les angles dans le tir d'artillerie. Une branche de la famille s'est fixée à Gênes à la fin du XVIIIe s.
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## Voodoo Correlations and Correlation Picking
Ex post selection based on correlation has been a long-standing issue at this blog (and has been independently discussed at other blogs from time to time – Luboš, Jeff Id and David Stockwell have all written on it independently. The issue came back into focus with Mann 2008, in which there is industrial strength correlation picking. While the problem is readily understood (other than by IPCC scientists), it’s hard to find specific references. Even here, surprisingly, my mentions of this have mostly been passim – in part, because I’d worked on this in pre-blog days. We mention the issue in our PNAS comment, using Stockwell (AIG News, 2006) as a reference as I mentioned before.
“Spurious” correlations are very familiar to someone familiar with the stock market, whereas people coming from applied math and physics seem to be much quicker to reify correlations and much less wary about the possibility of self-deception.
Reader Jonathan brings to our attention an interesting new study entitled “Voodoo Correlations in Social Neuroscience” which was discussed in Nature here
The problem seems to be highly similar to ex post selection of proxies by correlation. Vul et al write (and touch on other issues familiar to CA readers):
The implausibly high correlations are all the more puzzling because social-neuroscience method sections rarely contain sufficient detail to ascertain how these correlations were obtained. We surveyed authors of 54 articles that reported findings of this kind to determine the details of their analyses. More than half acknowledged using a strategy that computes separate correlations for individual voxels, and reports means of just the subset of voxels exceeding chosen thresholds. We show how this non-independent analysis grossly inflates correlations, while yielding reassuring-looking scattergrams. This analysis technique was used to obtain the vast majority of the implausibly high correlations in our survey sample. In addition, we argue that other analysis problems likely created entirely spurious correlations in some cases. We outline how the data from these studies could be reanalyzed with unbiased methods to provide the field with accurate estimates of the correlations in question. We urge authors to perform such reanalyses and to correct the scientific record.
In their running text, they observe:
in half of the studies we surveyed, the reported correlation coefficients mean almost nothing, because they are systematically inflated by the biased analysis.
They illustrate “voodoo correlations” with one more example of spurious correlation (echoing our reconstruction of temperature with principal components of tech stock prices:
It may be easier to appreciate the gravity of the non-independence error by transposing it outside of neuroimaging We (the authors of this paper) have identified a weather station whose temperature readings predict daily changes in the value of a specific set of stocks with a correlation of r=-0.87. For 50.00, we will provide the list of stocks to any interested reader. That way, you can buy the stocks every morning when the weather station posts a drop in temperature, and sell when the temperature goes up. Obviously, your potential profits here are enormous. But you may wonder: how did we find this correlation? The figure of -.87 was arrived at by separately computing the correlation between the readings of the weather station in Adak Island, Alaska, with each of the 3315 financial instruments available for the New York Stock Exchange (through the Mathematica function FinancialData) over the 10 days that the market was open between November 18th and December 3rd, 2008. We then averaged the correlation values of the stocks whose correlation exceeded a high threshold of our choosing, thus yielding the figure of -.87. Should you pay us for this investment strategy? Probably not: Of the 3,315 stocks assessed, some were sure to be correlated with the Adak Island temperature measurements simply by chance – and if we select just those (as our selection process would do), there was no doubt we would find a high average correlation. Thus, the final measure (the average correlation of a subset of stocks) was not independent of the selection criteria (how stocks were chosen): this, in essence, is the non-independence error. The fact that random noise in previous stock fluctuations aligned with the temperature readings is no reason to suspect that future fluctuations can be predicted by the same measure, and one would be wise to keep one’s money far away from us, or any other such investment advisor9. 9 See Taleb (2004) for a sustained and engaging argument that this error, in subtler and more disguised form, is actually a common one within the world of market trading and investment advising. Nature’s summary states: They particularly criticize a ‘non-independence error’, in which bias is introduced by selecting data using a first statistical test and then applying a second non-independent statistical test to those data. This error, they say, arises from selecting small volumes of the brain, called voxels, on the basis of their high correlation with a psychological response, and then going on to report the magnitude of that correlation. “At present, all studies performed using these methods have large question marks over them,” they write. The scientists under criticism say that the criticisms do not matter because they do appropriate adjustments: Appropriate corrections ensure that the correlations between the selected voxels and psychological responses are likely to be real, and not noise, Interestingly, these criticisms are said to have an “iconoclastic tone” and to have been widely covered in blogs, much to the annoyance of the scientists defending their correlations. Nature: The iconoclastic tone have attracted coverage on many blogs, including that of Newsweek. Those attacked say they have not had the chance to argue their case in the normal academic channels. “I first heard about this when I got a call from a journalist,” comments neuroscientist Tania Singer of the University of Zurich, Switzerland, whose papers on empathy are listed as examples of bad analytical practice. “I was shocked — this is not the way that scientific discourse should take place.” ### 39 Comments 1. Eve N. Posted Jan 16, 2009 at 10:29 AM | Permalink Cargo cult science strikes again! I’m sorry if I sound facetious, but I think every scientist has to read Mr. Feynman’s speech on the subject. *especially* climate scientists who seem to lack that certain thing which Mr. Feynman spoke of. Read It Here • Peter D. Tillman Posted Jan 16, 2009 at 11:33 AM | Permalink Re: Eve N. (#1), [Feynman discusses a meta-experiment on how to be smarter than a rat, in rat-in-maze experiments] I looked into the subsequent history of this research. The next experiment, and the one after that, never referred to Mr. Young. They never used any of his criteria of putting the corridor on sand, or being very careful. They just went right on running rats in the same old way, and paid no attention to the great discoveries of Mr. Young, and his papers are not referred to, because he didn’t discover anything about the rats. In fact, he discovered all the things you have to do to discover something about rats. But not paying attention to experiments like that is a characteristic of cargo cult science. Quite familiar story for CA regulars! Steve, keep after the rat-runners…. Best for 2009, Pete Tillman 2. Posted Jan 16, 2009 at 10:32 AM | Permalink Not sure if related, but I was reading this http://news.bbc.co.uk/2/hi/health/7825890.stm just before the post appeared 🙂 3. Tom Gray Posted Jan 16, 2009 at 10:33 AM | Permalink From a Nature article as qouted in the posting: Those attacked say they have not had the chance to argue their case in the normal academic channels. “I first heard about this when I got a call from a journalist,” comments neuroscientist Tania Singer of the University of Zurich, Switzerland, whose papers on empathy are listed as examples of bad analytical practice. “I was shocked — this is not the way that scientific discourse should take place.” Note that phrase “Those attacked”. No one is being “attacked” here. A scientific practice is being subject to informed criticism. The phrase reveals a great deal about the scientific community and the revelation is not flattering. 4. Ryan O Posted Jan 16, 2009 at 10:34 AM | Permalink The veracity of a criticism is independent of the source. Who cares if it comes from a blog? If the criticism is correct, then it is correct. . And Richard Feynman rulez. 5. Sam Urbinto Posted Jan 16, 2009 at 10:40 AM | Permalink Instead of calling them voxels, perhaps they should be called loa or vodun. 6. Hu McCulloch Posted Jan 16, 2009 at 10:45 AM | Permalink I always caution my econometrics students against “Data Mining”, which I interpret as searching through a long list of potential variables and combinations thereof for sets that appear to have significant correlations using critical values that are only valid for single tests. I am therefore disturbed to find that the SAS catalog has an entire section of manuals and packages that automate “Data Mining” for the researcher. Perhaps these procedures correct the critical values for the effects of the multiple tests on test size, but I doubt it. Wikipedia’s approving article on Data Mining makes no mention of size correction. 7. JD Long Posted Jan 16, 2009 at 10:47 AM | Permalink Eve, great Feynman reference! My colleagues have suggested that at meetings we have coconut halves and if anyone breaks out in Cargo Cult Science we make them wear the coconut halves like a radio headset. The only problem is dealing with the religious battles between those who see something as CCS and those who don’t! 8. GTFrank Posted Jan 16, 2009 at 11:23 AM | Permalink from Wired July 2008, I think “The End of Science” “The quest for knowledge used to begin with grand theories. Now it begins with massive amounts of data. Welcome to the Petabye Age.” “The End of Theory: The Data Deluge Makes the Scientific Method Obsolete” “But faced with massive data, this approach to science — hypothesize, model, test — is becoming obsolete. Consider physics: Newtonian models were crude approximations of the truth (wrong at the atomic level, but still useful). A hundred years ago, statistically based quantum mechanics offered a better picture — but quantum mechanics is yet another model, and as such it, too, is flawed, no doubt a caricature of a more complex underlying reality. The reason physics has drifted into theoretical speculation about n-dimensional grand unified models over the past few decades (the “beautiful story” phase of a discipline starved of data) is that we don’t know how to run the experiments that would falsify the hypotheses — the energies are too high, the accelerators too expensive, and so on” “There is now a better way. Petabytes allow us to say: “Correlation is enough.” We can stop looking for models. We can analyze the data without hypotheses about what it might show. We can throw the numbers into the biggest computing clusters the world has ever seen and let statistical algorithms find patterns where science cannot.” • Mark T. Posted Jan 16, 2009 at 11:51 AM | Permalink Re: GTFrank (#8), I read that this summer and cried. Mark 9. Hoi Polloi Posted Jan 16, 2009 at 11:23 AM | Permalink More Voodoo Correlations 10. Kenneth Fritsch Posted Jan 16, 2009 at 11:36 AM | Permalink “Spurious” correlations are very familiar to someone familiar with the stock market, whereas people coming from applied math and physics seem to be much quicker to reify correlations and much less wary about the possibility of self-deception. Your comment is much in line with what I have observed over the recent years in participating in blogs dealing with investment strategies where posters with hard science backgrounds seem to have a mental block when it comes to the dangers of data snooping. I sometimes think it might be the difficulty in separating the deterministic from the stochastic parts of the model. Posters were, of course, able to show that some of these data snooped strategies worked out-of- sample (for the relatively short time periods that they existed out-of-sample) as they came up with hundreds of them that were published (out of many thousands or more generated, snooped and discarded) and instead settling for the tendency of some to work by mere chance they would go into detailed analysis of why their snooped strategies worked. There was actually a poster on one web site called datasnooper, who explained, in great detail and with many examples and in an articulate manner, the dangers of data snooping. He was a whiz at statistics and a financial analyst. He had to fight off some very nasty efforts to discredit him on the site and finally gained a huge following of people who were willing and able to understand what he was attempting, in good faith, to teach them. The leaders on the site were very hesitant to recognize his efforts and appeared to go out of their way to ignore his comments. Once the data snooping has been used for finding a correlation (or investment strategy) it could, in this layperson’s view, be theoretically and approximately corrected for the data snooping by way of the Bonferroni correction, but that correction requires keeping track of the n (from the excerpt below) which in most cases is very difficult to impossible to do in practice as one must include all the explicit and implicit hypothesis considered. At least, the Bonferroni correction excercise allows one to see what that correction can do to the probabilty of a statistical test — providing, of course, that one is able to honestly record all the implicit and explicit hypothesis considered in finding a pet theory, conjecture or strategy. The Bonferroni correction is a safeguard against multiple tests of statistical significance on the same data falsely giving the appearance of significance, as 1 out of every 20 hypothesis-tests is expected to be significant at the α = 0.05 level purely due to chance. Furthermore, the probability of getting a significant result with n tests at this level of significance is 1-0.95n (1-probability of not getting a significant result with n tests). http://en.wikipedia.org/wiki/Bonferroni_correction 11. Demesure Posted Jan 16, 2009 at 12:00 PM | Permalink The stocks prediction method based on thermometer/stock correlation should be marketed with Mannian speak. It would be rendered obscure enough to sell like crazy to subprimes brokers. 12. Carl G Posted Jan 16, 2009 at 12:53 PM | Permalink #6: HU, I have both SAS and Enterprise Miner, and I don’t think that data mining is necessarily a problem. Suppose I have a dataset of 10,000,000 addresses and I am trying to market. An overfit model on a portion of that dataset will predict well at all on another portion of that dataset. The easy solution is this: make hundreds of models (different types of models, different variables, and/or different parameters) that fit well to a subset of the data. Then, on another large subset of the data, select the model that fits the 2nd dataset the best. The principle is that a model that is overfit on the calibration data will certainly predict horribly on the validation set, but a more modest calibration model will fit the validation set better. Finally, the model is used on a third subset of the data. This data, not used at all in selecting variables or setting parameters, is the best possible gauge of how the model will do on other datasets. It is from predictions on this dataset that confidence statistics should come. If there’s something wrong with the above method, I don’t see it. I think data mining has gotten a bad rap, especially as an exploratory tool, when it’s really poor validation methodology (or none at all)that is the problem. • Kenneth Fritsch Posted Jan 16, 2009 at 1:13 PM | Permalink Re: Carl G (#14), Perhaps data mining and data snooping should be contrasted, but I think there are dangers in both. I think of data mining as the process that you describe above as opposed to data snooping as I described in my previous post. Your methods prescribe an out-of-sample test that works within limits but depends on the experimeter not peeking at the “virgin” data or making multitudes of runs in searching for a fit in the in-sample and out-of-sample periods. I would also think that some well-reasoned a priori assumptions would complement the data mining approach and eliminate some rather obviously spurious correlations. Now I will let Hu give you an expert and professional answer. 13. Carl G Posted Jan 16, 2009 at 1:02 PM | Permalink #14: 2nd line, I meant to say: “An overfit model on a portion of that dataset will *NOT* predict well at all on another portion of that dataset” 14. Craig Loehle Posted Jan 16, 2009 at 1:08 PM | Permalink There is a difference between data mining for marketing on mailing addresses and in science. If your data mining finds that a certain zip code/income level/whatever buys more of certain products, that relationship will probably remain stable since the same people live there and the zip has the same income level from one year to the next. In stocks, the selected stocks based on spurious relations may well continue to go up or down for a while, and thus for short-term trading you might make money, but then things change…In other areas of science, not so much. 15. Jonathan Posted Jan 16, 2009 at 1:30 PM | Permalink My own thread on Climate Audit – now I can die happy 🙂 I should give full credit to my wife (an avid reader but rare poster) who spotted the Nature article over breakfast this morning. 16. Hu McCulloch Posted Jan 16, 2009 at 1:32 PM | Permalink Re Kenneth Fritsch #11, Thanks, Ken (Kenneth?) for the reference to the Wikipedia Bonferonni article! Unfortunately, your quote didn’t cut and paste the exponent correctly. Your comment had The Bonferroni correction is a safeguard against multiple tests of statistical significance on the same data falsely giving the appearance of significance, as 1 out of every 20 hypothesis-tests is expected to be significant at the α = 0.05 level purely due to chance. Furthermore, the probability of getting a significant result with n tests at this level of significance is 1-0.95n (1-probability of not getting a significant result with n tests). The last sentence should have read, using ^ for exponentiation since I can’t get superscripts to work without going into LaTeX, Furthermore, the probability of getting a significant result with n tests at this level of significance is 1-0.95^n (1-probability of not getting a significant result with n tests). In fact, this equation is the basis of what is known as the Šidàk correction: If n independent tests are performed at size α*, the probability that none will falsely reject the null is (1-α*)^n. In order for this to equal 1-α, where α is the probability that at least one of the tests will falsely reject its null, we must have α* = 1-(1-α)^(1/n). This adjustment is exact, but only for independent tests. For example, if we wanted the final test size α to be .05 and ran 5 independent tests, the Šidàk adjusted test size would be 1-.95^(1/5) = .0102. The Bonferroni adjustment itself (which is referenced in the Wikipedia paragraph preceding the one Ken quotes) is based on Bonferroni’s Inequality, which here states that whether or not the tests are independent, α* ≥ α/n. For example, if we want α = .05, we may have to set the individual test size α* as low as .05/5 = .01 in order to achieve it. This is the worst case scenario, but for small tests sizes, isn’t much different than the Šidàk adjustment which is based on independence, so it makes sense to just use the simple Bonferroni adjustment and rest assured that it is conservative. The best-case scenario is based on the additional inequality α ≥ α*. When this holds with equality, no adjustment is required at all, but it only applies when the individual tests are perfectly correlated, i.e. they are replications of exactly the same test with the same data. Hence it’s not very relevant, even if it’s the basis for Mann’s 2008 Pick-Two procedure. The Bonferrroni and Šidàk adjustments are discussed in an article by Hervé Abdi from the Encyclopedia of Measurement and Statistics cited by the Wiki article and online at http://www.utdallas.edu/~herve/Abdi-Bonferroni2007-pretty.pdf. Unfortunately, Abdi does not make it clear that Šidàk is exact for independent tests, while Bonferroni is a worst-case inequality. Of course, using either of these adjustments potentially loses a lot of efficiency, since a rectangular acceptance region is ordinarily not optimal. With Gaussian errors, if possible we would want to explicitly measure the correlations and use elliptical regions governed by a χ^2 or F critical value. • John A Posted Jan 16, 2009 at 5:00 PM | Permalink Hu, it’s pretty easy to use LaTeX here – just surround your text with the tex tags (the button is on the quicktags line and works the same way as bold or blockquote). Just don’t add the$signs because they are not required. $a^* = 1-(1-\alpha)^\frac{1}{n}$ $\chi ^2$ • Kenneth Fritsch Posted Jan 16, 2009 at 6:56 PM | Permalink Thanks, Ken (Kenneth?) Hu, you can call me Kenneth or you can call me Ken or you can call me Kenny, just don’t call me Kenny boy. Thanks for the link to the Abdi article and your more complete explanation of the Bonferroni correction. 17. Steve Geiger Posted Jan 16, 2009 at 1:38 PM | Permalink Craig L., you write “. In stocks, the selected stocks based on spurious relations may well continue to go up or down for a while, and thus for short-term trading you might make money” I thought the issue was that given enough chances (i.e., different stocks) one could find a reasonable correlation between a few of them and, say, the price of tea in China (Chinese tea stocks not withstanding, of course ;-). Thus, if that is the issue, I would see no benifit (short term or otherwise) to buy any stocks based on the seemingly great correlation to (fill in the parameter). 18. Posted Jan 16, 2009 at 2:01 PM | Permalink Parametric studies are a good way to handle a problem that you don’t understand. In the world of academia correlations can point to where more study could lead to understanding. However, recommending changes in the real world based on correlations that are not understood on the basis of first principles is reaching too far. My favorite example is the neural net that correlated tanks with clouds and concluded that tanks must be present if it’s cloudy. http://neil.fraser.name/writing/tank/ As founding CTO for Cernium (http://www.cernium.com/) my work involved the analysis of large noisy data sets of visual primitives and extracting high level recognition data such as “car” or “person”. There were many parameters most of which were same for all classes of target. My motto was “Correlation should make me curious not convinced”. If I was unable to understand the first principles of why a visual primitive should indicate an object class I did not use it. First principles are always correct. Correlations that hold up for one data set (the past for example) may not be relevant to another data set (the future in the case of climate). • Kenneth Fritsch Posted Jan 16, 2009 at 6:48 PM | Permalink If I was unable to understand the first principles of why a visual primitive should indicate an object class I did not use it. First principles are always correct. Correlations that hold up for one data set (the past for example) may not be relevant to another data set (the future in the case of climate). I recall some of the data snooping that was used in formulating some investment models using various past performance criteria and then looking at the square and cubed roots of that criteria. After obtaining an unbelievable in-sample performance with the cubed root of the criteria some of the rather convoluted explanations that were derived after the fact would almost sound convincing – if one did not recognize the dangers of data snooping. 19. henry Posted Jan 16, 2009 at 2:22 PM | Permalink I suppose this is better in this thread: The American Statistical Association (ASA) invites applications and nominations for the position of editor of the Statistical Analysis and Data Mining. Further info at amstat.org I wonder if any Climate Scientists are members of the ASA. Sounds like their kind of journal. 20. Hu McCulloch Posted Jan 16, 2009 at 2:41 PM | Permalink Carl G writes in #14, #6: HU, I have both SAS and Enterprise Miner, and I don’t think that data mining is necessarily a problem. Suppose I have a dataset of 10,000,000 addresses and I am trying to market. An overfit model on a portion of that dataset will predict well at all on another portion of that dataset. The easy solution is this: make hundreds of models (different types of models, different variables, and/or different parameters) that fit well to a subset of the data. Then, on another large subset of the data, select the model that fits the 2nd dataset the best. The principle is that a model that is overfit on the calibration data will certainly predict horribly on the validation set, but a more modest calibration model will fit the validation set better. Validation out of sample can help, but is not a cure-all. Suppose you start with 400 models that in truth have no explanatory power. We would expect to find that 20 of them are significant at the 5% level using the initial data set. Of these we would expect 1 to be significant at the 5% level using the validation data set, so it is not true that a false model will “certainly predict horribly on the validation set” as you state. It would make more sense to me to apply all 400 models to the combined data set, but to use a Bonferroni-adjusted test size of .05/400 = .000125 on the individual models to obtain a final test size of .05. (This corresponds, with large samples, to t values greater than about 3.84. This is not unusually high for a clearly valid model.) The final model(s) can then be double-checked for stability across subsamples with a Chow (switching regression) test, if desired. Finally, the model is used on a third subset of the data. This data, not used at all in selecting variables or setting parameters, is the best possible gauge of how the model will do on other datasets. It is from predictions on this dataset that confidence statistics should come. Again, with sufficient initial models, even this two-stage validation with a third data set could give misleading results. Suppose one had three decades of annual data on 8,000 tree ring series, and tried to use them to model the following year’s return on the S&P 500 stock index. Of these, 400 are expected to be significant at the 5% level in the first decade. Then, 20 of these survivors are expected to be significant at the 5% level when verified with the second decade of data. And 1 of the 20 finalists is expected to be significant when re-verified with the third decade of data. (Of course, the realization will not be 1 necessarily, but just a random number with expectation 1.) I sure wish I knew which tree that was before this past year’s bear market! Is it a Gaspé cedar, a Graybill BCP, or some linear combination thereof? 21. Steve McIntyre Posted Jan 16, 2009 at 2:59 PM | Permalink Almagre tree 2007-31 predicted the market decline, well ahead of the actual event giving ample opportunity to trade profitably. 22. Jerry M Posted Jan 16, 2009 at 4:01 PM | Permalink I also use SAS and Enterprise Miner for forecasting and predictive modeling. Yesterday I presented a model I developed for another group in my company. They were very excited and want more, but my time is limited. I was asked, “Well if you can’t do it, can you just tell one of our analysts what you did and they can just plug in new numbers?”. No, no I can’t. One needs to know statistics and what they are capable of. 23. Hu McCulloch Posted Jan 16, 2009 at 5:00 PM | Permalink Henry #22 reports, The American Statistical Association (ASA) invites applications and nominations for the position of editor of the Statistical Analysis and Data Mining. The new journal’s webpage is at http://www3.interscience.wiley.com/journal/112701062/home. The very title sets my teeth on edge, sort of like “The ASA Journal of Cherry Picking” would. But given that this is coming from the ASA, perhaps they are cognizant of the potential perils of data mining, and are primarily concerned with how to compensate correctly for it, eg with the Bonferroni or Šidàk adjustment? Or maybe it’s really just a spoof, like the famous Journal of Irreproducible Results, or the Sokol hoax? 24. per Posted Jan 16, 2009 at 6:59 PM | Permalink interesting post. It is probably germane to note that this paper, and the story, are not unique. This one has got a lot of press, perhaps due to the title, which is snappy. However, this is a common problem. See: Nature 454, 682-685 (2008) Amyotroph Lateral Scler. 2008;9(1):4-15 Scott et al. Scott et al followed up a mouse model used in over 50 publications, all of which purportedly showed statistically significant effects on prolonging lifespan in the mouse. Little bit of poor control of variables, bit of chucking out data you don’t like, and the use of small group size; and gee whiz, you have 50 papers. Small group sizes are key, because you can get a big effect size by random chance; you don’t publish the uninformative experiments but you do get a decent number of false positives. When you do the experiment properly, you cannot repeat the “positive” studies. One of the interesting aspects is the Nature overview, which quotes one researcher “There just aren’t the resources now to do really large, well-powered mouse studies”. That’s interesting, ‘cos there was all the money spent on enabling 50 different publications (plus unpublished failed replications), plus the resulting clinical trials in humans, and that has all gone down the drain because the mouse studies are (according to scott’s study) not replicable. So 50 different publications is going to be ~~$5 million, plus the clinical studies, which will easily double that.
there is also the rather invidious issue that many of these researchers who got these fantastic papers got follow-on grant money, prestige, etc.
per
• DeWitt Payne
Posted Jan 16, 2009 at 7:26 PM | Permalink
Re: per (#30),
Sounds like the old quality management saw: “There’s not enough resources to do it right the first time, but there’s always enough to do it over again.” Poorly designed trials can keep a project alive for years.
• MarkB
Posted Jan 16, 2009 at 9:10 PM | Permalink
Re: per (#30),
I saw something similar when I was in grad school. People use poor data sets, or limited experimental design. You point out their error – carefully, because you can’t make enemies out of faculty – and the respose is “It’s not optimal, but it’s all we’ve got in this field, so we need to use it to solve the problem.” That’s the good careerist answer, but of course the truth is that if you are unable to do it right, then you have to right to do it. Wrong is wrong, and if that means you have to find some other way to get published, so be it.
25. Steve McIntyre
Posted Jan 16, 2009 at 9:28 PM | Permalink
Little bit of poor control of variables, bit of chucking out data you don’t like, and the use of small group size; and gee whiz, you have 50 papers. Small group sizes are key, because you can get a big effect size by random chance
Sounds like the Team multiproxy literature with a bunch of studies with 6-15 “proxies”, all with bristlecones and/or Yamal, plus Tornetrask, ….
26. Posted Jan 17, 2009 at 2:06 PM | Permalink
For some reason, this part of the Sokal link thread, above, stood out:
http://en.wikipedia.org/wiki/Rosenhan_experiment
Researchers claimed to hear voices and were admitted to institutions. When the phony patients wrote their daily research notes, genuine patients quickly caught on that the researchers were sane. Staff, on the other hand, interpreted the note making as compulsive behaviour, a further symptom of insanity. The only way some of them could get released was to admit that they were insane. Irony²!
There’s a parallel here, somewhere…
27. EW
Posted Jan 17, 2009 at 2:22 PM | Permalink
“There just aren’t the resources now to do really large, well-powered mouse studies”.
And there’s also a trend to limit the use of laboratory animals or skip some levels of animal testing in the experiments, because of all this “green” legislative, especially in the EU. This, of course, can lead to half-baked experiments with totally insufficient controls.
28. Posted Mar 6, 2009 at 2:21 PM | Permalink
I thought this might amuse http://www.xkcd.com for more like it..
29. Dr Justin Marley
Posted Jan 1, 2010 at 7:23 AM | Permalink
Hi,
Any chance of some constructive feedback on a video I put together about the above study?
Regards
Justin
30. Matthew Lieberman
Posted Mar 24, 2010 at 3:22 PM | Permalink
For anyone interested, there was a public debate on Voodoo Correlations last fall at the Society of Experimental Social Psychologists between Piotr Winkielman (one of the authors on the Voodoo paper) and myself (Matt Lieberman). The debate has been posted online.
http://www.scn.ucla.edu/Voodoo&TypeII.html
31. tz2026
Posted May 30, 2013 at 4:43 PM | Permalink
Speaking of Voodoo, anyone else notice as soon as they started burning witches, the Medieval Warm Period ended? And that when we stopped burning witches temperatures started climbing?
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# Are GARCH models dependent on the returns forecasting model?
Hi Quantitative Fiance Stack Exchange,
It's my first go at GARCH models so please give me a chance with my phrasing.
I understand that GARCH models are used to forecast volatility. The GARCH(1,1) takes the form:
$$\sigma^2_t=\alpha+\beta_1\epsilon_{t-1}+\beta_2\sigma^2_{t-1}$$
I understand the lagged term $\sigma^2_{t-1}$ makes up the AR part of GARCH. However, I also understand the error term $\epsilon_{t-1}$ is dependent on the forecasting model. Consider, forecasting returns using one of the two models:
$$\hat{y_t}=\gamma\cdot y_{t-1}+\epsilon_t$$
and
$$\hat{y_t}=\theta\cdot x_{t-1}+\epsilon_t$$
Each model gives a different error term, which I believe is calculated as $\epsilon_t=y_t-\hat{y_t}$. So for the above models, error terms are $\epsilon_t=y_t-\gamma\cdot y_{t-1}$ and $\epsilon_t=y_t-\theta\cdot x_{t-1}$
Hence, is my understanding correct that calculating $\beta_1$ and $\beta_2$ of the GARCH(1,1) model depends on which forecasting model we're using?
Thank you for the help, Donny
I also understand the error term $\varepsilon_{t-1}$ is dependent on the forecasting model.
Yes, it is. The error term $\varepsilon_t$ in the GARCH model is coming from the full distributional model of $y_t$. The full model is \begin{aligned} y_t &= \mu_t + \varepsilon_t, \\ \varepsilon_t &= \sigma_t \xi_t, \\ \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \xi_t &\sim i.i.d(0,1), \end{aligned} where $\mu_t$ is the conditional mean of $y_t$, $\sigma_t^2$ is the conditional variance of $y_t$ and $d$ is some probability distribution with zero mean and unit variance.
If you are not sure which conditional mean model is best for $y_t$, you may end up with a few alternative models characterized by the conditional means $\mu_{1,t}, \mu_{2,t}, \dots$. The the corresponding error terms will differ across the models and will be $\varepsilon_{1,t} = y_t-\mu_{1,t}, \varepsilon_{2,t} = y_t-\mu_{2,t}, \dots$. This will affect the parameter estimates of the conditional variance model, just as you said.
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# Organizing genome engineering for the gigabase scale
## Abstract
Genome-scale engineering holds great potential to impact science, industry, medicine, and society, and recent improvements in DNA synthesis have enabled the manipulation of megabase genomes. However, coordinating and integrating the workflows and large teams necessary for gigabase genome engineering remains a considerable challenge. We examine this issue and recommend a path forward by: 1) adopting and extending existing representations for designs, assembly plans, samples, data, and workflows; 2) developing new technologies for data curation and quality control; 3) conducting fundamental research on genome-scale modeling and design; and 4) developing new legal and contractual infrastructure to facilitate collaboration.
## Introduction
Engineering the entire genome of an organism promises to enable large-scale changes to its organization, function, and interactions with its environment, with broad potential for impacts across science, industry, medicine, and society1. The past several decades have seen remarkable progress in our capability to synthesize DNA and modify genomes2,3,4. Since Khorana created the first synthetic gene 40 years ago5, our capability to construct DNA sequences has doubled, approximately every 3 years (Fig. 1a), progressing from plasmids in the early 1990’s6,7, viruses in the early 2000’s8, and gene clusters in the mid-2000’s9,10, to the first bacterial chromosome in 200811,12. Recently, several groups have re-engineered the 4 Mb genomes of Escherichia coli13,14 and Salmonella typhimurium15, and the Synthetic Yeast (Sc 2.0) project16,17 has nearly completed re-engineering an 11.4 Mb genome for Saccharomyces cerevesiae18. Looking ahead, in 2016 leaders from academia and industry formed Genome Project-Write1 to initiate the engineering of the gigabase genomes of higher-order eukaryotes. The goals of the GP-Write consortium include engineering a virus-resistant, ultra-safe human-derived cell line for pharmaceutical production19.
## From engineering genes to engineering genomes
Moving to the gigabase scale poses major technological and scientific challenges. Challenges related to DNA synthesis and editing have been discussed extensively in the literature20,21,22,23. Significant attention has also been devoted to the challenges of modeling24,25, designing17,26,27, and testing28 genomes. Less attention, however, has been devoted to the technologies, repositories, standards, and other resources needed to integrate these tasks into a cohesive workflow.
We contend that workflow integration is a first-class problem for gigabase-scale genome engineering. Over the last 40 years, the number of authors of pioneering genome engineering projects has risen markedly with genome size, suggesting that the complexity of genome engineering is also scaling with the size of the genome (Fig. 1b). If these trends continue, engineering a gigabase genome would be projected to become possible in ~2050 and require a team with the capabilities of around 500 investigators. To manage projects of such complexity without massive teams, we advocate for the development of an ecosystem of tools, services, automation, and other resources, which could enable a modestly sized team of bioengineers to indirectly access the equivalent capabilities of hundreds of people. To this end, we have examined the emerging design–build–test–learn workflow for genome engineering, identifying key interfaces and making recommendations for the adoption or development of technologies, repositories, standards, and frameworks.
## An emerging workflow for genome engineering
Recently, a number of groups have proposed or developed workflows for organism engineering3,18,27,28,29,30,31,32, converging toward a common engineering cycle consisting of the four stages shown in Fig. 2. These stages are (1) Design: bioengineers use models and design heuristics to specify a genome with an intended phenotype; (2) Build: genetic engineers construct the desired DNA sequence in a target organism; (3) Test: experimentalists assay molecular and behavioral phenotypes of the engineered organism; (4) Learn: modelers analyze the discrepancies between the desired and observed phenotypes to develop improved models and design heuristics. The process is repeated until an organism with the desired phenotype is identified. This incremental approach enables engineering despite our incomplete understanding of the complexities of biology.
The inner loop in Fig. 2 indicates the workflow used by many current genome engineering projects, which have primarily focused on “top-down” refactoring of existing genomes, e.g., by rewriting codons or reducing genomes to essential sequences. In the longer term, one of the key aims of synthetic biology is to engineer organisms that have novel phenotypes by “bottom-up” assembly of modular parts and devices33. At a much smaller scale, organism engineers are already beginning to use this approach to engineer novel metabolic pathways for commercial production of high-value chemicals34,35,36. For gigabase genome engineering, this approach will likely require more complex workflows that utilize more sophisticated design tools, phenotypic assays, data analytics, and models (outer loop of Fig. 2).
Executing these multistep workflows requires exchanging a wide range of materials, information, and other resources between numerous tools, people, institutions, and repositories. The design phase must communicate genome designs to the build phase, the build phase must deliver DNA constructs and cell lines to the test phase, the test phase must transmit measurements to the learn phase, the learn phase must provide models and design heuristics to the design phase, and workflows must be applied to coordinate the interaction and execution of tools across all of these stages.
In addition to these technical challenges, genome engineering must also address a number of safety, security, legal, contractual, and ethical issues. Throughout genome engineering workflows, bioengineers must pay careful attention to biosafety, biosecurity, and cybersecurity. To execute genome engineering workflows across multiple institutions, bioengineers must navigate materials transfer agreements, copyrights, patents, and licenses.
Every aspect of this genome engineering workflow must be scaled up to handle gigabase genomes. Ultimately, much or all of each step should be automated, and each interface between steps should be formalized to facilitate machine reasoning, removing the ad hoc and human-centric aspects of genome engineering as much as possible. In many cases, this can be facilitated by adopting or extending solutions from smaller-scale genome engineering, as well as solutions from related fields such as systems biology, genomics, genetics, bioinformatics, software engineering, database engineering, and high-performance computing. Other challenges of gigabase genome engineering, however, are likely to require the development of novel systems or additional fundamental research.
## Identifying and closing gaps in the state of the art
In this section, we discuss the integration challenges identified in the previous section, reviewing the state of the art in technologies and standards with respect to the emerging needs of gigabase genome engineering. Instead of focusing on specific evolving protocols and methods, which are likely to advance rapidly, we consider the information that must be communicated to enable protocols or methods to be composed into a comprehensive workflow. Through this analysis, we identify critical gaps and opportunities, where additional technologies and standards would facilitate workflows that can effectively deliver gigabase engineered genomes. Table 2 summarizes the potential solutions that we have identified, which are detailed in the following subsections.
## Genome refactoring and design
Current genome engineering projects have focused primarily on refactoring genomes while preserving their cellular function. For example, three recent projects have involved eliminating nonessential elements27, reordering genes17, and inserting metabolic pathways37. At this level, two critical challenges for scaling are accessing well-annotated source genomes and representing and exchanging designs for modified genomes. More complex changes of organism function will pose additional challenges related to composing parts to produce novel cellular functions.
Currently, genome design generally involves modifying pre-existing organism sequences, such as those available in the public archives of the International Nucleotide Sequence Database Collaboration (INSDC)38, which currently contains ~$$1{0}^{5}$$ bacterial genomes and hundreds of eukaryotic genomes39,40,41,42,43. Functional annotation is key, as genome engineers will need to consider tissue-specific expression patterns, regulatory elements, structural elements, replication origins, clinically significant sites of DNA recombination and instability, etc. The consistency of annotations is a key challenge, as many genomes have been annotated by different toolchains that produce significantly different annotations. For example, the human reference genomes generated by the RefSeq and GENCODE projects have notable differences44,45 with likely engineering consequences, such as ability to predict loss-of-function from interaction with alternative splicings. Much of this knowledge is also dispersed among different resources, though annotations can be integrated with the aid of services such as NCBI Genome Viewer46, WebGestalt47, and DAVID48. For moving to the gigabase scale, improved annotation APIs will be valuable, as would estimates of the confidence and reliability of annotations, such as the RefSeq database does with the Evidence and Conclusion Ontology49.
The gigabase scale poses challenges for the representation and exchange of genome designs as well. Common formats such as GenBank and EMBL are monolithic in their treatment of sequences, which makes it difficult to integrate or harmonize editing across multiple concurrent users, and can even cause difficulties in simply transferring the data. Two formats better suited for genome engineering are the Generic Feature Format (GFF) version 3 and the Synthetic Biology Open Language (SBOL) version 250. GFF3 allows hierarchical organization of sequence descriptions (e.g., genes may be organized into clusters, and clusters into chromosomes), uses the Sequence Ontology51 to annotated sequences, and has already been used in the Sc 2.0 genome engineering project18. SBOL 2 is also routinely used for hierarchical description of edited genomes52 and can interoperate with GFF3 (though GFF3 only represents a subset of SBOL)53. SBOL provides a richer design-centric language, including support for variants, libraries, and partial designs (e.g., identifying genes in a cluster, but not yet particular variants or cluster arrangement), other elements and cellular functions (e.g., proteins, metabolic pathways, regulatory interactions). SBOL also interoperates with models encoded in the Systems Biology Markup Language (SBML)54,55. Both GFF3 and SBOL, however, would benefit from more stable specifications of sequence positions within chromosomes, as sequence index is fragile to changes and sequence uncertainties. SBOL supports (and GFF3 could be extended to support) expression of nonstandard bases and sequence modifications in an enhanced sequence encoding language such as BpForms56.
Representations of genome designs also need to express design constraints and policies, such as removal of restriction sites, separation of overlapping features, replacement of codons, and optimization for DNA synthesis. Projects such as Sc 2.0 have implemented this with a combination of guidelines for human hand-editing and custom software tools, and DNA synthesis providers provide interfaces to check for manufacturability constraints. At the gigabase scale, however, it will be beneficial to adopt more powerful and expressive languages for describing design policies, such as rule-based ontologies57,58, and to include assembly and transformation plans in design representations to simplify adjustments for manufacturability. JGI’s BOOST tool provides a prototype in this direction59. SBOL is well-suited for this task, though GenBank and GFF3 could also, at least in principle, be extended to encode such information.
Modeling will become increasingly important as genome engineering moves beyond refactoring and recoding into more complex changes to an organism’s function. Genome-scale metabolic models60,61 and whole-cell models62 can be constructed by combining biochemical and genomic information from multiple databases, such as BioCyc63 and the SEED64. Models will also need to predict the behavior of organisms that are composed of separately characterized genetic parts, devices, pathways, and genome fragments. Substantial fundamental research still needs to be conducted to make such models practical at the gigabase scale.
## Building engineered genomes
Technology and protocols for building engineered genomes are advancing rapidly, with potential paths to the gigabase scale discussed, for example, in ref. 1 and ref. 23. Depending on the specific host and intended function of the engineered organism, there are numerous potential approaches and protocols for DNA synthesis, assembly, and delivery. Currently, there is an unmet need for guidance on best practices for measuring, tracking, and sharing information regarding engineered genomes and intermediate samples.
Manipulating DNA during assembly offers ample opportunities for reduced yield, breakage, error, and other sources of uncertainty in achieving the designed DNA sequence. Protocols and commercial kits to assemble shorter DNA fragments into larger constructs often involve amplification, handling, purification, transformation, or other storage and delivery steps that can increase uncertainty in the quality and quantity of the DNA. Assembled DNA may also include added sequences that are not biologically active, as in the case for some methods using restriction enzymes, or scars, such as occur may occur with Golden Gate Assembly65 or MoClo66. Gibson Assembly67 is scarless, but the yield and specific results may depend on the secondary structure of the DNA fragments. Thus, in addition to sequence information, workflows will likely need extended representations that can also track the full range of information likely to affect assembly products, including DNA secondary structure, assembly method, sequences required for assembly and their location along the DNA molecule (e.g., landing pads or sequences for compatibility with protocol-hosting strains of E. coli or yeast), and intended epigenetic modifications. The results verifying both intermediate and final sequence onstruction are typically produced in the FASTQ format68, which is generally sufficient for smaller constructs. To operate on large-scale genomes, however, more comprehensive descriptions of a genome and its variations may be made with representations such as GVF69 or SBOL70.
Suitable options for the delivery of large, assembled DNA constructs and whole genomes are generally lacking. The yield of existing processes, such as electrical and chemical transformation or genome transplantation, could be improved significantly to increase their utility, and a broader range of approaches should be developed for use with any organism and cell type. This may also require identifying new cell-free environments or cell-based chassis for assembling and manipulating DNA that also have compatibility with genome packaging and delivery systems into host organisms. To facilitate such development, delivery protocols and their associated information regarding number of biological and technical replicate experiments, methods, measurements, etc. should be available in a machine-readable format. This should include information regarding the host cell, such as its genotype, which is often not fully verified. The adoption of best practices from industrial biomanufacturing settings and implementation of laboratory information management systems (LIMS) could provide a path forward toward integrating appropriate measurements, process controls, and information handling, as well as the tracking and exchange of samples. Advancing the use of automation to support the build step of the genome engineering workflow requires evaluating which steps may reduce costs and speed results, the availability of automated methods, ways to effectively share those methods and adapt them across platforms and manufacturers, and ways to more simply integrate and tune automated workflows.
## Testing the function of engineered genomes
Strain fitness and other phenotypes can be assessed via a wide range of biochemical and omics measurements, the details of which are beyond the scope of this discussion. In all cases, however, collaborating organizations will need to agree on specific measurements, along with control and calibration measurements, to ensure that the results can be compared and used across the participating laboratories.
DNA constructs are often evaluated for their associated growth phenotypes to determine the nature and extent of unexpected consequences for cell function and fitness due to the revised genome sequence. Engineered cell lines should also be evaluated for robustness to changes in the environmental context that the cells are likely to experience during typical use in the intended application, as well as stability over relevant timescales to evolution or adaptation. This is complicated by the need for shared definitions and measurements for fitness, metabolic burden, and other phenotypic properties.
Standard protocols, reference cell lines, and the use of experimental design are examples of tools available to increase the rigor and confidence in conclusions that can be drawn from testing. It will likely also be useful to develop standards and measurement assurance for testing engineered genomes. Such foundations can be used to help identify relationships between genotype and phenotype or determine the contributions of biological stochasticity and measurement uncertainty to the overall variability in a measured trait, though comprehensive methods of this sort are likely to require significant fundamental research.
Calibration of biological assays aids in comparing results both within a single laboratory and across different laboratories. Recent studies, for example, for fluorescence71,72, absorbance73, and RNAseq74 measurements, demonstrate the possibility of realizing scalable and cost-effective comparability in biological measurements. Organism engineering is likely to be facilitated by the development of additional calibrated measurement methods and absolute quantitation of an organism’s properties.
Establishing shared representations and practices for metadata, process controls, and calibration will also be critical. Automation-assisted integration and comparison of the data, metadata, process controls, and calibration across laboratories will facilitate both the testing process and learning through modeling and simulation. Some existing ontologies can be leveraged for this purpose, such as the Experimental Conditions Ontology75 (ECO), the Experimental Factor Ontology76 (EFO), and the Measurement Method Ontology75 (MMO). In addition, appropriate LIMS tooling and curation assistance software (e.g., RightField77) will be vital for enabling such metadata to be created consistently, correctly, and in a timely fashion, by limiting the required input from human investigators.
## Learning systematically from test results
As genome engineering affects systems throughout an organism, comprehensive models are needed that can help to both predict and interpret the relationship between genotype and phenotype. Although some models have been constructed for a whole cell62 or whole organism78, developing and tuning such models is extremely challenging. To scale to gigabase genomes, it will be valuable develop improved capabilities for creating, calibrating, and verifying models.
The first challenge in learning from the data is discovering and marshaling the data needed. Partial solutions exist, such as the workflow model introduced in SBOL 2.250, and ontologies such as the Open Biological and Biomedical Ontology79, the Experimental Factor Ontology76, the Systems Biology Ontology80, and phenotype ontologies81,82. These will need to be integrated and extended to cover the full range of needs for genome engineering.
Automation-assisted generation and verification of models at scale, however, still have many open fundamental research challenges, including addressing the combinatorial complexity of biology and the multiple scales between genomes and organismal behavior, high-performance simulation of large models, model verification, and representation of model semantic meaning and provenance24,25.
Until we have comprehensive predictive models, engineers will likely rely on ad hoc combinations of predictive models of parts of organisms, data-driven models, and heuristic design rules. For example, constraint-based models are often used in metabolic engineering34, PSORTb83 can be used to help target proteins to specific compartments, and GC-content optimization can be used to improve host compatibility84. Gigabase-scale genome engineering will require applying many such models simultaneously, and thus will benefit from adopting existing standard formats designed to facilitate biological model sharing and composition, such as SBML85, CellML86, NeuroML87, and other standards in the Computational Modeling in Biology Network (COMBINE)88. Large numbers of models in these formats can already be found in public databases, such as BioModels89, the NeuroML database90, Open Source Brain91, and the Physiome Model Repository92. Similarly, repositories such as Kipoi93 and the DockerHub repository94 can already be used to share data-driven models. Further extensions to such formats, however, will be valuable for automating the learning process, including associating semantic meaning with model components, capturing the provenance of model elements (e.g., data sources, assumptions, and design motivations), and capturing information about their predictive capabilities and applicable scope.
To increase automation in learning such models from data, it will likely be valuable to develop new repositories of models of individual biological parts that can be composed into models of entire organisms95,96; new methods for generating model variants that explain new observations by incorporating models of additional parts, alternative kinetic laws, or alternative parameter values; and new model selection techniques for nonlinear multiscale models97.
## Coordination and sharing in complex workflows
Tasks in isolation are not enough: efficient operation of the design–build–test–learn cycle for engineering gigabase genomes will require coordinating all of the numerous heterogeneous tasks discussed into clear, cohesive, reproducible workflows98,99 for software interactions, for laboratory protocols, and for management of tasks and personnel. Automating workflows also provides opportunities to implement best practices for cybersecurity, cyberbiosecurity, and biosecurity.
For integrating informational tasks, computational workflow engines enable specification, reproducible execution, and exchange of complex workflows involving multiple software programs and computing environments. Current workflow tools include both general tools, such as the Common Workflow Language (CWL)100, the Dockstore101 and MyExperiment102 sharing environments, and the PROV ontology for tracking information provenance103 (which is already being applied to link design–build–test–learn cycles in SBOL50). There are also a number of bioinformatics-focused engines, including Cromwell104, Galaxy105, NextFlow106, and Toil107. These can be readily adopted for gigabase engineering through steps such as including CWL files in COMBINE archives108, developing REST or other programmatic interfaces for databases used in genome engineering, containerization109 of genome engineering computational tools, and depositing these containers to a registry such as DockerHub94. Other enhancements likely to be useful include the development of graphical workflow tools for genome engineering, an ontology for annotating the semantic meaning of workflow tasks, and the application of issue tracking systems, such as GitHub issues110 or Jira111, to help coordinate teams on the complex tasks involved in designing genomes that require human intervention.
For experimental protocols, a number of technologies have already been developed to automate and integrate experimental workflows as well. Laboratory automation systems can greatly improve both reproducibility and efficiency112 and can also be integrated with LIMS113 to help track workflows and reagent stocks. A number of automation languages and systems have been developed, including Aquarium114, Antha115, and Autoprotocol116. Although these have not been widely adopted, they have been successfully applied to genetic engineering (e.g., ref. 117), and gigascale genome engineering would benefit from standardization and integration of such systems for application to build and test protocols.
Once links are established across different portions of a workflow, unified access to information in databases for various institutions and stages of the workflow can be accomplished using standard federation methods and any of the various mature open tools for database management systems (DBMS). Scalable sharing would be further enhanced by adoption of the FAIR (findable, accessible, interoperable, reproducible) data management principles118, which puts specific emphasis on automation friendliness of data sharing. Repositories that support these principles and are applicable to genome engineering include FAIRDOMHub119, Experimental Data Depot (EDD)120, and SynBioHub121.
## Contracts, intellectual property, and laws
Large-scale genome engineering also poses novel challenges in coordinating legal and contractual interactions. When using digital information, both humans and machines need to know the accompanying copyright and licensing obligations. Systematic licensing regimes have been developed for software by the Open Source Initiative (OSI) and other software organizations122 and for media and other content with the Creative Commons (CC) family of licenses123, both of which readily allow either a user or a machine to determine if a digital object can be reused, if its reuse is prohibited, or if more complicated negotiation or determination is required. Such systems can be applied to much of the digital information in genome engineering. Care will need to be taken, however, regarding sensitive personal information and European Union database protection rights, which these do not address.
Transfer of physical biological materials was first standardized in 1995 with NIH’s Uniform Biological Materials Transfer Agreement (UBMTA), which is used extensively by organizations such as Addgene. Broader and more compatible systems have been developed in the form of the Science Commons project124 and the OpenMTA125. There are still significant open problems regarding compliance with local regulatory and legal systems, however, particularly when materials cross international borders. Moreover, material transfer agreements generally do not address the intellectual property for materials, which is typically governed through patent law. No publicly available system yet supports automation for patent licensing. Development of automation-friendly intellectual property management might be supported by defining tiered levels that are simultaneously intelligible for the common user, legal experts, and computer systems—though establishing which material or usages can be classified into which tiers may be a difficult process of legal interpretation. Effective use in automation-assisted workflows will also require recording information about which inputs are involved in the production of results, using mechanisms such as the PROV ontology103.
Finally, organizations will also need to manage the level of exposure of information, whether due to issues of privacy, safety, publication priority, or other similar concerns. Again, no current system exists, but a basis for developing one may be found in the cross-domain information sharing protocols that have been developed in other domains126,127.
## Recommendations and outlook
In summary, scaling up to gigabase genomes presents a wide range of challenges (Table 2). We observe that these challenges cluster into four general themes, each with a different set of needs and paths for development.
The first theme is representing and exchanging designs, plans, data, metadata, and knowledge. Managing information for gigabase genome design requires addressing many challenges regarding scale, representation, and standards. Relatively mature technologies exist to address most individual needs, as well as to assist with the integration of workflows. The practical implementation of effective workflows will require significant investment in building infrastructure and tools that adopt these technologies, including domain-specific extensions and refinements.
The second theme is sharing and integrating data quality and experimental measurements. Sharing and integrating information arising from measurements of biological material poses significant challenges. It remains unclear what information would be advantageous to share, given the difficulty of obtaining and interpreting measurements of biological systems and the expense and unfavorable scaling of data curation. However, effective integration depends on associating reproducible measurement data with well-curated knowledge and metadata in compatible representations. A number of potential solutions exist for each of these, but significant investment will be needed to investigate how the state of the art can be extended to address these needs.
The third theme is integration of modeling and design at the gigabase scale. Considerable challenges surround efforts to develop a deeper understanding of the relationship between genotype and phenotype, regarding both the interpretation of experimental data and the application of that data to create and validate models, which may be applied in computer-assisted design. Long-term investment in fundamental research is needed, and the suite of biological systems of varying complexity, from cell-free systems to minimal and synthetic cells to natural living systems, may offer suitable experimental platforms for learning the relationship between genotype and phenotype.
Finally, the fourth theme is technical support for Ethical, Legal, and Societal Implications (ELSI) and Intellectual Property (IP) at scale. At the gigabase scale, computer-assisted workflows will be necessary to manage contracts, intellectual property, materials transfers, and other legal and societal interactions. Such workflows will need to be developed by interdisciplinary teams involving experts in law, ELSI issues, software engineering, and knowledge representation. Moreover, it will be critical to address these issues early, to minimize the potential for problematic entanglements associated with the reuse of resources.
In short, engineering gigabase-scale genomes presents significant challenges that will require coordinated investment to overcome. Because many other areas of bioscience face similar challenges, solutions to these challenges will likely also benefit the broader bioscience community. Importantly, the challenges of scale, integration, and lack of knowledge faced in genome engineering are not fundamentally different in nature than those that have been overcome previously in other engineering ventures, such as aerospace engineering and microchip design, which required organizing humans and sharing information across many institutions over time. Thus, we expect to be able to adapt solutions from these other fields for genome engineering.
Investment in capabilities for genome engineering workflows is critical to move from a world in which genome engineering is a heroic effort to one in which genome engineering is routine, safe, and reliable. Investment in workflows for genome engineering will support and enable a vast number of projects, including many not yet conceived, as was the case for reading the human genome. As workflow technologies improve, we anticipate that the trends of expanding team size will eventually reverse, enabling high-fidelity whole-genome engineering at a modest cost and supporting a wide range of medical and industrial applications.
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## Acknowledgements
This work was supported, in part, by NIH awards P41-EB023912 and R35-GM119771 and by NSF awards 1548123 and 1522074. We thank the reviewers for legal insights for Section “Contracts, intellectual property, and laws”, and thank Nicola Hawes for help illustrating Fig. 2. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of these funding agencies or the U.S. Government. This document does not contain technology or technical data controlled under either U.S. International Traffic in Arms Regulation or U.S. Export Administration Regulations. Certain commercial equipment, instruments, or materials are identified to adequately specify experimental procedures. Such identification neither implies recommendation nor endorsement by the National Institute of Standards and Technology nor that the equipment, instruments, or materials identified are necessarily the best for the purpose.
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Bartley, B.A., Beal, J., Karr, J.R. et al. Organizing genome engineering for the gigabase scale. Nat Commun 11, 689 (2020). https://doi.org/10.1038/s41467-020-14314-z
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• ### Engineered yeast genomes accurately assembled from pure and mixed samples
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# zbMATH — the first resource for mathematics
## Fitting, Melvin Chris
Compute Distance To:
Author ID: fitting.melvin-c Published as: Fitting, M.; Fitting, M. C.; Fitting, Melvin; Fitting, Melvin C. Homepage: http://www.melvinfitting.org/ External Links: MGP · Wikidata · dblp · GND
Documents Indexed: 110 Publications since 1969, including 15 Books
all top 5
#### Co-Authors
94 single-authored 2 Thalmann, Lars 2 Voronkov, Andrei 1 Artemov, Sergei 1 Ben-Jacob, Marion 1 Fitting, Greer 1 Georgatos, Konstantinos 1 Kuznets, Roman 1 Marek, V. Wiktor 1 Mendelsohn, Richard L. 1 Müller, Berndt 1 Nerode, Anil 1 Orłowska, Ewa S. 1 Ramanujam, Rohit Sunkam 1 Rayman, Brian 1 Salvatore, Felipe 1 Smullyan, Raymond Merrill 1 Truszczyński, Mirosław 1 Ye, Ruili
all top 5
#### Serials
11 Annals of Pure and Applied Logic 8 Studia Logica 7 Notre Dame Journal of Formal Logic 7 Journal of Logic and Computation 6 Annales Societatis Mathematicae Polonae. Series IV 6 The Journal of Logic Programming 4 The Journal of Symbolic Logic 4 Fundamenta Informaticae 3 Theoria 2 Journal of Philosophical Logic 2 Theoretical Computer Science 2 Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 2 Oxford Logic Guides 2 Synthese Library 1 Synthese 1 Journal of Automated Reasoning 1 Bulletin of the European Association for Theoretical Computer Science (EATCS) 1 Journal of Applied Non-Classical Logics 1 Annals of Mathematics and Artificial Intelligence 1 Journal of Applied Logic 1 Cambridge Tracts in Mathematics 1 Studies in Logic and the Foundations of Mathematics 1 Studies in Fuzziness and Soft Computing 1 Trends in Logic – Studia Logica Library 1 Studies in Logic (London) 1 Texts in Mathematics 1 Cadernos de Lógica e Computação 1 Outstanding Contributions to Logic
all top 5
#### Fields
95 Mathematical logic and foundations (03-XX) 29 Computer science (68-XX) 6 General and overarching topics; collections (00-XX) 4 History and biography (01-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Mathematics education (97-XX)
#### Citations contained in zbMATH
80 Publications have been cited 1,295 times in 923 Documents Cited by Year
Proof methods for modal and intuitionistic logics. Zbl 0523.03013
Fitting, Melvin
1983
A Kripke-Kleene semantics for logic programs. Zbl 0589.68011
Fitting, Melvin
1985
Bilattices and the semantics of logic programming. Zbl 0757.68028
Fitting, Melvin
1991
First-order modal logic. Zbl 1025.03001
Fitting, Melvin; Mendelsohn, Richard L.
1998
First-order logic and automated theorem proving. 2nd ed. Zbl 0848.68101
Fitting, Melvin
1996
Intuitionistic logic, model theory and forcing. Zbl 0188.32003
Fitting, M. C.
1969
The logic of proofs, semantically. Zbl 1066.03059
Fitting, Melvin
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First-order logic and automated theorem proving. Zbl 0692.68002
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Many-valued modal logics. Zbl 0745.03018
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Fixpoint semantics for logic programming a survey. Zbl 1002.68023
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Kleene’s three valued logics and their children. Zbl 0804.03016
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Kleene’s logic, generalized. Zbl 0744.03025
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Bilattices are nice things. Zbl 1157.03308
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Bilattices and the theory of truth. Zbl 0678.03028
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First-order modal tableaux. Zbl 0648.03004
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Prefixed tableaus and nested sequents. Zbl 1241.03021
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Tableau methods of proof for modal logics. Zbl 0184.28102
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The family of stable models. Zbl 0798.68096
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Types, tableaus, and Gödel’s God. Zbl 1038.03001
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Notes on the mathematical aspects of Kripke’s theory of truth. Zbl 0588.03003
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Logic programming on a topological bilattice. Zbl 0647.68096
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First-order intensional logic. Zbl 1061.03024
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Partial models and logic programming. Zbl 0629.68090
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1986
Justification logics, logics of knowledge, and conservativity. Zbl 1173.03013
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Term-modal logics. Zbl 0992.03026
Fitting, Melvin; Thalmann, Lars; Voronkov, Andrei
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Tableaus for many-valued modal logic. Zbl 0837.03017
Fitting, Melvin
1995
Nested sequents for intuitionistic logics. Zbl 1327.03006
Fitting, Melvin
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Realizations and LP. Zbl 1221.03020
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Set theory and the continuum problem. Zbl 0888.03032
Smullyan, Raymond M.; Fitting, Melvin
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A quantified logic of evidence. Zbl 1133.03008
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Modal logics between propositional and first-order. Zbl 1017.03008
Fitting, Melvin
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Interpolation for first order S5. Zbl 1009.03013
Fitting, Melvin
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Metric methods: Three examples and a theorem. Zbl 0823.68017
Fitting, Melvin
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Many-valued non-monotonic modal logics. Zbl 0978.03518
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Fundamentals of generalized recursion theory. Zbl 0597.03028
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Modal interpolation via nested sequents. Zbl 1369.03103
Fitting, Melvin; Kuznets, Roman
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Reasoning with justifications. Zbl 1166.03006
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Destructive modal resolution. Zbl 0724.03011
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The pure logic of necessitation. Zbl 0819.03011
Fitting, Melvin C.; Marek, V. Wiktor; Truszczyński, Mirosław
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Model existence theorems for modal and intuitionistic logics. Zbl 0286.02060
Fitting, Melvin
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A tableau proof method admitting the empty domain. Zbl 0177.01102
Fitting, Melvin
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Logics with several modal operators. Zbl 0188.31801
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Realization using the model existence theorem. Zbl 1403.03030
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The realization theorem for S5 a simple, constructive proof. Zbl 1317.03024
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A theory of truth that prefers falsehood. Zbl 0880.03003
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Negation as refutation. Zbl 0716.68024
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A deterministic PROLOG fixpoint semantics. Zbl 0592.68021
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An embedding of classical logic in S4. Zbl 0219.02011
Fitting, M.
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Justification logic. Reasoning with reasons. Zbl 07056607
Artemov, Sergei; Fitting, Melvin
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Modal logics, justification logics, and realization. Zbl 1400.03040
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Justification logics and hybrid logics. Zbl 1215.03032
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Bisimulations and Boolean vectors. Zbl 1083.03025
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Intensional logic – beyond first order. Zbl 1048.03014
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Barcan both ways. Zbl 0993.03016
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A simple propositional $$\text{S}5$$ tableau system. Zbl 0972.03017
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leanTAP revisited. Zbl 0909.03013
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Enumeration operators and modular logic programming. Zbl 0608.68005
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Possible world semantics for first-order logic of proofs. Zbl 1345.03036
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2014
Incompleteness in the land of sets. Zbl 1152.03003
Fitting, Melvin
2007
Realizations and LP. Zbl 1132.03325
Fitting, Melvin
2007
Herbrand’s theorem for a modal logic. Zbl 0947.03024
Fitting, Melvin
1999
Foreword: A tribute to Professor Helena Rasiowa. Zbl 0922.01035
Fitting, Melvin
1999
A modal Herbrand theorem. Zbl 0863.68102
Fitting, Melvin
1996
Pseudo-Boolean valued Prolog. Zbl 0667.68110
Fitting, Melvin
1988
Computability theory, semantics, and logic programming. Zbl 0613.68019
Fitting, Melvin
1987
Subformula results in some propositional modal logics. Zbl 0397.03011
Fitting, Melvin
1978
Paraconsistent logic, evidence, and justification. Zbl 1417.03192
Fitting, Melvin
2017
Proving completeness for nested sequent calculi. Zbl 1261.03085
Fitting, Melvin
2011
FOIL axiomatized. Zbl 1114.03014
Fitting, Melvin
2006
Beyond two: Theory and applications of multiple-valued logic. Zbl 1015.00007
Fitting, Melvin (ed.); Orłowska, Ewa (ed.)
2003
First-order alethic modal logic. Zbl 1111.03303
Fitting, Melvin
2002
Databases and higher types. Zbl 0983.68055
Fitting, Melvin
2000
Modality and databases. Zbl 0963.03028
Fitting, Melvin
2000
Higher-order modal logic – a sketch. Zbl 0955.03030
Fitting, Melvin
2000
Introduction. Zbl 0972.03542
Fitting, Melvin
1999
On quantified modal logic. Zbl 0936.03019
Fitting, Melvin
1999
A program to compute Gödel-Löb fixpoints. Zbl 1028.03509
Fitting, Melvin
1996
A tableau system for propositional S5. Zbl 0314.02039
Fitting, Melvin
1977
Justification logic. Reasoning with reasons. Zbl 07056607
Artemov, Sergei; Fitting, Melvin
2019
Paraconsistent logic, evidence, and justification. Zbl 1417.03192
Fitting, Melvin
2017
Realization using the model existence theorem. Zbl 1403.03030
Fitting, Melvin
2016
Modal logics, justification logics, and realization. Zbl 1400.03040
Fitting, Melvin
2016
Modal interpolation via nested sequents. Zbl 1369.03103
Fitting, Melvin; Kuznets, Roman
2015
Nested sequents for intuitionistic logics. Zbl 1327.03006
Fitting, Melvin
2014
Possible world semantics for first-order logic of proofs. Zbl 1345.03036
Fitting, Melvin
2014
Prefixed tableaus and nested sequents. Zbl 1241.03021
Fitting, Melvin
2012
The realization theorem for S5 a simple, constructive proof. Zbl 1317.03024
Fitting, Melvin
2011
Fitting, Melvin
2011
Proving completeness for nested sequent calculi. Zbl 1261.03085
Fitting, Melvin
2011
Justification logics and hybrid logics. Zbl 1215.03032
Fitting, Melvin
2010
Realizations and LP. Zbl 1221.03020
Fitting, Melvin
2009
Reasoning with justifications. Zbl 1166.03006
Fitting, Melvin
2009
Justification logics, logics of knowledge, and conservativity. Zbl 1173.03013
Fitting, Melvin
2008
A quantified logic of evidence. Zbl 1133.03008
Fitting, Melvin
2008
Incompleteness in the land of sets. Zbl 1152.03003
Fitting, Melvin
2007
Realizations and LP. Zbl 1132.03325
Fitting, Melvin
2007
Bilattices are nice things. Zbl 1157.03308
Fitting, Melvin
2006
FOIL axiomatized. Zbl 1114.03014
Fitting, Melvin
2006
The logic of proofs, semantically. Zbl 1066.03059
Fitting, Melvin
2005
First-order intensional logic. Zbl 1061.03024
Fitting, Melvin
2004
Bisimulations and Boolean vectors. Zbl 1083.03025
Fitting, Melvin
2003
Intensional logic – beyond first order. Zbl 1048.03014
Fitting, Melvin
2003
Beyond two: Theory and applications of multiple-valued logic. Zbl 1015.00007
Fitting, Melvin (ed.); Orłowska, Ewa (ed.)
2003
Fixpoint semantics for logic programming a survey. Zbl 1002.68023
Fitting, Melvin
2002
Types, tableaus, and Gödel’s God. Zbl 1038.03001
Fitting, Melvin
2002
Modal logics between propositional and first-order. Zbl 1017.03008
Fitting, Melvin
2002
Interpolation for first order S5. Zbl 1009.03013
Fitting, Melvin
2002
First-order alethic modal logic. Zbl 1111.03303
Fitting, Melvin
2002
Term-modal logics. Zbl 0992.03026
Fitting, Melvin; Thalmann, Lars; Voronkov, Andrei
2001
Databases and higher types. Zbl 0983.68055
Fitting, Melvin
2000
Modality and databases. Zbl 0963.03028
Fitting, Melvin
2000
Higher-order modal logic – a sketch. Zbl 0955.03030
Fitting, Melvin
2000
Barcan both ways. Zbl 0993.03016
Fitting, Melvin
1999
A simple propositional $$\text{S}5$$ tableau system. Zbl 0972.03017
Fitting, Melvin
1999
Herbrand’s theorem for a modal logic. Zbl 0947.03024
Fitting, Melvin
1999
Foreword: A tribute to Professor Helena Rasiowa. Zbl 0922.01035
Fitting, Melvin
1999
Introduction. Zbl 0972.03542
Fitting, Melvin
1999
On quantified modal logic. Zbl 0936.03019
Fitting, Melvin
1999
First-order modal logic. Zbl 1025.03001
Fitting, Melvin; Mendelsohn, Richard L.
1998
leanTAP revisited. Zbl 0909.03013
Fitting, Melvin
1998
A theory of truth that prefers falsehood. Zbl 0880.03003
Fitting, Melvin
1997
First-order logic and automated theorem proving. 2nd ed. Zbl 0848.68101
Fitting, Melvin
1996
Set theory and the continuum problem. Zbl 0888.03032
Smullyan, Raymond M.; Fitting, Melvin
1996
A modal Herbrand theorem. Zbl 0863.68102
Fitting, Melvin
1996
A program to compute Gödel-Löb fixpoints. Zbl 1028.03509
Fitting, Melvin
1996
Tableaus for many-valued modal logic. Zbl 0837.03017
Fitting, Melvin
1995
Kleene’s three valued logics and their children. Zbl 0804.03016
Fitting, Melvin
1994
Metric methods: Three examples and a theorem. Zbl 0823.68017
Fitting, Melvin
1994
The family of stable models. Zbl 0798.68096
Fitting, Melvin
1993
Many-valued modal logics. II. Zbl 0772.03006
Fitting, Melvin
1992
Many-valued non-monotonic modal logics. Zbl 0978.03518
Fitting, M.
1992
The pure logic of necessitation. Zbl 0819.03011
Fitting, Melvin C.; Marek, V. Wiktor; Truszczyński, Mirosław
1992
Bilattices and the semantics of logic programming. Zbl 0757.68028
Fitting, Melvin
1991
Many-valued modal logics. Zbl 0745.03018
Fitting, Melvin C.
1991
Kleene’s logic, generalized. Zbl 0744.03025
Fitting, Melvin
1991
First-order logic and automated theorem proving. Zbl 0692.68002
Fitting, Melvin
1990
Destructive modal resolution. Zbl 0724.03011
Fitting, Melvin
1990
Bilattices and the theory of truth. Zbl 0678.03028
Fitting, Melvin
1989
Negation as refutation. Zbl 0716.68024
Fitting, Melvin
1989
First-order modal tableaux. Zbl 0648.03004
Fitting, Melvin
1988
Logic programming on a topological bilattice. Zbl 0647.68096
Fitting, Melvin
1988
Pseudo-Boolean valued Prolog. Zbl 0667.68110
Fitting, Melvin
1988
Enumeration operators and modular logic programming. Zbl 0608.68005
Fitting, Melvin
1987
Computability theory, semantics, and logic programming. Zbl 0613.68019
Fitting, Melvin
1987
Notes on the mathematical aspects of Kripke’s theory of truth. Zbl 0588.03003
Fitting, Melvin
1986
Partial models and logic programming. Zbl 0629.68090
Fitting, Melvin
1986
A Kripke-Kleene semantics for logic programs. Zbl 0589.68011
Fitting, Melvin
1985
A deterministic PROLOG fixpoint semantics. Zbl 0592.68021
Fitting, Melvin
1985
Proof methods for modal and intuitionistic logics. Zbl 0523.03013
Fitting, Melvin
1983
Fundamentals of generalized recursion theory. Zbl 0597.03028
Fitting, Melvin
1981
Subformula results in some propositional modal logics. Zbl 0397.03011
Fitting, Melvin
1978
A tableau system for propositional S5. Zbl 0314.02039
Fitting, Melvin
1977
Model existence theorems for modal and intuitionistic logics. Zbl 0286.02060
Fitting, Melvin
1974
Tableau methods of proof for modal logics. Zbl 0184.28102
Fitting, Melvin
1972
A tableau proof method admitting the empty domain. Zbl 0177.01102
Fitting, Melvin
1971
An embedding of classical logic in S4. Zbl 0219.02011
Fitting, M.
1971
Intuitionistic logic, model theory and forcing. Zbl 0188.32003
Fitting, M. C.
1969
Logics with several modal operators. Zbl 0188.31801
Fitting, M.
1969
all top 5
#### Cited by 1,034 Authors
26 Fitting, Melvin Chris 14 Kuznets, Roman 13 Wansing, Heinrich Theodor 11 Shramko, Yaroslav V. 10 Avron, Arnon 9 Arieli, Ofer 9 Benzmüller, Christoph Ewald 9 Demri, Stéphane P. 9 Petrukhin, Yaroslav Igorevich 8 Artemov, Sergei 8 Denecker, Marc 7 Peltier, Nicolas 7 Straccia, Umberto 6 Beckert, Bernhard 6 Bogaerts, Bart 6 De Cock, Martine 6 Dubois, Didier 6 Gabbay, Dov M. 6 Indrzejczak, Andrzej 6 Kamide, Norihiro 6 Leszczyńska-Jasion, Dorota 6 Liau, Churn-Jung 6 Massacci, Fabio 6 Pym, David J. 6 Rivieccio, Umberto 6 Subrahmanian, V. S. 6 Szałas, Andrzej 6 Yu, Junhua 5 Baratella, Stefano 5 Baumgartner, Peter 5 Braüner, Torben 5 Demey, Lorenz 5 Hähnle, Reiner 5 Hitzler, Pascal 5 Kurokawa, Hidenori 5 Marek, V. Wiktor 5 Martins, Manuel António 5 Nerode, Anil 5 Ono, Hiroakira 5 Rümmer, Philipp 5 Schockaert, Steven 5 Seda, Anthony Karel 5 Studer, Thomas 5 Truszczyński, Mirosław 5 Vennekens, Joost 5 Wintein, Stefan 4 Benevides, Mario R. F. 4 Blackburn, Patrick 4 Ciucci, Davide 4 Delahaye, Jean-Paul 4 Dyckhoff, Roy 4 Ésik, Zoltán 4 Giordani, Alessandro 4 Kreitz, Christoph 4 Madeira, Alexandre 4 Maruyama, Yoshihiro 4 Milnikel, Robert Saxon 4 Movsisyan, Yuri Movses 4 Muskens, Reinhard A. 4 Pacuit, Eric 4 Ramanayake, Revantha 4 Rodríguez, Ricardo Oscar 4 Rondogiannis, Panos 4 Rönnedal, Daniel 4 Schmidt, Renate A. 4 Shangin, Vasily 4 Smessaert, Hans 4 Vermeir, Dirk 4 Wijesekera, Duminda 4 Willard, Dan E. 4 Woltzenlogel Paleo, Bruno 4 Yamasaki, Susumu 3 Akama, Seiki 3 Areces, Carlos 3 Baaz, Matthias 3 Balbiani, Philippe 3 Belardinelli, Francesco 3 Bidoit, Nicole 3 Bobillo, Fernando 3 Bochman, Alexander 3 Bruynooghe, Maurice 3 Bucheli, Samuel 3 Cerrato, Claudio 3 Cintula, Petr 3 Cornelis, Chris 3 Davidova, Diana S. 3 Dean, Walter 3 Degtyarev, Anatoli Ivanovich 3 Donini, Francesco M. 3 Fan, Tuan-Fang 3 Ferguson, Thomas Macaulay 3 Font, Josep Maria 3 Gargov, George K. 3 Gerla, Giangiacomo 3 Giunchiglia, Fausto 3 Godo, Lluís 3 Governatori, Guido 3 Hájek, Petr 3 Hasan, Osman 3 Hölldobler, Steffen ...and 934 more Authors
all top 5
#### Cited in 89 Serials
93 Studia Logica 72 Theoretical Computer Science 51 Annals of Pure and Applied Logic 40 Journal of Philosophical Logic 40 Journal of Applied Non-Classical Logics 37 Journal of Automated Reasoning 30 Artificial Intelligence 25 The Journal of Symbolic Logic 21 Annals of Mathematics and Artificial Intelligence 21 Journal of Applied Logic 20 Fuzzy Sets and Systems 18 Information and Computation 17 International Journal of Approximate Reasoning 16 Synthese 15 Notre Dame Journal of Formal Logic 15 Logica Universalis 13 Journal of Logic, Language and Information 12 Journal of Symbolic Computation 12 Logic and Logical Philosophy 12 The Review of Symbolic Logic 10 Archive for Mathematical Logic 10 The Bulletin of Symbolic Logic 9 Theory and Practice of Logic Programming 7 Bulletin of the Section of Logic 6 New Generation Computing 6 Formal Aspects of Computing 6 The Journal of Logic and Algebraic Programming 5 Algebra Universalis 5 Journal of Computer and System Sciences 5 International Journal of Computer Mathematics 5 Mathematical Logic Quarterly (MLQ) 5 Theory of Computing Systems 5 Soft Computing 4 Information Sciences 4 Journal of Differential Equations 4 Publications of the Research Institute for Mathematical Sciences, Kyoto University 4 History and Philosophy of Logic 4 International Journal of Intelligent Systems 4 ACM Transactions on Computational Logic 4 Logical Methods in Computer Science 3 Lithuanian Mathematical Journal 3 Journal of Computer Science and Technology 3 MSCS. Mathematical Structures in Computer Science 3 RAIRO. Informatique Théorique et Applications 3 Erkenntnis 2 International Journal of Theoretical Physics 2 Information Processing Letters 2 Algebra and Logic 2 Cybernetics and Systems Analysis 2 Formal Methods in System Design 2 Journal of Mathematical Sciences (New York) 2 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 Journal of Logical and Algebraic Methods in Programming 1 Acta Informatica 1 Archiv für Mathematische Logik und Grundlagenforschung 1 Computers & Mathematics with Applications 1 Journal of the Franklin Institute 1 Mathematical Notes 1 Moscow University Mathematics Bulletin 1 Periodica Mathematica Hungarica 1 The Mathematical Intelligencer 1 Demonstratio Mathematica 1 International Journal of Game Theory 1 The Journal of Mathematical Sociology 1 Journal of Pure and Applied Algebra 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Rendiconti del Seminario Matematico della Università di Padova 1 Transactions of the American Mathematical Society 1 Cybernetics 1 JETAI. Journal of Experimental & Theoretical Artificial Intelligence 1 International Journal of Foundations of Computer Science 1 Games and Economic Behavior 1 Indagationes Mathematicae. New Series 1 Applied Categorical Structures 1 Sbornik: Mathematics 1 Mathematical Communications 1 Topoi 1 Journal of Intelligent and Fuzzy Systems 1 Fixed Point Theory and Applications 1 Journal of Shanghai Jiaotong University (Science) 1 Proceedings of the Steklov Institute of Mathematics 1 Journal of Fixed Point Theory and Applications 1 International Journal of Semantic Computing 1 Science China. Information Sciences 1 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 Symmetry 1 Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika 1 Frontiers of Computer Science 1 European Journal of Mathematics
all top 5
#### Cited in 28 Fields
698 Mathematical logic and foundations (03-XX) 434 Computer science (68-XX) 47 Order, lattices, ordered algebraic structures (06-XX) 20 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 14 General algebraic systems (08-XX) 12 History and biography (01-XX) 10 General and overarching topics; collections (00-XX) 9 Category theory; homological algebra (18-XX) 9 General topology (54-XX) 4 Biology and other natural sciences (92-XX) 3 Ordinary differential equations (34-XX) 3 Information and communication theory, circuits (94-XX) 2 Operator theory (47-XX) 2 Convex and discrete geometry (52-XX) 2 Quantum theory (81-XX) 2 Operations research, mathematical programming (90-XX) 2 Systems theory; control (93-XX) 1 1 Combinatorics (05-XX) 1 Field theory and polynomials (12-XX) 1 Commutative algebra (13-XX) 1 Associative rings and algebras (16-XX) 1 Group theory and generalizations (20-XX) 1 Integral equations (45-XX) 1 Geometry (51-XX) 1 Differential geometry (53-XX) 1 Statistics (62-XX) 1 Relativity and gravitational theory (83-XX)
#### Wikidata Timeline
The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.
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# mms&convergence seminar
The mms&convergence seminar is devoted to Sectional/Ricci lower bounds, Gromov-Hausdorff and Intrinsic Flat convergence and, in general, metric measure spaces.
In 2020 I organized it on my own and since 2021 Daniele Semola became a co-organizer.
The schedule for Spring 2022 was announced ---> here <---
Thanks to all the attendants and speakers for your participation
Dec 10, 2021. Guofang Wei. Examples of Ricci limit spaces with non-integer Hausdorff dimension
Dec 3, 2021. Paul Creutz. Video: Area minimizing surfaces for singular boundary values
Nov 26, 2021. Jikang Wang. Video: Ricci limit spaces are semi-locally simply connected
Nov 19, 2021. Mathias Braun. Video: Vector calculus for tamed Dirichlet spaces
Nov 12, 2021. David Bate. Characterising rectifiable metric spaces using tangent spaces
Nov 5, 2021. Ivan Violo. Video: Rigidity and almost-rigidity of the Sobolev inequality under lower Ricci curvature bounds
Oct 29, 2021. Max Hallgren. Video: Ricci Flow with a Lower Bound on Ricci Curvature
Oct 22, 2021. Mattia Fogagnolo. Video: Minkowski inequalities in manifolds with nonnegative Ricci curvature
Oct 15, 2021. Eva Kopfer. Optimal transport and homogenization
Oct 8, 2021. Sara Farinelli. Video: The size of the nodal set of Laplace eigenfunctions in singular spaces via optimal transport
Oct 1, 2021. Man-Chun Lee. Video: d_p convergence and epsilon-regularity theorems for entropy and scalar curvature lower bound
Sept 24, 2021. Enrico Pasqualetto. Video: The role of test plans in metric measure geometry
Sept 17, 2021. Qin Deng. Video: Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
October 9, 2020:
Daniele Semola (Scuola Normale Superiore)
Rectifiability of RCD(K,N) spaces via delta-splitting maps
Abstract: The theory of metric measure spaces verifying the Riemannian-Curvature-Dimension condition RCD(K,N) has attracted a lot of interest in the last years. They can be thought as a non smooth counterpart of the class of Riemannian manifolds with Ricci curvature bounded from below by K and dimension bounded from above by N.
In this talk, after providing some background and motivations, I will describe a simplified approach to the structure theory of these spaces relying on the so-called delta-splitting maps. This tool, developed by Cheeger-Colding in the study of Ricci limits, has revealed to be extremely powerful also more recently in the study of RCD spaces.
The seminar is based on a joint work with Elia Brue' and Enrico Pasqualetto.
October 2, 2020:
Flavia Santarcangelo (SISSA)
Independence of synthetic Curvature Dimension conditions on transport distance exponent
Abstract: The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound \$K\$ joint with an upper bound \$N\$ on the dimension.
Their condition, called the Curvature-Dimension condition and denoted by \$\mathsf{CD}(K,N)\$, is formulated in terms of a modified displacement convexity of an entropy functional along \$W_{2}\$-Wasserstein geodesics. In a joint work with A. Akdemir, F. Cavalletti, A. Colinet and R. McCann, we show that the choice of the squared-distance function as transport cost does not influence the theory. In particular, by denoting with \$\mathsf{CD}_{p}(K,N)\$ the analogous condition but with the cost given by the \$p^{th}\$ power of the distance, we prove that \$\CD_{p}(K,N)\$ are all equivalent conditions for any \$p>1\$ --- at least in spaces whose geodesics do not branch.
Following the strategy introduced in the work by Cavalletti-Milman, we also establish the local-to-global property of \$\mathsf{CD}_{p}(K,N)\$ spaces.
Finally, we will present a result obtained in collaboration with F. Cavalletti and N. Gigli that, combined with the one previously described, allows to conclude that for any \$p\geq1\$, all the \$\mathsf{CD}_{p}(K,N)\$ conditions, when expressed in terms of displacement convexity, are equivalent, provided the space \$X\$ satisfies the appropriate essentially non-branching condition.
September 25, 2020: Andrea Mondino (University of Oxford)
Abstract: In the seminar I will present a recent work joint with S. Suhr (Bochum) giving an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result, together with independent work by McCann on lower bounds for Lorentzian Ricci Curvature, gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.
September 11, 2020:
Danka Lučić (Jyväskylä)
Techniques for proving infinitesimal Hilbertianity
Abstract: A metric space is said to be "universally infinitesimally Hilbertian" if, when endowed with any arbitrary Radon measure, its associated 2-Sobolev space is Hilbert. For instance, all (sub)Riemannian manifolds and CAT(K) spaces have this property. In this talk, we will illustrate three different strategies to prove the universal infinitesimal Hilbertianity of the Euclidean space, which is the base case and where all the known approaches work.
The motivations come, among others, from the study of rectifiable metric measure spaces, of metric-valued harmonic maps, and of variational problems (such as models representing low-dimensional elastic structures).
September 4, 2020:
Gilles Carron
Euclidean heat kernel rigidity
Video
Abstract : This is joint work with David Tewodrose (Cergy). I will explain that a metric measure space with Euclidean heat kernel are Euclidean. An almost rigidity result comes then for free, and this can be used to give another proof of Colding's almost rigidity for complete manifold with non negative Ricci curvature and almost Euclidean growth.
July 3, 2020:
Null Distance and Convergence of Warped Product Spacetimes
Abstract: The null distance was introduced by Christina Sormani and Carlos Vega as a way of turning a spacetime into a metric space. This is particularly important for geometric stability questions relating to spacetimes such as the stability of the positive mass theorem. In this talk, we will describe the null distance, present properties of the metric space structure, and examine the convergence of sequences of warped product spacetimes equipped with the null distance. This is joint work with Annegret Burtscher.
June 26, 2020:
Ricci limit spaces : An introduction to the tools of Cheeger-Jiang-Naber's work
The goal of this expository talk is to explain parts of the work of J. Cheeger, W. Jiang and A. Naber: https://arxiv.org/abs/1805.07988 . For a converging, non-collapsing sequence of Riemannian manifolds with a uniform Ricci lower bound, they proved that singular strata of the limit space are rectifiable. Some of the key tools in the proof include quantitative stratification, which was first introduced in previous work of Cheeger-Naber, and new related volume estimates, together with a precise study of neck regions. After a brief review of Cheeger-Colding theory, the talk will focus on explaining the notions of quantitative stratifications, neck regions and their role in the proof.
June 12, 2020:
Title: Applications of needle decomposition for metric measure spaces
Abstract: In this talk I show how one can formulate and prove the Heintze-Karcher inequality in the context of nonsmooth spaces that satisfy a Ricci curvature bound in the sense of Lott, Sturm and Villani. As a by-product one obtains a notion of mean curvature for the boundary of Borel sets in such spaces. My approach is based on the needle decomposition method introduced for this framework by Cavalletti and Mondino.
June 5, 2020:
Title: A volume comparison theorem for characteristic numbers
Abstract: We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of any characteristic number of a Riemannian manifold M is bounded proportional to the volume, i.e. bounded by Cvol(M) where C depends only on the characteristic number, the dimension of M, and both bounds. The proof relies on the definition of a connection for an harmonic Hölder regular metric tensor as they appear for instance as Gromov-Hausdorff limits of Riemannian manifolds
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# Biblatex and mla-paper making weird headings
I'm trying to use mla-paper and biblatex with MLA style in the same document so I can have a managed bibliography but with the whole document in MLA. I seem to have it working almost properly, but it produces a bibliography heading that is larger than the title of the document. I'm getting this error:
Package Fancyhdr Warning: \headheight is too small (12.0pt):
Make it at least 14.49998pt.
We now make it that large for the rest of the document.
This may cause the page layout to be inconsistent, however.
But I'm not really sure what to do with it. It seems like it's making the bibliography heading bigger than I'd like it to. How can I fix this?
Either a direct solution or more general tips on better ways to use MLA with BibTex and LaTeX would be great. Thanks.
Update
Here's a full minimal example of what I'm doing and a link to an image of what I'm getting.
\documentclass[12pt,letterpaper]{article}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=mla]{biblatex}
\usepackage{ifpdf}
\usepackage{mla}
\bibliography{../Bibliography/Bibliography}
\begin{document}
\begin{mla}{Stephen}{Searles}{Schultz}{Queering American History}{\today}{Week 7}
Test sentence. \autocite{Somerville:2005fk}.
\begin{center}
\printbibliography
\end{center}
\end{mla}
\end{document}
Along with a standard BibTex file with that cite key, of course. It's citing properly as you'll see, so that's not the problem.
-
We will want a full minimal example of your code – Joseph Wright Feb 15 '11 at 22:20
The warning message has nothing to do with the size of the bibliography heading. fancyhdr merely tells you that it can't put the header text (“Searles 1” in your example) in the 12pt space you reserve for it with \setlength{\headheight}{12pt}. See also tex.stackexchange.com/questions/2394/… – Caramdir Feb 16 '11 at 4:37
Oh whoops. That line should have been taken out (reflected above now). I added that trying to fix that message. It, clearly, did not work, but I forgot to remove it. – Stephen Searles Feb 16 '11 at 5:56
When I follow the advice of that answer and fancyhdr, the message doesn't go away. I guess I have two problems then... – Stephen Searles Feb 16 '11 at 6:01
The main problem you have is the size of your "Works Cited" part. This is because the label is being generated by the biblatex-mla and not by the mla package. To fix this you could enclose the bibliography in the mla's workscited environment and redefine biblatex's bibheading to nothing. Alternatively, you can simply redefine bibheading to what the mla package defines it as, which is simpler:
\documentclass[12pt,letterpaper]{article}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=mla]{biblatex}
\usepackage{ifpdf}
\usepackage{mla}
\bibliography{../Bibliography/Bibliography}
\begin{document}
\begin{mla}{Stephen}{Searles}{Schultz}{Queering American History}{\today}{Week 7}
Test sentence. \autocite{Somerville:2005fk}.
\begin{center}
\printbibliography
\end{center}
\end{mla}
\end{document}
The mla package leaves a lot to be desired. It should use the setspace package for linespacing, instead of simply using \linespread. It also has a manual hack \tab` for indenting paragraphs after sections, among other things. But if it is more or less working for you, that's ok. I don't know of an equivalent package for this kind of use.
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I know this may seem like a late reply, but I think it would help people who are still seeking a good answer to this question. I found this package online called MLA13 that does everything for you. I used it in quite a few of my papers already. The thing that's good about it is that it uses your .bib files and formats everything according to MLA standards.
The website for this is:
Documentation: http://jackson13.info/mla13/Documentation.pdf
This package formats the header according to mla style by centering it and making it of font size 12.
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What are the positive and negative effects of the ozone layer?
Having ozone ( ${O}_{3}$) in the troposphere (the layer of the atmosphere we live in) can be damaging to vegetation and respiratory systems of many mammals. Additionally ozone is a main contributor to smog found in dense urban areas.
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# What does 'energy dissipation' in the description of the Kolmogorov Microscales in fluid dynamics mean?
I've asked about microscales before, and thinking more about the problem turned up more basic questions. Consider the length scale of a microscale: $$\eta = \left(\frac{\nu^3}{\epsilon}\right)^\frac{1}{4}$$ With $\nu$ being the viscosity, and $\epsilon$ the energy dissipation. I understand this to mean the energy transfer from turbulent kinetic flow to thermal energy (at this scale). The unit of $\epsilon$ is Watts. What's the reference volume?
-
The unit of $\epsilon$ is not Watts, it's Watts per kilogram. That is it is a dissipation in the unit of mass, to get dissipation density (per volume), do $\rho \epsilon$.
For who might be interested [W / kg] is actually [m^2 / s^3], and the units of the dynamic viscosity $\nu$ are squared meters per second [m^2 / s]
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# Topics on Test #2
1) LHopital’s Rule
2) Related Rates
3) Implicit Differentiation
4) Optimizations
5) Anti-derivatives
6) Curve Sketching
7) Extreme Values
8) Monotonicity
9) Shape Of Graphs
These are the topics I can think of on the top of my head.
This entry was posted in Homework, Uncategorized. Bookmark the permalink.
### 5 Responses to Topics on Test #2
1. Kate Poirier says:
Looks pretty good to me! Keep in mind that I organized some things a little differently from how your text and WebWork did. For example, your items 6, 7, 8, and 9 all fall under the heading “shapes of graphs” for me. This includes the list of all possible features of a graph that can be obtained by analyzing the formula for the function itself, as well as its first and second derivatives.
A good exercise would be to generate a sub-list of topics that fall under this broader “shape of graphs” heading. Either as a comment on this post, or as a new post altogether, I’d like one of you (Jimmy and/or anyone else) to provide a list of every feature of the graph of a function (that you’ve learned about in this course) that you can use to give a “reasonably accurate” sketch of the graph. Hint: I gave you a list in class way back when we started talking about these topics.
I will definitely be asking you at least one question on your next test where you have to sketch a graph by hand using each of the items on this sub-list. Whether I ask you for the individual features or not, this sub-list is your to-do list when it comes to sketching graphs of functions. So you’ll need to check off all of the items, even if the instructions for a question are simply, “sketch the graph of the function.”
2. Maloney says:
A list for sketching:
Domain, x intercepts, y intercept, end behavior (horizontal asymptotes etc.)
Discontinuities: vertical asymptotes, removable discontinuities, etc? + nearby behavior.
f(x) Sign analysis: Sign of the function over intervals.
f ‘ (x) Sign analysis: critical points, Local Maxima/Minima.
f ” (x) Sign analysis: Inflection points and concavity.
Bonus: Consider symmetry to save work: even f(-x) = f(x); odd f(-x) = -f(x).
3. Kate Poirier says:
This looks good! The classification of discontinuities and the behavior of a function near a discontinuity are essentially two sides of the same coin. Remember that studying limits as x approaches a discontinuity from either side will tell you what kind of discontinuity you’re dealing with.
More specifically…
If the limit from either side is $\pm \infty$, then the graph has a vertical asymptote; the $+$ or $-$ sign will tell you whether the function shoots up or down.
If the limits from either side are both finite, but not equal, then you have a jump discontinuity.
If the limits from either side are finite and equal, but do not equal the value of the function at the point, then you have a removable discontinuity (hole in the graph).
If either the left-hand or right-hand limit fails to exist (neither finite nor infinite), then there’s a different kind of discontinuity. An example is the function $f(x)=sin(\frac{1}{x})$ near $x=0$.
4. Ismail Akram says:
So if study this list, I’m all good for the test?
5. Kate Poirier says:
Hi Izzy, I just saw this now. My answer would have been yes! After taking the test, I hope that’s your answer too.
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# From vertex operator algebras to conformal nets and back
Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, Mihaly Weiner
March 04, 2015
We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra $V$ a conformal net $\mathcal{A}_V$ acting on the Hilbert space completion of $V$ and prove that the isomorphism class of $\mathcal{A}_V$ does not depend on the choice of the scalar product on $V$. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra $V$, the map $W\mapsto \mathcal{A}_W$ gives a one-to-one correspondence between the unitary subalgebras $W$ of $V$ and the covariant subnets of $\mathcal{A}_V$. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known $c=1$ unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörss gives back the strongly local vertex operator algebra $V$ from the conformal net $\mathcal{A}_V$ and give conditions on a conformal net $\mathcal{A}$ implying that $\mathcal{A}= \mathcal{A}_V$ for some strongly local vertex operator algebra $V$.
Keywords:
conformal qft, vertex operator algebras
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# Convexity on a direction
In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have:
A function $f$ defined on a triangle $T$ is said to be convex in the direction $u$ provided
$$\frac{f(w_3)-f(w_2)}{|w_3-w_2|} \ge \frac{f(w_2)-f(w_1)}{|w_2-w_1|}$$
for all ordered sets of points $w_1, w_2, w_3$ in $T$ lying on a line pointing in the direction of $u$. We say that $f$ is convex on $T$ provided it is convex in all directions. As is well known from calculus, if $f$ has two derivatives in the direction $u$, then this definition of convexity in the direction $u$ is equivalent to $D_u^2f(v)\ge 0$, all $v\in T$.
But All thing that we have about function convexity briefly is here: http://en.wikipedia.org/wiki/Convex_function.
I can't make a connection between definition of convex function and convexity of a function on a certain direction, also both of them with second directional derivative of function on a direction. How we can proof that usual definition of convex function and "We say that $f$ is convex on $T$ provided it is convex in all directions." are equivalence? What's proof of "If $f$ has two derivatives in the direction $u$, then this definition of convexity in the direction $u$ is equivalent to $D^2_uf(v)\ge 0$, all $v\in T$."?
A link to a comprehensive source is pleasured.
You have two different questions.
Question 1 :
($f$ is convex) $\Leftrightarrow$ ($f$ is convex all the directions)
Question 2 :
If $f$ has two derivatives, then ($f$ convex in the direction $u$) $\Leftrightarrow$ ($D^{2}_{u} (f) (v) \geq 0$, $\forall v \in T$)
For the first question.
One only needs to recall the definition of convexity.
$f$ is called convex if $\forall x_{1} ,x_{2} \, \forall t \in [0,1] : f (t x_{1} + (1-t)x_{2}) \leq t \, f (x)_{1} + (1-t) \, f (x_{2})$
$(\Rightarrow)$ : We choose an arbitrary direction $u$ and three points $w_{1}$, $w_{2}$ and $w_{3}$ aligned in this direction and in this order. We define the number $t$ as
$$t = \frac{|w_{3} - w_{2}|}{|w_{1} - w_{3}|}$$ Since the points are aligned in the order $w_{1}$, $w_{2}$ and $w_{3}$, we have $t \in [0,1]$. Moreover, we note that we have $w_{2} = t \, w_{1} + (1-t) \, w_{3}$. Applying the usual definition of convexity, we obtain:
$$f (w_{2}) \leq \frac{|w_{3} - w_{2}|}{|w_{1} - w_{3}|} f (w_{1}) + \frac{|w_{1} - w_{2}|}{|w_{1} - w_{3}|} f (w_{3})$$
so that we have
$$\frac{|w_{1} - w_{3}|}{|w_{1} - w_{2}| |w_{3} - w_{2}|} \, f (w_{2}) \leq \frac{f(w_{1})}{|w_{1} - w_{2}|} + \frac{f (w_{3})}{|w_{3} - w_{2}|}$$
The final step of the calculation is to note that $|w_{1} - w_{3}| = |w_{1} - w_{2}| + |w_{2} - w_{3}|$ since the points are aligned in the good order, and after rearrangement, we find back the expression for the definition of the convexity in the direction $u$.
($\Leftarrow$) : Completely similar. One just needs to redo this reasoning in the other way. (Is it clear or do you need more details ?)
For the second question.
The idea is that as soon as you have given yourself a direction $u$, you are back to the case of a $1D$ function. Indeed, for a given direction $u$ and a given starting point $v_{0}$, you need to study the convexity of the function
\begin{aligned} f_{u , v_{0}} : &\mathbb{R} \to X \\ & t \mapsto f (v_{0} + t \, u) \end{aligned} The usual criteria of convexity for twice differentiable function imposes you $f_{u , v_{0}} '' \geq 0$. The final remark is to note that $f_{u , v_{0}} '' (t) = D^{2}_{u} f \, (v_{0} + t u)$, so that the equivalence of the two properties is immediate.
• OK, I got it, thanks @jibe. Aug 5, 2014 at 18:05
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# Evaluate the indefinite integral. {eq}\int {{{\cos }^8}\left( \theta \right)\sin \left( \theta \right)} \ d\theta {/eq}
## Question:
Evaluate the indefinite integral.
{eq}\int {{{\cos }^8}\left( \theta \right)\sin \left( \theta \right)} \ d\theta {/eq}
## U-Substitution Method:
In calculus integral, the u-substitution method to solve complicated integral problems (both definite and indefinite integrals). Here, we convert the integral into a normal integral problem (which is easy to solve) by using the substitution method. It means we substitute {eq}u {/eq} instead of a part of the integrand, and we also change the differential value of the integral in terms of the new variable. It is also known by different names such as integration by substitution, the reverse chain rule, and change of variables.
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# On the determinant of an odd, continuous Galois representation.
In his paper, Duke paper, Serre consider continuous, odd Galois representation $\rho: G_{\mathbb{Q}}\longrightarrow GL_{n}(\overline{\mathbb{F}}_{p})$ where $p$ is a rational prime. Roughly, (I don't understand much French except for the help from Google translation) Serre claims (section 1.3) that
$\det(\rho(Frob_{l})) = \epsilon(Frob_{l})\omega^{k}(Frob_{l})$ for all prime $l\nmid pN$ where $N$ is defined as the level of the representation (with an explicit formula given in the paper) and $\epsilon$ is a Dirichlet character and $k$ is some positive integer.
This seems to be standard since other papers cited it without reproving and I could not find any reference for the proof. In particular, my questions are:
1) Where can I find a proof for this.
2) What exactly is $\epsilon$, in some paper, there is the claim that $\epsilon$ is the unique quadratic character mod $p$ ramified only at $p$, and I do not understand where this comes from?
3) How can one finds $k$.
For motivation, I think $\det(\rho(Frob_{l}))$ is an important invariant to compute since, for example, it appears in the attachment equation that associates these representations with modular forms.
Thanks in advance for any insight.
-
$\det (\rho)$ is a one dimensional rep of the absolute Galois group of the rationals, i.e., it is a character. All such characters can be described by class field theory or, more simply, by the Kronecker-Weber theorem. So is a Dirichlet character and, by the hypotheses, its conductor divides $pN$. Factor it as a character of conductor $p$ (that will be $\epsilon$) times a character of conductor $N$. The latter is a power of the cyclotomic character and $k$ is defined to be that power. The bit about quadratic character must be under additional hypotheses.
Edit: I got $p$ and $N$ switched above. The character of conductor $N$ is $\epsilon$. The character of conductor $p$ is a power of the cyclotomic character because $(\mathbb{Z}/p)^*$ is cyclic.
Thank you very much for the answer. Could you be more specific on why "The latter is a power of the cyclotomic character" and what (normally seen) additional hypothesis required to have $\epsilon$ as a quadratic character? Can you give a reference for what source (book?) using class field theory to describe those characters? Thanks. – T.B. Jan 20 '11 at 0:30
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Does diffusion happens for the photons from higher concentration to lower concentration
I understand that it is the random motion of the molecules that causes them to move from an area of high concentration to an area with a lower concentration and the diffusion will continue until the concentration gradient has been eliminated.
My question is, do light photons behave the same or not?
For example, if I put two light sources of the same power in a vacuum at a finite distance, will I find that it's the same probability of finding a photon when the distance measured from the sources is the same.
• Not likely. For a diffusion motion, the momentum of a particle should be randomiized (scattered) within a reasonable samll legth scale and time scale. It is not likely to be applicable to the photon case. – ytlu Mar 19 at 8:39
• Looking at a cloud outside my window. The light I see is photons that have entered the cloud and diffused back out. – John Doty Mar 19 at 17:32
• @JohnDoty Should scattering be considered the same as diffusion? – BioPhysicist Mar 19 at 17:41
• @BioPhysicist Diffusion is a consequence of scattering. – John Doty Mar 19 at 17:45
No. The light waves (and the associated photons) will just keep on going in a straight line away from each source. In the case of a source emitting equally in all directions, the light intensity falls with distance squared from a given source, so for two such sources in otherwise empty space the net intensity at some point $$\bf r$$ is $$P = \frac{P_A}{({\bf r} - {\bf r}_a)^2} + \frac{P_B}{({\bf r} - {\bf r}_b)^2}$$ if the sources are at $${\bf r}_a$$ and $${\bf r}_b$$ and $$P_A$$, $$P_B$$ is the intensity at unit distance from each source respectively.
• You should clarify that $\frac{1}{r^2}$ is for a point source, or for a source sufficiently distant that it can be treated as such – Carl Witthoft Mar 19 at 13:20
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# Difference between revisions of "1976 IMO Problems/Problem 4"
## Problem
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
## Solution
Since $3*3=2*2*2+1$, 3's are more efficient than 2's. We try to prove that 3's are more efficient than anything:
Let there be a positive integer $x$. If $3$ is more efficient than $x$, then $x^3<3^x$. We try to prove that all integers greater than 3 are less efficient than 3:
When $x$ increases by 1, then the RHS is multiplied by 3. The other side is multiplied by $\dfrac{(x+1)^3}{x^3}$, and we must prove that this is less than 3 for all $x$ greater than 3.
$\dfrac{(x+1)^3}{x^3}<3\Rightarrow \dfrac{x+1}{x}<\sqrt[3]{3}\Rightarrow 1<(\sqrt[3]{3}-1)x$
$\dfrac{1}{\sqrt[3]{3}-1}
Thus we need to prove that $\dfrac{1}{\sqrt[3]{3}-1}<4$. Simplifying, we get $5<4\sqrt[3]{3}\Rightarrow 125<64*3=192$, which is true. Working backwards, we see that all $x$ greater than 3 are less efficient than 3, so we try to use the most 3's as possible:
$\dfrac{1976}{3}=658.6666$, so the greatest product is $\boxed{3^{658}\cdot 2}$.
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• Rohini M Godbole
Articles written in Pramana – Journal of Physics
• Resolved photon processes
We review high-energy scattering processes that are sensitive to the hadronic structure of the photon, describing theoretical predictions as well as recent experimental results. These processes include deep-inelastic electron-photon scattering ate+e colliders; and the production of jets, heavy quarks and isolated photons in the collision of real photons ate+e colliders, as well as in photon-photon collisions atep colliders. We also comment on minijet based calculations of totalγp andγγ cross-sections, and discuss the possibility that future lineare+e colliders might produce very large photon fluxes due to the beamstrahlung phenomenon; in the most extreme cases, we predict more than one hadronicγγ event to occur at every bunch crossing.
• Higgs and SUSY searches at future colliders
In this talk, I discuss some aspects of Higgs searches at future colliders, particularly comparing and contrasting the capabilities of LHC and next linear collider (NLC), including the aspects of Higgs searches in supersymmetric theories. I will also discuss how the search and study of sparticles other than the Higgs can be used to give information about the parameters of the minimal supersymmetric Standard Model (MSSM).
• Working group report: Collider Physics
This is summary of the activities of the working group on collider physics in the IXth Workshop on High Energy Physics Phenomenology (WHEPP-9) held at the Institute of Physics, Bhubaneswar, India in January 2006. Some of the work subsequently done on these problems by the subgroups formed during the workshop is included in this report.
• CP violation in supersymmetry, Higgs sector and the large hadron collider
In this talk I discuss some aspects of CP violation (CPV) in supersymmetry (SUSY) as well as in the Higgs sector. Further, I discuss ways in which these may be probed at hadronic colliders. In particular I will point out the ways in which studies in the $$\widetilde\chi ^ \pm ,\widetilde\chi _2^0$$ sector at the Tevatron may be used to provide information on this and how the search can be extended to the LHC. I will then follow this by a discussion of the CP mixing induced in the Higgs sector due to the above-mentioned CPV in the soft SUSY breaking parameters and its effects on the Higgs phenomenology at the LHC. I would then point out some interesting aspects of the phenomenology of a moderately light charged Higgs boson, consistent with the LEP constraints, in this scenario. Decay of such a charged Higgs boson would also allow a probe of a light (≲50 GeV), CP-violating (CPV) Higgs boson. Such a light neutral Higgs boson might have escaped detection at LEP and could also be missed at the LHC in the usual search channels.
• # Pramana – Journal of Physics
Volume 94, 2020
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
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Volume 282 - 38th International Conference on High Energy Physics (ICHEP2016) - Beyond the Standard Model
A simplified model for dark matter interacting primarily with gluons.
G. Mendiratta,* R. Godbole, A. Shivaji, T.M.P. Tait
*corresponding author
Full text: pdf
Pre-published on: February 06, 2017
Published on: April 19, 2017
Abstract
We consider a simple renormalizable model providing a UV completion for dark matter whose interactions with the Standard Model are primarily via the gluons. The model consists of scalar dark matter interacting with scalar colored mediator particles. A novel feature is the feature that (in contrast to more typical models containing scalar dark matter) the colored scalars typically decay into multi-quark final states, with no associated missing energy. We construct this class of models and examine associated phenomena related to dark matter annihilation, scattering with nuclei, and production at colliders. We compare the results obtained from effective field theory (EFT) with a loop-induced calculations for the collider processes and show that EFT is not applicable for a large parameter space where mediator mass is comparable to the cuts on missing energy. We calculate the bounds from from $\sqrt{s}= 8$ and $13$ TeV data and show the expected reach of $\sqrt{s}= 14$ TeV LHC and $100$ TeV FCC in constraining or discovering the model.
DOI: https://doi.org/10.22323/1.282.0133
How to cite
Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.
Open Access
Copyright owned by the author(s) under the term of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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# GMAT Math : Cylinders
## Example Questions
← Previous 1 3
### Example Question #1 : Calculating The Surface Area Of A Cylinder
What is the surface area of a cylinder with a radius of 7 and a height of 3?
Explanation:
All we really need here is to remember the formula for the surface area of a cylinder.
### Example Question #501 : Geometry
The height of a cylinder is twice the circumference of its base. The radius of the base is 9 inches. What is the surface area of the cylinder?
Explanation:
The radius of the base is 9 inches, so its circumference is times this, or inches. The height is twice this, or inches.
Substitute in the formula for the surface area of the cylinder:
square inches
### Example Question #1 : Cylinders
Calculate the surface area of the following cylinder.
(Not drawn to scale.)
Explanation:
The equation for the surface area of a cylinder is:
we plug in our values: to find the surface area
### Example Question #1 : Cylinders
Calculate the surface area of the following cylinder.
(Not drawn to scale.)
Explanation:
The equation for the surface area of a cylinder is
We plug in our values into the equation to find our answer.
Note: we were given the diameter of the cylinder (10), in order to find the radius we had to divide the diameter by two.
### Example Question #1 : Calculating The Surface Area Of A Cylinder
A cylinder has a height of 9 and a radius of 4. What is the total surface area of the cylinder?
Explanation:
We are given the height and the radius of the cylinder, which is all we need to calculate its surface area. The total surface area will be the area of the two circles on the bottom and top of the cylinder, added to the surface area of the shaft. If we imagine unfolding the shaft of the cylinder, we can see we will have a rectangle whose height is the same as that of the cylinder and whose width is the circumference of the cylinder. This means our formula for the total surface area of the cylinder will be the following:
### Example Question #501 : Geometry
Grant is making a canister out of sheet metal. The canister will be a right cylinder with a height of mm. The base of the cylinder will have a radius of mm. If the canister will have an open top, how many square millimeters of metal does Grant need?
Explanation:
This question is looking for the surface area of a cylinder with only 1 base. Our surface area of a cylinder is given by:
However, because we only need 1 base, we can change it to:
We know our radius and height, so simply plug them in and simplify.
### Example Question #7 : Calculating The Surface Area Of A Cylinder
Find the surface area of a cylinder whose height is and radius is .
Explanation:
To find the surface area of a cylinder, you must use the following equation.
Thus,
### Example Question #1 : Cylinders
A right circular cylinder has bases of radius ; its height is . Give its surface area.
Explanation:
The surface area of a cylinder can be calculated from its radius and height as follows:
Setting and :
or
### Example Question #1 : Calculating The Volume Of A Cylinder
What is the volume of a cone with a radius of 6 and a height of 7?
Explanation:
The only tricky part here is remembering the formula for the volume of a cone. If you don't remember the formula for the volume of a cone, you can derive it from the volume of a cylinder. The volume of a cone is simply 1/3 the volume of the cylinder. Then,
### Example Question #1 : Calculating The Volume Of A Cylinder
What is the volume of a sphere with a radius of 9?
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To build these speed tests, and run their correctness tests, execute the following commands starting in the build directory : cd speed/sacado make check_speed_sacado VERBOSE=1 You can then run the corresponding speed tests with the following command ./speed_sacado speed seed where seed is a positive integer. See speed_main for more options.
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# A question regarding the algebraic closure of a field
I have slight problems understanding a thing about algebraic closures of fields. It seems to me that any algebraic closure $C$ of a field $K$ is a Galois extension, but I read that this is not true. Following are the definitions I use:
An extension $F/K$ is Galois if the fixed field of $Aut_K F$ is $K$.
Note that this definition is equivalent to the definition of F being simultaneously normal and separable over K but I want to use this one as it is this one that I have problems with.
Following is a theorem from my book:
Let $\sigma : K \rightarrow L$ be an isomorphism of fields. If $S$ is a set of polynomials in $K[x]$, $F$ a splitting field of $S$, and $S'$ a set of corresponding polynomials in $L[x]$, $P$ a splitting field of $S'$ then $\sigma$ extends to $F \cong P$.
All right. So my argument is as follows. Take any $u \in C \setminus K$. Then $u$ is algebraic with a nontrivial minimal polynomial, and at least another root $v \in C$. It is elementary knowledge that $K(u) \cong K(v)$. C is an algebraic closure, and therefore splitting field of all irreducible polynomials over $K(u)$ and $K(v)$. Then by the theorem, I should be able to extend the isomorphism to a K-automorphism of $C$ that switches $u,v$. These were chosen arbitrarily in $C$, hence there is a $K$-monomorphism affecting nontrivially every element of $C \setminus K$, hence by the definition, extension is Galois.
Where does this reasoning fail?
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Strictly speaking in your definition of Galois you need to say algebraic extension. – Qiaochu Yuan Jun 15 '12 at 13:22
The reasoning fails at the last step when you assume $u \neq v$. Consider $K = \mathbb{F}_p(u^p)$.
More generally, if $K$ is not perfect it admits inseparable extensions and the algebraic closure cannot be Galois. The maximal Galois extension is the separable closure.
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# The Unapologetic Mathematician
## Symmetric Tensors
Wow, people are loving my zero-knowledge test. It got 1,743 views yesterday, thanks to someone redditing it. Anywho…
Today and tomorrow I want to take last Friday’s symmetrizer and antisymmetrizer and apply them to the tensor representations of $\mathrm{GL}(V)$, which we know also carry symmetric group representations. Specifically, the $n$th tensor power $V^{\otimes n}$ carries a representation of $S_n$ by permuting the tensorands, and this representation commutes with the representation of $\mathrm{GL}(V)$. Then since the symmetrizer and antisymmetrizer are elements of the group algebra $\mathbb{F}[S_n]$, they define intertwiners from $V^{\otimes n}$ to itself. The their images are not just subspaces on which the symmetric group acts nicely, but subrepresentations of symmetric and antisymmetric tensors — $S^n(V)$ and $A^n(V)$, respectively.
Now it’s important (even if it’s not quite clear why) to emphasize that we’ve defined these representations without ever talking about a basis for $V$. However, let’s try to get a better handle on what such a thing looks like by assuming $V$ has finite dimension $d$ and picking a basis $\{e_i\}$. Then we have bases for tensor powers: a basis element of the $n$th tensor power is given by an $n$-tuple of basis elements for $V$. We’ll write a general one like $e_{i_1}\otimes e_{i_2}\otimes...\otimes e_{i_n}$.
How does a permutation act on such a basis element? Well, basis elements are pure tensors, so the permutation $\pi$ simply permutes these basis tensorands. That is:
$\displaystyle\pi\left(e_{i_1}\otimes...\otimes e_{i_n}\right)=e_{i_{\pi(1)}}\otimes...\otimes e_{i_{\pi(n)}}$
So the space of symmetric tensors $S^n(V)$ is the image of $V^{\otimes n}$ under the action of the symmetrizer. And so it’s going to be spanned by the images of a basis for $V^{\otimes n}$, which we can calculate now. The symmetrizer is an average of all the permutations in the symmetric group, so we find
\displaystyle\begin{aligned}S\left(e_{i_1}\otimes...\otimes e_{i_n}\right)=\frac{1}{n!}\sum\limits_{\pi\in S_n}\pi\left(e_{i_1}\otimes...\otimes e_{i_n}\right)\\=\frac{1}{n!}\sum\limits_{\pi\in S_n}e_{i_{\pi(1)}}\otimes...\otimes e_{i_{\pi(n)}}\end{aligned}
Now we notice something here: if two basic tensors are related by a permutation of their tensorands, then the symmetrizer will send them to the same symmetric tensor! This means that we can choose a preimage for each basic symmetric tensor. Just use whatever permutation we need to put the $n$-tuple of tensorands into order. That is, always select $i_1\leq i_2\leq...\leq i_n$. Given any basic tensor, there is a unique permutation of its tensorands which is in this order.
As an explicit example, let’s consider what happens when we symmetrize the tensor $e_1\otimes e_2\otimes e_1$. First of all, we toss it into the proper order, since this won’t change the symmetrization: $e_1\otimes e_1\otimes e_2$. Now we write out a sum of all the permutations of the three tensorands, with the normalizing factor out front
\displaystyle\begin{aligned}\frac{1}{3!}(e_1\otimes e_1\otimes e_2+e_1\otimes e_2\otimes e_1+e_1\otimes e_1\otimes e_2+\\e_1\otimes e_2\otimes e_1+e_2\otimes e_1\otimes e_1+e_2\otimes e_1\otimes e_1)\end{aligned}
Some of these terms are repeated, since we have two copies of $e_1$ in this tensor. So we collect these together and cancel off some of the normalizing factor to find
$\displaystyle\frac{1}{3}e_1\otimes e_1\otimes e_2+\frac{1}{3}e_1\otimes e_2\otimes e_1+\frac{1}{3}e_2\otimes e_1\otimes e_1$
Now no matter how we rearrange the tensorands we’ll get back this same tensor again.
Tomorrow we’ll apply the same sort of approach to the antisymmetrizer.
December 22, 2008
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# When Math meets CS: Counting Perfect Matchings using Inclusion Exclusion
I'm sure you've all heard of buzz-words like NP-Completeness and the like. They sound hard and unapproachable, so today let's solve one of the non-classical NP-Hard problems (in exponential time of course).
A graph, $$G$$, is basically a set of points (called vertices) on a plane floating around. However, there are also a set of lines joining the various points together in order to relate them together somehow. These are called nodes and edges, to imagine them, you can think of the cliche social network graph with the dense set of edges connecting you to possibly thousands of friends and acquantences. Typically, graphs aren't really that dense and well-connected, and typically you'll only see very few edges floating around. More formally, we describe a graph $$G$$ by a pair of sets: $$V$$, standinf for vertices; $$E$$ standing for edges. Each edge is composed of a line and two vertices, and we may denote the edge going from $$u$$ to $$v$$ (both vertices) as $$u - v$$. (Note in this presentation, edges do not have a direction.)
Alt
bipartite graph
Now, the problem of constructing a perfect matching turns out to be relatively easy to do with pen and paper, and even more so by a computer (we won't go into that right now, wait until your first algorithm course). However, not satisfied with just one perfect matching, I want you to tell me how many perfect matchings there are in a bipartite graph! As far as we know, this is a really hard problem.
How would you solve this problem? A natural starting point would be to brute-force the solution; since the problem is hard, we can't expect a sub-exponential algorithm anyways! Fair point, let's start with a brute-force algorithm. Suppose your bipartite graph has $$m = |E|$$ edges and $$n = |V|$$ nodes (where $$n$$ is even by definition of $$G$$ being bipartite), then a perfect matching must be a set of $$\frac{n}{2}$$ edges. One way to solve this problem then is to enumerate all of the $${m \choose n/2}$$ such edges, test each one individually to see if it is a perfect matching, and count how many were actually perfect matchings. How long would this algorithm take? Well, suppose $$m = c\frac{n}{2}$$ for $$c \ge 1$$ (because otherwise we can't have a perfect matching) and since there are $${m \choose n/2}$$ such edges we have to consider, then it's possible to show that the number of edge sets we have to consider is approximately $$\sqrt{c}^n$$! For many applications, it's very likely that $$c$$ isn't small. For example, consider the realestate market, here, it's not at all unlikely that the average number of houses considered per potential buyer is greater than 4! Can we do better?
Since in this problem, the number of edges could be potentially up to $$O(m^2)$$, it's a very desirable property to construct an algorithm that is completely independent of $$m$$ the number of edges! It turns out that there's a really elegant algorithm (that can be further improved in fact) which runs in time $$O(2^n)$$, no matter how many edges there are in our graph. Here's how to do it.
Consider the inclusion-exclusion principle, and let's consider a generalization of it. Suppose we have a function $$f(X \subseteq V): \text{subssets}(V) \to \mathbb{N}$$ that assigns a natural value to each subset of $$V$$, then we let the zeta-transform of $$f(X)$$ to be $$\zeta f = g(V) = \sum_{X \subseteq V} f(X)$$. A generalization of the Mobius Inversion Theorem (lifted out of number theory and applied to complete partial orders) states that $g(V) = \sum_{X \subseteq V} f(X) \implies f(V) = \sum_{X \in V} (-1)^{|V - X|} g(X)$ which can be seen as a generalization of the inclusion/exclusion principle on arbitrary predicate function $f(X)$. Informally, you can easily see the relationship from the mobius inversion theorem to the inclusion-exclusion theorem, which states that $|V| = \sum_{X \in V} (-1)^{|V - X|} |X|$
Now, let's take a leap of faith together. Let $$R$$ be the right half of the vertices; define a semi-leftmatch to be a set of $$n/2$$ edges, each of which starts at exactly one vertex on the left side (so it is a perfect matching for only the left hand side). For example, this is a semi-leftmatch
<center>
Alt text
</center>
Now, define $$g(X \subseteq R)$$ to be the number of semi-leftmatches whose edges do not enter the right hand side vertices in $$X \subseteq R$$. For example, the above semi-leftmatch is also included in $$g(\{a\})$$. Then it turns out that $$g(X)$$ is the zeta transform of $$f(X)$$ that returns the number of perfect matchings when $$X = R$$, that is $g(R) = \zeta f(X) = \sum_{X \subseteq R} f(X) \implies f(R) = \sum_{X \subseteq R} (-1)^{(n/2 - |X|)} g(X)$ To see this is true, look at the following example:
<center>
Alt text
</center>
Here, we have $$|R|$$ = 3. Now, $$g({})$$ considers every possible semi-leftmatch, of which there are $$3 \times 2 \times 2$$ possibilities (the first left node has 3 edges, the second has 2, and the last also has 2). Of course, we also need to subtract off the non-matching semimatches. Notice that we have the invariant: if we exclude a node out of the right hand side for matching, then none of the semimatches generated are true matchings, because it's impossible to match $$n$$ left hand side nodes with $$n-1$$ right hand side nodes. Now, if you actually look closely, there are only 3 perfect matchings. If you do the mobius inversion on $$g$$, which you can compute by counting the number of semimatches (shown in the figure), you'll also get 12 - 4 - 2 - 4 + 1 + 0 + 0 + 0 = 3! (I neglected to include the cases where there are zero semimatches).
Therefore, in order to compute the number of perfect matchings, we just need $$2^{|R|}$$ computations of $$g(X)$$, which is easy:
function g(G,X)
remove from E all edges pointing to each node of X
remove from R all of X
return the product of the number of remaining edges coming out of each of the node on the left
end
that is, collapse $$G$$ into $$\hat G = (V, \hat E) = G \backslash X$$ by removing all of the nodes and edges connected to $$X$$, then $g(X) = \prod_{l \in L} \text{deg}_{\hat E}(l)$ where $$\text{deg}(v)$$ is the function that returns the number of edges connected to a vertex $$v$$. We can then use the mobius inversion transform above to compute $$f(R)$$ in $$O(\sqrt{2}^n)$$ time, irregardless of how many edges are present.
This concludes another presentation on using clever mathematics to solve problems in computer science. The algorithm derived here came from relatively recent research, and as of late, inclusion-exclusion has been an actively researched area in algorithm design. For those interested, I would recommend Fast polynomial-space algorithms using M ̈obius inversion: Improving on Steiner Tree and related problems
Note by Lee Gao
3 years ago
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I think I am somewhat confused. I couldn't read till the end because I lost in between. So in the bipartite graph you are allowing edges inside the same group also.. isn't it? Why cant we have perfect matching with c=1? Why isn't the total number of perfect matchings be equal to n! ? · 3 years ago
1. No a bipartite graph only allows edges from the left group into the right group. It has a characteristic look of two disjoint set of vertices.
2. Oops, I meant $$m = \frac{cn}{2}$$ and considering $$c \ge 1$$. The case of $$c = 1$$ is the trivial ladder shaped graph.
3. Recall that a matching matches one node in the left group into a unique node into the right. However, recall also that matchings can only be made on edges that are already included as part of our bipartite graph. We're not allowed to add in arbitrary edges. In order to construct $$n!$$ perfect matchings in a bipartite graph with $$2n$$ nodes ($$n$$ in the left and $$n$$ in the right), it better be the case that each perfect matching represents a mapping from the left group into some permutation on the right, which means that it must be possible to match ever node in the left with every node in the right: this is what we will call a complete bipartite graph. However, not all bipartite graphs are complete, the example I used up there cannot make $$n!$$ maximum matchings.
· 3 years ago
Thanks for the quick reply! Can you then fix that $$m=cn$$ in the article? Also now I think I understand this problem and I will read it neatly. :-)
Actually I work in related field (theory of complex networks) and I am actually writing a series about it on brilliant (https://brilliant.org/newsfeed/tag-feed/complexnetworks/). If you want, we can try to make our articles complementary! · 3 years ago
Thanks, I've made the changes :) if you find any other mistakes please let me know as well. I didn't get much of a chance to proof-read this article before posting it under time constraint last night.
I'm also interested in network theory, albeit because my background has been very combinatorial in nature, I enjoy analyzing game theoretic and combinatorial properties of network behavior: matching mechanisms, incentive properties, and algorithm design for various graph problems, but I haven't had much of a chance to look at the more analytic side on properties of networks themselves and have been attempting to look into these, which seems to be what your articles focus on. This seems like a great pairing then :) I would love to publish complementary articles on the subject :) · 3 years ago
if you like, we can communicate over email: snehalshekatkar@gmail.com · 3 years ago
Okay :) mine is lg342@cornell.edu · 3 years ago
Typo, second paragraph: "$$V$$, standinf for..." should be "standing for". · 3 years ago
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Nonnegative matrix and tensor factorization (nmf, ntf) with any beta divergence in matlab
The following Matlab project contains the source code and Matlab examples used for nonnegative matrix and tensor factorization (nmf, ntf) with any beta divergence. function [W,H,Q, Vhat] = betaNTF(V,K,varargin) %------------------------------------------------------------------ % simple beta-NTF implementation % % Decomposes a tensor V of dimension FxTxI into a NTF model : % V(f,t,i) = \sum_k W(f,k)H(t,k)Q(i,k) % % by minimizing a beta-divergence as a cost-functions.
Practical nmf ntf with beta divergence in matlab
The following Matlab project contains the source code and Matlab examples used for practical nmf ntf with beta divergence. ---------------------------------- Class name : NMF ---------------------------------- Implements NMF with any beta divergence, works on data with arbitrary number of channels.
Simple drums separation with nmf in matlab
The following Matlab project contains the source code and Matlab examples used for simple drums separation with nmf. ---------------------------------- This script illustrates the use of NMF for the extraction of the drums section in polyphonic music.
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Following up on the deep learning and toposes-post, I was planning to do something on the logic of neural networks.
Prepping for this I saw David Spivak’s paper Learner’s Languages doing exactly that, but in the more general setting of ‘learners’ (see also the deep learning post).
And then … I fell under the spell of $\mathbf{Poly}$.
Spivak is a story-telling talent. A long time ago I copied his short story (actually his abstract for a talk) “Presheaf, the cobbler” in the Children have always loved colimits-post.
Last week, he did post Poly makes me happy and smart on the blog of the Topos Institute, which is another great read.
If this is way too ‘fluffy’ for you, perhaps you should watch his talk Poly: a category of remarkable abundance.
If you like (applied) category theory and have some days to waste, you can binge-watch all 15 episodes of the Poly-course Polynomial Functors: A General Theory of Interaction.
If you are more the reading-type, the 273 pages of the Poly-book will also kill a good number of your living hours.
Personally, I have no great appetite for category theory, I prefer to digest it in homeopathic doses. And, I’m allergic to co-terminology.
So then, how to define $\mathbf{Poly}$ for the likes of me?
$\mathbf{Poly}$, you might have surmised, is a category. So, we need ‘objects’ and ‘morphisms’ between them.
Any set $A$ has a corresponding ‘representable functor’ sending a given set $S$ to the set of all maps from $A$ to $S$
$y^A~:~\mathbf{Sets} \rightarrow \mathbf{Sets} \qquad S \mapsto S^A=Maps(A,S)$
This looks like a monomial in a variable $y$ ($y$ for Yoneda, of course), but does it work?
What is $y^1$, where $1$ stands for the one-element set $\{ \ast \}$? $Maps(1,S)=S$, so $y^1$ is the identity functor sending $S$ to $S$.
What is $y^0$, where $0$ is the empty set $\emptyset$? Well, for any set $S$ there is just one map $\emptyset \rightarrow S$, so $y^0$ is the constant functor sending any set $S$ to $1$. That is, $y^0=1$.
Going from monomials to polynomials we need an addition. We add such representable functors by taking disjoint unions (finite or infinite), that is
$\sum_{i \in I} y^{A_i}~:~\mathbf{Sets} \rightarrow \mathbf{Sets} \qquad S \mapsto \bigsqcup_{i \in I} Maps(A_i,S)$
If all $A_i$ are equal (meaning, they have the same cardinality) we use the shorthand $Iy^A$ for this sum.
The objects in $\mathbf{Poly}$ are exactly these ‘polynomial functors’
$p = \sum_{i \in I} y^{p[i]}$
with all $p[i] \in \mathbf{Sets}$. Remark that $p(1)=I$ as for any set $A$ there is just one map to $1$, that is $y^A(1) = Maps(A,1) = 1$, and we can write
$p = \sum_{i \in p(1)} y^{p[i]}$
An object $p \in \mathbf{Poly}$ is thus described by the couple $(p(1),p[-])$ with $p(1)$ a set, and a functor $p[-] : p(1) \rightarrow \mathbf{Sets}$ where $p(1)$ is now a category with objects the elements of $p(1)$ and no morphisms apart from the identities.
We can depict $p$ by a trimmed down forest, Spivak calls it the corolla of $p$, where the tree roots are the elements of $p(1)$ and the tree with root $i \in p(1)$ has one branch from the root for any element in $p[i]$. The corolla of $p=y^2+2y+1$ looks like
If $M$ is an $m$-dimensional manifold, then you might view its tangent bundle $TM$ set-theoretically as the ‘corolla’ of the polynomial functor $M y^{\mathbb{R}^m}$, the tree-roots corresponding to the points of the manifold, and the branches to the different tangent vectors in these points.
Morphisms in $\mathbf{Poly}$ are a bit strange. For two polynomial functors $p=(p(1),p[-])$ and $q=(q(1),q[-])$ a map $p \rightarrow q$ in $\mathbf{Poly}$ consists of
• a map $\phi_1 : p(1) \rightarrow q(1)$ on the tree-roots in the right direction, and
• for any $i \in p(1)$ a map $q[\phi_1(i)] \rightarrow p[i]$ on the branches in the opposite direction
In our manifold/tangentbundle example, a morphism $My^{\mathbb{R}^m} \rightarrow y^1$ sends every point $p \in M$ to the unique root of $y^1$ and the unique branch in $y^1$ picks out a unique tangent-vector for every point of $M$. That is, vectorfields on $M$ are very special (smooth) morphisms $Mu^{\mathbb{R}^m} \rightarrow y^1$ in $\mathbf{Poly}$.
A smooth map between manifolds $M \rightarrow N$, does not determine a morphism $My^{\mathbb{R}^m} \rightarrow N y^{\mathbb{R}^n}$ in $\mathbf{Poly}$ because tangent vectors are pushed forward, not pulled back.
If instead we view the cotangent bundle $T^*M$ as the corolla of the polynomial functor $My^{\mathbb{R}^m}$, then everything works well.
But then, I promised not to use co-terminology…
Another time I hope to tell you how $\mathbf{Poly}$ helps us to understand the logic of learners.
Published in geometry math stories
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G = C5×Dic3⋊4D4order 480 = 25·3·5
Direct product of C5 and Dic3⋊4D4
Series: Derived Chief Lower central Upper central
Derived series C1 — C6 — C5×Dic3⋊4D4
Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C10×C3⋊D4 — C5×Dic3⋊4D4
Lower central C3 — C6 — C5×Dic3⋊4D4
Upper central C1 — C2×C10 — C5×C22⋊C4
Generators and relations for C5×Dic34D4
G = < a,b,c,d,e | a5=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 404 in 188 conjugacy classes, 86 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Dic34D4, S3×C20, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, D4×C20, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C15×C22⋊C4, S3×C2×C20, Dic3×C2×C10, C10×C3⋊D4, C5×Dic34D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C4×D4, C2×C20, C5×D4, C22×C10, S3×C2×C4, S3×D4, D42S3, S3×C10, C22×C20, D4×C10, C5×C4○D4, Dic34D4, S3×C20, S3×C2×C10, D4×C20, S3×C2×C20, C5×S3×D4, C5×D42S3, C5×Dic34D4
Smallest permutation representation of C5×Dic34D4
On 240 points
Generators in S240
(1 58 46 34 22)(2 59 47 35 23)(3 60 48 36 24)(4 55 43 31 19)(5 56 44 32 20)(6 57 45 33 21)(7 231 219 207 195)(8 232 220 208 196)(9 233 221 209 197)(10 234 222 210 198)(11 229 217 205 193)(12 230 218 206 194)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 121 109 97 85)(74 122 110 98 86)(75 123 111 99 87)(76 124 112 100 88)(77 125 113 101 89)(78 126 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)(109 110 111 112 113 114)(115 116 117 118 119 120)(121 122 123 124 125 126)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)(145 146 147 148 149 150)(151 152 153 154 155 156)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)(181 182 183 184 185 186)(187 188 189 190 191 192)(193 194 195 196 197 198)(199 200 201 202 203 204)(205 206 207 208 209 210)(211 212 213 214 215 216)(217 218 219 220 221 222)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)
(1 67 4 70)(2 72 5 69)(3 71 6 68)(7 181 10 184)(8 186 11 183)(9 185 12 182)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 82 22 79)(20 81 23 84)(21 80 24 83)(25 90 28 87)(26 89 29 86)(27 88 30 85)(31 94 34 91)(32 93 35 96)(33 92 36 95)(37 102 40 99)(38 101 41 98)(39 100 42 97)(43 106 46 103)(44 105 47 108)(45 104 48 107)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 118 58 115)(56 117 59 120)(57 116 60 119)(61 126 64 123)(62 125 65 122)(63 124 66 121)(127 190 130 187)(128 189 131 192)(129 188 132 191)(133 198 136 195)(134 197 137 194)(135 196 138 193)(139 202 142 199)(140 201 143 204)(141 200 144 203)(145 210 148 207)(146 209 149 206)(147 208 150 205)(151 214 154 211)(152 213 155 216)(153 212 156 215)(157 222 160 219)(158 221 161 218)(159 220 162 217)(163 226 166 223)(164 225 167 228)(165 224 168 227)(169 234 172 231)(170 233 173 230)(171 232 174 229)(175 238 178 235)(176 237 179 240)(177 236 180 239)
(1 137 17 130)(2 136 18 129)(3 135 13 128)(4 134 14 127)(5 133 15 132)(6 138 16 131)(7 121 236 120)(8 126 237 119)(9 125 238 118)(10 124 239 117)(11 123 240 116)(12 122 235 115)(19 146 26 139)(20 145 27 144)(21 150 28 143)(22 149 29 142)(23 148 30 141)(24 147 25 140)(31 158 38 151)(32 157 39 156)(33 162 40 155)(34 161 41 154)(35 160 42 153)(36 159 37 152)(43 170 50 163)(44 169 51 168)(45 174 52 167)(46 173 53 166)(47 172 54 165)(48 171 49 164)(55 182 62 175)(56 181 63 180)(57 186 64 179)(58 185 65 178)(59 184 66 177)(60 183 61 176)(67 194 74 187)(68 193 75 192)(69 198 76 191)(70 197 77 190)(71 196 78 189)(72 195 73 188)(79 206 86 199)(80 205 87 204)(81 210 88 203)(82 209 89 202)(83 208 90 201)(84 207 85 200)(91 218 98 211)(92 217 99 216)(93 222 100 215)(94 221 101 214)(95 220 102 213)(96 219 97 212)(103 230 110 223)(104 229 111 228)(105 234 112 227)(106 233 113 226)(107 232 114 225)(108 231 109 224)
(1 130)(2 131)(3 132)(4 127)(5 128)(6 129)(7 123)(8 124)(9 125)(10 126)(11 121)(12 122)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)
G:=sub<Sym(240)| (1,58,46,34,22)(2,59,47,35,23)(3,60,48,36,24)(4,55,43,31,19)(5,56,44,32,20)(6,57,45,33,21)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,67,4,70)(2,72,5,69)(3,71,6,68)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,82,22,79)(20,81,23,84)(21,80,24,83)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,94,34,91)(32,93,35,96)(33,92,36,95)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,106,46,103)(44,105,47,108)(45,104,48,107)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,118,58,115)(56,117,59,120)(57,116,60,119)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,190,130,187)(128,189,131,192)(129,188,132,191)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,202,142,199)(140,201,143,204)(141,200,144,203)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,214,154,211)(152,213,155,216)(153,212,156,215)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,226,166,223)(164,225,167,228)(165,224,168,227)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,238,178,235)(176,237,179,240)(177,236,180,239), (1,137,17,130)(2,136,18,129)(3,135,13,128)(4,134,14,127)(5,133,15,132)(6,138,16,131)(7,121,236,120)(8,126,237,119)(9,125,238,118)(10,124,239,117)(11,123,240,116)(12,122,235,115)(19,146,26,139)(20,145,27,144)(21,150,28,143)(22,149,29,142)(23,148,30,141)(24,147,25,140)(31,158,38,151)(32,157,39,156)(33,162,40,155)(34,161,41,154)(35,160,42,153)(36,159,37,152)(43,170,50,163)(44,169,51,168)(45,174,52,167)(46,173,53,166)(47,172,54,165)(48,171,49,164)(55,182,62,175)(56,181,63,180)(57,186,64,179)(58,185,65,178)(59,184,66,177)(60,183,61,176)(67,194,74,187)(68,193,75,192)(69,198,76,191)(70,197,77,190)(71,196,78,189)(72,195,73,188)(79,206,86,199)(80,205,87,204)(81,210,88,203)(82,209,89,202)(83,208,90,201)(84,207,85,200)(91,218,98,211)(92,217,99,216)(93,222,100,215)(94,221,101,214)(95,220,102,213)(96,219,97,212)(103,230,110,223)(104,229,111,228)(105,234,112,227)(106,233,113,226)(107,232,114,225)(108,231,109,224), (1,130)(2,131)(3,132)(4,127)(5,128)(6,129)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)>;
G:=Group( (1,58,46,34,22)(2,59,47,35,23)(3,60,48,36,24)(4,55,43,31,19)(5,56,44,32,20)(6,57,45,33,21)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,67,4,70)(2,72,5,69)(3,71,6,68)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,82,22,79)(20,81,23,84)(21,80,24,83)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,94,34,91)(32,93,35,96)(33,92,36,95)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,106,46,103)(44,105,47,108)(45,104,48,107)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,118,58,115)(56,117,59,120)(57,116,60,119)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,190,130,187)(128,189,131,192)(129,188,132,191)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,202,142,199)(140,201,143,204)(141,200,144,203)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,214,154,211)(152,213,155,216)(153,212,156,215)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,226,166,223)(164,225,167,228)(165,224,168,227)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,238,178,235)(176,237,179,240)(177,236,180,239), (1,137,17,130)(2,136,18,129)(3,135,13,128)(4,134,14,127)(5,133,15,132)(6,138,16,131)(7,121,236,120)(8,126,237,119)(9,125,238,118)(10,124,239,117)(11,123,240,116)(12,122,235,115)(19,146,26,139)(20,145,27,144)(21,150,28,143)(22,149,29,142)(23,148,30,141)(24,147,25,140)(31,158,38,151)(32,157,39,156)(33,162,40,155)(34,161,41,154)(35,160,42,153)(36,159,37,152)(43,170,50,163)(44,169,51,168)(45,174,52,167)(46,173,53,166)(47,172,54,165)(48,171,49,164)(55,182,62,175)(56,181,63,180)(57,186,64,179)(58,185,65,178)(59,184,66,177)(60,183,61,176)(67,194,74,187)(68,193,75,192)(69,198,76,191)(70,197,77,190)(71,196,78,189)(72,195,73,188)(79,206,86,199)(80,205,87,204)(81,210,88,203)(82,209,89,202)(83,208,90,201)(84,207,85,200)(91,218,98,211)(92,217,99,216)(93,222,100,215)(94,221,101,214)(95,220,102,213)(96,219,97,212)(103,230,110,223)(104,229,111,228)(105,234,112,227)(106,233,113,226)(107,232,114,225)(108,231,109,224), (1,130)(2,131)(3,132)(4,127)(5,128)(6,129)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240) );
G=PermutationGroup([[(1,58,46,34,22),(2,59,47,35,23),(3,60,48,36,24),(4,55,43,31,19),(5,56,44,32,20),(6,57,45,33,21),(7,231,219,207,195),(8,232,220,208,196),(9,233,221,209,197),(10,234,222,210,198),(11,229,217,205,193),(12,230,218,206,194),(13,61,49,37,25),(14,62,50,38,26),(15,63,51,39,27),(16,64,52,40,28),(17,65,53,41,29),(18,66,54,42,30),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84),(73,121,109,97,85),(74,122,110,98,86),(75,123,111,99,87),(76,124,112,100,88),(77,125,113,101,89),(78,126,114,102,90),(127,175,163,151,139),(128,176,164,152,140),(129,177,165,153,141),(130,178,166,154,142),(131,179,167,155,143),(132,180,168,156,144),(133,181,169,157,145),(134,182,170,158,146),(135,183,171,159,147),(136,184,172,160,148),(137,185,173,161,149),(138,186,174,162,150),(187,235,223,211,199),(188,236,224,212,200),(189,237,225,213,201),(190,238,226,214,202),(191,239,227,215,203),(192,240,228,216,204)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108),(109,110,111,112,113,114),(115,116,117,118,119,120),(121,122,123,124,125,126),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144),(145,146,147,148,149,150),(151,152,153,154,155,156),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180),(181,182,183,184,185,186),(187,188,189,190,191,192),(193,194,195,196,197,198),(199,200,201,202,203,204),(205,206,207,208,209,210),(211,212,213,214,215,216),(217,218,219,220,221,222),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240)], [(1,67,4,70),(2,72,5,69),(3,71,6,68),(7,181,10,184),(8,186,11,183),(9,185,12,182),(13,78,16,75),(14,77,17,74),(15,76,18,73),(19,82,22,79),(20,81,23,84),(21,80,24,83),(25,90,28,87),(26,89,29,86),(27,88,30,85),(31,94,34,91),(32,93,35,96),(33,92,36,95),(37,102,40,99),(38,101,41,98),(39,100,42,97),(43,106,46,103),(44,105,47,108),(45,104,48,107),(49,114,52,111),(50,113,53,110),(51,112,54,109),(55,118,58,115),(56,117,59,120),(57,116,60,119),(61,126,64,123),(62,125,65,122),(63,124,66,121),(127,190,130,187),(128,189,131,192),(129,188,132,191),(133,198,136,195),(134,197,137,194),(135,196,138,193),(139,202,142,199),(140,201,143,204),(141,200,144,203),(145,210,148,207),(146,209,149,206),(147,208,150,205),(151,214,154,211),(152,213,155,216),(153,212,156,215),(157,222,160,219),(158,221,161,218),(159,220,162,217),(163,226,166,223),(164,225,167,228),(165,224,168,227),(169,234,172,231),(170,233,173,230),(171,232,174,229),(175,238,178,235),(176,237,179,240),(177,236,180,239)], [(1,137,17,130),(2,136,18,129),(3,135,13,128),(4,134,14,127),(5,133,15,132),(6,138,16,131),(7,121,236,120),(8,126,237,119),(9,125,238,118),(10,124,239,117),(11,123,240,116),(12,122,235,115),(19,146,26,139),(20,145,27,144),(21,150,28,143),(22,149,29,142),(23,148,30,141),(24,147,25,140),(31,158,38,151),(32,157,39,156),(33,162,40,155),(34,161,41,154),(35,160,42,153),(36,159,37,152),(43,170,50,163),(44,169,51,168),(45,174,52,167),(46,173,53,166),(47,172,54,165),(48,171,49,164),(55,182,62,175),(56,181,63,180),(57,186,64,179),(58,185,65,178),(59,184,66,177),(60,183,61,176),(67,194,74,187),(68,193,75,192),(69,198,76,191),(70,197,77,190),(71,196,78,189),(72,195,73,188),(79,206,86,199),(80,205,87,204),(81,210,88,203),(82,209,89,202),(83,208,90,201),(84,207,85,200),(91,218,98,211),(92,217,99,216),(93,222,100,215),(94,221,101,214),(95,220,102,213),(96,219,97,212),(103,230,110,223),(104,229,111,228),(105,234,112,227),(106,233,113,226),(107,232,114,225),(108,231,109,224)], [(1,130),(2,131),(3,132),(4,127),(5,128),(6,129),(7,123),(8,124),(9,125),(10,126),(11,121),(12,122),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)]])
150 conjugacy classes
class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M ··· 10T 10U ··· 10AB 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 20AG ··· 20AV 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 6 6 2 2 2 2 2 3 3 3 3 6 6 6 6 1 1 1 1 2 2 2 4 4 1 ··· 1 2 ··· 2 6 ··· 6 4 4 4 4 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4 4 ··· 4
150 irreducible representations
dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C10 C10 C20 S3 D4 D6 D6 C4○D4 C4×S3 C5×S3 C5×D4 S3×C10 S3×C10 C5×C4○D4 S3×C20 S3×D4 D4⋊2S3 C5×S3×D4 C5×D4⋊2S3 kernel C5×Dic3⋊4D4 Dic3×C20 C5×Dic3⋊C4 C5×D6⋊C4 C15×C22⋊C4 S3×C2×C20 Dic3×C2×C10 C10×C3⋊D4 C5×C3⋊D4 Dic3⋊4D4 C4×Dic3 Dic3⋊C4 D6⋊C4 C3×C22⋊C4 S3×C2×C4 C22×Dic3 C2×C3⋊D4 C3⋊D4 C5×C22⋊C4 C5×Dic3 C2×C20 C22×C10 C30 C2×C10 C22⋊C4 Dic3 C2×C4 C23 C6 C22 C10 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 8 4 4 4 4 4 4 4 4 32 1 2 2 1 2 4 4 8 8 4 8 16 1 1 4 4
Matrix representation of C5×Dic34D4 in GL4(𝔽61) generated by
34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
1 1 0 0 60 0 0 0 0 0 60 0 0 0 0 60
,
11 0 0 0 50 50 0 0 0 0 50 0 0 0 0 50
,
1 0 0 0 60 60 0 0 0 0 11 3 0 0 0 50
,
1 0 0 0 0 1 0 0 0 0 11 3 0 0 21 50
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,60,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[11,50,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[1,60,0,0,0,60,0,0,0,0,11,0,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,11,21,0,0,3,50] >;
C5×Dic34D4 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3\rtimes_4D_4
% in TeX
G:=Group("C5xDic3:4D4");
// GroupNames label
G:=SmallGroup(480,760);
// by ID
G=gap.SmallGroup(480,760);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,891,226,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
𝔽
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{}
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# Edwards, Tangherlini, Selleri transformations and their inverse
1. Nov 4, 2008
### bernhard.rothenstein
Edwards, Tangherlini, Selleri propose synchrony parameter dependent transformation equations we have discussed here. Call them direct transformations. They also their inverse version. As I see they are not used. Is there a special reason for that. Are they of interest?
Thanks
2. Nov 5, 2008
### DrGreg
There's no need to quote an inverse transformation unless the argument you are making requires such a transformation. I think I've probably seen inverse Selleri/Tangherlini transforms somewhere, though I can't remember where. It's quite easy to calculate.
The Edwards transform is between two arbitrarily synced systems and so is "its own inverse" in the sense that the Lorentz transform is "its own inverse", you just need to change the values of the parameters. (E.g. v to -v in the Lorentz case.)
3. Nov 6, 2008
### bernhard.rothenstein
My problem is with the Selleri transformation There in I the clocks are standard synchronized whereas in I using the so called external synchronization. If we know the direct transformations the inverse ones are not obtainable by the rule which works in the case when in both frames standard clock synchronization takes place i.e. change the sign of V and interchange the primed with unprimed same physical quantities.
As allways respect and thanks.
4. Nov 6, 2008
### DrGreg
If you know the direct transformation, finding the inverse is just mathematical algebra, in this case, solving two simultaneous equations, or, equivalently, inverting a 2x2 matrix.
Changing the sign of V etc won't work because whereas the forward transform is from isotropic to anistropic coordinates, the reverse (inverse) transform is from anisotropic to istropic coordinates, so you would not expect the "same" formula to apply. If should also be pointed out that V is measured within the isotropic coordinates (as dx/dt for the "moving" observer relative to the "stationary" observer). The velocity V' of the "stationary" observer relative to the "moving" observer as measured in the "moving" observer's anisotropic coordinates (dx'/dt') will not be -V.
$$x' = \gamma(x - Vt)$$
$$t' = t/\gamma$$
has inverse
$$x = (x' + \gamma^2 Vt')/\gamma$$
$$t = \gamma t'$$
from which it follows that
$$V' = -\gamma^2 V$$
So the rule in this case is to replace V by $-\gamma^2 V$ and $\gamma$ by $1/\gamma$.
5. Nov 6, 2008
### bernhard.rothenstein
Thanks
I know that considering the relative positions of the I and I' reference frame from I at a given time t I can derive the say direct transformation for the space coordinates of an event
taking into account length contraction. If I consider the same situation from I at a time t' and taking into account length contraction I can derive the inverse transformation of the space coordinate. Combining the two equations I can derive the direct and the inverse transformations for the time coordinates of the same event.
Do you know a way to do the same thing but starting with the time dilation?
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Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''
### Parametric inference for discretely observed non-ergodic diffusions
Auteur(s):
Code(s) de Classification MSC:
Résumé: We consider a multidimensional diffusion process $X$ whose drift and diffusion coefficients depend respectively on a parameter $\la$ and $\te$. This process is observed at $n+1$ equally--spaced times $0,\De_n,2\De_n,\ldots,n\De_n$, and $T_n=n\De_n$ denotes the length of the observation window''. We are interested in estimating $\la$ and/or $\te$. Under suitable smoothness and identifiability conditions, we exhibit estimators $\lan$ and $\ten$, such that the variables $\rn~(\ten-\te)$ and $\sqrt{T_n}~(\lan-\la)$ are tight, as soon as $\De_n\to0$ and $T_n\to\infty$. When $\la$ is known, we can even drop the assumption $T_n\to\infty$. The novelty is that these results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.
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View Single Post
2020-10-23, 08:20 #19
ONeil
Dec 2017
F016 Posts
Quote:
Originally Posted by Viliam Furik That is not a misquote, you are simply talking nonsense. Primes are not composite, that is their definition. If you mean the Mersenne numbers... Those can be composite. Mersenne primes are a very rare special case of those numbers.
This is what I said
Primes that make numbers from 2^p-1 which are not prime have factors that are prime, sorry for the confusion.
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How To Use Total Concentration Breathing
This is the use of essential oils on hot compress, in diffusers, or in hot water for inhalation. Psychological research shows how you can easily and totally relax in a matter of seconds, using a method learn by playing this. 1152 at 10 o C to 0. Using cannabidiol sublingually probably works best if you don’t like the idea of vaping or smoking. All aerobic creatures need oxygen for cellular respiration, which uses the oxygen to break down foods for energy and produces carbon dioxide as a waste product. The droplets ( orwooya ) are/is trapped. Feel the Rhythm. Deep Breathing Exercise Basic Routine. I have found that at least 100 to 200-liter reservoir is needed for most 30-60 minute exercise with oxygen therapy (EWOT) sessions. Closest would be the breathing pattern people use when running to prevent cramps and exhaustion, breathing in on one step and out on the next. The various forms of deep relaxation do, however, share some similar qualities. Safe breathing–even with extremely high H 2S concentrations. The videos are not intended to be a substitute for professional medical advice, diagnosis or treatment. How to Use This DIY Calming Oil Blend. In addition, data from healthy subjects were collected during coughs, swallows, tidal breathing, and breathing to total lung capacity (TLC) and residual volume (RV). Question: During a typical breathing cycle the {eq}\rm CO_2 {/eq} concentration in the expired air rises to a peak of 4. Haku breathing Haku breathing is a very short, forceful exhalation, repeated several times. The first step in this process is breathing in air, or inhaling. The inspiratory capacity (IC) is the amount of air that can be inhaled after the. The Relaxator Breathing Retrainer - Reduce Stress and Increase Concentration; The Sleep Tape - Get Deeper Sleep With Your Rebirthing and Holotropic breathing are two other popular ways of using breathing to change the state of mind and improve one's health. The total number of MP inhaled by the manikin over 24 hours reached up to 272 MP (L3S3). Brown and Dr. Another option is to use a white noise machine to save you from total silence, which can actually distract some adults with ADHD more than loud. Inhale deeply through your nose, pushing the hand on your stomach outward. Concentration problems, when present to an excessive degree, are also characteristic of certain physical and. In mindfulness, the meditator may still use breathing or music as aids to focus the mind, but concentration precludes judgment: it involves mere non-judging awareness or acceptance of the experience as just a matter of being. The revised equation also separates the organic mass into large and fine mode fractions using the total mass (if the total concentration is above 20 μg/m 3, all of it is assumed to be in the large mode. Here’s an example of how a full formula works: Now, compare 7. Respiration: the entire process of gas exchange. ately before use. 1 ppm between 2016 and 2017. It is one of the most important gases on the earth because plants use it to produce carbohydrates in a process called photosynthesis. Devices Used With Heliox Regulators The most readily available component specifically de-signed for use with heliox, the regulator, is required to transition the compressed gas into the breathing system. Five unused samples of each different filter were tested. 9%, while in the expired air is 15. Lengthen Your Exhale. It is the sum of the expiratory reserve volume, tidal volume, and inspiratory reserve volume. Carbon dioxide, a waste product, goes out of the body. Check your answers if you have time at the end. A simple protocol of 30 breaths, twice daily is sufficient to significantly improve performance. FRC is calculated from the total volume of N 2 exhaled and the change in alveolar N 2 from the beginning to the end of the test (Figure 4-2 and open-circuit method); (B) Closed-circuit equipment used for He dilution FRC determination includes a directional breathing circuit with a volume-based spirometer, He analyzer, CO 2 absorber, O 2 source. The main component of Earth's atmosphere, like that of Saturn's moon Titan, is nitrogen, and the other abundant element is oxygen. Ineffective Breathing Pattern: Inspiration and/or expiration that does not provide adequate ventilation. Asymptomatic carriers transmit SARS-CoV-2 with normal breathing. Breathing out, I am aware of my whole body. When breathing through your nose, you're more likely to be present and connected to your body, she explains. Here is a basic routine. The compressor control monitors the fill level and informs the operator in good time if the condensate needs to be emptied. On average, canned oxygen costs just under $50 a unit - a cost that would exceed$1,160 per day if you relied on it for constant use, and more than \$426,000 a year. It is present in the Earth's atmosphere at a low concentration and. MUC is determined by the assigned protection factor of the respirator or class of respirators and the exposure limit of the hazardous substance. “Mindfulness of breathing is especially good as a concentration method for use during travel and during the times when one is restlessly expecting a bus or a train. Where there are higher concentrations of contaminants, however, an atmosphere-supplying respirator such as the positive-pressure SAR offers better protection. The benefit to using shots is that you get a higher concentration of testosterone in one sitting, this makes positive (and potentially negative) side effects easier to identify. Inhale slowly and deeply through your nose to the count of four. This real-time simulation displays the CO2 emissions of every country in the world, as well as their birth and death rates. The various forms of deep relaxation do, however, share some similar qualities. FiO2 is defined as the concentration of oxygen that a person inhales. ? It is also used for scuba diving. A 2017 study published in the International Journal of Exercise Science showed breathing through your nose makes your heart rate higher than breathing through your mouth, even at the same pace. The alveolar ventilation rate is a critical physiological variable as it is an important factor in determining the concentrations of oxygen and carbon dioxide in functioning alveoli. A respiratory cycle is one sequence of inspiration and expiration. Nausea and general discomfort may also appear during the first few days. Postures, breathing, mantras and many other techniques to focus concentration all need to be considered when learning how to meditate. Rapid, shallow breathing, also called tachypnea, occurs when you take more breaths than normal in a given minute. Always seek. Using this product is secure, and keeping in mind, the audience of this product, the manufactures of this product have made sure to make the product in such a way that is easy to consume for all the age gap. For example, you may imagine a peaceful setting and then focus on controlled, relaxing breathing, slowing your heart rate, or feeling different physical. The technique has played a key role in the series and is fundamentally the basis for such powerful moves as Tanjiro's Water Breathing, though Zenitsu Agatsuma also used Total Concentration Breathing to keep Older Brother Spider's poison from becoming fatal. Lavender tea for stress relief. Many people suffer from afflictions such as asthmatic conditions, lung. Put another way, your lungs are at least as important to your body's ability to experience ongoing cleansing and detoxification as your digestive tract and kidneys. Some of the more serious respiratory symptoms would take even longer to surface, needing anywhere from 24 to 48 hours to appear. Hinokami Kagura Breathing (ヒノカミ神楽 (かぐら) の呼 (こ) 吸 (きゅう) Hinokami Kagura no Kokyu) is a unique form of Total Concentration Breathing passed down for generations in the Kamado Family. More: 3 Drills for Open Water Swimming. Note to reader: This fact sheet is intended to provide general awareness and education on a specific chemical agent. surfactant a lipoprotein mixture continually secreted into the alveolar air spaces. Sip and whiff at the same time!. Daoist breathing exercises are breathing practices designed to activate the diaphragm muscle, expand the lungs, and invoke the body's innate relaxation response. This ppm result is to be compared with the PEL without adjustment for temperature and pressure at the sampling site. , humidifier, PEEP valve) to be used during the case. Young amphibians, like tadpoles, use gills to breathe, and they don’t leave the water. Organism, organ system, organ, tissue, cell c. Suppose you have an equilibrium established between four substances A, B, C and D. Shallow breathing limits the diaphragm's range of motion. Relaxation and Positive Imagery: Combining simple relaxation techniques such as deep breathing with positive visual imagery helps the brain to improve or learn new skills. This is in contrast with Obstructive Sleep Apnea, where the ventilatory drive persists but airflow ceases because of the obstruction of the upper airway. It pushes air into the lungs by keeping the lungs open and thus allowing more oxygen to enter. Be sure to inhale into your belly, not your chest. Carbon dioxide is a gas consisting of one part carbon and two parts oxygen. The 10% Happier app , based on a book by Dan Harris, a co-anchor of "Nightline," takes a different approach to keep you coming back, by introducing the user to meditation with a personal coach. To begin, one assumes lotus pose or any comfortable pose with the spine erect. Blood/Breath Alcohol Concentration (BAC) is the amount of alcohol in the bloodstream or on one’s breath. There can be no wasted movement during its application in combat, and this makes it difficult to master. HAP offers an option to select a "space usage" type, based on the list of space usage types found in Table 6-1 in the standard. These videos do not provide medical advice and are for informational purposes only. (OTCPK:EHMEF) Q1 2020 Earnings Conference Call May 7, 2020 11:00 ET Company Participants Hal Khouri - Executive Vice President & Chief Financial. Elemental analysis of the tissue was used to estimate the growth conversion efficiency. That can make you feel short of breath and anxious. This game will teach you how to get rid of nagging thoughts and to concentrate on here and now. The chorioallantoic membrane (CAM) of the developing chicken embryo is an established model that is used in biomedical research in a multitude of different applications 1. Typically, these sensors are used to provide a hazardous condition threshold alarm set to 5% or 10% of the LEL concentration of the gases or vapors being measured. While it will increase total hardness (the calcium component), and temporarily increase the total alkalinity (the hydroxide component), its primary effect is to raise the pH of mud and water quickly and dramatically. It is practiced with your back straight and constantly looking forward. RESULTS: Switching from facial to tracheostomy delivery increased lung dose with nebulizer (all breathing patterns). Now do the exact same breathing technique as you did before, placing the cotton ball just beneath your nose. Ten minutes after you finish a big meal (it doesn't have to be a total pig-out!), sit comfortably in a quiet place, resting one hand on your stomach. Pulmonary Tuberculosis: Used to be called consumption. Additional use has shown to provide further blood pressure reductions, and there is no danger of using the device too much. You’ll see your child’s concentration and sequencing improve the more they play, which is a great reward for both of you. Close your mouth and inhale quietly through your nose to a mental count of four. concentration, centering, control, breathing, precision, and flow," adds Rich. hypnotic (s) and opioid (s) can be used and opioid-free techniques are described. When you feel these emotions in your body, focus on your breathing and force yourself to breathe at a normal, controlled rate. Every day I try a new technique on reducing stress, better concentration,Increasing your energy level,and many more chapters. Learn about the breathing process and its effects on the entire body in this vibrantly illustrated book filled with interesting text and colorful photographs. These can be measured to aid in the definitive diagnosis, quantification and monitoring of disease. Ventilation means getting fresh air into your home or workplace. The more often you use the pool, the more often you should reach for the swimming pool shock. Gas molecules move from a region of high concentration to a region of low concentration. Abdominal Breathing. Other sensors do not provide tactile stimulation when hypoxia is detected. Breathing in, I am aware of my whole body. When using a semi-closed system, the oxygen flow rate must exceed the patient's oxygen consumption. The benefit to using shots is that you get a higher concentration of testosterone in one sitting, this makes positive (and potentially negative) side effects easier to identify. The minimum oxygen concentration in the air required for human breathing is 19. Relative to carbon dioxide the other greenhouse gases together comprise about 27. Research suggests that between 9 and 30 percent of those who use marijuana may develop. Slow and deep breathing stimulates stretch-induced inhibitory signals and hyperpolarizes currents propagated in cells, leading to synchronization of neural elements in the heart, lungs, limbic system, and cortex. Inspiratory muscles can be trained, resulting in increased strength, endurance, reduced blood lactate concentration and RPE. A, Total hip arthroplasty. Edexcel Certificate Edexcel International GCSE. Others recommend experimenting with a longer inhale than exhale—2:1 for faster running, 4:3 or 3:2 for easier running—to see what feels more. Where there are higher concentrations of contaminants, however, an atmosphere-supplying respirator such as the positive-pressure SAR offers better protection. 6% had an oxygen saturation level of 95% and 42. 3% (see above). 1 ppm from 2017, similar to the increase of 2. Ted Slampyak. As with breathing, the mindset is essential in the practice of Zen meditation. Tissue, cell, organ, organ system, organism b. If not for a friends recommendation of "Total Breathing" my running would have stopped before I ever got started. The affected individual's trachea and oropharynges may need to be suctioned to remove particles of food or other substances. For example, the Henry's Law Constant for benzene ranges from 0. waking up a lot. Psychological research shows how you can easily and totally relax in a matter of seconds, using a method learn by playing this. High concentrations of ozone in air, when people are not present , are sometimes used to help decontaminate an unoccupied space from certain chemical or biological contaminants or odors (e. A Central Sleep Apnea (CSA) is defined as a cessation of breathing of at least ten seconds duration in the absence of a ventilatory drive. Asymptomatic carriers transmit SARS-CoV-2 with normal breathing. Below are 4 breathing exercises I try to do on a regular basis. Expand this section. This force is used in breathing to hold the moist surfaces of the pleural membranes together. Record breathing rate after hyperventilation has been completed. The idea is to shift our attention onto the sensations produced in our bodies while we breathe. Might just be me, but Total Concentration Water Breathing sounds way cooler than, Full Focus Breath of Water, the latter sounds like a direct translation that doesn't carry the same feeling as MIZU NO KOKYU: ZEN SHUCHU KOKYU!!!. It is practiced with your back straight and constantly looking forward. Sensors are typically placed upstream of the oxygen breathing system, which makes it hard to detect problems in connections, hoses, mask fit, etc. Do not pursue them or escape fight from them. * Inhale and exhale solely through the nose. Conventional Rubber Goods: 4” Breathing Tube, Large Nasal Inhaler and 3 Liter Breathing Bag Assembly Procedure: 1. Relaxation has been defined as a psychological strategy used by sports performers to help manage or reduce stress-related emotions (e. Perform Leak Check of the Breathing System. Your Concentration Training Program: 11 Exercises That Will Strengthen Your Attention In this series on mastering your attention, we have emphasized the fact that attention is not just the ability to focus on a single task without being distracted, but in fact is comprised of several different elements that must be effectively managed. If we multiply this result by 6 L/min (normal minute ventilation), it is equal to 336 ml of oxygen per. Water Breathing (水 (みず) の呼 (こ) 吸 (きゅう) Mizu no Kokyū) is one of the five main forms of Total Concentration Breathing and utilizes the element of water in the user's swordsmanship. We demonstrated that 18 F-FDG uptake in lung cancer would be affected by high concentration oxygen breathing. Air is the most common, and only natural, breathing gas. If it is ordinary breathing (kwiitsya), the virus may go for up to 1-2 meters from an infected person. from the breathing of oneself or others, possibly unlocking abilities related to the affinity and enhancing existing powers. , natural gas compressor station s, natural gas processing plants, condensate tank batteries, crude petroleum liquid storage facilities, etc. Patricia L. When the abdominal wall excursion during inspiration, expiration, or both do not maintain optimum ventilation for the individual, the nursing diagnosis Ineffective Breathing Pattern is one of the issues nurses need to focus on. Sometimes called belly breathing or diaphragmatic breathing. Some patients saw a reduction of severity and duration of flare-up symptoms with continued vitamin D use. If you have used CBD for a while, and are not happy with the results, take a close look at the brand. It's a potent reminder of how even a few seconds of focused breathing can cause you to stop what you're doing and change your mood. In gas chroma-tography (GC) mode. Aerosol deposition through HFNC was less than 2% but higher than drug delivery with the Bubble CPAP. It incorporates client education and breathing retraining exercises which aim to: improve the breathing volume, rate, and rhythm improve posture and promote correct use of the diaphragm and breathing muscles. This type of breathing may also help detoxify the blood, relax the mind, reduce stress, and increase concentration. Does breathing contribute to CO2 buildup in the atmosphere? Posted on 26 September 2010 by climatesight. We release this gas from a solution of 28% Sodium Chlorite (now called MMS1 by Jim Humble) by "activating" it with an equal amount (drops, or 1/2 oz. Normal range for arterial carbon dioxide tension. Total Petroleum Hydrocarbons (TPH) is a term used to describe a broad family of several hundred chemical compounds that originally come from crude oil. But don't worry: it's fun too!. Partial pressure is a measure of the concentration of the individual components in a mixture of gases. 26 atm (atmospheres). Asked in Health. Easy-to-position breathing system. A copy of the The Facts About Ammonia (General Information) is available in Adobe Portable Document Format (PDF, 63 KB, 3pg. The idea is to shift our attention onto the sensations produced in our bodies while we breathe. Do not pursue them or escape fight from them. New Study Confirms that Dog Ownership Increases Longevity. If you want to sit cross-legged on the ground, go for it. It's been said that the greatest power of the human mind is its ability to concentrate on one thing for an extended period of time. Blood/Breath Alcohol Concentration (BAC) is the amount of alcohol in the bloodstream or on one’s breath. hypnotic (s) and opioid (s) can be used and opioid-free techniques are described. Check your answers if you have time at the end. In addition to your weekly or semi-weekly treatments, you may want to perform an extra pool shock under certain circumstances, such as after: heavy pool use (like a pool party) a severe rainstorm or damaging winds (especially if your pool collected debris). Total Concentration Breathing increases the user's capabilities beyond that of a normal human through advanced breathing forms. The requirements for compressed air hose units are governed by EN 14593. (Chapt 18) Atmospheric Pressure = 760 mm Hg at sea = Breathing Pulmonary Function Tests use Spirometer. But allergic rhinitis often can be easily treated and self-managed. Lung Function Tests & breathing tests This page is designed to outline the reasons why you have been advised to have breathing tests and how to prepare for them. It’s been said that the greatest power of the human mind is its ability to concentrate on one thing for an extended period of time. In general, it should be large enough to include both the breathing zone of the exposed person and the emitting source. Energy Booster. Then exhale for a total of at least six seconds first from the chest, then ribs, then belly, pulling the lower ribs together at the end (this contracts the muscle of the diaphragm which helps you breathe even deeper. Types of Breathing Machines By Richard Nilsen A breathing machine is a piece of equipment that facilitates breathing in the case of low oxygen levels in the blood stream. This is known as obstructive sleep apnea. A face mask is worn and supplies the needed pressure to keep the patient breathing in and out. Can be used for as long as the battery lasts, or for the recommended interval between calibrations, whichever is less. This latent period played havoc. , or New York City. On the exhale, let the breath go first from the upper chest, then the ribcage, then the belly. 5 in another region downwind. After the subtraction, only the "changing absorbance signal" is left, and this. This frustrating and. Anhydrous ammonia gas is lighter than air and will rise, so that generally it dissipates and does not settle in low-lying areas. Flame Ionization Detector (FID) with Gas Chromatography Option Many organic gases and vapors. Gas Exchange in Humans [back to top] In humans the gas exchange organ system is the respiratory or breathing system. It does not hydrolyze when it enters water, and. Even with exercise induced asthma I feel like this is a better choice as the breathes for seem to fill the lungs better even at a faster pace. The breath is a vehicle for deepening concentration and revealing quiet sources of joy. A slight positive pressure in the facepiece throughout the breathing cycle is used in the pressure-demand, compressed-air breathing apparatus to eliminate any inward leakage. This can be due to: tighter ratio / higher concentration of solids or over-extraction. Unlike other methods of cannabis consumption, flower doesn’t have a standard dosing structure. Featured Supporters. The primary trigger for breathing is the rising CO2 level in the blood rather than the falling O2 level, but hyperventilating flushes out CO2 and this low level lead to narrowing of the blood vessels that supply blood to the brain. By the day of the empty chair, April 19, the total Covid inpatient population at Michigan Medicine had been dropping for the fifth day running, from a high of 229 on April 15 to 193. For the OPC results, the effects of the use of a face shield, the cough aerosol particle size distribution, and the distance from the coughing to the breathing simulator on the total volume of particles inhaled were compared by means of an analysis of variance (ANOVA), with these three parameters treated as fixed effects. This force is used in breathing to hold the moist surfaces of the pleural membranes together. Surprisingly, you may find this step requires some concentration initially. After a few minutes of deep breathing, you focus on one part of the body or group of muscles at a time and mentally releasing any physical tension you feel there. If you’d 2. Other sensors do not provide tactile stimulation when hypoxia is detected. Digestive system: abdominal pain, diarrhea, fever, nausea and vomiting. But research has shown that it. therefore, the respiratory protection device must be certified for use in explosive environments. Calculate the total quantity and the total days supply for the following Rx: Albuterol 2. 1% Prolonged exposure can affect powers of concentration 5000 ppm 0. Repeated samples, at higher or lower total air volume, can be collected to estimate the magnitude of oil mist concentration. Set up an equation using Dalton’s law, rearrange the equation to solve for the pressure of just the hydrogen gas, plug in your numbers, and solve: So, the partial pressure of hydrogen gas trapped in the tube is 98. Carbon dioxide is a gas consisting of one part carbon and two parts oxygen. Peatlands represent a globally important carbon stock. 2 : How to Get What You Want; How to Be a Master Leader by Correct Concentration and Breathing Combined by Edward Thomas Eiklor (2013, Hardcover) at the best online prices at eBay! Free shipping for many products!. for a total of about three hours. Haku breathing Haku breathing is a very short, forceful exhalation, repeated several times. This force is used in breathing to hold the moist surfaces of the pleural membranes together. Based on recent studies, dogs have been found to not only bring us joy and fulfillment but also increase longevity. I learned it in two of my Vipassana retreats over the past two years. Albuterol concentration was determined via spectrophotometry (276 nm). To measure oxygen saturation using a pulse oximeter, make sure the patient’s finger or other site is warm, as coldness can cause poor blood flow and result in an inaccurate reading. Other conditions treated with hyperbaric oxygen therapy include serious infections, bubbles of air in your blood vessels, and wounds that won't. The droplets ( orwooya ) are/is trapped. Use yoga as a tool for teaching yourself how to breathe naturally 3. When I’m micro-managing and obsessing over details, I know I’m in my. ? It is also used for scuba diving. People with narcolepsy usually feel rested after waking, but then feel very sleepy throughout much of the day. Deep Breathing Exercise Basic Routine. Increasing oxygen content. The VERTICUS breathing air compressors are available in versions for 225 and/or. People with miner's lung have a lower concentration of oxygen in their blood than healthy people. To properly use Total Concentration Breathing, the user must concentrate, relax their upper half and brace their lower half before taking a long breath. The use of masks under strenuous conditions can be quite uncomfortable and laborious. One of the downsides is that testosterone levels peak after day 2-3 of the injection and then slowly fall back down over the next 4-5 days. Blood Alcohol Concentration; Blood Alcohol Concentration. You’ll see your child’s concentration and sequencing improve the more they play, which is a great reward for both of you. Ted Slampyak. It is a necessary raw material for most plant life, which remove carbon dioxide from air using the process of photosynthesis. Electrical PEEP control; Breathing valve is disassembly and can be disinfected. Then, the total days supply will be 50 vials divided by 2 vials which is 25 days. Serum is a skincare product you can apply to your skin after cleansing but before moisturizing with the intent of delivering powerful ingredients directly into the skin. FRC is calculated from the total volume of N 2 exhaled and the change in alveolar N 2 from the beginning to the end of the test (Figure 4-2 and open-circuit method); (B) Closed-circuit equipment used for He dilution FRC determination includes a directional breathing circuit with a volume-based spirometer, He analyzer, CO 2 absorber, O 2 source. He used it to develop mindfulness and concentration. This allows oxygen to reach every. Download it once and read it on your Kindle device, PC, phones or tablets. For smaller particles (0. Cheyne–Stokes respiration is an abnormal pattern of breathing characterized by progressively deeper, and sometimes faster, breathing followed by a gradual decrease that results in a temporary stop in breathing called an apnea. It is hence imperative that the air inside the mask is as fresh as possible and the act of breathing is easy. Finding time to relax can be very difficult for most people. If you’ve heard of a therapeutic oxygen tent, you are seeing Dalton’s law of partial pressure put to work. This makes it easier to dip a long handled brush into the solution. Mark the time on your watch when you start. The slow cycle returns carbon to the atmosphere through volcanoes. As long as you don't use toxic gasses, you can replace the nitrogen and "other" with other gasses, like Helium, as long as you keep the partial pressure of oxygen near 0. What is the Average DLcoSb value?. As with breathing, the mindset is essential in the practice of Zen meditation. The Buddha's instructions indicate that sitting with a straight back is the best position for anapanasati. Feel the Rhythm. Calculate the partial pressure of the {eq}\rm CO_2 {/eq} at. Sit somewhere comfortable, and close your eyes. gov for more information. I used a non re-breather mask with 5L O2 with a Pt who was mottled, RR 28-32, O2Sat 88. Daily pranayama trains the lungs and improves the capacity of respiratory system immensely. Moisture levels in compressed breathing air6. The MUC usually. It describes what happens when our heart rate variability, blood pressure, and brainwave function come into a coherent frequency. Air-Breathing Aqueous Sulfur Flow Battery Concept The anolyte (left) is an aqueous polysulfide solution within which the working ions (here, Li + or Na + ) carry out the sulfur redox reaction. Zhuoyu High Concentration Household Hydrogen Inhaler Machine Hydrogen Breathing Machine , Find Complete Details about Zhuoyu High Concentration Household Hydrogen Inhaler Machine Hydrogen Breathing Machine,High Concentration Hydrogen Inhaler Machine,2019 New Hydrogen-rich Water Generator Portable Hydrogen Inhalation Machine High Con,Molecular Hydrogen Inhalation Machine from Gas Generation. Essential oils are a popular natural remedy for sinus congestion, stuffiness, and a blocked nose. Oxygen is the most important energy source for the cells. Rest for 3 minutes. Airborne Concentration or Condition of Use. Safe breathing–even with extremely high H 2S concentrations. In this blog, we look at how an argon alarm can keep you safe. Mind Body Soul 582,557 views. There are breathing techniques that accomplish the stated objective that opens up the lungs fully and maximises the oxygen in the lungs and bloodstream; but aside from giving a boost in clarity, maybe a bit of energy, and in some cases being used. Five unused samples of each different filter were tested. A Compressed air hose unit is a type of non-freely portable breathing apparatus. 48 M, which is spoken as "zero point forty-eight molarity" or "zero point forty-eight molar. The pulse oximeter is able to use some clever mathematics to extract the "changing absorbance" signal from the total signal, as will be described. Other sensors do not provide tactile stimulation when hypoxia is detected. Psychological research shows how you can easily and totally relax in a matter of seconds, using a method learn by playing this. Essential:The breathing system must a) deliver the gases from the machine to the alveoli in the same concentration as set and in the shortest possible time; b) effectively eliminate carbon-dioxide; c) have minimal apparatus dead space; andd) have low resistance. Our mission is to help people overcome mental and emotional health issues and live fuller, happier lives. Hyperbaric oxygen therapy involves breathing pure oxygen in a pressurized room or tube. General: Do not use compressed air to remove fly ash. Belly breathing allows runners to increase the total usable lung capacity. Parenting is an expensive business, especially in the early days when you need to. The 10% Happier app , based on a book by Dan Harris, a co-anchor of “Nightline,” takes a different approach to keep you coming back, by introducing the user to meditation with a personal coach. Try to draw it right down to your stomach. This technique blends breath focus with progressive muscle relaxation. Our body needs oxygen to obtain energy to fuel all our living processes. Mechanism of Breathing ribs and sternum lowered rib sternum vertebral column Ribs swing down and decrease volume of thorax • The volume of your thoracic cavity decreases. 1 Respiratory Volumes and Capacities Tidal Volume (TV): Volume of air inspired or expired during a normal respiration. the time it takes the inspired concentration to match the set vaporiser output concentration. On the deepest working dives, at depths greater than 600 m, ambient pressure is greater than 6100 kPa and the divers breathe gas mixtures containing about 2% oxygen to avoid acute oxygen toxicity. For instance, it is. To him, the point is to use the training to build mental strength and slow the brain from grinding on other issues. ) If you deliver 1. Arm 3 (HBO): treated with oxygen at 2. An average adult has about 600 million alveoli, giving a total surface area of about 100m², so the area is huge. This is good because there is a duality to oxygen, and while it is certainly necessary for life, excessive amounts will prematurely accelerate oxidative damage and aging. Oxygen is the most important energy source for the cells. Severe substance use disorders are also known as addiction. are significantly smaller. Then hyperventilate by breathing deeply and forcefully at the rate of about 1 breath/4 sec for about 30 seconds. Marijuana use can lead to the development of a substance use disorder, a medical illness in which the person is unable to stop using even though it's causing health and social problems in their life. concentration, centering, control, breathing, precision, and flow," adds Rich. With different patterns of breathing, you can fall in love, you can hate someone, you can feel the whole spectrum of feelings just by changing your breathing. The exercise takes. Easy mounting of breathing system on both sides. Join a local yoga gym, and begin learning additional ways to control your breathing. I am amazed at how much there is to breathing,and how much people take it for granted. Breathing normally, you inspire about a half a liter per breath, and this increases significantly during exercise. Non-rebreathers feature a swivel connector for optimum patient comfort and uninterrupted oxygen flow. He then shows you how to increase your BOLT score by using light breathing exercises and learning how to simulate high-altitude training, a technique used by Navy SEALs and professional athletes to help increase endurance, weight loss, and vital red blood cells to dramatically improve cardio fitness. Pulmonary Tuberculosis: Used to be called consumption. The whole point is the virus particles not being able to get through in the first place, i wouldn't have to replace the filter until it's saturated, which when just breathing normal air, is a very long time (considering these things are made to be used in heavy particulate situations). 5 mg/3 mL 1 breathing treatment q6-8h #25 vials. , or New York City. 1% Prolonged exposure can affect powers of concentration 5000 ppm 0. 7 pounds per square inch (psi). When all of these aspects of the Noble Path — virtue, concentration, and discernment — are brought together fully mature within the heart, you gain insight into all aspects of the breath, knowing that "Breathing this way gives rise to good mental states; breathing that way gives rise to bad mental states. Sit somewhere comfortable, and close your eyes. No , Did you know that your regular breathing pattern is very likely screwing up your body and its functions in tons of different ways? Yup. obtained using the Moore's test rig and the TSI test rig are comparable, enabling the filters to be put into the same rank order [4]. 21 being the fraction of inspired oxygen (FiO2) of room air. When you feel these emotions in your body, focus on your breathing and force yourself to breathe at a normal, controlled rate. Pilates involves precise moves and specific breathing techniques. In this problem, you will explore how this works. To properly use Total Concentration Breathing, the user must concentrate, relax their upper half and brace their lower half before taking a long breath. Once you are full of breath, seal your left nostril with the ring finger of the same hand, keeping your right nostril closed, and hold the breath for a moment. Using breathing exercises to increase energy. Breathing exercises, or focusing on slow, regular and sometimes deep breathing, are helpful ways for you to manage stress and improve your health. A breathing gas is a mixture of gaseous chemical elements and compounds used for respiration. Most of the time, the air in the atmosphere contains the proper amount. Hinokami Kagura Breathing (ヒノカミ神楽 (かぐら) の呼 (こ) 吸 (きゅう) Hinokami Kagura no Kokyu) is a unique form of Total Concentration Breathing passed down for generations in the Kamado Family. Recipe developer and Gaiambassador, Madelana Escudero, shares some wellness tips on staying healthy, hydrated and feeling your absolute best leading up to the big day. While breathing deeply, you release too much CO2 from your lungs, leading to constriction of your blood vessels, and making you a bit light-headed. This is a prospective, comparative, blinded, and randomized clinical trial with 2. Then draw the graph. Serum is particularly suited to this task because it is made up of smaller molecules that can penetrate deeply into the skin and deliver a very high concentration of active. When this concentration rises during a bout of activity, for example, nerve impulses are automatically sent to the diaphragm and rib muscles that increase the rate and the depth of breathing. In a study involving 323 mouth-breathing patients who used supplemental oxygen, 34. The 2-to-1 Exercise. TEDx Talks Recommended for you. 26 atm (atmospheres). FRC is calculated from the total volume of N 2 exhaled and the change in alveolar N 2 from the beginning to the end of the test (Figure 4-2 and open-circuit method); (B) Closed-circuit equipment used for He dilution FRC determination includes a directional breathing circuit with a volume-based spirometer, He analyzer, CO 2 absorber, O 2 source. the start of the procedure, the concentration in lungsandspirometerat theend, andthevolume ofthegasesin thespirometerareknown. Mouth breathing and over breathing are closely linked to snoring, and sometimes sleep apnea. Did you know that you take between 18 and 26,000 breaths every day? You require 88 lbs. A 1130 series O 2 Conserver Testing System (Hans Rudolph Inc, Shawnee, KS, USA) was used to obtain oxygen pulse volumes, durations, and delays for both the 2. Brown and Dr. Does breathing contribute to CO2 buildup in the atmosphere? Posted on 26 September 2010 by climatesight. Parenting is an expensive business, especially in the early days when you need to. The mechanics of breathing will be different and might be more like our digestive tract, another system with an entry and exit. Essential:The breathing system must a) deliver the gases from the machine to the alveoli in the same concentration as set and in the shortest possible time; b) effectively eliminate carbon-dioxide; c) have minimal apparatus dead space; andd) have low resistance. The exercise takes. Characterizing Exhaled Airflow from Breathing and Talking About 22 countries were found to have TB infections with a total of 1. Pulmonary ventilation is the act of breathing, which can be described as the movement of air into and out of the lungs. Can be used for as long as the battery lasts, or for the recommended interval between calibrations, whichever is less. Feel the air fill your lungs, one section at a time, until your lungs are completely full and the air moves into your abdomen. This will help ground and settle kids. The main features are shown in this diagram. Normal ranges for oxygen saturation (SaO2 and SpO 2) and PO 2 (PaO 2) in the blood at sea level. When it comes to what is most important in affecting involuntary breathing, you can see that CO2 levels tend to affect breathing much more than oxygen levels. There are breathing techniques that accomplish the stated objective that opens up the lungs fully and maximises the oxygen in the lungs and bloodstream; but aside from giving a boost in clarity, maybe a bit of energy, and in some cases being used. The advanced B-CONTROL MICRO is more powerful and ready to communicate with the B-APP for remotely controlling and monitoring the compressor. To make up 2 gallons of the bleach solution, use 1½ cups of bleach. 4 kg, sum of 4 skinfolds = 54. Death may occur when the muscles controlling breathing become ineffective, causing asphyxiation. Regain Concentration and Energy. About breathingearth. Even with exercise induced asthma I feel like this is a better choice as the breathes for seem to fill the lungs better even at a faster pace. Adderall is the brand name of a prescription medication used to treat attention deficit hyperactivity disorder (ADHD) in children and adults. Using this formula, a diver breathing EAN 35 will have a maximum operating depth of 99 feet, or 4 atm, when using the recreational limit. It has a Workplace Exposure Limit of 5000ppm. The Buteyko breathing technique can improve symptoms and reduce bronchodilator use but does not appear to change bronchial responsiveness or lung function in patients with asthma. 13 Most Commonly Used Essential Oils and How to Use Them. The total volumes of the breathing circuits (internal volume of the anesthesia breathing circuit + internal volume of the breathing hoses) are 3. Mineral or silicone oil which occasionally are used for compressor lubrication do not fluoresce under ultraviolet light. What creates this vast difference is the Hashiras ability to master the Full Focus Breathing or the Total Concentration Breathing at all times. This force is used in breathing to hold the moist surfaces of the pleural membranes together. save hide report. Using your imagination, imagine each inbreath as coming from exact (180 0 ) opposite sides, simultaneously on a straight line into your body. 5 in another region downwind. Do exercises like diaphragmatic, pursed lips, alternate nostril, and equal breathing. Ventilation: Use local exhaust ventilation to remove airborne fly ash from work areas when feasible. 8 How can rapid deep breathing hyperventilation lead to a feeling of light from 3515 3515 at Louisiana State University. Position breathing bag as shown and slide opening in breathing bag over outside diameter of bag mount. The results of the investigation are shown in the table. By the day of the empty chair, April 19, the total Covid inpatient population at Michigan Medicine had been dropping for the fifth day running, from a high of 229 on April 15 to 193. Join a local yoga gym, and begin learning additional ways to control your breathing. While the concentration of oxygen in canned air is high (95 percent), the cost is even higher. Does breathing contribute to CO2 buildup in the atmosphere? Posted on 26 September 2010 by climatesight. The sig says: use 1 vial in nebulizer twice daily The patient will use 2 vials per day. Concentration, focus and discipline are key to mastering the Art of Breathing. At 130 m depth in the northern sector of the North Sea oil field, the ambient pressure is 1400 kPa, so the breathing mixture used contains 10% oxygen. As with breathing, the mindset is essential in the practice of Zen meditation. are significantly smaller. Your body responds to this slight decrease by producing two hormones: glucagon and epinephrine. Influenza epidemics were found to cause about 47,200 deaths each year in the United States Gupta, J. A body scan can help boost your awareness of the mind-body connection. Mind Body Soul 582,557 views. Total Concentration Thunder Breathing grants the user the ability to utilize the. Marijuana use can lead to the development of a substance use disorder, a medical illness in which the person is unable to stop using even though it's causing health and social problems in their life. Another benefit of Belly Breathing is to help runners get rid of side stitches. 8 N MP m −3. waking up a lot. When a pMDI was used, lung dose was unchanged or increased for the 50- and 155-mL and decreased for the 300-mL breathing pattern. " Coherent breathing, Ujjayi breathing, and Breathing with visualization. Work = Pressure x Volume. I couldnt find a regular mask and my thought was that he was mouth breathing so a canula wouldnt have done much good. A breathing gas is a mixture of gaseous chemical elements and compounds used for respiration. Download it once and read it on your Kindle device, PC, phones or tablets. Now i'm not an expert so i'd love to hear from one. When it comes to what is most important in affecting involuntary breathing, you can see that CO2 levels tend to affect breathing much more than oxygen levels. Anapanasati is the meditation breathing technique which was taught by the Buddha. The Buteyko breathing technique can improve symptoms and reduce bronchodilator use but does not appear to change bronchial responsiveness or lung function in patients with asthma. The breath is a vehicle for deepening concentration and revealing quiet sources of joy. Chlorine Dioxide is a greenish-yellow gas that readily dissolves in water. Then, hold your breath for as long as you can and record below. Miner's lung is a disease caused by breathing in dust in coal mines. Now do the exact same breathing technique as you did before, placing the cotton ball just beneath your nose. The various forms of deep relaxation do, however, share some similar qualities. Breathing to relax. In Exam Mode: All questions are shown and the results, answers and rationales (if any) will only be given after you’ve finished the quiz. NCLEX Exam: Respiratory System Disorders (60 Questions) Congratulations - you have completed NCLEX Exam: Respiratory System. Mind Body Soul 582,557 views. Sometimes called belly breathing or diaphragmatic breathing. The position you adopt when breathing before a freedive also impacts how much oxygen you have in your system. Spontaneous Breathing during General Anesthesia Prevents the Ventral Redistribution of Ventilation as Detected by Electrical Impedance Tomography: A Randomized Trial You will receive an email whenever this article is corrected, updated, or cited in the literature. Gas exchange is more difficult for fish than for mammals because the concentration of dissolved oxygen in water is less than 1%, compared to 20% in air. A basic Physiology 1 experiment to demonstrate this consisted of a student breathing recycled air filtered through lime water to remove the CO2. RESULTS: Switching from facial to tracheostomy delivery increased lung dose with nebulizer (all breathing patterns). Organ system, organism, organ. As with breathing, the mindset is essential in the practice of Zen meditation. The amount of He in the spirometer is known at the beginning of the test (concentration × volume = amount). Using this procedure, H2S meters are typically set to alarm at 10 ppm and action is taken to reduce exposure, usually by leaving the area or by using SCBA equipment when air concentrations go. To make up 2 gallons of the bleach solution, use 1½ cups of bleach. People with fever, cough and difficulty breathing should seek medical attention. it also helps to expand the lung in all directions. Work of Breathing. Record breathing rate after hyperventilation has been completed. Amphibians have evolved multiple ways of breathing. In survey mode, detects the total concentration of many organic gases and vapors. Optimal Breathing Kit and Turbooxygen. Breathing problems in children are a huge worry to parents. Keep reading to learn about more ways to treat aspiration pneumonia now. One such breathing exercise is the 4-7-8 method, also called the relaxing breath exercise. Concentration definition is - the act or process of concentrating : the state of being concentrated; especially : direction of attention to a single object. In serum, the total calcium concentration is approximately 8. Oxygen levels are especially important for people with chronic obstructive pulmonary disease, emphysema and chronic bronchitis. For the lowest and highest temperature treatments, the short term response of respiration rate to measurement at the three growth carbon dioxide concentrations was also determined. Natalia and Sergei Lapa Breathing is one of the most important and most “instant” of all the vital functions of the body and yet the understanding of it, let alone the correction and therapeutic use of it, in complementary medicine is far from sufficient. The theory behind using vitamin D is that it helps reduce inflammation, which is a key issue in COPD. It is not known if Toujeo is safe and effective in children under 6 years of age. Maximum Use Concentration (MUC) means the maximum atmospheric concentration of a hazardous substance from which an employee can be expected to be protected when wearing a respirator. Anapanasati, "mindfulness of breathing", or breath meditation is a core contemplative practice of Buddhism. Some amphibians retain gills for life. the time it takes the inspired concentration to match the set vaporiser output concentration. This, it is not the total mass of carbon within the carbon cycle and can change over time. 1% Prolonged exposure can affect powers of concentration 5000 ppm 0. The majority of people only breathe superficially, using only the top part of their lungs or one-sixth of the capacity of their lungs. Delivers high concentration of oxygen; Reservoir bag assures oxygen supply to meet variable breathing patterns and tidal volumes. 1 Answer How does Charle's law. I have addressed power breathing in many of my writings; please review Muscle Media's back issues or my books. So to perform haku breathing is to focus all of your mind and body, and throw everything into the exhalation. Oxygenation: the process of getting oxygen to blood and cells. The good news, is once you master this trick, you wont use any extra energy. Aerobic exercises like running, jogging, and dancing can also. When using a semi-closed system, the oxygen flow rate must exceed the patient's oxygen consumption. And I’ve got 15 different schools of total concentration breathing styles that will do just the job. Spontaneous Breathing during General Anesthesia Prevents the Ventral Redistribution of Ventilation as Detected by Electrical Impedance Tomography: A Randomized Trial You will receive an email whenever this article is corrected, updated, or cited in the literature. maximal breathing capacity: [ kah-pas´ĭ-te ] the power to hold, retain, or contain, or the ability to absorb; usually expressed numerically as the measure of such ability. In 1977 I moved to a high desert spiritual. If the total concentration of atmospheric particulates is low, particulate filter air-purifying respirators can provide protection for long periods without the need to replace the filter. The Price of an Oxygen Bar. Repeat a few times. Six Steps to Refresh. Finall y, I would like to review the use of the nasal cannula and the FiO 2 that it provides. This chart from NASA shows the relationship between O2 concentration, atmospheric pressure, and how comfortable a human will be breathing that atmosphere. Brown and Dr. The 2-to-1 breathing exercise is simple: You exhale for twice as long as you inhale on a single breath. Work to overcome airway resistance is usually very small, except during exercise or in athsmatics. To him, the point is to use the training to build mental strength and slow the brain from grinding on other issues. A cementless prosthesis allows porous ingrowth of bone. Belly breathing is a simple breathing exercise you can use for stress relief. Hinokami Kagura Breathing (ヒノカミ神楽 (かぐら) の呼 (こ) 吸 (きゅう) Hinokami Kagura no Kokyu) is a unique form of Total Concentration Breathing passed down for generations in the Kamado Family. The remaining fingers should be stretched, but relaxed. Albuterol concentration was determined via spectrophotometry (276 nm). Magali "Super app that helps me everyday Olivier Uses - Stress relief - Calm down anxiety attacks - Cardiac coherence - Sleep aid: fight insomnia by focusing on deep breathing - Relaxation - Meditation sessions and sophrology exercises - Concentration improvement Features - Breathing rate between 1 and 15 cycles/min, adapted to a wide range of. , manganese compounds, tellurium compounds. It is an oxidizing agent, able to transfer oxygen to a variety of substrates, while gaining one or more electrons via oxidation-reduction (). Your total attention should be given to your breathing. Recipe developer and Gaiambassador, Madelana Escudero, shares some wellness tips on staying healthy, hydrated and feeling your absolute best leading up to the big day. Gas exchange is the intake of oxygen and the excretion of carbon dioxide at the lung surface. PM stands for particulate matter (also called particle pollution): the term for a mixture of solid particles and liquid droplets found in the air. This breathing exercise may help to clear the mind, relax the body, and improve focus. They help relieve breathing and moisturize the mucous membrane. When the abdominal wall excursion during inspiration, expiration, or both do not maintain optimum ventilation for the individual, the nursing diagnosis Ineffective Breathing Pattern is one of the issues nurses need to focus on. Total Petroleum Hydrocarbons (TPH) is a term used to describe a broad family of several hundred chemical compounds that originally come from crude oil. It acts in much the same way as organophosphate insecticides, block nerve endings from allowing muscles to stop contracting. End-expiratory apnea will remain a few seconds to start new inspiration cycle. The Power of Deep Breathing Level: Middle School (6-8) Timeframe: 30 minutes Concepts: • Attention • Breath • Coping Skills To control the breathing is to control the mind. The Facts About Ammonia General Information. Using the thumb of your dominant hand, block your right nostril and inhale through your left nostril only. Expand this section. They become "shallow breathers," using only a small portion of their lungs' capacity. Your body responds to this slight decrease by producing two hormones: glucagon and epinephrine. This leads to reduced cell oxygenation: the driving force of all chronic diseases. Belly breathing is a simple breathing exercise you can use for stress relief. from the breathing of oneself or others, possibly unlocking abilities related to the affinity and enhancing existing powers. Thecalculation assumesanearly even mixture of nitrogen through the lung-spirometer closed circuit at the end of the re-breathing period. HAP offers an option to select a "space usage" type, based on the list of space usage types found in Table 6-1 in the standard. Do you use the Total Concentration Breathing technique nonstop?" Maybe you wont believe me, but It's literally the technique, I keep my upper abs flexed around the clock all day, morning, noon and night while asleep, and while I sit, walk and talk, while I play, read and eat and that for the past 9 years non-stop (except few days when i get sick). During quiet breathing, the diaphragm and external intercostal muscles work at different extents, depending on the situation. HOW BREATHING PROPERLY CAN CHANGE YOUR LIFE - Steve Maxwell on London Real. In this relaxation technique, you use both visual imagery and body awareness to reduce stress. However, peat surface levels - and thus soil depths - change (“bog breathing”) largely as a response to peat shrinkage and expansion in relation to water table changes. Mark the time on your watch when you start. Concentration (using breathing techniques to "reset" is important for staying focused or we call it tennis meditation) Breathing is what we call an involuntary action, which means we do it without having to think about it or worry about it until it is compromised, then we freak out and so we should!.
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# The CDF of determinant of wishart distribution
Assuming $$b>0$$, $$mathbf{A} in mathbb{C}^{mtimes n}$$ with $$mleq n$$ and each element of $$mathbf{A}$$ is i.i.d. $$mathcal{CN}(0,1)$$ distributed, how to obtain $$mathbb{P} left( detleft( mathbf{A}mathbf{A}^H right) < b right)$$?
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The Carothers solution for a wedge loaded by a concentrated couple at its vertex is known to be valid for the wedge angles $2α<2α*≈257 deg$ only. Moreover, for $π<2α<2α*$ it exists for antisymmetric loading only. The more realistic model of the concentrated couple of the arbitrary orientation is examined by the approach of Dundurs-Markenscoff. It is shown that the Carothers type solution holds for the edge angles $2α<π.$[S0021-8936(00)00402-5]
1.
Uflyand, Y. S., 1967, The Integral Transforms in the Problems of Theory of Elasticity, Gostechizdat, Leningrad (in Russian).
2.
Markenscoff
,
X.
,
1994
, “
Some Remarks on the Wedge Paradox and Saint-Venant’s Principle
,”
ASME J. Appl. Mech.
,
61
, pp.
519
523
.
3.
Markenscoff
,
X.
, and
Paukshto
,
M.
,
1998
, “
The Wedge Paradox and a Correspondence Principle
,”
Proc. R. Soc. London, Ser. A
,
454
, pp.
147
154
.
4.
Markenscoff
,
X.
, and
Paukshto
,
M.
,
1995
, “
The Correspondence Between Cavities and Rigid Inclusions in Three-Dimensional Elasticity and the Cosserat Spectrum
,”
Int. J. Solids Struct.
,
32
, pp.
431
438
.
5.
Sternberg
,
E.
, and
Koiter
,
V.
,
1958
, “
The Wedge Under a Concentrated Couple: A Paradox in the Two-Dimensional Theory of Elasticity
,”
ASME J. Appl. Mech.
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581
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6.
Belonosov, S. M., 1962, The Basic Plane Static Problems of Theory of Elasticity for Simply Connected and Double Connected Domains, Siberia Department AS USSR., Novosibirsk, Russia (in Russian).
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Mathematics
OpenStudy (anonymous):
what would the domain of x/2-x be?
OpenStudy (anonymous):
did you mean, x/(2-x) or (x/2) - x
OpenStudy (anonymous):
is this $\frac{x}{x-2}$? if so all real numbers except 2
OpenStudy (anonymous):
because you still cannot divide by zero
OpenStudy (anonymous):
oh so do you only look at the denominator to figure out the domain?
OpenStudy (anonymous):
for this one.... $k(x)=\sqrt{x/2-x}$
OpenStudy (anonymous):
Domain really only considers two major elements: denominators of fractions cannot be zero (divide by 0 problem), and even radicals (such as square roots, 4th roots, etc) cannot be taken of negative values. For the problem of x/(2-x), the answer of all reals except 2 is correct. For the radical problem, you must also consider that the radicand must be greater than or equal to zero, or solve this inequality: $x/2 - x \ge 0$
OpenStudy (anonymous):
$x \le -2$
OpenStudy (anonymous):
Eliza27: Not correct for the radical problem proposed by desireeee11. Try x = 0 in the radical expression...it will not cause a domain violation. (k(0) = 0) Perhaps an algebra error?
OpenStudy (anonymous):
yes you only look at the denominator if you have a rational function
OpenStudy (anonymous):
Oh yeah I though the 0 was a 1 thanks
OpenStudy (anonymous):
if you have a radical inside must $\geq 0$
OpenStudy (anonymous):
OpenStudy (anonymous):
but if it is really $k(x)=\frac{x}{x-2}$\] then solution is more complicated
OpenStudy (anonymous):
i mean $k(x)=\sqrt{\frac{x}{x-2}}$
OpenStudy (anonymous):
then you have to make sure that $\frac{x}{x-2}\geq 0$
OpenStudy (anonymous):
solution would be $(\infty,0]\cup (2,\infty)$
OpenStudy (anonymous):
thank you
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# Fullscreen functionality
This topic is 1824 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hey guys,
I am using C++ with the Allegro 5 library for my project, and I have come up against something that I cant find the solution to, it seems easy so perhaps someone could tell me the answer.
At the moment I am creating a display matching my screen resolution, 1920 x 1080, which gives me an illusion of fullscreen - but it is not true fullscreen as the various windows bars still overlap it.
My question is this: how can I enable proper fullscreen functionality?
And a follow up, for computers that wont support 1920 x 1080, can I force the program, when it runs, to choose the highest supported resolution instead?
Cheers,
Toshi
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When you're calling al_set_new_display_flags (assuming you are), are you specifying the ALLEGRO_FULLSCREEN flag?
i.e.
al_set_new_display_flags(someFlags | ALLEGRO_FULLSCREEN);
Where someFlags is whatever flags you want other than fullscreen, such as ALLEGRO_OPENGL, ALLEGRO_DIRECT3D, or ALLEGRO_OPENGL_3_0.
EDIT:
I found this question on allegro 5 and creating a fullscreen resolution, which has the same resolution as the user's monitor, which you might be interested in.
Edited by pinebanana
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Thanks pinebanana, I had a cursory look through the documentation but my search-fu failed me...
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# Products classification by name
I am a beginner with machine learning, and I'm trying to build a model to classify products by category according to the words present in the product name. My goal is to predict the category of some new product, just by observing the categories for existing products.
For example, having the following products:
PRODUCT CATEGORY
soap bar johnsons green leaves bath
strawberry soap soft bath
spoon hercules medium kitchen
soap dish plastic medium bath
[...]
My first thought is to group the words (tokens) present in each product, indicating the designated category and the occurrences count (to be used as a weight). So, for this sample, I have:
WORD CATEGORY COUNT
soap bath 3
medium bath 1
medium kitchen 1
bar bath 1
johnsons bath 1
Having this, I could be able to train a model, and use it to classify a new product.
For example, having a new product hands liquid soap 120oz, it could be classified as bath, because it contains the word soap, which have a strong weight for the bath category.
In other case, the new product medium hammer could be classified as bath or kitchen , according the occurrence of the word medium in the training set.
So, my doubts are:
• Am I going to the correct approach?
• What is the best algorithm to be used in this case?
• How can I apply this using Weka?
• Do you know anything about zero-shot learning and word embedding algorithms? – Alireza Zolanvari Mar 15 '19 at 7:59
• @alirezazolanvari no, but I'll search for. – elias Mar 15 '19 at 13:25
• @elias how many CATEGORY do you have ? – mujjiga Mar 17 '19 at 20:52
• @mujjiga about 3.000 – elias Mar 18 '19 at 16:43
• @elias do you mean 3 or 3000 ? :) – mujjiga Mar 18 '19 at 17:51
I think, and have done similar problem too, that this problem can be solved in this way:
1. Generate NGrams
2. Create 1 hot encoding matrix
3. Pass to Naive Bayes or Random forest
It would automatically count the words count (you can apply TFIDF too) and based on that weightage will be calculated.
Examples:
If you have enough data and reasonable number of classes, you can definitely train your model. The grouping of words that you have done is similar to an approach called bag-of-words model. You can use that to build a classifier using Naive Bayes or SVM etc. On a different note, you can also look at the KNN algorithm because it looks fit for your use case. You can have a look at this paper
You can also try Tfidf Vectorizer from Sklearn which would be helpful in your case, As Tfidf vectorization inherently is able to learn and differentiate between the frequently occurring words and rarely occurring words by calculating the product of term frequency and inverse document frequency. Check here for more details. On top of this featurization, You can try Naive Bayes as it's pretty fast and seems to work well for text data as it uses with conditional probabilities. Use performance metrics such as confusion matrix to get a better sense of what is happening as accuracy is not a good measure when your data is imbalance. Hope it helps
This should be doable with pre-trained word vectors + document/sentence vectors. Tutorial : https://medium.com/scaleabout/a-gentle-introduction-to-doc2vec-db3e8c0cce5e
All Product labels with "similar meanings" should cluster in a short distance.
After product name ha been transformed into a vector, vector can be fed into a logistic regression classifier (Or a shallow neural network).
The steps are the following:
1. Prepare your dataset. Put everything in a dataframe. Divide it in train and test (or even train, cv and test). Use of the order of 10k samples for the test set, or 10-20%, whatever is smaller. Consider using https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_test_split.html
2. Encode your input. You can convert the input, which is a string, to a bag of words, or to a TFIDF. Consider using https://scikit-learn.org/stable/modules/generated/sklearn.feature_extraction.text.TfidfVectorizer.html.
3. Instantiate and train your model. You can use for example a simple logistic regression model, for example https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LogisticRegression.html.
4. Test the performance of the model. Use the test set to understand how well your model is doing. You can use for example the accuracy (https://scikit-learn.org/stable/modules/generated/sklearn.metrics.accuracy_score.html) or the precision and recall for each class (https://scikit-learn.org/stable/modules/generated/sklearn.metrics.precision_score.html and https://scikit-learn.org/stable/modules/generated/sklearn.metrics.recall_score.html). Understand well what they mean (https://en.wikipedia.org/wiki/Precision_and_recall).
Your pseudocode should be something like:
-> Divide train and test set. Output: X_train, y_train, X_test, y_test
-> Instantiate tfidf and the desired model (e.g. logistic regression).
-> Fit tfidf with X_train (e.g. use .fit_transform) and get the X_train_transformed
-> Fit the model (e.g. using .fit) with X_train_transformed and y_train
-> Use X_test to get a prediction y_pred of the model (first pass it through tfidf and then through the model object, e.g. using .predict)
-> Use y_pred and y_test to get some metrics to understand the performance of the model.
Hope this works for you.
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# Derivative of a particular function
Under what conditions is the derivative of $$f(z) = (z-(x_1+iy_1))(z-(x_2+iy_2))(z-(x_3+iy_3))$$ equal to $3(z^2-13)$ where $i$ is the imaginary number? When I put the equation in Wolfram it's a huge mess and I am wondering if there's an easier simplification or mathematical point to keep in mind.
-
what are you differentiating with respect to? $z$? What are the $x$s and $y$s? – Robert Mastragostino May 26 '12 at 23:39
Yes, with respect to z. Here is one sample solution: tinyurl.com/d9bm6m3 and the x's and y's are just integer values (cartesian coordinates) – John Smith May 26 '12 at 23:40
I would suggest to denote $z_k=x_k+iy_k$, $k=1,2,3$ to make it more concise. Then $$f\left(z\right)=\left(z-z_{1}\right)\left(z-z_{2}\right)\left(z-z_{3}\right)$$ $$f'\left(z\right)=\left(z-z_{2}\right)\left(z-z_{3}\right)+\left(z-z_{1}\right)\left(z-z_{3}\right)+\left(z-z_{1}\right)\left(z-z_{2}\right)$$ $$3z^{2}-2z\left(z_{1}+z_{2}+z_{3}\right)+z_{2}z_{3}+z_{1}z_{3}+z_{1}z_{2}=3z^{2}-39$$ Now comparing coefficients on both sides and separating real and imaginary parts we obtain: $$x_{1}+x_{2}+x_{3}=0$$ $$y_{1}+y_{2}+y_{3}=0$$ $$x_{2}x_{3}-y_{2}y_{3}+x_{1}x_{3}-y_{1}y_{3}+x_{1}x_{2}-y_{1}y_{2}=-39$$ $$x_{2}y_{3}+x_{3}y_{2}+x_{1}y_{3}+x_{3}y_{1}+x_{1}y_{2}+x_{2}y_{1}=0$$ Which leaves $x_3$, $y_3$ arbitrary
For the derivative to be $3z^2-39$, your $f$ will have to be $f(z) = z^3-39z+c$ where $c$ is an arbitrary complex constant. Solve this cubic equation to get your $x_j+iy_j$, but it will be messy.
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# Recent developments in the proof of fermat's last theorem
It's been 20 years since fermat's last theorem was proved by Andrew Wiles.
Has there been any simplification in proof in the last 20 years?
What I do only know is that different proofs of faltings's theorem were given by Vojta and Bombieri.
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InfoQ Homepage News ELIoT: Distributed Programming for the Internet of Things
# ELIoT: Distributed Programming for the Internet of Things
ELIoT (Extensible Language for the Internet of Things) is a simple and small programming language aiming to make distributed programming easier. A program in ELIoT may appear as a sigle program, but it actually runs on different computers, so, e.g., a variable or function declared on one computer is transparently used on another.
Among ELIoT goals are enabling configuration and control of swarms of small devices. As an example, in less than 20 lines of code, you can have two temperature sensors talk one another and, in case the difference in temperature between them is greater than a given threshold, inform a remote controlling application. This is expressed by the following single code snippet:
invoke "sensor2.corp.net",
every 1.1s,
send_temps sensor1_temp, temperature
send_temps T1:real, T2:real ->
if abs(T1-T2) > 2.0 then
show_temps T1, T2
show_temps T1:real, T2:real ->
write "Temperature on sensor1 is ", T1, " and on sensor2 ", T2, ". "
if T1>T2 then
writeln "Sensor1 is hotter by ", T1-T2, " degrees"
else
writeln "Sensor2 is hotter by ", T2-T1, " degrees"
At the heart of ELIoT are a few primitives, such as:
• tell: it sends a program asynchronously
• ask: it sends a program synchronously and waits for the result
• invoke: it sends a program and opens a bi-directional channel.
InfoQ has talked with ELIoT’s creator, Christophe de Dinechin.
Could you explain what led you to create ELIoT? What was the need to create a new language for distributed programming?
ELIoT is the convergence of three ideas:
1. A little over a decade of research in language design, with XL and its derivative Tao3D. The objective of my research is to create a language that is flexible enough to adapt to practically any usage scenario. To validate those ideas, I purposely selected use cases where other programming languages do not fare well, like describing interactive 3D documents (Tao3D) or distributed systems (ELIoT). I also validated the extensibility postulate by implementing in a library most of what other programming languages implement in the compiler/interpreter (e.g. loops, tests, arithmetic, optimizations etc). So the first key idea is “an extensible programming language”.
2. Discussions with engineers in the embedded space, notably from Intel, who gave me the idea of a language with a minimal footprint, but capable of dealing easily with a multiplicity of sensors or actuators. In embedded systems, languages like Lua are quite useful. But the value of small systems lies beyond the computations they can do locally. For example, an Apple Watch requires an iPhone to be useful, and most iPhone applications derive value from cloud-based servers. Hence the idea in ELIoT to exchange tiny program fragments very easily. And curiously, not that many languages do that well. So the second idea is “distributed programming with an eye on embedded systems”.
3. Discussions with former colleagues at HP who are designing something called The Machine, based on memristors (fully persistent memory). They are designing a clean-sheet operating system called Carbon. I know nothing about Carbon, but that got me thinking about what I would like to see in a clean-sheet OS. A recurring theme was: “how do you program that thing?”. Machines today are very different from back when Unix was designed, with heterogeneous CPU/GPUs/XPUs. Communication and storage became as important as computations. So I started dreaming about what I’d like in a system language for a machine made of 10000 CPUs each with a different architecture. And here again, I saw the same recurring theme of being able to send computations along with their data (“compute along the way”). So the third idea is “exchanging programs and data transparently, at the language level”.
In summary:
• An extensible programming language
• for distributed programming (with a focus on embedded systems)
• by exchanging data and programs transparently at the language level.
What problem does ELIoT specifically try to solve?
Let’s be honest: I’m first and foremost doing it for fun and personal development. If it happens to be useful, I’m thrilled, but in that case, the journey is really its own reward. And by that metric, ELIoT is getting closer and closer to my ideal programming language. For example, I’m relatively happy with ELIoT’s definition of complex numbers. In my biased opinion, it’s really short and to the point, and I’m pretty convinced it can someday be optimized to really good machine code. Of note, being designed to please me, myself and I makes ELIoT very “different”, and some people just hate that. It certainly does not look like C or Java and never will…
But with respect to what I saw as an unfulfilled need, it’s really: how do you control a small fleet of sensors and actuators easily, not knowing ahead of time what kind of data you are going to ask or what actions you are going to require. That led to the desire for a small, lightweight language applicable to embedded systems, that could be extended easily, as well as be used as an extension language for applications, and that would make distributed programming really easy. Several languages had one or the other property. None I know has them all, at least to my satisfaction.
Languages that had an influence on the design include Lisp (programs = data), Lua (small footprint), Python (as an extension language), Erlang (communication between processes), Pure (pattern matching), BASIC (no stinking parentheses in the syntax), and dozens of others, e.g. Ada, Hop, Haskell, C/C++, Java, Occam, XC to cite just a few. ELIoT tries to borrow what I see as the most salient strengths from each of these languages.
Could you explain how ELIoT works its “magics”?
ELIoT is a fully homoiconic language, i.e. programs and data are the same thing. All programs and all data in ELIoT (and its ancestor XL) are represented by a single data structure, the “tree” (an abstract syntax tree, aka parse tree). A recent redesign of the language compiler/interpreter ensured that all non-transient interpreter data structures were themselves represented using the same trees. That includes the symbol table, i.e. the way to tell that the value of variable ‘X’ is 28 in the current context.
Once you have built a language with these properties (and that’s the big effort, I don’t think it can ever happen for C or C++ for example), then the “magic” becomes very simple: you simply send over the wire a serialized format representing the parse tree for your program, combined with the parse tree representing its symbol table. The other side receives that data, reconstructs a symbol table and a program parse tree from it, and executes the program in the transmitted context.
There are a few details that matter, e.g. filtering which symbols are sent over the wire. In the current implementation, we stop at the source file level, i.e. you don’t send anything from other files. Built-in definitions and imported modules are assumed to also exist on the other side, possibly with a different implementation.
Finally, what do you envision for ELIoT’s future?
ELIoT comes with great ambitions. But I also made the deliberate decision to involve the open-source community long before it’s complete, while it’s still tiny and weak. I want this thing to have as many contributors as possible. In other words, I don’t have a preconceived notion of where it’s going. Hopefully, the community will take it places I would never have thought of. I’d be happy to have it taking pictures of Pluto one day ;-)
According to Slashdot and other people who contacted me, there is a big gap in security. And granted, the security model is not defined yet, mostly because a person I was discussing with at Intel is a big security guru and I wanted to discuss various ideas with him first. In any case, security is probably what I will focus on next. But there are a few other open issues for those who want to help.
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Acid catalysed hydration of allene
In the acid catalysed hydration of allene, acetone is obtained as the major product. This is because of electrophilic addition of $\ce{H+}$ across a double bond to give a vinyl carbocation, and subsequent addition of water and then tautomerism to yield the ketone. What I'm confused about is that why does not an allyl carbocation form, which would be resonance stabilised?(leading to allyl alcohol?)
• – Mithoron May 20 '15 at 19:27
Good question! As the following figure illustrates, there are 2 ways we can protonate an allene.
The top line in the drawing starts by protonating at the central allene carbon. This generates a very unstable primary carbocation. Initially, this primary carbocation is not stabilized by the adjacent double bond. It is not until after this carbocation rotates 90° that its p orbital is lined up with the p orbitals in the double bond and allylic stabilization can occur. Since there is no immediate allylic stabilization when the carbocation is formed, and since it is a primary carbocation, the pathway leading to it is very high energy.
On the other hand, protonation at the terminal allene carbon (bottom line of figure) produces a relatively stabilized secondary carbocation. Formation of a secondary carbocation is a much lower energy process. Therefore, protonation at the terminal allene carbon is lower energy and is what will occur.
Edit: Response to OP's comment
I always thought that a positive charge appearing on sp3 carbon (which then becomes sp2) is more stable than a positive charge appearing on sp2 carbon
Generally we think of vinylic carbocations as high energy. That is true, but what is meant by "vinylic" carbocation is different from what we have here. Here there is an empty p orbital. In a vinylic carbocation the empty orbital is not a p orbital, but rather something like $\ce{sp^2}$ or $\ce{sp}$ hybridized.
is the secondary carbocation more stable because of inductive effect, the rotational barrier in the primary carbocation, or both
The secondary carbocation is more stable for the same reasons that we have the tertiary > secondary > primary carbocation stability order.
• Inductive effects, $\ce{sp^3}$ and $\ce{sp^2}$ carbons are electron releasing towards an $\ce{sp}$ carbon due to electronegativity differences
• Resonance effects from hyperconjugation involving hydrogens on the carbon next to the carbocation center.
• OMG!! I had accepted the statement made by OP when I was told about this nature in allenes (long back). This is a wonderful explanation. +1000! – user223679 May 20 '15 at 17:26
• @ron I always thought that a positive charge appearing on sp3 carbon(which then becomes sp2) is more stable than a positive charge appearing on sp2 carbon(which then becomes sp)(even though sp3 carbon may be primary.). In this case, is the secondary carbocation more stable because of inductive effect, the rotational barrier in the primary carbocation, or both? – Abhishek May 20 '15 at 17:41
• @Abhishek I've edited my answer to address these points. – ron May 20 '15 at 17:57
• @ron Understood it all perfectly well now, thanks – Abhishek May 20 '15 at 18:18
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1. May 16, 2006
### jet10
integral identity
if we have $$\int dt f(t) = \int dt g(t)$$ where both integrals are indefinite integrals, can we immediately conclude that f(t) = g(t) ? I know this doesn't work with definite integrals.
Last edited: May 16, 2006
2. May 16, 2006
### vsage
If the two integrals are equivalent, then this implies to me at every t the shapes of f(t) and g(t) are equivalent. It does not work with definite integrals because it's entirely possible for two functions to have the same integral over a certain interval but have entirely different shapes.
3. May 16, 2006
### CarlB
Write it out in complete for example:
$$\int^xdt\;f(t) = \int^x dt\;g(t).$$
In other words, write the indefinite integrals as definite integrals. Now apply the fundamental theorem of calculus and you will find out that yes, indeed, f = g.
Carl
4. May 18, 2006
thanks! carl
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# Show that for a.e. $x\in[0,1]$, $\sum_{n=1}^\infty\sum_{k=1}^n\dfrac{1}{n^\gamma}\dfrac{1}{\sqrt{|x-\frac{k}{2^n}|}}$ converges.
One point to add, $$\gamma>2$$. My approach is to show its complement, i.e $$x$$ where the series is divergent, is measure zero by Borel-Cantelli lemma. Let $$A_{j,m}:=\{\sum_{n=1}^m\sum_{k=1}^n\dfrac{1}{n^\gamma}\dfrac{1}{\sqrt{|x-\frac{k}{2^n}|}}\geq j\}$$. The complement now becomes, $$\cap_{j\geq1}\cap_{m\geq N_j}A_{j,m}.$$ It is not the form where Borel-Cantelli is applicable.
Thank you!
$$\int_0^1 \sum_{n=1}^\infty\sum_{k=1}^n \frac{1}{n^\gamma}\frac{1}{\sqrt{|x-\frac{k}{2^n}|}}dx = \sum_{n=1}^\infty\frac{1}{n^\gamma}\sum_{k=1}^n \int_0^1 \frac{1}{\sqrt{|x-\frac{k}{2^n}|}}dx = \sum_{n=1}^\infty \frac{1}{n^\gamma} \sum_{k=1}^n 2\frac{k}{2^n}[\sqrt{\frac{2^{2n}}{k^2}-\frac{2^n}{k}}+\sqrt{\frac{2^n}{k}}] \le \sum_{n=1}^\infty \frac{1}{n^\gamma}\sum_{k=1}^n 2\frac{k}{2^n}2\frac{2^n}{k} = \sum_{n=1}^\infty \frac{4}{n^{\gamma-1}} < \infty.$$
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# Honeybee pupal length assessed by CT-scan technique: effects of Varroa infestation, developmental stage and spatial position within the brood comb
## Abstract
Honeybee pupae morphology can be affected by a number of stressor, but in vivo investigation is difficult. A computed tomography (CT) technique was applied to visualize a comb’s inner structure without damaging the brood. The CT scan was performed on a brood comb containing pupae developed from eggs laid by the queen during a time window of 48 hours. From the CT images, the position of each pupa was determined by recording coordinates to a common reference point. Afterwards, every brood cell was inspected in order to assess the developmental stage of the pupa, the presence of Varroa destructor, the number and progeny of foundress mites. Using data on 651 pupae, the relationships between varroa infestation status, developmental stage and spatial position of the pupa within the brood comb, and its length were investigated. Pupae at 8 post-capping days were shorter than pupae at 7 post-capping days. Pupae in infected cells were significantly shorter than those in varroa-free cells and this effect was linked both to mite number and stage and to the position in the comb. Overall, the results suggest that the CT-scan may represent a suitable non-invasive tool to investigate the morphology and developing status of honeybee brood.
## Introduction
In recent years, honeybee colony losses have been recorded throughout Europe and the World1,2,3. While a multitude of causative factors for this phenomenon have been extensively debated, now infestation with the invasive ectoparasitic mite Varroa destructor is considered one of the most significant causes for colony losses4. The mites depend on honey bee brood for reproduction, and the reproductive cycles of host and parasite are tightly linked to each other5. Within the isolated and protected environment of a capped cell, the reproducing mites and their offspring feed on the developing honey bee pupae. While the native host Apis cerana has evolved a multitude of behavioral adaptations to limit the damage inflicted by the parasite, heavy mite infestation in colonies of A. mellifera causes severe damage, typically associated with secondary virus infections and a complex of symptoms known as varroosis, and will eventually lead to colony collapse. At honeybee individual level, it was reported that varroa infestation causes weight loss and reduced life span6,7,8,9. Moreover, it was reported that multiple infestation of mites in one cell can cause shrinkage of the bee abdomen and increase the risk of developing deformed wings10.
The alteration of honeybee pupae morphology including size and length can be considered of value to assess the negative effects of mite infestation of the colony6,7,8,9. Current methods for varroa load assessment in the brood, as for instance opening a random sample of capped brood cells (n = 200) and measuring the percentage of infested cells, are invasive, partially or totally destructive and time consuming11. For research purpose it is important to develop innovative and non-invasive methods to assess the brood mite infestation degree of a colony. Among the currently available imaging diagnostic techniques, computed tomography (CT) imaging technique employs x-rays to produce cross-sectional images (slices) of a scanned object, allowing the visualization of its inner structures without inherent damages to live tissues and materials. In particular, µCT is commonly employed for the 3D visualization of inner structures on a small scale, i.e. for morphological investigation of invertebrates12. Benchtop µCT systems provide high penetrating power and high resolution images, but scanning typically takes some hours to be completed and suffers for sample size limitations12. On the other hand, medical CT devices are optimized for qualitative viewing of larger organisms and objects, providing much lower resolution but also less harmful radiation and reduced scanning time compared to µCT13. In this study, medical CT and image analysis approach coupled with brood manual inspection was used to clarify the relationship between Varroa destructor infestation status and pupa length, taking into consideration other factors such as the spatial position of the pupa within the comb and its developmental stage. Also, the distribution of infected cells throughout the brood area of the comb was investigated.
## Results
Figure 1 shows the development from larval (Day 10) to pupal (Day 17) stage of the brood cells from five randomly selected sections of the comb. In total, despite the medical CT radiation dose applied on day 10, 105 out of 107 cells (98,1%) correctly molted into pupae as expected following the normal development pattern of honeybees14.
A total of 2466 pupae were inspected for presence of varroa mite in their cells and the corresponding lengths were measured from the CT images. One-hundred two out of 2466 cells were infested by the mite, corresponding to a 4.1% total true brood infestation of the analyzed comb. Figure 2 summarizes the results from χ2 test by presenting the observed and expected frequencies of varroa mites in a contingency table. The association between presence and absence of varroa and the position of the cells in the twelve sections was statistically significant (χ2 = 75.41, DF = 11, P < 0.001). Moreover, considering the distribution of the presence of varroa within each section, the two central ones showed more varroa mites than expected (section 6: χ2 = 39.95, DF = 1, P < 0.001; section 7: χ2 = 4.49, DF = 1, P = 0.03). Besides, less mites than expected were observed in sections 8 (χ2 = 5.30, DF = 1, P = 0.02), 9 (χ2 = 4.04, DF = 1, P = 0.04), and 10 (χ2 = 7.85, DF = 1, P < 0.01).
The two central sections contained 651 cells whose 58 were parasitized resulting in a partial brood infestation of 8.9%. This value of brood infestation was higher compared to total brood infestation rate reported above (4.1%).
Results from each of the three statistical models showed that the stage of the pupae, the position in the brood area (i.e. the two central squares analyzed) as well as varroa mites had significant effects on the length of the pupae (P < 0.001). Each model showed that pupae at stage 8 were significantly shorter than pupae at stage 7. Statistically significant difference was also found between the length of pupae in square 6 and square 7. The pupae analyzed in square 6 were longer than pupae in square 7; this result could be explained by the fact that square 6 was facing the entrance of the hive, which was orientated to South and probably exposed to higher temperatures.
Table 1 reports the Least Square (LS) means of the length of the bee pupae estimated with each of the three models considering the variable varroa (V) in three separated categories: presence/absence of the mite, number of foundress mites and total number of mites found within the cell. LS means from the first model showed that the presence of varroa mite significantly affected the length of the pupal stage by a reduction of 0.35 mm (from 10.54 mm to 10.19 mm) which represents approx. the 3% of the average varroa mite free pupa length in our sample. In the second model the effect of varroa was considered as the number of foundress mites found in the analyzed cells. LS means for the length of pupae hosting one foundress mite was 10.20 mm and was significantly shorter compared to varroa free pupae (10.54 mm). The length of pupae parasitized by two or more foundress mites was 10.08 mm and significantly shorter than varroa-free pupae, but not significantly shorter than pupae with one foundress mite. Results from the third fitted model showed that the length of the pupae was significantly shortened also by the presence of more than three individuals within the same cell.
## Discussion
The CT technology is increasingly used in scientific research about insects, and particularly the µCt scan and the 3D Phase-contrast X-ray computed tomography have been performed for anatomical studies and for the analysis of internal pathogens of honeybee individuals12,15,16. In this study, the length of developing pupae within intact brood using medical CT-scan technology was carried out. This would be a relevant new tool to allow morphological measurements of honey bee’s developing stages without uncapping the cells during in vivo studies. Pupa is the developmental stage of honeybee during which the insect is referred as quiescent and still. For this reason, we exclude that movement of the individuals are a potential source of artifacts in the CT images. Previous published observations carried out under laboratory conditions, confirmed that in the period of time between the prepupal ecdysis and the pupal ecdysis, the insects lay still on their back17. Moreover, the applied radiation dose did not seem to affect the normal development of brood from larval to pupal stage (Fig. 1c). The spatial distribution of V. destructor in the studied comb showed that varroa mites preferentially invaded cells in the inner brood area rather than infesting evenly the brood cells. This could suggest a preference of varroa mites for central brood areas, where temperatures are known to be kept slightly higher and more constant by worker honeybees compared to the periphery of the combs, even if different results are reported for varroa mites in tropical environment, where the development of the parasite seems to occur at a lower temperature compared to that in the brood18,19. The findings about higher infestation rate of the central sections of the comb also confirmed the importance of random sampling of manually inspected cells during brood mite monitoring.
Our results showed that the length of the pupae was influenced by the developmental stage, by the position within the brood comb area and by the parasite load. The length of pupae at stage 7 and stage 8 (post-capping days) was negatively affected by the presence of the mite, and became shorter the more mite individuals were present in a cell. Such an inverse relationship between the length of the pupa and the number of affecting mites could be linked to the nutritional behavior of the parasites on the developing honeybees. Indeed, varroa mites during their reproductive stage within the brood cell pierce the cuticle and feed on the developing honey bee5. Considering that the size of the pupae can be correlated with its weight, the above results agree with previous studies on the effect of parasitization on the weight of honeybees at their emergence6,7,9.
From the perspective point of view, our study suggests that CT-imaging could become a fast and non-destructive approach to explore the developing status of the honey bee brood stages. Medical CT-scan cost is clearly lower compared to micro-CT scan and has fallen significantly over the past few years. Besides, medical CT-scan application is increasing not only in clinical settings but also in animal production and industrial systems13. It is also worth remembering that unlike what happens in the current clinical practice, for honeybee colonies the medical CT scanner could host simultaneously up to 36 combs/scan, thus allowing the monitoring of several colonies by one scan.
## Methods
The experiment was carried out in June 2018 at the Faculty of Veterinary Medicine, University of Milano, Via dell’Università n. 6, Lodi, Italy. Pupae from one brood comb were analyzed. The brood comb belonged to a honeybee colony in good health status and headed by naturally mated queen. At the beginning of the experiment (Day 0), the queen was caged on an empty comb and released after 48 hours (Day 2). This procedure permitted to obtain a comb hosting eggs within a range of maximum two days’ age difference. The queen was caged in order to obtain the most coeval individuals within a comb to minimize any variation that could possibly arise from the presence of different developmental stages of the honey bee. Moreover, from a practical point of view, the choice was made to be able to foresee the age of developing insects under study. After queen release, the brood comb was put back into the colony to allow the further development of brood under natural condition. Then, the comb was subjected to two CT scans on Day 10 and Day 17, respectively. At the time of the second scan, a population of pupae aged between stages 7-days and 8-days after capping should be expected20. Before each scan, the comb was extracted from its colony and put into a polystyrene hive nucleus for immediate CT scan at the close Veterinary Faculty Hospital. The images were acquired with a 16-slices CT scanner (GE Brightspeed®, GE Healthcare Milano – Italy), using a high resolution filter. Scanning parameters were set as follows: kV = 120, mA = 250, slice thickness = 0.625 mm, pitch = 0.9375. During the scans, a collection of 1529 and 1452 images was acquired on Day 10 and on Day 17, respectively. After the first scan on Day 10, the comb was put back in the colony. On Day 17, the comb was subjected to the second CT scan and stored afterwards at −20 °C until manual inspection.
### Image analysis
The acquired CT scans were visualized with image viewer Weasis (version 2.0.5), a free software which permits to handle DICOM files (Digital Imaging and COmmunications in Medicine). The length of each pupa was assessed using the measuring tool provided by the software on a selected group of images. For the most accurate measurement as possible, successive images of the same individual were considered in order to carry out the measurement on the one showing the widest slice of tomographic volume of the pupa. Moreover, the exact coordinates of the measured pupa in the comb were extracted using Weasis (version 2.0.5) and the original position within the comb determined by tracing back the coordinates to a common reference point (i.e., the top left part of the comb). This permitted to classify the spatial position of every cell in an imaginary array considered during statistical analysis (Fig. 2).
### Comb manual inspection
In order to assess the developmental stage of the pupa and the infestation of varroa mite, each cell of the comb was individually and manually inspected. The wax cap of each cell was opened with a scalpel and the pupa was extracted using a pair of tweezers. The age of the pupae and the presence of varroa mites were recorded according to Büchler et al.20. In addition, when any mites were found, the number of foundress mites (i.e., adult females with offspring) and the number of progeny were recorded.
### Statistical analysis
For the analysis of the length of the developing honeybee pupae, different factors were considered. Firstly, the position of pupae within the brood was taken into account by sub-setting the brood area of both sides of the analyzed comb into 12 uniform squares by a grid containing 12 sections (3 rows by 4 columns, see Fig. 2). Secondly, the age of the pupae within each cell was considered as a variation factor. Lastly, the effect on the length of the pupae given by the presence of varroa in the cell was tested considering three different categories: i. Mite presence or absence; ii. Number of foundress mites; and iii. Total number of mite’s individuals found in the cell.
To test the relationship between the presence of V. destructor and the position of the cells within the 12 sections of the brood comb, a χ2 analysis was performed. This permitted to assess if varroa mite was distributed in a uniform way within the brood comb.
We chose to analyze the length of pupae situated in the two central sections of the comb assuming that such area shares a slightly higher and more constant temperature, which can influence the size of the developing insect18,21,22,23.
The following fixed model was fitted to data, using PROC GLM of SAS®24:
$${y}_{ijkl}=\mu +{S}_{i}+{A}_{j}+{V}_{k}+{e}_{ijkl}$$
where µ is the overall mean oh the length of the pupa, S refers to ith section of brood in the comb (i = 1,2), A is the jth developmental age of the pupae (j = 1,2), V is the kth effect of varroa in the cell, and e is the random error term of the lth observation (l = 1, 651).
As regards to the effect of varroa, firstly V was fitted as a binary factor indicating the presence or absence of varroa within the cell (k = 0,1). Secondly the number of foundress mites was considered, where V term varied between 0, 1 foundress mite and more than one founder (k = 1,3). Lastly, the effect of the total number of mites within the cell (foundress, son and daughters) was assessed considering V ranging from 0 to 5 individuals (k = 1, 6), where cells with 2 mites were pooled with cells with 1 mite and cells with more than 5 mites where pooled with cells with 5 individuals.
Least square (LS) means were separated by pair-wise t-test and Bonferroni adjustment was applied. Mean separation for main effects were performed on least square mean using PDIFF option of SAS® 19. Statistical differences were declared at P < 0.05.
## Data Availability
Raw data were generated at the Faculty of Veterinary Medicine, University of Milano. Derived data supporting the findings of this study are available from the corresponding author [E.F.] on request.
## References
1. 1.
Oldroyd, B. P. What’s killing American honey bees? PLoS Biol. 5, e168 (2007).
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vanEngelsdorp, D., Underwood, R., Caron, D. & Hayes, J. An estimate of managed colony losses in the winter of 2006–2007: a report commissioned by the apiary inspectors of America. Am. Bee J. 147, 599–603 (2007).
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Neumann, P. & Carreck, N. L. Honey bee colony losses. J. Apic Res. 49, 1–6 (2010).
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Anderson, D. L. & Trueman, J. W. H. Varroa jacobsoni (Acari: Varroidae) is more than one species. Exp. Appl. Acarol. 24, 165–189 (2002).
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Rosenkranz, P., Aumeier, P. & Ziegelmann, B. Biology and control of Varroa destructor. J. Invertebr. Pathol. 103, S96–S119 (2010).
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De Jong, D., De Jong, P. H. & Goncalves, L. S. Weight loss and other damage to developing worker honeybees from infestation with Varroa jacobsoni. J. Apic. Res. 21, 165–167 (1982).
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Schneider, P. & Drescher, W. Einfluss der Parasitierung durch die Milbe Varrroa Jacobsoni oud. Auf das Schlupfgewicht, die Gewichtsentwicklung, die Entwicklung der Hypopharynxdrüsen und die Lebensdauer von Apis mellifera L. Apidologie 18, 101–110 (1987).
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Colin, M. E., Fernandez, P. G. & Ben Hamida, T. Varoosis, Bee Disease Diagnosis. Option Méditerranéennes 25, 121–142 (1999).
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Bowen-Walker, P. L. & Gun, A. The effects of the ectoparasitic mite, Varroa destructor on adult worker honeybee (Apis mellifera) emergence weights, water, protein, carbohydrate, and lipid levels. Entomol. Exp. Appl. 101, 207–217 (2001).
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Ritter, W. & Akratanakul, P. Parasitic bee mites in Honeybee diseases and pests: a practical guide 11–15 (FAO, 2006).
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Dietemann, V. et al. Standard methods for varroa research. J. Apic. Res. 52, 1–54 (2013).
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Poinapen, D. et al. Micro-CT imaging of live insects using carbon dioxide gas-induced hypoxia as anesthetic with minimal impact on certain subsequent life history traits. BMC Zool. 2, https://doi.org/10.1186/s40850-017-0018-x (2017).
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du Plessis, A., le Roux, S.G. & Guelpa A. Comparison of medical and industrial X-ray computed tomography for non destructive testing. Case Studies in Nondestructive Testing and Evaluation 6, 17–25 (2016).
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Cameron Jay, S. The Development of Honeybees in their. Cells, J. Apic. Res. 2, 117–134 (1963).
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Alba, T. & Alba, A. Comparing micro-CT results of insects with classical anatomical studies: The European honey bee (Apis mellifera Linnaeus, 1758) as a benchmark (Insecta: Hymenoptera, Apidae), https://microscopyanalysis.com/article/january_19/comparing_classical_anatomical_studies_of_insects (2019).
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Stevanovic, K., Giovenazzo, P. & Webb, M. A. Synchrotron imaging of intact honeybees affected by nosema IEEE MIT Undergraduate Research Technology Conference (URTC), https://doi.org/10.1109/URTC.2016.8284083 (2016).
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Cameron Jay, S. Prepupal and Pupal Ecdyses of the Honeybee. J. Apic. Res. 1, 14–18 (1962).
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Becher, M. A. & Moritz, R. F. A. A new device for continuous temperature measurement in brood cells of honeybees (Apis mellifera). Apidologie 40, 577–584 (2009).
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Rosenkranz, P. Temperaturpräferenz der Varroa-Milbe und Stocktemperaturen in Bienenvölkern an Tropenstandorten (Acarina: Varroidae/Hymenoptera: Apidae). Entomol. Gener. 14(2), 123–132 (1988).
20. 20.
Büchler, R., Costa, C., Mondet, F., Kezic, N. & Kovacic, M. Screening for low varroa mite reproduction (SMR) and recapping in European honeybees. Research Network for Sustainable Bee Breeding, https://dev.rescol.org/rnsbbweb/wp-content/uploads/2017/11/RNSBB_SMR-recapping_protocol_2017_09_11.pdf (2017).
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Büns, M. & Ratte, H. T. The combined effect of temperature and food consumption on body weight, egg production and developmental time in Chaoborus crystallinus de Geer (Diptera: Chaoboridae). Oecologia 88, 470–476 (1991).
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Sibly, R. M. & Atkinson, D. How rearing temperature affects optimal adult size in ectotherms. Funct. Ecol. 8, 486–493 (1994).
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Petz, M., Stabentheiner, A. & Crailsheim, C. Respiration of honeybee larvae in relation to age and ambient temperature. J. Comp. Physiol. B174, 511–518 (2004).
24. 24.
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## Author information
Authors
### Contributions
E.F. and M.M. conceived the experiment and designed the study. E.F., M.M. and L.N. performed the experiments. M.D.G. and M.E.A. performed CT-scan. E.F. and R.R. performed statistical analyses. E.F. wrote the main manuscript. All authors reviewed the manuscript.
### Corresponding author
Correspondence to Elena Facchini.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Facchini, E., Nalon, L., Andreis, M.E. et al. Honeybee pupal length assessed by CT-scan technique: effects of Varroa infestation, developmental stage and spatial position within the brood comb. Sci Rep 9, 10614 (2019). https://doi.org/10.1038/s41598-019-46474-4
• Accepted:
• Published:
• ### CT-supported analysis of the destructive effects of Varroa destructor on the pre-imaginal development of honey bee, Apis mellifera
• Sándor Keszthelyi
• , Tamás Sipos
• , Ádám Csóka
• & Tamás Donkó
Apidologie (2020)
|
{}
|
# Plus One Computer Science Chapter Wise Questions and Answers Chapter 2 Data Representation and Boolean Algebra
Students can Download Chapter 2 Data Representation and Boolean Algebra Questions and Answers, Plus One Computer Science Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.
## Kerala Plus One Computer Science Chapter Wise Questions and Answers Chapter 2 Data Representation and Boolean Algebra
### Plus One Data Representation and Boolean Algebra One Mark Questions and Answers
Question 1.
___________ is a collection of unorganized fact.
Data
Question 2.
Data can be organized into useful ____________
Information
Question 3.
___________ is used to help people to make decision.
Information
Question 4.
Processing is a series of actions or operations that convert inputs into __________
Output
Question 5.
The act of applying information in a particular context or situation is called ____________
Knowledge
Question 6.
What do you mean by data processing?
Data processing is defined as a series of actions or operations that converts data into useful information.
Question 7.
(b) 12
(3) 17
(d) Adeline aged 17 years is in class 12.
(d) Adeline aged 17 years is in class 12. This is information. The others are data.
Question 8.
Raw facts and figures are known as _______
data
Question 9.
Processed data is known as _______
Information
Question 10.
Which of the following helps us to take decisions?
(a) data
(b) information
(c) Knowledge
(d) intelligence
(b) information
Question 11.
Manipulation of data to get information is known as ___________
Data processing
Question 12.
Arrange the following in proper order
Process, Output, Storage, Distribution, Data Capture, Input.
1. Data Capture
2. Input
3. Storage
4. Process
5. Output
6. Distribution
Question 13.
Pick the odd one out and give reason:
(a) Calculation
(b) Storage
(c) Comparison
(d) Categorization
(b) Storage. It is one of the data processing stage the others are various operations in the stage Process
Question 14.
Information may act as data. State true or False.
False
Question 15.
Complete the Series.
1. (101)2, (111)2, (1001)2, ……….
2. (1011)2, (1110)2, (10001)2, ………
1. 1011, 1101
2. 10101, 10111
Question 16.
What are the two basic types of data which are stored and processed by computers?
Characters and number
Question 17.
The number of numerals or symbols used in a number system is its _______________
Base
Question 18.
The base of decimal number system is ________
10
Question 19.
MSD is ________
Most Significant Digit
Question 20.
LSD is _________
Least Significant Digit
Question 21.
Consider the number 627. Its MSD is _________
6
Question 22.
Consider the number 23.87. Its LSD is __________
7
Question 23.
The base of Binary number system is ___________
2
Question 24.
What are the symbols used in Binary number system?
0 and 1
Question 25.
Complete the following series,
(101)2, (111)2, (1001)2
1011, 1101
Question 26.
State True or False. In Binary, the unit bit changes either from 0 to 1 or 1 to 0 with each count.
True
Question 27.
The base of octal number system is ________
8
Question 28.
Consider the octal number given below and fill in the blanks.
0, 1, 2, 3, 4, 5, 6, 7, __
10
Question 29.
The base of Hexadecimal number system is ________
16
Question 30.
State True or False.
In Positional number system, each position has a weightage.
True
Question 31.
In addition to digits what are the letters used in Hexadecimal number system.
A(10), B(11), C(12), D(13), E(14), F(15)
Question 32.
Convert (1110.01011)2 to decimal.
1110.01011 = 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 + 0 × 2 – 1 + 1 × 2 – 2 + 0 × 2 – 3 + 1 × 2 – 4 + 1 × 2 – 5
= 8 + 4 + 2 + 0 + 0 + 0.25 + 0 + 0.0625 + 0.03125
= (14.34375)10
Question 33.
1 KB is bytes.
(a) 25
(b) 210
(c) 215
(d) 220
(b) 210
Question 34.
The base of hexadecimal number system is ________.
16
Question 35.
A computer has no _________
(a) Memory
(b) l/o device .
(c) CPU
(d) IQ
(d) IQ
Question 36.
Pick the odd man out.
(AND, OR, NAND, NOT)
NOT
Question 37.
Select the complement of X + YZ.
(a) $$\bar{x}+\bar{y}+\bar{z}$$
(b) $$\bar{x} .\bar{y}+\bar{z}$$
(c) $$\bar{x} \cdot(\bar{y}+\bar{z})$$
(d) $$\bar{x}+\bar{y} \cdot \bar{z}$$
(c) $$\bar{x} \cdot(\bar{y}+\bar{z})$$
Question 38.
Select the expression for absorption law.
(a) a + a = a
(b) 1 + a = 1
(c) o . a = 0
(d) a + a . b = a
(a) a + a . b = a
Question 39.
What is the characteristic of logical expression?
Logical expressions yield either true or false values
Question 40.
Name the table used to define the results of Boolean operations.
Truth Table
Question 41.
According to ________ law, $$\bar{x}+\bar{y}=\overline{x y}$$ and $$\overline{x y}=\bar{x}+\bar{y}$$
De Morgan’s law
Question 42.
A NOR gate is ON only when all its inputs are
(a) ON
(b) Positive
(c) High
(d) OFF
(d) OFF
Question 43.
The only function of a NOT gate is __________
Invert an output signal
Question 44.
NOT gate is also known as _________
Inverter
Question 45.
What is the relation between the following statements.
x + 0 = x and x . 1 = x
One is the dual of the other expression.
Question 46.
The algebra used to solve problems in digital systems is called __________
Boolean Algebra
Question 47.
Pick the one which is not a Basic Gate.
(AND, OR, XOR, NOT)
XOR
Question 48.
Select the universal gates from the list. (NAND, NOR, NOT, XOR)
NAND, NOR
Question 49.
Which is the final stage in data processing?
Distribution of information is the final stage in data processing
Question 50.
Fill up the missing digit.
(41)8 = ( )16
• Step 1: Divide the number into one each and write down the 3 bits equivalent.
• Step 2: Then divide the number into group of 4 bits starting from the right then write its equivalent hexa decimal.
Question 51.
Real numbers can be represented in memory by using __________
Exponent and Mantissa
Question 52.
Consider the number 0.53421 x 10-8 Write down the mantissa and exponent.
Mantissa: 0.53421
Exponent: -8
Question 53.
Characters can be represented in memory by using _________
ASCII Code
Question 54.
ASCII Code of A’ is __________
(100 0001)2 = 65
Question 55.
ASCII Code of’a’ is __________
(110 0001)2 = 97
Question 56.
Define the term ‘bit’?
A bit stands for Binary digit. That means either 0 or 1.
Question 57.
Find MSD in the decimal number 7854.25
Because it has the most weight
Question 58.
ASCII stands for __________.
American Standard Code for Information Interchange
Question 59.
List any two image file formats.
BMP, GIF
Question 60.
Name the operator which performs logical multiplication.
AND
Question 61.
Name a gate which is ON when all its inputs are OFF .
NAND or NOR
Question 62.
Specify the laws applied in the following cases.
1. a (b + c) = ab + ac
2. (a + b) + c = a + (b + c)
1. Distribution law
2. Associative law
Question 63.
Pick the correct Boolean expression from the following.
(a) $$A +\bar{A}=163.$$
(b) $$\text { A. } \bar{A}=1$$
(c) $$A \cdot \overline{A B}=A + B$$
(d) A + AB = A
(a) & (d)
Question 64.
1’s complement of the binary number 110111 is _________
Insert 2 zeroes in the left hand side to make the binary number in the 8 bit form 00110111
To find the 1’s complement, change all zeroes to one and all ones to zero. Hence the answer is 11001000
### Plus One Data Representation and Boolean Algebra Two Mark Questions and Answers
Question 1.
Why do we store information?
Normally large volume of data has to be given to the computer for processing so the data entry may be taken more days, hence we have to store the data. After processing these stored data, we will get information as a result that must be stored in the computer for future references.
Question 2.
What is source document.
Acquiring the required data from all the sources for the data processing and by using this data design a document, that contains all relevant data in proper order and format. This document is called source document.
Question 3.
Briefly explain data, information and processing with real life example.
Consider the process of making coffee. Here data is the ingredients – water, sugar, milk and coffee powder
Information is the final product i.e. Coffee Processing is the series of steps to convert the ingredients into final product, Coffee. That is mix the water,sugar and milk and boil it. Finally pour the coffee powder.
Question 4.
ASCII is used to represent characters in memory. Is it sufficient to represent all characters used in the written languages of the world? Propose a solution. Justify.
No It is not sufficient to represent all characters used in the written languages of the world because it is a 7 bit code so it can represent 27 = 128 possible codes. To represent all the characters Unicode is used because it uses 4 bytes, so it can represent 232 possible codes.
Question 5.
The numbers in column A have an equivalent number in another number system of column B.
Find the exact matvh
A B (12)8 (1110)2 F16 25 (19)16 10 (11)8 (13)16 (17)8 9
A B 12 10 F (17)8 (19)16 25 (11)8 9
Question 6.
1. Name various number systems commonly used in computers.
2. Include each of the following numbers into all possible number systems
123 569 1101
1. The number system are binary, octal, decimal and hexa decimal
2. All possible number systems are
• 123 Octal, decimal and hexa decimal
• 569 Decimal, hexa decimal
• 1101 Binary, Octal, Decimal, Hexa decimal
Question 7.
Fill up the missing digit. (Score 2)
If (220)a = (90)b then (451)a = ( )10
It contains 2 & 9, so a and b 2, b 8. The values of a can be 8 or 10. The values of b can be 10 or 16. L.H.S > R.H.S. a < b and a b also.
The possible values of a and b are given below.
Question 8.
Convert (106)10 = ( )2?
Question 9.
Convert (106)10 = ( )8?
Question 10.
(106)10 = ( )16?
Question 11.
Convert (55.625)10 = ( )2?
First convert 55, for this do the following
Write down the remainders from bottom to top.
(55)10 = (110111 )2
Next convert 0.625, for this do the following.
Write down the remainder from top to bottom. So the answer is
(55.625)10 = (110111.101)2
Question 12.
Convert (55.140625)10 = ( )8?
First convert 55, for this do the following.
Write down the remainders from bottom to top.
(55)10 = (67)8
Next convert 0.140625, for this do the following.
Write down the remainders from top to bottom. So the answer is
(55.140625)10 = (67.11 )8
Question 13.
Convert (55.515625)10 = ( )16
First convert 55, for this do the following.
Write down the remainders from bottom to top.
ie. (55)10 = (37)16
Next convert .515625
(55.515625)10 = (37.84)16
Question 14.
Convert (101.101)2 = ( )10?
101.101 = 1 × 22 + 0 × 21 + 1 × 20 + 1 × 2-1 + 0 × 2-2 + 1 × 2-3 = 4 + 0 + 1 + 1/2 + 0 + 1/8 = 5 + 0.5 + 0.125
(101.101)2 = (5.625)10
Question 15.
Convert (71.24)8 = ( )10?
71.24 = 7 × 81 + 1 × 80 + 2 × 8-1 + 4 × 8=2
= 56 + 1 + 2/8 + 4/82
= 57 + 0.25 + 0.0625 (71.24)8
(71.24)8 = (57.3125)10
Question 16.
Convert (AB.88)16 = ( )10?
= 160 + 11 + 0.5 + 0.03125
(AB.88)16 = (171.53125)10
Question 17.
Convert (1011)2 = ( )8?
Step I: First divide the number into groups of 3 bits starting from the right side and insert necessary zeroes in the left side.
0 0 1 | 0 1 1
Step II: Next write down the octal equivalent.
So the answer is (1011)2 = (13)8.
Question 18.
Convert (110100)2 = ( )16
• Step I: First divide the number into groups of 4 bits starting from the right side and insert necessary zeroes in the left side.
• Step II: Next write down the hexadecimal equivalent.
So the answer is (110100)2 = (34)16.
Question 19.
(72)8 = ( )2?
Write down the 3 bits equivalent of each digit.
So the answer is (72)8 = (111010)2.
Question 20.
Convert (AO)16 = ( )2 ?
Write down the 4 bits equivalent of each digit
So the answer is (AO)16 = (1010 0000)2.
Question 21.
Convert (67)8 = ( )16?
Step I: First convert this number into binary equivalent for this do the following
Step II: Next convert this number into hexadecimal equivalent for this do the following.
So the answer is (67)8 = (37)16
Question 22.
Convert (A1)16 = ( )8?
Step I: First convert this number into binary equivalent. For this do the following
Step II: Next convert this number into octal equivalent. For this do the following.
So the answer is (A1)16 = ( 241)8.
Question 23.
What is the use of the ASCII Code?
ASCII means American Standard Code for Information Interchange. It is a 7 bit code. Each and every character on the keyboard is represented in memory by using ASCII Code.
eg: A’s ASCII Code is 65 (1000001), a’s ASCII Code is 97 (1100001)
Question 24.
Pick invalid numbers from the following.
1. (10101)8
2. (123)4
3. (768)8
4. (ABC)16
1. (10101)8 – Valid
2. (123)4 – Valid
3. (768)8 – Invalid. Octal number system does not contain the symbol 8
4. (ABC)16 – Valid
Question 25.
Convert the decimal number 31 to binary.
(31)10 = (11111)2
Question 26.
Find decimal equivalent of (10001 )2
= 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20
= 16 + 0 + 0 + 0 + 1
= (17)10
Question 27.
If (X)8 =(101011 )2 then find X.
Divide the binary number into groups of 3 bits and write down the corresponding octal equivalent.
X = 53
Question 28.
Fill the blanks:
(a) (………..)2 = (AB)16
(b) (——D—–)16 = (1010 1000)2
(c) 0.2510 = (—–)2
Write down the 4bit equivalent of each digit
(a) (………..)2 = (AB)16
= (10101011)2
(b) (——D—–)16 = (1010 1000)2
(A D 8)16 =(1010 1101 1000)2
(c) 0.2510 = (—–)2
0.2510 = (0.01)2
Question 29.
Which is the MSB of representation of -80 in SMR?
It is 1 because In SMR if the number is negative then the MSB is 1.
Question 30.
Write 28.756 in Mantissa exponent form.
28.756 = .28756 × 100
= .28756 × 102
= .28756 E + 2
Question 31.
Represent -60 in 1’s complement form.
+60 = 00111100
Change all 1 to 0 and all 0 to 1 to get the 1’s complement.
-60 is in 1’s complement is 11000011
Question 32.
Define Unicode.
The limitations to store more characters is solved by the introduction of Unicode. It uses 16 bits so 216 = 65536 characters(i.e, world’s all written language characters) can store by using this.
Question 33.
Substract 1111 from 10101 by using 2’s complement method.
To subtract a number from another number find the 2’s complement of the subtrahend and add it with the minuend. Here the subtrahend is 1111 and minuend is 5 bits. So insert a zero. So subtrahend is 01111. First take the 1’s complement of subtrahend and add 1 to it
Here is a carry. So ignore the carry and the result is +ve.
Question 34.
You purchased a soap worth Rs. (10010)2 and you gave Rs. (10100)2 and how much rupees will you get back in binary.
Substract (10010)2 from (10100)2
You will get rupees (10)2
Question 35.
Draw the logic circuit diagram for the following Boolean expression.
Question 36.
Simplify the expression using basic postulates and laws of Boolean algebra.
1. $$\bar{x}+x \cdot \bar{y}$$
2. $$x(y+y . z)+y(\bar{x}+x z)$$
1. $$\bar{x}+\bar{y}$$
2. y
Question 37.
Show $$A(\bar{B}+C)$$ using NOR gates only.
Question 38.
The following statement Demorgan’s theorem of Boolean algebra. Identify and state ‘Break the line, change the sign’.
Demorgan’s theorems,
Demorgan’s first theorem,
$$\overline{x + y}$$ = $$\bar{x} . \bar{y}$$
Demorgan’s second theorem,
$$\overline{x – y}$$ = $$\bar{x} + \bar{y}$$
Question 39.
Prove algebraically that
$$x \cdot y + x \cdot \bar{y} \cdot z$$ = x . y + x . z
$$x \cdot y + x \cdot \bar{y} \cdot z$$ = $$x(y+\bar{y} . z)$$
= x . (y + z) = x . y + x . z
Hence proved.
Question 40.
State which of the following statements are logical statements.
(a) AND is a logical operator
(c) Go to class
(d) Sun rises in the west.
(e) Why are you so late?
(a) and (d) are logical statements because these statements have a truth value which may be true or false.
Question 41.
Express the integer number -39 in sign and magnitude represnetation.
First find the binary equivalent of 39 for this do the following
In sign and magnitude representation -39 in 8 bit form is (10100111)2.
Question 42.
1. Which logic gate does the Boolean expression $$\overline{\mathrm{AB}}$$ represent?
2. Some NAND gates are given. How can we construct AND gate, OR gate and NOT gate using them?
1. NAND
2. AND gate
Question 43.
Perform the following number conversions.
1. (110111011.11011)2 = (….)8
2. (128.25)10= (…..)8
1. 110111011.11011
Step 1: Insert a zero in the right side of the above number and divide the number into groups of 3 bits as follows
110 111 011 . 110 110
Step 2: Write down the corresponding 3 bit binary equivalent of each group
6 7 3 .6 6
Hence the result is (673.66)8
2. It consists of 2 steps.
Step 1: First convert 128 into octal number for this do the following
Write down the remainders from bottom to top.
(128)10 = (200)8
Step 2: Then convert .25 into octal number for this do the following
(0.25)10 = (0.2)8.
Combine the above two will be the result.
(128.25)10= (200.2)8
Question 44.
Represent -38 in 2’s complement form.
+38 = 00100110
First take the 1 ’s complement for this change all 1 to 0 and all 0 to 1
2’s complement of -38 is (11011010)8.
### Plus One Data Representation and Boolean Algebra Three Mark Questions and Answers
Question 1.
Differentiate manual data processing and electronic data processing?
In manual data processing human beings are the processors. Our eyes and ears are input devices. We get data either from a printed paper, that can be read using our eyes or heard with ears. Our brain is the processor and it can process the data, and reach in a conclusion known as result. Our mouth and hands are output devices.
In electronic data processing the data is processing with the help of a computer. In a super market, key board and hand held scanners are used to input data, the CPU process the data, monitor and printers (Bill) are output devices.
Question 2.
Complete the series.
1. 3248, 3278 ,3328, …., ….
2. 5678, 5768, 605s, ……, …..
1. (324)8 = 3 × 82 + 2 × 81 + 4 × 80
= 3 × 64 + 2 × 8 + 4 × 1
= 192 + 16 + 4 = (212)10
(327)8 = 3 × 82 + 2 × 81 + 7 × 80
= 192 + 16 + 7 = (215)10
(332)8= 3 × 82 + 3 × 81 +2 × 80
= 192 + 24 + 2 = (218)10
So the missing terms are (221)10, (224)10 we have to convert this into octal.
So the missing terms are (335)8, (340)8
2. (567)8 = 5 x 82 + 6 x 81 + 7 x 80
= 5 x 64 + 6 x 8 + 7 x 1
= 320 + 48 + 7 = (375)10
(576)8 = 5 x 82 + 7 x 81 + 6 x 80
= 320 + 56 + 6 = (382)10
(605)8 = 6 x 82 + 0 x 81 + 5 x 80
= 6 + 64 + 0 + 5
= 384 + 0 + 5 = (389)10
So the missing terms are (396)10, (403)10 we have to convert this into octal.
So the missing terms are (614)8, (623)8
Question 3.
Fill up the missing digits.
1. (4……)8 = (……110)2
2. (…….7……)8 = (100…….110)2
Consider the following
1. 4…….100 and 110……….6
So (46)8 = (100 110)2
2. 100…….4
7……111
110………6
So (476)8 = (100 111 110)2
Question 4.
Fill up the missing numbers.
1. (A…….)16 = (……..1001)2
2. (…….B…….)16 = (1000………1111)2
Consider the following:
1. A……..1010
1001………9
So (A9)16 = (1010 1001)2
2. B…….1011
1000………8
1111………F
So (8BF)16 = (1000 1011 1111)2
Question 5.
Complete the Series.
2. 14A9, 14AF, 14B5, …….., ……
1. Consider the sequence
Here the ‘numbers’ are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11,——–
Similarly 6ADF & 6AE1 is 2
So Add 2 to 6AE1 we will ge 6AE3 Then add 2 to 6AE3 we will get 6AE5 Therefore the missing terms 6AE3, 6AE5
2. Consider the sequence.
14A9, 14AF, 14B5, ———
The difference between 14A9 and 14AF is 6.
The normal sequence is
The difference between 14AF and 14B5 is also 6.
The normal sequence is
Similarly the next 6 terms in the sequence are given below.
Similarly the next 6 terms are
So the missing terms are 14BB and 14C1
Question 6.
Find the octal numbers corresponding to the following numbers using shorthand method.
Step 1: Write down the 4 bit binary equivalent of each digit
Step 2: Divide this number into groups of 3 bits starting from the right and write down the octal equivalent.
Step 1: Write down the 4 bit binary equivalent of each digit.
Step 2: Divide this number into groups of 3 bits starting from the right and write down the octal equivalent.
Question 7.
If (126)x = (56)y, then find x and y.
L.H.S contains 2 & 6, so x ≠ 2
R.H.S contains 5 & 6, so y ≠ 2
L.H.S > R.H.S
So x < y and x ≠ y also The possible values of x and y are given below.
Case I:
Let x = 8 then y = 10
It is grater than (56)10
so when x = 8 then y ≠ 10
case II:
let x = 8 and y = 16
Question 8.
If (102)x = (42)y then (154)x = ( )y.
L.H.S contains 2, so x ≠ 2
R.H.S contains 5 & 4, so y ≠ 2
L.H.S > R.H.S
So x < y and x ≠ y also
The possible values of x and y are given below.
case I:
let x = 8 and y = 10
So when x = 8 then y ≠ 10
case II:
let x = 8 and y = 16
So x = 8 and y = 16
then we have fo find the hexadecimal equivalent of (154)8 For this first convert this into binary thus again convert it into hexadecimal. First write down the 3 bit equivalent of 154.
Then divide this number into groups of 4 bits starting from the right and write down the hexa decimal equivalent.
so the result is (154)8 = (6C)16
Question 9.
Fill up the missing digit.
If (121)a = (441)b then (121)b = ( )10
L.H.S. contains 2, so a ≠ 2
R.H.S. contains 4, so b ≠ 2
L.H.S. < R.H.S. So a > b and a b also.
Hence the values of a can be 10 or 16.
The values of b can be 8 or 10.
The possible values of a and b are given below.
Case I:
Let a = 16 and b = 10
(121 )16 = (289)10, so b ≠ 10
Case II:
Let a = 16 and b = 8
(121)16 = (289)10
(441)8 = 4 × 82 + 4 × 81 + 1 × 80
= 256 +32 + 1
= (289)10.
So a = 16 and b = 8.
Then (121)8 = 1 × 82 + 2 × 81 + 1 × 80
= 64 + 16 + 1 = (81)10
Question 10.
Fill up the missing digit. (Score 3)
If (128)a = (450)b then (16)a = ( )10
L.H.S. contains 2 & 8, so a 2 and a ≠ 8.
R.H.S. contains 4 and 5, so b ≠ 2.
L.H.S. < R.H.S. so a > b and a ≠ b also.
The possible values of a and b are given below.
Case I:
a = 16 and b = 8
(128)16 = (296)10
(450)8 = (296)10 So a = 16 and b = 8.
Then (16)16 = 1 × 16 + 6 × 160 = (22)10
Question 11.
Fill up the missing digit.
(3A.6D)16 = ( )8
Step I: Write down the 4 bits equivalent of each digit.
Step II: Divide this number into groups of 3 bit starting from the right side of the left side of the decimal point and starting from the left side of the right side of the decimal point.
So 00/111/010.011/011/010
Step III: Write the octal equivalent of each group. SO we will get. (72.332)8.
(3A.6D)16 = (72.332)8
Question 12.
What are the various ways to represent integers in computer?
There are three ways to represent integers in computer. They are as follows:
1. Sign Magnitude Representation (SMR)
2. 1’s Complement Representation
3. 2’s Complement Representation
1. SMR:
Normally a number has two parts sign and magnitude, eg: Consider a number +5. Here + is the sign and 5 is the magnitude. In SMR the most significant Bit (MSB) is used to represent the sign. If MSB is 0 sign is +ve and MSB is 1 sign is – ve.
eg: If a computer has word size is 1 byte then
Here MSB is used for sign then the remaining 7 bits are used to represent magnitude. So we can represent 27 = 128 numbers. But there are negative and positive numbers. So 128 + 128 = 256 number. The numbers are 0 to +127 and 0 to -127. Here zero is repeated. So we can represent 256 – 1 = 255 numbers.
2. 1 ‘s Complement Representation: To get the 1’s complement of a binary number, just replace every 0 with 1 and every 1 with 0. Negative numbers are represented using 1’s complement but +ve number has no 1’s complement.
eg: To find the 1’s complement of 21 +21 = 00010101
To get the 1 ‘s complement change all 0 to 1 and all 1 to 0.
-21 = 11101010
1’s complement of 21 is 11101010
3. 2’s Complement Representation: To get the 2’s complement of a binary number, just add 1 to its 1’s complement +ve number has no 2’s complement.
eg: To find the 2’s complement of 21 +21 =00010101
First take the 1’s complement for this change all 1 to 0 and all 0 to 1
2’s complement of 21 is 1110 1011
Question 13.
Write short notes about Unicode (3)
It is like ASCII Code. By using ASCII, we can represent limited number of characters. But using Unicode we can represent all of the characters used in the written languages of the world.
eg: Malayalam, Hindi, Sanskrit .
Question 14.
Match the following.
1. (106)10 a. (171.53125)10 2. (71.24)8 b. (6a)16 3. (AB.88)16 c. (20)8 4. (10)16 d. (10000000)2 5. (128)10 e. (10)16 6. (16)10 f. (57.3125)10
1 – b, 2 – f, 3 – a, 4 – c, 5 – d, 6 – e
Question 15.
Find the largest number in the list.
1. (1001)2
2. (A)16
3. (10)8
4. (11)10
Convert all numbers into decimal
1. (1001)2 = 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20
= 8 + 0 + 0 + 1
= (9)10
2.) (A)16 = (10)10
3. (10)8 = 1 × 81+0 × 80
= (8)10
So the largest number is 4 – (11)10
Question 16.
Subtract 10101 from 1111 by using 2’s complement method.
To subtract a number from another number find the 2’s complement of the subtrahend and add it with the minuend. Here subtrahend is 10101 and minuend is 1111 First take the 1’s complement of subtrahend and add 1 to it.
1’s complement of 10101 is 01010 add 1 to it
Here is no carry. So the result, is -ve and take the 2’s complement of 11010 and put a -ve symbol. So 1’s complement of 11010 is 00101 add 1 to this
So the result is -00110
Question 17.
Mr. Geo purchased (10)2 kg sugar @Rs. (110 10)2 and (1010)2 kg Rice @Rs. (10100)2. So how much rupees he has to pay in decimal.
Convert each into decimal number system multiply and sum it up.
(10)2 = (2)10
(11010)2 = 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 21
= 16 +8 + 0 +2 + 0
= (26)10
(1010)2 = 1 × 23 + 0 × 22 + 1 × 21 +0 × 20
= 8 + 0 + 2 + 0
= (10)2
(10100)2 = 1 × 24 + 0 × 23 + 1 × 22 +0 × 21 + 0 × 20
= 16 + 0 + 4 + 0 + 0
= (20)2
therfore 2 × 26 + 10 × 20
= 52 + 200
= 252
So Mr. Geo has to pay Rs. 252/-
Question 18.
Mr. Vimal purchased a pencil @ Rs. (101)2, a pen @ Rs. (1010)2 and a rubber @ Rs. (10)2. So how much rupees he has to pay in decimal.
Add 101 + 1010 + 10
then convert (10001)2 into decimal
(10001)2 = 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20
= 16 + 0 + 0 + 0 + 1
= (17)10
So Mr. Vimal has to pay Rs. 17/-
Question 19.
Mr. Antony purchased 3 books worth Rs. a total of (1100100)2. Atlast he returned a book worth Rs. (11001)2. So how much amount he has to pay for the remaining two books in decimal number sys¬tem.
then convert (1001011 )2 into decimal
(1001011)2 = 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20
=64 + 0 + 0 + 8 + 0 + 2 + 1
= (75)10
So he has to pay Rs. 75/-
Question 20.
Mr. Leones brought two products from a super market a total of Rs. (11010010)2 and he got a dicount of Rs. (1111)2 So how much he has to pay for this products in decimal number system.
Substract (1111)2 from (11010010)2
then convert (11000011)2 into decimal
(11000011)2 = 1 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20
=128 + 64 + 0 + 0 + 0 + 0 + 2 + 1
= (195)10
Question 21.
A textile showroom sells shirts with a discount of Rs. (110010)2 on all barads. Mr. Raju wants to buy a shirt worth Rs. (11111000)2. So after discount how much amount he has to pay in decimal.
Substract (110010)2 from (11 111 000)2
then convert (11000110)2 into decimal
(11000110)2 = 1 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20
= 128 + 64 + 0 + 0 + 0 + 4 + 2 + 0
= (198)10
Question 22.
Mr. Lijo purchased a product worth Rs. (1110011)2 and he has to pay VAT @ Rs. (1100)2. Then calculate the total amount he has to pay in decimal.
then convert (1111111)2 into decimal
(1111111)2 = 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20
= 64 + 32 + 16 + 8 + 4 + 2 + 1
= (127)10
Question 23.
By using truth table, prove the following laws of Boolean Algebra.
1. Idempotent law
2. Involution law
1. A + A = A
A = A = A
2. (A1)1 = A
Question 24.
Consider the logical gate diagram.
1. Find the logical expression for the circuit given.
2. Find the compliment of the logical expression.
3. Draw the circuit diagram representing the compliment.
1. $$(x+\bar{y}) \cdot z$$
2. $$(\bar{x} . y)+\bar{z}$$
3.
Question 25.
Draw the logic circuit diagram for the following Boolean expression.
$$A \cdot(\bar{B} + \bar{C})+\bar{A} \bar{B} \bar{C}$$
Question 26.
Consider a bulb with three switches x, y and z. Write the Boolean expression representing the following states.
1. All the switches x, y and z are ON
2. x is ON and y is OFF or Z is OFF
3. Exactly one switch is ON.
1. xy . z
2. $$x \bar{y}+\bar{z}$$
3. $$x . \bar{y} . \bar{z}+\bar{x} . y . \bar{z}+\bar{x} . \bar{y} . \bar{z}$$
Question 27.
Match the following.
A B i. Idem potent law a. x + (y + z)=(x + y)+z ii. Involution law b. x + xy = x iii. Complementarity law c. x + y = y + x iv. Commutative law d. xx- 0 v. Absorption law e. x = x vi. Associative law f. x + x = x
i – f, ii – e, iii – d, iv – c, v – b, vi – a
Question 28.
Explain the principle of duality.
It states that, starting with a Boolean relation, another Boolean relation can be derived by
1. Changing each OR sign (+) to a AND sign (.)
2. Changing each AND sign (.) to an OR sign (+)
3. Replacing each 0 by 1 and each 1 by 0.
The relation derived using the duality principle is called the dual of the original expression,
eg: x + 0 = x is the dual of x . 1 = x
Question 29.
Draw the circuit diagram for $$F=A \bar{B} C+\bar{C} B$$ using NAND gate only.
$$F=A \bar{B} C+\bar{C} B$$
= (A NAND (NOT B) NAND C) NAND ((NOT C) NAND B)
Question 30.
Draw a logic diagram for the function f = YZ + XZ using NAND gates only.
f = YZ + XZ
= (Y NAND Z) NAND (X NAND Z)
Question 31.
How do you make various basic logic gates using NAND gates.
1. AND operation using NAND gate,
A.B = (A NAND B) NAND (A NAND B)
2. OR operation using NAND gate,
A + B = (A NAND A) NAND (B NAND B)
3. NOT operation using NAND gate,
NOT A = (A NAND A)
Question 32.
Which of the following Boolean expressions are correct? Write the correct forms of the incorrect ones.
1. A + A1 = 1
2. A + 0 = A
3. A . 1 = A
4. A . A1 = 1
5. A + A . B = A
6. A . (A + B) = A
7. A + 1=1
8. $$(\overline{\mathrm{A} . \mathrm{B}})=\overline{\mathrm{A}} . \overline{\mathrm{B}}$$
9. A + A1B = A + B
10. A + A = A
11. A + B . C = (A+B) . (B+C)
1. Correct
2. Correct
3. Correct
4. Wrong, A . A1 = 0
5. Correct
6. Correct
7. Correct
8. Wrong $$\overline{\mathrm{A} . \mathrm{B}}=\overline{\mathrm{A}}+\overline{\mathrm{B}}$$
9. Correct
10. Correct
11. Wrong, A + B . C = (A + B) . (A + C)
Question 33.
Prove algebraically that (x + y)’ . (x’ + y’) = x’ . y’
LHS = (x + y)’ . (x’ + y’)
= (x’ . y’) . (x’ . y’)
= x’ . y’ . x’ + x’ . y’ . y’
= x’ . y’ + x’ . y’
= x ‘. y’ = RHS
Hence proved.
Question 34.
Give the complement of the following Boolean Expression.
1. (A + B) . (C + D)
2. (P + Q) + (Q + R) . (R + P)
3. (B + D’) . (A + C’)
1. ((A+B) . (C+D))1 = (A+B)’ + (C+D)’
= A’ . B’ + C’ . D’
2. ((P+Q) + (Q+R) . (R.P))’ = (P+Q) ‘. ((Q+R) . (R+P))’
= P’ . Q’ . (Q+R)’ + (R+P)’
= P’ . Q’ . (Q’ . R’ + R’ . P’),
3. ((B+D’).(A+C’))’ = (B+D’)’0 + (A+C’)’
= B’ . D” + A’ . C”
= B’ . D + A’ . C
Question 35.
State and prove the idempotent law using truth table. Idempotent law
Idempotent law states that
1. A + A = Aand
2. A . A = A Proof
1. A + A = A
Truth table is as follows:
ie. A + A = A as it is true for both values of A. Hence proved.
2. A . A = A
Truth table is as follows:
ie. A . A = A itself. It is true for both values of A. Hence proved.
Question 36.
State the Absorption laws of Boolean algebra with the help of truth tables.
Absorption law states that
A + A . B = A and A . (A + B) = A
Proof:
The Truth table of the expression A + A . B=A is as follows.
Here both columns A and A + A . B are identical. Hence proved.
For A . (A + B) = A, the truth table is as follows:
Both columns A & A . (A + B) are identical. Hence proved
Question 37.
State Demorgen’s laws. Prove anyone with truth table method.
Demorgan’s first theorem states that (A + B)’ = A’ . B’
ie. the complement of sum of two variables equals product of their complements,
The second theorem states that (A . B)’ = A’ + B’
ie. The complement of the product of two variables equals the sum of the complement of that variables.
Proof:
Truth table of first one is as follows:
From the truth table the columns of both (A + B)’ and A’ . B’ are identical. Hence proved.
Question 38.
Fill in the blanks:
1. (0.625)10 = (……….)2
2. (380)10 = (……..)16
3. (437)8 = (………)2
1. (0.101)2
2. (17C)16
3. (100 011 111)2
Question 39.
What do you mean by universal gates? Which gates are called Universal gates? Draw their symbols.
OR
Construct a logical circuit for the Boolean expression $$\bar{a} \cdot b+a \cdot \bar{b}$$. Also write the truth table.
Universal gates:
By using NAND and NOR gates only we can create other gate hence these gates are called Universal gate.
NAND gate:
NOR gate:
Truth table:
Logical circute:
Question 40.
Computers uses a fixed number of bits to respresent data which could be a number, a character, image, sound, video etc. Explain the various methods used to represent characters in memory.
Representation of characters.
1. ASCII(American Standard Code for Information Interchange):
It is 7 bits code used to represent alphanumeric and some special characters in computer memory. It is introduced by the U.S. government. Each character in the keyboard has a unique number.
eg: ASCII code of ‘a’ is 97.
When you press ‘a’ in the keyboard , a signal equivalent to 1100001 (Binary equivalent of 97 is 1100001) is passed to the computer memory. 27 = 128, hence we can represent only 128 characters by using ASCII. It is not enough to represent all the characters of a standard keyboard.
2. EBCDIC(Extended Binary Coded Decimal Interchange Code):
It is an 8 bit code introduced by IBM(International Business Machine). 28 = 256 characters can be represented by using this.
3. ISCII(Indian Standard Code for Information Interchange):
It uses 8 bits to represent data and introduced by standardization committee and adopted by Bureau of Indian Standards(BIS).
4. Unicode:
The limitations to store more characters is solved by the introduction of Unicode. It uses 16 bits so 216 = 65536 characters (i.e, world’s all written language characters) can store by using this.
Question 41.
Draw the logic circuit for the function
$$f(a, b, c)=a . b . c+\bar{a} . b+a . \bar{b}+a . b . \bar{c}$$
OR
Prove algebrically.
$$x . y+x . \bar{y} . z=x . y .+x . z$$
OR
$$x \cdot y+x \cdot \bar{y} \cdot z=x \cdot(y+\bar{y} \cdot z)$$
= x . (y + z) = x . y + x . z
Hence proved.
Question 42.
Following are the numbers in various number systems. Two of the numbers are same. Identify them:
1. (310)8
2. (1010010)2
3. (C8)16
4. (201)10
OR
Consider the following Boolean expression:
(B’ + A)’ = B . A’
Identify the law behind the above expression and prove it using algebriac method.
1. (310)8 = 3 * 82 + 1 * 81 + 0 * 80
= 192 + 8 + 0
= (200)10
2. (1010010)2 = 1 × 26+ 0 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20
= 64 + 0 + 16 + 0 + 0 + 2 + 0
= (82)10
3. (C8)16 = C × 16 + 8 × 160
= 12 × 16 + 8 × 1
= 192 + 8
= (200)10
Here (a) (310)8 and (C8)16 are same
OR
This is De Morgan’s law (B’ + A’) = (B’)’ . A’
= B . A’
Hence it is proved
Question 43.
Find the decimal equivalent of hexadecimal number (2D)16. Represent this decimal number in 2’s complement form using 8 bit word length.
Convert (2D)16 to binary number for this write down the 4 bit binary equivalent of each number
(2D)16 = (00101101 )2
First find the 1’s complement of (00101101 )2 and add 1 to it
Hence 2’s complement is (11010011)2
Question 44.
Answer any one question from 15(a) and 15(b).
1. Draw the logic circuit for the Boolean expression:
$$(A+\overline{B C})+\overline{A B}$$
2. Using algebraic method prove that
$$\bar{Y} \cdot \bar{Z}+\bar{Y} \cdot Z+Y \cdot Z+Y=1$$
1.
OR
2. L.H.S. = $$\bar{Y} \cdot \bar{Z}+\bar{Y} \cdot Z+Y Z+Y$$
= $$\bar{y} \cdot(\bar{z}+z)+y \cdot(z+1)$$
= $$\bar{y}. 1+\bar{y} \cdot 1=y \cdot y=1$$
Question 45.
With the help of a neat circuit diagram, prove that NAND gate is a universal gate.
1. AND operation using NAND gate,
A . B = (A NAND B) NAND (A NAND B)
2. OR operation using NAND gate,
A + B = (A NAND A) NAND (B NAND B)
3. NOT operation using NAND gate,
NOT A = (A NAND A)
Question 46.
Boolean expression:
$$(A+\overline{B C})+\overline{A B}$$
OR
Using algebraic method, prove that
$$\bar{Y} \cdot \bar{Z}+\bar{Y} \cdot Z+Y \cdot Z+Y=1$$
OR
= Y . Z + Y . Z + Y . Z + Y
= Y . (Z + Z) + Y . (Z + 1)
= Y . 1 + Y. 1
= Y + Y
= 1
Hence the result.
### Plus One Data Representation and Boolean Algebra Five Mark Questions and Answers
Question 1.
Explain the components of Data processing.
Data processing consists of the techniques of sorting, relating, interpreting and computing items of data in orderto convert meaningful information. The components of data processing are given below.
1. Capturing data: In this step acquire or collect data from the user to input into the computer.
2. Input: It is the next step. In this step appropriate data is extracted and feed into the computer.
3. Storage: The data entered into the computer must be stored before starting the processing.
4. Processing/Manipulating data: It is a laborious work. It consists of various steps like computations, classification, comparison, summarization, etc. that converts input into output.
5. Output of information: In this stage we will get the results as information after processing the data.
6. Distribution of information: In this phase the information(result) will be given to the concerned persons/computers.
Question 2.
Define computer. What are the characteristics?
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# Area of a trapezoid from the area of triangles created by the diagonals
In the trapezoid $ABCD$ where $AB||CD$, $S$ is the point of intersection of the diagonals. What is the area of the trapezoid if the area of the triangles ADS and ABS is respectively 5 and 7?
I know that the areas of the ADS and the CBS are equal, but I don't know how to get the area of the CDS, even though I know it is similar to the ABS. I suppose you could use the equation from finding the area of a trapezoid using only 2 of the 4 triangles that makes up its interior but I don't know the name of the theorem and would prefer to not use it. Instead, I would like to apply the law of cosines or of sines, but I have no idea how to do it here.
triangles ADS and ABS have same height (drawn from A). so $DS:BS = 5:7$, similarly $CS:AS=5:7$, triangles ABS and CDS are similar (same angles) So $DC:AB = 5:7$. Note that the height of CDS and ABS (drawn from S) are also in the ratio of $5:7$, so area of CDS =$(\dfrac57)(\dfrac57)$ multiplied by area of ABS that is $(\dfrac57)(\dfrac57)7 = \dfrac{25}7$.
• it would be great if you used mathjax to format your answer . here is a quick tutorial on how to use it Jun 5 '18 at 18:57
• @The Integrator, Thanks I am trying to use it. Jun 5 '18 at 19:14
• Note: you can type fractions using \frac{}{} for example \frac{a}{b} gives $\frac{a}{b}$ Jun 5 '18 at 19:19
• @The Integrator, thanks! Jun 6 '18 at 20:38
This is what I got. I used similar triangles and ratio of lengths to get the area ratio.
Area $\Delta$ADS = Area$\Delta$BCS ($\Delta$ABD $-7 = \Delta$ABC$– 7 = 5)$
$\frac{1}{2}AB(h-x) = 7$
$\frac{1}{2}AB(h) = 12$
$\frac{(h-x)}{h} = \frac{7}{12}$
$7h = 12h – 12x$
$5h = 12x$
$x = \frac{5h}{12}$
Therefore $h-x = \frac{7h}{12}$
Because $\Delta$CDS is similar to $\Delta$ABS (three angles the same), then the ratio of their lengths is $\frac{5}{7}$
The ratio of their areas is $\frac{25}{49}$. The area of $\Delta$CDS is therefore $7(\frac{25}{49}) = \frac{25}{7}$
The area of the trapezoid is therefore $5+5+7+\frac{25}{7} = 20\frac{4}{7}$
• it would be great if you used mathjax to format your answer . here is a quick tutorial on how to use it Jun 5 '18 at 20:15
• @The Integrator. My answer was done in MS Word as it includes a graphic and is a picture file. I know how to use mathjax but that doesn't work in Word. I guess I should do the extra work and do the actual calculation in the answer space. Thanks for the reminder. Jun 5 '18 at 20:21
• you could just include the graphic but type the calculations, would be good Jun 5 '18 at 20:25
Here is a solution without using the law of sines and cosines.
Denote by $PQ$ the distance between two points $P$ and $Q$ Notice that the are of the trapezoid is given by $$\frac{AB + CD }{2} \cdot r,$$
$r$ being the orthogonal distance between $AB$ and $CD$ as shown in the picture.
The Triangle $ABD$ has an area of $7+5$, thus we get $$\frac{AB}{2} \cdot r = 12.$$
Therefore $r = \frac{24}{AB}$. Plugging this into $\frac{CD}{2}\cdot r$, we get $$\frac{CD}{2}\cdot r = \frac{CD \cdot 12}{AB}$$
By the intercept theorem $$\frac{CD}{AB} = \frac{SD}{BS} = \frac{SD}{BS} \cdot \frac{h/2}{h/2} = \frac{5}{7},$$ where $h$ is the orthogonal distance between $A$ and $BD$.
Thus
$$\frac{AB + CD }{2} \cdot r = \frac{AB}{2} \cdot r + \frac{CD}{2}\cdot r = \frac{AB}{2} \cdot r + \frac{CD \cdot 12}{AB} = 12 + 12 \cdot \frac{5}{7} =\frac{144}{7}.$$
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### April 2017
This is part 27 of Categories for Programmers. Previously: Ends and Coends. See the Table of Contents.
So far we’ve been mostly working with a single category or a pair of categories. In some cases that was a little too constraining. For instance, when defining a limit in a category C, we introduced an index category `I` as the template for the pattern that would form the basis for our cones. It would have made sense to introduce another category, a trivial one, to serve as a template for the apex of the cone. Instead we used the constant functor `Δc` from `I` to `C`.
It’s time to fix this awkwardness. Let’s define a limit using three categories. Let’s start with the functor `D` from the index category I to C. This is the functor that selects the base of the cone — the diagram functor.
The new addition is the category 1 that contains a single object (and a single identity morphism). There is only one possible functor `K` from I to this category. It maps all objects to the only object in 1, and all morphisms to the identity morphism. Any functor `F` from 1 to C picks a potential apex for our cone.
A cone is a natural transformation `ε` from `F ∘ K` to `D`. Notice that `F ∘ K` does exactly the same thing as our original `Δc`. The following diagram shows this transformation.
We can now define a universal property that picks the “best” such functor `F`. This `F` will map 1 to the object that is the limit of `D` in C, and the natural transformation `ε` from `F ∘ K` to `D` will provide the corresponding projections. This universal functor is called the right Kan extension of `D` along `K` and is denoted by `RanKD`.
Let’s formulate the universal property. Suppose we have another cone — that is another functor `F'` together with a natural transformation `ε'` from `F' ∘ K` to `D`.
If the Kan extension `F = RanKD` exists, there must be a unique natural transformation `σ` from `F'` to it, such that `ε'` factorizes through `ε`, that is:
`ε' = ε . (σ ∘ K)`
Here, `σ ∘ K` is the horizontal composition of two natural transformations (one of them being the identity natural transformation on `K`). This transformation is then vertically composed with `ε`.
In components, when acting on an object `i` in I, we get:
`ε'i = εi ∘ σ K i`
In our case, `σ` has only one component corresponding to the single object of 1. So, indeed, this is the unique morphism from the apex of the cone defined by `F'` to the apex of the universal cone defined by `RanKD`. The commuting conditions are exactly the ones required by the definition of a limit.
But, importantly, we are free to replace the trivial category 1 with an arbitrary category A, and the definition of the right Kan extension remains valid.
## Right Kan Extension
The right Kan extension of the functor `D::I->C` along the functor `K::I->A` is a functor `F::A->C` (denoted `RanKD`) together with a natural transformation
`ε :: F ∘ K -> D`
such that for any other functor `F'::A->C` and a natural transformation
`ε' :: F' ∘ K -> D`
there is a unique natural transformation
`σ :: F' -> F`
that factorizes `ε'`:
`ε' = ε . (σ ∘ K)`
This is quite a mouthful, but it can be visualized in this nice diagram:
An interesting way of looking at this is to notice that, in a sense, the Kan extension acts like the inverse of “functor multiplication.” Some authors go as far as use the notation `D/K` for `RanKD`. Indeed, in this notation, the definition of `ε`, which is also called the counit of the right Kan extension, looks like simple cancellation:
`ε :: D/K ∘ K -> D`
There is another interpretation of Kan extensions. Consider that the functor `K` embeds the category I inside A. In the simplest case I could just be a subcategory of A. We have a functor `D` that maps I to C. Can we extend `D` to a functor `F` that is defined on the whole of A? Ideally, such an extension would make the composition `F ∘ K` be isomorphic to `D`. In other words, `F` would be extending the domain of `D` to `A`. But a full-blown isomorphism is usually too much to ask, and we can do with just half of it, namely a one-way natural transformation `ε` from `F ∘ K` to `D`. (The left Kan extension picks the other direction.)
Of course, the embedding picture breaks down when the functor `K` is not injective on objects or not faithful on hom-sets, as in the example of the limit. In that case, the Kan extension tries its best to extrapolate the lost information.
## Kan Extension as Adjunction
Now suppose that the right Kan extension exists for any `D` (and a fixed `K`). In that case `RanK -` (with the dash replacing `D`) is a functor from the functor category `[I, C]` to the functor category `[A, C]`. It turns out that this functor is the right adjoint to the precomposition functor `-∘K`. The latter maps functors in `[A, C]` to functors in `[I, C]`. The adjunction is:
`[I, C](F' ∘ K, D) ≅ [A, C](F', RanKD)`
It is just a restatement of the fact that to every natural transformation we called `ε'` corresponds a unique natural transformation we called `σ`.
Furthermore, if we chose the category I to be the same as C, we can substitute the identity functor `IC` for `D`. We get the following identity:
`[C, C](F' ∘ K, IC) ≅ [A, C](F', RanKIC)`
We can now chose `F'` to be the same as `RanKIC`. In that case the right hand side contains the identity natural transformation and, corresponding to it, the left hand side gives us the following natural transformation:
`ε :: RanKIC ∘ K -> IC`
This looks very much like the counit of an adjunction:
`RanKIC ⊣ K`
Indeed, the right Kan extension of the identity functor along a functor `K` can be used to calculate the left adjoint of `K`. For that, one more condition is necessary: the right Kan extension must be preserved by the functor `K`. The preservation of the extension means that, if we calculate the Kan extension of the functor precomposed with `K`, we should get the same result as precomposing the original Kan extesion with `K`. In our case, this condition simplifies to:
`K ∘ RanKIC ≅ RanKK`
Notice that, using the division-by-K notation, the adjunction can be written as:
`I/K ⊣ K`
which confirms our intuition that an adjunction describes some kind of an inverse. The preservation condition becomes:
`K ∘ I/K ≅ K/K`
The right Kan extension of a functor along itself, `K/K`, is called a codensity monad.
The adjunction formula is an important result because, as we’ll see soon, we can calculate Kan extensions using ends (coends), thus giving us practical means of finding right (and left) adjoints.
## Left Kan Extension
There is a dual construction that gives us the left Kan extension. To build some intuition, we’ll can start with the definition of a colimit and restructure it to use the singleton category 1. We build a cocone by using the functor `D::I->C` to form its base, and the functor `F::1->C` to select its apex.
The sides of the cocone, the injections, are components of a natural transformation `η` from `D` to `F ∘ K`.
The colimit is the universal cocone. So for any other functor `F'` and a natural transformation
`η' :: D -> F'∘ K`
there is a unique natural transformation `σ` from `F` to `F'`
such that:
`η' = (σ ∘ K) . η`
This is illustrated in the following diagram:
Replacing the singleton category 1 with A, this definition naturally generalized to the definition of the left Kan extension, denoted by `LanKD`.
The natural transformation:
`η :: D -> LanKD ∘ K`
is called the unit of the left Kan extension.
As before, we can recast the one-to-one correspondence between natural transformations:
`η' = (σ ∘ K) . η`
in terms of the adjunction:
`[A, C](LanKD, F') ≅ [I, C](D, F' ∘ K)`
In other words, the left Kan extension is the left adjoint, and the right Kan extension is the right adjoint of the precomposition with `K`.
Just like the right Kan extension of the identity functor could be used to calculate the left adjoint of `K`, the left Kan extension of the identity functor turns out to be the right adjoint of `K` (with `η` being the unit of the adjunction):
`K ⊣ LanKIC`
Combining the two results, we get:
`RanKIC ⊣ K ⊣ LanKIC`
## Kan Extensions as Ends
The real power of Kan extensions comes from the fact that they can be calculated using ends (and coends). For simplicity, we’ll restrict our attention to the case where the target category C is Set, but the formulas can be extended to any category.
Let’s revisit the idea that a Kan extension can be used to extend the action of a functor outside of its original domain. Suppose that `K` embeds I inside A. Functor `D` maps I to Set. We could just say that for any object `a` in the image of `K`, that is `a = K i`, the extended functor maps `a` to `D i`. The problem is, what to do with those objects in A that are outside of the image of `K`? The idea is that every such object is potentially connected through lots of morphisms to every object in the image of `K`. A functor must preserve these morphisms. The totality of morphisms from an object `a` to the image of `K` is characterized by the hom-functor:
`A(a, K -)`
Notice that this hom-functor is a composition of two functors:
`A(a, K -) = A(a, -) ∘ K`
The right Kan extension is the right adjoint of functor composition:
`[I, Set](F' ∘ K, D) ≅ [A, Set](F', RanKD)`
Let’s see what happens when we replace `F'` with the hom functor:
`[I, Set](A(a, -) ∘ K, D) ≅ [A, Set](A(a, -), RanKD)`
and then inline the composition:
`[I, Set](A(a, K -), D) ≅ [A, Set](A(a, -), RanKD)`
The right hand side can be reduced using the Yoneda lemma:
`[I, Set](A(a, K -), D) ≅ RanKD a`
We can now rewrite the set of natural transformations as the end to get this very convenient formula for the right Kan extension:
`RanKD a ≅ ∫i Set(A(a, K i), D i)`
There is an analogous formula for the left Kan extension in terms of a coend:
`LanKD a = ∫i A(K i, a) × D i`
To see that this is the case, we’ll show that this is indeed the left adjoint to functor composition:
`[A, Set](LanKD, F') ≅ [I, Set](D, F'∘ K)`
Let’s substitute our formula in the left hand side:
`[A, Set](∫i A(K i, -) × D i, F')`
This is a set of natural transformations, so it can be rewritten as an end:
`∫a Set(∫i A(K i, a) × D i, F'a)`
Using the continuity of the hom-functor, we can replace the coend with the end:
`∫a ∫i Set(A(K i, a) × D i, F'a)`
We can use the product-exponential adjunction:
`∫a ∫i Set(A(K i, a), (F'a)D i)`
The exponential is isomorphic to the corresponding hom-set:
`∫a ∫i Set(A(K i, a), A(D i, F'a))`
There is a theorem called the Fubini theorem that allows us to swap the two ends:
`∫i ∫a Set(A(K i, a), A(D i, F'a))`
The inner end represents the set of natural transformations between two functors, so we can use the Yoneda lemma:
`∫i A(D i, F'(K i))`
This is indeed the set of natural transformations that forms the right hand side of the adjunction we set out to prove:
`[I, Set](D, F'∘ K)`
These kinds of calculations using ends, coends, and the Yoneda lemma are pretty typical for the “calculus” of ends.
## Kan Extensions in Haskell
The end/coend formulas for Kan extensions can be easily translated to Haskell. Let’s start with the right extension:
`RanKD a ≅ ∫i Set(A(a, K i), D i)`
We replace the end with the universal quantifier, and hom-sets with function types:
`newtype Ran k d a = Ran (forall i. (a -> k i) -> d i)`
Looking at this definition, it’s clear that `Ran` must contain a value of type `a` to which the function can be applied, and a natural transformation between the two functors `k` and `d`. For instance, suppose that `k` is the tree functor, and `d` is the list functor, and you were given a `Ran Tree [] String`. If you pass it a function:
`f :: String -> Tree Int`
you’ll get back a list of `Int`, and so on. The right Kan extension will use your function to produce a tree and then repackage it into a list. For instance, you may pass it a parser that generates a parsing tree from a string, and you’ll get a list that corresponds to the depth-first traversal of this tree.
The right Kan extension can be used to calculate the left adjoint of a given functor by replacing the functor `d` with the identity functor. This leads to the left adjoint of a functor `k` being represented by the set of polymorphic functions of the type:
`forall i. (a -> k i) -> i`
Suppose that `k` is the forgetful functor from the category of monoids. The universal quantifier then goes over all monoids. Of course, in Haskell we cannot express monoidal laws, but the following is a decent approximation of the resulting free functor (the forgetful functor `k` is an identity on objects):
`type Lst a = forall i. Monoid i => (a -> i) -> i`
As expected, it generates free monoids, or Haskell lists:
```toLst :: [a] -> Lst a
toLst as = \f -> foldMap f as
fromLst :: Lst a -> [a]
fromLst f = f (\a -> [a])```
The left Kan extension is a coend:
`LanKD a = ∫i A(K i, a) × D i`
so it translates to an existential quantifier. Symbolically:
`Lan k d a = exists i. (k i -> a, d i)`
This can be encoded in Haskell using GADTs, or using a universally quantified data constructor:
`data Lan k d a = forall i. Lan (k i -> a) (d i)`
The interpretation of this data structure is that it contains a function that takes a container of some unspecified `i`s and produces an `a`. It also has a container of those `i`s. Since you have no idea what `i`s are, the only thing you can do with this data structure is to retrieve the container of `i`s, repack it into the container defined by the functor `k` using a natural transformation, and call the function to obtain the `a`. For instance, if `d` is a tree, and `k` is a list, you can serialize the tree, call the function with the resulting list, and obtain an `a`.
The left Kan extension can be used to calculate the right adjoint of a functor. We know that the right adjoint of the product functor is the exponential, so let’s try to implement it using the Kan extension:
`type Exp a b = Lan ((,) a) I b`
This is indeed isomorphic to the function type, as witnessed by the following pair of functions:
```toExp :: (a -> b) -> Exp a b
toExp f = Lan (f . fst) (I ())
fromExp :: Exp a b -> (a -> b)
fromExp (Lan f (I x)) = \a -> f (a, x)```
Notice that, as described earlier in the general case, we performed the following steps: (1) retrieved the container of `x` (here, it’s just a trivial identity container), and the function `f`, (2) repackaged the container using the natural transformation between the identity functor and the pair functor, and (3) called the function `f`.
## Free Functor
An interesting application of Kan extensions is the construction of a free functor. It’s the solution to the following practical problem: suppose you have a type constructor — that is a mapping of objects. Is it possible to define a functor based on this type constructor? In other words, can we define a mapping of morphisms that would extend this type constructor to a full-blown endofunctor?
The key observation is that a type constructor can be described as a functor whose domain is a discrete category. A discrete category has no morphisms other than the identity morphisms. Given a category C, we can always construct a discrete category |C| by simply discarding all non-identity morphisms. A functor `F` from |C| to C is then a simple mapping of objects, or what we call a type constructor in Haskell. There is also a canonical functor `J` that injects |C| into C: it’s an identity on objects (and on identity morphisms). The left Kan extension of `F` along `J`, if it exists, is then a functor for C to C:
`LanJ F a = ∫i C(J i, a) × F i`
It’s called a free functor based on `F`.
In Haskell, we would write it as:
`data FreeF f a = forall i. FMap (i -> a) (f i)`
Indeed, for any type constructor `f`, `FreeF f` is a functor:
```instance Functor (FreeF f) where
fmap g (FMap h fi) = FMap (g . h) fi```
As you can see, the free functor fakes the lifting of a function by recording both the function and its argument. It accumulates the lifted functions by recording their composition. Functor rules are automatically satisfied. This construction was used in a paper Freer Monads, More Extensible Effects.
Alternatively, we can use the right Kan extension for the same purpose:
`newtype FreeF f a = FreeF (forall i. (a -> i) -> f i)`
It’s easy to check that this is indeed a functor:
```instance Functor (FreeF f) where
fmap g (FreeF r) = FreeF (\bi -> r (bi . g))```
Next: Enriched Categories.
## The Free Theorem for Ends
In Haskell, the end of a profunctor `p` is defined as a product of all diagonal elements:
`forall c. p c c`
together with a family of projections:
```pi :: Profunctor p => forall c. (forall a. p a a) -> p c c
pi e = e```
In category theory, the end must also satisfy the edge condition which, in (type-annotated) Haskell, could be written as:
`dimap f idb . pib = dimap ida f . pia`
for any `f :: a -> b`.
Using a suitable formulation of parametricity, this equation can be shown to be a free theorem. Let’s first review the free theorem for functors before generalizing it to profunctors.
## Functor Characterization
You may think of a functor as a container that has a shape and contents. You can manipulate the contents without changing the shape using `fmap`. In general, when applying `fmap`, you not only change the values stored in the container, you change their type as well. To really capture the shape of the container, you have to consider not only all possible mappings, but also more general relations between different contents.
A function is directional, and so is `fmap`, but relations don’t favor either side. They can map multiple values to the same value, and they can map one value to multiple values. Any relation on values induces a relation on containers. For a given functor `F`, if there is a relation `a` between type `A` and type `A'`:
`A <=a=> A'`
then there is a relation between type `F A` and `F A'`:
`F A <=(F a)=> F A'`
We call this induced relation `F a`.
For instance, consider the relation between students and their grades. Each student may have multiple grades (if they take multiple courses) so this relation is not a function. Given a list of students and a list of grades, we would say that the lists are related if and only if they match at each position. It means that they have to be equal length, and the first grade on the list of grades must belong to the first student on the list of students, and so on. Of course, a list is a very simple container, but this property can be generalized to any functor we can define in Haskell using algebraic data types.
The fact that `fmap` doesn’t change the shape of the container can be expressed as a “theorem for free” using relations. We start with two related containers:
```xs :: F A
xs':: F A'```
where `A` and `A'` are related through some relation `a`. We want related containers to be `fmap`ped to related containers. But we can’t use the same function to map both containers, because they contain different types. So we have to use two related functions instead. Related functions map related types to related types so, if we have:
```f :: A -> B
f':: A'-> B'```
and `A` is related to `A'` through `a`, we want `B` to be related to `B'` through some relation `b`. Also, we want the two functions to map related elements to related elements. So if `x` is related to `x'` through `a`, we want `f x` to be related to `f' x'` through `b`. In that case, we’ll say that `f` and `f'` are related through the relation that we call `a->b`:
`f <=(a->b)=> f'`
For instance, if `f` is mapping students’ SSNs to last names, and `f'` is mapping letter grades to numerical grades, the results will be related through the relation between students’ last names and their numerical grades.
To summarize, we require that for any two relations:
```A <=a=> A'
B <=b=> B'```
and any two functions:
```f :: A -> B
f':: A'-> B'```
such that:
`f <=(a->b)=> f'`
and any two containers:
```xs :: F A
xs':: F A'```
we have:
```if xs <=(F a)=> xs'
then F xs <=(F b)=> F xs'```
This characterization can be extended, with suitable changes, to contravariant functors.
## Profunctor Characterization
A profunctor is a functor of two variables. It is contravariant in the first variable and covariant in the second. A profunctor can lift two functions simultaneously using `dimap`:
```class Profunctor p where
dimap :: (a -> b) -> (c -> d) -> p b c -> p a d```
We want `dimap` to preserve relations between profunctor values. We start by picking any relations `a`, `b`, `c`, and `d` between types:
```A <=a=> A'
B <=b=> B'
C <=c=> C'
D <=d=> D'
```
For any functions:
```f :: A -> B
f' :: A'-> B'
g :: C -> D
g' :: C'-> D'```
that are related through the following relations induced by function types:
```f <=(a->b)=> f'
g <=(c->d)=> g'```
we define:
```xs :: p B C
xs':: p B'C'```
The following condition must be satisfied:
```if xs <=(p b c)=> xs'
then (p f g) xs <=(p a d)=> (p f' g') xs'
```
where `p f g` stands for the lifting of the two functions by the profunctor `p`.
Here’s a quick sanity check. If `b` and `c` are functions:
```b :: B'-> B
c :: C -> C'```
than the relation:
`xs <=(p b c)=> xs'`
becomes:
```xs' = dimap b c xs
```
If `a` and `d` are functions:
```a :: A'-> A
d :: D -> D'
```
then these relations:
```f <=(a->b)=> f'
g <=(c->d)=> g'```
become:
```f . a = b . f'
d . g = g'. c```
and this relation:
`(p f g) xs <=(p a d)=> (p f' g') xs'`
becomes:
`(p f' g') xs' = dimap a d ((p f g) xs)`
Substituting `xs'`, we get:
`dimap f' g' (dimap b c xs) = dimap a d (dimap f g xs)`
and using functoriality:
```dimap (b . f') (g'. c) = dimap (f . a) (d . g)
```
which is identically true.
## Special Case of Profunctor Characterization
We are interested in the diagonal elements of a profunctor. Let’s first specialize the general case to:
```C = B
C'= B'
c = b```
to get:
```xs = p B B
xs'= p B'B'```
and
```if xs <=(p b b)=> xs'
then (p f g) xs <=(p a d)=> (p f' g') xs'
```
Chosing the following substitutions:
```A = A'= B
D = D'= B'
a = id
d = id
f = id
g'= id
f'= g```
we get:
```if xs <=(p b b)=> xs'
then (p id g) xs <=(p id id)=> (p g id) xs'
```
Since `p id id` is the identity relation, we get:
`(p id g) xs = (p g id) xs'`
or
`dimap id g xs = dimap g id xs'`
## Free Theorem
We apply the free theorem to the term `xs`:
`xs :: forall c. p c c`
It must be related to itself through the relation that is induced by its type:
`xs <=(forall b. p b b)=> xs`
for any relation `b`:
`B <=b=> B'`
Universal quantification translates to a relation between different instantiations of the polymorphic value:
`xsB <=(p b b)=> xsB'`
Notice that we can write:
```xsB = piB xs
xsB'= piB'xs```
using the projections we defined earlier.
We have just shown that this equation leads to:
`dimap id g xs = dimap g id xs'`
which shows that the wedge condition is indeed a free theorem.
## Natural Transformations
Here’s another quick application of the free theorem. The set of natural transformations may be represented as an end of the following profunctor:
`type NatP a b = F a -> G b`
```instance Profunctor NatP where
dimap f g alpha = fmap g . alpha . fmap f```
The free theorem tells us that for any `mu :: NatP c c`:
`(dimap id g) mu = (dimap g id) mu`
which is the naturality condition:
`mu . fmap g = fmap g . mu`
It’s been know for some time that, in Haskell, naturality follows from parametricity, so this is not surprising.
## Acknowledgment
I’d like to thank Edward Kmett for reviewing the draft of this post.
## Bibliography
1. Bartosz Milewski, Ends and Coends
2. Edsko de Vries, Parametricity Tutorial, Part 1, Part 2, Contravariant Functions.
3. Bartosz Milewski, Parametricity: Money for Nothing and Theorems for Free
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# Show Reference: "A Gentle Tutorial of the {EM} Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models"
A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models (April 1998) by Jeff A. Bilmes
@techreport{bilmes-1998,
abstract = {We describe the maximum-likelihood parameter estimation problem and how the {ExpectationMaximization} ({EM}) algorithm can be used for its solution. We first describe the abstract
form of the {EM} algorithm as it is often given in the literature. We then develop the {EM} parameter estimation procedure for two applications: 1) finding the parameters of a mixture of
Gaussian densities, and 2) finding the parameters of a hidden Markov model ({HMM}) (i.e.,
the {Baum-Welch} algorithm) for both discrete and Gaussian mixture observation models.
We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor.},
author = {Bilmes, Jeff A.},
institution = {International Computer Science Institute},
keywords = {algorithmic, bayes, learning, math, probability, unsupervised-learning},
month = apr,
posted-at = {2012-01-06 11:07:32},
priority = {2},
title = {A Gentle Tutorial of the {EM} Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models},
url = {http://facartes.unal.edu.co/ggonzalez/ml/bilmes98gentle.pdf},
year = {1998}
}
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{}
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# $\cot{(90^°)}$ value
The cot value when angle of a right triangle equals to $90^°$ is called cot of angle $90$ degrees and it is written as $\cot{(90^°)}$ mathematically in sexagesimal system.
$\cot{(90^°)} \,=\, 0$
The exact value of $\cot{(90^°)}$ is zero mathematically.
## Alternative form
The $\cot{(90^°)}$ is written in different ways in alternative form. In other words, it is written as $\cot{\Big(\dfrac{\pi}{2}\Big)}$ in circular system and also written as $\cot{(100^g)}$ in centesimal system.
$(1) \,\,\,$ $\cot{\Big(\dfrac{\pi}{2}\Big)} \,=\, 0$
$(2) \,\,\,$ $\cot{(100^g)} \,=\, 0$
### Proof
You learnt that the value of $\cot{\Big(\dfrac{\pi}{2}\Big)}$ is zero exactly. Now, it is time to learn how the value of $\cot{(100^g)}$ is zero exactly in trigonometry.
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# What is the origin of λ for empty string?
I usually use the symbol $\varepsilon$ for empty string (empty word or null string). But I know some people use $\lambda$ instead of $\varepsilon$.
I think $\varepsilon$ is derived from the word "Empty". However I don't know what's the origin of $\lambda$.
In automata theory, there is the epsilon transition of automata, and it's also said to be the lambda transition. For example, JFLAP software uses $\lambda$ for the label of epsilon transitions by default.
I googled on the origin and searched cs.stackexchange, but I couldn't find. Does anyone know a reference that describes this?
The German Wikipedia claims that $\lambda$ comes from "leer", which means "empty" in German. That seems plausible, as German used to be one of the major languages in mathematics.
Chomsky used $I$ as the empty string (or actually as the identity element for string concatenation) in his early papers. Some people in combinatorics still use $1$ as the empty string, with the same justification.
• 1 is particularly nice when you're defining Regular Expressions algebraically. 1 is the empty string, 0 is the empty language, concatenation is $\cdot$ and union is $+$, and you get a Semi-ring. $*$ makes things a bit more complicated though. – jmite Oct 19 '16 at 18:21
• Thank you for the answer! It seems to be plausible, so I'm searching for the reference. Since the article of Formal Language of Wikipedia says the origin of FL is Gottlob Frege's Begriffsschrift (1879), I read the translated version of it today, but it doesn't seem to use the λ notation. Another historical paper Recursive Unsolvability of a Problem of Thue by Emil Post (1947) doesn't, either. Therefore I keep searching. Anyway, thanks for the big help :) – nekketsuuu Oct 20 '16 at 11:13
Probably the notation originates from the "Finnish school".
My copy of 'Formal Languages' by Arto Salomaa (Academic Press, ACM monograph series, 1973) uses $\lambda$ for the empty string. And so does his 1969 book 'Theory of Automata' (Pergamon Press).
We move back. The classic 'Finite Automata and Their Decision Problems' by M.O. Rabin and D.Scott (April 1959) have the notation (capital) $\Lambda$ for "the empty tape with no symbols" (where a tape is a finite sequence of symbols).
One of the early people to write on finite automata was Trakhtenbrot and he used a symbol much like $\Lambda$ but typeset as $\land$ (as in his book with Barzdin, 1970, my russian is lousy but I recognize $\land p= p\land=p$).
• IIRC, in Principia Mathematica (around 1910), Russell used $\Lambda$ for the empty set. I have no idea if this is somehow related. – chi Oct 20 '16 at 14:21
• @chi The books of Knuth are using $\Lambda$ for the nil-pointer. There must be a history of having $\Lambda$ mean "nothing". Might be related with the other answer where it is suggested that it stands for "Leer" or empty in German. – Hendrik Jan Oct 28 '16 at 23:15
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Postman Cheat Sheet
PostmanREST APISOAP API
This Postman Cheat Sheet is based on the official documentation page of Postman (which is available in the below link) and from the overall knowledge on Postman −
https://learning.postman.com/docs/getting-started/introduction/
a. Variables
All the variables can be set up manually from the GUI of Postman and they have a defined scope. The values of the variables can also be set with the help of scripts written under the Pre-request Script or Tests tab.
The variables can be added in the request URL, Headers and Body in the format as {{<variable name>}}.
Usage in request URL −
https://{{domain}}about/{{id}}
X-{{key}}:value
Usage in request Body −
{"registration_id": "{{Id}}", "firstname": "Postman"}
b. Global Variables
The Global variables are used when we need to send data to other requests. The script to add a Global variable can be included either in the Tests or Pre-request Script tab in Postman.
To set a Global variable −
pm.globals.set('<name of Global variable>', '<value of variable>')
To get the value of a Global variable −
pm.globals.get('<name of Global variable>')
To delete a Global variable, the script is −
pm.globals.unset('<name of Global variable>')
To delete all Global variable, the script is −
pm.globals.clear()
c. Collection Variables
Collection variables are a good alternative to the Global and Environment variables. They can also be used for URLs/ authentication credentials in case there is only one Environment. The script to add a Collection variable can be included either in the Tests or Pre-request Script tab in Postman.
To set a Collection variable −
pm.CollectionVariables.set('<name of variable>', '<value of variable>')
To get the value of a Collection variable −
pm.CollectionVariables.get('<name of Collection variable>')
To delete a Collection variable, the script is−
pm.CollectionVariables.unset('<name of Collection variable>')
d. Environment Variables
The Environment variables are used for a particular Environment. They are a good alternative to the Global variables as they have a limited scope. The script to add a Global variable can be included either in the Tests or Pre-request Script tab in Postman.
Environment variables are used to hold variables specific for an Environment, URLs, and to send data to other requests.
To set an Environment variable −
pm.environment.set('<name of Environment variable>', '<value of variable>')
To get an Environment variable −
pm.environment.get('<name of Environment variable>')
To delete an Environment variable −
pm.environment.unset('<name of Environment variable>')
To delete all Environment variables, the script is −
pm.environment.clear()
To get the name of the active Environment, the script is −
pm.environment.name
e. Data Variables
Data variables are used for execution of a particular iteration in a Collection Runner or Newman. They are mainly used where there are multiple data-sets from a CSV/JSON file.
To get the value of a Data variable −
pm.iterationData.get('<name of Data variable>')
f. Local Variables
Local variables can be accessed within a request or while executing via Collection Runner/Newman. These variables are removed by default once the request has been executed.
To set a Local variable −
pm.variables.set('<name of Local variable>', '<value of variable>')
To get a Local variable −
pm.variables.get('<name of Local variable>')
g. Dynamic Variables
Dynamic variables can be used with strings to generate dynamic and distinct data.
Example of dynamic variable in JSON body −
{"email": "test.{{\$timestamp}}@gmail.com"}
h. Debugging Variables
Launch the Postman Console and add console.log in the scripts under the Pre-request Scripts or Tests tab to debug a variable.
console.log(pm.globals.get('< name of Global variable >')
i. Assertions
Assertions are added under the block with pm.test callback.
pm.test("Response Status Code", function () {
pm.response.to.have.status(201)
})
j. Skipping Tests
To skip a test, we have to add pm.test.skip. The skipped tests are visible on the reports.
pm.test.skip("Response Status Code", function () {
pm.response.to.have.status(201)
})
k. Failing Tests
In Postman, we can explicitly fail a test without adding an Assertion.
pm.expect.fail('Scenario is failed');
Published on 25-Jun-2021 13:21:54
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# Python – Sort by Maximum digit in Element
PythonServer Side ProgrammingProgramming
When it is required to sort by maximum digit in element, a method is defined that uses ‘str’ and ‘max’ method to determine the result.
Below is a demonstration of the same −
## Example
Live Demo
def max_digits(element):
return max(str(element))
my_list = [224, 192, 145, 18, 3721]
print("The list is :")
print(my_list)
my_list.sort(key = max_digits)
print("The result is :")
print(my_list)
## Output
The list is :
[224, 192, 145, 18, 3721]
The result is :
[224, 145, 3721, 18, 192]
## Explanation
• A method named ‘max_digits’ is defined that takes element as a parameter, and converts it into a string, and then gets the maximum of it, and returns this as output.
• Outside the method, a list is defined and displayed on the console.
• The list is sorted using ‘sort’ method and the key is specified as the previously defined method.
• This is the output that is displayed on the console.
Published on 06-Sep-2021 08:49:59
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# The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?
Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) \cup I_{K[y]}(B)\rangle\rangle?$$ Here $I_{K[x]}(A)=\{f\in K[x]; f(a)\!=\!0\text{ for all }a\!\in\!A\}$ is the vanishing ideal of set $A$, and $\langle\langle\ldots\rangle\rangle$ is the ideal, generated by $\ldots$. The inclusion $\supseteq$ is easy, but I don't see how to show $\subseteq$.
If this equality does not hold, how else then can I prove (knowing $$K[x,y]/\langle\langle\mathfrak{a},\mathfrak{b}\rangle\rangle \,\cong\, K[x]/\mathfrak{a}\otimes_KK[y]/\mathfrak{b}$$ from this post), that there is an isomorphism of $K$-algebras (coordinate rings) $$K[A\!\times\!B] \,\cong\, K[A]\otimes_K K[B]?$$
-
This is well-known and can be found in any some introductions to classical algebraic geometry, in the section about products. Anyway, here is my favorite proof. I assume that $k$ is algebraically closed (otherwise it is wrong).
We have $V(I(A \times B))=A \times B = (A \times \mathbb{A}^n) \cap (\mathbb{A}^m \times B) = V(I(A) \cup I(B))$, thus $I(A \times B) = \sqrt{I(A) + I(B)}$. We have to prove that $I(A) + I(B)$ is a radical ideal, or equivalently that $k[x,y]/(I(A)+I(B))=k[X]/I(A) \otimes k[Y]/I(B)$ is reduced (i.e. $0$ is the only nilpotent element).
Lemma. If $k$ is an algebraically closed field and $R,S$ are reduced $k$-algebras, then $R \otimes_k S$ is reduced.
Proof: A colimit argument shows that we may assume that $R$ is of finite type over $k$. Since $R$ is reduced, the intersection of all prime ideals is $0$, which equals the intersection of all maximal ideals since $R$ is jacobson. This gives an embedding $R \hookrightarrow \prod_{\mathfrak{m} \in \mathrm{Spm}(R)} R/\mathfrak{m}$, where $R/\mathfrak{m}=k$. This induces an embedding of $k$-algebras
$$R \otimes_k S \hookrightarrow (\prod_{\mathfrak{m} \in \mathrm{Spm}(R)} k) \otimes_k S \hookrightarrow \prod_{\mathfrak{m} \in \mathrm{Spm}(R)} (k \otimes_k S) = \prod_{\mathfrak{m} \in \mathrm{Spm}(R)} S.$$ Therefore $R \otimes_k S$ is a subring of a product of reduced algebras, therefore also reduced. $~\square$
If $k$ is not algebraically closed, the Lemma fails, even for fields. In fact, for a polynomial $f \in k[x]$ with splitting field $L$ the tensor product $k[x]/(f) \otimes_k L$ is isomorphic to the product of the algebras $L[x]/(x-\alpha)^{v_{\alpha}}$, where $\alpha$ runs through the roots of $f$ and $v_\alpha$ is its multiplicity. This algebra is reduced iff $v_\alpha=1$ for all $\alpha$ iff $f$ is separable. For example, $\mathbb{F}_p(t) \otimes_{\mathbb{F}_p(t^p)} \mathbb{F}_p(t) = \mathbb{F}_p(t)[x]/(x-t)^p$ is not reduced. The Lemma also fails when $k$ has characteristic zero, but then there is no counterexample for fields.
By the way, the isomorphism $k[A \times B] \cong k[A] \otimes_k k[B]$ holds almost by definition for affine schemes $A,B$. In this context the Lemma translates to the statement that the product of two reduced $k$-schemes is again reduced.
-
What books for Classical Algebraic Geometry do you suggest? Hassett, Introduction to Algebraic Geometry, Beltrametti & Carletti & Gallarati, Lectures on Curves, Surfaces and Projective Varieties, Cox & Little & O'Shea, Using Algebraic Geometry, Perrin, Algebraic Geometry, Holme, A Royal Road to Algebraic Geometry, don't have what I need. – Leon Feb 4 '13 at 19:40
Ok I agree, it is not so easy to find in the literature. I've also looked at the books by Eisenbud-Harris, Ueno, Griffiths-Harris, Hartshorne, Shafarevich, Gathmann, but nowhere it is proved that $I+J \subseteq k[x,y]$ is a radical ideal when $I \subseteq k[x]$ and $J \subseteq k[y]$ are radical ideals. Sometimes it is argued that $\sqrt{I+J}$ is prime, which of course suffices to prove that (classical defined) affine varieties have products. But this does not suffice to prove that the coordinate ring is just the tensor product, it only shows that it is the tensor product mod nilpotents. – Martin Brandenburg Feb 4 '13 at 23:56
Anyway, a more modern account can be found in the book by Görtz-Wedhorn, Proposition 5.49. It is essentially the proof I gave above. I haven't even found it in EGA ... – Martin Brandenburg Feb 4 '13 at 23:57
Thank you for your help. The Görtz-Wedhorn book looks really good. Just one more question, does there hold $K[A\sqcup B]\cong K[A]\times K[B]$, i.e. does $K[-]$ send coproducts to products and vice versa? Do we need $K$ to be algebraically closed for both statements? – Leon Feb 5 '13 at 0:43
Yes $k[A \sqcup B] = k[A] \times k[B]$ holds for all affine schemes $A,B$. And $k[A \times B] = k[A] \otimes_k k[B]$ holds for all affine schemes, since $k[-]$ has a left adjoint. But in the context of classical algebraic geometry, where everything is reduced, one has to assume that $k$ is algebraically closed. – Martin Brandenburg Feb 5 '13 at 0:54
Let us assume that $K$ is algebraically closed.
If we have two $K$-algebras $C$ and $D$, there are canonical morphisms $$C\overset{\alpha}{\longrightarrow} C\otimes_KD\overset{\beta}{\longleftarrow} D$$ defined by $\alpha(c)=c\otimes 1_D$ and $\beta(d)=1_C\otimes d$. If you are given two morphisms of $K$-algebras $\phi:C\to E$ and $\psi:D\to E$ to a third $K$-algebra $E$, then the universal property of tensor product (in the category of $K$-algebras) tells you that there exists a unique morphism of $K$-algebras $q:C\otimes_KD\to E$ such that $\phi=q\circ\alpha$ and $\psi=q\circ\beta$. In other words, in the category of $K$-algebras, $C\otimes_KD$ together with the arrows $\alpha$ and $\beta$ satisfies the universal property above.
Now, there is a duality of categories between the category of $K$-algebras of finite type and the category of affine varieties. This duality is given by the functor Spec. So take $$\textrm{Spec (universal property of } C\otimes_KD),$$ after having assumed that $C$ and $D$ are of finite type (hence of the kind $C:=K[A]$ and $D:=K[B]$ as in your question). The universal property becomes a new universal property in the category of affine varieties, and now the universal object is $\textrm{Spec}(C\otimes_KD)$ with the two projections (the images $\textrm{Spec}(\alpha)$ and $\textrm{Spec}(\beta)$). But this is the universal property of $\textrm{Spec }C\times_K\textrm{Spec }D$, so that $$\textrm{Spec}(C\otimes_KD)\cong \textrm{Spec }C\times_K\textrm{Spec }D=A\times_KB.$$ Now, to recover $K[A\times_KB]\cong K[A]\otimes_KK[B]=:C\otimes_KD$ it is enough to go through the duality in the other direction: since the inverse of Spec is $K[-]$ (the functor "global sections", or "take the coordinate ring of"), just take global sections of the last displayed formula to get what you want.
[Note that the identity you wrote between the ideals follows now as an obvious corollary.]
-
For the category of affine schemes in $\mathbb{A}^n_K$ and the category of finitely generated reduced (commutative unital) $K$-algebras to be equivalent, need $K$ be algebraically closed? Furthermore, I think I've read that the first category is also equivalent to the category of finitely generated reduced (commutative unital) rings. Is this true? – Leon Feb 4 '13 at 14:06
For the algebraically closed hypothesis, you are right. I will correct my answer. And for the rest, as far as I know, it depends whether your definition of variety includes reducedness. I think the right correspondences are: rings, with affine schemes; and (reduced) $K$-algebras of finite type, with (reduced) affine varieties over $K$. – Brenin Feb 4 '13 at 14:37
I don't think that abstract nonsense can answer the question (only the second one for affine schemes). Even for schemes it has a content (see my answer). The question was obviously asked for varieties in the classical sense, which are therefore reduced. Then one has to show that there is a product of varieties (more generally reduced schemes, doesn't really matter). – Martin Brandenburg Feb 4 '13 at 14:56
Thank you for you time! – Leon Feb 5 '13 at 0:33
Dear Martin, thank you for your comment. Can you please explain to me why the equivalence between reduced algebras of finite type and affine varieties (in the classical sense, i.e. reduced) does not "include" the result that the ideal of the product is itself radical? Sorry to bother you but I want to understand this point. – Brenin Feb 5 '13 at 16:07
At the request of Martin Brandenburg, here is the result which may help you:
Let $k$ be a field and $V\subseteq \Bbb{A}_k^n$, $W\subseteq \Bbb{A}_k^m$ affine algebraic sets ( the proof I believe will also work if $V,W$ are affine varieties). Then $$I(V \times W )= I(V) + I(W).$$ By $I(V)$ we mean now the extension of the ideal $I(V)$ in the polynomial ring $k[x_1,\ldots,x_{m+n}]$.
Proof: See the discussion in my question here.
Using this result we can now prove the following theorem.
Theorem (Coordinate ring of a product): Let $V,W$ be as before. Then as $k$ - algebras we have $$k[V \times W] \cong k[V] \otimes_k k[W].$$
Proof: Let us write $I = \mathcal{I}(V)$ and $J = \mathcal{I}(W)$ and define $R = k[x_1,\ldots,x_n]$ and $S = k[x_{n+1},\ldots,x_{m+n}]$. Then by the lemma above, we have that $\mathcal{I}(V \times W) = I^e + J^e$ where the superscript denotes ideal extension in the ring $T = R \otimes_k S$, the polynomial algebra in $m+n$ variables. Now by the usual extension of scalars process we have $$I^e + J^e = I \otimes_k S +R \otimes_k J.$$ Thus to prove the theorem we need to prove that
$$\frac{T}{I \otimes_k S +R \otimes_k J} \cong \frac{R}{I} \otimes_k \frac{S}{J}.$$
The only tricky part in the proof is to construct a well-defined $k$ - algebra homomorphism $$f : \frac{R}{I} \times \frac{S}{J} \to \frac{ T}{I \otimes_k S + R \otimes_k J}$$
We can define one by declaring that $f$ sends $(\bar{a},\bar{b})$ to $a\otimes b \mod{(I \otimes_k S + R \otimes_k J)}$. Is this well defined? Suppose $(\bar{a},\bar{b}) = (\bar{c},\bar{d})$. Then notice that $$\begin{eqnarray*} a \otimes b - c\otimes d &=& a \otimes b + c\otimes b - c\otimes b - c\otimes d \\ &=& (a-c) \otimes b + c \otimes (b-d)\\ &=& 0 \mod{(I \otimes_k S+ R \otimes_k J)}\end{eqnarray*}$$ because $a -c \in I$ and $b-d \in J$. Thus we see that $f$ is well-defined and leave the rest of the details for your to fill in, including constructing an inverse map. If you have any questions, I can add more details.
-
Concerning the second part: It is not tricky at all, when you use the Yoneda Lemma - the proof is just one line. – Martin Brandenburg Feb 7 '13 at 16:38
@MartinBrandenburg As a beginner, the proof is going to be low-level. But I am very interested in the Yoneda Lemma now. I don't have to keep defining maps all over the place and check well-definiteness! – user38268 Feb 7 '13 at 16:44
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Homework Help: Please check my work (differential equation)
1. Apr 12, 2010
darryw
1. The problem statement, all variables and given/known data
ty' + 2y = sin t (no initial conditions given)
2. Relevant equations
3. The attempt at a solution
ty' + 2y = sin t
y' + (2/t)y = sin t / t
mu(x) = e^integ(2/t) = t^2
(t^2)y)' = integ t sin t
(t^2)y = tsin t - t cos t + c
y = (sin t - cos t)/ t + c/(t^2) (This is my solution)
thanks for any help
2. Apr 12, 2010
tiny-tim
Hi darryw!
(have an integral: ∫ and a mu: µ and try using the X2 tag just above the Reply box )
Your equations are ok down to …
… the first line of course should be (t^2)y)' = t sin t (without the ∫) ,
but more seriously your integration by parts has come out wrong …
check it by differentiating, and you'll see how to fix it.
3. Apr 12, 2010
darryw
i knew it! this is an ongoing problem for me.. I always have problem escaping the integration loop when i have something like ∫t cos t.. (or even worse: ∫e^t cos t
as i understand it, the idea is to integrate up to a certain point and then subtract the integrals identity from left hand side, so then you cancel the integrals. When i did that i ended up with tsin t - t cos t. Can you offer any help/tips so i dont have to write out the whole long integration process? thanks
4. Apr 12, 2010
tiny-tim
he he
the official method for ∫ fg dx is to integrate g only, giving [f(∫ g dx)], and then subtract the integral of f' time that: ∫ { f'(∫ g dx)} dx
but my method (only works for easy cases) is to make a guess (in this case, -tcost), differentiate it (-cost + tsint), and then integrate whatever's over (-cost)
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