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https://independencenumber.wordpress.com/2012/08/09/coding-and-theory/ | # Coding and Theory
Patrick and I have a long list of $\alpha$-properties and $\alpha$-bounds to code. Independence Number Theory be viewed as whatever properties and invariants we’ve already coded. So we can generate graphs which are “difficult” with respect to the coded theory. In our current minimal configuration, a 5-cycle with a chord popped out as difficult. This graph has a vertex v such that the graph induced on N[v] is complete. It can be argued that such a vertex is in some maximum independent set. This graph can be reduced. So the graph tells us what $\alpha$-property to code next.
Addendum: this process has continued. With the limited theory coded so far, a graph that was “almost KE” popped out, so that property was added, then a graph that was claw-free, so that property was added. Then the following graph popped out:
Notice that if you remove edge 6-0 you get a KE graph. We call the property where removal of some edge yields a KE graph, “almost-edge-KE”. It can be argued that the independence number of an almost-edge-KE graph G can be efficiently computed. Either G has a non-empty KE part, or every non-empty independent set has more neighbors, and $\alpha(G),\frac{n(G)}{2}$. It can then be argued that, if n is odd, so n=2k+1, then $\alpha(G)=k$, and if n is even, so n=2k, then $\alpha=k-1$ | 2017-12-18 07:02:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6930370330810547, "perplexity": 672.044072950896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948609934.85/warc/CC-MAIN-20171218063927-20171218085927-00228.warc.gz"} |
http://www.computmath.com/szjs/CN/10.12288/szjs.1981.1.55 | • 论文 •
### 从航摄照片求任意地面点高程的新算法
1. 中国科学院计算中心
• 出版日期:1981-01-20 发布日期:1981-01-20
### THE NEW ALGORITHM FOR OBTAINING THE HEIGHT OF ARBITRARY POINT ON THE GROUND
1. Fan Tian-wei Computing Center, Academia Sinica
• Online:1981-01-20 Published:1981-01-20
This paper discusses two new algorithms for obtaining the height of an arbitrary point onthe ground directly through photographs. They Can be efficiently used to compute the terraincorrection of gravity and operate an orthophoto with contours. The programming for the newprocedures is much simpler and clearer than the algorithms available. The memory spacerequired has been reduced to the minimum, and the amount of operations is more economical,especially in the case of terrain correction of gravity. This paper gives also some numerical examples for two mathematical models of terrainby means of fabricating photographs.
()
[1] P. Jarratt, D. Nudds, The Computer Journal, 8: 1(1965) . [2] D. E. Muller, Math. Tables Aids Comput., 10(1956) ,208-215． [3] 冯康等编,数值计算方法,国防工业出版社,北京,1978． [4] A. D. Maude, "Interpolation-meinlg for graph plotters" J. Computer 16: 1(1973) .
No related articles found! | 2021-12-01 02:57:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3358592689037323, "perplexity": 5804.755941361588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964359082.78/warc/CC-MAIN-20211201022332-20211201052332-00054.warc.gz"} |
https://www.illustrativemathematics.org/MD | #### 1.MD.A.1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.
• No tasks yet illustrate this standard.
#### 1.MD.B.3. Tell and write time in hours and half-hours using analog and digital clocks.
• No tasks yet illustrate this standard.
#### 2.MD.A.2. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
• No tasks yet illustrate this standard. | 2017-06-25 00:14:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2522895038127899, "perplexity": 3026.9096513968084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320368.57/warc/CC-MAIN-20170624235551-20170625015551-00225.warc.gz"} |
https://hal.univ-lorraine.fr/hal-02894451 | # Existence and nonexistence results for a weighted elliptic equation in exterior domains
Abstract : We consider positive solutions to the weighted elliptic problem \begin{equation*} -\mbox{div} (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\mbox{in $\mathbb{R}^N \backslash {\overline B}$},\quad u=0 \;\; \mbox{on $\partial B$}, \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question for $N':=N+\theta>2$ and $p > 0$. In particular, for $N' > 2$ and $\tau:=\ell-\theta >-2$, it is shown that for $0< p \leq p_s:=\frac{N'+2+2 \tau}{N'-2}$, the only nonnegative solution to the problem is $u \equiv 0$. This nonexistence result is new, even for the classical case $\theta = \ell = 0$ and $\frac{N}{N-2} < p \leq \frac{N+2}{N-2}$, $N \geq 3$. The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.
Keywords :
Document type :
Journal articles
Cited literature [16 references]
https://hal.univ-lorraine.fr/hal-02894451
Contributor : Dong Ye <>
Submitted on : Wednesday, July 8, 2020 - 10:50:13 PM
Last modification on : Monday, July 20, 2020 - 11:19:38 AM
Long-term archiving on: : Monday, November 30, 2020 - 4:46:19 PM
### File
GHY_Nonexistence_ZAMP20.pdf
Files produced by the author(s)
### Citation
Zongming Guo, Xia Huang, Dong Ye. Existence and nonexistence results for a weighted elliptic equation in exterior domains. Zeitschrift für Angewandte Mathematik und Physik, Springer Verlag, 2020, 71 (4), ⟨10.1007/s00033-020-01338-0⟩. ⟨hal-02894451⟩
Record views | 2021-01-24 19:01:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6536037921905518, "perplexity": 1087.5779388345309}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703550617.50/warc/CC-MAIN-20210124173052-20210124203052-00300.warc.gz"} |
https://www.physicsforums.com/threads/is-this-series-convergent-or-divergent.728329/ | # Is this series convergent or divergent.
1. Dec 14, 2013
### nothingkwt
1. The problem statement, all variables and given/known data
Ʃ ne(-n2)
2. Relevant equations
3. The attempt at a solution
I used the ratio test and wanted to know if the way I did it is correct or not
|a(n+1) / a(n)|
n+1 (e(-n2 -2n-1)) / n e(-n2)
Now e-n^2 cancels and we get
limn→∞ n+1/n * 1/(e2n)(e)
After you take the limits you get (1)*0 = 0 < 1 so it's convergent
2. Dec 14, 2013
### pasmith
3. Dec 14, 2013
### Curious3141
Sure, this works. But this sum looks almost purpose-built for the integral test. Try it (if you're allowed to use it).
4. Dec 14, 2013
### nothingkwt
I am actually but I just wanted to see if the ratio test worked
Thanks for the replies!
5. Dec 14, 2013
### Ray Vickson
You really need to learn how to write in ASCII---in particular, you need to use brackets. Your first equation AS WRITTEN says
$$\left|\frac{a_{n+1}}{a_n}\right| = n +1 \frac{e^{-n^2-2n-1}}{n} e^{-n^2}.$$
$$\lim_{n \to \infty} n + \frac{1}{n} \frac{1}{e^{2n}} e$$
I hope these are not what you mean. I hope you intended the first one to be
$$\left|\frac{a_{n+1}}{a_n}\right| = \frac{(n+1)e^{-n^2-2n-1}}{n e^{-n^2}},$$
etc. To make sure this happens you need parentheses!
6. Dec 15, 2013
### nothingkwt
Yeah I'm not very good with ASCII I'm still learning how to use them. | 2018-02-21 21:22:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44284582138061523, "perplexity": 1660.5426854041564}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813803.28/warc/CC-MAIN-20180221202619-20180221222619-00330.warc.gz"} |
https://ask.openstack.org/en/questions/101135/revisions/ | # Revision history [back]
### Why my vm is not responding to ARP request
Hi guys,
Here is a quick descritpion of my setup.
A CentOS7 VM and liberty distro, I used packstack and my config is "all-in-one". I've created a router, an external network and an internal network.
I've created a cirros image, and a test vm.
I've added the security rules for ping and ssh.
When I ping my vm with it's internal adress ( provided dynamically at creation ), I've got no answer either from my default namespace or from the router namespace.
With tcpdump, I monitored the tap interface and I see arp request, but the vm doesn't answer.
> nova list tells me the instance is ACTIVE and running.
And if I use virsh screenshot I have:
SeaBIOS (versio xxxxxxx)
Machine UUID xxxxxxxxxx
iPXE (http://ipxe.org) 00.03.0 C980 PCI2.10 PnP PMM
console.log is empty and my vm seems to be stuck in the boot step, but nova list and logs tell me that everything is fine.
I found a post on the net describing the same problem and the guy said that its image wasn't created correcly.
But I followed basic steps:
> wget http://download.cirros-cloud.net/0.3.4/cirros-0.3.4-x86_64-disk.img
> glance image-create --name "cirros-0.3.4-x86_64" --file cirros-0.3.4-x86_64-disk.img --disk-format qcow2 --container-format bare --visibility public --progress
> nova boot --flavor m1.tiny --image cirros-0.3.4-x86_64 --key-name tapaas-key --nic net-id=cc615845-dc51-41e7-b470-10e00c4b5518 vm-cirros-0.3.4-x86_64
I tried we other version of cirros and other images. But it's the same. At first I've created a floating ip and if I ping this address, it is correctly translated in the internal address, because I see the ARP request.
I'm running out of ideas.
So If someone could help me, I'd be thankful.
Charles.
### Why my vm is not responding to ARP request
Hi guys,
Here is a quick descritpion of my setup.
A CentOS7 VM and liberty distro, I used packstack and my config is "all-in-one". I've created a router, an external network and an internal network.
I've created a cirros image, and a test vm.
I've added the security rules for ping and ssh.
When I ping my vm with it's internal adress ( provided dynamically at creation ), I've got no answer either from my default namespace or from the router namespace.
With tcpdump, I monitored the tap interface and I see arp request, but the vm doesn't answer.
> nova list tells me the instance is ACTIVE and running.
And if I use virsh screenshot I have:
SeaBIOS (versio xxxxxxx)
Machine UUID xxxxxxxxxx
iPXE (http://ipxe.org) 00.03.0 C980 PCI2.10 PnP PMM
console.log is empty and my vm seems to be stuck in the boot step, but nova list and logs tell me that everything is fine.
I found a post on the net describing the same problem and the guy said that its image wasn't created correcly.
But I followed basic steps:
> wget http://download.cirros-cloud.net/0.3.4/cirros-0.3.4-x86_64-disk.img
> glance image-create --name "cirros-0.3.4-x86_64" --file cirros-0.3.4-x86_64-disk.img --disk-format qcow2 --container-format bare --visibility public --progress
> nova boot --flavor m1.tiny --image cirros-0.3.4-x86_64 --key-name tapaas-key --nic net-id=cc615845-dc51-41e7-b470-10e00c4b5518 vm-cirros-0.3.4-x86_64
I tried we other version of cirros and other images. But it's the same. At first I've created a floating ip and if I ping this address, it is correctly translated in the internal address, because I see the ARP request.
I'm running out of ideas.
So If someone could help me, I'd be thankful.
Charles.
!!!!! EDITING !!!!!
Hi,
Yes I agree with you this is more a boot instance problem than a network problem. I tried to load the image with VirtualBox, but it failed at boot it was not recognize as a valid image I used this command:
VBoxManage convertfromraw --format VDI vdisk_raw.img vdisk.vdi
I tried glance image-download and diff ... same image. I tried other versions of cirros images, but with the same result. I have access to VNC console and I've got the same message as in the screenshoot ( stuck with the iPxe message ).
Charles.
### Why my vm is not responding to ARP request
Hi guys,
Here is a quick descritpion of my setup.
A CentOS7 VM and liberty distro, I used packstack and my config is "all-in-one". I've created a router, an external network and an internal network.
I've created a cirros image, and a test vm.
I've added the security rules for ping and ssh.
When I ping my vm with it's internal adress ( provided dynamically at creation ), I've got no answer either from my default namespace or from the router namespace.
With tcpdump, I monitored the tap interface and I see arp request, but the vm doesn't answer.
> nova list tells me the instance is ACTIVE and running.
And if I use virsh screenshot I have:
SeaBIOS (versio xxxxxxx)
Machine UUID xxxxxxxxxx
iPXE (http://ipxe.org) 00.03.0 C980 PCI2.10 PnP PMM
console.log is empty and my vm seems to be stuck in the boot step, but nova list and logs tell me that everything is fine.
I found a post on the net describing the same problem and the guy said that its image wasn't created correcly.
But I followed basic steps:
> wget http://download.cirros-cloud.net/0.3.4/cirros-0.3.4-x86_64-disk.img
> glance image-create --name "cirros-0.3.4-x86_64" --file cirros-0.3.4-x86_64-disk.img --disk-format qcow2 --container-format bare --visibility public --progress
> nova boot --flavor m1.tiny --image cirros-0.3.4-x86_64 --key-name tapaas-key --nic net-id=cc615845-dc51-41e7-b470-10e00c4b5518 vm-cirros-0.3.4-x86_64
I tried we other version of cirros and other images. But it's the same. At first I've created a floating ip and if I ping this address, it is correctly translated in the internal address, because I see the ARP request.
I'm running out of ideas.
So If someone could help me, I'd be thankful.
Charles.
!!!!! EDITING !!!!!Edit1:
Hi,
Yes I agree with you this is more a boot instance problem than a network problem. I tried to load the image with VirtualBox, but it failed at boot it was not recognize as a valid image I used this command:
VBoxManage convertfromraw --format VDI vdisk_raw.img vdisk.vdi
I tried glance image-download and diff ... same image. I tried other versions of cirros images, but with the same result. I have access to VNC console and I've got the same message as in the screenshoot ( stuck with the iPxe message ).
Charles.
Edit2:
"If your deployment uses QEMU or KVM, we recommend using the images in qcow2 format. The most recent 64-bit qcow2 image as of this writing is cirros-0.3.4-x86_64-disk.img."
I have a reference plateform ( someone else installed it but I cannot contact him ) and the cirros is a qcow2.
Here is the infos about the image I use:
# qemu-img info cirros-0.3.0-x86_64-disk.img
image: cirros-0.3.0-x86_64-disk.img
file format: qcow2
virtual size: 39M (41126400 bytes)
disk size: 9.3M
cluster_size: 65536
Format specific information:
compat: 0.10
### Why my vm is not responding to ARP request
Hi guys,
Here is a quick descritpion of my setup.
A CentOS7 VM and liberty distro, I used packstack and my config is "all-in-one". I've created a router, an external network and an internal network.
I've created a cirros image, and a test vm.
I've added the security rules for ping and ssh.
When I ping my vm with it's internal adress ( provided dynamically at creation ), I've got no answer either from my default namespace or from the router namespace.
With tcpdump, I monitored the tap interface and I see arp request, but the vm doesn't answer.
> nova list tells me the instance is ACTIVE and running.
And if I use virsh screenshot I have:
SeaBIOS (versio xxxxxxx)
Machine UUID xxxxxxxxxx
iPXE (http://ipxe.org) 00.03.0 C980 PCI2.10 PnP PMM
console.log is empty and my vm seems to be stuck in the boot step, but nova list and logs tell me that everything is fine.
I found a post on the net describing the same problem and the guy said that its image wasn't created correcly.
But I followed basic steps:
> wget http://download.cirros-cloud.net/0.3.4/cirros-0.3.4-x86_64-disk.img
> glance image-create --name "cirros-0.3.4-x86_64" --file cirros-0.3.4-x86_64-disk.img --disk-format qcow2 --container-format bare --visibility public --progress
> nova boot --flavor m1.tiny --image cirros-0.3.4-x86_64 --key-name tapaas-key --nic net-id=cc615845-dc51-41e7-b470-10e00c4b5518 vm-cirros-0.3.4-x86_64
I tried we other version of cirros and other images. But it's the same. At first I've created a floating ip and if I ping this address, it is correctly translated in the internal address, because I see the ARP request.
I'm running out of ideas.
So If someone could help me, I'd be thankful.
Charles.
Edit1:
Hi,
Yes I agree with you this is more a boot instance problem than a network problem. I tried to load the image with VirtualBox, but it failed at boot it was not recognize as a valid image I used this command:
VBoxManage convertfromraw --format VDI vdisk_raw.img vdisk.vdi
I tried glance image-download and diff ... same image. I tried other versions of cirros images, but with the same result. I have access to VNC console and I've got the same message as in the screenshoot ( stuck with the iPxe message ).
Edit2:
"If your deployment uses QEMU or KVM, we recommend using the images in qcow2 format. The most recent 64-bit qcow2 image as of this writing is cirros-0.3.4-x86_64-disk.img."
I have a reference plateform ( someone else installed it but I cannot contact him ) and the cirros is a qcow2.
Here is the infos about the image I use:
# qemu-img info cirros-0.3.0-x86_64-disk.img
image: cirros-0.3.0-x86_64-disk.img
file format: qcow2
virtual size: 39M (41126400 bytes)
disk size: 9.3M
cluster_size: 65536
Format specific information:
compat: 0.10
Edit3:
@ doka: my problem is that I cannot access to a vnc console, at the time I wrote the title of my post I wasn't sure about the problem, now it looks like my vm is not booting properly, hence no responde to arp request.
I checked libvirt logs too and used virsh to test if my vm is running, and everything looks fine, exept that I cannot connect to the vm, and I have the message "seaBIOS" that make me thing that the vm is stuck at boot waiting something. | 2019-08-18 20:32:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22137674689292908, "perplexity": 10316.568332760517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313996.39/warc/CC-MAIN-20190818185421-20190818211421-00541.warc.gz"} |
https://golem.ph.utexas.edu/category/2010/11/over_4500_entries_in_the_nlab.html | ## November 26, 2010
### Over 4500 Entries in the nLab, and Its 2nd Birthday!
#### Posted by David Corfield
guest post by Zoran Škoda
The first entry of the n-Category Lab wiki appeared on November 28, 2008 02:18:19 marking the creation of the nLab; hence this Sunday morning it is 2 years old! Three days before the birthday its nominal count of pages reached 4500. This may be a proper moment to celebrate its wide usability already at its tender age and even more to invite people to use it more, and if possible to contribute. Like in our earlier update from May, we would like to point to some highlights in $n$Lab. But I somewhat run out of steam to dwell this time on the content and will rather outline some improvements in the content organization of the $n$Lab which may make it more attractive to you.
In my impression, in its first year, the $n$Lab was focused on our daily research needs and central areas of our interest: category theory, including higher, topoi, homotopy theory, topology, sheaves, stacks, simplicial objects, descent, cohomology (including differential), foundations and categorical aspects of physics. I have received signals from some users of $n$Lab that they do not contribute because they “do more concrete things”, say Lie algebras, representation theory, mathematical physics and so on and feel they do not wish to write about categories. But this is a misunderstanding: more stuff in related areas is very welcome and we need contributors telling us the story in nearby areas of algebra, mathematical physics, differential geometry and so on (of course, not that far an area that we can not understand, appreciate and connect to).
Seeing that we started in rather self-centered areas of categorical mathematics, it was difficult, in the first year, for a newcomer, to navigate through nLab and find out what interesting things (s)he can find there. Hence, in the last several months a great amount of activity was centered not only in creating new content but also around new lists/tables of pointers of content in particular fields of interest. In this vein, Urs has been very enthusiastic in adding floating tables of contents. For example, each entry in topos theory, e.g. subobject classifier has a floating table of contents for Topos theory on its right-hand side. It is a pull-down menu which helps you navigate through entries in the subject of the particular table. Even the HomePage uses one floating table, to help the newcomer, and there are top level tables for mathematics and for physics. One can navigate top-down to some subsubjects from there.
Another organizational change in the last year or so is that most discussions which would earlier take place in query boxes in $n$Lab entries shifted to the nForum, which is well structured for many-purpose activities, thanks to the software and maintenance care of Andrew Stacey. Some of the discussions are about spams, bugs, writing, future policies and software, while some are about mathematical research, where Todd, Toby, Urs, Domenico, Jim, Tim, Mike, John, David and others explain extensively to each other their insights. Some of the longer among such discussions happen in the Atrium section, especially under Mathematics, Physics and Philosophy.
I have personally made the effort to connect $n$Lab to the external sources of information and replaced the previous Online Resources page by a network of several resource pages including:
• top page math resources
• page math blogs (previous Online Resources) with list of blogs and wikis
• page math institutions with a selected list of top world institutes and links to AMS-maintained and some other lists of math societies and departments
• page math archives with a list of main archives of online content in areas of our interest; with few exceptions (like MathSciNet), subscription sites are avoided and free archives preferred
• page math resources by individuals where there are some extremely rich pages of individuals in the prime areas of $n$Lab (there are hundreds of pages on individual mathematicians in $n$Lab, here we list just few particularly useful links)
• page books and reviews in mathematical physics with selected list of major research level books in mathematical physics
While in the first year the physics entries were dwarfed by the mathematical part of $n$Lab, now we do have a non-negligeable physics content. We recommend an outline of the nPOV in physics in the entry (mainly by Urs Schreiber)
Some picture of $n$Lab’s physics content can be accessed from the links at the top page physicscontents.
Posted at November 26, 2010 1:42 PM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2310
### Re: Over 4500 entries in the nLab and its 2nd birthday!
This is great! Happy birthday, $n$Lab. Having 4,500 articles already is really stunning. It also happens to be my birthday today, too — I wish I’d gained as much knowledge in the past 2 years as the $n$Lab has :).
Posted by: Jamie Vicary on November 26, 2010 7:31 PM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
Happy birthday, Jamie and $n$Lab. And more importantly, congratulations on your great acquired wisdom (both of you).
Posted by: Tom Leinster on November 26, 2010 9:59 PM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
Happy birthday, Jamie! Any birthday wishes for something you’d like to see at the nLab?
Posted by: Todd Trimble on November 26, 2010 10:15 PM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
I wish the $n$Lab had existed 30 years ago, when I starting to learn mathematics — not for the math that’s on the $n$Lab now, but so I could type in what I learned as I learned it! Instead, after the internet was invented, I resorted to This Week’s Finds.
I hope students visit the $n$Lab and add more explanations of mathematics to the pages there as they learn it. It may seem a bit intimidating, but I think anyone of good intent and reasonable intelligence will find the $n$Lab crew eager for help, and eager to help.
Right now I’m busy learning about ecology, climate change, climate modeling, energy technology, and the kinds of applied math needed in these subjects. As I learn things, I’m typing them into the Azimuth Project. This is a wiki running on the same software as the nLab, hosted on the same server, and managed by the same person: Andrew Stacey. Just as the nLab has its nForum, the Azimuth Project has its Azimuth Forum. There was no need to reinvent the wheel: this system works well!
Right now you can see some applied math pages here on the Azimuth Project. So far it’s mainly Tim van Beek, David Tweed and Graham Young who have been writing these pages. We’ve been thinking a lot about stochastic resonance as a possible explanation for how the Milankovitch cycles get amplified to produce glacial cycles. But that’s just one of many topics I want to learn about — and in practical terms, not even one of the most important. There’s so much to do. If anyone reading this is an applied mathematician, please join the fun — either on the Azimuth Project, or the nLab, or both!
Anyway, while I can’t resist talking about the Azimuth Project, my main point here is that the nLab has succeeded to the point where it seems like a system worth copying in other subjects.
Posted by: John Baez on November 27, 2010 6:11 AM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
Happy birthday nLab!
4500 in two years is an incredible amount, given the average information density of a page.
John said:
If anyone reading this is an applied mathematician, please join the fun — either on the Azimuth Project, or the nLab, or both!
On the Azimuth project the biggest topic today is climate modelling, where we try to understand the basic stuff, discuss some papers and explain the mathematical tools. An example is the prediction of the next glacial inception. There is a lot about mathematical statistics, of course, but people interested in, for example, functional analysis, could help out by expanding the page on wavelets (as a tool for time series analysis).
Posted by: Tim van Beek on November 28, 2010 2:06 PM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
John wrote:
Right now you can see some applied math pages here on the Azimuth Project. So far it’s mainly Tim van Beek, David Tweed and Graham Young who have been writing these pages.
I did notice before that you stole Tim van Beek away from the $n$Lab and made him do labor at Azimuth instead. I see that with much regret: he used to be our best AQFT contributor.
I count that as the second big bad effect that climate change already has on the $n$Lab. I hope you guys get that problem sorted out soon and come back to do something less profane.
Posted by: Urs Schreiber on November 29, 2010 4:24 PM | Permalink | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
The nlab clearly celebrated a little too heavily and this morning has woken up feeling decidedly worse for wear. Indeed, after staggering home from the pub, it appears to have gone in to the wrong house with the effect that (until it finds its house keys again), it is at http://nlab.mathforge.org.
Seriously, something’s gone wrong with the nameservers on the ncatlab.org domain which means that the nlab (and the other personal webs) can’t be accessed via ncatlab.org. The backup address is: http://nlab.mathforge.org, and similar for the personal webs.
Remarks:
1. We’ll get the ncatlab.org domain back in the right place just as soon as we can.
2. The nlab.mathforge.org links are not a temporary replacement, they are and have always been an alternative route to the nlab.
3. The azimuth project is not affected by this.
Posted by: Andrew Stacey on November 28, 2010 7:31 PM | Permalink | Reply to this
### Pay the piper
That’s what happens when you don’t pay your domain registration fees …
At least your domain has been parked, rather than snapped-up by an online purveyor of catfood (or whatever).
Posted by: Jacques Distler on November 28, 2010 11:26 PM | Permalink | PGP Sig | Reply to this
### Re: Over 4500 entries in the nLab and its 2nd birthday!
the lab is back at http://ncatlab.org/nlab/
Posted by: Urs Schreiber on December 3, 2010 10:11 PM | Permalink | Reply to this
Read the post Jet Categories at the nForum
Weblog: The n-Category Café
Excerpt: nLab activity
Tracked: October 21, 2013 10:30 AM
Post a New Comment | 2015-04-01 09:25:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 19, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34424498677253723, "perplexity": 1967.3009883720638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131304412.34/warc/CC-MAIN-20150323172144-00135-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/890097/what-is-the-position-of-exclusive-or-in-order-of-precedence-for-logical-connec | # What is the position of “exclusive or” in order of precedence for logical connectives?
In propositional logic the order of precedence I have found for the logical connectives is
1. $\neg$
2. $\land$
3. $\lor$
4. $\Rightarrow$
5. $\Leftrightarrow$
Where do I have to put the exclusive or $\dot\lor$ in the above list?
Note: I also asked this question on this the German forum matheboard.de two days ago. Because I did not get an answer there and because I also did not find an answer in the internet, I also want to ask this question here. I hope, this is okay.
The exclusive-or can be thought of as "compound connective": a connective that involves $\lor$, $\land$ and $\lnot$ in any one of its forms: $$p \dot\lor q \equiv (p \lor q) \land \lnot(p \land q)\equiv (p \lor q)\land (\lnot p \lor \lnot q) \equiv (p \land \lnot q) \lor (\lnot p \land q)$$
Then the table of precedence you've posted covers exclusive-or, implicitly at least, as can be seen in the "expanded" versions of the exclusive-or.
• For example: Where do I have to put the parentheses in the statement $A \lor B \,\dot\lor\, C \land D$? – Stephan Kulla Aug 7 '14 at 15:35
• That is ambiguous, if I've ever seen ambiguous. – amWhy Aug 7 '14 at 15:40
• Based on the reasoning here, I'd be inclined to suggest it's last in your list, following the biconditional, since it can be thought of as the negation of the biconditional. So for your example, $(A \lor B)\dot\lor(C\land D)$. But this is just speculation. – amWhy Aug 7 '14 at 15:51
• Would you agree, if we say, that there is no clear answer to this question? – Stephan Kulla Aug 7 '14 at 15:53
• Yes, I'd agree with that!. – amWhy Aug 7 '14 at 15:54
'Exclusive or' ($\;\mathop{\dot \lor}\;$) is the negation of equivalence ($\;\Leftrightarrow\;$): $$(p \mathop{\dot \lor} q) \;\Leftrightarrow\; \lnot(p \Leftrightarrow q)$$ Therefore personally I prefer to write equivalence as $\;\equiv\;$ and 'exclusive or' as $\;\not\equiv\;$, to make the duality clear in the symbols.
And when using those symbols, of course, I prefer to use the same precedence for both operators, and write e.g. $\;A \lor B \;\not\equiv\; C \land D\;$ without any parentheses.
Finally, note that not only is $\;\equiv\;$ associative, but also $\;\not\equiv\;$ is associative, and they are even mutually associative, so that we can write: $$p \not\equiv q \;\equiv\; \lnot(p \equiv q)$$ without ambiguity.
On the same topic, I also dislike the precedence difference between $\;\land\;$ and $\;\lor\;$, as shown in your table. One reason is the strong duality between both (which is hidden even more in boolean algebra notation). But more importantly, a convention which makes $\;A \land B \lor C\;$ different from $\;A \land (B \lor C)\;$ can only lead to confusion, and instead it should be our goal as math writers to make the job as easy as possible on the math readers.
Therefore personally I never write $\;A \land B \lor C\;$, but instead always $\;(A \land B) \lor C\;$.
All of the above I did not invent myself, but I learned it from Dijkstra/Feijen/Scholten/Gries/Schneider. See the article E.W. Dijkstra, "The notational conventions I adopted, and why" (on-line), and also the books E.W. Dijkstra and C.S. Scholten, "Predicate Calculus and Program Semantics" and D. Gries and F.B. Schneider, "A Logical Approach to Discrete Math".
And all this goes to show that precedence is a convention: one can get used to any convention, there are arguments for and against each one, and you don't "have to" put every operator in a specific place in the precedence list.
In case of doubt for a specific operator, you can always use parentheses so that the reader does not have to worry. Dijkstra also taught me to use more whitespace around lower precedence operators, as in $$p \;\equiv\; q \;\equiv\; p \land q \;\equiv\; p \lor q$$ That also helps the reader in parsing, by suggesting the precedence.
• I like brackets as well. However note when we use the + and • notation for or and and, it becomes clear that we can do algebraic tricks. e.g. a•b + a•c ≡ a ( b + c ), just like regular algebra. – ctrl-alt-delor Nov 5 '15 at 21:47
• @richard Yes. But note that in that case we also have $\;(a+b) • (a+c) \;\equiv\; a + (b•c)\;$ which is really unclear and counter-intuitive, which for me shows that using $\;+\;$ and $\;•\;$ for 'or' and 'and' is more trouble than it seems worth at first sight. – MarnixKlooster ReinstateMonica Nov 5 '15 at 22:10
• Sorry my example was bad, I realised as I slept (following posting it). What I should have said is 1+1=2=1, 1+0 =1, 0+0=0, 0•0=0, 0•1=0, 1•1=1. Except for 2=1 it is like regular algebra. – ctrl-alt-delor Nov 6 '15 at 21:55 | 2021-03-03 21:31:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7864494323730469, "perplexity": 686.3512966159702}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178367790.67/warc/CC-MAIN-20210303200206-20210303230206-00602.warc.gz"} |
https://ashishkumarletslearn.com/sequences-and-series-class-11-maths/ | “Respect Is The Key Determinant Of High-Performance Leadership. How Much People Respect You Determines How Well They Perform.”
By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ….k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.
Let a1, a2, a3, … be the sequence, then the sum expressed as a1 + a2 + a3 + … is called series. A series is called finite series if it has got finite number of terms.
An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P. Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the nth term of the A.P. is given by
an= a + (n – 1) d.
The sum Sn of the first n terms of an A.P. is given by
$$S_n= \frac{n}{2}(2a + (n – 1) d)$$
The arithmetic mean A of any two numbers a and b is given by $$\frac{a + b}{2}$$ i.e., the sequence a, A, b is in A.P.
A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by an= arn – 1 The sum Sn of the first n terms of G.P. is given by $$S_n= \frac{a(r^n – 1)}{(r – 1)} = \frac{a(1 – r^n)}{(1 – r)}, {\rm if } r \ne 1$$.
The geometric mean (G.M.) of any two positive numbers a and b is given by $$\sqrt{ab}$$ i.e., the sequence a, G, b is G.P. | 2020-04-08 18:47:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8987167477607727, "perplexity": 281.4664830115592}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371821680.80/warc/CC-MAIN-20200408170717-20200408201217-00483.warc.gz"} |
https://www.physicsforums.com/threads/simple-derivative-question.108497/ | Simple derivative question
1. Jan 28, 2006
Chocolaty
What is the derivative of:
tan^4(3x)
The teacher wrote 4TAN^3(x) + sec^2(3x)(3) but i think he made a mistake.
I know that tan^4(3x) = tan(3x)^4 and that this equals 4tan(3x) in algebra
When I derive this i get the following: sec^2(3x)(3) i'm not sure what to do with the coefficient 4. Can someone give me the exact derivative formula to work this term out so I can copy it down plz.
thanks
2. Jan 28, 2006
Chocolaty
nevermind i got it. You need to use the generalized power rule:
[u(x)]^n => n*u^(n-1)*u'
so
tan(3x)^4 => 4*tan(3x)^3*sec^2(3x)(3)
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Have something to add? | 2017-01-23 06:46:15 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8026280999183655, "perplexity": 1995.6098457810697}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282140.72/warc/CC-MAIN-20170116095122-00441-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://bkms.kms.or.kr/journal/list.html?Vol=20&Num=1&mod=vol&book=BKMS&aut_box=Y&sub_box=Y&pub_box=Y | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
<< Previous Issue Bulletin of the Korean Mathematical Society (Vol. 20, No. 1) Next Issue >>
Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 5—67
On the Frum-Ketkov type fixed point theorems Sehie Park, Won Kyu Kim Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 5—8
Tables of $D$-classes in the semigroup $B_n$ of the binary relations on a set $X$ with $n$-elements Jin Bai Kim Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 9—13
Holomorphic functions with positive real part on complete circular domains Kyong T. Hahn Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 15—25
A note on the algebra $A_E(\Gamma)$ Sungwoo Suh Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 27—29
Derivation modules of group rings and integers of cyclotomic fields I. Y. Chung Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 31—36
A Malcev-admissible mutation of an alternative algebra Hyo Chul Myung Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 37—43
The finite dimensional prime rings Kwangil Koh Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 45—49
A generalization of an inequality of Li and Zhong, and its geometric application Dong Pyo Chi, Sang Moon Kim, Sung Ki Kim, Il Hae Lee, and Sa Ge Lee Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 51—54
On the Reidemeister numbers and Nielsen numbers of the (eventually abelian) fiber-preserving maps Soon-Kyu Kim and Jingyal Pak Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 55—63
Direct product and direct sum of a local ring Byoung-Song Chwe Bull. Korean Math. Soc. 1983 Vol. 20, No. 1, 65—67 | 2018-12-17 19:36:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3656856417655945, "perplexity": 1763.9572184718206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829115.83/warc/CC-MAIN-20181217183905-20181217205905-00530.warc.gz"} |
https://codegolf.stackexchange.com/questions/156100/quartic-summation | # Guidelines
Given two non-negative integers, find the sum of both numbers... to the power of 4.
### Examples
2, 3 -> 97 (2^4 + 3^4 = 97)
14, 6 -> 39712 (14^4 + 6^4 = 39712)
0, 25 -> 390625 (0^4 + 25^4 = 390625)
### Rules
• You will only ever receive two non-negative integers as input (bonus points if you get it to work with negative integer input).
• You can take the input however you would like (arrays, list, two separate arguments, comma-separated string)
• You can either output your answer as a number or a string.
• As this is code golf, the shortest answer in bytes will win.
• You say two positive integers, but one of the test cases contains a 0. Also, there's not many languages that would have trouble with negative integers, given the result is the same as if they were positive. – Jo King Feb 19 '18 at 10:08
• I've downvoted this challenge for the following reason: it is a fairly trivial, yet non-essential (such as Hello, World!) and I doubt that any interesting answers will be produced. – caird coinheringaahing Feb 19 '18 at 12:09
• Downvoted for the same reason as @cairdcoinheringaahing. – Yytsi Feb 19 '18 at 15:54
• Upvoted because sometimes trivial challenges are fun – FantaC Feb 19 '18 at 18:11
• @cairdcoinheringaahing Just because it is trivial, does not mean it is a bad challenge? Take a look at the CP-1610 answer, I would definitely call that interesting. It has produced interesting solutions. – Amorris Feb 19 '18 at 19:22
# CP-1610 assembly, 30 DECLEs = 38 bytes
Let's try this on a processor lacking a multiply instruction. This code is intended to be run on an Intellivision.
CP-1610 instructions are encoded with 10-bit values, known as 'DECLE' s. This subroutine is 30 DECLEs long, starting at $4808 and ending at$4825.
Takes input in registers R0 and R3. Saves the result in R2.
ROMW 10 ; use 10-bit ROM
ORG $4800 ; map program at address$4800
4800 02B8 000E MVII #14, R0 ; example call
4802 02BB 0006 MVII #6, R3
4804 0004 0148 0008 CALL addX4Y4
4807 0017 DECR PC ; loop forever
4809 0004 0148 001A CALL square ; compute R2 = R0^2
480C 0004 0148 0019 CALL square2 ; compute R2 = R2^2
480F 0272 PSHR R2 ; push this result on the stack
4810 0098 MOVR R3, R0 ; compute R2 = R3^2
4811 0004 0148 001A CALL square
4814 0004 0148 0019 CALL square2 ; compute R2 = R2^2
4817 02F2 ADD@ R6, R2 ; add this result to the intermediate one
4818 02B7 PULR PC ; return
4819 0090 square2 MOVR R2, R0 ; copy R2 to R0
481A 0081 square MOVR R0, R1 ; copy R0 to R1
481B 01D2 CLRR R2 ; initialize R2 = result
481C 0200 0002 B halve ; start by halving R1
481F 0048 loop SLL R0 ; double R0
4820 0079 halve SARC R1 ; halve R1
4821 0221 0004 BC add ; was the LSB set?
4823 022C 0005 BNEQ loop ; is R1 now equal to zero?
4825 00AF JR R5 ; return
### Example run
Running the above code (with R0 = 14 and R3 = 6) gives:
> b 4807
Set breakpoint at $4807 > r Hit breakpoint at$4807
0900 0000 9B20 0006 01FE 4817 02F1 4807 S-----iq DECR R7
^^^^
R2 is set to $9B20, which is 39712 in decimal. # Perl, 12 bytes Includes +1 for p Works for 1 or more numbers each given on a separate line on STDIN (echo 2; echo 3) | perl -pe '$\+=$_**4}{' # Haskell, 11 bytes sum.map(^4) This is a function that takes the parameters as a list. Try it online! • Same bytecount: a#b=a^4+b^4 – ბიმო Feb 19 '18 at 15:02 # ><>, 11 bytes :*:*$:*:*+n
Try it online!
Takes values through the -v flag. Dupe and multipy, dupe and multiply, and repeat with the other value before adding the two together and printing.
# Japt, 3 bytes
Takes input as an array of integers; can handle negatives and more than 2 integers at a time. Add N at the beginning to take input as individual integers.
xp4
Try it
## Explanation
p4 raises each element to the power of 4 and x reduces by addition.
# Pyt, 2 bytes
⁴Ʃ
Try it online!
Takes input as a list.
• Knew there would be a language with a built-in for **4, congrats. – ETHproductions Feb 20 '18 at 16:50
# APL (Dyalog Unicode), 5 bytesSBCS
Anonymous tacit prefix function. Takes a list as argument. The list may have any length and contain any numbers, even complex ones.
+.*∘4
Try it online!
+.* is a variant on matrix product, +.× as follows: a b+.×c d is (a×c)+(b×d) and a b+.×c is (a×c)+(b×c). So a b+.*c is (a*c)+(b*c). * is power.
∘4 curry four as right argument. This results in a monadic function (a*4)+(b*4).
## Retina, 9 bytes
.+
****
_
Try it online!
Input should be linefeed-separated.
### Explanation
.+
****
* is Retina's repetition operator. It has implicit operands $& and _, respectively, so the substitution pattern is short for $&*$&*$&*$&*_. It's also right-associative, if the regex matches a decimal number n, this generates a string of n4 underscores (i.e. a unary representation of the fourth power of n). _ To sum the two results and convert the sum back to decimal, we simply count the number of underscores in the string. # C (gcc), 0 + 24 bytes Compile this code main(){ printf("%d\n",x(2,3)); printf("%d\n",x(14,6)); printf("%d\n",x(0,25)); } With this flag: -D=x(a,b)a*a*a*a+b*b*b*b # Python 3, 20 bytes lambda x,y:x**4+y**4 • You can remove the assignment to f as this isn't a recursive function. – Shaggy Feb 19 '18 at 11:56 • @Shaggy Maybe I'm confused but how else would my answer take any input? Wouldn't I then be defining something that is essentially 'lost' right after being interpreted? It's not like I can feed input in via args, for example. Or it is normal in codegolf that those bytes 'f=' are not counted? – healeycodes Feb 19 '18 at 12:27 • What I meant was: you don't need to include it in your byte count. Anonymous functions/lambdas are valid. – Shaggy Feb 19 '18 at 13:00 # MATL, 3 bytes K^s Try it online! This can handle more than two input values, as well as negative inputs. ### Explanation: Fasten your seat belts, this might blow your mind! % Implicit input K % Push literal 4 ^ % Raise each element of the input vector to the 4th power s % Sum Also works: 4^s % Push 4 and raise input to it, then sum UUs % Square input twice, then sum # 05AB1E, 3 bytes 4mO Try it online! Explanation 4m # raise each to the power of 4 O # sum # Julia, 11 bytes a$b=a^4+b^4
Try it online!
# Julia, 12 bytes
!a=sum(a.^4)
Try it online!
# J, 7 6 bytes
1#.^&4
Try it online!
Works for lists with arbitrary length
^&4 - each item of the list to the 4-th power
1#. - sum of all 4-th powers by base-1 conversion
Try it online!
# J, 7 bytes
+/ .^4:
This is a variant of the matrix product, analogue of Adám's APL solution
Try it online!
• 6 bytes: 1#.^&4 – Adám Feb 19 '18 at 10:34
• @Adám Thanks, I forgot to try to add up the numbers by base-1 conversion. – Galen Ivanov Feb 19 '18 at 11:09
# C (gcc), 26 bytes
f(a,b){a=a*a*a*a+b*b*b*b;}
Try it online!
## C, C++ => 29 bytes
-1 byte thanks to Jonathan Frech
#define Q(a,b)a*a*a*a+b*b*b*b
Test cases :
#include <stdio.h>
int main() {
printf("Q(%d,%d) = %d\n", 2, 3, Q(2, 3));
printf("Q(%d,%d) = %d\n", 14, 6, Q(14, 6));
printf("Q(%d,%d) = %d\n", 0, 25, Q(0, 25));
}
• Could you not drop the space in ) a? – Jonathan Frech Feb 19 '18 at 11:07
# Triangularity, 31 bytes
...)...
..IEM..
.)4s^}.
u......
Try it online!
# R, 16 bytes
pryr::f(x^4+y^4)
Try it online!
## Ruby, 16 bytes
->a,b{a**4+b**4}
Try It Online!
# Perl 6, 9 bytes
*⁴+*⁴
Try it online!
• @EsolangingFruit The problem is that along with * WhateverCode lambdas there are also ** HyperWhatever lambdas; so it would have to be written as * **4+* **4, or else it would would be seen as ** *4+** *4 which doesn't work. ((**⁴)(2,3) results in (16,18)) – Brad Gilbert b2gills Feb 19 '18 at 17:59
• @BradGilbertb2gills I guess that makes sense. Why am I surprised Perl supports Unicode superscript exponents? – Esolanging Fruit Feb 20 '18 at 1:01
# Python 3, 20 bytes
lambda a,b:a**4+b**4
Try it online!
# JavaScript (ES7), 15 bytes
Does exactly what it says on the tin.
a=>b=>a**4+b**4
### Test cases
let f =
a=>b=>a**4+b**4
console.log(f(2)(3))
console.log(f(14)(6))
console.log(f(0)(25))
# IBM/Lotus Notes Formula, 17 bytes
@Sum(@Power(a;4))
Field formula that takes its input from a multi-value numeric field on the same form. Works because for most Formula functions, if a list is given as input then the formula will apply the given function recursively to every item in the list.
There is no TIO for Notes but here's a screenshot of one of the test cases:
# Excel, 10 bytes
=A1^4+B1^4
Nothing to see here.
# Jelly, 3 bytes
*4S
Try it online!
L,4^$4^+ Try it online! ## Perl 6, 15 bytes {$^a**4+\$^b**4}
An anonymous lambda. Try It Online!
# Java 8, 21 bytes
a->b->a*a*a*a+b*b*b*b
Try it online.
# APL+WIN, 5 bytes
Prompts for screen input of a two element vector. Seems to work for negative inputs
+/⎕*4
# Python 2, 30 28 bytes
lambda i:sum(x**4for x in i)
Try it online!
Alternative approach for Python (albeit slightly longer than the more obvious one). Takes input as a list of integers.
-2 with thanks to @JonathanAllan
• You can drop the [ and ] – Jonathan Allan Feb 19 '18 at 14:21
• @JonathanAllan - I was sure I'd tried that and got a syntax error but it seems to work now. Thanks! – ElPedro Feb 19 '18 at 15:25 | 2019-06-27 04:23:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2598232924938202, "perplexity": 7337.543218338207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000613.45/warc/CC-MAIN-20190627035307-20190627061307-00097.warc.gz"} |
https://www.geomag.nrcan.gc.ca/mag_fld/magref-en.php | Magnetic reference field models
Magnetic reference field models provide an easy way to calculate magnetic declination and other components of the magnetic field. A reference field model is a mathematical algorithm whose parameters are based on an analysis of magnetic monitoring satellites either over the entire world or a part of the world. Spherical harmonic analysis is the most common method used for producing global models. The International Geomagnetic Reference Field (IGRF) and the World Magnetic Model (WMM) are the most commonly used models for navigational purposes. Models are traditionally updated every five years. The Canadian Geomagnetic Reference Field (CGRF) is a model of the magnetic field over the Canadian region for the time period 1985-2010. It was produced using denser data over Canada than were used for the IGRF, and because the analysis was carried out over a smaller region, the CGRF can reproduce smaller spatial variations in the magnetic field than can the IGRF.
Since magnetic field models such as the IGRF and CGRF are approximations to observed data, a value of declination computed using either of them is likely to differ somewhat from the "true" value at that location. It is generally agreed that the IGRF achieves an overall accuracy of better than 1° in declination; the accuracy is better than this in densely surveyed areas such as Europe and North America, and worse in oceanic areas such as the south Pacific. The accuracy of the CGRF, in southern Canada, was about 0.5°. The accuracy of all models is worse in the Arctic near the North Magnetic Pole.
Magnetic field models are used to calculate magnetic declination and other components by means of computer programs such as the magnetic declination and magnetic field calculators.
Spherical harmonics
In 1838 the German mathematician and magnetician Frederick Gauss developed a method of representing the magnetic field in terms of a converging series whose terms were functions of latitude, longitude and radial distance from the centre of the earth. In modern notation, the representation is:
$V=a\sum _{n=1}^{\mathrm{N max}}{\left(\frac{a}{r}\right)}^{n+1}\left[{g}_{n}^{m}\mathrm{cos}\left(m\phi \right)+{h}_{n}^{m}\mathrm{sin}\left(m\phi \right)\right]{P}_{n}^{m}\left(\theta \right)$
• φ refers to longitude
• θ refers to latitude
• r is the radial distance
• n is the degree of the term
• m is the order of the term
• V is called the scalar potential
The ${P}_{n}^{m}$ are called associated Legendre polynomials which look very much like distorted sine waves. The ${g}_{n}^{m}$ and ${h}_{n}^{m}$ are called Gauss coefficients which are determined through a least-squares analysis of a world-wide distribution of magnetic observations.
In theory the series goes to infinity; in practice some maximum degree, Nmax is chosen so that the series is able to reproduce the observed field to the desired resolution and accuracy. For example, for the IGRF, Nmax = 13. To reproduce the field originating within the core of the Earth requires Nmax = 15. To reproduce crustal anomalies visible in magnetic data at satellite altitudes requires Nmax = 80.
The magnetic field components (X, Y and Z) can be calculated from the scalar potential through the following derivatives:
$X=\frac{1}{r}\frac{\partial V}{\partial \theta }$ $Y=\frac{1}{rsin\theta }\frac{\partial V}{\partial \phi }$ $Z=\frac{\partial V}{\partial r}$ | 2020-09-21 15:33:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7628126740455627, "perplexity": 694.7871362638257}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400201826.20/warc/CC-MAIN-20200921143722-20200921173722-00029.warc.gz"} |
https://jira.lsstcorp.org/browse/DM-17169?focusedCommentId=225402&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel | # Crosstalk sources dataset does not cleanly fit Gen3 butler concept
XMLWordPrintable
## Details
• Type: Improvement
• Status: Done
• Resolution: Done
• Fix Version/s: None
• Component/s: None
• Labels:
• Story Points:
18
• Team:
Data Release Production
## Description
For cameras with inter-CCD crosstalk, the crosstalkSources dataset is produced from data on other images. This does not cleanly fit in the Gen3 butler concept, as the required data is not necessarily known before running the butler query.
This does not prevent ci_hsc from working, as HSC has only intra-CCD CT, which does not require additional data.
## Activity
Hide
Jim Bosch added a comment - - edited
One promising idea for how to solve this (discussed recently with Christopher Waters and Nate Lust) would be to provide a way for a concrete PipelineTaskConnections implementation to customize a Prerequisite lookup. Nominally, this would take the form of a callable that is given a Registry instance, the data ID of the quantum, and the DatasetType of the prerequisite, and returns a list of DatasetRefs; the default implementation would just call Registry.queryDatasets, as the code in pipe_base does now. The DECam ISR customization would return all (existing) raws for that exposure, and use DeferredDatasetHandle to only load those it actually needs in runQuantum. One catch here is that we would like it to be possible for only the DECam ISR to do this, which may imply that we want this callback to be something we can configure somehow. A generic-ISR option for "hasMultiDetectorCrosstalk" that only DECam would enable (for now) might be easier, and would make sense.
That would also provide a workaround for the problem with jointcal QG generation (DM-21904): it would let us declare all of jointcal's inputs as prerequisites, and use the same customization hook to query for all detectors that in any exposure that overlaps the tract, rather than just the exposure-detector combinations that do. The converse is also true: a more complete solution to the jointcal problem on DM-21904 is likely to provide a solution for DECam crosstalk as well, but I'm going to assume we'll do this ticket first since it's probably easier, even though the changes here may ultimately be obsoleted by those on DM-21904.
I'm assigning this to Nate Lust as the first part of the work will be in the Connections APIs and the QG-generation logic; I expect he'll at least collaborate with Christopher Waters on the IsrTask changes and testing with DECam.
Show
Jim Bosch added a comment - - edited One promising idea for how to solve this (discussed recently with Christopher Waters and Nate Lust ) would be to provide a way for a concrete PipelineTaskConnections implementation to customize a Prerequisite lookup. Nominally, this would take the form of a callable that is given a Registry instance, the data ID of the quantum, and the DatasetType of the prerequisite, and returns a list of DatasetRefs; the default implementation would just call Registry.queryDatasets , as the code in pipe_base does now. The DECam ISR customization would return all (existing) raws for that exposure, and use DeferredDatasetHandle to only load those it actually needs in runQuantum . One catch here is that we would like it to be possible for only the DECam ISR to do this, which may imply that we want this callback to be something we can configure somehow. A generic-ISR option for "hasMultiDetectorCrosstalk" that only DECam would enable (for now) might be easier, and would make sense. That would also provide a workaround for the problem with jointcal QG generation ( DM-21904 ): it would let us declare all of jointcal's inputs as prerequisites, and use the same customization hook to query for all detectors that in any exposure that overlaps the tract, rather than just the exposure-detector combinations that do. The converse is also true: a more complete solution to the jointcal problem on DM-21904 is likely to provide a solution for DECam crosstalk as well, but I'm going to assume we'll do this ticket first since it's probably easier, even though the changes here may ultimately be obsoleted by those on DM-21904 . I'm assigning this to Nate Lust as the first part of the work will be in the Connections APIs and the QG-generation logic; I expect he'll at least collaborate with Christopher Waters on the IsrTask changes and testing with DECam.
Hide
Jim Bosch added a comment -
Minor comments on the PR. Only major concern (which may be a question for Christopher Waters) is whether we need another ticket to actually use this functionality in ISR as per the original ticket description. I had misremembered this as being for fringes, not crosstalk, and while it still may enable cleanups for fringes, if we actually need it to fix crosstalk we should make sure that work does eventually get done.
Show
Jim Bosch added a comment - Minor comments on the PR. Only major concern (which may be a question for Christopher Waters ) is whether we need another ticket to actually use this functionality in ISR as per the original ticket description. I had misremembered this as being for fringes, not crosstalk, and while it still may enable cleanups for fringes, if we actually need it to fix crosstalk we should make sure that work does eventually get done.
Hide
Christopher Waters added a comment -
I think we should have another ticket for using this in ISR. I think decam CT is the only place that requires this, but being able to test that would be good (and I think this would bring gen3 ISR to feature parity with gen2).
Show
Christopher Waters added a comment - I think we should have another ticket for using this in ISR. I think decam CT is the only place that requires this, but being able to test that would be good (and I think this would bring gen3 ISR to feature parity with gen2).
Hide
Ticket scope expanded to unblock jointcal/fgcm gen3 conversion and adding testing infrastructure to ci_hsc_gen3
Show
Yusra AlSayyad added a comment - Ticket scope expanded to unblock jointcal/fgcm gen3 conversion and adding testing infrastructure to ci_hsc_gen3
## People
• Assignee:
Nate Lust
Reporter:
Christopher Waters
Reviewers:
Jim Bosch
Watchers:
Christopher Waters, Jim Bosch, John Parejko, Krzysztof Findeisen, Yusra AlSayyad
0 Vote for this issue
Watchers:
5 Start watching this issue
## Dates
• Created:
Updated:
Resolved: | 2020-08-12 16:07:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44685670733451843, "perplexity": 3393.210583692445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738905.62/warc/CC-MAIN-20200812141756-20200812171756-00425.warc.gz"} |
http://serverfault.com/questions/187859/iis7-is-gzipping-files-but-not-serving-the-gzipped-version | # IIS7 is gzipping files but not serving the gzipped version
By following a number of helpful blog posts I have configured IIS to gzip my static files. I have even enabled Failed Request Tracing and filtered to the 200 status code, and I can see the successful compression events taking place as well as the finished headers, which look like this:
Headers="Content-Type: text/css
Content-Encoding: gzip
Last-Modified: Mon, 04 Oct 2010 17:35:08 GMT
Accept-Ranges: bytes
ETag: "02ef37cea63cb1:0"
Vary: Accept-Encoding
Server: Microsoft-IIS/7.5
X-Powered-By: ASP.NET
"
However, when I test in Fiddler and Firefox the Content-Encoding header is missing, and the file is not gzipped.
This is a similar issue to this question which was never resolved.
IIS is generating the gzipped files which I can see in C:\inetpub\temp\IIS Temporary Compressed Files .
Does anyone know how I can troubleshoot this?
-
Please see my answer to this question. It may help: serverfault.com/questions/505788/… – ianbeks Jul 23 at 9:29 | 2014-12-28 07:42:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1787804365158081, "perplexity": 5252.362136093711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447556094.122/warc/CC-MAIN-20141224185916-00095-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/308211/generating-irreducible-representations-of-a-simple-lie-algebra-with-schur-functo | # Generating Irreducible representations of a simple lie algebra with Schur functors
Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep_V(\mathfrak{g}) \subset Rep(\mathfrak{g})$ to be the smallest symmetric monoidal, idempotent complete, abelian subcategory with duals (so, closed under tensor products, retracts, direct sums and duals) which contains $V$.
Question: Does there always exist an irreducible $V$ for which $Rep_V(\mathfrak{g}) \cong Rep(\mathfrak{g})$? When it exists, is there a unique minimal one (in terms of the order on the weights) such $V$ (up to dualizing)? If not is there a unique self-dual such representation? If it doesn't exist, what is the minimal dimensional $V$ (possibly reducible) which satisfies this condition? Is it unique in some sense?
I'm interested in the question for all $\mathfrak{g}$ of type $A,B,C$ and $D$ (the exceptionals are a luxury). I think the standard representation $V$ in the case of type $A$ generates the entire category in this sense so that the answer is positive for this case but i'm not sure about any of the other cases.
• The irrep V won't be close to unique, even for SL_3, where the irrep with highest weight 2omega_1+omega_2 is also a tensor generator. Aug 14, 2018 at 2:09
• @PeterMcNamara Right, at least its the unique minimal one in the sense that $2 \omega_1 + \omega_2 > \omega_1+ \omega_2$ in the partial ordering of the weights. Aug 14, 2018 at 7:11
• @PeterMcNamara Isn't it also minimal in the sense of dimensions $dim V_{\omega_1+\omega_2} \lt dim V_{2 \omega_1 + \omega_2}$? Aug 14, 2018 at 7:18
• @PeterMcNamara Won't even $\omega_1$ suffice there? We get $\omega_2$ in the tensor square as the alternating part (or just by taking the dual, since the OP also required closure under duals). Aug 14, 2018 at 8:32
• Yes omega_1 suffices here. This was mentioned by the OP in their last paragraph, which is why I gave a different example to show non-uniqueness. Aug 14, 2018 at 9:48
You can replace "simple Lie algebra over $\mathbb{C}$" with "simply connected, simple compact Lie group" and the category of representations will be equivalent as a tensor category.
It's a standard result that a representation of a compact group with finite center is a tensor generator if and only if it is faithful, and at least for a simple Lie group, such an irrep is faithful if and only its highest weight generates the weight lattice modulo the root lattice. In particular, one exists if and only if the center is cyclic. There is a unique minuscule fundamental irrep for each nontrivial coset mod the root lattice, which will be a generator if that coset generates the quotient.
• In $A_n$, this means $\bigwedge{}^m\mathbb{C}^n$ for $m$ and $n$ relatively prime.
• In $B_n$, the Spin group has center $C_2$, so the spin representation is the unique minimal tensor generator.
• In $C_n$, the symplectic group has center $C_2$, so the vector representation is the unique minimal tensor generator.
• In $D_{2n+1}$, the center is $C_4$, so either of the two spin representations is a minimal generator.
The only case where the cyclic property fails is $D_{2n}$, where the weight lattice mod the root lattice is the Klein 4-group. There are 3 minuscule fundamentals (given by the vector rep and the two spin reps) but none is a tensor generator.
• Great! This is exactly what I wanted! Is there a good reference for these statements (especially for "faithful iff generates the center" and "tensor generator iff faithful")? Maybe its in Fulton and Harris? In any case just to make sure I understand, why do you state the statement that a representation is a tensor generator iff it is faithful only for compact groups? Does this fail in general for finite dimensional representations of complex connected simple lie groups? Your answer gives the result only for the simply connected ones but it seems weird for this statement to fail in general. Aug 14, 2018 at 17:24
I assume that you mean here full subcategory, not just subcategory. Assuming that, let $V$ be an irreducible representation of $\mathfrak{g}$ of dimension $n$ which is also $\textit{faithfull}$. This means that the resulting lie algebra map $\mathfrak{g}\to \mathfrak{sl}_n$ is injective. We thus get an injective map of associative algebras $$U(\mathfrak{g})\to U(\mathfrak{sl}_n).$$ We will think of this map as inclusion. Notice that due to the PBW basis, $U(\mathfrak{sl}_n)$ is free over $U(\mathfrak{g})$. Let $W$ be any simple representation of $\mathfrak{g}$. It will be enough to show that $W$ is contained in a direct summand of a direct sum of representation of the form $V^{a,b}:=V^{\otimes a}\otimes (V^*)^{\otimes b}$. Consider now the $\mathfrak{sl}_n$ representation $W':=U(\mathfrak{sl}_n)\otimes_{U(\mathfrak{g})} W$. This representation is generated by a single element (since $W$ is), and is therefore a quotient of $U(\mathfrak{sl}_n)$. Since $U(\mathfrak{sl}_n)$ has a filtration by finite dimensional representations which are direct sums of direct summands of $V^{a,b}$, the same is true for $W'$ (using here the semisimplicity of $\mathfrak{sl}_n$). By restricting back to $\mathfrak{g}$, the same is true for $W'$ considered as a representation of $\mathfrak{g}$. But as $\mathfrak{g}$-representations, $W'$ is a subrepresentation of $W$ (because of the PBW-basis). This implies that $W$ is a subrepresentation of a direct summand of $\oplus V^{a_i,b_i}$, which is what we wanted to show.
• Just to make sure I understand it seems we don't really need $V^*$ since we have $\bigwedge^{n-1} V \cong V^*$. Does that seem right to you? Secondly why is $W^{'}$ a subrepresentation of $W$? I don't see how to use the PBW basis here... Aug 14, 2018 at 17:47
• 2. $W$ is a subrepresentation of $W'$ due to the following reason: let $a_1=1,a_2,\ldots$ be a $free$ basis of $U(\mathfrak{sl}_n)$ over $U(\mathfrak{g})$ (which exists by PBW), Then as a $\mathfrak{g}$ representation $W'$ can be written as $a_1\otimes W\bigoplus \oplus_{i=2}^{\infty}a_i\otimes W$. The first part is isomorphic with $W$. Aug 15, 2018 at 9:37
• Both of the answers were great and contributed independently to answer the question. I had to choose 1 answer to accept though so I chose the one where the final solution was layed out. Thanks for your input! Aug 16, 2018 at 6:40 | 2022-09-27 08:48:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9204860925674438, "perplexity": 190.5326112087204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334992.20/warc/CC-MAIN-20220927064738-20220927094738-00388.warc.gz"} |
https://ibn.idsi.md/ro/vizualizare_articol/111135/datacite | Ways and Reserves of Increasing the Efficiency of Electric Power Transmission Lines
Articolul precedent Articolul urmator 481 0 SM ISO690:2012MAEVSKY, Dmitry; BESARAB, Aleksandr; MAEVSKAYA, Elena J.; BERZAN, Vladimir; SAVIELIEV, Artem. Ways and Reserves of Increasing the Efficiency of Electric Power Transmission Lines. In: Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer EngineeringTCSET 2020. Ediția a 15-a, 25-29 februarie 2020, Lviv - Slavske. Lviv-Slavske, Ukraine: Institute of Electrical and Electronics Engineers Inc., 2020, pp. 618-622. EXPORT metadate: Google Scholar Crossref CERIF DataCiteDublin Core
Ediția a 15-a, 2020
Lviv - Slavske, Ukraine, 25-29 februarie 2020
Ways and Reserves of Increasing the Efficiency of Electric Power Transmission Lines
DOI: https://doi.org/10.1109/TCSET49122.2020.235506
Pag. 618-622
Maevsky Dmitry1, Besarab Aleksandr1, Maevskaya Elena J.1, Berzan Vladimir2, Savieliev Artem1 1 Odessa National Polytechnic University,2 Institute of Power Engineering Disponibil în IBN: 3 octombrie 2020
Rezumat
The article is devoted to the analysis of losses in functioning transmission lines of electric energy and identifying opportunities for reduction of these losses. It is shown that on average in the world in transmission lines about 5% of generated electricity is lost, which in absolute values is more than 1300 TWh. Based on real data from 15 power lines of Ukraine, it was revealed that losses in them are on average 13.3%, and in absolute values - 19.1 TWh. To reduce these losses, it is proposed to use the operation of power lines in a matched load mode. The modeling carried out for this mode showed the possibility of reducing losses to 1.78%. To achieve matched load mode, it is proposed to use compact controlled high-voltage power lines. The principles of operation of these lines are considered and ways of their practical implementation using IoT technologies are identified.
Cuvinte-cheie
coefficient of efficiency, controlled high-voltage power line, smatched load mode, power efficiency, power line
DataCite XML Export
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<description xml:lang='en' descriptionType='Abstract'><p>The article is devoted to the analysis of losses in functioning transmission lines of electric energy and identifying opportunities for reduction of these losses. It is shown that on average in the world in transmission lines about 5% of generated electricity is lost, which in absolute values is more than 1300 TWh. Based on real data from 15 power lines of Ukraine, it was revealed that losses in them are on average 13.3%, and in absolute values - 19.1 TWh. To reduce these losses, it is proposed to use the operation of power lines in a matched load mode. The modeling carried out for this mode showed the possibility of reducing losses to 1.78%. To achieve matched load mode, it is proposed to use compact controlled high-voltage power lines. The principles of operation of these lines are considered and ways of their practical implementation using IoT technologies are identified.</p></description>
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</resource> | 2022-06-26 12:08:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5095905661582947, "perplexity": 6039.998979365404}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103205617.12/warc/CC-MAIN-20220626101442-20220626131442-00374.warc.gz"} |
https://tex.stackexchange.com/tags/siunitx/hot | # Tag Info
7
You need to set input-decimal-markers and input-ignore correctly: \documentclass{article} \usepackage{siunitx} \sisetup{input-decimal-markers = ., input-ignore ={,}} \begin{document} \SI{10,000.42}{\metre} \end{document}
5
You need to get rid of the unnecessary and, in fact, destructive mode of encasing the numbers in an S column in curly braces. While you're at it, do get rid of the \$ symbols in the S-type columns as well. \documentclass{article} \usepackage{siunitx} % for 'S' column type \usepackage{booktabs} % for well-spaced horizontal rules % Commands to add space ...
4
cause of your problem seems to be use of @{\extracolsep{\fill}} you define three columns, but use only two ... I would not use tabular* table environment, in my opinion, the result is not nice as solution to your problem I put the column headers into p columns with width equal to half the text width in my MWE, I remove all unnecessary code, repair definition ...
4
Here are three more possibilities: Since your table is too tall to fit into the textblock and you specified that you didn't want to chang the margins, I'd suggest the use of longtable. This might especially come in handy if your caption gets longer than a single line or if you decide to add more rows to your table. In the second example, I have allowed ...
4
with siunitx and tabularx (that table is fit into text area): \documentclass[12pt, a4paper]{article} \usepackage[format=hang, font={small,it}, labelfont={bf,it}, labelsep=space]{caption} \captionsetup[table]{labelsep=space,justification=raggedright,singlelinecheck=off} \usepackage{siunitx} \usepackage{booktabs, tabularx} ...
3
The time of loading is certainly intentional. The binary-units option code (for true) is: binary-units / true .code:n = { \AtBeginDocument { \__siunitx_load_binary: } }, which as the code suggests, does basically \AtBeginDocument{\input{siunitx-binary.cfg}} (with a little more work to have the file be loaded as a package). It isn't clear why this is ...
3
I would use \thead for the column heads, instead of two rows. Your S columns were not aligned on the decimal dot, because your numbers were surrounded by pairs of braces, which means they were ordinary data, which had to be centred. I added the caption package to have a more sensible spacing between caption and table. Your strut commands are needless if you ...
2
It is possible, but have to use dcolumn instead of siunitx, below are the modified tags: \documentclass{standalone} \usepackage{pgfplotstable} \usepackage{dcolumn}% \newcolumntype{d}[1]{D{.}{.}{#1}}% \begin{document} \begin{tabular}{d{2,3} d{2,3}} a & b \\ -0.135 & \pgfmathprintnumber{-0.135} \\ 0.1 & \...
2
too long that I can fit to comment: Welcome to TeX.SE! you define 22 column, but use only 8! your table nicely fit into text width (well, after repairing @{\hskip\tabcolsep\extracolsep\fill} to more correct @{\hspace*{\tabcolsep}\extracolsep{\fill}} so I can't figured out what is your problem ... after done small changes to your code, I obtain the ...
1
Building upon Zarko's answer, here is a variant that also aligns the contents of the first column with respect to the \rightarrow. However, I have replaced tabular* with tabular since there is a quite large amount of horizontal white space in the table. I have added some \addlinespace commands to provide more structure to the table. \documentclass{article} \...
1
Here is part of your table. I have used tabularx for automated line breaks in the last column, threeparttablex for the formatting of the tablenotes, added some sisetup options nd removed unnecessary \multicolumn commands as well as empty rows. Adapting the rest of the table is now up to you. \documentclass{article} \usepackage{geometry} \usepackage{...
1
To have something fill to one page using the MWE, perhaps using the \usepackage{fullpage} would be helpful in achieving that goal. \documentclass[12pt, a4paper]{article} \usepackage[format=hang,font={small,it},labelfont={bf,it},labelsep=space]{caption} \captionsetup[table]{labelsep=space,justification=raggedright,singlelinecheck=off} \usepackage{siunitx} ...
1
The problem here really is that your units are very long. TeX doesn't want to do the offending line break in your MWE because it would be a bad one (you would get an Underfull \hbox). In order to get around this, I you could try to make your units shorter. First off, your unit definition is a bit over-complicated. You can just use \DeclareSIUnit{\tco}{{t_{\...
1
As Joseph Wright explains in his comment, siunitx does not attempt to handle this, because human judgement is needed in deciding what the numbers mean and how to round them. If you want to automate this, you will have to use a different tool.
1
I encountered the same issue when using LaTeX (not XeLaTeX): The current version of siunitx (2018-05-27) is incompatible with the current version of FiraSans (2019-06-06). More precisely, using, for instance, “book” as the regular (medium) fontseries and/or using “semibold” as the default bold fontseries confuses siunitx to the extent that numbers that ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-08-22 11:41:27 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9632734060287476, "perplexity": 2706.2614282332756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317113.27/warc/CC-MAIN-20190822110215-20190822132215-00366.warc.gz"} |
http://math.stackexchange.com/questions/481733/meaning-of-the-rank-of-a-map-of-free-modules | # Meaning of the Rank of a Map of Free Modules?
I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one:
"Suppose that $J:R^t \rightarrow R^r$ is a map of free modules over a ring $R$ whose rank is less than or equal to $c$, as for the Jacobian matrix of an ideal of codimension $c$..." (Chapter 16.7, Page 407)
I'm not sure what "rank" stands for in this generality (where the image need not be free). Vanishing of minors?
-
You might check out section 1.4 of Bruns and Herzog's "Cohen Macaulay Rings", starting at page 20. Definition 1.4.2 and Proposition 1.4.3 might be helpful. – mbrown Sep 2 '13 at 3:20
Presumably it means the largest $k$ such that the induced map $\Lambda^k(J) : \Lambda^k(R^t) \to \Lambda^k(R^r)$ on exterior powers doesn't vanish. (This is a coordinate-free restatement of a condition on vanishing of minors.) At least, that would be my guess. Does the rest of the statement make sense with this interpretation?
I would have to brush up on my exterior powers (and I will!) to see if your interpretation fits as well, but I do think mbrown's lines up pretty nicely with what Eisenbud is doing. The $R$s that Eisenbud actually deals with wind up being domains, so it seems reasonable that he could mean the rank of the map induced by localization at 0 (he does mention vanishing of minors though). – Cass Sep 2 '13 at 4:04
@Cass: if $R$ is a domain, the two conditions should be equivalent. – Qiaochu Yuan Sep 2 '13 at 4:28 | 2016-02-10 22:36:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8266582489013672, "perplexity": 392.20664255208874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701160822.87/warc/CC-MAIN-20160205193920-00130-ip-10-236-182-209.ec2.internal.warc.gz"} |
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• ### $450 Tuition Credit & Official CAT Packs FREE December 15, 2018 December 15, 2018 10:00 PM PST 11:00 PM PST Get the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) # Math Revolution Approach (DS) new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 09 Nov 2016, 00:36 (ex 20) (function) In(on) the xy coordinate plane, what is the area of a circle K? 1) The circle has (2,1) as its center 2) The circle passes through a point (3,4) ==>In the original condition, since the circle is on the coordinate plane, there are 3 variables (the center (a,b)) and becomes the radius r), so E is highly likely to be the answer. Through 1) & 2), you get the radius r=√(〖(2-3)〗^2+〖(1-4)〗^2 )=√(1+9)=√10, and the area of the circle is π(√10)^2=10π. Since the answer is unique, it is sufficient. The answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
### Show Tags
10 Nov 2016, 00:18
There is an arithmetic sequence An where n is a positive integer such that An+1=An+3, what is the value of A10?
1) A1=3.
2) A3+A6=27
==>In the original condition, there are 2 variables (An+1, An), and 1 equation (An+1=An+3), so D is highly likely to be the answer. 1)=2), so you get A10=30, hence unique, and sufficient. The answer is D.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 11 Nov 2016, 00:06 If x/300=y/100, x=? 1) x+y=400 2) y=100 ==>In the original condition, there are 2 variables (x,y) and 1 equation (x/300=y/100), so D is highly likely to be the answer. Hence 1)=2), the remainder of both is x=300, so it is unique, and sufficient. The answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
### Show Tags
14 Nov 2016, 00:32
Is |x-1|>|x-3|?
1) x>2
2) x>1
==>If you modify the original condition and the problem, the sign of inequality does not change even if you square both sides. In other words, |x-1|>|x-3|?,Is |x-1|^2>|x-3|^2?, (x-1)^2>(x-3)^2?, and if you expand this, you get x^2-2x+1>x2-6x+9?, 6x-2x>9-1?, 4x>8?. That is, you get x>2?. 1) is yes, and sufficient. The answer is A.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 14 Nov 2016, 00:33 For all integers k, the function g is defined by g(k) = ak+4, where a is a constant. What is the value of g(1)? (1) g(3) = 7 (2) g(-1) =3 If you modify the original condition and the problem, you get g(1)=a×1+4=a+4=?. Since you get 1 variable(a), and in order for it to match with the number of equation it needs one, hence 1 for 1), and 1 for 2). Thus, there is high chance that D is the answer. In the case of 1), from g(3)=3a+4=7 you get a=1, therefore unique and sufficient. In the case of 2), from g(-1)=a(-1)+5=-a+4=3 you get a=1, therefore unique and sufficient. Answer: D. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
### Show Tags
17 Nov 2016, 03:16
m=?
1) 3^3m-1=9
2) 5^2m+1=125
==> In the original condition, there is 1 variable (m), and therefore D is most likely to be the answer.
In case of con 1), from 3^3m-1=9=3^2, you get 3m-1=2 and m=1, and hence it is sufficient.
In case of con 2), from 5^2m+1=125=5^3 and 2m+1=3, you get m=1, and hence it is sufficient.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 18 Nov 2016, 00:23 If x and y are integers greater than 1 and x>y, what are the values of x and y? 1) x+y=14 2) xy=33 ==> In the original condition, there are 2 variables (x, y) and therefore C is most likely to be the answer. By solving con 1) and con 2), you get x=11 and y=3. However, since this question is an integer question, one of the key questions, if you apply CMT 4 (A), in case of con 1), from x=11, y=3 or x=10 y=4, it is not unique and not sufficient, and in case of con 2), you only get x=11 and y=3, and hence it is unique and sufficient. Therefore, the answer is B. Answer: B _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Updated on: 21 Nov 2016, 23:39
Is ab+cd>0?
1) ac+db>0
==> In the original condition, there are 4 variables (a, b, c, d). Therefore, E is most likely to be the answer. By solving con 1) and con 2), (a, b, c, d) = (3, -4, -1, -1) is no, and (1,1,1,1) is yes. Therefore, the answer is E.
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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Originally posted by MathRevolution on 21 Nov 2016, 00:59. Last edited by MathRevolution on 21 Nov 2016, 23:39, edited 1 time in total. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 21 Nov 2016, 01:01 What is the standard deviation of 3 products’ prices? 1) The sum of any two prices of these 3 products is$1,000
2) At least one of them is 500
==> In the original condition, there are 3 variables. Therefore, E is most likely to be the answer. For con 1), the sum of any 2 prices of the products is always 1,000, so if you add the three equations a+b=1,000, b+c=1,000, and c+a=1,000 together, you get 2(a+b+c)=3,000 and a+b+c=1,500 each, and using substitution, a=b=c=500. Thus, the standard deviation is 0, hence it is sufficient. Therefore, the answer is A. This question is CMT 4 (A).
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 23 Nov 2016, 00:19 If x is positive integer, is √2x an integer? 1) x/2 is an integer squared 2) x/200 is an integer squared ==> In the original condition, there is 1 variable (x). Therefore, D is most likely to be the answer. For con 1), from x=2t^2, you get √2x = √(2*2t^2 ) =2t, hence yes, it is sufficient. For con 2), from x=200s^2, you get √2x = √(2*200s^2 ) =20s, hence yes, it is sufficient. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
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24 Nov 2016, 01:41
If a+b is even, is b an integer?
1) a-b=integer
2) a+3b=even
==> In the original condition, there are 2 variables (a, b) and 1 equation (a+b=even). Therefore, D is most likely to be the answer.
For con 1), a=b=1 is yes, and a=5/2, b=1/2 is no, hence it is not sufficient.
For con 2), a+3b=a+b+2b=even, even+2b=even, 2b=even, b=integer, hence yes, it is sufficient. Therefore, the answer is B.
This type of question is a 5051-level question related to CMT 4 (A).
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 25 Nov 2016, 01:20 1 The question below is also a 5051-level question, a typical integer question. When a positive integer n has 6 different factors, n=? 1) n has 2 prime factors 2) n<18 ==> In the original condition, in case of standard deviation questions, there is 1(n) variable, and therefore D is most likely to become the answer. In case of con 1), you get $$n=2^23^1, 2^13^2,$$ hence it is not unique and not sufficient. In case of con 2), you get only $$n=2^23^1$$, hence it is unique and sufficient. Therefore, the answer is B. Answer: B _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
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28 Nov 2016, 00:34
Is x<0?
1) $$x^5$$<0
2)$$x^5$$+x+1=0
==> In the original condition, there is 1 variable, and therefore D is most likely to be the answer.
In case of con 1), from $$x^5$$<0, if you divide both sides by $$x^4$$, you get x<0, hence yes, it is sufficient.
In case of con 2), it cannot be factored, and this is when you use CMT 4 (B). In other words, if you get A and B too easily, then consider D. Since$$x^5$$+x=-1, x($$x^4$$+1)=-1, and x=-1/$$(x^4$$+1) always agree with x4+1>0, and hence yes, it is sufficient. Therefore, the answer is D.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 28 Nov 2016, 00:36 If $$a^3b^4c^5<0$$, is abc<0? 1) c<0 2) b>0 ==> If you modify the original condition and the question, when you divide both sides of $$a^3b^4c^5<0$$ by $$a^2b^4c^4,$$ you get ac<0, and the question abc<0? becomes b>0?. From con 2) b>0, it becomes yes, hence it is sufficient. Therefore, the answer is B. Answer: B _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
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30 Nov 2016, 02:08
If m and n are positive integers, is $$n^m-n$$ divisible by 6?
1) m=3
2) n=2
==> In the original condition, there are 2 variables (m,n), and in order to match the number of variables to the number of equations, there must be 2 equations. Therefore, C is most likely to be the answer. By solving con 1) and con 2), from$$2^3-2=6$$, you get yes, and hence it is sufficient. The answer is C. However, this question is an integer question, one of the key questions, so you need to apply CMT 4. For con 1), from $$n^3-n=(n-1)n(n+1)$$, it is the multiple of the three consecutive integers, which always becomes the multiple of 6, hence yes, it is sufficient. For con 2), from n=2 and m=3 yes, m=2 no, and hence it is not sufficient. Therefore, the answer is A.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 01 Dec 2016, 00:57 |a-b|=b-a? 1) a<b 2) ab<0 and $$a^2b$$=1 ==> If you modify the original condition and question, to get |A|=-A, you need to get A≤0?, so it becomes |a-b|=b-a=-(a-b)?, a-b<0?, a<b?. Thus, for con 1), it is always yes, hence sufficient, and for con 2), if you apply CMT 4 (B), it also becomes a<b, hence yes, it is sufficient. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
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02 Dec 2016, 00:32
If x is a positive integer greater than 1, is 1/x a terminating decimal?
1) x has 3 as a factor
2) x is a factor of 81
==> If you modify the original condition and the question, to get 1/x as the terminating decimal, only 2 or 5 can be the prime factors of denominator x. However, for con 2), in order for x to become factors of 81=$$3^4$$, only 3 can be the prime factor of x, hence no, it is sufficient. Therefore, the answer is B.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 05 Dec 2016, 00:34 If 5 different positive integers have 3 as its median, is the average (arithmetic mean) of them greater than 5? 1) The greatest integer of them is 16 2) The smallest integer of them is 1 ==> If you modify the original condition and the question, the sum of 5 integers>585=25?, and so there are 5 variables and 1 equation. Therefore, E is most likely to be the answer. However, if the question is “greater than”, you need to find the least value. By solving con 1) and con 2), The least value of the sum becomes 1+2+3+4+16=26>25 yes, hence it is sufficient. The answer is C. However, this question is a key question, so you need to apply CMT 4 (A). For con 1), the least value of the sum=1+2+3+4+16=26>25, hence yes, it is sufficient. For con 2), 1+2+3+4+5=15<25 is no, 1+2+3+10+30=46>25 is yes, hence it is not sufficient. Therefore, the answer is A. This question, related to CMT 4 (A), is 5051-level question in current GMAT. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: Math Revolution Approach (DS) [#permalink]
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05 Dec 2016, 00:37
x=?
1) 1.1x=-1.2x
2) $$1.1x^2$$=-$$1.2x^2$$
==> In the original condition, there is 1 variable (x), and in order to match the number of variables to the number of equations, there must be 1 equation as well. Therefore, D is most likely to be the answer.
For con 1), you get from 2.3x=0 to x=0, hence it is unique and sufficient, and
For con 2), you get from 2.3x2=0 to x=0, hence it is also unique and sufficient. Therefore, the answer is D.
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution Approach (DS) [#permalink] ### Show Tags 07 Dec 2016, 00:33 What is the median of the 9 numbers? 1) The smallest 5 numbers of them are less than or equal to 10. 2) The largest 5 numbers of them are more than or equal to 10. ==> In the original condition, since there are 9 variables, E is likely to be the answer. Through 1) & 2), in order of size, the first 5 numbers are smaller than 10 or the same. Also, the last 5 numbers are bigger than 10 or the same. Hence, the fifth number should be always 10. Then, the fifth number becomes median and median=10, which is unique and sufficient. Hence, the answer is C. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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http://qcof.pixelzoom.it/pytorch-zero-nan.html | isnan() function on CPU tensors, but I think a native torch. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. Then, a scalar variable is logged called "input_var" which logs the variance of the input images, calculated via the _calculate_variance function - this. 1075 nan [torch. Now that we can binned values, we have a binary value for each latitude in California. If infs/NaN gradients are encountered in a given iteration, scaler. canglan211 回复qq_40941984: 没有解决，可能是mac环境的问题，所以我换liunx了 大约一个月之前 回复 qq_40941984 你好，我也遇到这个问题，请问你解决了吗. And then, so long as there's zero or less than or equal to zero, the neural network doesn't care how much further negative it is. array([[1, 1e40]], dtype=numpy. The trainer object will also set an attribute interrupted to True in such cases. nanやinfが発生している原因の一つとしてsoftmax_cross_entropyのxとして絶対値が大きな値が入っている可能性があります nanが出る例 In [80]: x = chainer. hamiltorch: a PyTorch Python package for sampling What is hamiltorch? hamiltorch is a Python package that uses Hamiltonian Monte Carlo (HMC) to sample from probability distributions. The input type is tensor and if the input contains more. PyTorch provides a convenient way to build networks like this where a tensor is passed sequentially through operations, nn. Method for down/re-sampling, default. preprocessing. If keepdims is true, the reduced dimensions are retained with length 1. LightGBM can use categorical features as input directly. First, the namelist() function retrieves all the members of the archive – in this case there is only one member, so we access this using the zero index. The function remquo has a behavior identical to this function. roc_auc_score (y_true, y_score, average='macro', sample_weight=None, max_fpr=None, multi_class='raise', labels=None) [source] ¶ Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores. Weights start out as NaN (Pytorch) I am trying to build a regression model with 4 features and an output. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4. This repository is forked from great work pytorch-yolo2 of @github/marvis, but I couldn't upload or modify directly to marvis source files because many files were. PyTorch 中内存泄漏的典型现象就是数据并不大，但 GPU 的内存已经被占满，而且 GPU 的利用率（ut… PyTorch 教程 • 2020年4月11日 242 阅读 图神经网络（GNN）教程 – 用 PyTorch 和 PyTorch Geometric 实现 Graph Neural Networks. backoff_factor (float, optional, default=0. I add a param multi_gpu to params. Parameters. Hey @dusty-nv, it seems that the latest release of NCCL 2. It may not have the widespread adoption that TensorFlow has -- which was initially released well over a year prior, enjoys the. For example, maybe your PyTorch is under CUDA10 but LibTorch is under CUDA9. 4354e-5 etc. That's why most material is so dry and math-heavy. append(obj) Parameters. Steps to handle missing data :. 3 release, PyTorch 1. The main benefit of Numpy is its powerful N-dimensional array object that it provides. If we set all the weights to be zero, then all the the neurons of all the layers performs the same calculation, giving the same output and there by making the whole deep net useless. In our case, we'll use Scikit- Learn's MinMaxScaler and scale our dataset to numbers between zero and one. RLlib: Scalable Reinforcement Learning¶ RLlib is an open-source library for reinforcement learning that offers both high scalability and a unified API for a variety of applications. Recently a new activation function named Exponential Linear Unit or its widely known name ELU was introduced. When you append the array using the axis =1, then append operation are done along the rows of the matrix a and b. This is a quick guide to setup Caffe2 with ROCm support inside docker container and run on AMD GPUs. EuclidesDB is a multi-model machine learning feature database that is tightly coupled with PyTorch and provides a backend for including and querying data on the model feature space. Comma-separated value (CSV) files are files that contain data from a table listed in plain text form, such as email contact details. Chainer extension to prune unpromising trials. The following are code examples for showing how to use torch. , to produce batches for training/validation. It is used for deep neural network and natural language processing purposes. tensor type which cannot be modified after defined. 2, has added the full support for ONNX Opset 7, 8, 9 and 10 in ONNX exporter, and have also enhanced the constant folding pass to support Opset 10 Avoid kernel launches for zero-sized tensor inputs. BoTorch settings. Problem 2: The images aren't zero-centered and variance-normalized. If we want to find a column that have at least one nonzero (any) value, this will help:. In this paper, we implement the Stochastic Damped LBFGS (SdLBFGS) for stochastic non-convex optimization. nan that can be assigned (or compared) to torch tensors for testing purposes. Unless keepdims is true, the rank of the tensor is reduced by 1 for each entry in axis. isnan(A)] = 0 The function isnan produces a bool array indicating where the NaN values are. Today, at the PyTorch Developer Conference, the PyTorch team announced the plans and the release of the PyTorch 1. ReLUs aren’t a magic bullet since they can “die” when fed with values less than zero. , increase the right-hand side), the optimal value of the problem will increase. In the pytorch docs, it says for cross entropy loss: input has to be a Tensor of size (minibatch, C) Does this mean that for binary (0,1) prediction, the input must be converted into an (N,2) t. This is useful when using recurrent layers which may take variable length input. Part 4 of the tutorial series on how to implement a YOLO v3 object detector from scratch using PyTorch. 1 examples (コード解説) : テキスト分類 – TorchText IMDB (LSTM, GRU) 翻訳 : (株)クラスキャット セールスインフォメーション 作成日時 : 08/14/2018 (0. 0 callbacks查看完整目录。 8. 从PyTorch的设计原理上来说，在每次进行前向计算得到pred时，会产生一个用于梯度回传的计算图，这张图储存了进行back propagation需要的中间结果，当调用了. In this post, I'll use PyTorch to create a simple Recurrent Neural Network (RNN) for denoising a signal. all()] Out[43]: col0 col1 col2 col4 col6 row0 24. Zero initialization serves no purpose. 0 15 NaN row3 92. As PyTorch ages, I expect the gap here will converge to zero. NumPy has standard trigonometric functions which return trigonometric ratios for a given angle in radians. PyTorch - NumPy Bridge. One means keep the required information and Zero means ignore the unnecessary information. Assuming margin to have the default value of 0, if y and (x1-x2) are of the same sign, then the loss will be zero. Right : The same image with the channels aligned. In this paper, we take an orthogonal but complementary approach by providing a model-agnostic vector representation for time, called Time2Vec, that can be easily imported into many. In the pytorch docs, it says for cross entropy loss: input has to be a Tensor of size (minibatch, C) Does this mean that for binary (0,1) prediction, the input must be converted into an (N,2) t. The open-source code, called darknet, is a neural network framework written in C and CUDA. preprocessing. PyTorch Lecture 05: Linear Regression in the PyTorch way by Sung Kim. The neural net does not perform symmetry-breaking. The floating part of the name floating point refers to the fact that the decimal point can “float”; that is, it can support a variable number of digits before and after the decimal point. Dealing with the few-shot problem in GAN training. 在本教程中，我们将在PyTorch中实现这种模型。 (zero padded ）。 如果我们 (NaN）或者越过梯度陡峭的悬崖。. If set it to True, the net will autocheck and replace all nan/inf in gradients to zero. For common (base-10) logarithms, see log10. 1 examples (コード解説) : テキスト分類 – TorchText IMDB (LSTM, GRU) 翻訳 : (株)クラスキャット セールスインフォメーション 作成日時 : 08/14/2018 (0. Description. The following are all floats: 1. pearsonr¶ scipy. a NaN (Not a Number), as presented by 'numpy. Hey @dusty-nv, it seems that the latest release of NCCL 2. 批标准化通俗来说就是对每一层神经网络进行标准化 (normalize) 处理, 我们知道对输入数据进行标准化能让机器学习有效率地学习. Transcript: This video will show you how to fill a PyTorch tensor with a certain scalar by using the PyTorch fill operation. mask_zero: Whether or not the input value 0 is a special "padding" value that should be masked out. get_value(borrow=True),dtype=theano. This comes from NumPy, and is a great example of why learning NumPy is worth your time. bmm(X)) cholesky_cpu: For batch 51100: U(22,22) is zero, singular U Since they are few f. ,1]), requires_grad=True) div = Variable(torch. As we learned above, this is a tuple that represents the shape of the DataFrame, i. isnan() function on CPU tensors, but I think a native torch. Problem 2: The images aren't zero-centered and variance-normalized. Pyro is a universal probabilistic programming language (PPL) written in Python and supported by PyTorch on the backend. NumPy KEY We'll use shorthand in this cheat sheet arr - A numpy Array object IMPORTS Import these to start import numpy as np LEARN DATA SCIENCE ONLINE Start Learning For Free - www. rolling(self, window, min_periods=None, center=False, win_type=None, on=None, axis=0, closed=None) [source] ¶ Provide rolling window calculations. Lets see an example which normalizes the column in pandas by scaling. For some matrices on my batch I'm having an exception due the matrix being singular. Today, at the PyTorch Developer Conference, the PyTorch team announced the plans and the release of the PyTorch 1. 6805 10003 2 1 The author is skeptical of the safety and reli If Google maps can' t keep up with road constru 3 724227034 True golden 240 NaN 2 0. snoop with torchsnooper. Find us on. Along with the ease of implementation in Pytorch , you also have exclusive GPU (even multiple GPUs) support in Pytorch. This PR implements support for generalized LU factorization that is required for various algorithms such as PCA (see issue #8049). 4 adds additional mobile support including the ability to customize build scripts at a fine-grain level. 3 release, PyTorch 1. Membership Operators. Publish Your Trinket!. Other operations, like reductions, often require the dynamic range. You should decide which mean or std you need to use for the training and test datasets. This is my note for reading PyTorch's JIT source. Tensor to convert a Python list object into a PyTorch tensor. Qingyu Yin, Yu Zhang, Wei-Nan Zhang, Ting Liu, and William Yang Wang, "Deep Reinforcement Learning for Chinese Zero Pronoun Resolution", to appear in Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (ACL 2018), full paper, Melbourne, Australia, July 15-20, 2018, ACL. abs_tol is the minimum absolute tolerance – useful for comparisons near zero. If A is a vector, then normalize operates on the entire vector. The dual variable for x-y >= 1 is 2. import numpy as np A[np. 1的源代码把num_batches_tracked参数删掉的，这就非常不建议了。 10. , setting batch as the first entry of its shape;. Machine learning is taught by academics, for academics. rand(500, ) train_data = lgb. The original github depository is here. LightGBM can use categorical features as input directly. On plotting the score it will be. Another reason is division by zero or taking the logarithm of zero. As far as I understood from the […]. Books at Amazon. 注意点として、nanの回避はできますが、学習がうまくいくかは保証できません。 環境としては. The fact that the dual variable is non-zero also tells us that if we tighten x-y >= 1, (i. 数据本身，是否存在Nan，可以用numpy. There’s a lot of NaN values floating around in various columns which would not make our model particularly happy. First, by initializing the Hessian at each step using an identity matrix, the algorithm converges better than original algorithm. one of the packages that you just can’t miss when you’re learning data science, mainly because this library provides you with an array data. Values 0 and 1, are between 34 and 35. Trainer (logger=True, print_nan_grads - Prints gradients with nan values Example: # default used by the Trainer trainer = Trainer (print_nan_grads = False) weights_summary - Prints a summary of the weights when training begins. Natural Language Processing with PyTorch: Build Intelligent Language Applications Using Deep Learning | Delip Rao, Brian McMahan | download | B–OK. all()] Out[43]: col0 col1 col2 col4 col6 row0 24. obj − This is the object to be appended in the list. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. Compared to vanishing gradients, exploding gradients is more easy to realize. The data preparation process can involve three steps: data selection, data preprocessing and data transformation. To handle this case, we replace values equal to 0. DataParallel. check_inf_or_nan_grad - Small value for avoiding zero division($$\epsilon$$). PyTorch is a python based library built to provide flexibility as a deep learning development platform. 996, Test Error: 90. When slicing in pandas the start bound is included in the output. Just like NumPy, PyTorch overloads a number of python operators to make PyTorch code shorter and more readable. Sign up Why GitHub? Features → Code review; Project management. SLSQP [1-2] is a sequential quadratic programming (SQP) optimization algorithm written by Dieter Kraft in the 1980s. I'm currently attempting to install it to my Jetson TX2, because I have been wanting this for some time. glorot_normal(seed=None) Glorot normal initializer, also called Xavier normal initializer. We explain what is under the hood of the GraphConv module. Python | PyTorch acos() method. has_inf_or_nan, which determines the presence of Nan or inf in any in-between tensors, which are neither inputs nor outputs. 7068, Test Accuracy: 0. This kind of loss is very well studied and also pretty stable if the framework of choice is handling possible overflows, for instance in PyTorch, one should use log_softmax instead of softmax. It is the same data, just accessed in a different order. 从PyTorch的设计原理上来说，在每次进行前向计算得到pred时，会产生一个用于梯度回传的计算图，这张图储存了进行back propagation需要的中间结果，当调用了. The input to this transformer should be an array-like of integers or strings, denoting the values. The second thing to take care of is that if any element is exactly 0. This allows mobile developers to optimize library size by only including the operators used by their models and, in the process, reduce their on device. indexcolumn, Grouper, array, or list of the previous. All Versions. array([[1, 1e40]], dtype=numpy. The AutoGrad, PyTorch, and TensorFlow extensions are not loaded automatically to not enforce a dependency on all three frameworks. a small value gets rounded to 0). 4 adds additional mobile support including the ability to customize build scripts at a fine-grain level. I am a data scientist with a decade of experience applying statistical learning, artificial intelligence, and software engineering to political, social, and humanitarian efforts -- from election monitoring to disaster relief. These operations when carried out in a large. 注意点として、nanの回避はできますが、学習がうまくいくかは保証できません。 環境としては. initializers. We can choose an arbitrary value for term, but generally is chosen, as it shifts all of elements in the vector to negative to zero, and negatives with large exponents saturate to zero rather than the infinity, avoiding overflowing and resulting in nan. The trainer will catch the KeyboardInterrupt and attempt a graceful shutdown, including running callbacks such as on_train_end. RTX2080Tiを2枚使って、PyTorchでMixed Precision、FP16による訓練の高速化、精度とのトレードオフを計測してみました。高速化はできましたが、GPUのチューニングがかなり奥深くて大変だったことがわかりました. The natural logarithm is the base-e logarithm: the inverse of the natural exponential function ( exp ). If bins is a sequence, it defines the bin edges, including the left edge of the first bin and the right. Parameters: x (n,) array or sequence of (n,) arrays. PyTorch and NumPy allow setting certain elements of a tensor using boolean masks. new_tensor (data, dtype=None, device=None, requires_grad=False) → Tensor. 0 CMake version: version 3. 数据本身，是否存在Nan，可以用numpy. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. As HMC requires gradients within its formulation, we built hamiltorch with a PyTorch backend to take advantage of the available automatic differentiation. 🚀 We have just released PyTorch v1. Defined the loss, now we'll have to compute its gradient respect to the output neurons of the CNN in order to backpropagate it through the net and optimize the defined loss function tuning the net parameters. Each window will. Image Classification - Quick Start¶. Title: Cross-lingual Zero- and Few-shot Hate Speech Detection Utilising Frozen Transformer Language Models and AXEL Authors: Lukas Stappen , Fabian Brunn , Björn Schuller Subjects: Computation and Language (cs. I would also propose a constant torch. pivot_table (data, values=None, index=None, columns=None, aggfunc='mean', fill_value=None, margins=False, dropna=True, margins_name='All', observed=False) → 'DataFrame' [source] ¶ Create a spreadsheet-style pivot table as a DataFrame. Also following is a short head of the data along with some of its columns and values. It is a lazy learning algorithm since it doesn't have a specialized training phase. bmm(X)) cholesky_cpu: For batch 51100: U(22,22) is zero, singular U Since they are few f. PyTorch Lecture 06: Logistic Regression by Sung Kim. ]]]) since zero is larger than all the input tensor elements. So, here's an attempt to create a simple educational example. training in pytorch. Problem 2: The images aren't zero-centered and variance-normalized. Sort index. When it encounters an assert statement, Python evaluates the accompanying. 1075 nan [torch. 6609 while for Keras model the same score came out to be 0. a : numpy array from which it needs to find the maximum value. A callable: A function that returns a PyTorch Module. tensor type which cannot be modified after defined. By complementarity this implies that x-y is 1, which we can see is true. 0 SP Biggin Date Distance. Ensemble의 효과를 낼 수 있음. To get started, take a look over the custom env example and the API documentation. As PyTorch ages, I expect the gap here will converge to zero. Graph attention network¶. Right : The same image with the channels aligned. script and torch. Assignment Operators. * A tuple (features, labels): Where features is a. CL) ; Machine Learning (cs. cholesky(Xt. This TensorRT 7. Solver class represents a stochastic gradient descent based optimizer for optimizing the parameters in the computation graph. Building Caffe2 for ROCm¶. For some matrices on my batch I'm having an exception due the matrix being singular. If we want to find a column that have at least one nonzero (any) value, this will help:. Unless keepdims is true, the rank of the tensor is reduced by 1 for each entry in axis. Assignment Operators. Problem 2: The images aren't zero-centered and variance-normalized. The slicing op is one of the overloaded operators that can make indexing tensors very easy: pythonz = x[begin:end] # z = torch. It returns nan if the input does not lie in the range [-1, 1]. * tensor creation ops (see Creation Ops). integration. 0 S Biggin 3 Abbotsford 18/659 Victoria St 3 u NaN VB Rounds 4 Abbotsford 5 Charles St 3 h 1465000. 21,807 This means we're going to have to add almost twenty-two thousand columns to your table, which brings up the Curse of Dimensionality - adding this many columns means we're going to need a lot more data for our model to work and will increase our computation time significantly. The zero_grad() method clears up the accumulated gradients, which the optimizer uses to find better parameters. 3 release, PyTorch 1. PyTorch helps to focus more on core concepts of deep learning unlike TensorFlow which is more focused on running optimized model on production system. This is useful when using recurrent layers which may take variable length input. nan that can be assigned (or compared) to torch tensors for testing purposes. Weights start out as NaN (Pytorch) I am trying to build a regression model with 4 features and an output. Fix flip() shape bug in CPU. (selecting the data, processing it, and transforming it). We explain what is under the hood of the GraphConv module. The Titanic challenge hosted by Kaggle is a competition in which the goal is to predict the survival or the death of a given passenger based on a set of variables describing him such as his age, his sex, or his passenger class on the boat. indexcolumn, Grouper, array, or list of the previous. If no inf/NaN gradients are encountered, scaler. datasets as dsets import torchvision. In this quick start, we'll use the task of image classification to illustrate how to use AutoGluon's APIs. DataParallel. 0 15 NaN row3 92. In standard MPC, the controller plans for a sequence of actions at each timestep, and only executes the first of the planned actions. strip('an') removed an leading the. nandiagroinitiative. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. Conv2d and nn. Source: Convolutional Deep Belief Networks on CIFAR-10. Linear respectively. Sign up Why GitHub? Features → Code review; Project management. By adopting tensors to express the operations of a neural network is useful for two a two-pronged purpose: both tensor calculus provides a very compact formalism and parallezing the GPU computation very easily. It returns infinity with the same sign as the argument, if the argument is infinite. They are from open source Python projects. If set it to True, the net will autocheck and replace all nan/inf in gradients to zero. 1 cuda100 -c pytorch Also, just by safety, clear again the cache: rm -rf ~/. How to debug neural networks. 从PyTorch的设计原理上来说，在每次进行前向计算得到pred时，会产生一个用于梯度回传的计算图，这张图储存了进行back propagation需要的中间结果，当调用了. x is not removed since it is at the middle of the string (whitespaces lead and trail the string) string. io LEARN DATA SCIENCE ONLINE Start Learning For Free - www. Consider using check_numerics ops. Check out the newest release v1. Show last n rows. We start by creating data in 60 timesteps and converting it into an array using NumPy. A PyTorch Neural Network for price prediction (Linear Regression) using loss_SGD, loss_Momentum, loss_RMSprop, loss_Adam CUDA PyTorch tensors Prepare the Tensors Visualize Loss Graph using Visdom¶ Data Output Execution Info Log Comments. Tensors are generally allocated into the Computer's RAM and processed. See Premade Estimators for more information. OneHotEncoder ¶ class sklearn. script and torch. Each window will. 1的源代码把num_batches_tracked参数删掉的，这就非常不建议了。 10. Also you should. With this additional. training in pytorch. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width. Viewed 197k times 77. rolling(self, window, min_periods=None, center=False, win_type=None, on=None, axis=0, closed=None) [source] ¶ Provide rolling window calculations. Updated CPUInfo with relevant fixes. 1 cuda100 -c pytorch Also, just by safety, clear again the cache: rm -rf ~/. Note: You should convert your categorical features to int type before you construct Dataset. I started using Pytorch two days ago, and I feel it is much better than Tensorflow. 前回の記事（VGG16をkerasで実装した）の続きです。 今回はResNetについてまとめた上でpytorchを用いて実装します。 ResNetとは 性能 新規性 ResNetのアイディア Bottleneck Architectureによる更なる深化 Shortcut connectionの実装方法 実装と評価 原論文との差異 実装 評価 環境 データの用意 画像の確認 学習 結果. Conditional statements are handled by IF statements in Python. _Trainer__attach_dataloaders ( model , train_dataloader=None , val_dataloaders=None , test_dataloaders=None ) [source] ¶. To use TorchSnooper, you just use it like using PySnooper. Also you should. Just like NumPy, PyTorch overloads a number of python operators to make PyTorch code shorter and more readable. isnan() function on CPU tensors, but I think a native torch. append(obj) Parameters. In the binary case, the probabilities are calibrated using. Update 28 Feb 2019: I added a new blog post with a slide deck containing the presentation I did for PyData Montreal. Lastly, If both variable values are numbers, they are considered equal if both are not NaN (Not a Number) and are the same value. size: A 1-D int32 Tensor of 2 elements: new_height, new_width. A place to discuss PyTorch code, issues, install, research. resize_ (*sizes, memory_format=torch. io LEARN DATA SCIENCE ONLINE Start Learning For Free - www. Tensor to convert a Python list object into a PyTorch tensor. For positional arguments. PyTorch 官网; 要点 ¶. Nan Cao: Intelligent Big Data Visualization Lab: Tongji University, Shanghai, China: Training the model to evaluate the quality of automatically generated poster. It may not have the widespread adoption that TensorFlow has -- which was initially released well over a year prior, enjoys the. By default, the returned Tensor has the same torch. Tensor mean &nd. Creating a tuple is as simple as putting different comma-separated values. Recently a new activation function named Exponential Linear Unit or its widely known name ELU was introduced. Use PyTorch clamp operation to clip PyTorch Tensor values to a specific range Type: We see that it's a torch. terminate_on_nan¶ (bool) - If set to True, will terminate training (by raising a ValueError) at the end of each training batch, if any of the parameters or the loss are NaN or +/-inf. pytorch中张丈量是什么意思呢？torch. Along with the ease of implementation in Pytorch , you also have exclusive GPU (even multiple GPUs) support in Pytorch. LG); Machine Learning (stat. import torch import torch. PyTorch helps to focus more on core concepts of deep learning unlike TensorFlow which is more focused on running optimized model on production system. 2), ssim & ms-ssim are calculated in the same way as tensorflow and skimage, except that zero padding is used during downsampling rather than symmetric padding (there is no symmetric padding in pytorch). import torch import torchvision. I add a param multi_gpu to params. Qingyu Yin, Yu Zhang, Wei-Nan Zhang, Ting Liu, and William Yang Wang, "Deep Reinforcement Learning for Chinese Zero Pronoun Resolution", to appear in Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (ACL 2018), full paper, Melbourne, Australia, July 15-20, 2018, ACL. One means keep the required information and Zero means ignore the unnecessary information. Author: Qi Huang, Minjie Wang, Yu Gai, Quan Gan, Zheng Zhang This is a gentle introduction of using DGL to implement Graph Convolutional Networks (Kipf & Welling et al. It can be used to solve nonlinear programming problems that minimize a scalar function: subject to general equality and inequality constraints: and to lower and upper bounds o. Return DataFrame index. Module Class in PyTorch 1:54 PyTorch MNIST: Load MNIST Dataset from PyTorch Torchvision Since the gradient in the flat region is close to zero, it is unlikely that training via stochastic gradient descent will. Avg Release Cycle. D uring gradient descent, as it backprop from the final layer back to the first layer, gradient values are multiplied by the weight matrix on each step, and thus the gradient can decrease exponentially quickly to zero. Just a quick recap on how slicing works with normal Python lists. bmm(X)) cholesky_cpu: For batch 51100: U(22,22) is zero, singular U Since they are few f. Understanding the difficulty of training deep feedforward neural networks Xavier Glorot Yoshua Bengio DIRO, Universit´e de Montr ´eal, Montr eal, Qu´ ´ebec, Canada Abstract Whereas before 2006 it appears that deep multi-layer neural networks were not successfully trained, since then several algorithms have been. The matrix objects inherit all the attributes and methods of ndarry. Use PyTorch clamp operation to clip PyTorch Tensor values to a specific range Type: We see that it's a torch. strip(' xoe') removed all whitespace,x, o, and e that lead or trailed the string. transforms as transforms import random import matplotlib. CL) ; Machine Learning (cs. Tensor是一种包含单一数据类型元素的多维矩阵。. Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores. The alternative hypothesis is that the predictor does contribute to the response. x can be loaded using this method. __init__() self. and gives the output in radian form. OS: Red Hat Enterprise Linux Workstation release 7. If an array is passed, it must be the same length as the data. reload_dataloaders_every_epoch¶ (bool) - Set to True to reload. The code for this tutorial is designed to run on Python 3. If you have a callback which shuts down compute resources, for example, you can conditionally. TensorFlowでDeep Learningを実行している途中で、損失関数がNaNになる問題が発生した。 Epoch: 10, Train Loss: 85. Weights can be set when needed: w = np. Using an L1 or L2 penalty on the recurrent weights can help with exploding gradients. Left: An image from the Prokudin-Gorskii Collection. This TensorRT 7. In this part, we will implement a neural network to classify CIFAR-10 images. It can be used to solve nonlinear programming problems that minimize a scalar function: subject to general equality and inequality constraints: and to lower and upper bounds o. 0 for instance), as the derivative of the square root is infinite in $0$, we will have a nan gradient. Tensor mean &nd. OneHotEncoder(categories='auto', drop=None, sparse=True, dtype=, handle_unknown='error') [source] ¶ Encode categorical features as a one-hot numeric array. Size of the moving window. cache/pykeops* We are getting closer to the solution!. Rounds x to an integral value, using the rounding direction specified by fegetround. Principal component analysis (PCA). DataParallel. The easiest way is by using add_check_numerics_ops : Control Flow. Scala combines object-oriented and functional programming in one concise, high-level language. It shows how you can take an existing model built with a deep learning framework and use that to build a TensorRT engine using the provided parsers. We cover implementing the neural network, data loading pipeline and a decaying learning rate schedule. PyTorchを使ってモデルをつくる！ ざっくりですがPyTorchを使うときには 1. It applies a rolling computation to sequential pairs of values in a list. The comparison results between pytorch-msssim, tensorflow and skimage can be found in the Tests section. PyTorch binaries were crashing on AWS Lambda and a few other niche systems, stemming from CPUInfo handling certain warnings as errors. Thank you for helping us improve the quality of Unity Documentation. This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. We will be using preprocessing method from scikitlearn package. 6805 10003 2 1 The author is skeptical of the safety and reli If Google maps can' t keep up with road constru 3 724227034 True golden 240 NaN 2 0. EmbeddingBag: fix NaN output when input is empty. Bitwise Operators. A confusion matrix is a table that is often used to describe the performance of a classification model (or "classifier") on a set of test data for which the true values are known. All Versions. This repository is forked from great work pytorch-yolo2 of @github/marvis, but I couldn't upload or modify directly to marvis source files because many files were. A place to discuss PyTorch code, issues, install, research. Share this post. It implements lots of algorithms for graph structure recovery (including algorithms from the bnlearn , pcalg packages), mainly based out of observational data. The return value can be used as a context manager within a with block; otherwise, you must call end_run() to terminate the current run. "4", it is a vector with "one hot" node and all the other nodes being zero i. The Keras model and Pytorch model performed similarly with Pytorch model beating the keras model by a small margin. The decision_function method of SVC and NuSVC gives per-class scores for each sample (or a single score per sample in the binary case). Download books for free. 2), ssim & ms-ssim are calculated in the same way as tensorflow and skimage, except that zero padding is used during downsampling rather than symmetric padding (there is no symmetric padding in pytorch). Utility class for generating batches of temporal data. A PyTorch Neural Network for price prediction (Linear Regression) using loss_SGD, loss_Momentum, loss_RMSprop, loss_Adam CUDA PyTorch tensors Prepare the Tensors Visualize Loss Graph using Visdom¶ Data Output Execution Info Log Comments. PyTorch models that are ConvNet-like and RNN-like (static graphs) can now be shipped to the ONNX format. RLlib: Scalable Reinforcement Learning¶ RLlib is an open-source library for reinforcement learning that offers both high scalability and a unified API for a variety of applications. Second, by performing direction normalization we could gain stable optimization procedure. If no inf/NaN gradients are encountered, scaler. Setting the weight of pad symbols to zero after softmax breaks the probability distribution, rows will no longer sum to one, so we need to ensure that the output of softmax is zero for these values by setting them to negative infinity beforehand. He suggests that the weight update should be in the order of 10−3. The Jetson Nano Developer Kit is an easy way to get started using Jetson Nano, including the module, carrier board, and software. Check out the newest release v1. Module은 입력 Tensor를 받고 출력 Tensor를 계산하는 한편, 학습 가능한 매개변수를 갖는 Tensor 같은 내부 상태(internal state)를 갖습니다. Bayesian Optimization in PyTorch. The research described in the paper Graph Convolutional Network (GCN) , indicates that combining local graph structure and node-level features yields. Although CSV files can be opened with many programs, like text editors, the data. 0 96 NaN row1 40. Today, at the PyTorch Developer Conference, the PyTorch team announced the plans and the release of the PyTorch 1. In Python, If Statement is used for decision making. pytorch / pytorch. Modify create_datasets(). To handle this case, we replace values equal to 0. Recommended Reading: Python Iterators, __iter__ and __next__ Return Value from zip (). Trainer (logger=True, print_nan_grads - Prints gradients with nan values Example: # default used by the Trainer trainer = Trainer (print_nan_grads = False) weights_summary - Prints a summary of the weights when training begins. Just like NumPy, PyTorch overloads a number of python operators to make PyTorch code shorter and more readable. In this paper, we take an orthogonal but complementary approach by providing a model-agnostic vector representation for time, called Time2Vec, that can be easily imported into many. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. In Python and most other OOP programming languages, multiplying two numbers by each other is a pretty straightforward process. 4-yolov3 : Yet Another Implimentation of Pytroch 0. The code for this tutorial is designed to run on Python 3. The neural net does not perform symmetry-breaking. PyTorch provides a convenient way to build networks like this where a tensor is passed sequentially through operations, nn. What about that accuracy? 83. To get started, we import PyTorch. 1 Autograd mechanics 3. courtesy: Machine Learning Memes for Convolutional Teens,facebook Some researchers are also against using neural nets in heavily important fields like autonomous cars and drones. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. 0 with a small epsilon = 1e-16. This is the goal. Reduces input_tensor along the dimensions given in axis. convolve¶ numpy. Next, let’s create a PyTorch. Publish Your Trinket!. Note the trailing zero is not required, although it is good style. This tutorial is broken into 5 parts: Part 1 (This one): Understanding How YOLO works. 4 from marvis/pytorch-yolo2. In this part, we will implement a neural network to classify CIFAR-10 images. A growing number of publishers have adopted the open access model of publishing, thereby allowing the dissemination of research results free of […]. NumPy is, just like SciPy, Scikit-Learn, Pandas, etc. lstm加三层感知器的神经网络，预测行人坐标轨迹，loss不下降是怎么原因？ 我用两层的lstm编码坐标，然后用三层感知器解码，预测后五帧的轨迹，用的是mse和adam，尝试了从0. Here you'll find current best sellers in books, new releases in books, deals in books, Kindle eBooks, Audible audiobooks, and so much more. CrossEntropyLoss()。其参数包括：weight，size_averaged，reduce weight参数通常默认值是0，如果你的训练样本很不均衡的话，可以设置其值。. array([[1, 1e40]], dtype=numpy. So, this is how you define the loss on a single triplet and the overall cost function for your neural network can be sum over a training set of these individual losses on different triplets. 406] and std = [0. It is free and open-source software released under the Modified BSD license. The various properties of linear regression and its Python implementation has been covered in this article previously. has_inf_or_nan, which determines the presence of Nan or inf in any in-between tensors, which are neither inputs nor outputs. Researcher, supervised by Prof. 数据本身，是否存在Nan，可以用numpy. Merge (style) [source] ¶ Module that takes two or more vectors and merges them produce a single vector. snoop with torchsnooper. _Trainer__attach_dataloaders ( model , train_dataloader=None , val_dataloaders=None , test_dataloaders=None ) [source] ¶. There’s a lot of NaN values floating around in various columns which would not make our model particularly happy. Integration¶ class optuna. 1 recognizes ARM CPUs. Pass axis=1 for columns. pivot_table (data, values=None, index=None, columns=None, aggfunc='mean', fill_value=None, margins=False, dropna=True, margins_name='All', observed=False) → 'DataFrame' [source] ¶ Create a spreadsheet-style pivot table as a DataFrame. I would also propose a constant torch. This function takes a scalar or array-like object and indicates whether values are valid (not missing, which is NaN in numeric arrays, None or NaN in object arrays, NaT in datetimelike). In the pytorch docs, it says for cross entropy loss: input has to be a Tensor of size (minibatch, C) Does this mean that for binary (0,1) prediction, the input must be converted into an (N,2) t. In our case, we'll use Scikit- Learn's MinMaxScaler and scale our dataset to numbers between zero and one. cache/pykeops* We are getting closer to the solution!. cur() // Break the loop if scale is unchanged or perfect, or if we've just had enough. " The Python package has added a number of performance improvements, new layers, support to ONNX, CUDA 9, cuDNN 7, and "lots of bug fixes" in the new version. Explore US Bikeshare Data. Active 2 years, 2 months ago. A floating point type variable is a variable that can hold a real number, such as 4320. Pratyaksha Jha. You can vote up the examples you like or vote down the ones you don't like. 如果把每一层后看成这种接受输入数据的模式, 那我们何不 “批标准化” 所有的层呢?. It returns positive infinity, if the argument passed is positive infinite. For case when classes are exclusive, you don't need to sum over them - for each sample only non-zero value is just $-log p(s \in c)$ for true class c. __init__() self. Remember to replace the pysnooper. If the second argument is positive or negative zero, then the result is 1. With this additional. isnan() function on CPU tensors, but I think a native torch. a NaN (Not a Number), as presented by 'numpy. However, I must warn: some scripts from the master branch of nccl git are commited with messages from previous releases, which is a yellow flag. step # ステージ4 50回毎にコストを表示. Must contain decimal numbers. One simple trick which can help a lot is to normalize the images by substracting their mean and then dividing their standard deviation. 5 answers to this question. You can vote up the examples you like or vote down the ones you don't like. pearsonr¶ scipy. Names are used to match variables. Update 28 Feb 2019: I added a new blog post with a slide deck containing the presentation I did for PyData Montreal. PyTorch Print Tensor - Print full tensor in PyTorch so that you can see all of the elements rather than just seeing the truncated or shortened version 2:27 Back to PyTorch Tutorial Lesson List. Topic Replies Activity; Help with indexing. You can also learn to visualize and understand what the attention mechanism has learned. import torch import torchvision. 1 Tasks We test our NTM implementation on three of the ve arti cial sequence learning tasks described in the original NTM paper [4]. Explore US Bikeshare Data. Sign up Why GitHub? Features → Code review; Project management. After doing a lot of searching, I think this gist can be a good example of how to deal with the DataParallel subtlety regarding different behavior on input and hidden of an RNN in PyTorch. The weights can take on the value of an "NaN" or between 0 & 1. title: pytorch中LSTM笔记 date: 2018-07-12 16:52:53 tags: - torch项目 categories: - pytorch. This TensorRT 7. N = normalize (A) returns the vectorwise z -score of the data in A with center 0 and standard deviation 1. OneHotEncoder(categories='auto', drop=None, sparse=True, dtype=, handle_unknown='error') [source] ¶ Encode categorical features as a one-hot numeric array. Linear Regression using PyTorch Linear Regression is a very commonly used statistical method that allows us to determine and study the relationship between two continuous variables. This is a quick guide to setup Caffe2 with ROCm support inside docker container and run on AMD GPUs. It implements lots of algorithms for graph structure recovery (including algorithms from the bnlearn , pcalg packages), mainly based out of observational data. The function isnan produces a bool array indicating where the NaN values are. hamiltorch is a Python package that uses Hamiltonian Monte Carlo (HMC) to sample from probability distributions. Description. If you want to log histograms of parameter values as well, you can pass log='all' argument to the watch method. If an array is passed, it must be the same length as the data. It can be used to solve nonlinear programming problems that minimize a scalar function: subject to general equality and inequality constraints: and to lower and upper bounds o. OK, so performance is on par, but WHY should I consider Docker for deep learning? To put it simply, you escape dependency hell. There's a lot of NaN values floating around in various columns which would not make our model particularly happy. 0 preview with many nice features such as a JIT for model graphs (with and without tracing) as well as the LibTorch, the PyTorch C++ API, one of the most important. EmbeddingBag: fix NaN output when input is empty. Dataset object: Outputs of Dataset object must be a tuple (features, labels) with same constraints as below. Books at Amazon. As HMC requires gradients within its formulation, we built hamiltorch with a PyTorch backend to take advantage of the available automatic differentiation. Suppose you are working with images. windowint, offset, or BaseIndexer subclass. First, by initializing the Hessian at each step using an identity matrix, the algorithm converges better than original algorithm. array : [array_like] Input array or object. How to debug neural networks. The package is based on Numpy, Scikit-learn, Pytorch and R. 7068, Test Accuracy: 0. If that succeeded you are ready for the tutorial, otherwise check your installation (see Installing Theano). nan similar to numpy. If it's a string, the string should contain decimal points. Pyro is a universal probabilistic programming language (PPL) written in Python and supported by PyTorch on the backend. ary = []; #make list of 0 length ary. Graph Convolutional Network¶. IntTensor of size 2x4] 可以用python. transforms as transforms import random import matplotlib. In this post you will discover two simple data transformation methods you can apply to your data in Python using scikit-learn. all()] Out[43]: col0 col1 col2 col4 col6 row0 24. I am a data scientist with a decade of experience applying statistical learning, artificial intelligence, and software engineering to political, social, and humanitarian efforts -- from election monitoring to disaster relief. With this additional. 이미지 크기는 28*28임. We can convert PyTorch tensors to numpy arrays and vice-versa pretty easily. It returns negative infinity, if the argument passed is positive zero or negative zero. A boolean array can by used to index an array of the same shape. loss value. PyTorch and NumPy allow setting certain elements of a tensor using boolean masks. PyTorch Lecture 05: Linear Regression in the PyTorch way by Sung Kim. As i know, my variables are run in theano. So we need to compute the gradient of CE Loss respect each CNN class score in. Here, Argument 0 is a string "Adam" and Argument 1 is a floating number 230. If you want to stop a training run early, you can press "Ctrl + C" on your keyboard. Tensor to convert a Python list object into a PyTorch tensor. trainlm is often the fastest backpropagation algorithm in the toolbox, and is highly recommended as a first-choice supervised algorithm, although it does require more memory than other algorithms. * A tuple (features, labels): Where features is a. Conditional Statement in Python perform different computations or actions depending on whether a specific Boolean constraint evaluates to true or false. You should decide which mean or std you need to use for the training and test datasets. Second, by performing direction normalization we could gain stable optimization procedure. The following are code examples for showing how to use torch. However, there is still some functionality which TensorFlow supports that PyTorch doesn't. An extension can alternatively be loaded via import lab. " The Python package has added a number of performance improvements, new layers, support to ONNX, CUDA 9, cuDNN 7, and "lots of bug fixes" in the new version. Since hamiltorch is based on PyTorch, we ensured that. With the imageFolder loaded, let's split the data into a 20% validation set and 10% test set; then pass it to DataLoader, which takes a dataset like you'd get from ImageFolder. Part 2 : Creating the layers of the network architecture. Python | PyTorch acos() method. The input to this transformer should be an array-like of integers or strings, denoting the values. Select row by label. If no inf/NaN gradients are encountered, scaler. 0 3 NaN a 0 1. CrossEntropyLoss() images, channels. Module은 입력 Tensor를 받고 출력 Tensor를 계산하는 한편, 학습 가능한 매개변수를 갖는 Tensor 같은 내부 상태(internal state)를 갖습니다. all()] Out[43]: col0 col1 col2 col4 col6 row0 24. Image Classification - Quick Start¶. This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. If bins is a sequence, it defines the bin edges, including the left edge of the first bin and the right. The Pearson correlation coefficient measures the linear relationship between two datasets. Debugging Neural Networks with PyTorch and W&B Using Gradients and Visualizations. We are releasing the C++ frontend marked as "API Unstable" as part of PyTorch 1. Code written in Pytorch is more concise and readable. trainlm is a network training function that updates weight and bias values according to Levenberg-Marquardt optimization. Pytorch【60天修炼计划】之第一阶段——入门：Kaggle实战-房价预测，灰信网，软件开发博客聚合，程序员专属的优秀博客文章阅读平台。. TorchSnooper is a PySnooper extension that helps you debugging these errors. latest_checkpoint. This is a quick guide to setup Caffe2 with ROCm support inside docker container and run on AMD GPUs. Strictly speaking, Pearson’s correlation requires that each dataset be normally distributed. 1 Tasks We test our NTM implementation on three of the ve arti cial sequence learning tasks described in the original NTM paper [4]. terminate_on_nan¶ (bool) - If set to True, will terminate training (by raising a ValueError) at the end of each training batch, if any of the parameters or the loss are NaN or +/-inf. trainlm is a network training function that updates weight and bias values according to Levenberg-Marquardt optimization. It validates the following: (i) none of the inputs contain NaN values (ii) the training data (train_X) is normalized to the unit cube (iii) the training targets (train_Y) are standardized (zero mean, unit var) No checks (other than the NaN check) are performed for observed variances (train_Yvar) at this point. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4. Today, at the PyTorch Developer Conference, the PyTorch team announced the plans and the release of the PyTorch 1. view() on when it is possible to return a view. Just don't know why, but when i train the net, the loss always become nan after several epoch. If we want to find a column that have at least one nonzero (any) value, this will help:. array([[1, 1e40]], dtype=numpy. io LEARN DATA SCIENCE ONLINE Start Learning For Free - www. Starting with an introduction to PyTorch, you'll get familiarized with tensors, a type of data structure used to calculate arithmetic operations and also learn how they operate. In our case, we'll use Scikit- Learn's MinMaxScaler and scale our dataset to numbers between zero and one. Output: a 0 1. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. 一个张量tensor可以从Python的list或序列构建： >>> torch. He suggests that the weight update should be in the order of 10−3. convolve¶ numpy. Zero-shot Learning via Simultaneous Generating and Learning Hyeonwoo Yu, Beomhee Lee; Ask not what AI can do, but what AI should do: Towards a framework of task delegability Brian Lubars, Chenhao Tan; Stand-Alone Self-Attention in Vision Models Niki Parmar, Prajit Ramachandran, Ashish Vaswani, Irwan Bello, Anselm Levskaya, Jon Shlens. The left image displays what a. Recently a new activation function named Exponential Linear Unit or its widely known name ELU was introduced. You can vote up the examples you like or vote down the ones you don't like. This means it is ready to be used for your research application, but still has some open construction sites that will stabilize over the next couple of releases. Besides its obvious scientific uses, NumPy can also be used as an efficient multi-dimensional container of generic data. Variable for chainer. I used the same preprocessing in both the models to be better able to compare the platforms. As HMC requires gradients within its formulation, we built hamiltorch with a PyTorch backend to take advantage of the available automatic differentiation. But something I missed was the Keras-like high-level interface to PyTorch and there was not much out there back then. All Versions. 从PyTorch的设计原理上来说，在每次进行前向计算得到pred时，会产生一个用于梯度回传的计算图，这张图储存了进行back propagation需要的中间结果，当调用了. This function takes a scalar or array-like object and indicates whether values are valid (not missing, which is NaN in numeric arrays, None or NaN in object arrays, NaT in datetimelike). Solved PyTorch CTCLoss become nan after several epoch. Find us on. preprocessing. Module Class in PyTorch 1:54 PyTorch MNIST: Load MNIST Dataset from PyTorch Torchvision Since the gradient in the flat region is close to zero, it is unlikely that training via stochastic gradient descent will. The new size for the images. 大家好，在实现自定义的语义分割的loss函数的时候，遇到了问题，请大家帮忙一下， 这个自定义的loss函数的做法是，根据真实label(batchsize,h,w)的每个pixel的对应的class值，在网络的输出的预测值（batch-size,num-class,h,w）中，选出class对应的那个预测值,得到的就是真实label的每个pixel的class对应的预测值. A boolean array can by used to index an array of the same shape. bmm(X)) cholesky_cpu: For batch 51100: U(22,22) is zero, singular U Since they are few f. Today, we’re extremely happy to announce Amazon SageMaker Debugger, a new capability of Amazon SageMaker that automatically identifies complex issues developing in machine learning (ML) training jobs. Since the try block raises an error, the except block will be executed. mask_zero: Whether or not the input value 0 is a special "padding" value that should be masked out. view() on when it is possible to return a view.
6hvfln2ybme 2p2ex7g0dv5rsk 4ksp5pyogcy gqymum7wkjeuizj 4zlw4jqt6u1 rv04ovstdb xlklru4ialso mtgr2ul0mv enz6837ya8 xg88pp5ynz2f6d r3henv0nq0yu6 60kd9btwf0vp6t ofcpuobeqb3ufkl e88fnv4s619ec stfmn8ywoy4g gynvy0623ebbo h8thrgavgglt1f6 0chzvbugncz z5xcxvvtdf lwiydcdo78y gp9ixk2hq6 qtz1jqooj0h 6m8kn9n8hbk8y 4qhj5iqxuc84 95rb7v52uh o9qyaxq9kxz8l smkzeg2mnqpqe b0cgrxw8bpbh xmxm7wt44bu35o | 2020-09-18 19:27:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2608387768268585, "perplexity": 3320.932864613321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400188841.7/warc/CC-MAIN-20200918190514-20200918220514-00110.warc.gz"} |
https://mathoverflow.net/questions/162705/countable-vs-ultra-negligible-sets | # Countable vs. ultra-negligible sets [duplicate]
A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us say that $A$ is ultra-negligible if for any sequence $(\epsilon_n)$ of positive numbers, there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\vert I_n\vert \leq \epsilon_n$ for all $n$ (for example, any countable set is ultra-negligible). Clearly, every ultra-negligible set is negligible. In his famous problem book, P. Halmos shows that the converse is not true : the triadic Cantor set is negligible but not ultra-negligible.
I am wondering if there are examples of ultra-negligible sets that are not countable ?
• You're right. Sorry for that! Apr 7, 2014 at 16:45
• To add more references, Arnold Miller has a paper Special subsets of the real line (e.g. see Theorem 13) at his web page (and some other papers about $\gamma$-sets might be related). Apr 8, 2014 at 1:13 | 2022-05-28 02:13:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9387984871864319, "perplexity": 272.12253952670136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663011588.83/warc/CC-MAIN-20220528000300-20220528030300-00317.warc.gz"} |
https://cmde.tabrizu.ac.ir/article_6540.html | A numerical study of electrohydrodynamic flow analysis in a circular cylindrical conduit using orthonormal Bernstein polynomials
Document Type : Research Paper
Authors
1 Department of Mathematics, Yazd University, Yazd, Iran
2 Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
Abstract
In this work, the nonlinear boundary value problem in electrohydrodynamics flow of a fluid in an ion-drag configuration in a circular cylindrical conduit is studied numerically. An effective collocation method, which is based on orthonormal Bernstein polynomials is employed to simulate the solution of this model. Some properties of orthonormal Bernstein polynomials are introduced and utilized to narrow down the computation of nonlinear boundary value problem to the solution of algebraic equations. Also, by using the residual correction process, an efficient error estimation is introduced. Graphical and tabular results are presented to investigate the influence of the strength of nonlinearity $\alpha$ and Hartmann electric number $Ha^2$ on velocity profiles. The significant merit of this method is that it can yield an appropriate level of accuracy even with large values of $\alpha$ and $Ha^2$. Compared with recent works, the numerical experiments in this study show a good agreement with the results obtained by using MATLAB solver bvp5c and its competitive ability.
Keywords
History
• Receive Date: 01 December 2016
• Revise Date: 26 September 2017
• Accept Date: 24 September 2017 | 2021-06-22 16:40:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4768471419811249, "perplexity": 993.4224061158728}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488519183.85/warc/CC-MAIN-20210622155328-20210622185328-00065.warc.gz"} |
https://msp.org/agt/2005/5-4/b13.xhtml | Volume 5, issue 4 (2005)
1 J Caruso, S Waner, An approximation to $\Omega^{n}\Sigma^{n}X$, Trans. Amer. Math. Soc. 265 (1981) 147 MR607113 2 F Cohen, Homology of $\Omega^{(n+1)}\Sigma^{(n+1)}X$ and $C_{(n+1)}X$, $n\gt 0$, Bull. Amer. Math. Soc. 79 (1973) MR0339176 3 A Dold, R Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. $(2)$ 67 (1958) 239 MR0097062 4 S Kallel, Spaces of particles on manifolds and generalized Poincaré dualities, Q. J. Math. 52 (2001) 45 MR1820902 5 J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972) MR0420610 6 D McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975) 91 MR0358766 7 P Salvatore, Configuration spaces with summable labels, from: "Cohomological methods in homotopy theory (Bellaterra, 1998)", Progr. Math. 196, Birkhäuser (2001) 375 MR1851264 8 G Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213 MR0331377 9 K Shimakawa, Configuration spaces with partially summable labels and homology theories, Math. J. Okayama Univ. 43 (2001) 43 MR1913872 | 2023-01-29 02:31:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8266585469245911, "perplexity": 2446.7837135505365}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499697.75/warc/CC-MAIN-20230129012420-20230129042420-00858.warc.gz"} |
https://www.thejournal.club/c/paper/123256/ | #### On the Integrality Gap of the Prize-Collecting Steiner Forest LP
##### Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, Jens Vygen
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs $\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{\pi_i\}_{i=1}^k$ for each terminal pair; the goal is to find a forest $F$ to minimize $c(F)+\sum_{i: (s_i,t_i)\text{ not connected in }F}\pi_i$. The Steiner forest problem can be viewed as the special case where $\pi_i=\infty$ for all $i$. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least $9/4$. This holds even for planar graphs. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than $4$. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP- approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most $1/3$ and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than $3$ using a direct iterative rounding method.
arrow_drop_up | 2021-04-14 12:27:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8603693246841431, "perplexity": 873.5412217218613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077810.20/warc/CC-MAIN-20210414095300-20210414125300-00007.warc.gz"} |
https://www.physicsforums.com/threads/implications-of-quantum-foundations-on-interpretations-of-relativity.978996/page-2 | # Implications of quantum foundations on interpretations of relativity
Gold Member
4) is a bit strange, but not and interpretation. It just adds something additional, that is completely unnecessary.
Maybe it's unnecessary within classical physics, but it appears in some versions of relativistic Bohmian mechanics.
You say "If the Bell theorem is interpreted as nonlocality of nature...", well what if it isn’t?
Then 1. is the most natural formulation of relativity.
Gold Member
For me the fact that space-time has the structure of Minkowski space is not an interpretation but a consequence.
What about block universe? Is that a consequence or an interpretation?
martinbn
Maybe it's unnecessary within classical physics, but it appears in some versions of relativistic Bohmian mechanics.
Then it is unrelated. In celestial mechanics it may be convenient to choose coordinates centered at the sun, but that is not an interpretation of classical mechanics.
Then 1. is the most natural formulation of relativity.
I don't understand this.
weirdoguy
martinbn
What about block universe? Is that a consequence or an interpretation?
I've seen different people to mean different things by block universe. What do you take it to mean?
Gold Member
I've seen different people to mean different things by block universe. What do you take it to mean?
The past, presence and future exist on an equal footing.
Michael Price
Gold Member
I don't understand this.
See the second paragraph in #3.
A. Neumaier
2019 Award
For me the fact that space-time has the structure of Minkowski space is not an interpretation but a consequence.
But this alleged fact is already false in general relativity.....
Gold Member
But this alleged fact is already false in general relativity.....
He said consequence of special relativity.
Summary: If the Bell theorem is interpreted as nonlocality of nature, then what does it tell us about the meaning of Einstein theory of relativity?
According to ether theories, there are absolute space and absolute time, but under certain approximations some physical phenomena obey effective laws of motion that look as if absolute space and time did not exist. The original Lorentz version of ether theory was ruled out by the Michelson-Morley experiment, but some more sophisticated versions of ether theory are still alive.
Sorry, but what is known as the Lorentz ether is simply equivalent to SR (and therefore an interpretation of SR) and therefore not ruled out by the Michelson-Morley experiment. And which versions you think about?
4. Spacetime+foliation interpretation. This interpretation posits that in addition to spacetime, there is some timelike vector field nμ(x)nμ(x) that defines a preferred foliation of spacetime, such that nμ(x)nμ(x) is orthogonal to the spacelike hypersurfaces of the foliation. This preferred foliation defines a preferred notion of simultaneity.
The Lorentz ether is here only a particular case, where the foliation is defined by a preferred inertial frame.
What different interpretations of QM can tell us about those interpretations of relativity? Which interpretations of relativity seem natural from the perspective of which interpretations of QM?
There is a quite simple general answer: All realistic as well as all causal interpretations require a preferred foliation. Here, "realistic" means that the EPR criterion of reality holds, and "causal" means a notion of causality which includes Reichenbach's common cause principle. This follows from variants of Bell's theorem, which use, beyond Einstein causality, only EPR realism resp. Reichenbach's common cause principle.
martinbn
The past, presence and future exist on an equal footing.
How is that specific to relativity? It seems like a general philosophical position. In fact it seems very non-relativistic in spirit. What is present in relativity? A choice of simultaneity convention? Which one?
What about block universe? Is that a consequence or an interpretation?
An interpretation. In interpretations with a preferred frame, that preferred frame also defines the presence objectively, and the relativity of simultaneity is reduced to an impossibility to identify the preferred frame by local observations.
How is that specific to relativity? It seems like a general philosophical position. In fact it seems very non-relativistic in spirit. What is present in relativity? A choice of simultaneity convention? Which one?
A philosophical position that assumes a block universe exists too, it is named fatalism. In fatalism, the future is predefined, thus, already existing in the same way as the present. In what I would simply name common sense, the future, as well as the past, have a different status, only what is present exists.
This difference is an objective one, a property of the world, not of observations of the world. Once the preferred frame cannot be identified by observation, it cannot be a choice by an observer. The observer can only guess which is the correct preferred frame (and the CMBR frame gives a quite plausible guess).
The preferred frame interpretations are, indeed, very non-relativistic in spirit. Relativistic symmetry holds only for some observable effects, it is not a fundamental symmetry, and in particular not a symmetry of space and time. This is what makes them much better compatible with similarly non-relativistic interpretations of quantum theory.
A class of interpretations of QT which depends on a preferred frame for extensions into the relativistic domain can be easily identified: If we look at the Schrödinger equation in the configuration space, it gives a continuity equation for the density ##\rho(q)##:
$$\partial_t \rho(q,t) + \partial_i ( \rho(q,t)v^i(q,t)) = 0.$$
All one needs is to give the corresponding ##\rho(q,t)v^i(q,t)## a physical interpretation, as a probability flow.
What about block universe? Is that a consequence or an interpretation?
A consequence. The block universe has always seemed, to me, a consequence of pre-Minkowski classical physics, which describes time as a fourth dimension. Nothing in SR (or GR) changes this.
I don't get your point 4, though. There is no preferred foliation for time after Einstein and Minkowski, and no preferred foliation problem, just as there is no preferred basis problem in quantum mechanics.
How is that specific to relativity? It seems like a general philosophical position. In fact it seems very non-relativistic in spirit. What is present in relativity? A choice of simultaneity convention? Which one?
Just as there is no preferred position, so there is no preferred time. Seems entirely relativistic to me. | 2020-09-22 15:23:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6271838545799255, "perplexity": 899.8359796284351}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400206133.46/warc/CC-MAIN-20200922125920-20200922155920-00676.warc.gz"} |
http://math.stackexchange.com/questions/239967/group-does-not-contain-any-elements-of-order-p2 | # Group does not contain any elements of order $p^2$?
I am trying to show that the group of $3 \times 3$ upper triangular matrices over the field $\mathbb{F}_p$ with diagonal entries 1 does not contain any elements of order $p^2$ when $p \geq 3$.
I've tried to argue by contradiction: suppose it had an element $g$ of order $p^2$, then the subgroup generated by $g$ has index $p$ so it is normal, and from there I would like to find an element $h$ of order $p$ whose cyclic subgroup has trivial intersection with $\langle g \rangle$. Then $G$ is the semi-direct product of $\langle g \rangle$ and $\langle h \rangle$, and I am hoping this will give me a contradiction by telling me that $G$ is abelian or something similar.
Any help is appreciated!
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To show that the group does not have any elements of order $p^2$, it suffices to show that every element has order at most $p$. Equivalently, it suffices to show that $x^p = 1$ for every element $x$. This can be done by direct calculation. Calculate $A^2, A^3, A^4, A^5, \ldots$ for $$A = \left( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} \right)$$
and you should see a pattern. Once you have a formula for $A^k$, the rest should be easy.
Thanks for the help! I did attempt to prove $x^p = 1$ by direct computation earlier, but did not go to high enough powers to see the pattern. – Jonas Nov 18 '12 at 19:24
The matrix $$A = \left( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} \right)$$ has minimum polynomial $(x-1)^3$ so in particular satisfies the polynomial $(x-1)^p = x^p - 1$ as long as $p \geq 3$. For $p=2$ (or fields of characteristic 2), when $a=c \neq 0$, then the matrix has order 4, not 2. For fields of characteristic 0, the group has no non-identity elements of finite order. | 2014-09-21 12:12:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8985929489135742, "perplexity": 50.14932592256361}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657135549.24/warc/CC-MAIN-20140914011215-00027-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://www.fightfinance.com/?q=478,477,151,542,278,353,407,525,295,526,575,577,554,221,466,531,467,473,444,445,443,515,44,37,60,190,500,501,251,252,250,502,532,579,580,1,2,4,5,6,7,8,481,9,10,11,12,13,14,15,16,17,479,517,518,528,264,201,289,148,488,356,36,40,441,348,341,280,299,462,18,20,21,23,26,27,28,30,31,32,33,34,35,38,41,43,51,52,54,55,59,61,63,65,260,261,262,272,288,331,346,616,133,159,620,257,96,64,108,132,134,135,136,137,144,146,147,149,150,152,156,157,158,161,162,165,166,167,168,169,170,171,172,175,179,181,182,183,184,185,186,189,194,196,198,199,200,203,372,373,581,583,290,131,265,222,107,128,129,130,509,510,227,229,254,279,490,508,456,363,482,446,447,451,352,270,465,358,354,364,457,463,330,505,211,249,374,259,160,298,459,138,153,163,193,213,230,233,266,287,328,332,220,485,350,176,225,512,552,576,496,613,454,480,533,534,574,521,530,210,282,362,405,453,455,522,523,527,538,188,208,209,223,224,226,291,296,342,359,360,368,504,511,555,619,658,506,238,349,486,492,377,369,375,78,115,94,559,111,285,293,557,556,563,703,80,112,326,71,79,93,627,672,673,674,410,114,302,418,119,243,569,449,448,568,625,708,709,710,712,714,419,655,656,657,659,660,661,662,663,664,665,666,667,669,617,618,626,629,707,711,713,691,715,717,724,100,242,416,90,86,72,92,98,248,117,695,696,698,699,700,701,702,284,81,82,73,294,558,562,379,67,507,89,413,84,411, | # Fight Finance
#### CoursesTagsRandomAllRecentScores
Total cash flows can be broken into income and capital cash flows. What is the name given to the income cash flow from owning shares?
An asset's total expected return over the next year is given by:
$$r_\text{total} = \dfrac{c_1+p_1-p_0}{p_0}$$
Where $p_0$ is the current price, $c_1$ is the expected income in one year and $p_1$ is the expected price in one year. The total return can be split into the income return and the capital return.
Which of the following is the expected capital return?
A share was bought for $30 (at t=0) and paid its annual dividend of$6 one year later (at t=1).
Just after the dividend was paid, the share price fell to $27 (at t=1). What were the total, capital and income returns given as effective annual rates? The choices are given in the same order: $r_\text{total}$ , $r_\text{capital}$ , $r_\text{dividend}$. For an asset price to double every 10 years, what must be the expected future capital return, given as an effective annual rate? Imagine that the interest rate on your savings account was 1% per year and inflation was 2% per year. After one year, would you be able to buy , exactly the as or than today with the money in this account? A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 3% pa. Inflation is expected to be 2% pa. All rates are given as effective annual rates. What are the property's expected real total, capital and income returns? The answer choices below are given in the same order. A stock has a real expected total return of 7% pa and a real expected capital return of 2% pa. Inflation is expected to be 2% pa. All rates are given as effective annual rates. What is the nominal expected total return, capital return and dividend yield? The answers below are given in the same order. Which of the following statements about cash in the form of notes and coins is NOT correct? Assume that inflation is positive. Notes and coins: When valuing assets using discounted cash flow (net present value) methods, it is important to consider inflation. To properly deal with inflation: (I) Discount nominal cash flows by nominal discount rates. (II) Discount nominal cash flows by real discount rates. (III) Discount real cash flows by nominal discount rates. (IV) Discount real cash flows by real discount rates. Which of the above statements is or are correct? How can a nominal cash flow be precisely converted into a real cash flow? You expect a nominal payment of$100 in 5 years. The real discount rate is 10% pa and the inflation rate is 3% pa. Which of the following statements is NOT correct?
What is the present value of a real payment of $500 in 2 years? The nominal discount rate is 7% pa and the inflation rate is 4% pa. On his 20th birthday, a man makes a resolution. He will put$30 cash under his bed at the end of every month starting from today. His birthday today is the first day of the month. So the first addition to his cash stash will be in one month. He will write in his will that when he dies the cash under the bed should be given to charity.
If the man lives for another 60 years, how much money will be under his bed if he dies just after making his last (720th) addition?
Also, what will be the real value of that cash in today's prices if inflation is expected to 2.5% pa? Assume that the inflation rate is an effective annual rate and is not expected to change.
The answers are given in the same order, the amount of money under his bed in 60 years, and the real value of that money in today's prices.
You're considering making an investment in a particular company. They have preference shares, ordinary shares, senior debt and junior debt.
Which is the safest investment? Which will give the highest returns?
Which business structure or structures have the advantage of limited liability for equity investors?
Who is most in danger of being personally bankrupt? Assume that all of their businesses' assets are highly liquid and can therefore be sold immediately.
Which of the following statements about book and market equity is NOT correct?
The below screenshot of Commonwealth Bank of Australia's (CBA) details were taken from the Google Finance website on 7 Nov 2014. Some information has been deliberately blanked out.
What was CBA's market capitalisation of equity?
The investment decision primarily affects which part of a business?
The financing decision primarily affects which part of a business?
Business people make lots of important decisions. Which of the following is the most important long term decision?
The expression 'you have to spend money to make money' relates to which business decision?
The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time.
What is the Net Present Value (NPV) of the project?
Project Cash Flows Time (yrs) Cash flow ($) 0 -100 1 0 2 121 If a project's net present value (NPV) is zero, then its internal rate of return (IRR) will be: The required return of a project is 10%, given as an effective annual rate. What is the payback period of the project in years? Assume that the cash flows shown in the table are received smoothly over the year. So the$121 at time 2 is actually earned smoothly from t=1 to t=2.
Project Cash Flows Time (yrs) Cash flow ($) 0 -100 1 11 2 121 A project has the following cash flows: Project Cash Flows Time (yrs) Cash flow ($) 0 -400 1 0 2 500
What is the payback period of the project in years?
Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $500 at time 2 is actually earned smoothly from t=1 to t=2. The below graph shows a project's net present value (NPV) against its annual discount rate. For what discount rate or range of discount rates would you accept and commence the project? All answer choices are given as approximations from reading off the graph. The below graph shows a project's net present value (NPV) against its annual discount rate. Which of the following statements is NOT correct? You have$100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume an equal amount now (t=0) and in one year (t=1) and have nothing left in the bank at the end (t=1).
How much can you consume at each time?
You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate. You wish to consume an equal amount now (t=0), in one year (t=1) and in two years (t=2), and still have$50,000 in the bank after that (t=2).
How much can you consume at each time?
Your neighbour asks you for a loan of $100 and offers to pay you back$120 in one year.
You don't actually have any money right now, but you can borrow and lend from the bank at a rate of 10% pa. Rates are given as effective annual rates.
Assume that your neighbour will definitely pay you back. Ignore interest tax shields and transaction costs.
The Net Present Value (NPV) of lending to your neighbour is $9.09. Describe what you would do to actually receive a$9.09 cash flow right now with zero net cash flows in the future.
An investor owns an empty block of land that has local government approval to be developed into a petrol station, car wash or car park. The council will only allow a single development so the projects are mutually exclusive.
All of the development projects have the same risk and the required return of each is 10% pa. Each project has an immediate cost and once construction is finished in one year the land and development will be sold. The table below shows the estimated costs payable now, expected sale prices in one year and the internal rates of returns (IRR's).
Mutually Exclusive Projects Project Costnow ($) Sale price inone year ($) IRR(% pa) Petrol station 9,000,000 11,000,000 22.22 Car wash 800,000 1,100,000 37.50 Car park 70,000 110,000 57.14
Which project should the investor accept?
An investor owns a whole level of an old office building which is currently worth $1 million. There are three mutually exclusive projects that can be started by the investor. The office building level can be: • Rented out to a tenant for one year at$0.1m paid immediately, and then sold for $0.99m in one year. • Refurbished into more modern commercial office rooms at a cost of$1m now, and then sold for $2.4m when the refurbishment is finished in one year. • Converted into residential apartments at a cost of$2m now, and then sold for $3.4m when the conversion is finished in one year. All of the development projects have the same risk so the required return of each is 10% pa. The table below shows the estimated cash flows and internal rates of returns (IRR's). Mutually Exclusive Projects Project Cash flownow ($) Cash flow inone year ($) IRR(% pa) Rent then sell as is -900,000 990,000 10 Refurbishment into modern offices -2,000,000 2,400,000 20 Conversion into residential apartments -3,000,000 3,400,000 13.33 Which project should the investor accept? How many years will it take for an asset's price to double if the price grows by 10% pa? How many years will it take for an asset's price to quadruple (be four times as big, say from$1 to $4) if the price grows by 15% pa? Jan asks you for a loan. He wants$100 now and offers to pay you back $120 in 1 year. You can borrow and lend from the bank at an interest rate of 10% pa, given as an effective annual rate. Ignore credit risk. Remember: $$V_0 = \frac{V_t}{(1+r_\text{eff})^t}$$ Will you or Jan's deal? Katya offers to pay you$10 at the end of every year for the next 5 years (t=1,2,3,4,5) if you pay her $50 now (t=0). You can borrow and lend from the bank at an interest rate of 10% pa, given as an effective annual rate. Ignore credit risk. Will you or Katya's deal? For a price of$13, Carla will sell you a share which will pay a dividend of $1 in one year and every year after that forever. The required return of the stock is 10% pa. Would you like to Carla's share or politely ? For a price of$6, Carlos will sell you a share which will pay a dividend of $1 in one year and every year after that forever. The required return of the stock is 10% pa. Would you like to his share or politely ? For a price of$102, Andrea will sell you a share which just paid a dividend of $10 yesterday, and is expected to pay dividends every year forever, growing at a rate of 5% pa. So the next dividend will be $10(1+0.05)^1=10.50$ in one year from now, and the year after it will be $10(1+0.05)^2=11.025$ and so on. The required return of the stock is 15% pa. Would you like to the share or politely ? For a price of$1040, Camille will sell you a share which just paid a dividend of $100, and is expected to pay dividends every year forever, growing at a rate of 5% pa. So the next dividend will be $100(1+0.05)^1=105.00$, and the year after it will be $100(1+0.05)^2=110.25$ and so on. The required return of the stock is 15% pa. Would you like to the share or politely ? For a price of$10.20 each, Renee will sell you 100 shares. Each share is expected to pay dividends in perpetuity, growing at a rate of 5% pa. The next dividend is one year away (t=1) and is expected to be $1 per share. The required return of the stock is 15% pa. Would you like to the shares or politely ? This annuity formula $\dfrac{C_1}{r}\left(1-\dfrac{1}{(1+r)^3} \right)$ is equivalent to which of the following formulas? Note the 3. In the below formulas, $C_t$ is a cash flow at time t. All of the cash flows are equal, but paid at different times. For a price of$129, Joanne will sell you a share which is expected to pay a $30 dividend in one year, and a$10 dividend every year after that forever. So the stock's dividends will be $30 at t=1,$10 at t=2, $10 at t=3, and$10 forever onwards.
The required return of the stock is 10% pa.
Would you like to the share or politely ?
For a price of $95, Sherylanne will sell you a share which is expected to pay its first dividend of$10 in 7 years (t=7), and will continue to pay the same $10 dividend every year after that forever. The required return of the stock is 10% pa. Would you like to the share or politely ? For a price of$100, Vera will sell you a 2 year bond paying semi-annual coupons of 10% pa. The face value of the bond is $100. Other bonds with similar risk, maturity and coupon characteristics trade at a yield of 8% pa. Would you like to her bond or politely ? For a price of$100, Carol will sell you a 5 year bond paying semi-annual coupons of 16% pa. The face value of the bond is $100. Other bonds with similar risk, maturity and coupon characteristics trade at a yield of 12% pa. Would you like to her bond or politely ? For a price of$100, Rad will sell you a 5 year bond paying semi-annual coupons of 16% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 6% pa. Would you like to the bond or politely ? For a price of$100, Andrea will sell you a 2 year bond paying annual coupons of 10% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 6% pa. Would you like to the bond or politely ? For a price of$95, Nicole will sell you a 10 year bond paying semi-annual coupons of 8% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 8% pa. Would you like to the bond or politely ? A credit card offers an interest rate of 18% pa, compounding monthly. Find the effective monthly rate, effective annual rate and the effective daily rate. Assume that there are 365 days in a year. All answers are given in the same order: $$r_\text{eff monthly} , r_\text{eff yearly} , r_\text{eff daily}$$ A three year bond has a face value of$100, a yield of 10% and a fixed coupon rate of 5%, paid semi-annually. What is its price?
Discounted cash flow (DCF) valuation prices assets by finding the present value of the asset's future cash flows. The single cash flow, annuity, and perpetuity equations are very useful for this.
Which of the following equations is the 'perpetuity with growth' equation?
A stock is expected to pay its next dividend of $1 in one year. Future annual dividends are expected to grow by 2% pa. So the first dividend of$1 will be in one year, the year after that $1.02 (=1*(1+0.02)^1), and a year later$1.0404 (=1*(1+0.02)^2) and so on forever.
Its required total return is 10% pa. The total required return and growth rate of dividends are given as effective annual rates.
Calculate the current stock price.
A stock just paid a dividend of $1. Future annual dividends are expected to grow by 2% pa. The next dividend of$1.02 (=1*(1+0.02)^1) will be in one year, and the year after that the dividend will be $1.0404 (=1*(1+0.02)^2), and so on forever. Its required total return is 10% pa. The total required return and growth rate of dividends are given as effective annual rates. Calculate the current stock price. The perpetuity with growth formula, also known as the dividend discount model (DDM) or Gordon growth model, is appropriate for valuing a company's shares. $P_0$ is the current share price, $C_1$ is next year's expected dividend, $r$ is the total required return and $g$ is the expected growth rate of the dividend. $$P_0=\dfrac{C_1}{r-g}$$ The below graph shows the expected future price path of the company's shares. Which of the following statements about the graph is NOT correct? The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation. $$P_0=\frac{d_1}{r-g}$$ A stock pays dividends annually. It just paid a dividend, but the next dividend ($d_1$) will be paid in one year. According to the DDM, what is the correct formula for the expected price of the stock in 2.5 years? The following is the Dividend Discount Model (DDM) used to price stocks: $$P_0=\dfrac{C_1}{r-g}$$ If the assumptions of the DDM hold, which one of the following statements is NOT correct? The long term expected: In the dividend discount model: $$P_0 = \dfrac{C_1}{r-g}$$ The return $r$ is supposed to be the: The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation. $$p_0 = \frac{d_1}{r - g}$$ Which expression is NOT equal to the expected dividend yield? Two companies BigDiv and ZeroDiv are exactly the same except for their dividend payouts. BigDiv pays large dividends and ZeroDiv doesn't pay any dividends. Currently the two firms have the same earnings, assets, number of shares, share price, expected total return and risk. Assume a perfect world with no taxes, no transaction costs, no asymmetric information and that all assets including business projects are fairly priced and therefore zero-NPV. All things remaining equal, which of the following statements is NOT correct? Your friend overheard that you need some cash and asks if you would like to borrow some money. She can lend you$5,000 now (t=0), and in return she wants you to pay her back $1,000 in two years (t=2) and every year after that for the next 5 years, so there will be 6 payments of$1,000 from t=2 to t=7 inclusive.
What is the net present value (NPV) of borrowing from your friend?
Assume that banks loan funds at interest rates of 10% pa, given as an effective annual rate.
A stock pays annual dividends which are expected to continue forever. It just paid a dividend of $10. The growth rate in the dividend is 2% pa. You estimate that the stock's required return is 10% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price? A stock is expected to pay the following dividends: Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0.00 1.00 1.05 1.10 1.15 ...
After year 4, the annual dividend will grow in perpetuity at 5% pa, so;
• the dividend at t=5 will be $1.15(1+0.05), • the dividend at t=6 will be$1.15(1+0.05)^2, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in three and a half years (t = 3.5)?
A fairly valued share's current price is $4 and it has a total required return of 30%. Dividends are paid annually and next year's dividend is expected to be$1. After that, dividends are expected to grow by 5% pa in perpetuity. All rates are effective annual returns.
What is the expected dividend income paid at the end of the second year (t=2) and what is the expected capital gain from just after the first dividend (t=1) to just after the second dividend (t=2)? The answers are given in the same order, the dividend and then the capital gain.
Estimate the US bank JP Morgan's share price using a price earnings (PE) multiples approach with the following assumptions and figures only:
• The major US banks JP Morgan Chase (JPM), Citi Group (C) and Wells Fargo (WFC) are comparable companies;
• JP Morgan Chase's historical earnings per share (EPS) is $4.37; • Citi Group's share price is$50.05 and historical EPS is $4.26; • Wells Fargo's share price is$48.98 and historical EPS is $3.89. Note: Figures sourced from Google Finance on 24 March 2014. Estimate Microsoft's (MSFT) share price using a price earnings (PE) multiples approach with the following assumptions and figures only: • Apple, Google and Microsoft are comparable companies, • Apple's (AAPL) share price is$526.24 and historical EPS is $40.32. • Google's (GOOG) share price is$1,215.65 and historical EPS is $36.23. • Micrsoft's (MSFT) historical earnings per share (EPS) is$2.71.
Source: Google Finance 28 Feb 2014.
You own a nice suit which you wear once per week on nights out. You bought it one year ago for $600. In your experience, suits used once per week last for 6 years. So you expect yours to last for another 5 years. Your younger brother said that retro is back in style so he wants to wants to borrow your suit once a week when he goes out. With the increased use, your suit will only last for another 4 years rather than 5. What is the present value of the cost of letting your brother use your current suit for the next 4 years? Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new suit when your current one wears out and your brother will not use the new one; your brother will only use your current suit so he will only use it for the next four years; and the price of a new suit never changes. Carlos and Edwin are brothers and they both love Holden Commodore cars. Carlos likes to buy the latest Holden Commodore car for$40,000 every 4 years as soon as the new model is released. As soon as he buys the new car, he sells the old one on the second hand car market for $20,000. Carlos never has to bother with paying for repairs since his cars are brand new. Edwin also likes Commodores, but prefers to buy 4-year old cars for$20,000 and keep them for 11 years until the end of their life (new ones last for 15 years in total but the 4-year old ones only last for another 11 years). Then he sells the old car for $2,000 and buys another 4-year old second hand car, and so on. Every time Edwin buys a second hand 4 year old car he immediately has to spend$1,000 on repairs, and then $1,000 every year after that for the next 10 years. So there are 11 payments in total from when the second hand car is bought at t=0 to the last payment at t=10. One year later (t=11) the old car is at the end of its total 15 year life and can be scrapped for$2,000.
Assuming that Carlos and Edwin maintain their love of Commodores and keep up their habits of buying new ones and second hand ones respectively, how much larger is Carlos' equivalent annual cost of car ownership compared with Edwin's?
The real discount rate is 10% pa. All cash flows are real and are expected to remain constant. Inflation is forecast to be 3% pa. All rates are effective annual. Ignore capital gains tax and tax savings from depreciation since cars are tax-exempt for individuals.
You own some nice shoes which you use once per week on date nights. You bought them 2 years ago for $500. In your experience, shoes used once per week last for 6 years. So you expect yours to last for another 4 years. Your younger sister said that she wants to borrow your shoes once per week. With the increased use, your shoes will only last for another 2 years rather than 4. What is the present value of the cost of letting your sister use your current shoes for the next 2 years? Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new pair of shoes when your current pair wears out and your sister will not use the new ones; your sister will only use your current shoes so she will only use it for the next 2 years; and the price of new shoes never changes. The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation. $$p_{0} = \frac{c_1}{r_{\text{eff}} - g_{\text{eff}}}$$ What is the discount rate '$r_\text{eff}$' in this equation? Your friend wants to borrow$1,000 and offers to pay you back $100 in 6 months, with more$100 payments at the end of every month for another 11 months. So there will be twelve $100 payments in total. She says that 12 payments of$100 equals $1,200 so she's being generous. If interest rates are 12% pa, given as an APR compounding monthly, what is the Net Present Value (NPV) of your friend's deal? A fixed coupon bond was bought for$90 and paid its annual coupon of $3 one year later (at t=1 year). Just after the coupon was paid, the bond price was$92 (at t=1 year). What was the total return, capital return and income return? Calculate your answers as effective annual rates.
The choices are given in the same order: $r_\text{total},r_\text{capital},r_\text{income}$.
Bonds X and Y are issued by the same US company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency. The only difference is that bond X and Y's coupon rates are 8 and 12% pa respectively. Which of the following statements is true? A European bond paying annual coupons of 6% offers a yield of 10% pa. Convert the yield into an effective monthly rate, an effective annual rate and an effective daily rate. Assume that there are 365 days in a year. All answers are given in the same order: $$r_\text{eff, monthly} , r_\text{eff, yearly} , r_\text{eff, daily}$$ A 180-day Bank Accepted Bill has a face value of$1,000,000. The interest rate is 8% pa and there are 365 days in the year. What is its price now?
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
$$P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}}$$
What would you call the expression $C_1/P_0$?
A share was bought for $20 (at t=0) and paid its annual dividend of$3 one year later (at t=1). Just after the dividend was paid, the share price was $16 (at t=1). What was the total return, capital return and income return? Calculate your answers as effective annual rates. The choices are given in the same order: $r_\text{total},r_\text{capital},r_\text{income}$. What is the NPV of the following series of cash flows when the discount rate is 5% given as an effective annual rate? The first payment of$10 is in 4 years, followed by payments every 6 months forever after that which shrink by 2% every 6 months. That is, the growth rate every 6 months is actually negative 2%, given as an effective 6 month rate. So the payment at $t=4.5$ years will be $10(1-0.02)^1=9.80$, and so on.
You really want to go on a back packing trip to Europe when you finish university. Currently you have $1,500 in the bank. Bank interest rates are 8% pa, given as an APR compounding per month. If the holiday will cost$2,000, how long will it take for your bank account to reach that amount?
Bonds A and B are issued by the same company. They have the same face value, maturity, seniority and coupon payment frequency. The only difference is that bond A has a 5% coupon rate, while bond B has a 10% coupon rate. The yield curve is flat, which means that yields are expected to stay the same.
Which bond would have the higher current price?
A wholesale glass importer offers credit to its customers. Customers are given 30 days to pay for their goods, but if they pay within 5 days they will get a 1% discount.
What is the effective interest rate implicit in the discount being offered? Assume 365 days in a year and that all customers pay on either the 5th day or the 30th day. All rates given below are effective annual rates.
A European company just issued two bonds, a
• 1 year zero coupon bond at a yield of 8% pa, and a
• 2 year zero coupon bond at a yield of 10% pa.
What is the company's forward rate over the second year (from t=1 to t=2)? Give your answer as an effective annual rate, which is how the above bond yields are quoted.
A two year Government bond has a face value of $100, a yield of 0.5% and a fixed coupon rate of 0.5%, paid semi-annually. What is its price? The following is the Dividend Discount Model (DDM) used to price stocks: $$P_0 = \frac{d_1}{r-g}$$ Assume that the assumptions of the DDM hold and that the time period is measured in years. Which of the following is equal to the expected dividend in 3 years, $d_3$? A project to build a toll road will take 3 years to complete, costing three payments of$50 million, paid at the start of each year (at times 0, 1, and 2).
After completion, the toll road will yield a constant $10 million at the end of each year forever with no costs. So the first payment will be at t=4. The required return of the project is 10% pa given as an effective nominal rate. All cash flows are nominal. What is the payback period? A stock pays semi-annual dividends. It just paid a dividend of$10. The growth rate in the dividend is 1% every 6 months, given as an effective 6 month rate. You estimate that the stock's required return is 21% pa, as an effective annual rate.
Using the dividend discount model, what will be the share price?
A three year project's NPV is negative. The cash flows of the project include a negative cash flow at the very start and positive cash flows over its short life. The required return of the project is 10% pa. Select the most correct statement.
A stock is expected to pay the following dividends:
Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0.00 1.15 1.10 1.05 1.00 ... After year 4, the annual dividend will grow in perpetuity at -5% pa. Note that this is a negative growth rate, so the dividend will actually shrink. So, • the dividend at t=5 will be $1(1-0.05) = 0.95$, • the dividend at t=6 will be $1(1-0.05)^2 = 0.9025$, and so on. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock? A stock is expected to pay the following dividends: Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0.00 1.15 1.10 1.05 1.00 ...
After year 4, the annual dividend will grow in perpetuity at -5% pa. Note that this is a negative growth rate, so the dividend will actually shrink. So,
• the dividend at t=5 will be $1(1-0.05) = 0.95$,
• the dividend at t=6 will be $1(1-0.05)^2 = 0.9025$, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in four and a half years (t = 4.5)?
The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time.
What is the Net Present Value (NPV) of the project?
Project Cash Flows Time (yrs) Cash flow ($) 0 -100 1 11 2 121 In Australia, domestic university students are allowed to buy concession tickets for the bus, train and ferry which sell at a discount of 50% to full-price tickets. The Australian Government do not allow international university students to buy concession tickets, they have to pay the full price. Some international students see this as unfair and they are willing to pay for fake university identification cards which have the concession sticker. What is the most that an international student would be willing to pay for a fake identification card? Assume that international students: • consider buying their fake card on the morning of the first day of university from their neighbour, just before they leave to take the train into university. • buy their weekly train tickets on the morning of the first day of each week. • ride the train to university and back home again every day seven days per week until summer holidays 40 weeks from now. The concession card only lasts for those 40 weeks. Assume that there are 52 weeks in the year for the purpose of interest rate conversion. • a single full-priced one-way train ride costs$5.
• have a discount rate of 11% pa, given as an effective annual rate.
Approach this question from a purely financial view point, ignoring the illegality, embarrassment and the morality of committing fraud.
The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.
Considering this, which of the following statements is NOT correct?
Question 65 annuity with growth, needs refinement
Which of the below formulas gives the present value of an annuity with growth?
Hint: The equation of a perpetuity without growth is: $$V_\text{0, perp without growth} = \frac{C_\text{1}}{r}$$
The formula for the present value of an annuity without growth is derived from the formula for a perpetuity without growth.
The idea is than an annuity with T payments from t=1 to T inclusive is equivalent to a perpetuity starting at t=1 with fixed positive cash flows, plus a perpetuity starting T periods later (t=T+1) with fixed negative cash flows. The positive and negative cash flows after time period T cancel each other out, leaving the positive cash flows between t=1 to T, which is the annuity.
\begin{aligned} V_\text{0, annuity} &= V_\text{0, perp without growth from t=1} - V_\text{0, perp without growth from t=T+1} \\ &= \dfrac{C_\text{1}}{r} - \dfrac{ \left( \dfrac{C_\text{T+1}}{r} \right) }{(1+r)^T} \\ &= \dfrac{C_\text{1}}{r} - \dfrac{ \left( \dfrac{C_\text{1}}{r} \right) }{(1+r)^T} \\ &= \dfrac{C_\text{1}}{r}\left(1 - \dfrac{1}{(1+r)^T}\right) \\ \end{aligned}
The equation of a perpetuity with growth is:
$$V_\text{0, perp with growth} = \dfrac{C_\text{1}}{r-g}$$
A share just paid its semi-annual dividend of $5. The dividend is expected to grow at 1% every 6 months forever. This 1% growth rate is an effective 6 month rate. Therefore the next dividend will be$5.05 in six months. The required return of the stock 8% pa, given as an effective annual rate.
What is the price of the share now?
A share was bought for $4 and paid an dividend of$0.50 one year later (at t=1 year).
Just after the dividend was paid, the share price fell to $3.50 (at t=1 year). What were the total return, capital return and income returns given as effective annual rates? The answer choices are given in the same order: $r_\text{total}$, $r_\text{capital}$, $r_\text{income}$ A 90-day$1 million Bank Accepted Bill (BAB) was bought for $990,000 and sold 30 days later for$996,000 (at t=30 days).
What was the total return, capital return and income return over the 30 days it was held?
Despite the fact that money market instruments such as bills are normally quoted with simple interest rates, please calculate your answers as compound interest rates, specifically, as effective 30-day rates, which is how the below answer choices are listed.
$r_\text{total}$, $r_\text{capital}$, $r_\text{income}$
Suppose you had $100 in a savings account and the interest rate was 2% per year. After 5 years, how much do you think you would have in the account if you left the money to grow? than$102, $102 or than$102?
There are many ways to write the ordinary annuity formula.
Which of the following is NOT equal to the ordinary annuity formula?
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
$$p_0= \frac{c_1}{r-g}$$
Which expression is equal to the expected dividend return?
Your poor friend asks to borrow some money from you. He would like $1,000 now (t=0) and every year for the next 5 years, so there will be 6 payments of$1,000 from t=0 to t=5 inclusive. In return he will pay you $10,000 in seven years from now (t=7). What is the net present value (NPV) of lending to your friend? Assume that your friend will definitely pay you back so the loan is risk-free, and that the yield on risk-free government debt is 10% pa, given as an effective annual rate. "Buy low, sell high" is a phrase commonly heard in financial markets. It states that traders should try to buy assets at low prices and sell at high prices. Traders in the fixed-coupon bond markets often quote promised bond yields rather than prices. Fixed-coupon bond traders should try to: A bond maturing in 10 years has a coupon rate of 4% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value of the bond is$100. What is its price?
A three year bond has a fixed coupon rate of 12% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value is $100. What is its price? Let the 'income return' of a bond be the coupon at the end of the period divided by the market price now at the start of the period $(C_1/P_0)$. The expected income return of a premium fixed coupon bond is: A 10 year bond has a face value of$100, a yield of 6% pa and a fixed coupon rate of 8% pa, paid semi-annually. What is its price?
An Australian company just issued two bonds paying semi-annual coupons:
• 1 year zero coupon bond at a yield of 8% pa, and a
• 2 year zero coupon bond at a yield of 10% pa.
What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
In Germany, nominal yields on semi-annual coupon paying Government Bonds with 2 years until maturity are currently 0.04% pa.
The inflation rate is currently 1.4% pa, given as an APR compounding per quarter. The inflation rate is not expected to change over the next 2 years.
What is the real yield on these bonds, given as an APR compounding every 6 months?
An Australian company just issued two bonds:
• A 1 year zero coupon bond at a yield of 10% pa, and
• A 2 year zero coupon bond at a yield of 8% pa.
What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
A 90-day Bank Accepted Bill (BAB) has a face value of $1,000,000. The simple interest rate is 10% pa and there are 365 days in the year. What is its price now? You want to buy an apartment worth$400,000. You have saved a deposit of $80,000. The bank has agreed to lend you the$320,000 as a fully amortising mortgage loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
Your credit card shows a $600 debt liability. The interest rate is 24% pa, payable monthly. You can't pay any of the debt off, except in 6 months when it's your birthday and you'll receive$50 which you'll use to pay off the credit card. If that is your only repayment, how much will the credit card debt liability be one year from now?
A stock was bought for $8 and paid a dividend of$0.50 one year later (at t=1 year). Just after the dividend was paid, the stock price was $7 (at t=1 year). What were the total, capital and dividend returns given as effective annual rates? The choices are given in the same order: $r_\text{total}$, $r_\text{capital}$, $r_\text{dividend}$. The following cash flows are expected: • 10 yearly payments of$60, with the first payment in 3 years from now (first payment at t=3 and last at t=12).
• 1 payment of $400 in 5 years and 6 months (t=5.5) from now. What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate? A text book publisher is thinking of asking some teachers to write a new textbook at a cost of$100,000, payable now. The book would be written, printed and ready to sell to students in 2 years. It will be ready just before semester begins.
A cash flow of $100 would be made from each book sold, after all costs such as printing and delivery. There are 600 students per semester. Assume that every student buys a new text book. Remember that there are 2 semesters per year and students buy text books at the beginning of the semester. Assume that text book publishers will sell the books at the same price forever and that the number of students is constant. If the discount rate is 8% pa, given as an effective annual rate, what is the NPV of the project? A three year corporate bond yields 12% pa with a coupon rate of 10% pa, paid semi-annually. Find the effective six month yield, effective annual yield and the effective daily yield. Assume that each month has 30 days and that there are 360 days in a year. All answers are given in the same order: $r_\text{eff semi-annual}$, $r_\text{eff yearly}$, $r_\text{eff daily}$. A 30-day Bank Accepted Bill has a face value of$1,000,000. The interest rate is 8% pa and there are 365 days in the year. What is its price now?
You want to buy an apartment priced at $500,000. You have saved a deposit of$50,000. The bank has agreed to lend you the $450,000 as a fully amortising loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments? A share just paid its semi-annual dividend of$10. The dividend is expected to grow at 2% every 6 months forever. This 2% growth rate is an effective 6 month rate. Therefore the next dividend will be $10.20 in six months. The required return of the stock is 10% pa, given as an effective annual rate. What is the price of the share now? The following cash flows are expected: • 10 yearly payments of$80, with the first payment in 3 years from now (first payment at t=3).
• 1 payment of $600 in 5 years and 6 months (t=5.5) from now. What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate? A 2 year government bond yields 5% pa with a coupon rate of 6% pa, paid semi-annually. Find the effective six month rate, effective annual rate and the effective daily rate. Assume that each month has 30 days and that there are 360 days in a year. All answers are given in the same order: $r_\text{eff semi-annual}$, $r_\text{eff yrly}$, $r_\text{eff daily}$. A 90-day Bank Accepted Bill has a face value of$1,000,000. The interest rate is 6% pa and there are 365 days in the year. What is its price?
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
$$p_0=\frac{d_1}{r_\text{eff}-g_\text{eff}}$$
Which expression is NOT equal to the expected capital return?
A share just paid its semi-annual dividend of $10. The dividend is expected to grow at 2% every 6 months forever. This 2% growth rate is an effective 6 month rate. Therefore the next dividend will be$10.20 in six months. The required return of the stock 10% pa, given as an effective annual rate.
What is the price of the share now?
A share was bought for $10 (at t=0) and paid its annual dividend of$0.50 one year later (at t=1). Just after the dividend was paid, the share price was $11 (at t=1). What was the total return, capital return and income return? Calculate your answers as effective annual rates. The choices are given in the same order: $r_\text{total}$, $r_\text{capital}$, $r_\text{dividend}$. For certain shares, the forward-looking Price-Earnings Ratio ($P_0/EPS_1$) is equal to the inverse of the share's total expected return ($1/r_\text{total}$). For what shares is this true? Use the general accounting definition of 'payout ratio' which is dividends per share (DPS) divided by earnings per share (EPS) and assume that all cash flows, earnings and rates are real rather than nominal. A company's forward-looking PE ratio will be the inverse of its total expected return on equity when it has a: A stock pays annual dividends. It just paid a dividend of$3. The growth rate in the dividend is 4% pa. You estimate that the stock's required return is 10% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price?
A project's net present value (NPV) is negative. Select the most correct statement.
A four year bond has a face value of $100, a yield of 6% and a fixed coupon rate of 12%, paid semi-annually. What is its price? A stock is expected to pay the following dividends: Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 8 8 8 20 8 ...
After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What is the current price of the stock?
A stock is expected to pay the following dividends:
Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 8 8 8 20 8 ... After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What will be the price of the stock in 5 years (t = 5), just after the dividend at that time has been paid? The following is the Dividend Discount Model used to price stocks: $$p_0=\frac{d_1}{r-g}$$ Which of the following statements about the Dividend Discount Model is NOT correct? You just signed up for a 30 year fully amortising mortgage loan with monthly payments of$2,000 per month. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 5 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
A project has the following cash flows. Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $250 at time 2 is actually earned smoothly from t=1 to t=2: Project Cash Flows Time (yrs) Cash flow ($) 0 -400 1 200 2 250
What is the payback period of the project in years?
A firm wishes to raise $20 million now. They will issue 8% pa semi-annual coupon bonds that will mature in 5 years and have a face value of$100 each. Bond yields are 6% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue?
A stock pays annual dividends. It just paid a dividend of $5. The growth rate in the dividend is 1% pa. You estimate that the stock's required return is 8% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price? A project's NPV is positive. Select the most correct statement: A five year bond has a face value of$100, a yield of 12% and a fixed coupon rate of 6%, paid semi-annually.
What is the bond's price?
A stock is expected to pay the following dividends:
Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 2 2 2 10 3 ... After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock? A stock is expected to pay the following dividends: Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 2 2 2 10 3 ...
After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in 5 years (t = 5), just after the dividend at that time has been paid?
The following is the Dividend Discount Model used to price stocks:
$$p_0=\frac{d_1}{r-g}$$
All rates are effective annual rates and the cash flows ($d_1$) are received every year. Note that the r and g terms in the above DDM could also be labelled as below: $$r = r_{\text{total, 0}\rightarrow\text{1yr, eff 1yr}}$$ $$g = r_{\text{capital, 0}\rightarrow\text{1yr, eff 1yr}}$$ Which of the following statements is NOT correct?
A project has the following cash flows:
Project Cash Flows Time (yrs) Cash flow ($) 0 -400 1 0 2 500 The required return on the project is 10%, given as an effective annual rate. What is the Internal Rate of Return (IRR) of this project? The following choices are effective annual rates. Assume that the cash flows shown in the table are paid all at once at the given point in time. A firm wishes to raise$8 million now. They will issue 7% pa semi-annual coupon bonds that will mature in 10 years and have a face value of $100 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat. How many bonds should the firm issue? A share pays annual dividends. It just paid a dividend of$2. The growth rate in the dividend is 3% pa. You estimate that the stock's required return is 8% pa. Both the discount rate and growth rate are given as effective annual rates.
Using the dividend discount model, what is the share price?
A stock is expected to pay the following dividends:
Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0 6 12 18 20 ... After year 4, the dividend will grow in perpetuity at 5% pa. The required return of the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock? A stock is expected to pay the following dividends: Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0 6 12 18 20 ...
After year 4, the dividend will grow in perpetuity at 5% pa. The required return of the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in 7 years (t = 7), just after the dividend at that time has been paid?
A stock is expected to pay the following dividends:
Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 0 6 12 18 20 ... After year 4, the dividend will grow in perpetuity at 5% pa. The required return of the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. If all of the dividends since time period zero were deposited into a bank account yielding 8% pa as an effective annual rate, how much money will be in the bank account in 2.5 years (in other words, at t=2.5)? You just signed up for a 30 year fully amortising mortgage loan with monthly payments of$1,500 per month. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
Which of the following statements is NOT correct? Borrowers:
Which of the following statements is NOT correct? Lenders:
A home loan company advertises an interest rate of 6% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given to four decimal places.
A semi-annual coupon bond has a yield of 3% pa. Which of the following statements about the yield is NOT correct? All rates are given to four decimal places.
Which of the below statements about effective rates and annualised percentage rates (APR's) is NOT correct?
Calculate the effective annual rates of the following three APR's:
• A credit card offering an interest rate of 18% pa, compounding monthly.
• A bond offering a yield of 6% pa, compounding semi-annually.
• An annual dividend-paying stock offering a return of 10% pa compounding annually.
All answers are given in the same order:
$r_\text{credit card, eff yrly}$, $r_\text{bond, eff yrly}$, $r_\text{stock, eff yrly}$
On his 20th birthday, a man makes a resolution. He will deposit $30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity. The bank account pays interest at 6% pa compounding monthly, which is not expected to change. If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment? You just agreed to a 30 year fully amortising mortgage loan with monthly payments of$2,500. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change. The below choices are given in the same order.
You want to buy an apartment worth $300,000. You have saved a deposit of$60,000.
The bank has agreed to lend you $240,000 as an interest only mortgage loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments? An 'interest payment' is the same thing as a 'coupon payment'. or ? An 'interest rate' is the same thing as a 'coupon rate'. or ? An 'interest rate' is the same thing as a 'yield'. or ? Calculate the price of a newly issued ten year bond with a face value of$100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid annually. So there's only one coupon per year, paid in arrears every year.
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid semi-annually. So there are two coupons per year, paid in arrears every six months. Which one of the following bonds is trading at a premium? An investor bought two fixed-coupon bonds issued by the same company, a zero-coupon bond and a 7% pa semi-annual coupon bond. Both bonds have a face value of$1,000, mature in 10 years, and had a yield at the time of purchase of 8% pa.
A few years later, yields fell to 6% pa. Which of the following statements is correct? Note that a capital gain is an increase in price.
Your main expense is fuel for your car which costs $100 per month. You just refueled, so you won't need any more fuel for another month (first payment at t=1 month). You have$2,500 in a bank account which pays interest at a rate of 6% pa, payable monthly. Interest rates are not expected to change.
Assuming that you have no income, in how many months time will you not have enough money to fully refuel your car?
Do you think that the following statement is or ? “Buying a single company stock usually provides a safer return than a stock mutual fund.”
Which of the following is NOT a synonym of 'required return'?
Which of the following equations is NOT equal to the total return of an asset?
Let $p_0$ be the current price, $p_1$ the expected price in one year and $c_1$ the expected income in one year.
In the 'Austin Powers' series of movies, the character Dr. Evil threatens to destroy the world unless the United Nations pays him a ransom (video 1, video 2). Dr. Evil makes the threat on two separate occasions:
• In 1969 he demands a ransom of $1 million (=10^6), and again; • In 1997 he demands a ransom of$100 billion (=10^11).
If Dr. Evil's demands are equivalent in real terms, in other words $1 million will buy the same basket of goods in 1969 as$100 billion would in 1997, what was the implied inflation rate over the 28 years from 1969 to 1997?
The answer choices below are given as effective annual rates:
A residential investment property has an expected nominal total return of 8% pa and nominal capital return of 3% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
What are the property's expected real total, capital and income returns? The answer choices below are given in the same order.
The below screenshot of Microsoft's (MSFT) details were taken from the Google Finance website on 28 Nov 2014. Some information has been deliberately blanked out.
What was MSFT's market capitalisation of equity?
The working capital decision primarily affects which part of a business?
Payout policy is most closely related to which part of a business?
The first payment of a constant perpetual annual cash flow is received at time 5. Let this cash flow be $C_5$ and the required return be $r$.
So there will be equal annual cash flows at time 5, 6, 7 and so on forever, and all of the cash flows will be equal so $C_5 = C_6 = C_7 = ...$
When the perpetuity formula is used to value this stream of cash flows, it will give a value (V) at time:
Two years ago Fred bought a house for $300,000. Now it's worth$500,000, based on recent similar sales in the area.
Fred's residential property has an expected total return of 8% pa.
He rents his house out for $2,000 per month, paid in advance. Every 12 months he plans to increase the rental payments. The present value of 12 months of rental payments is$23,173.86.
The future value of 12 months of rental payments one year ahead is $25,027.77. What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%? You own an apartment which you rent out as an investment property. What is the price of the apartment using discounted cash flow (DCF, same as NPV) valuation? Assume that: • You just signed a contract to rent the apartment out to a tenant for the next 12 months at$2,000 per month, payable in advance (at the start of the month, t=0). The tenant is just about to pay you the first $2,000 payment. • The contract states that monthly rental payments are fixed for 12 months. After the contract ends, you plan to sign another contract but with rental payment increases of 3%. You intend to do this every year. So rental payments will increase at the start of the 13th month (t=12) to be$2,060 (=2,000(1+0.03)), and then they will be constant for the next 12 months.
Rental payments will increase again at the start of the 25th month (t=24) to be $2,121.80 (=2,000(1+0.03)2), and then they will be constant for the next 12 months until the next year, and so on. • The required return of the apartment is 8.732% pa, given as an effective annual rate. • Ignore all taxes, maintenance, real estate agent, council and strata fees, periods of vacancy and other costs. Assume that the apartment will last forever and so will the rental payments. The boss of WorkingForTheManCorp has a wicked (and unethical) idea. He plans to pay his poor workers one week late so that he can get more interest on his cash in the bank. Every week he is supposed to pay his 1,000 employees$1,000 each. So $1 million is paid to employees every week. The boss was just about to pay his employees today, until he thought of this idea so he will actually pay them one week (7 days) later for the work they did last week and every week in the future, forever. Bank interest rates are 10% pa, given as a real effective annual rate. So $r_\text{eff annual, real} = 0.1$ and the real effective weekly rate is therefore $r_\text{eff weekly, real} = (1+0.1)^{1/52}-1 = 0.001834569$ All rates and cash flows are real, the inflation rate is 3% pa and there are 52 weeks per year. The boss will always pay wages one week late. The business will operate forever with constant real wages and the same number of employees. What is the net present value (NPV) of the boss's decision to pay later? Estimate the Chinese bank ICBC's share price using a backward-looking price earnings (PE) multiples approach with the following assumptions and figures only. Note that the renminbi (RMB) is the Chinese currency, also known as the yuan (CNY). • The 4 major Chinese banks ICBC, China Construction Bank (CCB), Bank of China (BOC) and Agricultural Bank of China (ABC) are comparable companies; • ICBC 's historical earnings per share (EPS) is RMB 0.74; • CCB's backward-looking PE ratio is 4.59; • BOC 's backward-looking PE ratio is 4.78; • ABC's backward-looking PE ratio is also 4.78; Note: Figures sourced from Google Finance on 25 March 2014. Share prices are from the Shanghai stock exchange. Which firms tend to have low forward-looking price-earnings (PE) ratios? Only consider firms with positive earnings, disregard firms with negative earnings and therefore negative PE ratios. Which firms tend to have high forward-looking price-earnings (PE) ratios? Which firms tend to have low forward-looking price-earnings (PE) ratios? Only consider firms with positive PE ratios. Private equity firms are known to buy medium sized private companies operating in the same industry, merge them together into a larger company, and then sell it off in a public float (initial public offering, IPO). If medium-sized private companies trade at PE ratios of 5 and larger listed companies trade at PE ratios of 15, what return can be achieved from this strategy? Assume that: • The medium-sized companies can be bought, merged and sold in an IPO instantaneously. • There are no costs of finding, valuing, merging and restructuring the medium sized companies. Also, there is no competition to buy the medium-sized companies from other private equity firms. • The large merged firm's earnings are the sum of the medium firms' earnings. • The only reason for the difference in medium and large firm's PE ratios is due to the illiquidity of the medium firms' shares. • Return is defined as: $r_{0→1} = (p_1-p_0+c_1)/p_0$ , where time zero is just before the merger and time one is just after. Which of the following statements about effective rates and annualised percentage rates (APR's) is NOT correct? A low-quality second-hand car can be bought now for$1,000 and will last for 1 year before it will be scrapped for nothing.
A high-quality second-hand car can be bought now for $4,900 and it will last for 5 years before it will be scrapped for nothing. What is the equivalent annual cost of each car? Assume a discount rate of 10% pa, given as an effective annual rate. The answer choices are given as the equivalent annual cost of the low-quality car and then the high quality car. You're advising your superstar client 40-cent who is weighing up buying a private jet or a luxury yacht. 40-cent is just as happy with either, but he wants to go with the more cost-effective option. These are the cash flows of the two options: • The private jet can be bought for$6m now, which will cost $12,000 per month in fuel, piloting and airport costs, payable at the end of each month. The jet will last for 12 years. • Or the luxury yacht can be bought for$4m now, which will cost $20,000 per month in fuel, crew and berthing costs, payable at the end of each month. The yacht will last for 20 years. What's unusual about 40-cent is that he is so famous that he will actually be able to sell his jet or yacht for the same price as it was bought since the next generation of superstar musicians will buy it from him as a status symbol. Bank interest rates are 10% pa, given as an effective annual rate. You can assume that 40-cent will live for another 60 years and that when the jet or yacht's life is at an end, he will buy a new one with the same details as above. Would you advise 40-cent to buy the or the ? Note that the effective monthly rate is $r_\text{eff monthly}=(1+0.1)^{1/12}-1=0.00797414$ Details of two different types of desserts or edible treats are given below: • High-sugar treats like candy, chocolate and ice cream make a person very happy. High sugar treats are cheap at only$2 per day.
• Low-sugar treats like nuts, cheese and fruit make a person equally happy if these foods are of high quality. Low sugar treats are more expensive at $4 per day. The advantage of low-sugar treats is that a person only needs to pay the dentist$2,000 for fillings and root canal therapy once every 15 years. Whereas with high-sugar treats, that treatment needs to be done every 5 years.
The real discount rate is 10%, given as an effective annual rate. Assume that there are 365 days in every year and that all cash flows are real. The inflation rate is 3% given as an effective annual rate.
Find the equivalent annual cash flow (EAC) of the high-sugar treats and low-sugar treats, including dental costs. The below choices are listed in that order.
Ignore the pain of dental therapy, personal preferences and other factors.
Which of the following statements is NOT equivalent to the yield on debt?
Assume that the debt being referred to is fairly priced, but do not assume that it's priced at par.
You want to buy a house priced at $400,000. You have saved a deposit of$40,000. The bank has agreed to lend you $360,000 as a fully amortising loan with a term of 30 years. The interest rate is 8% pa payable monthly and is not expected to change. What will be your monthly payments? You want to buy an apartment priced at$500,000. You have saved a deposit of $50,000. The bank has agreed to lend you the$450,000 as an interest only loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
A prospective home buyer can afford to pay $2,000 per month in mortgage loan repayments. The central bank recently lowered its policy rate by 0.25%, and residential home lenders cut their mortgage loan rates from 4.74% to 4.49%. How much more can the prospective home buyer borrow now that interest rates are 4.49% rather than 4.74%? Give your answer as a proportional increase over the original amount he could borrow ($V_\text{before}$), so: $$\text{Proportional increase} = \frac{V_\text{after}-V_\text{before}}{V_\text{before}}$$ Assume that: • Interest rates are expected to be constant over the life of the loan. • Loans are interest-only and have a life of 30 years. • Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates compounding per month. In Australia in the 1980's, inflation was around 8% pa, and residential mortgage loan interest rates were around 14%. In 2013, inflation was around 2.5% pa, and residential mortgage loan interest rates were around 4.5%. If a person can afford constant mortgage loan payments of$2,000 per month, how much more can they borrow when interest rates are 4.5% pa compared with 14.0% pa?
Give your answer as a proportional increase over the amount you could borrow when interest rates were high $(V_\text{high rates})$, so:
$$\text{Proportional increase} = \dfrac{V_\text{low rates}-V_\text{high rates}}{V_\text{high rates}}$$
Assume that:
• Interest rates are expected to be constant over the life of the loan.
• Loans are interest-only and have a life of 30 years.
• Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates (APR's) compounding per month.
Bonds A and B are issued by the same Australian company. Both bonds yield 7% pa, and they have the same face value ($100), maturity, seniority, and payment frequency. The only difference is that bond A pays coupons of 10% pa and bond B pays coupons of 5% pa. Which of the following statements is true about the bonds' prices? Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100) and maturity (3 years).
The only difference is that bond X and Y's yields are 8 and 12% pa respectively. Which of the following statements is true?
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100), maturity (3 years) and yield (10%) as each other. Which of the following statements is true? Which one of the following bonds is trading at par? The coupon rate of a fixed annual-coupon bond is constant (always the same). What can you say about the income return ($r_\text{income}$) of a fixed annual coupon bond? Remember that: $$r_\text{total} = r_\text{income} + r_\text{capital}$$ $$r_\text{total, 0 to 1} = \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0}$$ Assume that there is no change in the bond's total annual yield to maturity from when it is issued to when it matures. Select the most correct statement. From its date of issue until maturity, the income return of a fixed annual coupon: A firm wishes to raise$10 million now. They will issue 6% pa semi-annual coupon bonds that will mature in 8 years and have a face value of $1,000 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat. How many bonds should the firm issue? A four year bond has a face value of$100, a yield of 9% and a fixed coupon rate of 6%, paid semi-annually. What is its price?
Bonds X and Y are issued by the same company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency. The only difference is that bond X pays coupons of 6% pa and bond Y pays coupons of 8% pa. Which of the following statements is true? A 30 year Japanese government bond was just issued at par with a yield of 1.7% pa. The fixed coupon payments are semi-annual. The bond has a face value of$100.
Six months later, just after the first coupon is paid, the yield of the bond increases to 2% pa. What is the bond's new price?
A 10 year Australian government bond was just issued at par with a yield of 3.9% pa. The fixed coupon payments are semi-annual. The bond has a face value of $1,000. Six months later, just after the first coupon is paid, the yield of the bond decreases to 3.65% pa. What is the bond's new price? Bonds X and Y are issued by the same US company. Both bonds yield 6% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X pays coupons of 8% pa and bond Y pays coupons of 12% pa. Which of the following statements is true?
A project has the following cash flows. Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $105 at time 2 is actually earned smoothly from t=1 to t=2: Project Cash Flows Time (yrs) Cash flow ($) 0 -90 1 30 2 105
What is the payback period of the project in years?
A young lady is trying to decide if she should attend university or not.
The young lady's parents say that she must attend university because otherwise all of her hard work studying and attending school during her childhood was a waste.
What's the correct way to classify this item from a capital budgeting perspective when trying to decide whether to attend university?
The hard work studying at school in her childhood should be classified as:
Find Sidebar Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Sidebar Corp Income Statement for year ending 30th June 2013 $m Sales 405 COGS 100 Depreciation 34 Rent expense 22 Interest expense 39 Taxable Income 210 Taxes at 30% 63 Net income 147 Sidebar Corp Balance Sheet as at 30th June 2013 2012$m $m Cash 0 0 Inventory 70 50 Trade debtors 11 16 Rent paid in advance 4 3 PPE 700 680 Total assets 785 749 Trade creditors 11 19 Bond liabilities 400 390 Contributed equity 220 220 Retained profits 154 120 Total L and OE 785 749 Note: All figures are given in millions of dollars ($m).
The cash flow from assets was:
Why is Capital Expenditure (CapEx) subtracted in the Cash Flow From Assets (CFFA) formula?
$$CFFA=NI+Depr-CapEx - \Delta NWC+IntExp$$
A firm has forecast its Cash Flow From Assets (CFFA) for this year and management is worried that it is too low. Which one of the following actions will lead to a higher CFFA for this year (t=0 to 1)? Only consider cash flows this year. Do not consider cash flows after one year, or the change in the NPV of the firm. Consider each action in isolation.
Find the cash flow from assets (CFFA) of the following project.
Project Data Project life 2 years Initial investment in equipment $6m Depreciation of equipment per year for tax purposes$1m Unit sales per year 4m Sale price per unit $8 Variable cost per unit$3 Fixed costs per year, paid at the end of each year $1.5m Tax rate 30% Note 1: The equipment will have a book value of$4m at the end of the project for tax purposes. However, the equipment is expected to fetch $0.9 million when it is sold at t=2. Note 2: Due to the project, the firm will have to purchase$0.8m of inventory initially, which it will sell at t=1. The firm will buy another $0.8m at t=1 and sell it all again at t=2 with zero inventory left. The project will have no effect on the firm's current liabilities. Find the project's CFFA at time zero, one and two. Answers are given in millions of dollars ($m).
An investor bought a 10 year 2.5% pa fixed coupon government bond priced at par. The face value is $100. Coupons are paid semi-annually and the next one is in 6 months. Six months later, just after the coupon at that time was paid, yields suddenly and unexpectedly fell to 2% pa. Note that all yields above are given as APR's compounding semi-annually. What was the bond investors' historical total return over that first 6 month period, given as an effective semi-annual rate? What is the present value of a nominal payment of$1,000 in 4 years? The nominal discount rate is 8% pa and the inflation rate is 2% pa.
A firm is considering a business project which costs $10m now and is expected to pay a single cash flow of$12.1m in two years.
Assume that the initial $10m cost is funded using the firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa. Which of the following statements about net present value (NPV), internal rate of return (IRR) and payback period is NOT correct? You own a debt asset. Are you a or a ? A mining firm has just discovered a new mine. So far the news has been kept a secret. The net present value of digging the mine and selling the minerals is$250 million, but $500 million of new equity and$300 million of new bonds will need to be issued to fund the project and buy the necessary plant and equipment.
The firm will release the news of the discovery and equity and bond raising to shareholders simultaneously in the same announcement. The shares and bonds will be issued shortly after.
Once the announcement is made and the new shares and bonds are issued, what is the expected increase in the value of the firm's assets $(\Delta V)$, market capitalisation of debt $(\Delta D)$ and market cap of equity $(\Delta E)$? Assume that markets are semi-strong form efficient.
The triangle symbol $\Delta$ is the Greek letter capital delta which means change or increase in mathematics.
Ignore the benefit of interest tax shields from having more debt.
Remember: $\Delta V = \Delta D+ \Delta E$
What type of present value equation is best suited to value a residential house investment property that is expected to pay constant rental payments forever? Note that 'constant' has the same meaning as 'level' in this context.
You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate. You wish to consume twice as much now (t=0) as in one year (t=1) and have nothing left in the bank at the end. How much can you consume at time zero and one? The answer choices are given in the same order. You have$100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume half as much now (t=0) as in one year (t=1) and have nothing left in the bank at the end.
How much can you consume at time zero and one? The answer choices are given in the same order.
What is the present value of a nominal payment of $100 in 5 years? The real discount rate is 10% pa and the inflation rate is 3% pa. The following cash flows are expected: • 10 yearly payments of$80, with the first payment in 6.5 years from now (first payment at t=6.5).
• A single payment of $500 in 4 years and 3 months (t=4.25) from now. What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate? You are promised 20 payments of$100, where the first payment is immediate (t=0) and the last is at the end of the 19th year (t=19). The effective annual discount rate is $r$.
Which of the following equations does NOT give the correct present value of these 20 payments?
Assume that the Gordon Growth Model (same as the dividend discount model or perpetuity with growth formula) is an appropriate method to value real estate.
The rule of thumb in the real estate industry is that properties should yield a 5% pa rental return. Many investors also regard property to be as risky as the stock market, therefore property is thought to have a required total return of 9% pa which is the average total return on the stock market including dividends.
Assume that all returns are effective annual rates and they are nominal (not reduced by inflation). Inflation is expected to be 2% pa.
You're considering purchasing an investment property which has a rental yield of 5% pa and you expect it to have the same risk as the stock market. Select the most correct statement about this property.
You're the boss of an investment bank's equities research team. Your five analysts are each trying to find the expected total return over the next year of shares in a mining company. The mining firm:
• Is regarded as a mature company since it's quite stable in size and was floated around 30 years ago. It is not a high-growth company;
• Share price is very sensitive to changes in the price of the market portfolio, economic growth, the exchange rate and commodities prices. Due to this, its standard deviation of total returns is much higher than that of the market index;
• Experienced tough times in the last 10 years due to unexpected falls in commodity prices.
• Shares are traded in an active liquid market.
Your team of analysts present their findings, and everyone has different views. While there's no definitive true answer, whose calculation of the expected total return is the most plausible?
Assume that:
• The analysts' source data is correct and true, but their inferences might be wrong;
• All returns and yields are given as effective annual nominal rates.
Three years ago Frederika bought a house for $400,000. Now it's worth$600,000, based on recent similar sales in the area.
Frederika's residential property has an expected total return of 7% pa.
She rents her house out for $2,500 per month, paid in advance. Every 12 months she plans to increase the rental payments. The present value of 12 months of rental payments is$29,089.48.
The future value of 12 months of rental payments one year ahead is $31,125.74. What is the expected annual capital yield of the property? The perpetuity with growth formula is: $$P_0= \dfrac{C_1}{r-g}$$ Which of the following is NOT equal to the total required return (r)? The perpetuity with growth equation is: $$P_0=\dfrac{C_1}{r-g}$$ Which of the following is NOT equal to the expected capital return as an effective annual rate? A fairly priced unlevered firm plans to pay a dividend of$1 next year (t=1) which is expected to grow by 3% pa every year after that. The firm's required return on equity is 8% pa.
The firm is thinking about reducing its future dividend payments by 10% so that it can use the extra cash to invest in more projects which are expected to return 8% pa, and have the same risk as the existing projects. Therefore, next year's dividend will be $0.90. No new equity or debt will be issued to fund the new projects, they'll all be funded by the cut in dividends. What will be the stock's new annual capital return (proportional increase in price per year) if the change in payout policy goes ahead? Assume that payout policy is irrelevant to firm value (so there's no signalling effects) and that all rates are effective annual rates. A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 2.5% pa. Inflation is expected to be 2.5% pa. All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity. What are the property's expected real total, capital and income returns? The answer choices below are given in the same order. A low-growth mature stock has an expected nominal total return of 6% pa and nominal capital return of 2% pa. Inflation is expected to be 3% pa. All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity. What are the stock's expected real total, capital and income returns? The answer choices below are given in the same order. Total cash flows can be broken into income and capital cash flows. What is the name given to the cash flow generated from selling shares at a higher price than they were bought? Risk-free government bonds that have coupon rates greater than their yields: Find Trademark Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. Trademark Corp Income Statement for year ending 30th June 2013$m Sales 100 COGS 25 Operating expense 5 Depreciation 20 Interest expense 20 Income before tax 30 Tax at 30% 9 Net income 21
Trademark Corp Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 120 80 PPE Cost 150 140 Accumul. depr. 60 40 Carrying amount 90 100 Total assets 210 180 Liabilities Current liabilities 75 65 Non-current liabilities 75 55 Owners' equity Retained earnings 10 10 Contributed equity 50 50 Total L and OE 210 180
Note: all figures are given in millions of dollars ($m). Find UniBar Corp's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. UniBar Corp Income Statement for year ending 30th June 2013$m Sales 80 COGS 40 Operating expense 15 Depreciation 10 Interest expense 5 Income before tax 10 Tax at 30% 3 Net income 7
UniBar Corp Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 120 90 PPE Cost 360 320 Accumul. depr. 40 30 Carrying amount 320 290 Total assets 440 380 Liabilities Current liabilities 110 60 Non-current liabilities 190 180 Owners' equity Retained earnings 95 95 Contributed equity 45 45 Total L and OE 440 380
Note: all figures are given in millions of dollars ($m). Find Piano Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. Piano Bar Income Statement for year ending 30th June 2013$m Sales 310 COGS 185 Operating expense 20 Depreciation 15 Interest expense 10 Income before tax 80 Tax at 30% 24 Net income 56
Piano Bar Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 240 230 PPE Cost 420 400 Accumul. depr. 50 35 Carrying amount 370 365 Total assets 610 595 Liabilities Current liabilities 180 190 Non-current liabilities 290 265 Owners' equity Retained earnings 90 90 Contributed equity 50 50 Total L and OE 610 595
Note: all figures are given in millions of dollars ($m). Which one of the following will increase the Cash Flow From Assets in this year for a tax-paying firm, all else remaining constant? Cash Flow From Assets (CFFA) can be defined as: Find World Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. World Bar Income Statement for year ending 30th June 2013$m Sales 300 COGS 150 Operating expense 50 Depreciation 40 Interest expense 10 Taxable income 50 Tax at 30% 15 Net income 35
World Bar Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 200 230 PPE Cost 400 400 Accumul. depr. 75 35 Carrying amount 325 365 Total assets 525 595 Liabilities Current liabilities 150 205 Non-current liabilities 235 250 Owners' equity Retained earnings 100 100 Contributed equity 40 40 Total L and OE 525 595
Note: all figures above and below are given in millions of dollars ($m). Find Scubar Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. Scubar Corp Income Statement for year ending 30th June 2013$m Sales 200 COGS 60 Depreciation 20 Rent expense 11 Interest expense 19 Taxable Income 90 Taxes at 30% 27 Net income 63
Scubar Corp Balance Sheet as at 30th June 2013 2012 $m$m Inventory 60 50 Trade debtors 19 6 Rent paid in advance 3 2 PPE 420 400 Total assets 502 458 Trade creditors 10 8 Bond liabilities 200 190 Contributed equity 130 130 Retained profits 162 130 Total L and OE 502 458
Note: All figures are given in millions of dollars ($m). The cash flow from assets was: Which one of the following will decrease net income (NI) but increase cash flow from assets (CFFA) in this year for a tax-paying firm, all else remaining constant? Remember: $$NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c )$$ $$CFFA=NI+Depr-CapEx - ΔNWC+IntExp$$ A new company's Firm Free Cash Flow (FFCF, same as CFFA) is forecast in the graph below. To value the firm's assets, the terminal value needs to be calculated using the perpetuity with growth formula: $$V_{\text{terminal, }t-1} = \dfrac{FFCF_{\text{terminal, }t}}{r-g}$$ Which point corresponds to the best time to calculate the terminal value? Which one of the following will have no effect on net income (NI) but decrease cash flow from assets (CFFA or FFCF) in this year for a tax-paying firm, all else remaining constant? Remember: $$NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c )$$ $$CFFA=NI+Depr-CapEx - ΔNWC+IntExp$$ Find Ching-A-Lings Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013. Ching-A-Lings Corp Income Statement for year ending 30th June 2013$m Sales 100 COGS 20 Depreciation 20 Rent expense 11 Interest expense 19 Taxable Income 30 Taxes at 30% 9 Net income 21
Ching-A-Lings Corp Balance Sheet as at 30th June 2013 2012 $m$m Inventory 49 38 Trade debtors 14 2 Rent paid in advance 5 5 PPE 400 400 Total assets 468 445 Trade creditors 4 10 Bond liabilities 200 190 Contributed equity 145 145 Retained profits 119 100 Total L and OE 468 445
Note: All figures are given in millions of dollars (m). The cash flow from assets was: A method commonly seen in textbooks for calculating a levered firm's free cash flow (FFCF, or CFFA) is the following: \begin{aligned} FFCF &= (Rev - COGS - Depr - FC - IntExp)(1-t_c) + \\ &\space\space\space+ Depr - CapEx -\Delta NWC + IntExp(1-t_c) \\ \end{aligned} Does this annual FFCF or the annual interest tax shield? Read the following financial statements and calculate the firm's free cash flow over the 2014 financial year. UBar Corp Income Statement for year ending 30th June 2014m Sales 293 COGS 200 Rent expense 15 Gas expense 8 Depreciation 10 EBIT 60 Interest expense 0 Taxable income 60 Taxes 18 Net income 42
UBar Corp Balance Sheet as at 30th June 2014 2013 $m$m Assets Cash 30 29 Accounts receivable 5 7 Pre-paid rent expense 1 0 Inventory 50 46 PPE 290 300 Total assets 376 382 Liabilities Trade payables 20 18 Accrued gas expense 3 2 Non-current liabilities 0 0 Contributed equity 212 212 Retained profits 136 150 Asset revaluation reserve 5 0 Total L and OE 376 382
Note: all figures are given in millions of dollars ($m). The firm's free cash flow over the 2014 financial year was: Find the cash flow from assets (CFFA) of the following project. One Year Mining Project Data Project life 1 year Initial investment in building mine and equipment$9m Depreciation of mine and equipment over the year $8m Kilograms of gold mined at end of year 1,000 Sale price per kilogram$0.05m Variable cost per kilogram $0.03m Before-tax cost of closing mine at end of year$4m Tax rate 30%
Note 1: Due to the project, the firm also anticipates finding some rare diamonds which will give before-tax revenues of $1m at the end of the year. Note 2: The land that will be mined actually has thermal springs and a family of koalas that could be sold to an eco-tourist resort for an after-tax amount of$3m right now. However, if the mine goes ahead then this natural beauty will be destroyed.
Note 3: The mining equipment will have a book value of $1m at the end of the year for tax purposes. However, the equipment is expected to fetch$2.5m when it is sold.
Find the project's CFFA at time zero and one. Answers are given in millions of dollars ($m), with the first cash flow at time zero, and the second at time one. Find the cash flow from assets (CFFA) of the following project. Project Data Project life 2 years Initial investment in equipment$8m Depreciation of equipment per year for tax purposes $3m Unit sales per year 10m Sale price per unit$9 Variable cost per unit $4 Fixed costs per year, paid at the end of each year$2m Tax rate 30%
Note 1: Due to the project, the firm will have to purchase $40m of inventory initially (at t=0). Half of this inventory will be sold at t=1 and the other half at t=2. Note 2: The equipment will have a book value of$2m at the end of the project for tax purposes. However, the equipment is expected to fetch $1m when it is sold. Assume that the full capital loss is tax-deductible and taxed at the full corporate tax rate. Note 3: The project will be fully funded by equity which investors will expect to pay dividends totaling$10m at the end of each year.
Find the project's CFFA at time zero, one and two. Answers are given in millions of dollars (m). To value a business's assets, the free cash flow of the firm (FCFF, also called CFFA) needs to be calculated. This requires figures from the firm's income statement and balance sheet. For what figures is the balance sheet needed? Note that the balance sheet is sometimes also called the statement of financial position. To value a business's assets, the free cash flow of the firm (FCFF, also called CFFA) needs to be calculated. This requires figures from the firm's income statement and balance sheet. For what figures is the income statement needed? Note that the income statement is sometimes also called the profit and loss, P&L, or statement of financial performance. A firm has a debt-to-equity ratio of 25%. What is its debt-to-assets ratio? A company increases the proportion of debt funding it uses to finance its assets by issuing bonds and using the cash to repurchase stock, leaving assets unchanged. Ignoring the costs of financial distress, which of the following statements is NOT correct: Which one of the following will decrease net income (NI) but increase cash flow from assets (CFFA) in this year for a tax-paying firm, all else remaining constant? Remember: $$NI = (Rev-COGS-FC-Depr-IntExp).(1-t_c )$$ $$CFFA=NI+Depr-CapEx - \Delta NWC+IntExp$$ A young lady is trying to decide if she should attend university. Her friends say that she should go to university because she is more likely to meet a clever young man than if she begins full time work straight away. What's the correct way to classify this item from a capital budgeting perspective when trying to find the Net Present Value of going to university rather than working? The opportunity to meet a desirable future spouse should be classified as: A man has taken a day off from his casual painting job to relax. It's the end of the day and he's thinking about the hours that he could have spent working (in the past) which are now: Issuing debt doesn't give away control of the firm because debt holders can't cast votes to determine the company's affairs, such as at the annual general meeting (AGM), and can't appoint directors to the board. or ? One formula for calculating a levered firm's free cash flow (FFCF, or CFFA) is to use earnings before interest and tax (EBIT). \begin{aligned} FFCF &= (EBIT)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ \end{aligned} \\ Does this annual FFCF or the annual interest tax shield? One formula for calculating a levered firm's free cash flow (FFCF, or CFFA) is to use net operating profit after tax (NOPAT). \begin{aligned} FFCF &= NOPAT + Depr - CapEx -\Delta NWC \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC \\ \end{aligned} \\ Does this annual FFCF or the annual interest tax shield? A company issues a large amount of bonds to raise money for new projects of similar risk to the company's existing projects. The net present value (NPV) of the new projects is positive but small. Assume a classical tax system. Which statement is NOT correct? A firm has a debt-to-assets ratio of 50%. The firm then issues a large amount of debt to raise money for new projects of similar market risk to the company's existing projects. Assume a classical tax system. Which statement is correct? Your friend just bought a house for400,000. He financed it using a $320,000 mortgage loan and a deposit of$80,000.
In the context of residential housing and mortgages, the 'equity' tied up in the value of a person's house is the value of the house less the value of the mortgage. So the initial equity your friend has in his house is $80,000. Let this amount be E, let the value of the mortgage be D and the value of the house be V. So $V=D+E$. If house prices suddenly fall by 10%, what would be your friend's percentage change in equity (E)? Assume that the value of the mortgage is unchanged and that no income (rent) was received from the house during the short time over which house prices fell. Remember: $$r_{0\rightarrow1}=\frac{p_1-p_0+c_1}{p_0}$$ where $r_{0-1}$ is the return (percentage change) of an asset with price $p_0$ initially, $p_1$ one period later, and paying a cash flow of $c_1$ at time $t=1$. Which of the following statements about standard statistical mathematics notation is NOT correct? All things remaining equal, the variance of a portfolio of two positively-weighted stocks rises as: Two risky stocks A and B comprise an equal-weighted portfolio. The correlation between the stocks' returns is 70%. If the variance of stock A increases but the: • Prices and expected returns of each stock stays the same, • Variance of stock B's returns stays the same, • Correlation of returns between the stocks stays the same. Which of the following statements is NOT correct? All things remaining equal, the higher the correlation of returns between two stocks: An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 6% pa. • Stock A has an expected return of 5% pa. • Stock B has an expected return of 10% pa. What portfolio weights should the investor have in stocks A and B respectively? An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 12% pa. • Stock A has an expected return of 10% pa and a standard deviation of 20% pa. • Stock B has an expected return of 15% pa and a standard deviation of 30% pa. The correlation coefficient between stock A and B's expected returns is 70%. What will be the annual standard deviation of the portfolio with this 12% pa target return? What is the correlation of a variable X with itself? The corr(X, X) or $\rho_{X,X}$ equals: Mr Blue, Miss Red and Mrs Green are people with different utility functions. Each person has$500 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose $500. Each player can flip a coin and if they flip heads, they receive$500. If they flip tails then they will lose $500. Which of the following statements is NOT correct? Diversification is achieved by investing in a large amount of stocks. What type of risk is reduced by diversification? According to the theory of the Capital Asset Pricing Model (CAPM), total risk can be broken into two components, systematic risk and idiosyncratic risk. Which of the following events would be considered a systematic, undiversifiable event according to the theory of the CAPM? A fairly priced stock has an expected return equal to the market's. Treasury bonds yield 5% pa and the market portfolio's expected return is 10% pa. What is the stock's beta? Stock A has a beta of 0.5 and stock B has a beta of 1. Which statement is NOT correct? Which statement is the most correct? A stock's correlation with the market portfolio increases while its total risk is unchanged. What will happen to the stock's expected return and systematic risk? Assets A, B, M and $r_f$ are shown on the graphs above. Asset M is the market portfolio and $r_f$ is the risk free yield on government bonds. Which of the below statements is NOT correct? A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates. What do you think will be the stock's expected return over the next year, given as an effective annual rate? A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates. In the last 5 minutes, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. The risk free rate was unchanged. What do you think was the stock's historical return over the last 5 minutes, given as an effective 5 minute rate? A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates. Over the last year, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. So $r_{m} = (P_{0} - P_{-1})/P_{-1} = -0.01$, where the current time is zero and one year ago is time -1. The risk free rate was unchanged. What do you think was the stock's historical return over the last year, given as an effective annual rate? The CAPM can be used to find a business's expected opportunity cost of capital: $$r_i=r_f+β_i (r_m-r_f)$$ What should be used as the risk free rate $r_f$? A firm's WACC before tax would decrease due to: Which of the following statements about the weighted average cost of capital (WACC) is NOT correct? Project Data Project life 1 year Initial investment in equipment$8m Depreciation of equipment per year $8m Expected sale price of equipment at end of project 0 Unit sales per year 4m Sale price per unit$10 Variable cost per unit $5 Fixed costs per year, paid at the end of each year$2m Interest expense in first year (at t=1) $0.562m Corporate tax rate 30% Government treasury bond yield 5% Bank loan debt yield 9% Market portfolio return 10% Covariance of levered equity returns with market 0.32 Variance of market portfolio returns 0.16 Firm's and project's debt-to-equity ratio 50% Notes 1. Due to the project, current assets will increase by$6m now (t=0) and fall by $6m at the end (t=1). Current liabilities will not be affected. Assumptions • The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio. • Millions are represented by 'm'. • All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year. • All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates. • The project is undertaken by a firm, not an individual. What is the net present value (NPV) of the project? Your friend claims that by reading 'The Economist' magazine's economic news articles, she can identify shares that will have positive abnormal expected returns over the next 2 years. Assuming that her claim is true, which statement(s) are correct? (i) Weak form market efficiency is broken. (ii) Semi-strong form market efficiency is broken. (iii) Strong form market efficiency is broken. (iv) The asset pricing model used to measure the abnormal returns (such as the CAPM) is either wrong (mis-specification error) or is measured using the wrong inputs (data errors) so the returns may not be abnormal but rather fair for the level of risk. Select the most correct response: Fundamentalists who analyse company financial reports and news announcements (but who don't have inside information) will make positive abnormal returns if: The average weekly earnings of an Australian adult worker before tax was$1,542.40 per week in November 2014 according to the Australian Bureau of Statistics. Therefore average annual earnings before tax were $80,204.80 assuming 52 weeks per year. Personal income tax rates published by the Australian Tax Office are reproduced for the 2014-2015 financial year in the table below: Taxable income Tax on this income 0 –$18,200 Nil
$18,201 –$37,000 19c for each $1 over$18,200
$37,001 –$80,000 $3,572 plus 32.5c for each$1 over $37,000$80,001 – $180,000$17,547 plus 37c for each $1 over$80,000
$180,001 and over$54,547 plus 45c for each $1 over$180,000
The above rates do not include the Medicare levy of 2%. Exclude the Medicare levy from your calculations
How much personal income tax would you have to pay per year if you earned $80,204.80 per annum before-tax? Question 449 personal tax on dividends, classical tax system A small private company has a single shareholder. This year the firm earned a$100 profit before tax. All of the firm's after tax profits will be paid out as dividends to the owner.
The corporate tax rate is 30% and the sole shareholder's personal marginal tax rate is 45%.
The United States' classical tax system applies because the company generates all of its income in the US and pays corporate tax to the Internal Revenue Service. The shareholder is also an American for tax purposes.
What will be the personal tax payable by the shareholder and the corporate tax payable by the company?
A small private company has a single shareholder. This year the firm earned a $100 profit before tax. All of the firm's after tax profits will be paid out as dividends to the owner. The corporate tax rate is 30% and the sole shareholder's personal marginal tax rate is 45%. The Australian imputation tax system applies because the company generates all of its income in Australia and pays corporate tax to the Australian Tax Office. Therefore all of the company's dividends are fully franked. The sole shareholder is an Australian for tax purposes and can therefore use the franking credits to offset his personal income tax liability. What will be the personal tax payable by the shareholder and the corporate tax payable by the company? A company conducts a 1 for 5 rights issue at a subscription price of$7 when the pre-announcement stock price was $10. What is the percentage change in the stock price and the number of shares outstanding? The answers are given in the same order. Ignore all taxes, transaction costs and signalling effects. Question 625 dividend re-investment plan, capital raising Which of the following statements about dividend re-investment plans (DRP's) is NOT correct? Convert a 10% continuously compounded annual rate $(r_\text{cc annual})$ into an effective annual rate $(r_\text{eff annual})$. The equivalent effective annual rate is: Which of the following interest rate quotes is NOT equivalent to a 10% effective annual rate of return? Assume that each year has 12 months, each month has 30 days, each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. APR stands for Annualised Percentage Rate. A continuously compounded monthly return of 1% $(r_\text{cc monthly})$ is equivalent to a continuously compounded annual return $(r_\text{cc annual})$ of: An effective monthly return of 1% $(r_\text{eff monthly})$ is equivalent to an effective annual return $(r_\text{eff annual})$ of: Which of the following quantities is commonly assumed to be normally distributed? Project Data Project life 1 year Initial investment in equipment$6m Depreciation of equipment per year $6m Expected sale price of equipment at end of project 0 Unit sales per year 9m Sale price per unit$8 Variable cost per unit $6 Fixed costs per year, paid at the end of each year$1m Interest expense in first year (at t=1) $0.53m Tax rate 30% Government treasury bond yield 5% Bank loan debt yield 6% Market portfolio return 10% Covariance of levered equity returns with market 0.08 Variance of market portfolio returns 0.16 Firm's and project's debt-to-assets ratio 50% Notes 1. Due to the project, current assets will increase by$5m now (t=0) and fall by $5m at the end (t=1). Current liabilities will not be affected. Assumptions • The debt-to-assets ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio. • Millions are represented by 'm'. • All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year. • All rates and cash flows are real. The inflation rate is 2% pa. • All rates are given as effective annual rates. • The 50% capital gains tax discount is not available since the project is undertaken by a firm, not an individual. What is the net present value (NPV) of the project? The 'time value of money' is most closely related to which of the following concepts? Which of the following statements is NOT correct? Lenders: A stock's required total return will decrease when its: A home loan company advertises an interest rate of 9% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given with an accuracy of 4 decimal places. How much more can you borrow using an interest-only loan compared to a 25-year fully amortising loan if interest rates are 6% pa compounding per month and are not expected to change? If it makes it easier, assume that you can afford to pay$2,000 per month on either loan. Express your answer as a proportional increase using the following formula:
$$\text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1$$
A stock's total standard deviation of returns is 20% pa. The market portfolio's total standard deviation of returns is 15% pa. The beta of the stock is 0.8.
What is the stock's diversifiable standard deviation?
Which of the following interest rate labels does NOT make sense?
A firm has a debt-to-assets ratio of 20%. What is its debt-to-equity ratio?
What is the present value of real payments of $100 every year forever, with the first payment in one year? The nominal discount rate is 7% pa and the inflation rate is 4% pa. A company conducts a 10 for 3 stock split. What is the percentage increase in the stock price and the number of shares outstanding? The answers are given in the same order. A company conducts a 2 for 3 rights issue at a subscription price of$8 when the pre-announcement stock price was $9. Assume that all investors use their rights to buy those extra shares. What is the percentage increase in the stock price and the number of shares outstanding? The answers are given in the same order. The Australian cash rate is expected to be 2% pa over the next one year, while the US cash rate is expected to be 0% pa, both given as nominal effective annual rates. The current exchange rate is 0.73 USD per AUD. What is the implied 1 year USD per AUD forward foreign exchange rate? Which of the following is NOT a valid method for estimating the beta of a company's stock? Assume that markets are efficient, a long history of past data is available, the stock possesses idiosyncratic and market risk. The variances and standard deviations below denote total risks. A stock's required total return will increase when its: Who owns a company's shares? The: The Australian cash rate is expected to be 2% pa over the next one year, while the Japanese cash rate is expected to be 0% pa, both given as nominal effective annual rates. The current exchange rate is 100 JPY per AUD. What is the implied 1 year forward foreign exchange rate? Which of the following statements about yield curves is NOT correct? Convert a 10% effective annual rate $(r_\text{eff annual})$ into a continuously compounded annual rate $(r_\text{cc annual})$. The equivalent continuously compounded annual rate is: A continuously compounded semi-annual return of 5% $(r_\text{cc 6mth})$ is equivalent to a continuously compounded annual return $(r_\text{cc annual})$ of: An effective semi-annual return of 5% $(r_\text{eff 6mth})$ is equivalent to an effective annual return $(r_\text{eff annual})$ of: A bank quotes an interest rate of 6% pa with quarterly compounding. Note that another way of stating this rate is that it is an annual percentage rate (APR) compounding discretely every 3 months. Which of the following statements about this rate is NOT correct? All percentages are given to 6 decimal places. The equivalent: If a variable, say X, is normally distributed with mean $\mu$ and variance $\sigma^2$ then mathematicians write $X \sim \mathcal{N}(\mu, \sigma^2)$. If a variable, say Y, is log-normally distributed and the underlying normal distribution has mean $\mu$ and variance $\sigma^2$ then mathematicians write $Y \sim \mathbf{ln} \mathcal{N}(\mu, \sigma^2)$. The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue. Select the most correct statement: The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue. Let $P_1$ be the unknown price of a stock in one year. $P_1$ is a random variable. Let $P_0 = 1$, so the share price now is$1. This one dollar is a constant, it is not a variable.
Which of the below statements is NOT correct? Financial practitioners commonly assume that the shape of the PDF represented in the colour:
If a stock's future expected continuously compounded annual returns are normally distributed, what will be bigger, the stock's or continuously compounded annual return? Or would you expect them to be ?
A company selling charting and technical analysis software claims that independent academic studies have shown that its software makes significantly positive abnormal returns. Assuming the claim is true, which statement(s) are correct?
(I) Weak form market efficiency is broken.
(II) Semi-strong form market efficiency is broken.
(III) Strong form market efficiency is broken.
(IV) The asset pricing model used to measure the abnormal returns (such as the CAPM) had mis-specification error so the returns may not be abnormal but rather fair for the level of risk.
Select the most correct response:
Select the most correct statement from the following.
'Chartists', also known as 'technical traders', believe that:
A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever.
All rates are given as nominal effective annual returns. Assume that:
• His forecast is true.
• Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true.
• Ignore all costs such as taxes, agent fees, maintenance and so on.
• All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property.
• The non-monetary benefits of owning real estate and renting remain constant.
Which one of the following statements is NOT correct? Over time:
According to the theory of the Capital Asset Pricing Model (CAPM), total variance can be broken into two components, systematic variance and idiosyncratic variance. Which of the following events would be considered the most diversifiable according to the theory of the CAPM?
Treasury bonds currently have a return of 5% pa. A stock has a beta of 0.5 and the market return is 10% pa. What is the expected return of the stock?
Portfolio Details Stock Expected return Standard deviation Correlation Beta Dollars invested A 0.2 0.4 0.12 0.5 40 B 0.3 0.8 1.5 80
What is the beta of the above portfolio?
Which statement(s) are correct?
(i) All stocks that plot on the Security Market Line (SML) are fairly priced.
(ii) All stocks that plot above the Security Market Line (SML) are overpriced.
(iii) All fairly priced stocks that plot on the Capital Market Line (CML) have zero idiosyncratic risk.
Select the most correct response:
A firm changes its capital structure by issuing a large amount of debt and using the funds to repurchase shares. Its assets are unchanged. Ignore interest tax shields.
According to the Capital Asset Pricing Model (CAPM), which statement is correct?
The total return of any asset can be broken down in different ways. One possible way is to use the dividend discount model (or Gordon growth model):
$$p_0 = \frac{c_1}{r_\text{total}-r_\text{capital}}$$
Which, since $c_1/p_0$ is the income return ($r_\text{income}$), can be expressed as:
$$r_\text{total}=r_\text{income}+r_\text{capital}$$
So the total return of an asset is the income component plus the capital or price growth component.
Another way to break up total return is to use the Capital Asset Pricing Model:
$$r_\text{total}=r_\text{f}+β(r_\text{m}- r_\text{f})$$
$$r_\text{total}=r_\text{time value}+r_\text{risk premium}$$
So the risk free rate is the time value of money and the term $β(r_\text{m}- r_\text{f})$ is the compensation for taking on systematic risk.
Using the above theory and your general knowledge, which of the below equations, if any, are correct?
(I) $r_\text{income}=r_\text{time value}$
(II) $r_\text{income}=r_\text{risk premium}$
(III) $r_\text{capital}=r_\text{time value}$
(IV) $r_\text{capital}=r_\text{risk premium}$
(V) $r_\text{income}+r_\text{capital}=r_\text{time value}+r_\text{risk premium}$
Which of the equations are correct?
A firm can issue 5 year annual coupon bonds at a yield of 8% pa and a coupon rate of 12% pa.
The beta of its levered equity is 1. Five year government bonds yield 5% pa with a coupon rate of 6% pa. The market's expected dividend return is 4% pa and its expected capital return is 6% pa.
The firm's debt-to-equity ratio is 2:1. The corporate tax rate is 30%.
What is the firm's after-tax WACC? Assume a classical tax system.
Mr Blue, Miss Red and Mrs Green are people with different utility functions. Which of the statements about the 3 utility functions is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions. Which of the statements about the 3 utility functions is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions. Which of the statements about the 3 utility functions is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions.
Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions.
Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions.
Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?
Mr Blue, Miss Red and Mrs Green are people with different utility functions.
Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?
The following table shows a sample of historical total returns of shares in two different companies A and B.
Stock Returns Total effective annual returns Year $r_A$ $r_B$ 2007 0.2 0.4 2008 0.04 -0.2 2009 -0.1 -0.3 2010 0.18 0.5
What is the historical sample covariance ($\hat{\sigma}_{A,B}$) and correlation ($\rho_{A,B}$) of stock A and B's total effective annual returns?
Stock A and B's returns have a correlation of 0.3. Which statement is NOT correct?
Portfolio Details Stock Expected return Standard deviation Correlation Dollars invested A 0.1 0.4 0.5 60 B 0.2 0.6 140
What is the expected return of the above portfolio?
Portfolio Details Stock Expected return Standard deviation Covariance $(\sigma_{A,B})$ Beta Dollars invested A 0.2 0.4 0.12 0.5 40 B 0.3 0.8 1.5 80
What is the standard deviation (not variance) of the above portfolio? Note that the stocks' covariance is given, not correlation.
Which of the following statements about short-selling is NOT true?
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 16% pa.
• Stock A has an expected return of 8% pa.
• Stock B has an expected return of 12% pa.
What portfolio weights should the investor have in stocks A and B respectively?
What is the covariance of a variable X with itself?
The cov(X, X) or $\sigma_{X,X}$ equals:
Companies must pay interest and principal payments to debt-holders. They're compulsory. But companies are not forced to pay dividends to share holders. or ?
Here are the Net Income (NI) and Cash Flow From Assets (CFFA) equations:
$$NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c)$$
$$CFFA=NI+Depr-CapEx - \varDelta NWC+IntExp$$
What is the formula for calculating annual interest expense (IntExp) which is used in the equations above?
Select one of the following answers. Note that D is the value of debt which is constant through time, and $r_D$ is the cost of debt.
A firm has a debt-to-equity ratio of 60%. What is its debt-to-assets ratio?
A retail furniture company buys furniture wholesale and distributes it through its retail stores. The owner believes that she has some good ideas for making stylish new furniture. She is considering a project to buy a factory and employ workers to manufacture the new furniture she's designed. Furniture manufacturing has more systematic risk than furniture retailing.
Her furniture retailing firm's after-tax WACC is 20%. Furniture manufacturing firms have an after-tax WACC of 30%. Both firms are optimally geared. Assume a classical tax system.
Which method(s) will give the correct valuation of the new furniture-making project? Select the most correct answer.
There are many ways to calculate a firm's free cash flow (FFCF), also called cash flow from assets (CFFA).
One method is to use the following formulas to transform net income (NI) into FFCF including interest and depreciation tax shields:
$$FFCF=NI + Depr - CapEx -ΔNWC + IntExp$$
$$NI=(Rev - COGS - Depr - FC - IntExp).(1-t_c )$$
Another popular method is to use EBITDA rather than net income. EBITDA is defined as:
$$EBITDA=Rev - COGS - FC$$
One of the below formulas correctly calculates FFCF from EBITDA, including interest and depreciation tax shields, giving an identical answer to that above. Which formula is correct?
A firm is considering a new project of similar risk to the current risk of the firm. This project will expand its existing business. The cash flows of the project have been calculated assuming that there is no interest expense. In other words, the cash flows assume that the project is all-equity financed.
In fact the firm has a target debt-to-equity ratio of 1, so the project will be financed with 50% debt and 50% equity. To find the levered value of the firm's assets, what discount rate should be applied to the project's unlevered cash flows? Assume a classical tax system.
A firm plans to issue equity and use the cash raised to pay off its debt. No assets will be bought or sold. Ignore the costs of financial distress.
Which of the following statements is NOT correct, all things remaining equal? | 2021-04-17 07:59:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3487703502178192, "perplexity": 2476.8156267290833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038118762.49/warc/CC-MAIN-20210417071833-20210417101833-00361.warc.gz"} |
https://flobi.users.phpclasses.org/discuss/package/5166/thread/1/ | # Hi !! I Believe that in your case it is suficient .
Subject: Hi !! I Believe that in your case it... Package rating comment 2 José Filipe Lopes Santos 2009-02-25 12:56:01 2009-03-03 02:28:52
José Filipe Lopes Santos rated this package as follows:
Utility: Sufficient Good Sufficient
1. Hi !! I Believe that in your case it... Reply Report abuse
José Filipe Lopes Santos - 2009-02-25 12:56:01
Hi !!
I Believe that in your case it is suficient .. !!
But in general case this is poor .. !!
Why you dont use Template_PHPLIB package from the PEAR of the PHP ??
Is more best, simple. realiable, ...
`pear.php.net/package/HTML_Template_` ...
2. Re: Hi !! I Believe that in your case it... Reply Report abuse
shahnur - 2009-03-03 02:28:52 - In reply to message 1 from José Filipe Lopes Santos
Actually it's a basic version still now... I'm going to update it soon. http://pear.php.net/package/HTML_Template_PHPLIB/docs/1.4.0/ seemed not so good or easy to me.
For more information send a message to `info at phpclasses dot org`. | 2022-06-29 19:22:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8636910319328308, "perplexity": 9047.799515173228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00799.warc.gz"} |
http://listserv.tau.ac.il/cgi-bin/wa?A2=ind0110&L=ivritex&D=1&T=0&P=1390&F=P | On Wed, Oct 17, 2001, Tzafrir Cohen wrote about "bidi in twocolumns": > Hi > > I needed to typeset text in two columns and didn't want to manually break > columns and visually arange them. > > Here is a small modification to the multicol package to have bidi support > (note that this allows only two columns). This is an initial version. >...
Tzafrir - as far as I know (but I didn't test your specific patch) there's a problem with your patch, because the dvi of the two columns is _output_ at the wrong order: you still output the left column first and the right column after it, but now this means that the logically first column (in Hebrew that is the right column) is ouput second... This causes problems, if I remember correctly, in that that lines are also output in incorrect order to the table of contents. Try having many sections on a few pages, and see what I mean.
When I faced the same problem when I wrote my 2.09 Hebrew support, what I did instead is to keep the boxes output in the same order, but move them to the correct places by hfills. Here's my code: (of course, you'll need to modify it to LaTeX 2e)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Two Columns % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If unchanged, the two column setup in latex puts the columns in the % wrong direction for Hebrew (in which the first column should appear % on the right, and the second on the left). % % We should use the hebdocument flag to check if the document is a Hebrew one, % since checking the current paragraph is not the correct thing to do. % % \@outputdblcol (from latex.tex) is called from \@opcol and outputs a double % column page (after the second column. The first column is not output but % instead kept in a box). This is the macro we need to change.
% The first try to fix this (not a good fix!) was to switch the outputing % of the columns (once both columns are ready) in case of Hebrew. We used % hebdocument to check if the document is a Hebrew one (since checking % the current paragraph is not the correct thing to do). This worked, but % since the columns are output to dvi in reverse order, the table of contents % also got written in mixed up order, which was totally unacceptable, so this % fix was abandoned: % %\def\@outputdblcol{\if@firstcolumn \global\@firstcolumnfalse % \global\setbox\@leftcolumn\box\@outputbox % \else \global\@firstcolumntrue % \setbox\@outputbox\vbox{\hbox to\textwidth{% % \ifhebdocument % \hbox to\columnwidth{\box\@outputbox \hss}% % \hfil\vrule width\columnseprule\hfil % \hbox to\columnwidth{\box\@leftcolumn \hss}% % \else % \hbox to\columnwidth{\box\@leftcolumn \hss}% % \hfil\vrule width\columnseprule\hfil % \hbox to\columnwidth{\box\@outputbox \hss}% % \fi }}\@combinedblfloats % \@outputpage \begingroup \@dblfloatplacement \@startdblcolumn % \@whilesw\if@fcolmade \fi{\@outputpage\@startdblcolumn}\endgroup % \fi}
% The following definition is more eleborate, but outputs the column boxes % in the correct order. % % Note that @leftcolumn is a misnomer in case of hebrew - it will actually % become the right column. However, I chose not to waste another box variable % just for having better names. Perhaps I should have renamed @leftcolumn % to @thefirstcolumn or something.
\def\@outputdblcol{\if@firstcolumn \global\@firstcolumnfalse \global\setbox\@leftcolumn\box\@outputbox \else \global\@firstcolumntrue \setbox\@outputbox\vbox{\hbox to\textwidth{% \ifhebdocument \hbox to 0pt{\hbox to\textwidth{% \hss \hbox to\columnwidth{\box\@leftcolumn \hss}% }} \hfil\vrule width\columnseprule\hfil \hbox to 0pt{\hss\hbox to\textwidth{% \hbox to\columnwidth{\box\@outputbox \hss}% }} \else \hbox to\columnwidth{\box\@leftcolumn \hss}% \hfil\vrule width\columnseprule\hfil \hbox to\columnwidth{\box\@outputbox \hss}% \fi }}\@combinedblfloats \@outputpage \begingroup \@dblfloatplacement \@startdblcolumn \@whilesw\if@fcolmade \fi{\@outputpage\@startdblcolumn}\endgroup \fi}
-- Nadav Har'El | Wednesday, Oct 17 2001, 1 Heshvan 5762 [log in to unmask] |----------------------------------------- Phone: +972-53-245868, ICQ 13349191 |Disclaimer: The opinions expressed above http://nadav.harel.org.il |are not my own.
Back to: Top of message | Previous page | Main IVRITEX page
LISTSERV.TAU.AC.IL | 2015-04-26 15:40:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9689574241638184, "perplexity": 5656.060030118669}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246654687.23/warc/CC-MAIN-20150417045734-00034-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.chemedx.org/category/chemistry-topic/electrochemistry?page=4 | electrochemistry
Electrolysis of sodium nitrate
Electrolysis of a sodium nitrate solution produces oxygen at the anode and hydrogen at the cathode.
Electrolysis of magnesium nitrate
Electrolysis of a magnesium nitrate solution produces oxygen at the anode and hydrogen at the cathode.
Electrolysis of a sodium sulfate solution
An electric current is passed through a sodium sulfate solution. Gases are collected at the electrodes are tested for pH. | 2018-04-24 14:33:20 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8001255393028259, "perplexity": 5487.586558810263}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946721.87/warc/CC-MAIN-20180424135408-20180424155408-00615.warc.gz"} |
https://allenai.github.io/allennlp-docs/api/allennlp.models.semantic_parsing.atis.html | # allennlp.models.semantic_parsing.atis¶
class allennlp.models.semantic_parsing.atis.atis_semantic_parser.AtisSemanticParser(vocab: allennlp.data.vocabulary.Vocabulary, utterance_embedder: allennlp.modules.text_field_embedders.text_field_embedder.TextFieldEmbedder, action_embedding_dim: int, encoder: allennlp.modules.seq2seq_encoders.seq2seq_encoder.Seq2SeqEncoder, decoder_beam_search: allennlp.state_machines.beam_search.BeamSearch, max_decoding_steps: int, input_attention: allennlp.modules.attention.attention.Attention, add_action_bias: bool = True, training_beam_size: int = None, decoder_num_layers: int = 1, dropout: float = 0.0, rule_namespace: str = 'rule_labels', database_file='/atis/atis.db') → None[source]
Parameters: vocab : Vocabulary utterance_embedder : TextFieldEmbedder Embedder for utterances. action_embedding_dim : int Dimension to use for action embeddings. encoder : Seq2SeqEncoder The encoder to use for the input utterance. decoder_beam_search : BeamSearch Beam search used to retrieve best sequences after training. max_decoding_steps : int When we’re decoding with a beam search, what’s the maximum number of steps we should take? This only applies at evaluation time, not during training. input_attention: Attention We compute an attention over the input utterance at each step of the decoder, using the decoder hidden state as the query. Passed to the transition function. add_action_bias : bool, optional (default=True) If True, we will learn a bias weight for each action that gets used when predicting that action, in addition to its embedding. dropout : float, optional (default=0) If greater than 0, we will apply dropout with this probability after all encoders (pytorch LSTMs do not apply dropout to their last layer). rule_namespace : str, optional (default=rule_labels) The vocabulary namespace to use for production rules. The default corresponds to the default used in the dataset reader, so you likely don’t need to modify this. database_file: str, optional (default=/atis/atis.db) The path of the SQLite database when evaluating SQL queries. SQLite is disk based, so we need the file location to connect to it.
decode(output_dict: typing.Dict[str, torch.Tensor]) → typing.Dict[str, torch.Tensor][source]
This method overrides Model.decode, which gets called after Model.forward, at test time, to finalize predictions. This is (confusingly) a separate notion from the “decoder” in “encoder/decoder”, where that decoder logic lives in TransitionFunction.
This method trims the output predictions to the first end symbol, replaces indices with corresponding tokens, and adds a field called predicted_actions to the output_dict.
forward(utterance: typing.Dict[str, torch.LongTensor], world: typing.List[allennlp.semparse.worlds.atis_world.AtisWorld], actions: typing.List[typing.List[allennlp.data.fields.production_rule_field.ProductionRule]], linking_scores: torch.Tensor, target_action_sequence: torch.LongTensor = None, sql_queries: typing.List[typing.List[str]] = None) → typing.Dict[str, torch.Tensor][source]
We set up the initial state for the decoder, and pass that state off to either a DecoderTrainer, if we’re training, or a BeamSearch for inference, if we’re not.
Parameters: utterance : Dict[str, torch.LongTensor] The output of TextField.as_array() applied on the utterance TextField. This will be passed through a TextFieldEmbedder and then through an encoder. world : List[AtisWorld] We use a MetadataField to get the World for each input instance. Because of how MetadataField works, this gets passed to us as a List[AtisWorld], actions : List[List[ProductionRule]] A list of all possible actions for each World in the batch, indexed into a ProductionRule using a ProductionRuleField. We will embed all of these and use the embeddings to determine which action to take at each timestep in the decoder. linking_scores: torch.Tensor A matrix of the linking the utterance tokens and the entities. This is a binary matrix that is deterministically generated where each entry indicates whether a token generated an entity. This tensor has shape (batch_size, num_entities, num_utterance_tokens). target_action_sequence : torch.Tensor, optional (default=None) The action sequence for the correct action sequence, where each action is an index into the list of possible actions. This tensor has shape (batch_size, sequence_length, 1). We remove the trailing dimension. sql_queries : List[List[str]], optional (default=None) A list of the SQL queries that are given during training or validation.
get_metrics(reset: bool = False) → typing.Dict[str, float][source]
We track four metrics here:
1. exact_match, which is the percentage of the time that our best output action sequence matches the SQL query exactly.
2. denotation_acc, which is the percentage of examples where we get the correct denotation. This is the typical “accuracy” metric, and it is what you should usually report in an experimental result. You need to be careful, though, that you’re computing this on the full data, and not just the subset that can be parsed. (make sure you pass “keep_if_unparseable=True” to the dataset reader, which we do for validation data, but not training data).
3. valid_sql_query, which is the percentage of time that decoding actually produces a valid SQL query. We might not produce a valid SQL query if the decoder gets into a repetitive loop, or we’re trying to produce a super long SQL query and run out of time steps, or something.
4. action_similarity, which is how similar the action sequence predicted is to the actual action sequence. This is basically a soft measure of exact_match.
static is_nonterminal(token: str)[source] | 2019-02-18 22:37:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22943805158138275, "perplexity": 4661.801923913739}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247488490.40/warc/CC-MAIN-20190218220415-20190219002415-00433.warc.gz"} |
http://math.stackexchange.com/questions/777775/combinatorics-counting-in-two-ways | # Combinatorics - Counting in two ways? [duplicate]
This question already has an answer here:
What is the idea of proving a binomial identity by counting in two ways?
Could you please illustrate this with this example? Thank you very much.
$$\binom{2n}{n}= \sum_{k=0}^n {\binom nk}^2$$
(original screenshot)
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## marked as duplicate by Marc van Leeuwen, Sami Ben Romdhane, Claude Leibovici, Davide Giraudo, Mark BennetMay 2 '14 at 7:11
This becomes a special case of the Vandermonde identity if you replace one of the factors $\binom nk$ by $\binom n{n-k}$. Also, it is a duplicate of at least this question, if not of many others. – Marc van Leeuwen May 2 '14 at 5:28
The idea behind the technique is to perform the same task using procedures that are distinct. From each distinct procedure, we obtain a combinatorial expression for the number of ways that the task can be performed. Since the same task was performed, the combinatorial expressions are the same.
The most simple example is probably the procedure that results in selecting $k$ objects from a collection of $n$ distinct objects. We might perform this task by first placing the $n$ objects on a table, marking $k$ of them as the objects to select, and then move the $k$ marked objects to a separate pile. There are $\binom{n}{k}$ ways to do this. Alternatively, we might look at our pile, and mark $n-k$ objects to not select, and then move the remaining $k$ unmarked objects to a separate pile. There are $\binom{n}{n-k}$ ways to mark the $n-k$ objects to leave out of the collection. Since each procedure results in the same thing -- a collection of $k$ objects-- the number of ways to perform each procedure must be the same. Therefore, $$\binom{n}{k}=\binom{n}{n-k}$$
In your given example can think as follows. The left hand side $$\binom{2n}{n}$$ counts the number of ways to select $n$ objects from a collection of $2n$ objects. We could perform this task as above: place all objects on a table and mark $n$ of them to keep which gives the above expression. Alternatively, we could first split the pile of $2n$ objects into two separate piles of $n$ objects. To get the desired collection of $n$ objects, we could select $k$ from the first pile ($\binom{n}{k}$ ways), and the remaining $n-k$ from the second ($\binom{n}{n-k}$ ways). Thus, the number of ways to do so is: $$\sum_{k=0}^n\binom{n}{k}\binom{n}{n-k}$$ Therefore, $$\binom{2n}{n}=\sum_{k=0}^n\binom{n}{k}\binom{n}{n-k}$$ To get your desired result, simply use the identity established earlier to find that: \begin{align*} \binom{2n}{n}&=\sum_{k=0}^n\binom{n}{k}\binom{n}{n-k}\\ &=\sum_{k=0}^n\binom{n}{k}\binom{n}{k}\\ &=\sum_{k=0}^n\binom{n}{k}^2 \end{align*}
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Puntuation never goes after TeX's double dollar sign! – Mariano Suárez-Alvarez May 2 '14 at 2:22
@Mariano Gracias, was just typing along without thinking. – Scott H. May 2 '14 at 2:24
[EDIT]
The idea behind double counting arguments is to provide verbal proofs of combinatorial identities that give an intuitive understanding of why they should be so. By simply showing that two expressions measure different ways to count the same thing it becomes clear to even a novice mathematician that they must be equivalent.
On the left hand side: $\binom{2n}{n}$ is the count of ways to divide a set of distinct objects into two equal sized subsets.
On the right hand side: $\sum\limits_{k=0}^{n} \binom{n}{k}^2$ is the count of ways to redistribute objects between two equal sized sets as follows:
$\binom{n}{k}$ counts the ways to select $k$ objects form a set of $n$ distinct objects. Thus $\binom{n}{k}^2$ counts the ways to do this for two sets; which makes it the count of ways to transfer $k$ object from one set to the other in exchange for $k$ objects originally in the second set. The summation is thus the count of ways to exchange equal numbers of distinct objects between two equal sized sets, for all possible size of the exchanges.
It should be clear that one gets the same result by simply combining the two sets and counting all the ways to split them into equal sized sets.
Hence it follows that: $\dbinom{2n}{n}=\sum\limits_{k=0}^{n} \dbinom{n}{k}^2$
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An alternative proof is the following.
What is the coefficient of $x^n$ in $(1+x)^{2n}$?
Write this as $(1+x)^n(1+x)^n$. What is it now?
Note $$\sum_{k=0}^n\binom nk^2=\sum_{k=0}^n \binom nk\binom n{k -n}=\sum_{i+j=n}\binom ni\binom nj$$
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Let $a$ = number of ways of choosing $n$ animals from $n$ cats and $n$ dogs is $\binom{2n}{n}$.
Let $c(k)$ = number of ways of choosing $k$ cats from $n$ cats = $\binom{n}{k}$
Let $d(k)$ = number of ways of choosing $n-k$ dogs from $n$ dogs = $\binom{n}{n-k}$ = $\binom{n}{k}$
Thus, $e(k)$ = number ways of choosing $k$ cats and $n-k$ dogs = $c(k)*d(k) = {\binom nk}^2$
Obviously $a$ = Number of ways of choosing $k$ cats and $n-k$ dogs for all $k \in [0, n]$
Therefore, $a = \sum_{k=0}^n e(k) = {\binom nk}^2$. QED.
- | 2016-04-30 05:34:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7471655011177063, "perplexity": 240.79958741454288}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111612.51/warc/CC-MAIN-20160428161511-00026-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://otfried.org/courses/cs500/hw5.html | # Homework 5
#### Problem 1.
Let $$G=(V,E)$$ be a directed graph, let $$u,v,w$$ be three distinct nodes in $$G$$, and let $$k > 1$$. Give a three-line argument to prove the following claim: If there are $$k$$ edge-disjoint paths from $$u$$ to $$v$$, and there are $$k$$ edge-disjoint paths from $$v$$ to $$w$$, then there are $$k$$ edge-disjoint paths from $$u$$ to $$w$$.
#### Problem 2.
Give the dual of the following linear program.
$\begin{array}{rrl} \textrm{maximize} & 4x_1+x_2 +5x_3+3x_4& \\ \textrm{such that} & x_1-x_2-x_3+3x_4 & \le 1\\ & 5x_1+x_2+3x_3+8x_4 & \le 55\\ & -x_1+2x_2+3x_3-5x_4 & \le 3\\ & x_1,x_2,x_3x_4 & \ge 0 \end{array}$
#### Problem 3.
A restaurant has 880 abalones and 720 oysters. They offer two dishes: A dish for 20,000 Won (4 abalones, 1 oyster), and a dish for 15,000 Won (2 abalones, 3 oysters).
Formulate a linear program to determine the optimal number of the two dishes to be sold.
Give the dual of the linear program. Can you give an interpretation of the variables, constraints, and objective function?
#### Problem 4.
You are given the universe $$U = \{1, 2, 3, \ldots, n\}$$, and a family $$\mathcal{F} = \{B_1, B_2, \ldots, B_m\}$$ of $$m$$ subsets of $$U$$. Each element $$i \in U$$ has a weight $$w_i \geq 0$$. Your goal is to find a subset $$A \subseteq U$$ that contains an element of every $$B_j \in \mathcal{F}$$, and you want the total weight $$w(A) = \sum_{i \in A}w_i$$ to be as small as possible.
Let $$b = \max_{j} |B_{j}|$$ denote the maximum size (number of elements) of any of the sets $$B_1, B_2, \ldots, B_{m}$$. Give a polynomial time approximation algorithm for this problem that finds a set $$A$$ whose weight is at most $$b$$ times the minimum possible weight (in other words, your algorithm must be a $$b$$-approximation).
#### Problem 5.
Your company has a supplier providing aluminium bars of length 3.00m. You use those bars to provide pieces for window elements. Today, you have received an order for 300 pieces of length 0.50m, 130 pieces of length 1.00m, and 100 pieces of length 1.20m. How many bars of 3.00m length do you need to order from your supplier to make these pieces? (Note that any leftover aluminium from cutting a bar cannot be reused, so we really want to minimize the number of bars.)
Model this problem as a linear program. Clearly explain the meaning of your variables and constraints.
(You will have to hope the solution to the LP has integer values—ignore this issue.) | 2021-04-15 17:20:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5802164077758789, "perplexity": 465.0227208208443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038087714.38/warc/CC-MAIN-20210415160727-20210415190727-00262.warc.gz"} |
https://www.overleaf.com/latex/examples/using-luatex-to-run-software-installed-on-overleafs-servers/nnvfvnfhzwsy | # Using LuaTeX to run software installed on Overleaf’s servers
Author
Graham Douglas
View Count
1559
AbstractThis example shows how to use LuaTeX to run software tools and utilities installed on Overleaf’s servers, including how to capture, and then typeset, text that would normally be displayed in a terminal window. | 2019-02-22 10:30:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3905271589756012, "perplexity": 7225.858144900998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247515149.92/warc/CC-MAIN-20190222094419-20190222120419-00338.warc.gz"} |
https://www.physicsforums.com/threads/second-order-homogeneous-ode.639177/ | # Second order homogeneous ODE
1. Sep 26, 2012
### carlosbgois
1. The problem statement, all variables and given/known data
Find the values of $α$ for which all the solutions of $y''-(2α-1)y'+α(α-1)y=0$ (a) tend to zero and (b) are ilimited, when $t->∞$.
2. Relevant equations
$y''-(2α-1)y'+α(α-1)y=0 => (t)=Ae^{αt}+Be^{(α-1)t}$
3. The attempt at a solution
I found that the general solution to the problem is $y(t)=Ae^{αt}+Be^{(α-1)t}$, which I believe is correct. Then I said that (a) is verified for $t<1/2$ and (b) for $t>=1/2$, but the book's answer is (a) $t<0$ and (b) $t>1$.
What am I missing?
Thanks
2. Sep 26, 2012
### Simon Bridge
As you wrote it, the question asks for values of $\alpha$, not $t$.
So it looks like your (and the book's) answer is for a different question.
$y(t)\rightarrow 0$ for $t\rightarrow \infty$ when $\alpha < 0$ which makes y(t) a sum of decaying exponentials.
3. Sep 27, 2012
### carlosbgois
My mistake, I meant α where I said t. I understand what you said, but then what happens for 0<α<1 ?
Thanks
4. Sep 27, 2012
### Simon Bridge
Well then put that into y(t) ... you have two exponentials added together and $\alpha$ appears in the power.
What does "ilimited" mean? | 2017-12-12 15:16:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8373215198516846, "perplexity": 621.4523375725182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948517181.32/warc/CC-MAIN-20171212134318-20171212154318-00342.warc.gz"} |
https://testthat.r-lib.org/reference/expect_cpp_tests_pass.html | Test compiled code in the package package. A call to this function will automatically be generated for you in tests/testthat/test-cpp.R after calling use_catch(); you should not need to manually call this expectation yourself.
expect_cpp_tests_pass(package)
## Arguments
package The name of the package to test. | 2019-11-18 13:35:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9017978310585022, "perplexity": 1757.0725005666977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669795.59/warc/CC-MAIN-20191118131311-20191118155311-00304.warc.gz"} |
http://simbad.cds.unistra.fr/simbad/sim-ref?bibcode=2007MNRAS.381.1569H | other querymodes : Identifierquery Coordinatequery Criteriaquery Referencequery Basicquery Scriptsubmission TAP Outputoptions Help
2007MNRAS.381.1569H - Mon. Not. R. Astron. Soc., 381, 1569-1574 (2007/November-2)
Establishing the nature of companion candidates to X-ray-emitting late B-type stars.
HUBRIG S., MARCO O., STELZER B., SCHOLLER M. and HUELAMO N.
Abstract (from CDS):
The most favoured interpretation for the detection of X-ray emission from late B-type stars is that these stars have a yet undiscovered late-type companion (or an unbound nearby late-type star) that produces the X-rays. Several faint infrared objects at (sub)arcsecond separation from B-type stars have been uncovered in our earlier adaptive optics imaging observations, and some of them have been followed up with the high spatial resolution of the Chandra X-ray observatory, pinpointing the X-ray emitter. However, firm conclusions on their nature require a search for spectroscopic signatures of youth. Here we report on our recent ISAAC observations carried out in low-resolution spectroscopic mode. Equivalent widths have been used to obtain information on spectral types of the companions. All eight X-ray-emitting systems with late B-type primaries studied contain dwarf-like companions with spectral types later than A7. The only system in the sample where the companion turns out to be of early spectral type is not an X-ray source. These results are consistent with the assumption that the observed X-ray emission from late B-type stars is produced by an active pre-main-sequence companion star. | 2023-01-31 14:44:43 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8112358450889587, "perplexity": 4515.384038226863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499871.68/warc/CC-MAIN-20230131122916-20230131152916-00145.warc.gz"} |
http://blog.inf.ed.ac.uk/sxw/category/uncategorized/ | # Simon's Musings
## January 27, 2011
Filed under: Uncategorized — sxw @ 10:55 am
OpenAFS’s gerrit generally keeps pace with the latest releases from upstream, but I’ve been much more tardy about upgrading the Informatics not-a-service instance.
Last night, I took the plunge, and moved this from 2.1.1.1 to 2.1.6.1. This brings with it lots of shiny new features, in particular for access control. Full details are in the release notes.
## May 16, 2010
### Spotting trivial rebases in gerrit
Filed under: Uncategorized — sxw @ 4:51 pm
One of the pains with gerrit is that of rebasing. Once a change has gone through its review process, it is common to find that the tree has moved on since it was written, and the change requires updating before it can be applied. On many occasions, all that is required is a trivial rebase – just moving the patch to the latest version of the tree, without actually changing any of the code. However, whenever a new patch is uploaded, gerrit forgets the existing review and verification scores. In many cases, this is a good thing – you don’t want your “This looks great, submit it!” review attached to a piece of code that is substantially different from what you reviewed. With a trivial rebase, things are different – the code hasn’t changed, it probably doesn’t need re-reviewing, and dropping the existing review and verification information just means that information is lost when the commit is eventually applied.
So, Nasser Grainawi wrote a trivial rebase hook, which uses gerrit’s hook and command line features to identify these trivial rebases and reapply all of the review scores when it identifies one. I’ve enabled this hook for OpenAFS’s gerrit instance. It turns out that doing so isn’t quite as simple as it might appear, so I thought I’d document the hoops I jumped through.
### 1: Create an SSH key
The script will run as the gerrit user, so we need to create a ssh public key pair for it to use as that user. Just run ssh-keygen as normal, and put the key pair somewhere sensible.
### 2: Create a User
A role user needs to be created for the script to use when interrogating gerrit. You need this, despite the fact that the script also uses the super-user functionality obtained from the server’s private key when reapplying existing scores. The development version of gerrit has a nice simple command to create new role accounts, but we’re runnign the stable series, which hasn’t yet got this feature. So, we get to play directly with the SQL database.
I did::
insert into accounts (registered_on, account_id, full_name, preferred_email)
VALUES (now(), nextval('account_id'), 'Automated Hook Processor',
'<email>');
insert into account_ssh_keys (valid, account_id, seq, ssh_public_key)
VALUES ('Y', currval('account_id'), 1, '<ssh public key>');
insert into account_external_ids (account_id, external_id)
VALUES (currval('account_id'), 'username:hooks');
… substituting <email> with an email address that the hook processor can be attributed with, and <ssh public key> with the public portion of the key we generated earlier.
### 3: Configure the SSH client
Let the ssh client know how to talk to gerrit. The script uses ‘localhost’ as the hostname to talk to, so we need to do something like:
Host localhost
IdentityFile ~/.ssh/id_rsa
User hooks
### 4: Connect to Gerrit
In order to populate the known hosts file, we need to connect to gerrit, and accept the host key. Something like
ssh -p 29418 localhost
will accomplish this.
### 5: Make the new user an admistrator
Go into the gerrit front end, and make the user that we’ve just created an administrator
### 6: Test the rebase script
Testing the rebase script independently from gerrit will let you know if everything above has worked. You can run it from the command line, as the gerrit user:
GIT_DIR=<gitRepository> \
./trivial_rebase.py --change <changeId> \
--project <projectName> \
--commit <commitSHA1> \
--patchset <patchSetNumber> \
--private-key-path=$site_path/etc/ssh_host_dsa_key • gitRepository is the path to the bare git repository that gerrit saves its changes into. • changeId is the full changeID (something like I97e07a6e730df8ac480d295b4cf30b0695ace511) • commitSHA1 is the git SHA1 for the patchset that’s just bene uploaded • patchset is gerrit’s patchset number The best way to test this is to pick a change that has been trivially rebased, but which isn’t marked as such. Go to its page in the web interface, and get the changeID, the SHA1 of the newest patchest, and its number, from that page, and plug them in to this command. All being well, the change will be marked as a trivial rebase. If that doesn’t happen, then the logs in$site_path/logs/error_log will give an idea of what’s going on.
### 7: Write a wrapper
The trivial rebase script can’t be run directly from a gerrit hook, so we need to provide a wrapper script to fill in some of the gaps. The script OpenAFS is currently using is:
#!/usr/bin/perl
use strict;
use warnings;
use Getopt::Long;
my $gerritRoot="/srv/gerrit/cfg"; my$hookDir="/srv/gerrit/gerrit-hooks";
my $change; my$project;
my $branch; my$commit;
my $patchset; my$result = GetOptions("change=s" => \$change, "project=s" => \$project,
"branch=s" => \$branch, "commit=s" => \$commit,
"patchset=s" => \$patchset); system($hookDir."/trivial_rebase.py",
"--change", $change, "--project",$project,
"--commit", $commit, "--patchset",$patchset,
"--private-key-path", $gerritRoot."/etc/ssh_host_dsa_key"); This script needs to go into the hooks directory below your gerrit site path as a file called patchset-created. In order for gerrit to run the hook, it needs to be executable. ## April 8, 2010 ### FOSDEM 2010 Filed under: Uncategorized — sxw @ 4:12 pm Tags: From the better-late-than-never department I spent the first weekend of February at FOSDEM 2010, a pretty much unique conference of free software developers held yearly in Brussels. Whilst I emailed these notes around internally upon my return, it’s taken a while to get around to putting them up on my blog. In that time, though, FOSDEM have published the videos they made of most of these talks. Links to those videos are included below. FOSDEM is a completely free event, whose scale has to be seen to be believed – they have over 300 talks spread across the 2 days, and initial estimates suggested that there were approximately 10,000 delegates on Saturday (the wireless network had over 4000 active leases at one point). I split my time between attending main track sessions, listening to development room talks, and volunteering with conference support ### Bullet points for the pointy haired … • Mozilla don’t have any real interest in the enterprise space • Facebook are doing some amazing things, and actually talking about it. • Usable open source smartcard stacks are finally in today’s distributions. • Hadoop, and the NoSQL movement in general, are gaining a large amount of mindshare. If we’re not getting requests for Hadoop clusters now, it’s probably just a matter of time. • The pound is taking a serious beating. ### In the corridors I spoke to Chris Blizzard and Gervase Markham from Mozilla about our experiences with the move from Firefox 2->3. They think that the latest development versions will offer some improvement to the number of small disk writes, but admitted that the level of disk activity had never been a concern in benchmarking the application. Gervase had some helpful suggestions about things we could do to assist with this in the future. There’s a growing interest in Europe around smartcards. This is primarily being driven by the fact that a number of European governments are adding smartcards to their existing national ID cards, and people are wanting to exploit the authentication properties of these. One group were demonstrating an out-of-the box Ubuntu system using a smartcard for authentication, and document signing. Fedora apparently offers the same user experience. There are also powerful Open Source CA products becoming available, which would be very useful if we decide to go down the smartcard route at any point. ### Talks Brooks Davis gave an interesting talk about moving a large company (Aerospace, in the US) towards open source development models. He had some interesting observations on the uptake of new tools in big organisations, and on the reluctance to share code between development teams. Some of his horror stories were staggering – teams whose idea of revision control is hosting all of their source on a shared filesystem, and using a physical whiteboard to indicate who was modifying which files. Needless to say, code loss was common. Aerospace are rolling out an internal system based around Trac to encourage the use of revision control and other good sofware development practices company wide. Next, Richard Clayton gave a fascinating tour of ‘Evil on the Internet’. This was an eye-opening overview of the way in which criminals steal money over the internet, exploring the whole ecosystem, from those who phish for credentials, the mules they recruit to launder the money, and the many, many fake websites that they have to con you into participating. What was alarming was the realism of many of these sites – I’d become used to identifying phishing scams through poorly formed emails, and the plausibility of much of what he demonstrated was alarming. I spent much of the afternoon in the XMPP devroom. They’re doing some very interesting work both on integrating XMPP with web applications, and in terms of moving voice and video forwards with Jingle. I popped out of that devroom to listen to a talk by Mitchell Bakerabout Mozilla’s mission, and their view of how they can contribute to ensuring an open internet. Inspiring stuff, but sadly she did confirm that Mozilla as a whole don’t have an interest in the enterprise space. Due to lack of space (a common theme during the weekend, sadly) I couldn’t get into the Spacewalk talk. Spacewalk (http://www.redhat.com/spacewalk/ ) is the open source version of RedHat Network, and is designed to manage software updates across an enterprise wide deployment of Linux. There was a lot of buzz about this at FOSDEM, and it looks like something we should definitely investigate. On Sunday, I spent a some time in the Mozilla devroom, in particular speaking to Ludo about current, and forthcoming, changes to Thunderbird. We discussed the current state of both GSSAPI and LDAP support (my patch enabling GSSAPI for LDAP in Thunderbird was part of the 3.0 release). I was hoping to listen to some of the NoSQL talks too, but sadly that room was overflowing everytime I tried. I spoke with Peter St Andre (XMPP Foundation, now with Cisco) about improving XMPP’s Kerberos support – in particular how we can push support for domain based names into the SASL software stack. I will be following that up with him and Alexey Melnikov after the conference. I also met with some other OpenAFS developers, for a brief chat about the state of the tree, and the move towards the 1.6 release. On Sunday afternoon, I helped moderate the talks in the Janson auditorium, in the scalability conference track. Isabel Jost started with a fascinating talk about Apache Hadoop, which offers distributed petabyte scale data processing, using tools modelled on Google’s MapReduce paradigm. HDFS looks pretty much like an implementation of what’s publicly known about Google’s GFS, and Hadoop layers Map/Reduce on top of that. The talk provided a great overview, firstly of map reduce and its power, secondly of the flexibility of the Hadoop implementation, and finally an idea of the huge degrees to which it can scale. The next talk on this track was from Facebook. They started with some fairly blinding statistics – 8 billion minutes every day are spent on Facebook and 2.5 billion photos are uploaded each month. They provided an overview of their whole infrastructure, highlighting the various projects that make it tick. Their development language is PHP but due to issues with its speed, and memory use, they’ve developed hipohp, a static analyzer and translator which converts PHP into optimised C++. For logging, they used to use syslog, but their log volumes (~25 terabytes per day) melted down all of the syslog servers that they tried, so they moved towards ‘Scribe’ – which is now used by both themselves and Twitter, and offers massively scalable log storage. To process these logs, and for other data analysis tasks, Facebook are big Hadoop users. They’ve built Hive, which puts an SQL like layer on top of Hadoop’s syntax, with the aim of encouraging ease of use, and internal adoption. Their Hadoop cluster currently uses more than 80,000 compute hours per day. Facebook’s data store totals 160 billion photos and serves 1.2 million of them every second. Nobody’s NFS scales to this kind of data load – I/O bandwidth, rather than storage density, ends up being the limiting factor. They’ve built a new storage system called Haystack, which removes a lot of metadata information, and transfers data serving from 10 disk seeks per file served into a single operation. memcache is crucial to facebook’s interactive performance, but they have a bit of a love/hate relationship with it. They’ve done a fair bit of work on extending it, by adding features 64bit and multithreading support. Their permanent data storage is MySQL, which they like very much – simple, fast and very reliable. They keep their usage of it simple – they don’t do joins at the database layer, but combine datasets together in PHP. One very interesting observation from Facebook was that most of their interesting projects happen as “hack projects” where a very small group of engineers work intensively on an exciting idea – Haystack, for example, was built by 3 people. Finally, in the scalability track, there was a talk about Status.Net and identi.ca. Choice quote: “When web people talk about scalability – what they really mean is will it keep working” Status.Net is a twitter-like service which is available both as a hosted system (identi.ca) and as a locally installable server, designed for organisations who want to host their own. The talk dealt more with the overall architecture of Status.Net (formerly laconica), rather than detailing its scalability, but provided an interesting example of how various open source components could be stitched together to produce a compelling product, and a rallying cry for the importance of both libre web services, and the ownership of your own web content. To round off a very long couple of days, Greg K-H gave a whistle stop tour of writing your first patch for the Linux kernel. This was a very well delivered overview of a lot of complex topics, and was aimed at inspiring members of the audience to contribute, rather than being an in depth description of the kernel. It was a very well chosen end to an inspiring weekend ## February 17, 2010 ### Git Tip 2: Splitting up existing changes Filed under: Uncategorized — sxw @ 5:31 pm Tags: Today’s tip is pretty clearly spelled out in the manpage for git-rebase. But I find it so useful, I thought I would highlight it here. From time to time, you end up with a patchset in your history that contains more than it should. It may be that it contains files (or chunks) that have been added incorrectly, or that a review comment has suggested would be better split into multiple changes. Git’s all purpose rewriter of history - git rebase - can help with doing this. git rebase is a fantastic tool which can let you completely change the history of your project at the drop of a hat. I’ll be writing more about it in tips to come. As with all powerful tools, you need to use its power wisely. In effect, rebasing doesn’t rewrite your history, but creates an entirely new timeline, starting at some common point in the past. If you are working on a personal repository, then this isn’t an issue. If you are sharing your repository, you should think very carefully about rebasing it as your collaborators will have based their work on the old tip of your tree. When you rebase, you create a new branch, with a new tip, of which your collaborators will be completely unaware. How to split a change is described in detail in the git rebase manpage, so we’ll just recap it here. • Find the SHA1 hash of the change you wish to split – this is probably easiest done by running git log –oneline • Start up an interactive rebase, with the start point being the parent of the commit you wish to split. The git notation HASH^ - for exampledeadbeef^ will give you the parent of a commit. So, run: git rebase -i HASH^ • In the editor window that appears replace the word pick that appears beside the change you are picking with edit. This instructs git to stop the rebase operation at this point, and to allow you to modify this change. • Save the file, and exit your editor. Git will now start the rebase operation. • When it reaches the change you are modifying, git will pause the rebase, and return control. You can now modify this change – make a note of the SHA1 id it has stopped at, as this will come in handy later! • git reset HEAD^ will reset the index to the state of the parent commit. This gives you a working tree that contains the contents of the change you are splitting. • Use git add, and git commit to create as many commits as you wish from this tree. If you want your original commit message, then using git commit -C HASH will commit the current index, using the same message as that in HASH. • When you’re done, and have committed all of the fragments, use git rebase –continue to resume the rebasing process Providing all goes well, you’ll end up with a modified history, and a change that has been split into multiple parts. You’ll note that every change after the split now has a new SHA1 – to git, these are completely new changes, as they have a different history, and this is the whole reason why rebases can be dangerous. ## February 16, 2010 ### Git tip of the day #1 Filed under: Uncategorized — sxw @ 9:24 am Tags: This is the first of an occasional series of helpful hints and tips for the git revision control system – look for articles tagged with ‘git’ ## Cleaning your local commits Many of the projects I submit to are picky about both trailing, and embedded, whitespace. git show and git diff will let you see these and if you are pulling in a patchset from elsewhere, git apply --whitespace=fix will tidy them up for you. But, I’d always thought that doing it in your own tree was harder. However … git rebase -f --whitespace=fix origin/master … will rebase your current working tree (assuming that the branch point was origin/master), and also clean up any embeded whitespace problems along the way. Obviously this has all of the caveats that rebasing carries – you don’t want to do it on a tree that others are working from. But as a way of cleaning up local changes before pushing them into gerrit, it’s remarkably useful. ## January 24, 2010 ### GSSAPI Key Exchange for OpenSSH 5.3p1 Filed under: Uncategorized — sxw @ 12:00 pm Once again, far far later than I would have liked, I’ve produced a set of updated patches for OpenSSH 5.3p1. Compared to previous releases this one is pretty simple – a resolved merge conflict, and a few one line patches. However, I really need to get quicker at doing these – it’s 4 months since 5.3p1 appeared, and 5.4 will be just around the corner. The announcement email was: From the better-late-than-never-department, I’m pleased to announce the availability of my GSSAPI Key Exchange patches for OpenSSH 5.3p1. This is a pretty minor maintenance release – it contains a couple of fixes to take into account changes to the underlying OpenSSH code, and a compilation fix for when GSSAPI isn’t required. Thanks to Colin Wilson and Jim Basney for their bug reports. I’d like to thank the distributors who’ve been patiently waiting for me to get this done – sorry once again for the delay. Why? —- Whilst OpenSSH contains support for GSSAPI user authentication, this still relies upon SSH host keys to authenticate the server to the user. For sites with a deployed Kerberos infrastructure this adds an additional, unnecessary, key management burden. GSSAPI key exchange allows the use of security mechanisms such as Kerberos to authenticate the server to the user, removing the need for trusted ssh host keys, and allowing the use of a single security architecture. How? —- This patch adds support for the RFC4462 GSSAPI key exchange mechanisms to OpenSSH, along with adding some additional, generic, GSSAPI features. It implements: *) gss-group1-sha1-*, gss-group14-sha1-* and gss-gex-sha1-* key exchange mechanisms. (#1242) *) Support for the null host key type (#1242) *) Support for CCAPI credentials caches on Mac OS X (#1245) *) Support for better error handling when an authentication exchange fails due to server misconfiguration (#1244) *) Support for GSSAPI connections to hosts behind a round-robin load balancer (#1008) *) Support for GSSAPI connections to multi-homed hosts, where each interface has a unique name (#928) *) Support for cascading credentials renewal *) Support for the GSSAPIClientIdentity option, to allow the user to select which client identity to use when authenticating to a server. (bugzilla.mindrot.org bug numbers are in brackets) Where? —— As usual, the code is available from http://www.sxw.org.uk/computing/patches/openssh.html Two patches are available, one containing cascading credentials support, and one without. In addition, the quilt patch series that makes up this release is also provided, for those who wish to pick and choose! Cheers, Simon. ## January 17, 2010 ### Debugging a Mac OS X Kernel Panic Filed under: Uncategorized — sxw @ 12:40 pm Tags: , Adam Megacz posted to the openafs-info mailing list a kernel panic that he encountered with the 1.4.12 release candidate. Derrick and I debugged this over Jabber (with Derrick doing the kernel disassembly). I thought it might be useful to share the methods we used, and to record the command invocations for future googling. ## The Problem A Mac OS X kernel panic. Adam first posted the panic itself, then followed up with the results of running OpenAFS’s decode-panic tool. decode-panic basically drives gdb to turn an panic log with addresses into a readable backtrace with symbols – it needs to be run with the same kernel, and architecture, that created the panic in the first place. ## The Analysis The output from decode-panic looks something like: Panic Date: Interval Since Last Panic Report: 472905 sec Kernel Version: Darwin Kernel Version 10.2.0: Tue Nov 3 10:37:10 PST 2009; root:xnu-1486.2.11~1/RELEASE_I386 OpenAFS Version: org.openafs.filesystems.afs(1.4.12fc1) ============= add symbol table from file "/tmp/afsdebugLAjeJl/org.openafs.filesystems.afs.sym"? 0x21b2bd <panic+445>: mov 0x8011d0,%eax 0x2a7ac2 <kernel_trap+1530>: jmp 0x2a7ade <kernel_trap+1558> 0x29d968 <lo_alltraps+712>: mov %edi,%esp 0x4607e500 <afs_GetDCache+7832>: mov 0x64(%edx),%ebx 0x46078a18 <BPrefetch+144>: mov %eax,-0x3c(%ebp) 0x4607928d <afs_BackgroundDaemon+573>: jmp 0x460792cb <afs_BackgroundDaemon+635> 0x460e76a7 <afsd_thread+719>: call 0x2a013e <current_thread> 0x29d68c <call_continuation+28>: add$0x10,%esp
The line in bold is the point where the panic actually occurred (all of the stuff before it is related to the trap mechanism). The lines afterwards tell us how we arrived in afs_GetDCache – for this particular problem, we’re not concerned about the callers, so we can ignore these.
If we look at the raw panic log, we see:
CR0: 0x8001003b, CR2: 0x00000064, CR3: 0x00101000, CR4: 0x000006e0
EAX: 0x00100000, EBX: 0x00000000, ECX: 0x460870e2, EDX: 0x00000000
CR2: 0x00000064, EBP: 0x34cabf1c, ESI: 0x0bff4004, EDI: 0xffffffff
EFL: 0x00010297, EIP: 0x4607e500, CS: 0x00000004, DS: 0x0000000c
So, at the time of the panic, EDX was probably NULL. This implies that the bug is a NULL pointer dereference of some kind, and that we’re looking up something that’s 0×64 bytes into the structure that’s pointed at by EDX.
If we had a kernel with debugging symbols, then we could look directly at the C source which corresponds to afs_GetDCache+7832. Unfortunately, we didn’t have one to hand, so Derrick improvised. By disassembling the current kernel module, we can see exactly what code occurs at that location…
gdb --arch=i386 /var/db/openafs/etc/afs.kext/Contents/MacOS/afs
(gdb) disassemble afs_GetDCache
[ ... ]
0x00011500 <afs_GetDCache+7832>: mov 0x64(%edx),%ebx
0x00011503 <afs_GetDCache+7835>: call 0x0 <afs_atomlist_create>
0x00011508 <afs_GetDCache+7840>: cmp 0x0,%eax
0x0001150e <afs_GetDCache+7846>: je 0x1152c <afs_GetDCache+7876>
0x00011510 <afs_GetDCache+7848>: movl $0x8aa,0x8(%esp) 0x00011518 <afs_GetDCache+7856>: movl$0x8e438,0x4(%esp)
0x00011520 <afs_GetDCache+7864>: movl $0x8e080,(%esp) 0x00011527 <afs_GetDCache+7871>: call 0x642f8 <osi_AssertFailK> 0x0001152c <afs_GetDCache+7876>: movl$0x0,0x0
0x00011536 <afs_GetDCache+7886>: mov 0x0,%eax
0x0001153b <afs_GetDCache+7891>: mov %eax,(%esp)
0x0001153e <afs_GetDCache+7894>: call 0x0 <afs_atomlist_create>
0x00011543 <afs_GetDCache+7899>: mov %ebx,0x4(%esp)
0x00011547 <afs_GetDCache+7903>: mov -0xb8(%ebp),%ecx
0x0001154d <afs_GetDCache+7909>: mov %ecx,(%esp)
0x00011550 <afs_GetDCache+7912>: call 0x4df78 <rx_EndCall>
[ ... ]
(The call 0×0 is because we’re debugging a module that isn’t loaded. You can get a dump that fills in the blanks for these by doing the following. When kextutil asks for the load address, give it the one that the panic log said the OpenAFS module was loaded at, in this case 0x4606c000:
mkdir /tmp/symbols
kextutil -n -s /tmp/symbols /var/db/openafs/etc/afs.kext
cp -R /var/db/openafs/etc/afs.kext /tmp/symbols
gdb –arch=i386 /mach_kernel
(gdb) disassemble afs_GetDCache
)
We’re relatively lucky here, as near the panic location is a call to rx_EndCall(), the 2nd of 3 that occur in that GetDCache. Lining this up with the corresponding source code, the last call to rx_EndCall occurs at line 2219 of afs_dcache.c, which looks something like:
if (length > size) {
/* The fileserver told us it is going to send more data
* than we requested. It shouldn't do that, and
* accepting that much data can make us take up more
* cache space than we're supposed to, so error. */
code = rx_Error(tcall);
RX_AFS_GUNLOCK();
code1 = rx_EndCall(tcall, code);
RX_AFS_GLOCK();
tcall = (struct rx_call *)0;
code = EIO;
}
So, assuming that the compiler hasn’t reordered our code, we’ve got rx_Error(), followed by RX_GUNLOCK. It isn’t possible for the compiler to reorder over a lock, so it’s pretty likely that the ordering is as shown. Both of these are macros masquerading as functions. Looking at the simpler, rx_Error(), we have:
#define rx_Error(call) ((call)->error)
Now, if you recall, we panic’d because we were looking up something that has an offset of 0×64 from the base of the structure. We can verify that ‘error’ is stored 0×64 bytes into the rx_call structure by visually examining the structure definition or, if you’ve got a build tree from the same architecture, by running:
gdb src/libafs/MODLOAD/libafs.nonfs.o
(gdb) print &((struct rx_call*)0)->error
\$2 = (afs_int32 *) 0x64
All of this points pretty conclusively at a NULL value for ‘tcall’. Examining the code, there is a situation where that can occur, if we get a bogus length from the network, and as a result double free the rx call. It seems highly likely that this is the bug.
## December 22, 2009
### Building OpenAFS on OpenSolaris
Filed under: Uncategorized — sxw @ 9:21 pm
I had a few spare moments, and an OpenAFS bug report about building on OpenSolaris seemed like it would be relatively easy to fix. So, I decided to bring up an OpenSolaris VM and go about fixing it. This is a record of all of the steps from bare metal to building a version of OpenAFS on Solaris.
The bug report was against the newest OpenSolaris development snapshot (snv_129), so that’s what I aimed to install. Much of the trauma below is because I was aiming for this, rather than the last release, 2009.06
• Install a VM with 2009.06. This is straightforward – boot off the image, and select the install icon on the desktop
• Upgrade the VM to snv_129. I used the instructions at http://pkg.opensolaris.org/dev/en/index.shtml
which boil down to:
pfexec pkg set-publisher -O http://pkg.opensolaris.org/dev opensolaris.org
pfexec pkg image-update
• Marvel as the machine now panics on boot.
• Googling produced http://mail.opensolaris.org/pipermail/opensolaris-announce/2009-December/001343.html (to be fair, it is the release announcement), and the link to http://bugs.opensolaris.org/bugdatabase/view_bug.do?bug_id=6820576. Adding -B disable-pcieb=true to the kernel line in grub is sufficient to get the machine up and running.
• Now, the machine will actually boot, but attempts to log in remotely are doomed with errors like
sshd[8714]: error: session_pty_req: session 0 alloc failed
Fortunately, http://defect.opensolaris.org/bz/show_bug.cgi?id=12380 contains both the cause, and a workaround – once you’ve managed to log in as root on console, just
chmod 666 /dev/ptmx
• Now that we’ve finally got a machine that works, make sure and add -B disable-pcieb=true to the kernel lines of the grub configuration in /rpool/boot/grub/menu.lst otherwise it will just panic when you reboot it.
• OpenSolaris ships without a development environment, so first, lets get one of those
pfexec pkg install ss-dev
• Current OpenAFS sources are in ‘git’, so get the git command suite
pfexec pkg install SUNWgit
You should then be able to download (with git clone) and build the current OpenAFS ‘master’ branch. At least you will be able to once the bug fixes that were the reason for embarking on this waste of time get into the tree.
## July 26, 2009
### Making the OpenAFS client faster
Filed under: Uncategorized — sxw @ 1:41 pm
Tags: , , ,
During a course of a project here it became apparent that the Linux OpenAFS cache manager is slow when performing reads from the local disk. In this case, all of the data is already on the local disk, and the cache manager knows that the data is up to date. Naively, you would imagine that reading this data would take roughly the same time as if you were reading directly from the cache filesystem. However, that is not the case – in fact, reads appear to be more than twice as slow when fetched through the AFS cache manager, as compared to fetching the equivalent files from the local disk.
I’ve implemented modifications to the cache manager which attempt to reduce this speed deficit. These modifications can be broadly split into 5 sections
### Remove crref() calls
Pretty much every call into the OpenAFS VFS does a crref(), to get a reference to the users current credentials, despite the fact that this information isn’t always required. crref is relatively expensive – it acquires a number of locks in order to perform its copies, and can be a cause of unnecessary serialisation. By only calling crref when required we can gain a small, but measurable, performance increase
### Reduce the code path leading to a cache hit
In readpages, we perform a lot of setup operations before we discover whether the data we’re interested in is cached or not. By making the cached case the fast path, we can gain a performance increase for cache hits, without causing a noticable degradation for cache misses.
### Remove abstraction layers, and use native interfaces
The code currently uses operating system independent abstraction layers to perform the reads from the disk cache. These don’t know anything about the way in which Linux organises its virtual memory, and do a significant amount of extra, unnecessary work. For example, we use the ‘read’ system call to read in the data, rather than the significantly faster readpages(). As we’re being invoked through the AFS module’s readpages() entry point, we can guarantee that we’re going to be fetching a page off disk. Read() also gets called from a user, rather than kernel, memory context, adding to the overhead.
The next trick, then, is to make the readahead occur in the background. By having a background kernel thread which waits until each page of data is read from the cache, and then handles copying it over into corresponding AFS page, the business of reading and copying data from the cache can be hidden from the user.
## Conclusions
This set of changes actually makes a signifciant improvement to cache read speed. In simple tests where the contents of the cache are copied to /dev/null, the new cache manager is around 55% faster than the old one. Tests using Apache to serve data from AFS show significant (but slightly less dramatic, due to other overheads) performance improvements.
Sadly, the Linux Memory Management architecture means that we’re never going to obtain speeds equivalent to using the native filesystem directly. The architecture requires that a page of memory must be associated with a single filesystem. So, we end up reading a page from the disk cache, copying that page into the AFS page, and returning the AFS page to the user. Ideally, we’d be able to dispense with this copy and read directly into the AFS page by switching the page mappings once the read was complete. However, this isn’t currently an option, and the performance benefits obtained through the current approach are still significant.
## May 26, 2009
### Converting OpenAFS to git
Filed under: Uncategorized — sxw @ 12:46 pm
Tags: , ,
For a while now, there have been plans afoot to convert OpenAFS’s CVS repository to git. A number of attempts have been made, which have all stalled due to the complexity of the underlying problem, and issues with the existing tools. Previously, it was felt that the main hurdle to a successful conversion was OpenAFS’s use of ‘deltas’ to provide a changeset style interface on top of CVS. A delta is a collection of related changes, grouped using a comment in the CVS revision log. However, unlike a real changeset, there is no requirement that a delta’s changes be contiguous. A file may be modified by delta A, then by delta B, and then modified by delta A again. This makes it impossible to properly represent all deltas as single changesets. In addition, abuse of deltas within OpenAFS has caused some to span branch or tag points, again making it impossible to represent those deltas as a changeset without destroying the repository history. For many months now, people have been trying to produce conversion tools that achieve as close to a 1 to 1 correspondence between deltas and changesets as is possible, just leaving the troublesome cases as multiple commits.
Frustrated with the lack of progress of this approach, I decided to do a quick and dirty conversion, with the view to getting something completed by the start of coding for this year’s Summer of Code (which I’ve missed) and the yearly Best Practices Conference (which I might just make). I decided to not concern myself with merging deltas at all, but instead use cvsps and the existing git-cvsimport tool to produce a tree where the branch heads and all tag points matched, and which retained enough information to reconstruct deltas without forcing them to be single changesets. In order to be able to perform simple manipulations, I decided to create a perl script which would post-process the cvsps output before feeding it to git. I also created a tool which would check out every branch and tag from cvs, and compare them to the corresponding item in git, and report on any errors. Pretty straightforwards, I thought …
Unfortunately, I rapidly discovered that cvsps had significant problems with the OpenAFS repository. Many tags in CVS were simply not in the cvsps output, other tags (both those marked as FUNKY and INVALID, and those not) were in the wrong place and branchpoints were being incorrectly determined. Rather than get into cvsps’s internals, I ended up extending my post processing script to deal with these errors. It now performs a number of tasks:
Reordering inverted patchsets Some of cvsps’s output gets the patchset ordering wrong, such that a patchset that does fileA:1.2->1.3 comes before fileA:1.1->1.2. The script scans through all of the patchsets for this problem and swaps any that it finds.
Tag point determination Using the output from CVS’s rls command, it is possible to get the revision numbers of every file in a given tag. With this information, the set of patchsets from cvsps can be walked in order to identify the first patchset to satisify every revision contained within the tag. Unfortunately, cvsps patchsets aren’t correctly ordered, so this process also works out how to reorder the patch sets such that no patchsets with file revisions higher than those in the tag occur before the tag point. This reordering is carefully performed in order to not break any tag or branch points which we have already placed! In addition, cvsps sometimes merges commits which occur over a tag point, so we also need to split patchsets which contain both files with revisions before the tag point, and files with revisions after it.
Branch point determination The cvsps output incorrectly places many of OpenAFS’s branchpoints. Fortunately, many of these were marked by tags at the time they were created, and a hardcoded list of these is used to place some branch points in the correct position. For branches that don’t have a corresponding tag, a brute force approach is used. By examining all of the patchsets on the branch, it’s possible to determine the starting revision number of every file that’s modified on that branch – combining this with the contents of the branch head tag from cvs rls gives the equivalent of a tag marking the branchpoint. This can then be processed by the tag point algorithm to work out the correct position in which to place the branch point. This gives the patchset that the branch occurs after, rather than cvsps’s “Ancestor branch” field, which gives the first patchset on the branch. Ancestor branch is fundamentally flawed, as it doesn’t allow changes to occur on HEAD after a branchpoint is created, and before the first patch on that branch. As part of this process, git-cvsimport was modified to understand a new ‘post-patchset’ branch syntax
Unknown branch determination cvsps fails to place some patchsets on the correct branch. By comparing the revision numbers of files in these patchsets with those elsewhere in the list, the correct branch for all of these can be determined (this needs to be done in order that we can work out tag points, as well as being necessary for repository consistency)
We also clean up the output to deal with problems of our own making
Delta naming Whilst there is a style of delta name where the branch is given first, a number of deltas don’t conform to this style, and have the same name across multiple branches. All deltas are renamed such that they are something like STABLE14-delta-name-yyyy-mm-dd
Mistagged files In some places, tags have been incorrectly applied such that files on HEAD are tagged as part of a branch tag. The script contains manual overrides to fix these to tag a revision on the correct branch.
Finally, because having done all of the above I had a pretty good toolset for dealing with patchsets, I implemented support for merging deltas. This merges all bar 10o or so, out of 15,000 deltas into single patchsets. The remaining items are comprised of deltas which span tag or branch points (and which can never be merged) and deltas which contain conflicting changes to a single file (which it might be possible to merge, but which would require manual intervention). These deltas are tagged in a set of git references at refs/deltas/branches/<branch>/<delta>. We separate them from tags so that git-tags doesn’t have to deal with over 10,000 tags, and split them into branches to avoid directory size limitations.
The resulting git tree isn’t a perfect replica of the CVS repository. It has a number of issues which it’s going to be really difficult to fix, and which probably aren’t earth shattering
It contains additional files There are a number of places where a user has added additional directories to cvs. When another user has subsequently tagged their working directory for a release, they haven’t done a cvs update -d, and so these additional directories (and in a small number of cases, files) aren’t contained in the tag for that release. It’s impossible to create a patchset ordering which allows a git tag to not include these directories, so we end up with additional files in the git tag. I don’t think that this is a particular problem
It’s missing a tag There is a tag in the CVS repository (BP-openafs-rxkad-krb5-lha) which is so broken that it’s impossible to construct a patch set ordering that matches it. It is just omitted from the resulting git repository
One branch is bad The openafs-rxkad-krb5-lha branch was created by only branching certain files in the tree. This means that it’s impossible to create a git branch which mimics this one without creating a huge number of additional ‘pull-up’ patch sets. Whilst we include all of the changes that were made on this branch, the final branch state is very different from the one in CVS.
Some deltas are disjoint As discussed, some deltas cannot be merged into single patchsets. This is going to require a new wdelta style tool which understands how to merge these deltas.
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https://www.physicsforums.com/threads/question-in-solid-state.348332/ | # Question in solid state
1. Oct 23, 2009
### lovephy85
1. The problem statement, all variables and given/known data
A crystal has a basis of one atom per lattice point and a set of primitive translation vectors of
a = 3i, b = 3j, c = 1.5(i+j+k)
where i,j,k are unit vectors in the x,y,z directions of a Cartesian coordinate system. What is the Bravais lattice type of this crystal what is miller indices? and what are the volumes of the primitive and conventional unit cells?
2. Relevant equations
Primitive unit cell volume V = a . (b x c) but in this state a not perpendicular with b,c
3. The attempt at a solution
I'm slightly unsure about these Bravais lattices given the multiple permutations they can seem to take.
My assumption, as is that it's Hexagonal. However that also requires that , where gamma is the angle between a,b, alpha between b,c, beta between c,a. But that seems to contradict that the a,b vectors are in i,j directions, ie at 90 degrees. Am I missing something here!?
and miller indices 112
I've worked out the primitive unit cell volume to be 13.5, however I'm also at a loss how to calculate the conventional unit cell volume...
Any help would be hugely appreciated
2. Oct 23, 2009
### jdwood983
You have asked this in another thread, you shouldn't repost the same question
https://www.physicsforums.com/showthread.php?t=348243
Primitive unit cells contain only one lattice point, conventional unit cells do not. If you have the primitive unit cell volume, then the conventional unit cell volume is $$n$$ times bigger, where $$n$$ is the number of lattice points in the conventional unit cell.
Last edited: Oct 23, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook | 2018-02-22 21:11:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5910028219223022, "perplexity": 1083.658270273256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814290.11/warc/CC-MAIN-20180222200259-20180222220259-00612.warc.gz"} |
https://tex.stackexchange.com/questions/624272/creating-a-solution-label-for-an-exercise-using-declaretheoremstyle | # Creating a solution label for an exercise using \declaretheoremstyle
I am writing a book with exercises scattered throughout the text, for which solutions are presented at the end of each chapter.
I want the layout to be as follows:
An exercise environment is introduced inside the text. A solution to this exercise would later appear in the "Solution to exercises" section at the end of the chapter.
In addition, at the end of the exercise environment would be a page reference to the page where its solution is labeled.
A visual example (made artificially) is this:
And the last section of the chapter would be
Is it possible to do that using the amsthm and thmtools packages?
I would assume that the "Solution on page xx" string would be the value of the postfoothook key of the \declaretheoremstyle command of thmtools, but I don't know how to additionally insert a label into that value.
Maybe I am going too far, but I am also open for a way to do that using one of the exercise packages (xsim looks the most promising one). I just haven't stumbled upon this exact layout.
## EDIT:
Since the final document is to be typeset in LyX, I think that a better way to achieve my goal is not to work with xsim, since for referencing, LyX works with the basic building stones of LaTeX - label, pageref and their friends.
I think that an appropriate code to be inserted into postfoothook and heading would do the job, but this is where I am asking for help:
The exercises and solutions labels would be in the following format:
• ex:ch1:ratio-of-odd-numbers <-> sol:ex:ch1:ratio-of-odd-numbers
• ex:ch2:rolling-n-sided-die <-> sol:ex:ch2:rolling-n-sided-die
• ex:ch2:the-hotel-problem <-> sol:ex:ch2:the-hotel-problem
What I need is to refer to sol:⟨the exrecise label⟩ inside postfoothook in exercise, and to refer to ⟨the exrecise label⟩ in the heading of solution.
\begin{exercise}\label{ex:ch1:every-prime}
blah blah
\end{exercise}
for which a solution is in the format
\begin{solution}\label{sol:ex:ch1:every-prime}
Solution to blah blah.
\end{solution}
1. To produce the ref "Solution on page xx" I need to concatenate sol: with ⟨the exrecise label⟩ to produce
\pageref{sol:⟨the exrecise label⟩}
where, in this example, ⟨the exrecise label⟩ is ex:every-prime (to create \pageref{sol:ex:ch1:every-prime}).
1. To produce the header "Solution to Exercise xx" I need to remove the sol: part from the solution label to produce \ref{⟨the exrecise label⟩}, so in this example it would be, \ref{ex:ch1:every-prime}.
A MWE:
\documentclass{book}
\usepackage{fontspec,hyperref,amsthm,thmtools}
%%
\declaretheoremstyle[
postfoothook = \rightline{Solution on page \pageref{sol:⟨the exrecise label⟩}
]{exercise}
\declaretheorem[style=exercise]{exercise}
\declaretheoremstyle[
heading = Solution to Exercise \ref{⟨the solution label without "sol:"⟩},
numbered=no,
]{solution}
\declaretheorem[style=solution]{solution}
%%
\begin{document}
\chapter{Prime numbers}
Lorem ipsum
\section{Dolor sit amet}
\begin{exercise}[Euclid]{ex:ch1:every-prime}
Prove that for every prime $p$, there is a prime $p^\prime > p$. In particular, the list of primes, $2, 3, 5, 7, \ldots$ , is infinite.
\end{exercise}
\section{Solution to exercises}
\begin{solution}\label{sol:ex:ch1:every-prime}
The proof is straight forward. First note that...
\end{solution}
\end{document}
## Edit 2
As a follow up to muzimuzhi Z answer, I would like also to add an "Exercise with no solution" to my book. This environment may be named exercise-nosol and I want it to share the same counter with the exercise environment defined in the accepted answer.
How to do that?
That is, how to define the two environments (exercise with xsim, exercise-nosol with thmtool) so that they share the same counter?
I defined it using thmtools as follows:
\declaretheoremstyle[
postfoothook = \rightline{(No Solution) $\blacktriangleleft$}
]{exercise-nosol}
\declaretheorem[name=Exercise,numberwithin=chapter,style=exercise]{exercise}
output:
• I am writing a textbook in Lyx and deciding the package to use for exercises. I do not require the page numbers. I only need to print the answers at the end of the book. Can I use xsim? Please refer to the full question in tex.stackexchange.com/questions/631983/… Jan 29 at 22:44
An xsim attempt, based on the "Example 5: Crossreferencing between problems and answers" shown in package manual.
\documentclass{article}
\usepackage{amssymb}
\usepackage{xsim,lipsum,hyperref}
\DeclareExerciseEnvironmentTemplate{custom}
{%
\IfInsideSolutionTF
{\label{sol:\ExerciseID}}
{\label{ex:\ExerciseID}}%
\paragraph*
{%
\XSIMmixedcase{\GetExerciseName}%
\IfInsideSolutionTF
{
to \GetExerciseParameter{exercise-name}%
~\GetExerciseProperty{counter}%
~(From Page~\pageref{ex:\ExerciseID})
}
{%
~\GetExerciseProperty{counter}%
\GetExercisePropertyT{subtitle}{~(\PropertyValue)}
}%
}%
}
{%
\IfInsideSolutionF
{\par\leavevmode\hfill
Solution on page~\pageref{sol:\ExerciseID}~$\blacktriangleleft$}%
}
\xsimsetup{
exercise/template = custom ,
solution/template = custom ,
}
\begin{document}
\section{Exercises}
\begin{exercise}[subtitle=title]
\lipsum[1][2]
\end{exercise}
\begin{solution}
\lipsum[1][2]
\end{solution}
\begin{exercise}[subtitle=second title]
\lipsum[4]
\end{exercise}
\begin{solution}
\lipsum[4]
\end{solution}
\clearpage
\printsolutions
\end{document}
Update: Support no-solution exercises
To avoid possible incompatibilities, here is an xsim-only attempt.
A new command \IfSolutionEmptyTF is provided which uses xsim internal \xsim_attribute_get:nn. This might not be the most robust way.
I've tried xsim's \IfExistSolutionTF and it seems to not work as I expected. Also probably due to an xsim issue (see cgnieder/xsim#90), currently you have to provide an empty solution environment for each of no-solution exercises, like
\begin{solution} % no empty lines in env contents
\end{solution}
Full example
\documentclass{article}
\usepackage{amssymb}
\usepackage{xsim,lipsum,hyperref}
\ExplSyntaxOn
\cs_generate_variant:Nn \tl_if_empty:nTF {e}
\NewExpandableDocumentCommand{\IfSolutionEmptyTF}{}
{
\tl_if_empty:eTF
{
\exp_args:Ne \xsim_attribute_get:nn
{exercise!\ExerciseID} {solution-body}
}
}
\ExplSyntaxOff
\DeclareExerciseEnvironmentTemplate{custom-exercise}
{%
\label{ex:\ExerciseID}%
\paragraph*
{%
% typset "Exercise <num> (<subtitle>)"
\XSIMmixedcase{\GetExerciseName}%
~\GetExerciseProperty{counter}%
\GetExercisePropertyT{subtitle}{~(\PropertyValue)}
}%
}
{%
\par\leavevmode\hfill
% typeset "Solution on page <page>" or "(No solution)"
\ifcsname r@sol:\ExerciseID\endcsname
Solution on page~\pageref{sol:\ExerciseID}%
\else
(No Solution)%
\fi
~$\blacktriangleleft$%
}
\DeclareExerciseEnvironmentTemplate{custom-solution}
{%
\IfSolutionEmptyTF{}
{
\label{sol:\ExerciseID}
\paragraph*
{%
\XSIMmixedcase{\GetExerciseName}
to \GetExerciseParameter{exercise-name}%
~\GetExerciseProperty{counter}%
~(From Page~\pageref{ex:\ExerciseID})
}%
}
}{%
\par
}
\xsimsetup{
exercise/template = custom-exercise ,
solution/template = custom-solution ,
}
\begin{document}
\section{Exercises}
\begin{exercise}[subtitle=title]
\lipsum[1][2]
\end{exercise}
\begin{solution}
\lipsum[1][2]
\end{solution}
\begin{exercise}[subtitle=no solution]
\lipsum[3][1] aaa
\end{exercise}
% You still need to provide an empty solution environment.
\begin{solution}
\end{solution}
\begin{exercise}[subtitle=third title]
\lipsum[4][1-3]
\end{exercise}
\begin{solution}
\lipsum[4][4-6]
\end{solution}
\begin{exercise}[subtitle=fourth title]
\lipsum[4][1-3]
\end{exercise}
\begin{solution}
\lipsum[4][4-6]
\end{solution}
\clearpage
\printsolutions
\end{document}
• Thanks a lot for your answer. I prefer a way to achieve such a layout by typesetting the document in the LyX editor. I think it is too complicated to get it inside LyX with modules.
– tush
Nov 30, 2021 at 14:40
• I add my upvoted😃 Dec 6, 2021 at 14:28
• @tush The difficulty here is how to make a solution env aware of which exercise it is linked to. And I doubt LyX would preview such utilities successfully. Dec 7, 2021 at 0:56
• @muzimuzhiZ Thanks a lot for your answer! There is only one thing I would like to add to my book. Please see my addition to the question.
– tush
Dec 7, 2021 at 12:10
• @tush Answer updated. Dec 8, 2021 at 4:55
One approach would be to use the exsheets package which conveniently allows the placement of solutions at the end of a chapter with the \printsolutions[chapter] command. We can label each question (and solution) for page numbering and linking using the provided \CurrentQuestionID. I chose to label the questions Q:\CurrentQuestionID and the answers A:\CurrentQuestionID.
In order to style the questions and solutions similar to your example, we define a new command \problem which takes in the <subtitle><question text><solution> and passes it to exsheets's question and solution environment and handles the basic formatting.
Obviously the document needs to be compiled twice for the page references.
\documentclass{book}
\usepackage{wasysym}
\usepackage{exsheets}
\newcommand\problem[3]{
\begin{question}
\textbf{(#1)} \label{Q:\CurrentQuestionID} #2
\begin{flushright} Solution on page \pageref{A:\CurrentQuestionID} \LEFTarrow \end{flushright}
\end{question}
\begin{solution}[name = Solution to exercise]
\textbf{(From Page \pageref{Q:\CurrentQuestionID})}
\label{A:\CurrentQuestionID} #3
\end{solution}
}
\begin{document}
\chapter{Prime Numbers}
\section{Dolor sit amet}
\problem{Euclid}{
Prove that for every prime $p$, there is a prime $p^\prime > p$. In particular, the list of primes, $2, 3, 5, 7, \ldots$ , is infinite.
}{The proof is straight forward. First note that...}
\problem{Fermat}{
$a^n+b^n=c^n$}{The proof is too large to fit on this page}
\section{Solutions}
\printsolutions[chapter]
\end{document}
This is doable in LyX. Under Documents>Settings>LaTeX Preamble enter the preamble and while in that settings pane change documentclass to book (or something else with chapters. Then you just enter the \problem in LaTeX code in LyX as shown. The output is the same as above.
• Can I choose not to see the Latex code on your Lyx screen but to see equations in the usual Lyx UI way? Is Lyx fully compatible with xsim? Thank you! Jan 29 at 20:53
• I posted a new question related to the above in tex.stackexchange.com/questions/631983/… Jan 29 at 22:39
With exercise package:
\documentclass{book}
\usepackage{fontspec,amsthm,thmtools}
\usepackage{amsmath, amssymb}
\usepackage{chngcntr}
\usepackage{etoolbox}
\usepackage{mwe}% <--- for testing purpose only
\usepackage{xcolor}
\usepackage[pdfborder={0 0 0}]{hyperref}% you must load it before the exercise package
\usepackage{exercise}
%\counterwithin{Exercise}{chapter} This line if you want exercise 1.1, 1.2, etc. in chapter 1
\renewcounter{Exercise}[chapter]
\renewcommand{\ExerciseName}{Exercise}
\setlength{\ExerciseSkipBefore}{\baselineskip}
\setlength{\ExerciseSkipAfter}{0pt}
\AfterEndEnvironment{Exercise}{\hspace{\fill}
Solution on page \textcolor{red}{\pageref{\ExerciseLabel-Answer}} $\blacktriangleleft$\vspace{\baselineskip}\par}
\begin{document}
\chapter{Prime numbers}
Lorem ipsum
\section{Dolor sit amet}
\begin{Exercise}[title={Euclid},label={ex:ch1:every-prime}]
Prove that for every prime $p$, there is a prime $p^\prime > p$. In particular, the list of primes, $2, 3, 5, 7, \ldots$ , is infinite.
\end{Exercise}
\blindtext% <--- for testing purpose only
\begin{Exercise}[title={Another one},label={ex:ch1:second}]
Prove that ducks are the best.
\end{Exercise}
\blindtext% <--- for testing purpose only
\section{Solution to Exercises}
The proof is straight forward. First note that...
\blindtext% <--- for testing purpose only
The proof is straight forward. First note that...
\blindtext% <--- for testing purpose only
\chapter{Second chapter}
Lorem ipsum
\section{A section}
\begin{Exercise}[title={An exercise},label={ex:ch2:an-ex}]
Prove that something is true.
\end{Exercise}
\blindtext% <--- for testing purpose only
\begin{Exercise}[title={Another one again},label={ex:ch2:second}]
Prove that ducks are the best of all.
\end{Exercise}
\section{Solution to Exercises}
\blindtext% <--- for testing purpose only
The proof of the first.
\blindtext% <--- for testing purpose only
The proof of the second.
\end{document}
If the Section with the answers contains only the answers, you can also use answerdelayed and print all the answers together with \shipoutAnswer:
\documentclass{book}
\usepackage{fontspec,amsthm,thmtools}
\usepackage{amsmath, amssymb}
\usepackage{chngcntr}
\usepackage{etoolbox}
\usepackage{mwe}% <--- for testing purpose only
\usepackage{xcolor}
\usepackage[pdfborder={0 0 0}]{hyperref}% you must load it before the exercise package
%\counterwithin{Exercise}{chapter} This line if you want exercise 1.1, 1.2, etc. in chapter 1
\renewcounter{Exercise}[chapter]
\renewcommand{\ExerciseName}{Exercise}
\setlength{\ExerciseSkipBefore}{\baselineskip}
\setlength{\ExerciseSkipAfter}{0pt}
\AfterEndEnvironment{Exercise}{\hspace{\fill}
Solution on page \textcolor{red}{\pageref{\ExerciseLabel-Answer}} $\blacktriangleleft$}
\begin{document}
\chapter{Prime numbers}
Lorem ipsum
\section{Dolor sit amet}
\begin{Exercise}[title={Euclid},label={ex:ch1:every-prime}]
Prove that for every prime $p$, there is a prime $p^\prime > p$. In particular, the list of primes, $2, 3, 5, 7, \ldots$ , is infinite.
\end{Exercise}
The proof is straight forward. First note that...
\blindtext% <--- for testing purpose only
\begin{Exercise}[title={Another one},label={ex:ch1:second}]
Prove that ducks are the best.
\end{Exercise}
The proof is straight forward. First note that...
\blindtext% <--- for testing purpose only
\section{Solution to Exercises}
\chapter{Second chapter}
Lorem ipsum
\section{A section}
\begin{Exercise}[title={An exercise},label={ex:ch2:an-ex}]
Prove that something is true.
\end{Exercise}
The proof of the first.
\blindtext% <--- for testing purpose only
\begin{Exercise}[title={Another one again},label={ex:ch2:second}]
Prove that ducks are the best of all.
\end{Exercise}% | 2022-07-07 15:15:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8074327707290649, "perplexity": 3248.297901245183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104692018.96/warc/CC-MAIN-20220707124050-20220707154050-00128.warc.gz"} |
https://www.studyadda.com/solved-papers/solved-paper-economics-2012-delhi-set-iii_q3/647/329145 | • # question_answer Define an indifference map. Explain why an indifference curve to the right shows higher utility level.
Indifference map is a family or collection of indifference curves that depicts the different levels of satisfaction and preferences of a consumer. Each indifference curve in an indifference map depicts a particular level of satisfaction. Higher IC denotes higher level of satisfaction and lower IC denotes lower level of satisfaction. The above figure depicts an Indifference Map comprising of six indifference curves (from$I{{C}_{1}}$ to$I{{C}_{6}}$). As the consumer moves farther away from$I{{C}_{1}}$ to higher indifference curves the level of satisfaction derived by the consumer increases $I{{C}_{6}}$depicts the highest level of satisfaction. On the other hand, $I{{C}_{1}}$ depicts the lowest level of satisfaction. An indifference curve is downward sloping from left to right. It implies that a consumer cannot simultaneously have more of both the goods. An increase in the quantity of one good is associated with the decrease in the quantity of the other good. This is in accordance with the assumption of monotonic preferences & there is an inverse relation between good 1 & 2. | 2020-04-02 11:29:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8285115361213684, "perplexity": 891.3937677296191}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506959.34/warc/CC-MAIN-20200402111815-20200402141815-00361.warc.gz"} |
https://math.stackexchange.com/questions/3709863/does-every-inner-product-space-have-hilbert-completion | # Does every inner product space have Hilbert completion?
Does every inner product space have Hilbert completion which is a Hilbert space? If so, how can we define the inner product in the Hilbert space?
Yes, note that an inner product induces a norm by $$\|x\|:=\sqrt{\langle x,x\rangle}$$ and a norm is induced by an inner product iff it satisfies the the parallelogram law: $$2\|x\|^2+2\|y\|^2 \overset!= \|x+y\|^2+\|x-y\|^2\qquad \text{ for all x,y\in V}.$$
So if $$V$$ is equipped with an inner product let $$\overline{V}$$ denote the completion of $$V$$ wrt the norm induced by the inner product. If $$x,y\in \overline V$$ and $$x_n,y_n\in V$$ with $$x_n\to x$$, $$y_n\to y$$ you have that: $$2\|x\|^2+ 2\|y\|^2= \lim_n (2\|x_n\|^2+2\|y_n\|^2) =\lim_n( \|x_n+y_n\|^2+\|x_n-y_n\|^2)\\ = \|x+y\|^2+\|x-y\|^2$$ hence the norm on $$\overline V$$ still obeys the parallelogram law and as such $$\overline V$$ is a Hilbert space completion of $$V$$.
• What is the meaning of the $\overset!=$ operator in your first equation? – Carmeister Jun 8 '20 at 1:02 | 2021-04-16 17:18:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.992529034614563, "perplexity": 79.78786587801729}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038088245.37/warc/CC-MAIN-20210416161217-20210416191217-00044.warc.gz"} |
http://mathschallenge.net/full/quarter_square | ## Quarter Square
#### Problem
A square is split into four smallers squares and exactly two of these smaller squares are shaded. For example, the top left and bottom right squares could be shaded.
How many distinct ways can exactly two smaller square be shaded?
#### Solution
It can be seen that there are exactly six ways:
However, we could have arrived at this answer by a quite different method. Let us consider two distinct cases: the top left square is either shaded or unshaded.
(i) If the top left square is shaded then there are three remaining squares that could be shaded (see the top three diagrams).
(ii) If the top left square is unshaded then there are three remaining squares that could also be left unshaded, which means that the other two must be shaded (see the bottom three diagrams).
Therefore there are $3 + 3 = 6$ ways of shading exactly two smaller square.
We can use this method to explain why there are six ways of picking two numbers from $\{1,2,3,4\}$. We pick $1$ and one other number from $\{2,3,4\}$ or we do not pick $1$ and do not pick one other from $\{2,3,4\}$.
However, when we consider picking two numbers from $\{1,2,3,4,5\}$ we need to adapt the strategy slightly. We pick $1$ and one other number from $\{2,3,4,5\}$: 4 ways, or we do not pick $1$ and do not pick two from $\{2,3,4,5\}$. This second case is equivalent to picking two from $\{1,2,3,4\}$: 6 ways. Hence there are $4 + 6 = 10$ ways of picking two numbers from $\{1,2,3,4,5\}$.
Use the same method to explain why there are 15 ways of picking two from $\{1,2,3,4,5,6\}$.
How many ways can you pick two numbers from $\{1,2,3,4,5,6,7\}$?
Problem ID: 353 (23 Jul 2009) Difficulty: 1 Star
Only Show Problem | 2015-06-30 19:42:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8562073707580566, "perplexity": 294.4315406056245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094491.62/warc/CC-MAIN-20150627031814-00001-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/equation-of-motion-from-given-2d-potential.828753/ | # Equation of motion from given 2D Potential
1. Aug 21, 2015
1. The problem statement, all variables and given/known data
A particle of mass m moves in two dimensions under the following potential energy
function:
V($\vec{r}$) = ½ k (x2 + 4y2)
Find the resulting motion, given the initial condition at t=0:
x = a, y = 0, x' = 0, y' = vo
2. Relevant equations
F = ma = -dV/dr
3. The attempt at a solution
This will obviously involve a 2nd order diff eq, and there are enough initial conditions to solve for the unknown constants. If the potential were given with the r variable instead of x and y, it would be simpler. As such, I'm not sure how to take dV/dr when V is V(x,y) not V(r)...
If I use x = r Cosθ, y = r Sinθ, I can put it as:
V($\vec{r}$) = (3/2) k r2 Sin2θ
But now that I have θ in the formula, is it ok to take dV/dr as such and set it in F = -dV/dr?
2. Aug 21, 2015
### Noctisdark
You are doing it the wrong way, there are vector involved and $\frac{d}{d \vec r} = \nabla$ and everyone already says it: force is the gradient of the potential, can you work it out now ?
[Edit: In case I wasn't clear, $\nabla = \lt \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \gt$ ]
3. Aug 21, 2015
*facepalm* Thank you!
$\vec{F}$ = -k( x $\hat{x}$ + 4y $\hat{y}$)
Then set to m$\vec{a}$ = m($\ddot{x}$ $\hat{x}$ + $\ddot{y}$ $\hat{y}$), and compare associated vector components... yes?
4. Aug 21, 2015
X(t) = a $\cos$($\sqrt{k/m} t$)
Y(t) =$\frac{v_{o}}{\sqrt{4k/m}}\sin(\sqrt{4k/m} t)$ | 2018-03-19 13:55:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5969069600105286, "perplexity": 1223.7760452260675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257646914.32/warc/CC-MAIN-20180319120712-20180319140712-00654.warc.gz"} |
https://physics.stackexchange.com/questions/102707/renormalizing-ir-and-uv-divergences/102769 | # Renormalizing IR and UV divergences
In lectures on effective field theory the professor wanted to find the correction to the four point vertex in massless $\phi^4$ theory by calculating the diagram,
$\hspace{6cm}$
We consider the zero external momentum limit and denote $p$ as the momentum in the loop. Then we get, \begin{align} \int \frac{ d ^d p }{ (2\pi)^4}\frac{1}{p ^4 } & = \frac{ - i }{ 16 \pi ^2 } ( 4\pi ) \Gamma ( \epsilon ) \mu ^\epsilon \\ & = - \frac{ i }{ 16 \pi ^2 } \left( \frac{1}{ \epsilon _{ UV}} - \gamma + \log 4\pi - \log \mu ^2 \right) \\ & = \frac{ i }{ 16 \pi ^2 } \left( \frac{1}{ \epsilon _{ UV}} - \frac{1}{ \epsilon _{ IR}} \right) \end{align} where we introduced $\mu$ as an IR cut-off and then take $\log \mu ^2$ as a $\frac{1}{\epsilon_{IR}}$.
This is fine, however the professor then goes on to say that this diagram is zero since the two divergences cancel. Why would this be the case? The two divergences arise for completely different reasons. The UV divergence is due to a UV cutoff (possibly from new high energy particles arising at some high up scale) and the second is a consequence of studying a massless theory.
For more context the lecture notes are available here under Effective Field Theory (Eq. 4.17)
I think you misunderstood what the professor wanted to say. To understand this, let us evaluate the integral more thoroughly (your expressions contain some mistakes). If we use the dimensional regularization prescription $d\rightarrow d-2\epsilon$ and an additional mass scale $\mu$, we get for the integral in question the following result:
$$\int \frac{d^{d-2\epsilon}p}{(2\pi)^{d-2\epsilon}}\frac{1}{(p^2+\mu^2)^2}=\frac{\Gamma(2-d/2+\epsilon)}{(4\pi)^{d/2-\epsilon}}\mu^{-2(2-d/2+\epsilon)}.$$
For $d=4$ we get
$$\int\frac{d^{4-2\epsilon}p}{(2\pi)^{4-2\epsilon}}\frac{1}{(p^2+\mu^2)^2}=\frac{\Gamma(\epsilon)}{16\pi^2}\left(\frac{\mu^2}{4\pi}\right)^{-\epsilon}.$$
Expanding this at $\epsilon\rightarrow 0,$ we arrive at
$$\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^2+\mu^2)^2}\approx\frac{1}{16\pi^2}\left[\frac{1}{\epsilon}-\gamma+\log(4\pi)-\log(\mu^2)\right].$$
In the massless limit, i.e. $\mu\rightarrow 0$, the logarithm diverges. So what can we say about the nature about this divergence?
As can be concluded from powercounting, a positive $\epsilon$ corresponds to curing UV divergences, while a negative one cures IR divergences. First, let us assume that that we deal with UV divergences and identify $\epsilon=\epsilon_{UV}.$ What can we say about the remaining divergent term? We can observe that the whole integral has to vanish (which is proven earlier in the lecture), and this happens only when the divergent term is equal to minus the $1/\epsilon$ term, i.e.
$$\frac{1}{\epsilon_{UV}}=\gamma-\log(4\pi)+\log(\mu^2).$$
Next, let us assume at we are dealing with divergences from the infrared, and identify $\epsilon=\epsilon_{IR}.$ We now have to observe that evaluating the integral gives us just the same result, but with $\epsilon_{UV}$ and $\epsilon_{IR}$ exchanged. The condition for vanishing of the integral is now
$$\frac{1}{\epsilon_{IR}}=\gamma-\log(4\pi)+\log(\mu^2).$$
But the right hand side is just the same as in the condition for the UV! This means we actually get
$$\epsilon_{UV}=\epsilon_{IR}.$$
As the lecturer has pointed out, this can be interpreted as dimensional regularization "taming" both the UV and the IR simultaneously.
• I was just wondering how you can associate the $\epsilon$ pole as originating from an IR divergence in the first place? The scale $\mu^2$ is introduced precisely as a regulator for the IR divergence of the original integral. So doesn't this mean we use dim reg with $\epsilon >0$ to cure the UV divergence and an explicit IR cut off to cure the IR divergence (i.e in four dimensions with the cut off in place the integral is finite at low $p$ so no need for dim reg in this low momentum sector of the integral?) Thanks! – CAF Nov 19 '16 at 20:08
• On page 23 of this lecture note, $\int d^dp(p^2)^k=0$ only for $k>0,k\in \mathbb{Z}$. But at just the below of (4.17) about $\int d^dp\frac{1}{p^4}$, there goes "This integral is zero as mentioned above." Why is it zero? I think the vanishment of this integral isn't proven. – GotchaP Feb 18 '17 at 5:55 | 2019-08-26 01:30:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9931111335754395, "perplexity": 324.63038002214773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330913.72/warc/CC-MAIN-20190826000512-20190826022512-00185.warc.gz"} |
https://www.imomath.xyz/2022/03/2020-isl-a2.html | ### 2020 ISL #A2
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*}with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
The answer is $4$.
We first prove that $n \ge 4$. It suffices to show that $x^2y$ is not expressible. For the sake of contradiction, suppose it was. Plugging in $z=-x-y$ gives $\big(xy-(x+y)^2\big)Q(x,y,-x-y) + xy(-x-y)P(x,y,-x-y)=x^2y.$Since the RHS has degree $3$, we know that all non-degree $3$ terms on the LHS must be $0$. Solely considering the degree $3$ terms on the LHS, we have $(x^2+xy+y^2)(c_1x + c_2y) + xy(x+y)d_1 = x^2y$for some constants $c_1,c_2,d_1$. Expanding, we note that the coefficients of $x^2y$ and $xy^2$ are equal on the LHS, which is a contradiction. Thus, no $n\le 3$ work.
Now we show our construction for $n=4$. In other words, we must show that $n=4$ works. We will prove that any monomial of degree $n\ge 4$ is expressible. Let this monomial be $x^ay^bz^c$.
Case 1: $a,b,c\ge 1$. This is divisible by $xyz$, so just use $R$.
Case 2: $c=0$ and $a,b\ge 1$. (Other such cases follow by symmetry.)
Subcase 2.1: $a,b\ge 2$. Then$x^ay^b=x^{a-1}y^{b-1}(xy+yz+zx)-(x^{a-2}x^{b-1}-x^{a-1}y^{b-2})xyz.$
Subcase 2.2: $a=1$ and $b\ge 2$. (The case $b=1$ and $a\ge 2$ is analogous.) In fact, $b\ge 3$ since $xy^b$ must have degree at least $4$. Notice$xy^b=xy^{b-1}(x+y+z)-x^2y^{b-1} - xy^{b-1}z.$The second term is expressible from Subcase 2.1, and the last term is divisible by $xyz$, done.
Case 3: $c=0,b=0,a\ge 4$. (Other such cases follow by symmetry.) We want to construct $x^a$. Notice$x^a=(x+y+z)x^{a-1} - x^{a-1}y - x^{a-1}z.$The final two terms are expressible by Subcase 2.2, and we are done.
In all cases, $x^ay^bz^c$ is expressible, so we have proved that $n=4$ works.
We are done. $\square$
1. Anonymous3/21/2022
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hi | 2023-01-29 23:20:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9516183137893677, "perplexity": 278.62259476617487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499768.15/warc/CC-MAIN-20230129211612-20230130001612-00335.warc.gz"} |
https://biz.libretexts.org/Bookshelves/Law/Book%3A_Law_of_Commercial_Transactions/25%3A_Liability_and_Discharge/25.07%3A_Summary_and_Exercises | # 25.7: Summary and Exercises
## Summary
As a general rule, one who signs a note as maker or a draft as drawer is personally liable unless he or she signs in a representative capacity and either the instrument or the signature shows that the signing has been made in a representative capacity. Various rules govern the permutations of signatures when an agent and a principal are involved.
The maker of a note and the acceptor of a draft have primary contract liability on the instruments. Secondarily liable are drawers and indorsers. Conditions precedent to secondary liability are presentment, dishonor, and notice of dishonor. Under the proper circumstances, any of these conditions may be waived or excused.
Presentment is a demand for payment made on the maker, acceptor, or drawee, or a demand for acceptance on the drawee. Presentment must be made (1) at the time specified in the instrument unless no time is specified, in which case it must be at the time specified for payment, or (2) within a reasonable time if a sight instrument.
Dishonor occurs when acceptance or payment is refused after presentment, at which time a holder has the right of recourse against secondary parties if he has given proper notice of dishonor.
A seller-transferor of any commercial paper gives five implied warranties, which become valuable to a holder seeking to collect in the event that there has been no indorsement or the indorsement has been qualified. These warranties are (1) good title, (2) genuine signatures, (3) no material alteration, (4) no defenses by other parties to the obligation to pay the transferor, and (5) no knowledge of insolvency of maker, acceptor, or drawer.
A holder on presentment makes certain warranties also: (1) entitled to enforce the instrument, (2) no knowledge that the maker’s or drawer’s signature is unauthorized, and (3) no material alteration.
Among the ways in which the parties may be discharged from their contract to honor the instrument are the following: (1) payment or satisfaction, (2) tender of payment, (3) cancellation and renunciation, (4) impairment of recourse or of collateral, (5) reacquisition, (6) fraudulent and material alteration, (7) certification, (8) acceptance varying a draft, and (9) unexcused delay in presentment or notice of dishonor.
## EXERCISES
1. Howard Corporation has the following instrument, which it purchased in good faith and for value from Luft Manufacturing, Inc.
Figure 25.2
Judith Glen indorsed the instrument on the back in her capacity as president of Luft when it was transferred to Howard on July 15, 2012.
1. Is this a note or a draft?
2. What liability do McHugh and Luft have to Howard? Explain.
2. An otherwise valid negotiable bearer note is signed with the forged signature of Darby. Archer, who believed he knew Darby’s signature, bought the note in good faith from Harding, the forger. Archer transferred the note without indorsement to Barker, in partial payment of a debt. Barker then sold the note to Chase for 80 percent of its face amount and delivered it without indorsement. When Chase presented the note for payment at maturity, Darby refused to honor it, pleading forgery. Chase gave proper notice of dishonor to Barker and to Archer.
1. Can Chase hold Barker liable? Explain.
2. Can Chase hold Archer liable? Explain.
3. Can Chase hold Harding liable? Explain.
3. Marks stole one of Bloom’s checks, already signed by Bloom and made payable to Duval, drawn on United Trust Company. Marks forged Duval’s signature on the back of the check and cashed it at Check Cashing Company, which in turn deposited it with its bank, Town National. Town National proceeded to collect on the check from United. None of the parties was negligent. Who will bear the loss, assuming Marks cannot be found?
4. Robb stole one of Markum’s blank checks, made it payable to himself, and forged Markum’s signature on it. The check was drawn on the Unity Trust Company. Robb cashed the check at the Friendly Check Cashing Company, which in turn deposited it with its bank, the Farmer’s National. Farmer’s National proceeded to collect on the check from Unity. The theft and forgery were quickly discovered by Markum, who promptly notified Unity. None of the parties mentioned was negligent. Who will bear the loss, assuming the amount cannot be recovered from Robb? Explain.
5. Pat stole a check made out to the order of Marks, forged the name of Marks on the back, and made the instrument payable to herself. She then negotiated the check to Harrison for cash by signing her own name on the back of the instrument in Harrison’s presence. Harrison was unaware of any of the facts surrounding the theft or forged indorsement and presented the check for payment. Central County Bank, the drawee bank, paid it. Disregarding Pat, who will bear the loss? Explain.
6. American Music Industries, Inc., owed Disneyland Records over $340,000. As evidence of the debt, Irv Schwartz, American’s president, issued ten promissory notes, signing them himself. There was no indication they were obligations of the corporation, American Music Industries, Inc., or that Irv Schwartz signed them in a representative capacity, but Mr. Schwartz asserted that Disneyland knew the notes were corporate obligations, not his personally. American paid four of the notes and then defaulted, and Disneyland sued him personally on the notes. He asserted he should be allowed to prove by parol evidence that he was not supposed to be liable. Is he personally liable? Explain.Schwartz v. Disneyland Vista Records, 383 So.2d 1117 (Fla. App. 1980). 7. Alice Able hired Betty Baker as a bookkeeper for her seamstress shop. Baker’s duties included preparing checks for Able to sign and reconciling the monthly bank statements. Baker made out several checks to herself, leaving a large space to the left of the amount written, which Able noticed when she signed the checks. Baker took the signed checks, altered the amount by adding a zero to the right of the original amount, and cashed them at First Bank, the drawee. Able discovered the fraud, Baker was sent to prison, and Able sued First Bank, claiming it was liable for paying out on altered instruments. What is the result? 8. Christina Reynolds borrowed$16,000 from First Bank to purchase a used Ford automobile. Bank took a note and a secured interest in the car (the car is collateral for the loan). It asked for further security, so Christina got her sister Juanita to sign the note as an accommodation maker. Four months later, Christina notified Bank that she wished to sell the Ford for \$14,000 in order to get a four-wheel drive Jeep, and Bank released its security interest. When Christina failed to complete payment on the note for the Ford, Bank turned to Juanita. What, if anything, does Juanita owe?
## SELF-TEST QUESTIONS
1. Drawers and indorsers have
1. primary contract liability
2. secondary liability
3. no liability
4. none of the above
2. Conditions(s) needed to establish secondary liability include
1. presentment
2. dishonor
3. notice of dishonor
4. all of the above
3. A demand for payment made on a maker, acceptor, or drawee is called
1. protest
2. notice
3. presentment
4. certification
4. An example of an implied warranty given by a seller of commercial paper includes a warranty
1. of good title
2. that there are no material alterations
3. that signatures are genuine
4. covering all of the above
5. Under UCC Article 3, discharge may result from
1. cancellation
2. impairment of collateral
3. fraudulent alteration
4. all of the above | 2022-10-07 15:04:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45870113372802734, "perplexity": 8802.91670672931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338213.55/warc/CC-MAIN-20221007143842-20221007173842-00656.warc.gz"} |
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# [lno] Update the scalar evolutions algorithm
• From: Sebastian Pop <sebastian dot pop at cri dot ensmp dot fr>
• To: gcc-patches at gcc dot gnu dot org
• Date: Wed, 3 Mar 2004 17:10:29 +0100
• Subject: [lno] Update the scalar evolutions algorithm
Hi,
This patch updates the scalar evolution algorithm. The analysis is
about 3000 lines down from about 4500.
I've also included the pass of checks elimination on which I'm
working, even if the compiler does not bootstraps with it enabled,
because I'm using it for testing the behaviour of the analyzer: see
testsuite/.../ssa-chrec-04.c and others testcases. In order to
correct this optimization pass I have to modify the way the analyzer
deals with the types of the scalars it analyzes.
I'm attaching the documentation for both these passes. A higher level
presentation of the extraction algorithm is available as a set of
slides:
http://cri.ensmp.fr/~pop/gcc/feb04/slides.pdf
/*
Description:
This pass analyzes the evolution of scalar variables in loop
structures. The algorithm is based on the SSA representation,
and on the loop hierarchy tree. This algorithm is not based on
the notion of versions of a variable, as it was the case for the
previous implementations of the scalar evolution algorithm, but
it assumes that each defined name is unique.
A short sketch of the algorithm is:
Given a scalar variable to be analyzed, follow the SSA edge to
its definition:
- When the definition is a MODIFY_EXPR: if the right hand side
(RHS) of the definition cannot be statically analyzed, the answer
of the analyzer is: "don't know", that corresponds to the
conservative [-oo, +oo] element of the lattice of intervals.
Otherwise, for all the variables that are not yet analyzed in the
RHS, try to determine their evolution, and finally try to
evaluate the operation of the RHS that gives the evolution
function of the analyzed variable.
- When the definition is a condition-phi-node: determine the
evolution function for all the branches of the phi node, and
finally merge these evolutions (see chrec_merge).
- When the definition is a loop-phi-node: determine its initial
condition, that is the SSA edge defined in an outer loop, and
keep it symbolic. Then determine the SSA edges that are defined
in the body of the loop. Follow the inner edges until ending on
another loop-phi-node of the same analyzed loop. If the reached
loop-phi-node is not the starting loop-phi-node, then we keep
this definition under a symbolic form. If the reached
loop-phi-node is the same as the starting one, then we compute a
symbolic stride on the return path. The result is then the
symbolic chrec {initial_condition, +, symbolic_stride}_loop.
Examples:
Example 1: Illustration of the basic algorithm.
| a = 3
| loop_1
| b = phi (a, c)
| c = b + 1
| if (c > 10) exit_loop
| endloop
Suppose that we want to know the number of iterations of the
loop_1. The exit_loop is controlled by a COND_EXPR (c > 10). We
ask the scalar evolution analyzer two questions: what's the
scalar evolution (scev) of "c", and what's the scev of "10". For
"10" the answer is "10" since it is a scalar constant. For the
scalar variable "c", it follows the SSA edge to its definition,
"c = b + 1", and then asks again what's the scev of "b".
Following the SSA edge, we end on a loop-phi-node "b = phi (a,
c)", where the initial condition is "a", and the inner loop edge
is "c". The initial condition is kept under a symbolic form (it
may be the case that the copy constant propagation has done its
work and we end with the constant "3" as one of the edges of the
loop-phi-node). The update edge is followed to the end of the
loop, and until reaching again the starting loop-phi-node: b -> c
-> b. At this point we have drawn a path from "b" to "b" from
which we compute the stride in the loop: in this example it is
"+1". The resulting scev for "b" is "b -> {a, +, 1}_1". Now
that the scev for "b" is known, it is possible to compute the
scev for "c", that is "c -> {a + 1, +, 1}_1". In order to
determine the number of iterations in the loop_1, we have to
instantiate_parameters ({a + 1, +, 1}_1), that gives after some
more analysis the scev {4, +, 1}_1, or in other words, this is
the function "f (x) = x + 4", where x is the iteration count of
the loop_1. Now we have to solve the inequality "x + 4 > 10",
and take the smallest iteration number for which the loop is
exited: x = 7. This loop runs from x = 0 to x = 7, and in total
there are 8 iterations. In terms of loop normalization, we have
created a variable that is implicitly defined, "x" or just "_1",
and all the other analyzed scalars of the loop are defined in
function of this variable:
a -> 3
b -> {3, +, 1}_1
c -> {4, +, 1}_1
or in terms of a C program:
| a = 3
| for (x = 0; x <= 7; x++)
| {
| b = x + 3
| c = x + 4
| }
Example 2: Illustration of the algorithm on nested loops.
| loop_1
| a = phi (1, b)
| c = a + 2
| loop_2 10 times
| b = phi (c, d)
| d = b + 3
| endloop
| endloop
For analyzing the scalar evolution of "a", the algorithm follows
the SSA edge into the loop's body: "a -> b". "b" is an inner
loop-phi-node, and its analysis as in Example 1, gives:
b -> {c, +, 3}_2
d -> {c + 3, +, 3}_2
Following the SSA edge for the initial condition, we end on "c = a
+ 2", and then on the starting loop-phi-node "a". From this point,
the loop stride is computed: back on "c = a + 2" we get a "+2" in
the loop_1, then on the loop-phi-node "b" we compute the overall
effect of the inner loop that is "b = c + 30", and we get a "+30"
in the loop_1. That means that the overall stride in loop_1 is
equal to "+32", and the result is:
a -> {1, +, 32}_1
c -> {3, +, 32}_1
Example 3: Higher degree polynomials.
| loop_1
| a = phi (2, b)
| c = phi (5, d)
| b = a + 1
| d = c + a
| endloop
a -> {2, +, 1}_1
b -> {3, +, 1}_1
c -> {5, +, a}_1
d -> {5 + a, +, a}_1
instantiate_parameters ({5, +, a}_1) -> {5, +, 2, +, 1}_1
instantiate_parameters ({5 + a, +, a}_1) -> {7, +, 3, +, 1}_1
Example 4: Lucas, Fibonacci, or mixers in general.
| loop_1
| a = phi (1, b)
| c = phi (3, d)
| b = c
| d = c + a
| endloop
a -> (1, c)_1
c -> {3, +, a}_1
The syntax "(1, c)_1" stands for a PEELED_CHREC that has the
following semantics: during the first iteration of the loop_1, the
variable contains the value 1, and then it contains the value "c".
Note that this syntax is close to the syntax of the loop-phi-node:
"a -> (1, c)_1" vs. "a = phi (1, c)".
The symbolic chrec representation contains all the semantics of the
original code. What is more difficult is to use this information.
Example 5: Flip-flops, or exchangers.
| loop_1
| a = phi (1, b)
| c = phi (3, d)
| b = c
| d = a
| endloop
a -> (1, c)_1
c -> (3, a)_1
Based on these symbolic chrecs, it is possible to refine this
information into the more precise PERIODIC_CHRECs:
a -> |1, 3|_1
c -> |3, 1|_1
This transformation is not yet implemented.
Further readings:
You can find a more detailed description of the algorithm in:
http://icps.u-strasbg.fr/~pop/DEA_03_Pop.pdf
http://icps.u-strasbg.fr/~pop/DEA_03_Pop.ps.gz. But note that
this is a preliminary report and some of the details of the
algorithm have changed. I'm working on a research report that
updates the description of the algorithms to reflect the design
choices used in this implementation.
Fixmes:
FIXME taylor: This FIXME concerns all the cases where we have to
deal with additions of exponential functions: "exp + exp" or
"poly + exp" or "cst + exp". This could be handled by a Taylor
decomposition of the exponential function, but this is still
under construction (not implemented yet, or chrec_top).
The idea is to represent the exponential evolution functions
using infinite degree polynomials:
| a -> {1, *, 2}_1 = {1, +, 1, +, 1, +, ...}_1 = {1, +, a}_1
Proof:
\begin{eqnarray*}
\{1, *, t+1\} (x) &=& exp \left(log (1) + log (t+1) \binom{x}{1} \right) \\
&=& (t+1)^x \\
&=& \binom{x}{0} + \binom{x}{1}t + \binom{x}{2}t^2 +
\ldots + \binom{x}{x}t^x \\
&=& \{1, +, t, +, t^2, +, \ldots, +, t^x\} \\
\end{eqnarray*}
While this equality is simple to prove for exponentials of degree
1, it is still work in progress for higher degree exponentials.
*/
/*
Description:
Compute the scalar evolutions for all the scalar variables of a
condition expression, and based on this information performs a
proof. The condition is rewritten based on the result of this
proof.
Examples:
Example 1: A simple illustration of the algorithm.
Given the COND_EXPR "if (a < b)" with "a -> {2, +, 1}_1" and "b
-> {3, +, 1}_1", the proof consists in comparing these evolution
functions: is it always true for a given iteration x that "{2, +,
1}_1 (x) < {3, +, 1}_1 (x)"? The answer is yes, and the test of
the condition is consequently replaced by "1".
Further readings:
There is no further readings for the moment.
Based on the fact that this algorithm is similar to the Value
Range Propagation you can have a look at the corresponding
papers.
*/
Changelog:
* Makefile.in (OBJS-common): Add tree-elim-check.o.
(tree-chrec.o): Add dependence on tree-pass.h.
(tree-elim-check.o): New rule.
* tree-elim-check.c: New file.
* basic-block.h (edge_source, edge_destination): New inlined
functions.
* cfgloop.h (loop_nb_iterations): Added a comment on the use
of this accessor.
* common.opt (ftree-elim-checks): New flag.
* flags.h (flag_tree_elim_checks): Declared here.
* opts.c (decode_options): Set flag_tree_elim_checks to zero.
(common_handle_option): Add case OPT_ftree_elim_checks.
* timevar.def (TV_TREE_ELIM_CHECKS): Defined.
* toplev.c (flag_tree_elim_checks): Defined.
* tree-cfg.c (print_pred_bbs, print_succ_bbs, print_loop):
Modify the dumping style. Print nb_iterations.
* tree-chrec.c, tree-chrec.h, tree-scalar-evolution.c,
tree-scalar-evolution.h, tree-data-ref.c: New version.
* tree-fold-const.c (tree_fold_bezout): Define.
* tree-fold-const.h (tree_fold_int_round_div,
tree_fold_int_trunc_mod, tree_fold_int_ceil_mod,
tree_fold_int_floor_mod, tree_fold_int_round_mod): Removed, because
not used for the moment.
(chrec_merge_types): New function.
* tree-optimize.c (pass_scev_elim_checks): Register the pass.
* tree-pass.h (pass_scev_elim_checks): Declare the pass.
* tree-pretty-print.c (dump_generic_node): Print
PEELED_CHREC. Remove PERIODIC_CHREC.
* tree-vectorizer.c (): Modify the use of
analyze_scalar_evolution.
* tree.def (POLYNOMIAL_CHREC, EXPONENTIAL_CHREC): Store the
evolution loop in a third leaf instead of in TREE_TYPE.
TREE_TYPE is then used in storing the type of the chrec.
(PERIODIC_CHREC): Removed since it is not used for the moment.
(PEELED_CHREC): New node.
* doc/invoke.texi (fdump-tree-scev, fdump-tree-ddall): Correct
the name of these flags.
(ftree-elim-checks, fdump-tree-elck): Document.
/* Elimination of redundant checks.
Copyright (C) 2004 Free Software Foundation, Inc.
Contributed by Sebastian Pop <sebastian.pop@cri.ensmp.fr>
This file is part of GCC.
GCC is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation; either version 2, or (at your option) any later
version.
GCC is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
for more details.
You should have received a copy of the GNU General Public License
along with GCC; see the file COPYING. If not, write to the Free
Software Foundation, 59 Temple Place - Suite 330, Boston, MA
02111-1307, USA. */
/*
Description:
Compute the scalar evolutions for all the scalar variables of a
condition expression, and based on this information performs a
proof. The condition is rewritten based on the result of this
proof.
Examples:
Example 1: A simple illustration of the algorithm.
Given the COND_EXPR "if (a < b)" with "a -> {2, +, 1}_1" and "b
-> {3, +, 1}_1", the proof consists in comparing these evolution
functions: is it always true for a given iteration x that "{2, +,
1}_1 (x) < {3, +, 1}_1 (x)"? The answer is yes, and the test of
the condition is consequently replaced by "1".
Further readings:
There is no further readings for the moment.
Based on the fact that this algorithm is similar to the Value
Range Propagation you can have a look at the corresponding
papers.
*/
#include "config.h"
#include "system.h"
#include "coretypes.h"
#include "tm.h"
#include "errors.h"
#include "ggc.h"
#include "tree.h"
/* These RTL headers are needed for basic-block.h. */
#include "rtl.h"
#include "basic-block.h"
#include "diagnostic.h"
#include "tree-flow.h"
#include "tree-dump.h"
#include "timevar.h"
#include "cfgloop.h"
#include "tree-fold-const.h"
#include "tree-chrec.h"
#include "tree-data-ref.h"
#include "tree-scalar-evolution.h"
#include "tree-pass.h"
#include "flags.h"
static void remove_redundant_check (tree, bool);
static void try_eliminate_check (tree);
static void scan_all_loops_r (struct loop *);
/* Remove the check by setting the condition COND to VALUE. */
static void
remove_redundant_check (tree cond, bool value)
{
/* A dead COND_EXPR means the condition is dead. We don't change any
flow, just replace the expression with a constant. */
if (tree_dump_file && (tree_dump_flags & TDF_DETAILS))
fprintf (tree_dump_file, "Replacing one of the conditions.\n");
if (value == true)
COND_EXPR_COND (cond) = integer_one_node;
else
COND_EXPR_COND (cond) = integer_zero_node;
modify_stmt (cond);
}
/* If the condition TEST is decidable at compile time, then eliminate
the check. */
static void
try_eliminate_check (tree cond)
{
bool value;
tree test, opnd0, opnd1;
tree chrec0, chrec1;
unsigned loop_nb = loop_num (loop_of_stmt (cond));
if (tree_dump_file && (tree_dump_flags & TDF_DETAILS))
{
fprintf (tree_dump_file, "(try_eliminate_check \n");
fprintf (tree_dump_file, " (cond = ");
print_generic_expr (tree_dump_file, cond, 0);
fprintf (tree_dump_file, ")\n");
}
test = COND_EXPR_COND (cond);
switch (TREE_CODE (test))
{
case SSA_NAME:
/* Matched "if (opnd0)" ie, "if (opnd0 != 0)". */
opnd0 = test;
chrec0 = analyze_scalar_evolution (loop_nb, opnd0);
if (chrec_contains_undetermined (chrec0))
break;
chrec0 = instantiate_parameters (loop_nb, chrec0);
if (tree_dump_file && (tree_dump_flags & TDF_DETAILS))
{
fprintf (tree_dump_file, " (test = ");
print_generic_expr (tree_dump_file, test, 0);
fprintf (tree_dump_file, ")\n (loop_nb = %d)\n (chrec0 = ", loop_nb);
print_generic_expr (tree_dump_file, chrec0, 0);
fprintf (tree_dump_file, ")\n");
}
if (prove_truth_value_ne (chrec0, integer_zero_node, &value))
remove_redundant_check (cond, value);
break;
case LT_EXPR:
case LE_EXPR:
case GT_EXPR:
case GE_EXPR:
case EQ_EXPR:
case NE_EXPR:
opnd0 = TREE_OPERAND (test, 0);
opnd1 = TREE_OPERAND (test, 1);
chrec0 = analyze_scalar_evolution (loop_nb, opnd0);
if (chrec_contains_undetermined (chrec0))
break;
chrec1 = analyze_scalar_evolution (loop_nb, opnd1);
if (chrec_contains_undetermined (chrec1))
break;
chrec0 = instantiate_parameters (loop_nb, chrec0);
chrec1 = instantiate_parameters (loop_nb, chrec1);
if (tree_dump_file && (tree_dump_flags & TDF_DETAILS))
{
fprintf (tree_dump_file, " (test = ");
print_generic_expr (tree_dump_file, test, 0);
fprintf (tree_dump_file, ")\n (loop_nb = %d)\n (chrec0 = ", loop_nb);
print_generic_expr (tree_dump_file, chrec0, 0);
fprintf (tree_dump_file, ")\n (chrec1 = ");
print_generic_expr (tree_dump_file, chrec1, 0);
fprintf (tree_dump_file, ")\n");
}
switch (TREE_CODE (test))
{
case LT_EXPR:
if (prove_truth_value_lt (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
case LE_EXPR:
if (prove_truth_value_le (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
case GT_EXPR:
if (prove_truth_value_gt (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
case GE_EXPR:
if (prove_truth_value_ge (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
case EQ_EXPR:
if (prove_truth_value_eq (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
case NE_EXPR:
if (prove_truth_value_ne (chrec0, chrec1, &value))
remove_redundant_check (cond, value);
break;
default:
break;
}
break;
default:
break;
}
if (tree_dump_file && (tree_dump_flags & TDF_DETAILS))
fprintf (tree_dump_file, ")\n");
}
/* Compute the exit edges for all the loops. */
static void
scan_all_loops_r (struct loop *loop)
{
if (!loop)
return;
/* Recurse on the inner loops, then on the next (sibling) loops. */
scan_all_loops_r (inner_loop (loop));
scan_all_loops_r (next_loop (loop));
flow_loop_scan (loop, LOOP_EXIT_EDGES);
}
/* Walk over all the statements, searching for conditional statements.
A better way to determine the conditional expressions that are good
candidates for elimination would be needed. For the moment
systematically search the conditional expressions over the whole
function. */
void
eliminate_redundant_checks (void)
{
basic_block bb;
block_stmt_iterator bsi;
bb = BASIC_BLOCK (0);
if (bb && bb->loop_father)
{
scan_all_loops_r (bb->loop_father);
FOR_EACH_BB (bb)
{
struct loop *loop = bb->loop_father;
/* Don't try to prove anything about the loop exit
conditions: avoid the block that contains the condition
that guards the exit of the loop. */
if (!loop_exit_edges (loop)
|| edge_source (loop_exit_edge (loop, 0)) == bb)
continue;
for (bsi = bsi_start (bb); !bsi_end_p (bsi); bsi_next (&bsi))
{
tree expr = bsi_stmt (bsi);
switch (TREE_CODE (expr))
{
case COND_EXPR:
try_eliminate_check (expr);
break;
default:
break;
}
}
}
}
}
Attachment: testsuite.tar.gz
Description: Binary data
Attachment: scev3.diff.gz
Description: Binary data
Index Nav: Message Nav: [Date Index] [Subject Index] [Author Index] [Thread Index] [Date Prev] [Date Next] [Thread Prev] [Thread Next] | 2020-07-13 02:01:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5018956661224365, "perplexity": 12962.527961970432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657140746.69/warc/CC-MAIN-20200713002400-20200713032400-00138.warc.gz"} |
https://proofwiki.org/wiki/Definition:Geometry | # Definition:Geometry
## Definition
Geometry is a branch of mathematics which studies such matters as form, position, dimension and various other properties of ordinary space.
It has been suggested that geometry can be divided into $3$ main branches:
Metrical geometry, that is to say, what is understood as geometry proper
Projective geometry
Analytic geometry
## Also see
• Results about geometry can be found here.
## Historical Note
The Greek historian Herodotus visited Egypt in about $450$ BCE and reported that the annual flooding of the Nile would wipe out all the boundaries between fields.
It is suggested that the discipline of geometry may have arisen there, from the need for the rulers to re-establish those boundaries subsequently.
## Linguistic Note
The word geometry derives from the Greek words meaning Earth and measure, suggesting that the discipline arose from the need to measure land. | 2021-05-07 03:48:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4823647439479828, "perplexity": 1858.5175640128507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988774.96/warc/CC-MAIN-20210507025943-20210507055943-00175.warc.gz"} |
https://math.stackexchange.com/questions/3293654/can-any-linear-operator-be-written-as-composition-of-linear-maps | # Can any linear operator be written as composition of linear maps?
Let $$V$$ be a finite dimensional vector space over $$\mathbb{R}$$ and $$T:V\to V$$ be a linear map. Can you always write $$T= T_2\circ T_1$$ for some linear maps $$T_1:V\to W$$, $$T_2:W\to V$$, where $$W$$ is some finite dimensional vector space and such that
1. both $$T_1$$ and $$T_2$$ are onto
2.both $$T_1$$ and $$T_2$$ are one to one
1. $$T_1$$ is onto, $$T_2$$ is one to one
2. $$T_1$$ is one to one , $$T_2$$ is onto
One or more than one option are correct.
Do I have to show that any n×n matrix $$A$$ can be written as products of some n×m matrix $$B$$ and some m×n matrix $$C$$ such that one or more than one of the above options holds?
I have no clue then how to proceed.
Any help would be great. Thanks.
Edit: I think(after reading the question for almost 1 hour) that I have to choose different $$W$$ in each case otherwise the problem is absurd.
So , the first case and the 2nd case are false if I consider the zero transformation , whatever $$W$$ I choose, it doesn't matter .
I am working on case 3 and 4 , so if anyone comes up with anything please let me know.
• Is $W$ given? – Bernard Jul 15 '19 at 10:57
• What are your thoughts? Are you allowed to pick a different $W$ each time? – Joppy Jul 15 '19 at 10:59
• $W$ is exactly what I write above. and I wrote my thought above too. ..and but I don't know whether my thought is right or wrong. – suchanda adhikari Jul 15 '19 at 11:16
Hints:
1. The composition of two injective (resp. surjective) maps is injective (resp. surjective).
If $$W$$ is not given:
1. For case 3, think of $$\operatorname{Im}T$$.
2. For the last case, let $$E$$ any vector space. Look at $$W=V\oplus E$$.
• @ Bernard sir, I didn't understand your hints first, but now after spending a quality time with this problem I finally got your hints ...for the last option I realised that dim of $E$ has to be greater than or equal to dim of $V$. I want to know your opinion about that. – suchanda adhikari Jul 15 '19 at 18:30
• I don't think so, as there is always a canonical injection from each factor into their direct sum, and canonical projections from the direct sum onto each factor. This is independent of the dimensions.. I'm glad you've understood my hints, as they're intended to make the O.P. think about the problems, not to solve for him/her. – Bernard Jul 15 '19 at 18:37
• sir, sorry for disturbing you again, I understood what you said above but I didn't understand how it's gonna work in any arbitrary linear map $T$ . For example, suppose $T$ is the zero transformation then for the last case I have to find a suitable $W$ and a one one linear map $T_1$ and an onto linear map $T_2$ such that $T_2\circ T_1=T$ . I chose $W=E\oplus V$ where $E$ contains range of $T_1$ and then define $T_2$(x+v)=v for all x in E . Thus I can write $T_2\circ T_1=T$. So I don't understand how can I take any vector space $E$ . – suchanda adhikari Jul 16 '19 at 14:44
• Please sir if you please explain a little more. – suchanda adhikari Jul 16 '19 at 14:46
• You're right, I didn't explain enough this point. I only meant, as a hint, that, given a direct sum, we have canonical injections, and canonical projections, that you can choose $E$, so that $W=V\oplus E$ and linear maps $T_1, T_2$ satisfying the required conditions. For your example ($T=0$), just take $E=V$, $T_1 =$ the injection of the first factor, and $T_2=$ the second projection. – Bernard Jul 16 '19 at 14:58 | 2020-07-06 09:57:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8837874531745911, "perplexity": 206.49514886180975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890157.10/warc/CC-MAIN-20200706073443-20200706103443-00218.warc.gz"} |
https://math.eretrandre.org/tetrationforum/printthread.php?tid=204 | Additional super exponential condition - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Additional super exponential condition (/showthread.php?tid=204) Additional super exponential condition - bo198214 - 10/13/2008 I was just thinking about the following for an arbitrary super exponential $\text{sexp}$: We surely have for natural numbers m and n that $\text{sexp}(n+m)\ge \text{sexp}(n) ^ {\text{sexp}(m)}$ So why not demand this rule also for the super exponential extended to the reals? For a super logarithm the rule would be: $\text{slog}(x^y) \le \text{slog}(x) + \text{slog}(y)$ Note that this rule is not applicable to the left-bracketed super exponentials. Because from the rule it follows already that: $\text{sexp}(n)\ge \exp^{\circ n}(1)$ which is not valid for left bracketed super exponentials because they grow more slowly. I didnt verify the rule yet for our known tetration extensions. Do you think it will be valid? However I dont think that this condition suffice as a uniqueness criterion. But at least it would reduce the set of valid candidates. RE: Additional super exponential condition - andydude - 10/14/2008 bo198214 Wrote:For a super logarithm the rule would be: $\text{slog}(x^y) \le \text{slog}(x) + \text{slog}(y)$ This is certainly consistent. For example: $\text{slog}(e^y) \le \text{slog}(e) + \text{slog}(y)$ $\text{slog}(y) + 1 \le \text{slog}(e) + \text{slog}(y)$ $1 \le \text{slog}(e)$ $1 \le 1$ which is true. Andrew Robbins Extension by mean values - bo198214 - 10/21/2008 Ansus Wrote:By the way, I had an idea to extend hyper-operator based on the sequence of mean values:And how? I.e. what is $\text{mean}_4(a,b)$? RE: Additional super exponential condition - bo198214 - 10/21/2008 Ansus Wrote:$ \text{mean}_n(a,b)=\text{hroot}_{n+1}(\text{hyper}_n(a,b),2)$ Ya of course, but what is $\text{hyper}_4(a,b)$? You said you have an idea how to extend it to real $b$ via those means. RE: Additional super exponential condition - martin - 10/21/2008 I doubt this is an option to extend the mean value operations. In a given set of data (say a1, a2, ...), ordering is irrelevant for calculating a mean value. But a1^a2(^a3...an) is different from a2^a1(^a3...). Well, at least most of the time. | 2022-01-24 03:57:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 13, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6543070077896118, "perplexity": 1300.6450723733383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00581.warc.gz"} |
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A man bought 10 crates of oranges for $80 total. If he lost 2 of the crates, at what price would he have to sell each of the remaining crates in order to earn a total profit of 25 percent of the total cost? A.$10.00
B. $10.50 C.$12.00
D. $12.50 E.$120.00
[Reveal] Spoiler: OA
Last edited by Bunuel on 07 Jan 2013, 01:39, edited 1 time in total.
Edited the question and added the OA.
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Re: A man bought 10 crates of oranges for $80 total. If he lost [#permalink] ### Show Tags 05 Jan 2013, 05:02 Cost per crate is 8$.
Now he lost 2 crates out of 10 that means quantity lost is 20%. To recover loss from this loss he should increase price by 25% ( 2/10 loss in quantity => 2/8 rise in cost on remaining for break-even). Thus break-even price for remaining 8 crates is 10 $(8*125/100 or 8* 10/8). Now he wants 25% profit on top of this. So he should raise break-even price by 25%. So the answer is 12.5$
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Re: A man bought 10 crates of oranges for $80 total. If he lost [#permalink] ### Show Tags 05 Jan 2013, 09:23 10 Crates for$80
25% Profit over $80 = (5/4)*$80 = $100 Lost 2 Crates, So remaining Crates 10-2 = 8 Crates 8 Crates he has to sell for$100
so, 1 Crate for $100/8 =$12.5
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Re: A man bought 10 crates of oranges for $80 total. If he lost [#permalink] ### Show Tags 07 Jan 2013, 01:43 scoobee wrote: A man bought 10 crates of oranges for$80 total. If he lost 2 of the crates, at what price would he have to sell each of the remaining crates in order to earn a total profit of 25 percent of the total cost?
A. $10.00 B.$10.50
C. $12.00 D.$12.50
E. $120.00 The total cost is$80. In order to earn a total profit of 25% the total sales must be 80*1.25 = $100. Thus he must sell each of he remaining crates for 100/8 =$12.5.
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Re: A man bought 10 crates of oranges for $80 total. If he lost [#permalink] 07 Jan 2013, 01:43 Similar topics Replies Last post Similar Topics: David bought 13 BMW cars for a total price of 1,105,000 dollars. If he 3 08 Jun 2016, 05:09 1 Ralph bought a stereo priced at$120. If the total price he paid at th 2 07 Feb 2016, 10:50
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Display posts from previous: Sort by | 2016-08-31 02:42:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5723194479942322, "perplexity": 10204.839008685598}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471983553659.85/warc/CC-MAIN-20160823201913-00292-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/cauchy-sequence.576013/ | # Cauchy Sequence
## Homework Statement
Show that the sequence of real numbers defined by $x_{n + 1} = x_n + \frac{1}{x_n^2}, \, x_1 = 1$ is not a Cauchy sequence.
## Homework Equations
A sequence $\{ p_n \}$ is Cauchy if and only if, for all $\varepsilon > 0$, there exists an $N > 0$ such that $d(p_n, p_m) < \varepsilon$ for all $m, n > N$.
## The Attempt at a Solution
We can assume that $d$ is the usual metric on $\mathbb{R}$. I don't even see where to begin. I see that the sequence is monotonically increasing, so that
$1 = \frac{1}{x_1} > \frac{1}{x_2} > \frac{1}{x_3} > \dotsb.$
So
$1 = \frac{1}{x_1^2} > \frac{1}{x_2^2} > \frac{1}{x_3^2} > \dotsb.$
Related Calculus and Beyond Homework Help News on Phys.org
Well if you're trying to show that it is not Cauchy, state what it means for a sequence to not be Cauchy. That is where I would start :)
Well if you're trying to show that it is not Cauchy, state what it means for a sequence to not be Cauchy. That is where I would start :)
The sequence isn't Cauchy if there exists an $\varepsilon > 0$ such that for all $N > 0$ there exists $m, n > 0$ such that $d(x_m, x_n) \geq \varepsilon$.
...
still stuck
...
LCKurtz
Homework Helper
Gold Member
Do you have the theorem that a sequence is Cauchy if and only if it is convergent? And if so, what happens if you suppose $\lim_{x\rightarrow \infty} = L$?
Do you have the theorem that a sequence is Cauchy if and only if it is convergent? And if so, what happens if you suppose $\lim_{x\rightarrow \infty} = L$?
Yes, I can assume the sequence is Cauchy if and only if it is convergent. The definition of limits that we are using states that $\lim_{n \to \infty} x_n = L$ if and only if
$\forall \varepsilon > 0 \, \exists N > 0 \, s.t. \, \forall n > N \quad d(L, x_n) < \varepsilon$.
Now can I somehow use the fact that $x_n$ is increasing to say that $d(L, x_n)$ is always increasing? And hence it is greater than or equal to $\varepsilon$ for some $n$?
LCKurtz
Homework Helper
Gold Member
Yes, I can assume the sequence is Cauchy if and only if it is convergent. The definition of limits that we are using states that $\lim_{n \to \infty} x_n = L$ if and only if
$\forall \varepsilon > 0 \, \exists N > 0 \, s.t. \, \forall n > N \quad d(L, x_n) < \varepsilon$.
Now can I somehow use the fact that $x_n$ is increasing to say that $d(L, x_n)$ is always increasing? And hence it is greater than or equal to $\varepsilon$ for some $n$?
It's easier than that. If the sequence has a limit, what happens if you take the limit of both sides of your recursion?
It's easier than that. If the sequence has a limit, what happens if you take the limit of both sides of your recursion?
The limit of both sides of the recurrence should then equal the same thing, namely $L$. Then I would have that
$\lim_{n \to \infty} x_{n + 1} = \lim_{n \to \infty} x_n + \frac{1}{x_n^2} = L + \lim_{n \to \infty} \frac{1}{x_n^2} = L$.
Doesn't this simply prove that $\lim_{n \to \infty} \frac{1}{x_n^2} = 0$?
The limit of both sides of the recurrence should then equal the same thing, namely $L$. Then I would have that
$\lim_{n \to \infty} x_{n + 1} = \lim_{n \to \infty} x_n + \frac{1}{x_n^2} = L + \lim_{n \to \infty} \frac{1}{x_n^2} = L$.
Doesn't this simply prove that $\lim_{n \to \infty} \frac{1}{x_n^2} = 0$?
But then doesn't this imply $\lim_{n \to \infty} x_n^2 = \infty$? completing the problem?
Matterwave | 2020-01-21 16:41:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9633874297142029, "perplexity": 87.97010083742535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250604849.31/warc/CC-MAIN-20200121162615-20200121191615-00148.warc.gz"} |
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https://nrich.maths.org/584 | ### Diophantine N-tuples
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### Sixational
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.
# Janine's Conjecture
##### Age 14 to 16 Challenge Level:
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number cubed''; e.g., 3, 4, 5 gives 3 x 4 x 5 + 4 = 64, which is a perfect cube. Does this always work? Can you prove or disprove this conjecture? | 2020-04-03 18:20:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8315260410308838, "perplexity": 461.4422550407977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370515113.54/warc/CC-MAIN-20200403154746-20200403184746-00383.warc.gz"} |
https://leanprover-community.github.io/archive/stream/217875-Is-there-code-for-X%3F/topic/eq_iff_modeq_int.html | ## Stream: Is there code for X?
### Topic: eq_iff_modeq_int
#### Kevin Buzzard (May 16 2020 at 16:48):
lemma eq_iff_modeq_int {n : ℕ+} {a b : ℤ} : (a : zmod n) = b ↔ a ≡ b [ZMOD (n : ℕ)] := ...
I'm fixing up some old codewars kata and I find that this has disappeared from the library, in #2511. I need (a : zmod n) = b -> a % n = b % n. How am I supposed to get that now?
#### Kevin Buzzard (May 16 2020 at 17:14):
I need to learn about char_p?
#### Patrick Massot (May 16 2020 at 17:14):
No you don't, because you already learned it for the perfectoid project.
#### Kevin Buzzard (May 16 2020 at 17:22):
But n isn't prime!
#### Kevin Buzzard (May 16 2020 at 19:07):
lemma eq_iff_modeq_int {n : ℕ} {a b : ℤ} : (a : zmod n) = b ↔ a ≡ b [ZMOD (n : ℕ)] :=
by rw [eq_comm, ←sub_eq_zero, ←int.cast_sub, char_p.int_cast_eq_zero_iff (zmod n) n, int.modeq.modeq_iff_dvd]
#### Johan Commelin (May 16 2020 at 19:08):
But this doesn't use zmod n in an essential way, right?
#### Johan Commelin (May 16 2020 at 19:09):
So it should be a lemma about arbitrary R with char_p R n
#### Kevin Buzzard (May 16 2020 at 19:09):
All I care about is that my kata is compiling again right now
#### Mario Carneiro (May 16 2020 at 20:00):
@Johan Commelin you deleted a lemma? shame on you
#### Johan Commelin (May 16 2020 at 20:00):
Did I? Where? When?
#### Mario Carneiro (May 16 2020 at 20:01):
This was a lemma before the refactor, I can't find it now
Oops
#### Kevin Buzzard (May 16 2020 at 20:02):
eq_zero_iff_dvd_int got renamed to char_p.int_cast_eq_zero_iff and a similar story for nat, but the lemma in question just seemed to vanish completely.
#### Johan Commelin (May 16 2020 at 20:02):
I see... I guess I messed up with porting some of the zmodp stuff to zmod p.
My apologies
#### Kevin Buzzard (May 16 2020 at 20:03):
Johan is saying that we have a better lemma somewhere now
#### Johan Commelin (May 16 2020 at 20:03):
Hmm... except that maybe we don't have it...
#### Johan Commelin (May 16 2020 at 20:03):
I think I messed up
#### Johan Commelin (May 16 2020 at 20:06):
I will try to restore stuff next week... now I first need to rest.
#### Reid Barton (May 16 2020 at 20:07):
This is what you get for not PRing your katas to mathlib
exactly
#### Kevin Buzzard (May 16 2020 at 20:08):
/me deletes all of his Lean repos other than mathlib
Noooo!
#### Patrick Massot (May 16 2020 at 20:09):
Fortunately git is decentralized
#### Johan Commelin (May 16 2020 at 20:09):
The perfectoid repo is on the community account
#### Kevin Buzzard (May 17 2020 at 10:53):
Whilst proving example : (R : Type*) [ring R] (p : ℕ) [char_p R p] (a b : ℤ) : (a : R) = b ↔ a ≡ b [ZMOD p] I noticed the following thing:
theorem sub_eq_zero : a - b = 0 ↔ a = b
and
theorem modeq_iff_dvd : a ≡ b [ZMOD n] ↔ (n:ℤ) ∣ b - a
One a-b, one b-a. This meant that I had to add an eq_comm in the following proof:
import number_theory.quadratic_reciprocity
import data.int.modeq
lemma thing (R : Type*) [ring R] (p : ℕ) [char_p R p] (a b : ℤ) :
(a : R) = b ↔ a ≡ b [ZMOD p] :=
begin
rw eq_comm,
rw ←sub_eq_zero,
rw ←int.cast_sub,
rw char_p.int_cast_eq_zero_iff R p,
rw int.modeq.modeq_iff_dvd,
end
I think the above lemma is the correct thing to PR to mathlib (with perhaps a better proof) but this experience did make me wonder whether there should be a "convention" for the order of the subtractands when proving lemmas of this form.
#### Johan Commelin (May 18 2020 at 12:09):
I think the trouble with my refactor PR was that I completely rewrote the API for zmod. Hence a lot of auxilliary lemmas no longer made sense, or weren't needed. I guess this lemma was collateral damage. My apologies.
@Kevin Buzzard Do you want to PR it, or shall I do it?
#### Kevin Buzzard (May 18 2020 at 12:37):
I don't want to PR it, my PR's usually have to go through a few iterations and I am not looking at the mathlib github site at all at the minute
Ok, I'll do it.
#### Kevin Buzzard (May 18 2020 at 12:49):
Thanks. I'm trying to get another Lean post-doc and time is tight :-)
#### Johan Commelin (May 18 2020 at 13:33):
Is there a slick way of proving
import algebra.char_p
import data.int.modeq
lemma eq_iff_modeq (R : Type*) [semiring R] (p : ℕ) [char_p R p] (a b : ℕ) :
(a : R) = b ↔ a ≡ b [MOD p] :=
begin
wlog h : a ≤ b,
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le h,
rw [nat.modeq.modeq_iff_dvd' h, ← char_p.cast_eq_zero_iff R p,
end
Currently I can only think of going through zmod p...
#### Kenny Lau (May 18 2020 at 13:35):
have we fixed wlog?
#### Johan Commelin (May 18 2020 at 13:37):
No, I don't think so.
#### Johan Commelin (May 18 2020 at 13:37):
But that's not the hard part (-;
#### Johan Commelin (May 18 2020 at 13:37):
I can easily work aroun wlog, of course
#### Johan Commelin (May 18 2020 at 13:37):
Note that in this particular case it isn't slow (for a human)
#### Kenny Lau (May 18 2020 at 13:44):
import algebra.char_p
import data.int.modeq
set_option profiler true
-- elaboration of eq_iff_modeq took 1.41s
lemma eq_iff_modeq (R : Type*) [semiring R] (p : ℕ) [char_p R p] (a b : ℕ) :
(a : R) = b ↔ a ≡ b [MOD p] :=
begin
wlog h : a ≤ b,
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le h,
rw [nat.modeq.modeq_iff_dvd' h, ← char_p.cast_eq_zero_iff R p,
end
-- elaboration of eq_iff_modeq' took 222ms
lemma eq_iff_modeq' (R : Type*) [semiring R] (p : ℕ) [char_p R p] (a b : ℕ) :
(a : R) = b ↔ a ≡ b [MOD p] :=
begin
wlog h : a ≤ b := le_total a b using [a b, b a] tactic.skip,
swap, { rwa [eq_comm, nat.modeq, @eq_comm _ (a % p)] },
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le h,
rw [nat.modeq.modeq_iff_dvd' h, ← char_p.cast_eq_zero_iff R p,
` | 2021-05-17 16:08:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4854150712490082, "perplexity": 4628.1752395529675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00350.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/97844-unbiased-estimators.html | 1. ## Unbiased estimators
Suppose that $X_{1}, X_{2},...,X_{n}$ are independently and identically distributed variables in a random sample drawn from a population with mean $\mu=20$ and variance $\sigma^{2}=9$.
a) Determine the mean and variance of $Y=\frac{3X_{1}+2X_{2}+X_{3}+...+X_{n}}{n+3}$.
b) Construct your own unbiased estimator for the population mean, and calculate its variance.
I'm not entirely sure what I'm doing for a) and I don't know what to do for b) at all. Can anyone help?
2. Originally Posted by chella182
Suppose that $X_{1}, X_{2},...,X_{n}$ are independently and identically distributed variables in a random sample drawn from a population with mean $\mu=20$ and variance $\sigma^{2}=9$.
a) Determine the mean and variance of $Y=\frac{3X_{1}+2X_{2}+X_{3}+...+X_{n}}{n+3}$.
Use:
$E\left(\sum_{i=1}^n \lambda_i X_i\right)=\sum_{i=1}^n \lambda_i E(X_i)$
That will give you the mean straight off. Then as the $X_i$'s are iid:
$Var\left(\sum_{i=1}^n \lambda_i X_i\right)=\sum_{i=1}^n \lambda_i^2 Var(X_i)$
CB
3. Originally Posted by chella182
Suppose that $X_{1}, X_{2},...,X_{n}$ are independently and identically distributed variables in a random sample drawn from a population with mean $\mu=20$ and variance $\sigma^{2}=9$.
a) Determine the mean and variance of $Y=\frac{3X_{1}+2X_{2}+X_{3}+...+X_{n}}{n+3}$.
b) Construct your own unbiased estimator for the population mean, and calculate its variance.
I'm not entirely sure what I'm doing for a) and I don't know what to do for b) at all. Can anyone help?
For b) why not make like simple for yourself at select an estimator:
$m=X_1$
for $\mu$ ?
(The question does not require that the estimator be particularly good in whatever sense you might like to choose, only that it be unbiased)
CB
4. Originally Posted by CaptainBlack
Use:
$E\left(\sum_{i=1}^n \lambda_i X_i\right)=\sum_{i=1}^n \lambda_i E(X_i)$
That will give you the mean straight off. Then as the $X_i$'s are iid:
$Var\left(\sum_{i=1}^n \lambda_i X_i\right)=\sum_{i=1}^n \lambda_i^2 Var(X_i)$
CB
Okay, I know all of that, but would that make the mean $\frac{80+40n}{n+3}$?
5. Originally Posted by chella182
Okay, I know all of that, but would that make the mean $\frac{80+40n}{n+3}$?
$
Y=\frac{3X_{1}+2X_{2}+X_{3}+...+X_{n}}{n+3}
$
so:
$E(Y)=\frac{1}{n+3} \left(3E(X_1)+2E(X_2)+ E(X_3) + .. + E(X_n)\right)$ $=\frac{1}{n+3} \left(3(20)+2(20)+ (20) + .. + (20)\right)=20$
CB
6. Originally Posted by CaptainBlack
$
Y=\frac{3X_{1}+2X_{2}+X_{3}+...+X_{n}}{n+3}
$
so:
$E(Y)=\frac{1}{n+3} \left(3E(X_1)+2E(X_2)+ E(X_3) + .. + E(X_n)\right)$ $=\frac{1}{n+3} \left(3(20)+2(20)+ (20) + .. + (20)\right)=20$
CB
Sorry, I don't understand how you've gotten 20
7. ## Hidden sense
I think I track the hidden sense of your problem. They expect you to compare different population mean estimators including sample mean that looks like (X_1+...X_n)/n because it has the smallest variance among unbiased pop mean estimators (Gauss-Markov) provided X_i have identical variances. You can choose any combination g_1*X_1+...+g_n*X_n with g_1+...+g_n=1 as unbiased pop mean estimate. If they have the same means but different variances the optimal weights g_i are proportional to
1/(sigma_i)^2. Let's say e.g. g_i=1/(sigma_i)^2 / ( 1/(sigma_1)^2+...+1/(sigma_n)^2 )
8. Originally Posted by chella182
Sorry, I don't understand how you've gotten 20
$E(X_1)=E(X_2)= ... =E(X_n)=20$
Now count up how many expectations you have, there are $n$ terms but the first two add an extra $3 \times 20$ on so you have:
$E(Y)=\frac{(n+3)20}{n+3}=20$
CB
9. Originally Posted by CaptainBlack
$E(X_1)=E(X_2)= ... =E(X_n)=20$
Now count up how many expectations you have, there are $n$ terms but the first two add an extra $3 \times 20$ on so you have:
$E(Y)=\frac{(n+3)20}{n+3}=20$
CB
Ah right, yeah, thankyou. | 2017-08-17 22:07:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 36, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8019907474517822, "perplexity": 441.7995716164207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104160.96/warc/CC-MAIN-20170817210535-20170817230535-00003.warc.gz"} |
https://wiki.evoludo.org/index.php?title=EvoLudoLab:_Rock-Paper-Scissors_-_IBS | # EvoLudoLab: Rock-Paper-Scissors - IBS
Along the bottom of the applet there are several buttons to control the execution and the speed of the simulations - for details see the EvoLudo GUI documentation. Of particular importance are the parameters button and the data views pop-up list along the top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.
Color code: Rock Scissors Paper New rock New scissors New paper
Payoff code: Low High
## Individual based simulations (IBS)
The interior fixed point $$\hat x$$ is a stable focus of the replicator dynamics. Stochastic fluctuations arise in individual based simulations of populations with a finite size, $$N=1000$$. In the absence of mutations, the boundaries are absorbing and even though the interior fixed point is attracting, the population will eventually end up in one of the absorbing homogenous states with all rock, all paper or all scissors.
The parameters are $$s = 1.4$$ with a starting point is close to the attractor $$\hat x=(1/2, 1/3, 1/6)$$.
### Data views
Strategy - Population
Snapshot of the spatial arrangement of strategies.
Strategy 3D - Population
Snapshot of the spatial arrangement of strategies in 3D.
Strategy - Mean
Time evolution of the strategy frequencies.
Strategy - Simplex $$S_3$$
Trajectories of strategy frequencies shown in the simplex $$S_3$$. Double clicks in the interior of $$S_3$$ set the initial frequencies of strategies.
Fitness - Population
Snapshot of the spatial distribution of payoffs.
Fitness 3D - Population
Snapshot of the spatial distribution of payoffs in 3D.
Fitness - Mean
Time evolution of average population payoff bounded by the minimum and maximum individual payoff.
Fitness - Histogram
Payoff distribution of each strategy in population.
Degree - Histogram
Degree distribution in structured populations.
Console Log
Displays messages, warnings and errors reported by the simulation engine plus information on the applet/application.
## Game parameters
The list below describes only the parameters related to the rock-scissors-paper game and the population dynamics. Follow the link for a complete list and descriptions of all other parameters such as spatial arrangements or update rules on the player and population level.
Payoffs
general $$3\times 3$$ payoff matrix for the three strategic types $$R, S, P$$. The rock-paper-scissors game requires cyclic dominance between the three types.
Init Rock, Scissors, Paper
initial frequencies of rock, paper, and scissors types. If they do not add up to 100%, the values will be scaled accordingly. | 2019-01-17 12:50:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3542563319206238, "perplexity": 1511.6832396611555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658981.19/warc/CC-MAIN-20190117123059-20190117145059-00588.warc.gz"} |
https://socratic.org/questions/562a93b211ef6b1a79823009 | # One parental organism has the genotype "TT", and another has the genotype "Tt". What will the genotypes of their gametes be?
Parent organism a) would produce only one type of gamete, $\text{T}$.
Parent organism b) would produce two types of gametes, $\text{T}$ and $\text{t}$.
The parental diploid genotypes, such as $\text{TT}$ and $\text{Tt}$ will be separated into haploid gametes during meiosis. So the parent with the genotype $\text{TT}$ can only pass on the $\text{T}$ allele to all of its gametes. So it would produce only one type of gamete when it comes to the $\text{TT}$ genotype, which would be $\text{T}$. The parent with the genotype $\text{Tt}$ has two different alleles, and half of its gametes will have the $\text{T}$ allele, and half will have the $\text{t}$ allele. So in regard to the $\text{Tt}$ genotype, the parent organism will produce two different types of gametes, $\text{T}$ and $\text{t}$. | 2022-10-04 04:29:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 15, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7545632123947144, "perplexity": 1619.2596370207136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337473.26/warc/CC-MAIN-20221004023206-20221004053206-00556.warc.gz"} |
https://www.physicsforums.com/threads/stability-for-turbulent-boundary-layer.123251/ | Stability for turbulent boundary layer
1. Jun 8, 2006
alsey42147
concerning fluid flow past a body, i know that if the boundary layer is laminar then the flow becomes unstable when the Reynold's number of the flow is greater than the critical Reynold's number. is this also true when the boundary layer is turbulent?
tia
2. Jun 8, 2006
Staff: Mentor
I'm not quite sure what you are asking, but a turbulent boundary layer is more stable than a laminar one. Flow separation is less violent.
3. Jun 10, 2006
Clausius2
The turbulent boundary layer is unstable per-se. In a laminar boundary layer, as the Reynolds number increases, the flow becomes more unstable in the sense than any small perturbation in the flow field ends up growing above the critical Reynolds. The flow becomes unstable, all the perturbations are amplified, and after the transition to fully turbulent boundary layer the flow is unstable but statistically stable. The transition to instability of a laminar boundary layer is described mathematically as the classical eigenvalue problem so called Orr-Somerfeld. | 2017-03-01 18:17:31 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8482044339179993, "perplexity": 549.9995424735052}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501174215.11/warc/CC-MAIN-20170219104614-00497-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/why-dont-the-slits-collapse-the-wave-function.131634/ | # Why don't the Slits collapse the wave function?
1. Sep 11, 2006
### kliide
I found a layman's explanation of the wave characteristics of subatomic particles in the form of a "Dr.Quantum" video from "What the Bleep do we know?". Aside from the parapsychological junk in the last 2/3rds of the movie, the explanations of quantum properties seemed mostly accurate and concise. The way they described subatomic behavior was easy to digest from a layman's perspective.
One thing I don't understand is how the screen with the slits do not collapse the wave function until there is a measurement. Wouldn't bouncing off the sides of the slit constitute some sort of interaction?
What is it about the "observer" or measurement device that would interact differently with the photon than how the photon interacts with the slit?
2. Sep 11, 2006
### ZapperZ
Staff Emeritus
The issue here isn't the slit. The issue of importance here is the PATH, and that there are two paths that it can go through. So the superposition is the path it took through the slit. I can easily do the same type of experiment using a superconductor having a split path - you get the same type of "interference" pattern. This is what you get from Superconducting Quantum Interference Devices (SQUIDs).
Zz.
3. Sep 12, 2006
### kliide
Thanks for the quick reply. After some mulling over your response, I think I asked the wrong question, but it's been sorted out now and I appreciate the help.
Last edited: Sep 12, 2006
4. Sep 12, 2006
### J77
I hope I get this right, I to am an amateur but quantum enthusiast...
The wave is like a corkscrew, representing the momentum.
When this goes through the slits you have two corkscrews upon which changes of the phase give the differing interference patterns.
If you measure the position, you lose the corkscrew - the Fourier transform of which is the position: a Dirac delta function.
Therefore, you lose the interference pattern.
Someone please correct me if wrong - I'm only recalling what I've read recently in Penrose's books.
5. Sep 12, 2006
### jpr0
I think the question he is asking is why interaction with the slit doesn't constitute a measurement, whereas interaction with some kind of "detection" device does. What's the difference between interacting with the slit and the thing we use to measure the electron?
The double slits can be anything - they can be two impurities in parallel from which the electrons scatter, in which case the above question looks more reasonable.
I think the answer is because the slit (or impurities) has no internal degrees of freedom, so as the electron passes through the slit arrangement there is no randomization of its phase, and at the detector the parts of the wave function arriving from both paths have a definite phase relationship, which produces interference.
If you give the impurity (or slit) some internal degree of freedom, so that it can change its state upon interaction with the electron, then you will (may...) randomize the phase of the electron, and there will be no definite phase relationship between the two paths.
This is explained in a nice way in the book by Datta, "Electron transport in mesoscopic systems" (the experiment he talks about is an electron in an Ahronov-Bohm ring). The point is that if the impurity, or slit, has a internal degrees of freedom, then it's possible, by measuring the state of the impurity, to tell which path the electron took. Hence any interference would be destroyed.
This might be incorrect in some way, but it's what I understand from what I've read.
6. Sep 12, 2006
### ZapperZ
Staff Emeritus
The "slit" in the idealized case is simply an illustration of separate paths for the photon, electron, neutron, buckyball, etc. As I've said, that isn't the real issue. Now, if the slit happens to be a metallic device that can somehow detect things like electric field of an electron or a photon, THEN it is now a detector that can tell you if a photon, electron, or whatever, passed through it. This is now a different set up. In that case, yes, the slit will cause an interaction that is now part of the system to be considered.
Zz.
7. Sep 12, 2006
### jpr0
Hi,
I agree that the slits are there only to spatially seperate trajectories, and highlight the effect of interference, which is fine. But the question he was asking was
'What is it about the "observer" or measurement device that would interact differently with the photon than how the photon interacts with the slit?'
The answer here is that one destroys phase coherence and the other does not.
8. Sep 12, 2006
Staff Emeritus
I like this answer, but could you or someone else flesh it out a bit showing how the different actions on the wave function look?
9. Sep 12, 2006
### masudr
It's just like how you need phase coherence for there to be any useful interference. The reason for this is that to get Fraunhofer diffraction, we require the phase difference between two bits of wave to be a linear function of position, otherwise the Fourier integral doesn't give a nice answer.
10. Sep 15, 2006
### blackwizard
Bloody hell, its hard 2 figure out wat ye guys r talkin bout sometimes! Internal degrees of freedom?
Im just gonna ask if my view is right or not. Which is double slit does collapse the wavefunction. When the wavefunction hits the double slit plate it can collaspe to a point on the plate (particle hits the plate itself) or to the area covered by the two slits (particle passes through the slits). So in the second possibility only the part of the wavefunction that hit the plate collapsed and the remaining part, that emerges the other side is the shape of the slits. Then the 2 parts of the wavefunction (slit 1 & 2) spread out in2 each other to cause the interferance.
11. Sep 15, 2006
### ZapperZ
Staff Emeritus
Now was that simple enough?
Zz.
12. Sep 15, 2006
### blackwizard
13. Sep 15, 2006
### ZapperZ
Staff Emeritus
Tell me how to possibly do that without using "big words".
You freely used the terms "wavefunction" and "collapse" as IF they were "simple" words, where in reality, there's a LOT of physics that comes with such terminology. Those are not simply words that can be used weely neely. Everything in physics has CLEAR and unambiguous definition with underlying mathematical description.
I have no ability to explain that without using "big words", or without refering to physics papers that have been cited many times on here.
Zz.
14. Sep 16, 2006
### Chronos
OK, I'm tempted beyond restraint to toss a coin into the fountain. The wavefunction, in quantum speak, is a mathematical construct that explains 'how' a photon, electron or other quantum entity 'chooses' the path it takes to the target. The concept is perfectly logical when you accept it is a probability, not a deterministic prediction. Quantum entities, by definition, are fuzzy to begin with [the uncertainty principle thing], so it is not [at least to me] very surprising their paths also appear to be fuzzy. In the very unintuitive realm of quantum spacetime, photons, et. al., are free to be in more than one place at the same time - re: shooting electrons through a double slit one at a time.
15. Sep 18, 2006
### blackwizard
Ah, well that does make things clearer, thanks. If i picked you up right that is. I was tryin 2 understand wavefunctions as physical entities. Probly 2 used 2 bein able to visualise physics.
No idea, just hopin. Only started 2 learn bout Theoretical Physics a month ago
16. Sep 18, 2006
### Careful
Hi,
The question is not silly at all There is nothing in the theoretical construct of quantum mechanics which tells you when and how measurement occurs; at least not in those interpretations where the wave function is imagined to collapse such as in the copenhagen interpretation. As a rule of thumb you might use the reduction rule when a macroscopic apparatus is put in front of the setup (but in this case you are partially right, even the plate with the two slits will reflect a large part of the wavefunction and fringe effects at the slits will occur). But then again, you have macroscopic samples which might occur in superposition and claims exist that this superposition of macroscopic states can be observed (in other words, observation is not going to disturb the system too much)! Measurement is a tricky business and one should think carefully about it.
Careful
17. Sep 18, 2006
Staff Emeritus
I note that in http://realityconditions.blogspot.com/2006/09/on-price-and-penrose-on-time-asymmetry_18.html (which I have also posted about in the Philosophy of Science and Math Forum), it is claimed that all the cases where the wave function is supposed to collapse occur in statistical contexts where entropic time asymmetry is to be expected. The author cites decoherence in justification, and notes that there is no asymmetry when two electrons interact by exchanging a (virtual) photon.
What does anybody think of this idea? Would it have any bearing on the OP question?
18. Sep 19, 2006
### Chronos
I think this is a very good idea. A virtual photon represents the average of all possible superposition states that can be shared between the interacting electrons. Collapsing the wavefunction of one of the interacting electrons automatically collapses the other, the way I interpret matters.
19. Sep 19, 2006
### Careful
Hi, my first reaction would be a hmmmm, entropy of what ?'' For example, lets do a double slit experiment with electrons. The electron wave function will slow down in front of the plate with the slits resulting in electromagnetic radiation being created at uncertain times. Assuming the plate to be a perfect reflector, the entropy'' of the localized'' state particle + EM field goes down since radiation escapes (the expectation values of the Von Neumann entropy operator : entropy particle + EM field + universe should'' remain the same) --> measurement ?! I did not calculate this, but you might want to check if this radiation can also effect that part of the electron wave function which does not slow down and goes through the slit (it seems so to me). This also shows that only that part of it which is involved in strong'' interactions can be measured. However, it seems to me that the decoherence argument is far from sufficient either since (a) it does not help you with closed systems (b) there is a debate about the ontology of the density matrix (c) with a density matrix you can only compute statistical expectation values and you do not arrive at a single event interpretation in any case (hence it is FAPP). In case of our reflecting plate, the way out of course is that reflection is not constraining the wave package in any way. Anyway, measurement only occurs when the spatial wave function can be written as a sum over disjoint localized wave packages which (a) remain localized AND (b) more or less disjoint for a sufficient amount of time (the position basis is physically preferred). That is, the de Broglie current = more or less sum over currents. This explains why the reflected, nor the other part are measured when passing the double slit plate. This is different from the decoherence interpretation, I do not need to trace out the environment, I need to study the time evolution of the different spatial parts of the particle wave though! So, one should study in detail how the interaction with the final plate is going to decohere the different spatial parts of the interfering wave prior to measurement, I guess de Broglie has said something similar already in the 1920 ties. I remember disagreeing with Penrose too about this *particular* example of his when reading the book; I have more sympathy for the thoughts of Price (which I recommended already a few times here before on this site) though. Anyway, I can only repeat myself when saying make a decent theory of single events !!''.
Careful
Last edited: Sep 19, 2006
20. Sep 19, 2006
Staff Emeritus
Reading your argument Careful I am a little puzzled, you do not seem at all to have coupled with the author's point that the slits and the screen are both macroscopic objects, and you just can't assume in the real world that "the plate is a perfect reflector". The formalism you cite was all developed with just such toy models and the author claims that that, ignoring the unavoidable coupling to the statistical environment in real interactions is the reason that the formalism has the awkward time asymmetry appearing.
21. Sep 19, 2006
### Careful
Excuse me ? :grumpy: I treated this point with due care and explained how one can avoid these presumed difficulties, it seems you did not get that at all. And of course you can assume the plate with the slits to be a perfect conductor, that will do just fine for all practical purposes.
** The formalism you cite was all developed with just such toy models and the author claims that that, ignoring the unavoidable coupling to the statistical environment in real interactions is the reason that the formalism has the awkward time asymmetry appearing **
I explicitely commented upon the decoherence interpretation which is a working tool and is for example useless in the context of closed quantum systems. I even specifically referred to the latter again in the context of a de Broglie like interpretation which is actually superior to decoherence and does not need a conscious choice'' to one of the worlds at all - and no time asymmetry is involved here either. Moreover, the effective dynamics in the decoherence interpretation is not unitary, so no wonder you can make of an AND an (approximate) OR. Actually, although the entropy in the decoherence interpretation increases during the loss of coherence process, it decreases again due to Poincare recurrence times. Moreover, selection of one macroscopic possibility (that is measurement) is decreasing entropy (at least from this point of view, in the Copenhagen interpretation entropy remains constant after renormalization).
For the rest, I am commenting upon the question posed in this tread from my own point of view - there is no reason why I should agree (which you call to couple'') with what is written by someone else.
Careful
Last edited: Sep 19, 2006
22. Sep 19, 2006
### Hans de Vries
I have been playing with the following argument why there should be
interaction between the photon and the (two) slits even if the photon
doesn't "collapse" there.
Imagine the experiment floating in zero-g: The screen + arm holding the
plate with the two slits. The photon hitting the screen will give the setup
an arbitrary angular momentum depending on where it hits the screen...
Now this is not allowed. (Say we are shooting the photon along the
center of gravity line). Angular momentum should be conserved. The
only other place where momentum transfer could take place is, yes,
at the two slits. This is where the photon was "bend", but then, it
doesn't know it is bend because it hasn't collapsed yet at the screen.
The two interactions (at screen and slits) together would conserve
angular momentum, but the interaction at the slits is at odds with
many interpretations. The idea however is ofcourse to examine how
well the interpretations hold up...
Regards, Hans
Last edited: Sep 19, 2006
23. Sep 20, 2006
### DrChinese
When the setup is to detect a photon passing 2 slits: in the path integral view, you consider all possible ways that the photon could arrive there. Those possibilities give rise to interference, and a pattern results.
That does not mean that every photon emitted from the source made such a journey. And those "other photons" would have the property of having had their wave function collapsed elsewhere, and would not contribute to the interference pattern observed. So in a way, the slits could also act as the "observer" if a source photon was absorbed there and not re-emitted. This is simply a relatively less likely outcome.
In sum: The interference pattern represents the subset of photons that went through the slits. In effect, the detection mechanism is designed to be a fertile target for the source photons and the rest of the apparatus is not. That is the only distinction.
24. Sep 22, 2006
### nrqed
But that begs an obvious question. Any physical slit is composed of atoms. So why can we say that if there is a metallic device (i.e. a macroscopic measuring device) then we now have a detector and the interference patterm will disappear, and yet say that if there is no metallic device, there is no collapse of the wavefunction, even though the atoms in the slits should in principle be "detecting" whethere an electron passed or not. So what's the difference? Of course, in the second case the information is unavailable to me (at the macroscopic scale). But in the end, any actual measuring device is made of atoms, so at what point can we say that there is a collapse? When the atoms are arranged in just a specific way as to make a recording possible? What is the exact rule that dictates when a collapse will occur and when it won't?
Maybe I am being too dense but that is really not clear to me.
Cheers
25. Sep 22, 2006
### ueit
I've had this idea some time ago and I've even debated it on iidb forums. This is the first time I've seen it presented by someone else and this gives me hope that I'm up to something.
I've found yet another problem with this experiment. Two assumptions are made:
1) the wall does not interact with the particles
2) the wall is perfectly opaque.
It is easy to see that these assumptions are logically incompatible.
I think that the whole "collapse" thing is misleading. The wave function is a mathematical abstraction and that's why it can be non-local. We need an explanation in terms of what is real, a local-realistic model. I think it is pretty obvious that the particles interact at the slits and the real question is how this interaction produces the interference patterns. "When the wave collapses" is a non-question on par with "when the edges of a cube produce the volume inside the cube?". | 2018-07-21 21:52:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.649596631526947, "perplexity": 636.3030612390909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592778.82/warc/CC-MAIN-20180721203722-20180721223722-00234.warc.gz"} |
https://www.physicsforums.com/threads/shear-and-moments.465260/ | # Shear and moments
rsq_a
I can't seem to wrap my head around the signs of shears and moments when applied to beams. Consider a cantilever beam that goes from x = 0 to x = L (with positive deflection, or y(x), corresponding to a deflection upwards).
The standard equations tell us that
$$\text{Moment} = EI \frac{d^2 y}{dx^2}$$
$$\text{Shear} = EI \frac{d^3 y}{dx^3}$$
$$\text{Load} = EI \frac{d^4 y}{dx^4}$$
Now consider what happens when we change $$x = -x$$ (that is, we put our coordinate system so that the beam begins at x = 0 and goes to x = -L).
Why does that change the shear to negative, but keep the sign of the moments and loads the same?
$$\text{Curvature} = EI \frac{d^1 y}{dx^1}$$ | 2022-08-11 15:30:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6525826454162598, "perplexity": 351.8229052184749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00521.warc.gz"} |
https://www.physicsforums.com/threads/finding-point-where-gradient-is-the-greatest.610210/ | # Homework Help: Finding Point where Gradient is the Greatest
1. May 30, 2012
### mariya259
I have the function:
f(x,y)= x*(y^2)*e^-((x^2+y^2)/4)
I am not sure how to find the point where the gradient is the greatest.
The gradient I found after taking the partials is:
partial with respect to x: e^(-(x^2+y^2)/4)*((y^2)-.5(x^2)(y^2))
partial with respect to y:e^(-(x^2+y^2)/4)*(2yx-.5x(y^3))
What do I do next?
(I have also found min/max points for this function)
2. May 31, 2012
### algebrat
If you are trying to maximize the length of the gradient over the region, you might want to try squaring the gradient, so that it is a map from the plane to the real numbers, and use methods you have on that. (The gradient is longest where it's square is longest.)
3. May 31, 2012
### SammyS
Staff Emeritus
Hello mariya259. Welcome to PF !
You sure are having fun with this function, aren't you ?
It is easier to work with the square of the magnitude of the gradient, rather than with the magnitude of the gradient itself.
Get that by summing the squares of the two components which you have found.
It sure involves some messy algebra !
Last edited: May 31, 2012
4. May 31, 2012
### mariya259
Do you mean taking the square root of each of the partials squared to find the maginitude:
sqrt((e^(-(x^2+y^2)/4)*((y^2)-.5(x^2)(y^2)))^2 + (e^(-(x^2+y^2)/4)*(2yx-.5x(y^3)))^2) ?
5. May 31, 2012
### algebrat
You don't want to maximize, that, it'll be unnecessarily computationally complicated. If you square it, looking for critical points won't be as messy.
6. May 31, 2012
### mariya259
So only this part is what I need to do?
e^(-(x^2+y^2)/4)*((y^2)-.5(x^2)(y^2)))^2 + (e^(-(x^2+y^2)/4)*(2yx-.5x(y^3)))^2
7. May 31, 2012
### algebrat
Yes. Maximize the square of the gradient, to find the point where the gradient is longest.
8. May 31, 2012
### mariya259
I took the sum of squares. From what I understand now I need to set it equal to 0 and solve for critical points?
9. May 31, 2012
### Ray Vickson
No. You need to find the gradient of your new function (which is the square of the length of the gradient of the old function), then set this new gradient to zero. After all, is that not what you do when you want to maximize something?
RGV
10. May 31, 2012
### SammyS
Staff Emeritus
As algebrat said, if you work with the square of this, life will be easier. The maximum will occur in the same location for both this function and its square.
You can factor out a common factor to simplify matters a bit.
The square of the gradient as you gave it, which I think is correct, is:
$\displaystyle \left(e^{-(x^2+y^2)/4}\ ((y^2)-.5(x^2)(y^2))\right)^2 + \left(e^{-(x^2+y^2)/4}\ (2yx-.5x(y^3))\right)^2 =\left(e^{-(x^2+y^2)/4}\right)^2\left(((y^2)-.5(x^2)(y^2))^2+(2yx-.5x(y^3))^2\right)$
$\displaystyle =e^{-(x^2+y^2)/2}\left(16 x^2 y^2+4 y^4-12 x^2 y^4+x^4 y^4+x^2 y^6\right)/4$
That's still pretty messy.
It turns out that the gradient is maximum where you might expect --- midway between neighboring min & max .
11. May 31, 2012
### mariya259
Alright. To find the partials and critical points can I just use the function with the sum of squares or do I need to take the function of the square root(sum of squares)?
12. May 31, 2012
### Ray Vickson
This question was already answered. Anyway, instead of asking, just DO IT. If you do it both ways and compare the work involved, you will soon see which method is better. A good way to learn is to try thing yourself.
RGV
13. May 31, 2012
### mariya259
I understand you can use sum of squares instead of the sqrt of everything, I don't understand why you can do that. Wouldn't the square root change the answer of where the maximum is?
14. May 31, 2012
### SammyS
Staff Emeritus
Below is a graph of your function as given by WolframAlpha.
Below is a graph of the square ot the gradient you gave as graphed by WolframAlpha with relative maxima indicated in red.
#### Attached Files:
• ###### Function_xy.gif
File size:
9.1 KB
Views:
142
File size:
13.6 KB
Views:
140
15. May 31, 2012
### SammyS
Staff Emeritus
No.
If a > b ≥ 0, then a2 > b2. Right ? | 2018-07-21 17:44:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6711481213569641, "perplexity": 843.9893875908995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592650.53/warc/CC-MAIN-20180721164755-20180721184755-00087.warc.gz"} |
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# Cities A and B are in different time zones. A is located 3000 km east
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Cities A and B are in different time zones. A is located 3000 km east [#permalink]
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08 Jan 2020, 01:50
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Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.
Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
What is the plane’s cruising speed in km per hour?
(A) 500
(B) 550
(C) 600
(D) 700
(E) Cannot be determined
Are You Up For the Challenge: 700 Level Questions
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Re: Cities A and B are in different time zones. A is located 3000 km east [#permalink]
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08 Jan 2020, 01:50
Bunuel wrote:
Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.
Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
What is the plane’s cruising speed in km per hour?
(A) 500
(B) 550
(C) 600
(D) 700
(E) Cannot be determined
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Cities A and B are in different time zones. A is located 3000 km east [#permalink]
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09 Jan 2020, 00:14
Let the time taken for the flight from A to B to Tab and the time from B to A be Tba. Let the speed of the plane be S, and the time difference be Td.
Then from the information provided above, the following equations can be formed.
Tab=7-td --------(1)
Tba=4+td--------(2)
Sab = S+50 -----(3)
Sba = S-50 ------(4)
Since the plane would cover the same distance in a trip from A to B and B to A, Average Speed, Sav, =[(S+50)+(S-50)]/2 = 2S/2 = S.
Average time, Tav, = [(4+td)+(7-td)] = 11/2 = 5.5hrs.
Sav = S = Average distance Between A and B / Average Time = 3000/5.5 = 545 approximately 550km/hr
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Re: Cities A and B are in different time zones. A is located 3000 km east [#permalink]
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09 Jan 2020, 03:44
given the data ,
let speed of plane is S , t is difference in the time zone.
Time taken while going from B to A = 7hr - t
Time taken while going from A to B = 4hr + t
total time taken during travelling = 7 + 4 = 11
i.e $$\frac{3000}{s-50} + \frac{3000}{s+50} = 11$$ , by plugging in the s = 550kmph exactly.
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Re: Cities A and B are in different time zones. A is located 3000 km east [#permalink]
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09 Jan 2020, 05:59
T1 + T2 = 4 + 7 = 11 hours
$$T1=\frac{D1}{V1}=\frac{3000}{(S+50)}$$
$$T2= \frac{D2}{V2}=\frac{3000}{(S-50)}$$
$$\frac{3000}{(S+50)}+\frac{3000}{(S-50)}=11$$
At this stage you can either solve the equation (not recommended) or plug in the numbers. 550 is a perfect fit:
$$\frac{3000}{(500+50)}+\frac{3000}{(550-50)}=5+6=11$$
Re: Cities A and B are in different time zones. A is located 3000 km east [#permalink] 09 Jan 2020, 05:59
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# GWAS of mosaic loss of chromosome Y highlights genetic effects on blood cell differentiation
## Abstract
Mosaic loss of chromosome Y (mLOY) is frequently observed in the leukocytes of ageing men. However, the genetic architecture and biological mechanisms underlying mLOY are not fully understood. In a cohort of 95,380 Japanese men, we identify 50 independent genetic markers in 46 loci associated with mLOY at a genome-wide significant level, 35 of which are unreported. Lead markers overlap enhancer marks in hematopoietic stem cells (HSCs, P ≤ 1.0 × 10−6). mLOY genome-wide association study signals exhibit polygenic architecture and demonstrate strong heritability enrichment in regions surrounding genes specifically expressed in multipotent progenitor (MPP) cells and HSCs (P ≤ 3.5 × 10−6). ChIP-seq data demonstrate that binding sites of FLI1, a fate-determining factor promoting HSC differentiation into platelets rather than red blood cells (RBCs), show a strong heritability enrichment (P = 1.5 × 10−6). Consistent with these findings, platelet and RBC counts are positively and negatively associated with mLOY, respectively. Collectively, our observations improve our understanding of the mechanisms underlying mLOY.
## Introduction
Mosaic loss of chromosome Y (mLOY)1 is characterized by the presence of cells that have lost chromosome Y. mLOY is well described in samples from the bone marrow2 and can also be detected in peripheral blood cells3. Age4 and smoking5 are well established risk factors for mLOY, however, the broader mechanisms that influence mLOY, including the stage(s) of hematopoietic differentiation during which mLOY arises, are not fully known. Genetic analysis can provide biological insight into the basis of this phenomenon. The same array-based assays can provide information about single-nucleotide polymorphism (SNP) genotype and, from intensity data of genome-wide probes, allow inference of mosaic events6,7. This has allowed previous studies to identify some genetic determinants underlying mLOY. Zhou et al. reported that mLOY is associated with a variant in the TCL1A gene locus8. Wright et al. identified a total of 19 loci associated with mLOY9, including TCL1A and other genes involved in cell cycle regulation and DNA damage response. Previous studies examined European populations, so genetic studies from an East Asian population would expand our knowledge of the genetic architecture underlying mLOY, in turn allowing us to make deeper biological inferences about mLOY based on expanded genetic findings.
The clinical significance of mLOY is still unclear10. Previous studies reported that mLOY is associated with shorter life span and increased risk for cancer onset3,11,12. Other studies suggested that mLOY is associated with onset of acute myeloid leukemia (AML) or other hematological disorders13,14. However, Zhou et al. did not find significant differences in overall survival or cancer-specific survival in data from 5340 cancer cases8. No studies have analyzed the clinical significance of mLOY in Asian populations. Although the clinical significance of mLOY is not well established, clarifying the mechanisms of chromosome loss may provide a deeper understanding of clonal expansion during hematopoiesis, cellular ageing, and cancer development. Furthermore, there is a possibility that mLOY has a distinct clinical impact in Asian populations.
Here, we use genotype array data from 95,380 Japanese males enrolled in the Biobank Japan Project (BBJ) to study mLOY15. We examine the genetic architecture of mLOY, performing the genome-wide association study (GWAS) of this condition in an East Asian population. To infer biological mechanisms driving mLOY, we conduct downstream analyses using our GWAS results, making inferences based on overlap of associated variants and polygenic mLOY heritibility with functional elements to pinpoint the cell types and gene pathways involved in mLOY. Furthermore, we conduct association studies between mLOY and survival data of the participants to evaluate the clinical significance of mLOY in an Asian population.
## Results
### Estimation of mLOY from probe intensity data of DNA microarray
The data were generated in three separate batches due to advances in genotyping platforms during data collection. Detailed description of the samples is shown in Supplementary Table 1.
We applied a method similar to that used in Wright et al.9 to estimate mLOY in our data. We obtained logarithm of R ratio (LRR) probe intensity data across more than 1100 variants (depending on batch) in chromosome Y for each male subject. We used mean LRR (mLRR-Y) as a proxy for mean Y chromosome dosage in circulating blood cells of subjects (for details, see the Methods section). Hereafter, we use the term ‘mLOY GWAS’ to denote GWAS on mLRR-Y, as in Wright et al.9.
### Associations between mLOY and age or smoking
We observed a strong association between age at DNA collection and mLOY (1 year increase in age associated with 2.2% standard deviation (SD) decrease in mLRR-Y signal, P < 1.0 × 10−100, Wald test in linear regression analysis, Fig. 1a; Supplementary Fig. 1) explaining 9.6% of the variance in mLOY. We also observed a significant association between smoking and mLOY (Supplementary Fig. 2); smokers often have lower mean intensity of chromosome Y probes, indicating a higher fraction of cells with loss of chromosome Y (smokers associated with 4.6% SD decrease in mLRR-Y signal, P = 7.5 × 10−10, Wald test in linear regression analysis). These associations are in agreement with previous studies5.
### mLOY as a highly polygenic trait
Next, we imputed genotypes using genotyping array data after standard quality control (Methods) and a reference panel of whole-genome sequenced individuals, including 2504 individuals from the 1000 Genomes project phase 3 and 1,037 Japanese16,17 (Methods and Supplementary Note 1). We conducted a Bayesian mixed-model association study to identify mLOY susceptibility loci, controlling for age, smoking, genotyping batch, and disease status (for details, see the Methods section).
We found strong evidence for polygenicity of mLOY GWAS signals. Chi-squared statistics for association with mLOY deviated slightly from expected statistics in a quantile–quantile plot (lambda = 1.066, Supplementary Fig. 3). We evaluated polygenic effects on mLOY using linkage disequilibrium (LD) score regression18, with the use of ldsc software. This regression analysis suggested that mLOY is a highly polygenic trait and that departure of mean chi-squared statistics could be largely explained by polygenic effects (lambda genomic control 1.086 > intercept 1.019 in ldsc, Fig. 1b and Supplementary Note 2). Since LD score regression revealed minimal bias, we did not correct study statistics by genomic control19.
### Common genetic architecture of mLOY with Europeans
To understand our results in the context of what is known about the genetics of mLOY in Europeans, we compared our results to previous analyses. We found a total of 46 loci significantly associated with mLOY, 15 of which were reported by Wright et al.9 from analysis of data from the UK BioBank (Fig. 1c, Tables 1, 2; Supplementary Fig. 4). We found a consistent association trend (direction of effect of minor alleles) across all three batches for each of the 46 associated variants (Supplementary Table 2). Conditional analyses revealed that four loci, all of which were among those found in Wright et al.9, contained two previously undetected independent signals (Table 3; Supplementary Fig. 5). The 50 independent variants over 46 loci explained 3.2% of mLOY variance, 1.6% of which could be explained by the 35 new variants. We estimated SNP-heritability of mLOY to be 9.3% (see the Methods section).
We did not find significant associations in 4 of the 19 regions identified by Wright et al.9 (Supplementary Fig. 6). The most strongly associated variant (rs17758695) in the BCL2 gene locus, the top locus identified by Wright et al.9, is not polymorphic in the Japanese population (Supplementary Table 3). The top variants in the other three regions identified by Wright et al.9, but not here, were also very rare or not polymorphic in the Japanese population (Supplementary Table 3). Therefore relative invariance at these four loci in the Japanese population precludes identification of a statistically significant signal in this population.
To further assess similarities and differences in the genetic architecture of mLOY across populations, we examined associations of the 46 loci with mLOY in 205 thousand male participants in the full UK Biobank release (Supplementary Table 4). Thirty-nine of the 46 top variants were polymorphic, and had minor allele frequencies (MAF) >0.5% in the UK population. Thirty-seven of these 39 variants had the same effect direction in UK Biobank and BBJ (P = 2.8 × 10−9, binomial test). When considering only the 31 novel loci, this trend remained consistent; twenty-five variants were polymorphic and had MAF >0.5% in the UK population, and 23 out of the 25 variants had the same effect direction in the two studies (P = 1.9 × 10−5, binomial test), indicating strong genetic overlap in association with mLOY between the two populations.
### Involvement of hematopoietic stem cells with mLOY
In order to assess the cell types important for mLOY based on significant GWAS signals, we next analyzed the 46 lead variants in BBJ for overlap with cell-type-specific enhancer marks using Haploreg20. We found a strong overlap between the variants and enhancer marks present in hematopoietic stem cells (HSCs) (Fig. 2; Supplementary Table 5). This finding is in line with the theory that clonal expansion of HSCs lacking chromosome Y underlies mLOY, as has been shown for clonal hematopoiesis of HSCs bearing a point mutation21. The 31 new loci found in the current study accounted for this overlap. Two completely independent sets of variants (31 and 19 loci in Japan and UK, respectively) showed overlap with enhancer marks in HSCs (Supplementary Fig. 7). Enhancer marks in embryonic stem cells (ES cells) or induced pluripotent stem cells (iPS cells), both of which can differentiate into HSCs22, did not strongly overlap with mLOY GWAS significant signals. Thus the variants associated with mLOY overlap with cell type-specific epigenetic features emerging after differentiation into hematopoietic cells, rather than epigenetic features characterizing pluripotency itself.
To further infer the functional implications of our genetic results regarding mLOY, we next analyzed the polygenic architecture of mLOY, not restricting to GWAS significant signals. We partitioned heritability of mLOY according to functional annotations23 using ldsc (see the Methods section). Superenhancer, H3K27ac, and transcription start site (TSS) were the top three categories with positive and significant heritability enrichment (>2.5-fold enrichment, P ≤ 3.1 × 10−6, Supplementary Table 6), suggesting that gene expression regulation, especially by superenhancers, is involved in the mechanisms underlying mLOY. Deep involvement of superenhancers with gene expression regulation may characterize mLOY genetic architecture, considering low enrichment of superenhancers in other traits’ heritability23. We next conducted cell-group-specific heritability enrichment analysis using ldsc. This revealed that mLOY heritability was highly enriched in histone marks associated with the hematopoietic cell group (Fig. 3a). We next used ldsc to further analyze whether histone marks enriched in specific hematopoietic cell types, among a total of 220 cell types, could explain the heritability of mLOY. We found that histone marks specific for CD34-positive cells (HSCs and hematopoietic progenitor cells) showed the strongest heritability enrichment (Fig. 3b; Supplementary Table 7). The mLOY heritability enrichment in superenhancers, histone marks in the hematopoietic cell group, and CD34-positive cells were confirmed in the analyses with ldsc where we used different baseline annotations recently proposed to take into consideration difference in LD pattern of variants, information of synonymous/nonsynonymous coding variants, sequence age and conservation of variants across species24 (see Methods and Supplementary Note 3). Taken together, both genome-wide significant loci and polygenic signals suggested involvement of CD34-positive cells with mLOY.
### Heritability enrichment in hematopoietic progenitor cells
CD34 is expressed in hematopoietic progenitor cells, which include multipotent progenitor (MPP) and common myeloid progenitor (CMP), as well as HSC25. Therefore, to understand which differentiation stages are important for developing mLOY, we analyzed enrichment of mLOY heritability in chromatin regions that are open in various hematopoietic progenitor cells and lineages. We analyzed the GWAS results of mLOY with ldsc applied to specifically expressed genes (LDSC-SEG) using ATAC-seq data of hematopoietic progenitor cells and lineages at different differentiation stages26. Genes expressed in MPP and HSC, evidenced by open-chromatin state assessed by ATAC-seq, showed the strongest and the 2nd strongest heritability enrichment with mLOY, respectively (Fig. 3c; Supplementary Table 8 and Supplementary Note 3). We found decreasing heritability enrichment in genes expressed at subsequent differentiation stages; the further differentiated the cell type was from HSC and MPP, the less enrichment was observed. These results suggest that not only HSC but also MPP may be important differentiation stages at which genetic factors influence the development of mLOY.
### Enriched heritability in transcription factor binding sites
Previous studies have revealed that transcription factors (TFs) play a critical role in hematopoietic cell differentiation and function27. Thus, we hypothesized that overlap of genetic associations with TF-binding sites in various hematopoietic cells may provide information about lineage- or context-specific influence on mLOY. We used available chromatin immunoprecipitation sequencing (ChIP-seq) data to help identify TFs involved in mLOY. We obtained a total of 37 ChIP-seq data sets derived from CD34-positive cells or HSCs (for details, see the Methods section). Binding sites of three transcription factors, namely, FLI1, GATA2, and RUNX1, showed significant overlap with regions positively enriched in mLOY heritability (P ≤ 4.0 × 10−4, Fig. 3d). GATA2 and RUNX1 are TFs critical for HSC and hematopoiesis28,29, supporting the findings above. We focused on binding sites of FLI1, a TF important for megakaryocyte differentiation30, which showed the strongest heritability enrichment (P = 1.5 × 10−6, P-value for enrichment, and Supplementary Note 3). Binding sites of FLI1 that associated with mLOY heritability were independent from those of GATA2 and RUNX1 (P = 4.1 × 10−4, P-value for enrichment). Heritability enrichment in FLI1-binding sites was not seen in the 11 other cell types for which FLI1 ChIP-seq data were available, supporting the biological specificity of this result; heritability of mLOY is enriched in binding sites of FLI1 in CD34-positive cells (Supplementary Fig. 8). We searched for possible enrichment of mLOY heritability in TF-binding sites for any cell type. Notably, the FLI1-binding sites in CD34-positive cells showed the strongest heritability enrichment across all 2861 TF-cell type pairs available (see the Methods section and Supplementary Fig. 9). Among megakaryocyte/erythrocyte progenitors (MEP), common lymphoid progenitors (CLP) and granulocyte/macrophage progenitors (GMP) (which are similarly differentiated and lineage-committed progenitors, Fig. 3c), the highest heritability enrichment in expressed genes was observed for MEP. This is broadly consistent with heritability enrichment in binding sites of FLI1, which is critical for differentiation of MEP31. Importantly, ES cells and iPSC, which have potential to differentiate to HSCs, did not show strong heritability enrichment in TF-binding sites (357 ChIP-seq data, P > 0.05/357, P-value for enrichment, Supplementary Fig. 10), emphasizing that interactions in the nucleus of HSCs or hematopoietic progenitor cells which appear after differentiation to the hematopoietic lineage are important for mLOY.
### Associations between hematopoietic traits and mLOY
FLI1 drives MEP31 to develop into megakaryocytes and thus produce platelets rather than erythrocytes32,33. Therefore, we analyzed the association between mLOY and hematologic traits (including platelet count and red blood cell (RBC) count) using data from 57,987 subjects from whom complete blood counts (CBC) were available at the time of DNA collection. mLOY was positively associated with platelet count (i.e., higher platelet count, lower mLRR-Y, and higher mLOY; 1 SD increase in platelet count (66,000/ul) associated with 0.03 SD decrease in mLRR-Y signal, P = 7.6 × 10−14, Wald test in linear regression analysis) and WBC count (P = 4.2 × 10−15, Wald test in linear regression analysis, Supplementary Fig. 11). In contrast, we observed a nonsignificant negative association with RBC count (i.e., higher RBC count, higher mLRR-Y, and lower mLOY, Supplementary Fig. 11). The effect size of platelet count (and RBC) on mLOY was enhanced in individuals estimated to have a high fraction of cells with loss of chromosome Y (Fig. 3e).
These results are compatible with significant heritability enrichment in FLI1-binding sites. Since RUNX1 also regulates maturation of megakaryocytes and represses erythrocyte gene expression during megakaryocytic differentiation34, these results also support significant heritability enrichment of RUNX1 binding sites. We did not find significant genetic correlations between mLOY and platelet or RBC counts (Supplementary Table 9 and Supplementary Note 4). This suggests that the association between mLOY and platelet count may be derived from limited genetic components (likely including FLI1 and RUNX1) rather than from diverse polygenic effects or from shared nongenetic factors.
### Pathway analysis and Mendelian randomization of mLOY
Our results so far suggest that the genetic variants and polygenic associations of mLOY mostly overlap with functional elements in specific hematopoietic progenitor cells. However, the specific cellular pathways and gene-regulatory networks involved in mLOY remain uncertain. To address this, we conducted pathway analysis using PASCAL software35 and genetic correlation analyses with other quantitative and qualitative traits (see Methods). PASCAL takes not only GWAS significant loci but whole-genome signals into consideration. We found that gene scores calculated based on the GWAS signal of mLOY showed the strongest and most significant enrichment in cell cycle pathways and a mitotic pathway (Supplementary Table 10). This is compatible with the partial overlap of mLOY susceptibility genes and oncogenes9 as well as with the hypothesis that mitotic clonal expansion of an HSC lacking chromosome Y underlies lower mLRR-Y values. We also conducted genetic correlation analyses between mLOY and quantitative traits or malignancies with which we previously reported genetic associations36. We observed a significant genetic correlation between mLOY and aspartate transaminase (AST) (Supplementary Fig. 12 and Supplementary Table 9). A bidirectional Mendelian randomization (MR) approach suggests a causal effect of mLOY on increase of AST (P = 0.037, Supplementary Note 4).
### Associations between mLOY and clinical outcomes
Finally, we investigated the clinical significance of mLOY. We analyzed survival and mLOY in 54,887 BBJ participants followed at a maximum of 12 years after registry37 (for details, see Methods). There was no association between mLOY and overall survival (Fig. 4a). A previous study that found an association between mLOY and decreased survival used a different analytic approach, one which is strongly influenced by the small number of subjects who have a high fraction of cells lacking chromosome Y (Supplementary Note 5)3. Therefore, we tested whether subjects with a high fraction of cells lacking chromosome Y have adverse outcomes in the studied cohort. We found that subjects with lower mLRR-Y values (indicating a higher fraction of cells lacking chromosome Y) were more likely to have experienced an adverse outcome (Supplementary Fig. 13), which seems compatible with the previous finding3.
Genome instability, including chromosome rearrangements and loss, is a central event in development of some cancers, so genetic determinants that increase susceptibility to mLOY may influence cancer risk. We therefore analyzed whether the associations between mLRR-Y and mortality were mediated by malignances. We found a significant association between mLOY and death due to lung cancer (P = 0.0010, Cox proportional hazards regression, Hazard ratio 1.09 (95% CI: 1.04–1.15), Fig. 4b) after conditioning on covariates, including smoking. Subjects with a high fraction of mLOY consistently showed associations with death of lung cancer (Supplementary Fig. 14). However, we found that the association between mLOY and lung cancer was mostly contributed by individuals with a smoking history; the confounding effect of smoking on both lung cancer and mLOY is difficult to disentangle even conditioning on this covariate. When we extracted and analyzed subjects with detailed smoking information including smoking duration and quantity (at the expense of decreasing sample size), the effect size of mLOY on lung cancer was decreased, and the association was no longer significant (Supplementary Note 6). We did not find significant associations between mLOY and other cancer types or with all malignancies (Fig. 4b).
## Discussion
This is a large study to analyze mLOY in a non-European population. In accordance with the large study of mLOY in Europeans (Wright et al.9), we used the mean of LRR as a proxy of chromosome Y copy number. Using the mean, rather than the median, of LRR allows that some strongly deviated markers could unduly influence estimation of chromosome Y dosage. However, we found that the median and mean are strongly correlated (Spearman’s rho: 0.993) and lead to quite similar genetic association results (median LRR identifies 44 of the 46 loci identified using mean, Supplementary Note 7). The minimal difference in the two indices supports the rationale to use the method applied in this study.
Our analyses provide insights into the mechanisms underlying mLOY, suggesting that MPP and HSC are likely the primary cell types undergoing of clonal expansion after loss of chromosome Y. Genes expressed in hematopoietic lineages, especially HSC and MPP, seem to be critical for mLOY development. As our results suggest variants overlapping myeloid precursor functional regions are involved in the development of mLOY, it would be interesting to analyze subgroups of WBC in elderly populations to identify if myeloid populations contain more cells lacking chromosome Y than co-circulating lymphoid cells.
It is of note that while mLOY was detected by assessing nucleated leukocytes, platelet and RBC counts were associated with mLOY. The involvement was also supported by heritability enrichment of FLI1 and GATA2 bindings. This indicates that the molecular and cellular mechanisms underlying mLOY act before cell fate is restricted to certain terminally differentiated nucleated WBC types, in line with the importance of MPP and HSC shown by enrichment analyses of GWAS significant signals and polygenic signals. The association between WBC counts and mLOY in a nondose dependent manner may reflect higher chance to detect mLOY signals from probe intensity in subjects having more WBC.
Genetic correlation between AST and mLOY and the result of MR suggested a causal relationship between mLOY and increased AST (more mLOY leading to higher AST). Since recent studies have revealed that mosaic events frequently occur systemically38, this may suggest that mLOY also occurs in the liver and affects susceptibility to liver damage. It is of note that we did not observe an association between mLOY and ALT. This suggests that increase in AST level or ratio of AST and ALT in elderly populations could serve as a marker of mLOY. Especially given the cell-type specificity of the genetic factors we identified, determining the genetic landscape of mLOY in other tissues including the liver would be interesting.
Our study also analyzed association between mLOY and survival at an unprecedented scale. While we could replicate a trend of poor outcome in subjects with a high burden of mLOY, the clinical significance of mLOY was not very apparent in our study. Since this is a large study to analyze effects of mLOY on clinical outcomes in an Asian population, it is possible that the clinical significance of mLOY is not the same across all populations. However, it is also possible that mLOY is associated with onset of specific types of cancer and/or context-dependent survival, such as poor outcome in specific diseases, which could not be fully captured by our population cohort.
While different LD structure between Japanese and UK population made it difficult to quantify genetic similarities of mLOY between the two populations, we showed that significant variants found in the BBJ very often shared direction of effect on mLOY in the UKB. We tried to replicate 156 significant variants found in UKB, as a result, 92 out of the 100 variants which were polymorphic and passed QC showed shared direction in effect39, suggesting the strong overlap of susceptibility loci for mLOY between the two populations. Considering common genetic background of mLOY, susceptibility loci which are found significant only in one population suggest that the causal variants in these loci are polymorphic only in that population. In addition, taking advantage of genetic overlap seemed to help us to pinpoint shared causal variants of susceptibility loci.
Further in-depth analysis including trans-ethnic meta-analysis could reveal additional genetic components and context-specific clinical significance of mLOY. Because we showed mLOY shares genetic architecture across populations, trans-ethnic meta-analysis would be attractive to increase the number of associated loci and to obtain further biological insights from finer resolution functional inference. The association between mLOY and clinical outcomes should be replicated to resolve the clinical impacts of mLOY and develop management protocols for elderly populations. Disease-specific associations of mLOY should be analyzed in disease-specific data sets rather than population cohorts.
## Methods
### Subjects and genotype data
We used 183,899 subjects in the BBJ project for selection of samples to analyze for this study. Written informed consent was obtained from all the participants. The study was approved by the ethical committees in the Institute of Medical Science, the University of Tokyo and RIKEN Center for Integrative Medical Science. Most of the subjects had already been genotyped using genome-wide genotyping arrays40. In this study, we used genotype data from three different sets using four different arrays, namely, (1) HumanOmniExpressExome v1.0, (2) HumanOmniExpressExome v1.2, and (3) a combination of Illumina HumanOmniExpress v1.0 and Human Exome v1.0 or v1.1 BeadChips. A breakdown of the subjects and arrays is given in Supplementary Table 1.
### Estimation of mLOY
Using log-R ratio (LRR) generated by Illumina GenomeStudio software, mean LRR in Y chromosome (mLRR-Y) was calculated and used as a proxy of mLOY as follows. We extracted only male subjects and reclustered the subjects based on their probe signal intensities in variants on chromosome Y. This step was performed separately for each type of array. We did not use intensity data of Human Exome BeadChip to avoid batch effects within the same subjects. We obtained LRR for all variants on chromosome Y excluding the pseudoautosomal regions, and subjects and variants with missing rate higher than 5% (restricted to the Y chromosome) were excluded from mLRR-Y calculation. As a result, 1268, 1305, and 1162 variants remained for calculation of mLRR-Y in the three batches described above, respectively. We set LRR in variants with missing genotypes as missing. We then standardized LRR in each variant, and calculated mean of LRR in each subject across variants on chromosome Y. The mLRR-Y was standardized per batch and used as a signal reflecting mLOY in the subjects. We took this approach to avoid strong influence of limited number of variants with extreme values of LRR. To confirm that our data did not suffer from noise due to variants with extreme LRR, we also took median of LRR signals in each individuals among the variants and conducted GWAS with the use of median LRR instead of mLRR-Y.
### Quality control of subjects and SNPs
We excluded subjects found to be genetically identical to other subjects (321 samples), showing discrepancy between reported sex and inferred sex based on genotypes of variants on X chromosome (1245 samples), found to be outliers from the EAS (Japanese and Chinese) cluster (121 samples) in the analysis where we merged GWAS data with 1000 Genomes Project genotype data, pruned variants by excluding variants in moderate LD with other variants (with cutoff level of r2 of 0.5) and conducted principal component (PC) analysis to project subjects in PC1 and PC2 space, included in the reference panel mentioned below (939 samples) or showing a missing rate higher than 2% (0 sample at this stage). Finally, we obtained data of 95,380 males whose probe intensity data in the Y chromosome was available for subsequent analyses. SNPs with missing rates more than 1%, Hardy–Weinberg equilibrium P-values <1.0 × 10−6 or heterozygote calls <5 in each of the three arrays were excluded.
### Whole-genome imputation
All genome-scanned samples were merged before phasing and whole-genome imputation was conducted using a reference panel which was produced by combining genotype data from 2504 subjects from the 1000 Genomes project41 and whole-genome sequence data (x30) of 1037 Japanese subjects16 (see for detail, Supplementary Note 1). Eagle42 and Minimach343 softwares were used for phasing and imputation, respectively.
### Genome-wide association study
After whole-genome imputation, we tested 9,591,901 variants with squared Pearson correlation (r2) >0.3 (to ensure imputation accuracy) and minor allele frequency >0.005 for association with mLOY. We applied the Bayesian mixed-model using bolt-lmm44 (URL). Age, arrays, smoking, and disease status (prevalence >0.5% in the subjects) were included as covariates (Supplementary Table 11 and Supplementary Note 8). Since we used mixed model taking into account genetic correlation matrix among subjects, we did not include principal components in the covariates. The mean of smoking status was used for subjects with missing information. P-values of 5.0 × 10−8 was set as genome-wide significant level.
### Variance and heritability estimation of mLOY
We calculated variance explained by variants by the following formula:
$${\mathrm{Var}}_ - {\mathrm{explained}} = 2 \times {\mathrm{E}}^2 \times {\mathrm{af}} \times \left( {1 - {\mathrm{af}}} \right) \times {\mathrm{Var}}_ - {\mathrm{tot}}^{ - 1}$$
where Var_explained indicates variance explained by a SNP, E indicates beta in LMM, af indicates allele frequency of tested allele, and Var_tot indicates variance of mLOY (we set to 1 by data standardization).
We used bolt-lmm software to evaluate heritability of mLOY. Briefly, bolt calculates heritability of model SNPs (in this case, we provided genotyping array data) by restricted maximum likelihood estimation (REML) to compute variance of genetic components.
### Independent signals in a single locus
Two associated loci on the same chromosome were regarded as different if a genome-wide significant marker at one locus was at least 1 Mbp apart from those in the other locus. This definition was also applied to regard significant loci in the BBJ as the same as or different from those described by Wright et al.9. We conducted conditional analyses to confirm that two loci were independently associated with mLOY when the boundary of the two regions were <1 Mbp apart. If significant associations were observed in conditional analyses and adjacent loci were apart by >200 kbp but <1 Mbp, we regarded the two loci as different.
### Conditional analysis
We applied conditional analyses to identify independent signals in a single locus or adjacent loci. We incorporated dosages of a variant whose effect we would like to condition on as a covariate in linear mixed model. We applied the same threshold (P < 5.0 × 10−8) to the conditional analyses to define significant associations.
### Replication of the associations with the UKBB data
We obtained UKBB association results of the 46 top variants in the BBJ. In the UKBB data, 205,011 male subjects were analyzed, and presence of mLOY was determined by hidden Markov model together with phased B allele frequencies calculated by signal intensity data of markers in pseudo autosomal region (PAR)1 of genotyping arrays39. We compared risk allele between the two populations. We applied the same MAF threshold of 0.005 as the BBJ to the UKBB. The four secondary signals in the BBJ were not included, since top variants were different between the two populations and statistics in conditional analyses with the use of the same conditioned variants were not available.
### Functional annotation using Haploreg
We analyzed overlap between lead variants in significant loci associated with mLOY and enhancer marks in cells by Haploreg (URL) to infer cell types important for mLOY. Briefly, enrichment of lead SNPs for mLOY in enhancer histone marks in cell types was evaluated by comparing estimated overlap based on pruned variants in the 1000 Genomes Project with minor allele frequencies >0.05 in any population. The enrichment significance was computed by binomial test. Since Haploreg requires rs id of SNPs which is not available for Chr2:136879065, we used rs6751768, which is in strong LD with Chr2:136879065 (r2 = 0.80), for this locus. A significant level of associations was set based on Bonferroni’s correction.
### Pathway analysis
We conducted pathway analysis with the use of the PASCAL software35 (URL) which takes LD structure of nearby genes into consideration. To avoid overestimation of GWAS significant variants, we calculated gene score for pathway analysis by PASCAL by using the sum of statistics of variants in single genes rather than the maximum of the statistics. Bonferroni’s correction accounting for all of the pathways (REACTOME, KEGG, and BIOCARTA) was used to set a threshold of statistically significant enrichment (P < 0.05/1077).
### LD score regression to estimate confounding bias
To evaluate the polygenic effect on mLOY and assess confounding bias leading to inflation of median chi-square statistics, we used LD score regression analysis with the use of ldsc software18. We regarded intercept in LD score regression <1.05 or ldsc ratio (an index estimating confounding bias of GWAS statistics) <0.3 as no confounding bias18.
### Partitioning heritability for cell groups and cell types
We evaluated enrichment of heritability of histone marks in cell groups or detailed cell types in each cell group by conditioning on the full baseline model described in Finucane et al.23 by ldsc. Briefly, the full baseline model contains coding, intronic, UTR, promoter, enhancer regions, histone marks, open-chromatin regions, and their extended regions, none of which are specific to any cell types23. Although ldsc uses chi-square statistics which are calculated from different distribution from that of bolt-lmm44, we confirmed similar results of ldsc between bolt-lmm chi-square statistics and normal linear regression chi-square statistics (with the same covariates as bolt-lmm). Thus we showed results of analyses with the use of chi-square statistics in bolt-lmm. We also applied to the analyses another full baseline model of ldsc recently reported including information of LD-related annotations, synonymous, and nonsynonymous annotations, ancient sequence age, and conserved function across species24.
### Genetic correlation
We also used ldsc software to assess genetic correlation45 of mLOY and quantitative trait or malignancy susceptibility (Supplementary Note 4). We used 44 blood tests out of the 58 quantitative traits we previously reported36 and whose summary statistics are available (JENGER, see URL). We used GWAS of malignancy whose summary statistics are available in the BBJ36 (JENGER, see URL). While samples were overlapped between traits, a genetic correlation estimated by ldsc is shown not to be affected by sample overlap since sample overlap does not change LD score45.
### Mendelian randomization analysis
We conducted Mendelian randomization analysis to assess causal relationship between mLOY and AST which showed a significant genetic correlation with mLOY (Supplementary Note 4). We used a Generalized Summary-data-based Mendelian Randomization approach implemented in GCTA software46 (bidirectional Mendelian randomization with option –gsmr-direction 2).
### Partitioning heritability with the use of ChIP seq data
We used the ChIP-seq data in the previous report47. Raw human ChIP-seq data files in SRA format were obtained from the GEO database and were converted to FASTQ format using the fastq-dump function of SRA Toolkit. Each sequence read was aligned to the human hg19 genome using Bowtie2 version 2.2.5 with default parameters. Peaks were called using Model-based Analysis of ChIP-Seq (MACS) version 2.1 with default settings (q < 0.01). Through this analysis, we obtained a total of 2856 ChIP seq data. LD scores of transcription factor binding were constructed by extending 500 bp from the peaks in ChIP seq. We conducted partitioning heritability of GWAS data of mLOY by ChIP seq using ldsc. We first extracted 37 ChIP seq data, in which HSC was analyzed for transcription factor binding based on the results of enrichment analysis of significant variants in cell-specific enhancer marks and of cell-type or group analyses of partitioning heritability. We partitioned heritability of mLOY with the use of the LD scores of transcription factor bindings and the full baseline model by Finucane et al.23. In addition, we also analyzed the data with the use of the latest full baseline model described above.
### LDSC-SEG
To evaluate the important cell types among the CD34+ cells, we used LDSC-SEG48, a method recently developed to estimate heritability enrichment of gene expression (or open chromatin) in various tissues and cell types by taking advantage of tissue- and cell-specific eQTL data, which are available in the widest range of tissues and cell types. We used data from Corces et al.26 who generated ATAC-seq data of hematopoietic cell lineages, including CD34+ hematopoietic stem progenitor cells. We computed Japanese LD scores of cell-specific gene expression based on the ATAC-seq data by referring to European annotations for the analyses (we obtained very similar results regardless of the origin of LD scores). In this analysis, we used the full baseline model by Finucane et al.23 as baseline of LD scores. In addition, we also analyzed the data with the use of the latest full baseline model described above.
### Survival data in the BBJ
The living status (dead or alive) of a total of 141,612 BBJ subjects with one of 32 diseases was prospectively followed for more than 10 years after DNA collection37. If a death was reported, a detailed search was conducted to identify causes of death coded by ICD10 by accessing the national vital registration system at the Japanese Ministry of Health, Labor, and Welfare. Further details are written elsewhere37. We extracted cancers with >1000 deaths overall, and analyzed lung, colon, esophagus, gastric, pancreas, hematologic and prostate/testis cancer as specific malignancies. To analyze associations between mLOY and malignancy mortality, we excluded subjects having had malignancy at registry, without information of follow-up period or with follow-up <1 year to exclude potential undiagnosed cancers and other fatal diseases. As a result, 54,887 subjects with follow-up period of mean 8.0 years and standard deviation of 2.4 years remained for the analyses. During the follow-up periods, 12,410 deaths were observed. We used the survival package of R statistical software. We set subjects who did not die during follow-up as controls and evaluated associations between mLOY and mortality of overall or specific diseases. In addition to the use of mLRR-Y signals as a quantitative trait, we used subjects with the lowest 3, 5, 10, and 20% of mLRR-Y signals as categorical variables and compared with subjects whose mLRR-Y signals not reaching the lowest 20%. The 20% cutoff was determined by the latest UKBB mLOY study which reports 20% of male subjects in the UKBB were positive for mLOY39. We used Cox proportional hazards regression for survival analysis to estimate associations between mLOY and the cause of death adjusting for age at DNA collection, data batches, disease status, and smoking.
### Associations between mLOY and CBC data
We analyzed a total of 57,987 subjects for associations between mLOY and CBC data for inference of biological insights. We z-transformed CBC data to assess associations. Age, smoking, disease status, and data set were used as covariates in the association study. To test the hypothesis that subjects carrying high mLOY signals were strongly associated with CBC data, we took subjects carrying top 1, 3, 5, and 10% of mLOY signal (i.e., those with the lowest mean LRR of variants in chromosome Y). We divided all subjects in this study into two groups (binary traits), with or without top 1, 3, 5, and 10% of mLOY signals and associated the binary traits with the CBC data mentioned above together with covariates of age, smoking, disease status, data set with the use of logistic regression analysis.
### URL
For 1000 Genomes Project, see http://www.1000genomes.org/; for GWAS catalog, see https://www.ebi.ac.uk/gwas/; for LDSC and SEG-LDSC, see https://github.com/bulik/ldsc/; for PASCAL, see https://www2.unil.ch/cbg/index.php?title=Pascal; for Minimac, see https://genome.sph.umich.edu/wiki/Minimac; for Haploreg, see https://pubs.broadinstitute.org/mammals/haploreg/haploreg.php; for JENGER, see http://jenger.riken.jp/en/; for R, see https://www.r-project.org/; for Plink1.9, see https://www.cog-genomics.org/plink2; for bolt-lmm, see https://data.broadinstitute.org/alkesgroup/BOLT-LMM/; for LCR, see ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20130502/supporting/low_complexity_regions/hs37d5-LCRs.20140224.bed.gz.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
GWAS summary statistics of mLOY in the BBJ is available at RIKEN website (http://jenger.riken.jp/en/). While individual-level genetic data are not accessible, all other data contained in the article and its supplementary information are available upon reasonable request.
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## Acknowledgements
We deeply thank Dr. Nicholas Parrish for critically reviewing and editing the manuscript. We thank all of the staff in the BBJ project for their efforts to keep and handle samples. This research was conducted using the UK Biobank Resource under Application #19808. P.-R.L. was supported by NIH grant DP2 ES030554, a Burroughs Wellcome Fund Career Award at the Scientific Interfaces, the Next Generation Fund at the Broad Institute of MIT and Harvard, and a Glenn Foundation for Medical Research and AFAR Grants for Junior Faculty award. G.G. was supported by US Department of Defense Breast Cancer Research Breakthrough Award W81XWH-16-1-0316 and the Stanley Center for Psychiatric Research.
## Author information
Authors
### Contributions
C.T. and K.Y. designed the study. C.T. analyzed the data. C.T., P.R.L., G.G., J.P. and Y.K. wrote the paper. P.R.L., G.G. and J.P. contributed to the UKB association results. K.I., E.K., H.S. and T.O. made data of TF-binding sites. Y.M., M.H., K.M., Y.M., M.K. and Y.K. contributed to creation of the BBJ genetic data, clinical information, and follow-up data. K.I. and M.A. contributed to make an imputation reference panel. All authors critically reviewed and approved the final version of the paper.
### Corresponding authors
Correspondence to Chikashi Terao or Yoichiro Kamatani.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks Xia Jiang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Terao, C., Momozawa, Y., Ishigaki, K. et al. GWAS of mosaic loss of chromosome Y highlights genetic effects on blood cell differentiation. Nat Commun 10, 4719 (2019). https://doi.org/10.1038/s41467-019-12705-5
• Accepted:
• Published:
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Cellular and Molecular Life Sciences (2021) | 2021-04-20 23:49:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5997692346572876, "perplexity": 9538.835042413595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039491784.79/warc/CC-MAIN-20210420214346-20210421004346-00106.warc.gz"} |
http://mathoverflow.net/questions/61044/maximum-principle-for-weak-solutions/61052 | # Maximum principle for weak solutions
Hello,
maximum principles for parabolic PDEs seem to be well-known, if the solution is a priori C^2 (cf. Protter, Weinberger: Maximum principles in differential equations). However, what about weak solutions? To be specific, are there any maximum principles on the nonnegativity of solutions $u\in W^{1,p}(0,T;L^p(\Omega))\cap L^p(0,T;W^{2,p}(\Omega))$, $p\in(1,\infty)$, where $\Omega\subset R^n$ is a bounded domain? For given nonnegative initial data, does the solution remain positive, as long as it exists?
I assume yes, since there are numerous authors that use the results from Weinberger/Protter just for weak solutions. I would appreciate any hints on this topic.
Best regards, Marc
-
Solutions of parabolic equations like heat equation are smooth away from the boundary of the space-time domain. Therefore they obey the maximum principle exactly as in P.-W. – Denis Serre Apr 8 '11 at 11:45
Thank you for your reply. Actually im working on a heat equation with lower order perturbations whose coefficient functions are only continuous. Can one still expect smoothness of solutions in this situation? – Marc Apr 8 '11 at 13:25
Yes. See the references I gave, or any pde book. – András Bátkai Apr 8 '11 at 15:28
Yes. If you use operaor semigroups to represent the solutions, you can infer the positivity of the mild solutions (which are the same as the weak solutions) immediately.
There is an extensive treatment of positive semigroups in R. nagel (ed.): One-parameter semigroups of positive operators, Springer, 1986.
You can find a nice introduction, with a short discussion of this topic in Engel-Nagel: A short course on operator semigroups, Springer, 2006. ChapterVI.
Of course, there are less functional analytic arguments as well, but this is what I am familiar with
-
You might want to distinguish between maximum principles (which assert typically things like "the max of the solution is attained on the boundary / parabolic boundary of the set") and positivity, which assert things like "if the data are non-negative on the (parabolic) boundary, then so is the solution in the entire domain". The latter often can be shown with functional analytic techniques (see previous post).
As to maximum principles for generalized solutions, there is work by Jensen on viscosity solutions of fully nonlinear elliptic problems. And there is work in the 70s that extends maximum principles for elliptic equations to solutions in $W^{2,n}$ where $n$ is the spatial dimension (if I remember correctly).
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https://math.stackexchange.com/questions/3207039/true-falsethe-polynomial-x47x3%E2%88%9213x211x-has-exactly-one-real-root | # [True/False]The polynomial $x^4+7x^3−13x^2+11x$ has exactly one real root.
[True/False]The polynomial $$x^4+7x^3−13x^2+11x$$ has exactly one real root.
I want to solve it without drawing the graph. Here is my idea. Note that $$f(1)=1+7-13+11=6>0$$ and $$f(-1)=1-7-13-11=-30<0$$
So we have at least one real root. Now since degree is $$4$$ we have $$4$$ roots but rest three can not be complex as they occur in pairs, so we must have another real root.
So the statement is False
Am I right?
Thanks for reading and all the help.
• How do you know the root is not a double root? – Théophile Apr 29 '19 at 15:26
• There are at least two. $0$ and one real root of $x^3+7x^2-13x+11$. – Saucy O'Path Apr 29 '19 at 15:26
• You are right, but the root $x=0$ is even more obvious. – Yves Daoust Apr 29 '19 at 15:29
Yes, your result is correct (how to improve the actual argumentation is sufficiently discussed within the comments of this answer). It can be done in another by noting that $$f(x)=x^4+7x^3-12x^2+11x=x\cdot(x^3+7x^2-12x+11)=x\cdot g(x)$$ $$f(x)$$ has one real root at $$x=0$$. Now, $$g(x)$$ is of degree $$3$$ and every polynomial of degree $$3$$ has at least one real root by the Intermediate Value Theorem since $$\lim\limits_{x\to-\infty}g(x)=-\infty$$ and $$\lim\limits_{x\to\infty}g(x)=\infty$$.
Thus, $$f(x)$$ has at least two real roots.
• @StammeringMathematician $x^2(x^2+1)$ has a real root on the interval $[-1, 1]$. Then the exact same argument as the one you use in your answer will "prove" that there must be another root (there are three remaining roots, and we can't have that none of them are real), but there isn't one. – Arthur Apr 29 '19 at 15:35
• Whether we count multiple roots or not is technically up to the original problem author. Conventions vary. But yes, I would think that the intended interpretation is to count distinct roots. If the intended interpretation is to count with multiplicity, then you don't even need to look for roots in the first place. You just know that there can only be $0, 2$ or $4$ non-real roots, and therefore there cannot be exactly one real root. That's a bit boring, and it would seem rather pointless to give a concrete polynomial to work with. – Arthur Apr 29 '19 at 15:39 | 2020-08-11 13:08:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 14, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7035801410675049, "perplexity": 150.37953844706448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738777.54/warc/CC-MAIN-20200811115957-20200811145957-00151.warc.gz"} |
https://socratic.org/questions/how-do-you-factor-and-solve-16x-2-48x-27 | # How do you factor and solve 16x^2-48x= -27?
May 23, 2015
$y = 16 {x}^{2} - 48 x + 27$ = (x - p)(x - q). I use the new AC Method (Google, Yahoo Search) to factor trinomials.
Converted trinomial: y' = x^2 - 48x + 432 = (x - p')(x - q').
Find p' and q' by composing factor pairs of a.c = 432. Proceed: ...(8, 54)(12, 36). This last sum is 12 + 36 = 48 = -b. Then p' = -12 and q' = -36. Then, $p = \frac{p '}{a} = - \frac{12}{16} = - \frac{3}{4}$ and $q = \frac{q '}{a} = - \frac{36}{16} = - \frac{9}{4.}$
Finally: $y = \left(x - \frac{3}{4}\right) \left(x - \frac{9}{4}\right) = \left(4 x - 3\right) \left(4 x - 9\right)$
Check by developing: y = 16x^2 - 36x - 12x + 27. OK
Solving y = 0 --> (4x - 3) = 0 --> $x = \frac{3}{4}$
(4x - 9) = 0 ->$x = \frac{9}{4}$
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# Numerical Simulations of Electromigration and Stressmigration Driven Void Evolution in Solder InterconnectsOPEN ACCESS
[+] Author and Article Information
Purdue University, West Lafayette, IN 47907subramanya@purdue.edu
Ganesh Subbarayan1
Purdue University, West Lafayette, IN 47907ganeshs@purdue.edu
Lei Jiang
Intel Corporation,Portland, OR 97124lei.jiang@intel.com
Daniel Pantuso
Intel Corporation,Portland, OR 97124daniel.pantuso@intel.com
1
Corresponding author.
J. Electron. Packag 134(2), 020907 (Jun 11, 2012) (9 pages) doi:10.1115/1.4006707 History: Received September 20, 2011; Revised February 13, 2012; Published June 11, 2012; Online June 11, 2012
## Abstract
Understanding the effect of high current density on void formation and growth and relating the size of the void to the resulting electrical/mechanical failure is a critical need at the present time to ensure reliable functioning of flip-chip packages. In general, toward this end, the modeling and simulation of geometrical evolution of current induced voids have been relatively few. Simulations considering the coupled effects of mass transport through mechanisms of surface and bulk diffusion under the influence of electrical, thermal, and stress fields in solder joints leading to eventual electromigration failure do not appear to be common. In this study, we develop a phase field model for the evolution of voids under electrical, thermal, and stress fields in a flip-chip solder interconnect. We derive the equations of motion for the void accounting for energetic contributions from the active factors of surface energy, stress, and electric potential, considering both surface diffusion and transfer of the material through the bulk of the material. We describe the implementation of this model using a finite element code written in the PYTHON language, coupled with a commercial finite element solver from which we obtain the electrical, thermal, and stress fields driving the void motion. We demonstrate the implemented methodology through simulations of void evolution in flip-chip solder joints under the effects of mechanical/electrical fields and surface/bulk diffusion.
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## Introduction
The decrease in the size of the flip-chip solder joints has resulted in a corresponding increase in the current densities prevalent in the joints. This has led to an increase in the importance of failure due to electromigration in flip-chip solder joints [1]. Electromigration is caused by the momentum exchange between electrons and ions that forces material diffusion in the direction of electron flow. This leads to the formation of voids near the cathode that might lead to an open failure. There is also the likelihood of the formation of hillocks at the anode that might cause a shorting failure. The formation and motion of the voids are also driven by the existence of stress and temperature gradients. A unique problem that occurs in solder joints is the failure due to current crowding. The current density in the signal trace leading up to the joint is significantly higher than the current density in the solder joint. At the interface between the two, there is a region of the solder joint, where the current density is an order of magnitude higher than the rest of the joint (≈105 A cm−2 at the entrance versus ≈104 A cm−2 near the middle [1]). This causes the void formation to be very rapid near the current crowding region (see Fig. 1).
The computational simulation of the formation and propagation of voids in solder is a challenging problem. There are a multitude of interacting physical effects such as the electrical, thermal, and stress fields that determine the void shape and its rate of evolution. A significant attempt at modeling electromigration in solder joints has been through the use of damage mechanics to simulate the loss in load-bearing capacity of the solder joints [2-3]. These models enable one to find locations of void formation and in modeling the motion of very small, diffuse, voids, whose boundaries are not explicitly captured. These methods do not explicitly model the geometry of the voids, but infer the effects of the formed voids through the damage they cause to the load-bearing capacity. However, geometrically speaking, the removal of material from a region results in the formation of new surfaces. Further, energywise, every new surface formation has an associated surface energy cost [4] that needs to be accounted for in any complete simulation of the motion of voids.
An alternative approach to modeling electromigration has been to capture the divergence of atomic flux directly within a finite element code [5-6]. In these approaches, the modeling of motion and growth of larger voids appear to rely mostly on element deletion schemes, which approximate the boundary of the voids accurately in the limit of mesh refinement [6]. Furthermore, these methods also ignore the surface energy associated with void boundaries or surface diffusion along the void boundary, and rely on the underlying physics of electromigration being captured entirely in the bulk through the divergence of flux.
As the voids evolve, the underlying geometrical domain on which the boundary value problems are solved changes. One way to handle this is by explicitly tracking the motion of the void surface. There are very few attempts in the current literature aimed at explicitly modeling the evolution of electromigration-induced voids [7-8]. The main challenge with explicit geometry tracking methods is handling the constantly changing problem domain, which necessitates remeshing with a mesh-based numerical scheme such as the finite element method. Also, in an explicit scheme, the processes that cause topological changes such as the splitting of the voids require special treatment and are in general very difficult to model numerically.
An alternative to this method is to use a phase field model for the void boundary. In phase field models, the existence of a boundary is indicated by a change in an order parameter. Quantities that are defined along the surface of the boundary are smeared over a narrow region surrounding the boundary. The main advantage of using a phase field model is the ability to retain a fixed finite element mesh for the simulations even as the geometrical shapes of the voids evolve. Also, changes in the void connectivity such as the merging and splitting of voids are handled automatically and efficiently. This method has been used earlier for the modeling of void evolution in Al–Cu interconnects through surface diffusion [9].
In this study, we develop a phase field model for simulating the motion of pre-existing voids in flip-chip solder joints due to diffusive mass transport along the surface of the voids (and therefore preserve the volume of the void) as well as diffusive mass transport between the voids (and therefore do not preserve the volume of the individual voids). We implement a finite element solution methodology, including the coupling to a commercial finite element code that is used to estimate the electrical, thermal, and stress fields driving the void evolution.
The structure of the rest of the paper is as follows. We first describe the basis for our governing equation derived from a statement of the minimization of a free energy rate. We follow that with a description of the phase field model that is used for the solution of the void propagation problem. Finally, we illustrate the utility of this method by studying the motion of the void in a flip-chip solder joint, under general loading conditions.
## Theory
###### Minimization of Free Energy Rate
Considering the system shown in Fig. 2, the evolution of a near-equilibrium system can be described by the minimization of the dissipation or of a free energy rate. The free energy rate $Ψ·$ in the void evolution problem has three sources, namely, surface effects, electrical field, and due to mechanical load Display Formula
$Ψ·=Ψ·s+Ψ·e+Ψ·m$
(1)
In the above rate, the free energy of the void surface, Γ can be written as Display Formula
$Ψs=∫ΓγdΓ$
(2)
Thus, the rate of change of the surface free energy is given by [10] Display Formula
$Ψ·s=∫ΓγκvndΓ$
(3)
The surface free energy measures the energetic cost of adding or removing material from the surface of the void. The role played by surface energy in solids is slightly different from that in liquids. In liquids, the surface energy affects both the stress equilibrium of the material and the cost of adding and removing the material. However, in solids the surface energy does not play a significant role in the determination of the deformation equilibrium. The inclusion of a surface term in the expression for the energy makes it possible to model scale effects.
Assuming that a charge density ρc exists everywhere in the domain, the net electrical potential energy is given by ρc φe , where φe is the electric potential. The total electrical energy of the system at steady state is given by Display Formula
$Ψe=∫ΩρcφedΩ$
(4)
The rate of change of this quantity, assuming equilibrium is reached for electrical potential much faster than the rest of the system, is obtained by taking the material time derivative [11] of Eq. 4Display Formula
$Ψ·e=∫Ω∂(ρcφe)∂tdΩ+∫Γext(ρcφe)vndΓ+∫Γint[[(ρcφe)vn]]dΓ$
(5)
where [[]] is the jump in the quantities across the interface defined as [[a]] = a+ + a− , referring to the quantity on either side of the interface. Assuming that the material velocity is significantly slower than the time taken to reach steady state, we can assume that the system is always at steady electrical state. Hence, the spontaneous change term, $∂(ρcφe)∂t→0$. Also, if the external boundaries of the system are fixed, $∫Γext(ρcφe)vndΩ=0$. Further, for a material–void system, one side of the jump term can be ignored. Hence, the free energy rate due to electrical work can be approximated in electromigration-induced void evolution problems as Display Formula
$Ψ·e=∫Γint(ρcφe)vndΓ$
(6)
Similarly, the contribution to free energy from mechanical loads is Display Formula
$Ψm=∫ΩφmdΩ$
(7)
where φm is the strain energy density. To evaluate the mechanical energy contributions, from the balance of mechanical energy on the system, we get Display Formula
$∫Γt→·vdΓ=∫Ωσ:DdΩ$
(8)
where D is the rate of deformation tensor. Assuming linear elasticity (σ = E: ɛ), and therefore the strain energy density $φm=(12)ɛ:E:ɛ$, we have, after similar considerations for equilibrium of the stresses Display Formula
$∫Γt→·vdΓ=∫Ω∂φm∂t dΩ+∫ΓextφmvndΓ+∫Γint[[φmvn]]dΓ$
(9)
Arguing, as before, that the system reaches equilibrium instantly relative to the times involved in mass transport, we have $∂φm∂t→0$. Assuming fixed external boundaries, and no loads in the void, we get Display Formula
$Ψ·m=∫Γt→·vdΓ=∫ΓintφmvndΓ$
(10)
Combining Eqs. 3,6,10, we get Display Formula
$Ψ·=∫Γint[γκ+ρcφe+φm]vndΓ$
(11)
Following the arguments from Ref. [12], and introducing surface and bulk mobilities Ms and Mb , we get the following equations for the velocity of the front.
For surface diffusion, assuming a local conservation of mass Display Formula
$vn=∇Γ·Ms∇Γφ$
(12)
Display Formula
$φ=γκ+ρcφe+φm$
(13)
For the bulk diffusion, we assume that the mass is transferred only from one void surface to another. This leads to an inherently nonlocal equation for the bulk diffusion equations. The equation for the velocity of the void surface is given as Display Formula
$vn=-Mb[φ+λ]$
(14)
where λ is a Lagrange multiplier that enforces conservation of mass throughout the system. This constraint can be written as Display Formula
$∫ΓvndΓ=0$
(15)
Finally, the motion under coupled surface and bulk diffusion effects is thus described by Display Formula
$vn=∇Γ·Ms∇Γφ-Mb[φ+λ]$
(16)
This is similar to the Eq. (1.14) derived in Ref. [13]. In Ref. [13], the authors derive the equations using the principles of rational thermodynamics. While the first term in the two equations is the same, the interpretation of the second term differs. While the authors in Ref. [13] use the second and third terms as modeling evaporation and condensation effects from and to the surface, here, on the other hand, we use the terms to mean any rapid transport of material through the bulk of the material.
###### Diffuse Interface Equations
The equations derived above give us the evolution equations, based on a sharp interface description of the void interface. These have the problem of being difficult to handle when the shape of voids evolve or when there are changes in the topology of the voids since these would require remeshing of the domain. A related challenge is on utilizing the calculated void motion defined on the sharp boundary in a commercial finite element analysis software used to analyze the electrical/mechanical behavior of the system. A diffuse interface approximation for the void interface functions by smoothing the surface over a notional thickness of order ε and indicating the surface through the level sets of a phase field variable u that varies from − 1 inside the void to + 1 outside the void (see Fig. 3). The surface energy (Eq. 2) is written as Display Formula
$Ψs=1cF∫Ωγ[12ɛ|∇u|2+f(u)ɛ]dΩ$
(17)
where cF is a constant scaling parameter defined as in Ref. [14].
The gradient energy was first used in Ref. [15] to study spinodal decomposition in alloys. It was later extended to study other problems with moving boundaries in a diffuse sense, such as solidification [16]. The gradient energy allows the treatment of surface quantities in a diffuse volumetric sense. Locations with nonzero gradients indicate the presence of the interface. The normal to the interface can be inferred by $(∇u/∇u|∇u||∇u|)$. The term f(u) in Eq. 17 is a bulk free energy term, with two minima that correspond to the two phases, in this case, the material and the void. The presence of $(f(u)/f(u)ɛ)$ in the energy functional penalizes solutions that are not ±1.
A good choice for the potential function is the biquadratic function $f(u)=(1-u2)22$ shown in Fig. 4, $cF=43$, and the notional thickness of the interface is 2ɛ. Using this description, the equations for the motion of the void can be written in terms of partial differential equations in u. We have, for the surface diffusion problem Display Formula
$∂u∂t=∇Ms(u)∇φ$
(18)
Display Formula
$φ=1cFɛ[-γ(ɛ2∇2u+f′(u))+ρcφe+φm]$
(19)
where Ms (u) is restricted to the interfacial region where −1 ≤ u ≤ 1. The equation for diffusion through the bulk is given by Display Formula
$∂u∂t=-MbcFɛ[γ(ɛ2∇2u+f′(u))+ρcφe+φm+λ]$
(20)
Display Formula
$∫Ωu(t)dΩ=∫Ωu(0)dΩ$
(21)
The combined motion of the void surface due to both the surface and the bulk effects is obtained by adding Eqs. 18,20. Equation 18 is a specialized version of the Cahn–Hilliard equation [15], while Eq. 20 is a version of the Allen–Cahn equation with an additional constraint on the phase field variable [17].
## Numerical Solution
The equations for the void motion are solved in a two-step fashion solving first for the surface diffusion and then solving for the diffusion through the bulk. The partial differential equation for the surface diffusion is 4th order in space. Usually, these equations require C1 continuity across element boundaries when using finite element analysis. However, by splitting the governing equations into two 2nd order equations, we can solve these equations using C0 elements following the approach outlined in Ref. [18]. As there are no external fluxes, we can assume Neumann boundary conditions everywhere on the boundary.
In order to make the solution procedure robust, we scaled the length by the dimension of the solder joint, r, and the time by an arbitrary scale, $r4Msγ$. Further, we scaled the electric potential by the applied potential, and the strain energy with the maximum strain energy observed in the structure. On scaling, Eqs. 18,19,20,21 become Display Formula
$∂u∂t=∇2φ$
(22)
Display Formula
$φ=1cFɛ[-(ɛ2∇2u+f′(u))+(keφeφ0e+kmφmφ0m)]$
(23)
Display Formula
$∂u∂t=-kdcFɛ[(ɛ2∇2u+f′(u))+r2(keφeφ0e+kmφmφ0m)+λ]$
(24)
Display Formula
$∫Ωu(t)dΩ=∫Ωu(0)dΩ$
(25)
where $ke=ρcφ0eγ$, $km=φ0mγ$, and $kd=Mbr2Ms$. kd controls the relative effect of surface diffusion with respect to bulk diffusion. The other parameters control the relative effect of the quantities with respect to the motion due to the reduction of surface energy. Specifically, ke controls the relative effect between the electric potential and the surface energy, km controls the relative effect between the strain energy density and the surface energy, and kd controls the relative effect of bulk and surface diffusion velocities.
###### Discretized Forms of Governing Equations
Multiplying the equations for surface diffusion (Eq. 18) by a test function w and integrating by parts, we get Display Formula
$∫Ωw∂u∂tdΩ=∫Ω[-M(u)∇w∇φ]dΩ$
(26)
Display Formula
$∫ΩwφdΩ=1cFɛ∫Ω[γ(ɛ2∇w∇u-wf′(u))+wρcφe+wφm]dΩ$
(27)
Discretizing the above equations using C0 continuous linear triangular elements (for convenience, since it assures constant derivatives), and assuming that $dudt=ui+1-uiΔt$, we get Display Formula
$[A1cFγɛBΔtCA][φui+1]=[1cFɛA(ρcφe+φm+γf′(ui))Aun]$
(28)
Here, A is the lumped mass matrix for the system, $B→∫Ω∇Ni∇NjdΩ$ and $C→∫Ω∇NiM(ui)∇NjdΩ$. Similarly, for the bulk diffusion equations, by multiplying Eq. 20 by the same test function w, we get Display Formula
$∫Ωw∂u∂tdΩ= MbcFɛ∫Ω[-γ(-ɛ2∇w·∇u+wf′(u))+ wρcφe+wφm]dΩ+λ∫ΩwdΩ$
(29)
Display Formula
$∫ΩudΩ=∫Ωu0dΩ$
(30)
Again, discretizing using C0 linear triangular elements and using an Euler backward scheme, we get the following matrix equations: Display Formula
$[1ΔtA+MbcFγɛBaaT0][ui+1λ]=[MbcFɛA(ρcφe+φm+γf′(ui))∫Ωu0dΩ]$
(31)
Here, a are the diagonal terms of the lumped mass matrix A.
The coupled effect of bulk and surface diffusions are accounted for by solving the surface diffusion equation first and then solving the bulk diffusion equation. This is equivalent to solving Eq. 28 first and then solving Eq. 31 using the value of ui from the solution to Eq. 28.
The solutions for the strain energy, electric potential, and temperature values are obtained by solving the corresponding boundary value problems in a commercial finite element code, using C0 finite elements on the same mesh used for the solution of the phase field problem. The void is indicated by modifying the material properties for Young’s modulus E and electrical conductivity η as Display Formula
$E(u)=E0(1+u)2, η(u)=η0(1+u)2$
(32)
###### Challenges to Numerical Solution
As the Cahn–Hilliard equation is a fourth order nonlinear equation, time integration is difficult. Simple explicit solutions such as the forward Euler method are precluded as the time step required for numerical stability is ≈ch4 , where h is the size of the element and c is an arbitrary constant. Hence, a semi-implicit method is used for time integration. The mobility and the forces are treated explicitly and are computed and held constant from the end of the time step, while the phase field variable itself is treated implicitly and intreated for. A further restriction on the time-step size is imposed due to the nonconvex nature of f(u). This causes the solution to converge uniformly to either +1 or −1 for large values of time steps. In our studies, we found that the numerical time steps necessary for the convergence of the non-linear solver were on the order of 10−4 to 10−5 . As this problem is solely due to the numerical difficulties presented by the Cahn–Hilliard equation, the stress and electrical field solution was not recomputed at every time step, but only computed once every few steps of the solution of the diffusion equations. This was also beneficial in reducing the number of times that the commercial solver needed to be called for computing the electrical and stress fields.
###### Implementation
The boundary value problem to obtain the electric field values was solved using a commercial finite element code, using thermoelectric elements. Similarly, the stress problem was solved using plane stress elements. The finite element analyses relied on linear triangular elements. This was linked to the solver for the phase field equations that was written in PYTHON language using SCIPY and SCIPY [19] for the numerical data structures and solvers, respectively. The linear systems were solved using the UMFPACK sparse direct solvers. The code developed, pyPhase, consists of approximately 1200 lines of PYTHON code. The PYTHON code includes capabilities to read in information from commercial finite element tools and creating and running simulations. It has the capability of handling arbitrary geometries of solder joints and also assemblies that include multiple joints. The code is independent of the choice of commercial finite element software. The structure of the code is described in Fig. 5.
## Validation
To validate the model and its implementation, it was tested on rectangular line geometries. In a line that is under the influence of surface diffusion, circular voids would collapse to slits under axial loading, while under electrical loading, the void would move toward the cathode. This is shown in Figs. 67.
We also tested the solution of the coupled solution including both bulk and surface diffusion. In the absence of any external loading such as electric potential gradients or stress, the system is expected to evolve purely by reduction of the surface free energy. While under surface diffusion, multiple noncircular voids are expected to relax to separate circular voids, with bulk diffusion allowed, smaller voids are expected to lose volume and eventually disappear. This can be tested by varying the relative rate of surface and bulk diffusion.
In Fig. 8, the relative rate of surface and bulk diffusion kd is varied from 0.1 to 10 from left to right, as can be seen; at high values of kd , the bulk diffusion effect dominates and the smaller void disappears. On the other hand, when the value of kd is lowered, the area of individual voids is conserved and there is only a relaxation due to the surface diffusion equation.
## Flip-Chip Solder Joint Model
The 2D model (Fig. 9) was constructed in a commercial finite element tool, according to dimensions from Refs. [(20-21) ]. The material properties used were those of tin (Table 1). The chip side and board side pads were assumed to be made of copper. We assumed that the chip side was at ground, while the board side was maintained at 5 V. The solder joint itself was assumed to be under isothermal conditions. A spatially varying shear displacement was applied on the chip side to simulate the effect of the thermal expansion mismatch. This displacement varied from a value of 0% to 10% of the pad diameter across the diameter of the joint. The interface between the solder joint and the pads was treated as diffusion barrier. This prevents the void from propagating into the pads.
## Results and Discussion
Given below are the results of the simulations on the flip-chip solder joint. The problem was scaled and nondimensionalized for numerical stability as described in Eqs. 22,23,24,25. Specifically, we chose r, the characteristic length, to be the midsection diameter of the solder joint. As described earlier, we also used a normalized timescale $τ=r4Msγ$.
###### Surface Diffusion
We first demonstrate our model on examples that are limited solely to surface diffusion. In this case, Eq. 20 is deactivated and not solved. The first example illustrates the effect of strain energy alone on the void motion (Fig. 1). In the next example, we simulate the motion of a void surface due to the electrical effects alone (Fig. 1). We see that in these cases the strain energy density and electrical effects cause the void to evolve in opposite directions. The evolution under an electrical field leads quite quickly to a pancake shaped void that has the potential to cause electrical failure. This is due to the fact that the electric current pushes the particles away from the current crowding region. Hence, it is expected that the voids in the solder migrate to this location. For the case with strain energy driven diffusion, the reason for the direction of the void surface motion is that even though the mechanical loading is a horizontal shear force. The direction of the strain energy gradient is tangential to the surface of the void. This causes the void to tend to collapse in that direction (see Fig. 1).
###### Combined Bulk and Surface Diffusion
In real applications, both surface and bulk diffusions are significant. This leads to a void growth analogous to Ostwald ripening effect observed in heterogeneous microstructures where one void grows at the expense of another. Shown in Fig. 1 is a case where there is a significant diffusion through the bulk. It is observed that the system evolves by transferring the mass to the void closest to the current crowding corner. The farther void reduces in size, until it is completely dissolved. The reason for this location of the final void configuration is the same as the explanation for the position of the void for the case with only surface diffusion.
###### Simulating Electromigration in Assemblies of Solder Joints
The electromigration in real solder joints is subject to effects that are visible only in assemblies of solder joints. One of these is the effect of the direction of the current in the current crowding region on the propagation of voids. In this example, an assembly of two solder joints is considered. The dimensions and loading of this assembly are described in Fig. 1. The entire joint is loaded by applying a potential difference of 10 V across the two solder joints. Additionally, a shear force is simulated on the two solder joints by applying a deformation of 10% of the midsolder section diameter on the joints (13.2 μm) in each of the joints. By loading the assembly in this fashion, one is able to consider the effect of the direction of the current in the current crowding region on the motion of the voids since, for instance, the top left corner of the first joint and the top right corner of the second joint experience opposite current directions, while maintaining the mechanical load direction to be the same.
In the first case (Fig. 1), the effects of the strain energy driven diffusion are turned off by setting km = 0. The direction of the current in the left solder joint causes the void to move into the current crowding region and form a pancake like void in that region, while in the right joint, the direction of the current is reversed and the void is pushed into the solder joint. Figure 1 illustrates the effect of mechanical load alone on the evolution of the void, and provides the baseline for the study of the effect of strain energy distribution on the electromigration of the void. As before, to understand the reason for the evolution of the void into the bulk of the solder joint, it is illustrative to study the strain energy distribution in the solder joint (Fig. 1). In Fig. 1, the combined effects of both electromigration and stress-induced void evolution are illustrated. As can be seen, the strain energy gradient causes the void to evolve in a direction counter to the direction of the electromigration. In this case, it slows down the evolution of the void toward the current crowding region. This suggests that the severity of failure due to the electromigration of voids might depend significantly on the position of the solder joint in the package, depending on the type and severity of the mechanical loading seen by each joint.
## Summary
We demonstrated, in this work, a phase field method to simulate the motion of pre-existing voids in the flip-chip solder joints. We detailed the sharp interface equations for the motion of a void in a material evolving under the influence of surface energy, strain energy, and electric potential. This is related to diffuse interface equations that reduce in the limit of ɛ to the sharp interface equations, and tracks the motion of the interface. The equations presented here account for the surface diffusion and also the bulk diffusion between multiple voids. The numerical solution to these equations is implemented into a numerical framework that uses a commercial finite element code to solve for the stress and electrical potential fields. We use this system to study the dynamics of void motion in a two-dimensional model of a flip-chip solder joint assembly. We showed the effects of introducing a void in the current crowding region of the solder interconnect. It was found that the electromigration void evolution depends on the directionality of current, with a pancake void forming in the cathodic region and the void migrating into the interior of the joint away from the interface near the anodic region. Further, the stress-induced void migration direction is opposite to that of current-induced void migration, and as a result, failure is likely to depend on the location of the solder joint in the array, the presence of significant shear loading, and the direction of current. Electromigration in solders is a highly complicated process and while the model shown here gives us an indication to the likely direction of the motion of the void, the role of plasticity, and the initiation of voids needs to be accounted for in the models in a thermodynamically consistent way. This is currently under investigation.
## Acknowledgements
The authors are grateful for support from Intel Corporation. The authors would also like to thank Dr. Yu Xiao for discussions.
vn =
normal velocity at a point on the boundary of the void
u, f(u) =
phase field order parameter, phase field potential function
cF =
velocity scaling parameter
Ψ =
free energy
γ =
surface energy density
E =
Young’s modulus
η =
electrical conductivity
φe , φ0e =
electric potential in domain, applied electric potential
φm , φ0m =
strain energy density in domain, maximum strain energy density
κ =
mean curvature
ɛ =
notional interface thickness
ρc =
charge density
Ms =
surface diffusion mobility
Mb =
bulk diffusion mobility
kd =
scaling parameter accounting for the relative magnitudes of bulk and surface mobility
λ =
Lagrange multiplier for global conservation of mass
τ =
scaled time
## References
View article in PDF format.
## Figures
Figure 7
Collapse of circular void into a slit under axial tension
Figure 8
As the value of kd is increased, the effect of bulk diffusion is increased and the bulk diffusion starts to dominate
Figure 9
Schematic diagram of the solder joint over which the diffusion problem is solved
Figure 10
Example showing surface diffusion due to strain energy alone. Here, kd = 0, ke = 0, and km = 40.
Figure 14
Schematic showing the configuration and loading of an assembly with two solder joints. The shear load is exactly symmetric.
Figure 15
Evolution of voids in the left and right solder joints showing effect of the direction of the current on the motion of the void. Here, kd = 0, ke = 100, and km = 0.
Figure 16
Evolution of voids in the left and right solder joints showing effect of the strain energy gradient on the development of the void. Here, kd = 0, ke = 0, and km = 100.
Figure 17
Strain energy distribution around the void region. The extremely high and low values have been removed in order to show the distribution.
Figure 18
Evolution of voids in the left and right solder joints showing effect of the strain energy gradient and electrical potential on the development of the void. Here, kd = 0, ke = 100, and km = 100.
Figure 1
Formation of pancake voids in the current crowding region [7]
Figure 2
Schematic showing the geometry and boundary conditions needed to solve the electromigration problem
Figure 3
Diffuse description of the interface
Figure 4
The biquadratic potential used in the simulations
Figure 5
Schematic structure of the code
Figure 6
Migration of circular void under electric potential gradient
Figure 11
Example showing surface diffusion due to electric potential gradients alone. Here, kd = 0, ke = 40, and km = 0, as time progresses the void rearranges into a pancake formation, similar to that shown in Fig. 1.
Figure 12
Distribution of strain energy near the surface of the void
Figure 13
Example showing significant diffusion through the bulk of the material. Here, kd = 5, ke = 20, and km = 20.
## Tables
Table 1
Material properties used in the simulations
## Discussions
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Topic Collections | 2018-04-23 21:11:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 49, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46163061261177063, "perplexity": 708.6274250252587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946199.72/warc/CC-MAIN-20180423203935-20180423223935-00373.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_8_Problems/Problem_21&diff=prev&oldid=111078 | # Difference between revisions of "2014 AMC 8 Problems/Problem 21"
## Problem
The $7$-digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$
## Solution 1
The sum of a number's digits $\mod{3}$ is congruent to the number $\pmod{3}$. $74A52B1 \mod{3}$ must be congruent to 0, since it is divisible by 3. Therefore, $7+4+A+5+2+B+1 \mod{3}$ is also congruent to 0. $7+4+5+2+1 \equiv 1 \pmod{3}$, so $A+B\equiv 2 \pmod{3}$. As we know, $326AB4C\equiv 0 \pmod{3}$, so $3+2+6+A+B+4+C =15+A+B+C\equiv 0 \pmod{3}$, and therefore $A+B+C\equiv 0 \pmod{3}$. We can substitute 2 for $A+B$, so $2+C\equiv 0 \pmod{3}$, and therefore $C\equiv 1\pmod{3}$. This means that C can be 1, 4, or 7, but the only one of those that is an answer choice is $\boxed{\textbf{(A) }1}$.
## Solution 2
Since both numbers are divisible by 3, the sum of their digits has to be divisible by three. 7 + 4 + 5 + 2 + 1 = 19. In order to be a multiple of 3, A + B has to be either 2 or 5 or 8... and so on. We add up the numerical digits in the second number; 3 + 2 + 6 + 4 = 15. We then add two of the selected values, 5 to 15, to get 20. We then see that C = 1, 4 or 7, 10... and so on, otherwise the number will not be divisible by three. We then add 8 to 15, to get 23, which shows us that C = 1 or 4 or 7... and so on. In order to be a multiple of three, we select a few of the common numbers we got from both these equations, which could be 1, 4, and 7. However, in the answer choices, there is no 7 or 4 or anything greater than 7, but there is a 1, so $\boxed{\textbf{(A) }1}$ is your answer. | 2021-01-19 16:12:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 20, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47410210967063904, "perplexity": 78.27663777757157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519395.23/warc/CC-MAIN-20210119135001-20210119165001-00701.warc.gz"} |
https://yo-dave.com/2019/06/03/detecting-from-a-web-client-whether-a-particular-font-is-installed/ | # Detecting from a Web Client Whether a Particular Font is Installed
Sometimes, a web client just needs to know what font it is using. The use cases may vary, but when you need it, you need it.
In my own case, I have a tag editing component for my personal wiki. The component allows entry of any number of tags. It also changes the size of the input as the user types in the text, getting smaller when characters are deleted or larger when new characters are entered.
In order to make accurate measurements of the width of the text, the tag editor(s) need to know what font they are using.
That can be a surprisingly difficult question to answer.
One approach is to specify and use a web font. For various reasons that may not be possible. For example, the user may have turned off the ability of their browser to download web fonts because of a slow internet connection. Or perhaps their employer prohibits such downloads.
For my use, I typically specify a font-family in the CSS for components in my programs so that I have graceful fallback regardless of which OS and what version I am using. But the browser will not tell you explicitly which font it has selected to use from the suggestions you have provided. And without knowing exactly which font is in use, it is impossible to make accurate measurements of text to be displayed.
What to do?
## The Testing Concept
### An Algorithm
One idea that has been bouncing around for many years is to:
• Start by measuring the width of a sample string using a generic font. (The last time I checked, there were nine generic fonts supported: serif, sans-serif, monospace, cursive, fantasy, system-ui, emoji, math, and fangsong.)
• Then construct a font-family with the font of interest and the generic font.
• Render the same string with the font-family.
• If the font of interest is not installed, the width measurement will be identical to that produced by rendering with the generic font.
### Some Finer Points
In point of fact, you must test against more than one generic font since you might want to ask if a font that is used as a generic is installed. For example, if you want to know if Arial is installed and run this test, you would receive a false negative on a system where Arial is used as the generic sans-serif font.
In the code I use, I test against three generic fonts, serif, sans-serif, and monospace.
Also, what size should the sample string be and what letters should it consist of? One of the earliest implementations of this method is attributed to one “Lalit Patel” (http://www.lalit.org/lab/javascript-css-font-detect (broken link)). They propose a string consisting of wide letters like “M” and “W” and thin letters like “l” and “i”. They also suggest using a large font size. I have no reason to quibble with these choices. I use M. Patels suggestions of “mmmmmmmmmmlli” for the sample string and 72 pixels for the size.
### How to “Render” and “Measure” the Sample String ###
You may have noticed that I have been playing fast and loose with some terminology here. What does it mean to “render” the string? You don’t want to render to the screen; this should all happen invisibly in the background.
There have been different approaches but the one I prefer is to create an HTML canvas element (unconnected to the DOM), get its 2-dimensional context, set the font size and family, and use the measureText method to get the width.
### Which Font from a font-family Declaration is Actually Being Used?
The rule is that when a CSS font-family is applied to an element the actual font used is the first one that is found to be installed. For example, if a CSS rule is applied to body like so:
body {
font-family: Boojum, Woojum, Helvetica, Wanger, Banger, serif;
}
In all likelihood, “Helvetica” will be the selected font. Writing a function to figure out which font will be applied is trivial.
## Some Code
Here’s my implementation in ClojureScript:
;;;;
;;;; Utilities to help with determining what if particulare fonts are
;;;; installed on the client system.
;;;;
(ns cwiki-mde.font-detection
(:require [clojure.string :as string]))
(def measured-font-widths (atom {}))
(def generic-fonts [:monospace :serif :sans-serif])
;; Suggested by Lalit Patel
;; Website: http://www.lalit.org/lab/javascript-css-font-detect/ (broken)
(def test-string "mmmmmmmmmmlli")
(def test-size 72)
(def canvas (.createElement js/document "canvas"))
(def context (.getContext canvas "2d"))
(defn- measure-string-width
"Return the width in pixels required to render the string in the given
font at the given size."
[s font-name font-size]
(set! (.-font context) (str font-size "px " font-name))
(.-width (.measureText context s)))
(defn- init-generic-font-widths!
"Set the global variable containing the measured widths of the test
string at the test size for each of the test fonts."
[]
(mapv #(swap! measured-font-widths
merge
{% (measure-string-width test-string (name %) test-size)})
generic-fonts))
(init-generic-font-widths!)
(defn- measure-against-generic
"Measure the width of the test string using the font name and compare it
to the width when one of the generic fonts (given by the generic-key) is
used. Return true if the width does NOT match the width produced by the
generic font. Return false otherwise."
[font-name generic-key]
(let [family (str test-size "px " font-name ", " (name generic-key))]
(set! (.-font context) family)
(let [width (measure-string-width test-string family test-size)]
(not= width (generic-key @measured-font-widths)))))
(defn font-available?
"Return true if the font named is available on the system."
[font-name]
(reduce #(or %1 (measure-against-generic font-name %2)) false generic-fonts))
;; Usage:
;;
;; (println "(font-available? \"Calibri\"): " (font-available? "Calibri"))
;; => true on Windows, false on Mac
; (println "(font-available? \"Calibri Regular\"): " (font-available? "Calibri Regular"))
;; => false
;; (println "(font-available? \"Arial\"): " (font-available? "Arial"))
;; => true
;; (println "(font-available? \"Boojum\"): " (font-available? "Boojum"))
;; => false
;; (println "(font-available? \"Helvetica Neue\"): " (font-available? "Helvetica Neue"))
;; => false on Windows, true on Mac
;; (println "(font-available? \"Helvetica Newish\"): " (font-available? "Helvetica Newish"))
;; => false
(defn font-family->font-used
"Given a CSS font family, determine which is actually used. This is the
first name in the list that is actually installed. Return nil if none of
the fonts in the list are installed."
[font-family]
(let [names (mapv string/trim (string/split font-family #","))]
(some #(when (font-available? %) %) names)))
;; (println "Selected from headline font family: " (font-family->font-used "\"Century Gothic\", Muli, \"Segoe UI\", Arial, sans-serif"))
;; => Muli ; on my system
;; (println "Selected from body font family: " (font-family->font-used "Palatino, \"Palatino Linotype\", \"Palatino LT STD\", \"Book Antiqua\", Georgia, serif"))
;; => Palatino ; on my system
;; (println "Selected from fixed font family: " (font-family->font-used "Consolas, \"Ubuntu Mono\", Menlo, Monaco, \"Lucida Console\",\n \"Liberation Mono\", \"DejaVu Sans Mono\", \"Bitstream Vera Sans Mono\",\n \"Courier New\", monospace, serif"))
;; => "Ubuntu Mono" ; on my system
The two important public functions are:
• font-available? used to determine if a particular font is installed.
• font-family->font-used that will return the name of the font that will be used in the browser from a font-family declaration.
So far, this code has proven small, quick, and robust. I hope you find it useful. | 2019-08-18 18:03:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4103163182735443, "perplexity": 8379.562870287227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313987.32/warc/CC-MAIN-20190818165510-20190818191510-00355.warc.gz"} |
https://search.datacite.org/works/10.6092/ISSN.1973-2201/796 | ### A method of inclusion probability proportional to size selection
V. K. Gupta
The purpose of this article is to propose a procedure for selecting an inclusion probability proportional to size sample n. The simplicity in sample selection and in computing joint inclusion probabilities is achieved by using the combinatorial properties of balanced incomplete block designs. The proposed method gives the same second order inclusion probabilities as that of Midzuno-Sen's procedure. | 2018-07-23 17:50:13 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9173939228057861, "perplexity": 448.4108605424072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676599291.24/warc/CC-MAIN-20180723164955-20180723184955-00130.warc.gz"} |
https://www.compadre.org/nexusph/course/Example_The_impulse-momentum_theorem | # Example: The impulse-momentum theorem
## Understanding the situation
The Impulse-Momentum theorem restates Newton's second law so that it expresses what forces do to an object as changing a property of the object: its momentum, mv. For an object A, the law looks like this:
$$\Delta\overrightarrow{p}_A = \int_{t_i}^{t_f}\overrightarrow{F}^{net}_A dt$$
This says that forces acting on an object changes its momentum and the amount of change is proportional to the amount of time those forces act. We can read off easily from this that if the net force is 0 (all forces are balanced) the object's momentum will not change. (NOT that the momentum will be 0!) Let's see how this works in an example. Here's the simplest possible example.
## Presenting a sample problem
A box rests on an air table and can slide freely without friction. If a small frictionless puck is slid towards the box consider two situations: it bounces straight back with about the same velocity or it is captured. If the interaction times between the box and the puck are the same, which puck exerts a greater force on the box?
## Solving this problem
This is a rather trivial problem and doesn't seem very interesting. We'll see however, in the next problem that it has interesting implications.
This is a qualitative problem, but we can still use an equation to solve it: the Impulse-momentum theorem. The change in momentum of the puck is equal to the impulse it receives from the box. The box seems much bigger than the puck so lets ignore the box's motion at first. If the puck has mass $m$ and velocity $v$, in the captured case the magnitude of the change in momentum is about $mv$ -- it goes from $mv$ to 0.
In the bounces back case, the momentum of the puck goes from $mv$ to $-mv$, so the magnitude of the change is $2mv$. Remember that momentum is a vector quantity! It first decreases to 0, then decreases even further to negative values.
So the impulse received by the puck from the box is twice as big in the bounces back case as in the captured case. If the interaction times are the same (given) then the force the box exerts in the bounces back case is twice as big as in the captured case.
But that's the force of the box on the puck. What about the force of the puck on the box? Of course these two forces are related by Newton's 3rd law: In any interaction, the force that two objects exert on each other is equal and opposite. So if we know the force the box exerts on the puck, we know the force the puck exerts on the box.
## Some grander implications of this toy model problem
This trivial case can be imbedded in a much more interesting case: molecules hitting a wall. Again, we will take only a simple case -- a stream of molecules in a vacuum. But we will see later that the same reasoning will allow us to understand how a gas exerts pressure and to extract the physical meaning of the ideal gas law in terms of molecules. (See Kinetic theory: The ideal gas law.)
Suppose of stream of gas having cross sectional area A is traveling in a vacuum and is directed at a wall. If the density of molecules in the gas is n (number of molecules per cubic meter) and they are traveling with a speed v, what will be the average force that the molecules exert on the wall if (a) they stick to the wall, and (b) they bounce off the wall with the same speed they hit the wall with?
As is typical in any problem, there are assumptions hidden in the way the problem is stated and we have to figure out how to treat it. The wall is being bombarded by lots of little molecules. Each one that hits it will exert a sudden quick force on the wall and then so will the next, and the next, etc. So there will be lots of tiny little forces that vary quickly. The problem can't mean for us to calculate those -- there isn't enough information about the wall molecule interaction. But the fact that the problem uses a macroscopic word ("wall") and a microscopic word ("molecule") suggests that we might make some reasonable approximations.
So let's assume that we have lots of molecules in the gas and that they are moving fast. The word "average" suggests that we shouldn't focus on the individual fluctuations of the force but rather on the result of lots of molecules. Since "wall" implies much, much bigger than a molecule, let's assume that the wall doesn't move significantly when a molecule hits it. (A typical molecule has a mass on the order of 10-26 kg and a wall might have a mass of a few kgs.)
Each molecule that hits the wall changes its momentum. To get a force, we might use the impulse-momentum theorem. But that gives the force the wall exerts on the molecule. We want the force the molecule exerts on the wall! Of course these two forces are related by Newton's 3rd law: In any interaction, the force that two objects exert on each other is equal and opposite. So if we know the force the wall exerts on the molecule, we know the force the molecule exerts on the wall. Since the times during the interaction are equal, the impulse that the wall gives to the molecule must be equal and opposite to the impulse that the molecule gives to the wall.
This also resolves the time issue. On a time scale natural for the wall, lots of molecules will hit it. The impulse momentum theorem tells us the amount of impulse the wall must provide to a bunch of molecules in a certain time interval, Δt. This will then tell us the amount of impulse the molecules provide to the wall in that time. Since we are told what happens to the velocities of the molecules, we can figure out their momentum change. Then we can calculate the average force the molecules exert on the wall.
$$\langle \overrightarrow{F}_{wall \rightarrow molecules} \rangle \Delta t = \Delta\overrightarrow{p}_{molecules}$$
$$\langle \overrightarrow{F}_{wall \rightarrow molecules} \rangle \Delta t = \langle \overrightarrow{F}_{molecules \rightarrow wall} \rangle \Delta t = \Delta\overrightarrow{p}_{molecules}$$
This is a rather standard way to use the Impulse-Momentum theorem. If you know the momentum change in a time interval, you can infer the impulse and therefore something about the average forces during that interval.
Let's see how that works here. Consider case (a): the molecules stick to the wall. In that case, the molecule initially had momentum $mv$ and after the collision it basically stops. (Assuming that the wall doesn't recoil significantly. This of course depends on our assumptions about how big the wall is and how big the stream of gas is.) This means each molecule changes its momentum by an amount mv: from $mv$ to 0.
Now let's consider a time interval in which many molecules will hit the wall. In a time interval, $Δt$, how many will hit? To get this, look at the figure above. In a time interval, $Δt$, a molecule will move a distance $d = vΔt$. If we take our distance $d$ in the figure to be $vΔt$ then all the molecules in there will hit the wall and stick. How many is that? Well, we know the density and the volume of molecules hitting the wall is $A x d = AvΔt$. So the total number, $N$, hitting the wall in that time is the density times the volume or
$N$ = number hitting the wall in time $Δt = nAvΔt$
So since each molecule changes its momentum by mv, the total change in momentum of the molecules in that time is Nmv, which gives a force
$$\langle \overrightarrow{F}_{wall \rightarrow molecules} \rangle = \frac{\Delta\overrightarrow{p}_{molecules}}{\Delta t} = \frac{N mv}{\Delta t} = \frac{(nAv \Delta t) mv}{\Delta t} = nmAv^2$$
For case (b), if each molecule bounces back with the same speed as it entered it changes its momentum from $mv$ to $-mv$ a total change of $2mv$. Therefore, the result will be twice as big as if the molecule stuck to the wall.
Joe Redish 8/6/15
Article 397 | 2023-01-28 16:05:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6432760953903198, "perplexity": 277.7495909330546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499646.23/warc/CC-MAIN-20230128153513-20230128183513-00783.warc.gz"} |
https://scicoding.com/convolution-in-python-3-essential-packages/ | # 4 Ways to Calculate Convolution in Python
How to calculate convolution in Python. Here are the 3 most popular python packages for convolution + a pure Python implementation.
Convolution is one of the fundamental operations in signal processing. Similarly to cross-correlation, it can be used to analyze the similarity of two signals with different lags.
There are many applications to convolution. It is particularly popular in image processing and machine vision, where convolution has been in edge detection and similar applications. Convolution of an input signal and impulse response gives the output of a linear time-invariant system. This makes it a very common operation in electrical engineering.
Below you can see an illustration of the convolution between square and triangular pulses.
In the previous posts, we have also gone through related operations and their implementations in Python:
We will have a look at three packages for calculating convolution in Python + a pure Python implementation without any dependencies.
## Convolution the definition
Convolution is defined as the integral of the product of two signals (functions), where one of the signals is reversed in time. It is closely related to cross-correlation. In fact, it is cross-correlation after one of the signals has been reversed.
The definition is quite simple, you overlap the two signals with a given delay and correlate with the latter signal in reverse order. That is, one of the signals is reversed. We can write this for real-valued discrete signals as
$R_{fg}(l) = \sum_{n=0}^N f(n)g(n - l)$
In the following, you can see a simple animation highlighting the process. Notice how the triangle function is flipped before taking the cross-correlation, in the beginning, to reverse the input signal and perform convolution.
Definitions for complex, continuous, and random signals can be found, e.g., in Wikipedia.
## Data set and number of lags to calculate
Before going into the methods of calculating convolution, we need to have some data. We use two signals as our data sets. The first one is a square pulse and the second one is a triangular pulse.
def sig_square(x):
return 0 if x < 3 or x > 5 else 2
def sig_triag(x):
return 0 if x < 0 or x > 2 else x
# First signal (square pulse)
sig1 = [sig_square(x/100) for x in range(1000)]
# Seconds signal (triangle pulse)
sig2 = [sig_triag(x/100) for x in range(200)]
For the sake of simplicity, we consider a one-dimensional data set. All of the considered packages also work with 2D data. This provides an obvious application for image processing.
## Convolution: 3 essential packages + pure python implementation
Now, we are ready to dive into the different implementations of convolution. We begin with the Python-only implementation. This gives us the baseline. Then, we compare the results to 3 essential signal processing packages that provide their own implementation of convolution.
## Native python implementation
This is a Python-only method without any external dependencies for calculating the cross-correlation. For the sake of simplicity, we don't consider any padding. That is, we will consider only the lags when the signals overlap fully.
The output for the vanilla Python implementation should look like
## Numpy
When doing any numerical or scientific computation in Python, NumPy is usually the first package that will be imported. NumPy has a convenient implementation for convolution readily available.
You can choose the mode to handle partially overlapping signals, i.e., padding in the beginning and end of the signal. We use mode='valid', which only considers fully overlapping signals. This makes the comparison against our Python-only reference implementation easier.
The output of the NumPy implementation is identical to the Python-only implementation, which can be used to verify our implementation as well.
Check the NumPy reference manual for more examples.
## Scipy
SciPy is the go-to package for numerical analysis and particularly many signal processing-specific methods. Whenever NumPy is missing the method, the SciPy should be the next target to go for.
Convolution can be found in the scipy.signal package. This method offers the same choices for padding as NumPy. In addition, you can choose between directand fft implementation. Direct implementation follows the definition of convolution similar to the pure Python implementation that we looked at before. The fft-based approach does convolution in the Fourier domain, which can be more efficient for long signals.
''' SciPy implementation '''
import matplotlib.pyplot as plt
import scipy.signal as sig
conv = sig.convolve(sig1, sig2, mode='valid')
conv /= len(sig2) # Normalize
plt.plot(conv)
The output of the SciPy implementation is identical to the Python-only and NumPy implementations.
Check the SciPy reference manual for more examples.
## Astropy
Astropy is astronomy focused package and less common than the aforementioned packages. Nevertheless, it contains general-purpose functions such as convolution that are useful for other signal-processing domains as well.
Convolution in Astropy is meant to improve the SciPy implementation, particularly for scipy.ndimage. Some of these improvements include
• Proper treatment of NaN values (ignoring them during convolution and replacing NaN pixels with interpolated values)
• A single function for 1D, 2D, and 3D convolution
• Improved options for the treatment of edges
• Both direct and Fast Fourier Transform (FFT) versions
• Built-in kernels that are commonly used in Astronomy
Here, we are just using the 1D convolution. These improvements are more visible when taking the higher dimension convolution.
Below, you can see how to utilize astropy.convolution to our test data. One restriction that needs to be managed is that the kernel needs to have an odd length. Thus for even signal length, we will use zero padding to make the signal non-even in length.
Notice that we don't have explicit normalization here, as Astropy implementation has normalization on by default.
This will result in the following graph. Notice how the peak is shifted from the NumPy and SciPy versions. The shape of the convolution is identical.
Check the Astropy reference manual for more examples.
## Summary
We did not compare the performance of the convolution methods. If we ignore the potential performance differences, it is pretty much a convenience-choice between the packages. Use whatever package you are using anyways. If you are after a bit more advanced features astropy might be the choice for you. Otherwise, I would go with numpy or scipy. Pure Python version should be used only for reference or if you are really trying to cut back in the dependencies, as there is very likely performance consideration. | 2023-03-31 09:05:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5005031824111938, "perplexity": 971.5409551407716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949598.87/warc/CC-MAIN-20230331082653-20230331112653-00514.warc.gz"} |
https://doc.cgal.org/5.0.2/Cone_spanners_2/group__PkgConeSpanners2Ref.html | CGAL 5.0.2 - Cone-Based Spanners
Cone-Based Spanners Reference
Weisheng Si, Quincy Tse and Frédérik Paradis
This package provides functors for constructing two kinds of cone-based spanners: Yao graph and Theta graph, given a set of vertices on the plane and the directions of cone boundaries. Both exact and inexact constructions are supported. In exact construction, the cone boundaries are calculated using the roots of polynomials, thus avoiding the use of $$\pi$$, which cannot be represented exactly. In inexact construction, the cone boundaries are calculated using the approximate $$\pi$$ value defined in CGAL, which is still accurate enough for most applications. Moreover, for visualization purpose, this package provides a global function to generate the data and script files used by Gnuplot to plot the constructed graphs. This package also provides options for the Half Yao graph and the Half Theta graph.
Introduced in: CGAL 4.9
BibTeX: cgal:st-cbs-20a
## Functors
• CGAL::Compute_cone_boundaries_2
• CGAL::Construct_yao_graph_2
• CGAL::Construct_theta_graph_2
## Functions
• CGAL::gnuplot_output_2()
## Enumerations
• CGAL::Cones_selected
## Classes
class CGAL::Compute_cone_boundaries_2< Traits_ >
The functor for computing the directions of cone boundaries with a given cone number and a given initial direction. More...
class CGAL::Construct_theta_graph_2< Traits_, Graph_ >
A template functor for constructing Theta graphs with a given set of 2D points and a given initial direction for the cone boundaries. More...
class CGAL::Construct_yao_graph_2< Traits_, Graph_ >
A template functor for constructing Yao graphs with a given set of 2D points and a given initial direction for the cone boundaries. More...
## Enumerations
enum CGAL::Cones_selected { CGAL::EVEN_CONES = 0, CGAL::ODD_CONES = 1, CGAL::ALL_CONES = 2 }
An enum of the choice of cones in cone spanners. More...
## Functions
template<typename Graph >
void CGAL::gnuplot_output_2 (const Graph &g, const std::string &prefix)
Output a set of files used by Gnuplot to plot g. More...
## ◆ Cones_selected
enum CGAL::Cones_selected
#include <CGAL/Cone_spanners_enum_2.h>
An enum of the choice of cones in cone spanners.
Enumerator
EVEN_CONES
Select even cones.
ODD_CONES
Select odd cones.
ALL_CONES
Select all cones.
## ◆ gnuplot_output_2()
template<typename Graph >
void CGAL::gnuplot_output_2 ( const Graph & g, const std::string & prefix )
#include <CGAL/gnuplot_output_2.h>
Output a set of files used by Gnuplot to plot g.
The files that are generated for Gnuplot are: (1) prefix.v (vertex list) (2) prefix.plt (Gnuplot script), This script will read prefix.v as input to plot the vertex list. The edge list is also included in this script.
Notes: (1) If these files already exists, this function will overwrite these files. (2) Parallel and self-edges cannot be plotted.
Template Parameters
Graph The type of the graph to be plotted. For this function to work, the graph type must be boost::adjacency_list with CGAL::Point_2 as the VertexProperties.
Parameters
g A boost::adjacency_list graph with CGAL::Point_2 as the VertexProperties to be plotted prefix The prefix of the output files names
Examples:
Cone_spanners_2/theta_io.cpp. | 2022-05-24 21:49:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.537860095500946, "perplexity": 6109.337198653459}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00752.warc.gz"} |
https://myassignments-help.com/2022/09/09/solid-state-physics-dai-xie-physics7544/ | # 物理代写|固体物理代写Solid-state physics代考|PHYSICS7544
## 物理代写|固体物理代写Solid-state physics代考|Elastic moduli
We have so far introduced four different elastic parameters, namely the Young modulus, the Poisson ratio and the two Lamé coefficients. We previously introduced also the bulk modulus $B$ when we investigated thermal expansion in section 4.2.1. Since $B$ was defined as the inverse of the isothermal compressibility (see appendix $C$ ), that is it deals with volume variations, it is expected to be related to some elastic parameter. In order to elucidate this issue, let us consider a hydrostatic stress $T_{i j}=P \delta_{i j}$ (where $P$ is the macroscopic hydrostatic pressure) and insert it into the constitutive equation (5.39) so as to get
$$\left.\mathbb{S}=\frac{1}{3} \frac{1}{\lambda+\frac{2}{3} \mu} P\right]$$
The connection with equation (C.11) is established by defining
$$B=\lambda+\frac{2}{3} \mu,$$
so that
$$\left.\mathbb{S}=\frac{1}{3 B} P\right] \quad \rightarrow \quad \operatorname{Tr}(\mathbb{S})=\sum_i \epsilon_{i t}=\frac{\Delta V}{V}=\frac{P}{B}$$
which leads to the following definition
$$\frac{1}{B}=\frac{1}{V} \frac{\Delta V}{P}$$
representing the finite difference counterpart of equation (C.11). This result reconciles the thermodynamical and elastic treatment of deformations affecting the system volume and it allows us to recast the stress-strain relation of a homogeneous and isotropic linear elastic medium in the form
$\mathbb{T}=2 \mu \mathbb{S}+\left(B-\frac{2}{3} \mu\right) \operatorname{Tr}(\mathbb{S}) \rrbracket$
$\left.=3 B\left[\frac{1}{3} \operatorname{Tr}(\mathbb{S}) \rrbracket\right]+2 \mu\left[\mathbb{S}-\frac{1}{3} \operatorname{Tr}(\mathbb{S})\right]\right]$,
where the first and second term on the right-hand side are, respectively, named spherical part and deviatoric part of the stress tensor: they describe the hydrostatic volume variation and the change in shape of the solid body subject to $\mathbb{I}$.
## 物理代写|固体物理代写Solid-state physics代考|Thermoelasticity
We have so far implicitly assumed that the deformations are imposed to the system at zero temperature. While this assumption was useful to define a clean purely-elastic problem, we must duly generalise our theory to include stress actions applied at $T>0 \mathrm{~K}$ as well [6].
The starting point is of course the energy balance stated by the first law of thermodynamics (see equation (C.5)) which for a system with volume $V$ in equilibrium at temperature $T$ under some elastic action is written as
$$d \mathcal{U}=V \sum_{i j} T_{i j} d \epsilon_{i j}+T d S,$$
where the mechanical work contributing to the internal energy $\mathcal{U}$ has been written in terms of the stress tensor since we know that this latter describes any possible kind of volume and shape variation of the system. It is easy to reconcile equation (5.47) with the standard thermodynamical formulation by simply considering the case of a hydrostatic stress $T_{i j}=-P \delta_{i j}$, where $P$ is the applied pressure whose negative sign indicates that the mechanical action is compressive. We assume it to operate quasistatically, like anywhere else in the remaining of this chapter. By inserting the hydrostatic stress into equation (5.47) we get
\begin{aligned} d \mathcal{U} &=V \sum_{i j}\left(-P \delta_{i j}\right) d \epsilon_{i j}+T d S \ &=-P V \sum_i d \epsilon_{i i}+T d S \ &=-P V \frac{d V}{V}+T d S \ &=-P d V+T d S \end{aligned}
## 物理代写|固体物理代写Solid-state physics代考|Elastic moduli
$$\left.\mathbb{S}=\frac{1}{3} \frac{1}{\lambda+\frac{2}{3} \mu} P\right]$$
$$B=\lambda+\frac{2}{3} \mu,$$
$$\left.\mathbb{S}=\frac{1}{3 B} P\right] \quad \rightarrow \quad \operatorname{Tr}(\mathbb{S})=\sum_i \epsilon_{i t}=\frac{\Delta V}{V}=\frac{P}{B}$$
$$\frac{1}{B}=\frac{1}{V} \frac{\Delta V}{P}$$
## 物理代写|固体物理代写Solid-state physics代考|Thermoelasticity
$$d \mathcal{U}=V \sum_{i j} T_{i j} d \epsilon_{i j}+T d S,$$
$$d \mathcal{U}=V \sum_{i j}\left(-P \delta_{i j}\right) d \epsilon_{i j}+T d S \quad=-P V \sum_i d \epsilon_{i i}+T d S=-P V \frac{d V}{V}+T d S \quad=-P d V+T d S$$
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myassignments-help擅长领域包含但不是全部: | 2022-11-28 20:02:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9188372492790222, "perplexity": 810.6171452801557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710534.53/warc/CC-MAIN-20221128171516-20221128201516-00481.warc.gz"} |
http://www.aimsciences.org/article/doi/10.3934/dcds.2011.29.1197 | # American Institute of Mathematical Sciences
2011, 29(3): 1197-1204. doi: 10.3934/dcds.2011.29.1197
## Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups
1 Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, United States
Received January 2010 Revised May 2010 Published November 2010
Assume that $X$ is a hyperbolic basic set for $f:X\to X$. We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensions of $f$, those that are transitive are open and dense. The fibers are semidirect products of compact and nilpotent groups.
Citation: Viorel Niţică. Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1197-1204. doi: 10.3934/dcds.2011.29.1197
##### References:
[1] D. Z. Djoković, The union of compact subgroups of a connected locally compact group,, Math. Zeitschrift, 158 (1978), 99. doi: 10.1007/BF01320860. [2] M. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets,, Ergod. Theory Dynam. Systems, 25 (2005), 517. doi: 10.1017/S0143385704000355. [3] M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979). [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611. doi: 10.1017/S0143385702001554. [7] I. Melbourne, V. Niţică and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems,, Annales Henri Poincaré, 6 (2005), 725. doi: 10.1007/s00023-005-0221-0. [8] I. Melbourne, V. Niţică and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems,, Discrete Contin. Dynam. Systems, 14 (2006), 355. [9] I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems,, Ergod. Theory Dynam. Systems, 29 (2009), 1585. doi: 10.1017/S0143385708000886. [10] I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., (). [11] V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257. doi: 10.1017/S0143385704000471. [13] V. Niţică and A. Török, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one,, Topology, 40 (2001), 259. doi: 10.1016/S0040-9383(99)00060-9. [14] J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302. [15] T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869. doi: 10.2307/2154147.
show all references
##### References:
[1] D. Z. Djoković, The union of compact subgroups of a connected locally compact group,, Math. Zeitschrift, 158 (1978), 99. doi: 10.1007/BF01320860. [2] M. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets,, Ergod. Theory Dynam. Systems, 25 (2005), 517. doi: 10.1017/S0143385704000355. [3] M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979). [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611. doi: 10.1017/S0143385702001554. [7] I. Melbourne, V. Niţică and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems,, Annales Henri Poincaré, 6 (2005), 725. doi: 10.1007/s00023-005-0221-0. [8] I. Melbourne, V. Niţică and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems,, Discrete Contin. Dynam. Systems, 14 (2006), 355. [9] I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems,, Ergod. Theory Dynam. Systems, 29 (2009), 1585. doi: 10.1017/S0143385708000886. [10] I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., (). [11] V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257. doi: 10.1017/S0143385704000471. [13] V. Niţică and A. Török, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one,, Topology, 40 (2001), 259. doi: 10.1016/S0040-9383(99)00060-9. [14] J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302. [15] T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869. doi: 10.2307/2154147.
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https://projecteuclid.org/euclid.agt/1513797071 | ## Algebraic & Geometric Topology
### The Seidel morphism of Cartesian products
Rémi Leclercq
#### Abstract
We prove that the Seidel morphism of $(M×M′,ω⊕ω′)$ is naturally related to the Seidel morphisms of $(M,ω)$ and $(M′,ω′)$, when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with nontrivial image via Seidel’s morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [Int. Math. Res. Not. (2008) Art. ID rnn049].
#### Article information
Source
Algebr. Geom. Topol., Volume 9, Number 4 (2009), 1951-1969.
Dates
Accepted: 31 August 2009
First available in Project Euclid: 20 December 2017
https://projecteuclid.org/euclid.agt/1513797071
Digital Object Identifier
doi:10.2140/agt.2009.9.1951
Mathematical Reviews number (MathSciNet)
MR2550462
Zentralblatt MATH identifier
1176.57027
#### Citation
Leclercq, Rémi. The Seidel morphism of Cartesian products. Algebr. Geom. Topol. 9 (2009), no. 4, 1951--1969. doi:10.2140/agt.2009.9.1951. https://projecteuclid.org/euclid.agt/1513797071
#### References
• P Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. (2007) Art. ID rnm134, 56pp
• P Biran, L Polterovich, D Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003) 65–118
• C C Conley, E Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V I Arnol$'$d, Invent. Math. 73 (1983) 33–49
• M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• S Hu, F Lalonde, Anti-symplectic involution and Maslov indices, to appear in Trans. Amer. Math. Soc.
• F Lalonde, D McDuff, L Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369–385
• D McDuff, Quantum homology of fibrations over $S\sp 2$, Internat. J. Math. 11 (2000) 665–721
• A Pedroza, Seidel's representation on the Hamiltonian group of a Cartesian product, Int. Math. Res. Not. (2008) Art. ID rnn049, 19pp
• S Piunikhin, D Salamon, M Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171–200
• D Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology (Park City, UT, 1997)”, (Y Eliashberg, L Traynor, editors), IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 143–229
• P Seidel, $\pi\sb 1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046–1095
• E Witten, Topological sigma models, Comm. Math. Phys. 118 (1988) 411–449 | 2020-01-29 10:59:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6978006958961487, "perplexity": 2975.9198584057926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00106.warc.gz"} |
http://physics.stackexchange.com/tags/atoms/new | # Tag Info
0
L-S coupling is the appropriate way of description of small small electronic configuration ($Z\leq10$). In case of atoms having a large number of electrons, J-J coupling works.
3
As it turns out, the excited states of an atom are not quite eigenstates. That is, they're eigenstates of the atomic hamiltonian, but they are not eigenstates of the atom-plus-EM-field hamiltonian. How do we know? If you prepare them (excited atom, empty field) and then you leave them alone, they change, so they can't be eigenstates. What they are is ...
1
If I understand your question correctly, you are confusing between two definitions of "width". One is the spread of the spectral line and the other is the lifetime of the atom in excited state. Well, these two are related(as you guessed) by uncertainty principle. $\Delta E\,\Delta t = h/4\pi$. So if the time the atom spends in excited state is less, the ...
0
Random thermal motion (Brownian motion) allows the particles to become ergodically distributed in the their phase space. They scattering off each other and any other particles in the environment, this randomises the motion. Otherwise they would carry on in the same direction until acted on my a force. It is the randomisation nature of scattering events that ...
0
Nothing "causes" diffusion, it is a statistical process. The atoms in any system in thermal equilibrium are constantly moving with velocities of order $\langle v^2\rangle \simeq T/m$. This motion is randomized by collisions on some microscopic time scale $\tau$. The simplest case is a dilute gas, where $\tau\sim 1/(vn\sigma)$, where $n$ is the density of the ...
2
Quantum tunneling keeps the energy of the tunneling particle the same as it had within the potential . As you will find, putting realistic numbers for nuclear penetration in the calculator of the link the probabilities come out zero. So spontaneous "turn into a nuclear bomb " is out. Lets took "technically" : A human exposed to MeV radiation can he/she ...
3
The thing is, we're made of mostly stable matter of low atomic number. In a nuclear bomb, unstable nuclei split, releasing a number of energetic neutrons which strike other unstable nuclei, and the reactions chain uncontrollably. Splitting a small nucleus actually costs energy, so even if a carbon atom in your body did split, it would only split into ...
3
It also appeared to me that the higher you go, the harder it is to make an element. This is pretty much true. I have a tendency to be wordy and long in my posts, but I'll try to cover a few points as concise as possible. Ununoctium was created by "bombarding atoms of californium-249 with ions of calcium-48. This produced ununoctium-294, an ...
3
It's not a game of chance. Selecting the right nuclides for the heavy ion collisions is key and the detection requires extremely sensitive and well calibrated detectors. If you want to put an attribute on it then "art" would be far more fitting. You are correct, it does get harder for heavier nuclei. Practical applications? That's not a question for science, ...
1
There are no random motions in solids. All motions are highly correlated, you are just adding up a lot of modes at different frequencies, which looks like random motion if you are only looking at a single atom. It's not totally wrong to look at single particles being in random motion, though, since the Fourier transform of a lot of frequencies at arbitrary ...
3
The measurement of C-14 can be accurate. The interpretation of the measurement, in terms of the presumed age of the sample, is the thing that is subject to careful calibration. The assumption of carbon dating is that the ratio of C12/C13/C14 in the sample had some known value at the time the organism was alive. "Known" does not mean "constant over all ...
7
It depends on the mechanism by which the photon is emitted. Stimulated Emission: Yes, the emitted photon will inherit the characteristics of the photon that stimulated it, including its propagation direction. That's how lasers get coherent light. Spontaneous Emission: No, this should be random orientation. Another way to think about spontaneous emission is ...
12
In the general case, "no" as there is a angular momentum transfer involved (meaning there are preferred directions relative the original and/or final angular momentum of the atom). That said, for most matter at room temperature the atoms have random orientation so you can treat the answer as "yes" for experimental purposes. Now, I see that you are ...
Top 50 recent answers are included | 2015-10-06 04:00:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7019602060317993, "perplexity": 574.8372155825422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736678409.42/warc/CC-MAIN-20151001215758-00228-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://conferences.famnit.upr.si/event/11/page/1-contributed-talks | # Software Tools For Mathematics
24-28 September 2018
UP Famnit
UTC timezone
# Contributed Talks
## Tentative Schedule
Session 1 (Thursday, 27 September 2018)
• 11:30 Russ Woodroofe: The story of a figure
• 12:00 Alba Marina Málaga Sabogal: Tools for 3D printing mathematical surfaces
• 12:30 Mirza Krbezlija: Using Java and the CPLEX solver for solving integer linear programs
Chair: Matjaž Krnc
Problem Session (Thursday, 27 September 2018)
The problem session starts at 16:30.
Chair: Eric Gottlieb
Session 2 (Friday, 28 September 2018)
• 11:30 Matjaž Krnc: Pipelines in mathematical research
• 12:00 Alba Marina Málaga Sabogal: Reverse engineering your mathematical PDF files
• 12:30 Tomaž Pisanski: Computing Jacobian matrix of a graph with Sage and Python
Chair: Katja Berčič
Session 3 (Friday, 28 September 2018)
• 16:30 Katja Berčič: Databases of discrete mathematical objects and DiscreteZOO
• 17:00 Janoš Vidali: DiscreteZOO – a repository of combinatorial objects and its companion Sage package
Chair: Nino Bašić
## Abstracts
Databases of discrete mathematical objects and DiscreteZOO
Katja Berčič
Universidad Nacional Autónoma de México, Morelia, Mexico
Abstract: There are (at least) two interesting problems when it comes to databases of mathematical objects. On one hand, they should be useful to a researcher who wants to look up properties or references or to a researcher who is looking for a pattern or a counterexample. On the other hand, a typical researcher who produces a collection of objects, say graphs, does not have the resources to develop a platform such as The On-Line Encyclopedia of Integer Sequences or The L-functions and Modular Forms Database. One of the objectives of the DiscreteZOO project is to provide a reusable platform for databases of mathematical objects.
I will talk about how DiscreteZOO fits into the larger effort on managing mathematical knowledge, what problems one encounters in making a reusable platform for databases of mathematical objects and the current work on a database of maniplexes.
Using Java and the CPLEX solver for solving integer linear programs
Mirza Krbezlija
University of Primorska, Koper, Slovenia
Abstract: In my final project paper for the first bologna cycle, among other things, I studied the vector connectivity problem. Vector connectivity is a graph optimization problem whose optimal value can be computed by formulating it as an integer linear program. To solve the integer linear program and to find its optimal value I used the programming language Java together with the CPLEX solver library. I will present some difficulties I had while using both Java and the CPLEX solver together with the adjustments I had to do in order to get past those difficulties so I could obtain the desired results for the related topic.
Pipelines in mathematical research
Matjaž Krnc
University of Primorska, Koper, Slovenia
Abstract: In this talk we will go through several hands-on examples of my research projects where the CoCalc service was used extensively. We will discuss the benefits and drawbacks of the tools offered by the service. The emphasis will be on the pipelining several tools covered in our workshop, and gluing them together as needed.
Reverse engineering your mathematical PDF files
Alba Marina Málaga Sabogal
French Institute for Research in Computer Science and Automation, Paris, France
Abstract: How many times have you realized that you lost your LaTeX source files for that beautiful PDF paper? How many times did you needed to quote some formula from a paper and you had to type it all again by yourself? If you are a sighted person, you may want to recover LaTeX sources for convenience. But if you are a blind person, having access to a document with rich markup may be essential for your screen reader to understand the content of the document. That's what accessibility is about: to make easy for the helpful technology to work correctly.
In the project OPALINe we work on making electronic documents accessible again. Our main tool for that is the GROBID library. In this talk I will explain how to use GROBID together with the im2markup visual formula decompiler in order to get back a source file for a mathematical paper PDF.
Tools for 3D printing mathematical surfaces
Alba Marina Málaga Sabogal
French Institute for Research in Computer Science and Automation, Paris, France
Abstract: In this talk I will present a few software tools that you can use for 3D printing mathematical surfaces, like real algebraic surfaces, invariant surfaces in dynamics or minimal surfaces. Personnally the tools I use the most often are MathMod and Blender, but during the talk I will also present Functy, Surface Evolver, Antimony, Bertini_real, NetFabb and others.
Computing Jacobian matrix of a graph with Sage and Python
Tomaž Pisanski
University of Primorska, Koper, Slovenia & IMFM, Ljubljana, Slovenia
Abstract: We present the power of Sage and Python by exhibiting a short program for computing the Jacobian group of a graph.
Here is the program:
def Jacobian(g, verbose=False):
if verbose:
g.show()
lm = g.laplacian_matrix()
A = matrix(ZZ, lm)
AS = A.smith_form()
diag = AS[0].diagonal()
res = []
for x in diag:
if x > 1:
res.append(x)
return res
Using the above program one can easily compute the number of spanning trees of a connected graph.
DiscreteZOO – a repository of combinatorial objects and its companion Sage package
Janoš Vidali
University of Ljubljana, Ljubljana, Slovenia
Abstract: DiscreteZOO is a project combining a central repository of discrete objects, its website front-end and extensions for software packages like Sage. The repository contains certain precomputed properties to speed up the processes of filtering, searching and computation. For now, it can store graphs, with the groundwork already laid out for more combinatorial objects (such as maps, maniplexes, geometries, etc.).
In the talk we will show how one can interact with DiscreteZOO on the example of the censuses of vertex-transitive graphs by Royle, Conder, and Potočnik, Spiga & Verret. We will perform some example searches on the website, download a subset of the database and showcase some queries that can be run locally with the Sage package.
The story of a figure
Russ Woodroofe
University of Primorska, Koper, Slovenia
Abstract: I'll describe how I drew a complex figure for a recent paper. | 2019-12-11 21:34:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25459933280944824, "perplexity": 2303.5587960205157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00413.warc.gz"} |
https://wikimili.com/en/Gravity | # Gravity
Last updated
In physics, gravity (from Latin gravitas 'weight' [1] ) is a fundamental interaction which causes all things with mass or energy to be attracted (or gravitate) toward one another. Gravity is by far the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong interaction, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles. [2] However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.
## Contents
On Earth, gravity gives weight to physical objects, and the Moon's gravity causes tides in the oceans. Gravity also has many important biological functions, helping to guide the growth of plants through the process of gravitropism and influencing the circulation of fluids in multicellular organisms. Investigation into the effects of weightlessness has shown that gravity may play a role in immune system function and cell differentiation within the human body.
The gravitational attraction between the original gaseous matter in the Universe allowed it to coalesce and form stars which eventually condensed into galaxies, so gravity is responsible for many of the large-scale structures in the Universe. Gravity has an infinite range, although its effects become weaker as objects get farther away.
Gravity is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915), which describes gravity not as a force, but as the curvature of spacetime, caused by the uneven distribution of mass, and causing masses to move along geodesic lines. The most extreme example of this curvature of spacetime is a black hole, from which nothing—not even light—can escape once past the black hole's event horizon. [3] However, for most applications, gravity is well approximated by Newton's law of universal gravitation, which describes gravity as a force causing any two bodies to be attracted toward each other, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them.
Current models of particle physics imply that the earliest instance of gravity in the Universe, possibly in the form of quantum gravity, supergravity or a gravitational singularity, along with ordinary space and time, developed during the Planck epoch (up to 10−43 seconds after the birth of the Universe), possibly from a primeval state, such as a false vacuum, quantum vacuum or virtual particle, in a currently unknown manner. [4] Scientists are currently working to develop a theory of gravity consistent with quantum mechanics, a quantum gravity theory, which would allow gravity to be united in a common mathematical framework (a theory of everything) with the other three fundamental interactions of physics.
## History
### Ancient world
The nature and mechanism of gravity was explored by a wide range of ancient scholars. In Greece, Aristotle believed that objects fell towards the Earth because the Earth was the center of the Universe and attracted all of the mass in the Universe towards it. He also thought that the speed of a falling object should increase with its weight, a conclusion which was later shown to be false. [5] While Aristotle's view was widely accepted throughout Ancient Greece, there were other thinkers such as Plutarch who correctly predicted that the attraction of gravity was not unique to the Earth. [6]
Although he didn't understand gravity as a force, the ancient Greek philosopher Archimedes discovered the center of gravity of a triangle. [7] He also postulated that if two equal weights did not have the same center of gravity, the center of gravity of the two weights together would be in the middle of the line that joins their centers of gravity. [8]
In India, the mathematician-astronomer Aryabhata first identified gravity to explain why objects are not driven away from the Earth by the centrifugal force of the planet's rotation. Later, in the seventh century CE, Brahmagupta proposed the idea that gravity is an attractive force which draws objects to the Earth and used the term gurutvākarṣaṇ to describe it. [9] [10] [11] This research has led some people to claim that Brahmagupta, not Isaac Newton, was responsible for "discovering" gravity. [12] [13]
In the ancient Middle East, gravity was a topic of fierce debate. The Persian intellectual Al-Biruni believed that the force of gravity was not unique to the Earth, and he correctly assumed that other heavenly bodies should exert a gravitational attraction as well. [14] In contrast, Al-Khazini held the same position as Aristotle that all matter in the Universe is attracted to the center of the Earth. [15]
### Scientific revolution
In the mid-16th century, various European scientists experimentally disproved the Aristotelian notion that heavier objects fall at a faster rate. [16] In particular, the Spanish Dominican priest Domingo de Soto wrote in 1551 that bodies in free fall uniformly accelerate. [16] De Soto may have been influenced by earlier experiments conducted by other Dominican priests in Italy, including those by Benedetto Varchi, Francesco Beato, Luca Ghini, and Giovan Bellaso which contradicted Aristotle's teachings on the fall of bodies. [16] The mid-16th century Italian physicist Giambattista Benedetti published papers claiming that, due to specific gravity, objects made of the same material but with different masses would fall at the same speed. [17] With the 1586 Delft tower experiment, the Flemish physicist Simon Stevin observed that two cannonballs of differing sizes and weights fell at the same rate when dropped from a tower. [18] Finally, in the late 16th century, Galileo Galilei performed his famous Leaning Tower of Pisa experiment in order to show once again that balls of different weights would fall at the same speed. [19] Combining this knowledge with careful measurements of balls rolling down inclines, Galileo firmly established that gravitational acceleration is the same for all objects. [20] Galileo postulated that air resistance is the reason that objects with a low density and high surface area fall more slowly in an atmosphere.
In 1604, Galileo correctly hypothesized that the distance of a falling object is proportional to the square of the time elapsed. [21] This was later confirmed by Italian scientists Jesuits Grimaldi and Riccioli between 1640 and 1650. They also calculated the magnitude of the Earth's gravity by measuring the oscillations of a pendulum. [22]
### Newton's theory of gravitation
In 1684, Newton sent a manuscript to Edmond Halley titled De motu corporum in gyrum ('On the motion of bodies in an orbit'), which provided a physical justification for Kepler's laws of planetary motion. [23] Halley was impressed by the manuscript an urged Newton to expand on it, and a few years later Newton published a groundbreaking book called Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). In this book, Newton described gravitation as a universal force, and claimed that "the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve." This statement was later condensed into the following inverse-square law:
${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}$
where F is the force, m1 and m2 are the masses of the objects interacting, r is the distance between the centers of the masses and G is the gravitational constant.
Newton's Principia was well-received by the scientific community, and his law of gravitation quickly spread across the European world. [24] More than a century later, in 1821, his theory of gravitation rose to even greater prominence when it was used to predict the existence of Neptune. In that year, the French astronomer Alexis Bouvard used this theory to create a table modeling the orbit of Uranus, which was shown to differ significantly from the planet's actual trajectory. In order to explain this discrepancy, many astronomers speculated that there might be a large object beyond the orbit of Uranus which was disrupting Neptune's orbit. In 1846, the astronomers John Couch Adams and Urbain Le Verrier independently used Newton's law to predict Neptune's location in the night sky, and the planet was discovered there within a day. [25]
### General relativity
Eventually, astronomers noticed an eccentricity in the orbit of the planet Mercury which could not be explained by Newton's theory: the perihelion of the orbit was increasing by about 42.98 arcseconds per century. The most obvious explanation for this discrepancy was an as-yet-undiscovered celestial body (such as a planet orbiting the Sun even closer than Mercury), but all efforts to find such a body turned out to be fruitless. Finally, in 1915, Albert Einstein developed a theory of general relativity which was able to accurately model Mercury's orbit. [26]
In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of a force. Einstein began to toy with this idea in the form of the equivalence principle, a discovery which he later described as "the happiest thought of my life." [27] In this theory, free fall is considered to be equivalent to inertial motion, meaning that free-falling inertial objects are accelerated relative to non-inertial observers on the ground. [28] [29] In contrast to Newtonian physics, Einstein believed that it was possible for this acceleration to occur without any force being applied to the object.
Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. These straight paths are called geodesics. As in Newton's first law of motion, Einstein believed that a force applied to an object would cause it to deviate from a geodesic. For instance, people standing on the surface of the Earth are prevented from following a geodesic path because the mechanical resistance of the Earth exerts an upward force on them. This explains why moving along the geodesics in spacetime is considered inertial.
Einstein's description of gravity was quickly accepted by the majority of physicists, as it was able to explain a wide variety of previously baffling experimental results. [30] In the coming years, a wide range of experiments provided additional support for the idea of general relativity. [31] [32] [33] [34] Today, Einstein's theory of relativity is used for all gravitational calculations where absolute precision is desired, although Newton's inverse-square law continues to be a useful and fairly accurate approximation. [35]
## Modern understanding
In modern physics, general relativity remains the framework for the understanding of gravity. Physicists continue to work to find solutions to the Einstein field equations that form the basis of general relativity, while some scientists have speculated that general relativity may not be applicable at all in certain scenarios. [35]
Einstein discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime and are named after him. The Einstein field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of spacetime. A metric tensor describes a geometry of spacetime. The geodesic paths for a spacetime are calculated from the metric tensor.
### Solutions
Notable solutions of the Einstein field equations include:
### Tests
The tests of general relativity included the following: [36]
• General relativity accounts for the anomalous perihelion precession of Mercury. [37]
• The prediction that time runs slower at lower potentials (gravitational time dilation) has been confirmed by the Pound–Rebka experiment (1959), the Hafele–Keating experiment, and the GPS.
• The prediction of the deflection of light was first confirmed by Arthur Stanley Eddington from his observations during the Solar eclipse of 29 May 1919. [38] [39] Eddington measured starlight deflections twice those predicted by Newtonian corpuscular theory, in accordance with the predictions of general relativity. However, his interpretation of the results was later disputed. [40] More recent tests using radio interferometric measurements of quasars passing behind the Sun have more accurately and consistently confirmed the deflection of light to the degree predicted by general relativity. [41] See also gravitational lens.
• The time delay of light passing close to a massive object was first identified by Irwin I. Shapiro in 1964 in interplanetary spacecraft signals.
• Gravitational radiation has been indirectly confirmed through studies of binary pulsars. On 11 February 2016, the LIGO and Virgo collaborations announced the first observation of a gravitational wave.
• Alexander Friedmann in 1922 found that Einstein equations have non-stationary solutions (even in the presence of the cosmological constant). In 1927 Georges Lemaître showed that static solutions of the Einstein equations, which are possible in the presence of the cosmological constant, are unstable, and therefore the static Universe envisioned by Einstein could not exist. Later, in 1931, Einstein himself agreed with the results of Friedmann and Lemaître. Thus general relativity predicted that the Universe had to be non-static—it had to either expand or contract. The expansion of the Universe discovered by Edwin Hubble in 1929 confirmed this prediction. [42]
• The theory's prediction of frame dragging was consistent with the recent Gravity Probe B results. [43]
• General relativity predicts that light should lose its energy when traveling away from massive bodies through gravitational redshift. This was verified on Earth and in the Solar System around 1960.
### Gravity and quantum mechanics
An open question is whether it is possible to describe the small-scale interactions of gravity with the same framework as quantum mechanics. General relativity describes large-scale bulk properties whereas quantum mechanics is the framework to describe the smallest scale interactions of matter. Without modifications these frameworks are incompatible. [44]
One path is to describe gravity in the framework of quantum field theory, which has been successful to accurately describe the other fundamental interactions. The electromagnetic force arises from an exchange of virtual photons, where the QFT description of gravity is that there is an exchange of virtual gravitons. [45] [46] This description reproduces general relativity in the classical limit. However, this approach fails at short distances of the order of the Planck length, [44] where a more complete theory of quantum gravity (or a new approach to quantum mechanics) is required.
## Specifics
### Earth's gravity
Every planetary body (including the Earth) is surrounded by its own gravitational field, which can be conceptualized with Newtonian physics as exerting an attractive force on all objects. Assuming a spherically symmetrical planet, the strength of this field at any given point above the surface is proportional to the planetary body's mass and inversely proportional to the square of the distance from the center of the body.
The strength of the gravitational field is numerically equal to the acceleration of objects under its influence. [47] The rate of acceleration of falling objects near the Earth's surface varies very slightly depending on latitude, surface features such as mountains and ridges, and perhaps unusually high or low sub-surface densities. [48] For purposes of weights and measures, a standard gravity value is defined by the International Bureau of Weights and Measures, under the International System of Units (SI).
That value, denoted g, is g = 9.80665 m/s2 (32.1740 ft/s2). [49] [50]
The standard value of 9.80665 m/s2 is the one originally adopted by the International Committee on Weights and Measures in 1901 for 45° latitude, even though it has been shown to be too high by about five parts in ten thousand. [51] This value has persisted in meteorology and in some standard atmospheres as the value for 45° latitude even though it applies more precisely to latitude of 45°32'33". [52]
Assuming the standardized value for g and ignoring air resistance, this means that an object falling freely near the Earth's surface increases its velocity by 9.80665 m/s (32.1740 ft/s or 22 mph) for each second of its descent. Thus, an object starting from rest will attain a velocity of 9.80665 m/s (32.1740 ft/s) after one second, approximately 19.62 m/s (64.4 ft/s) after two seconds, and so on, adding 9.80665 m/s (32.1740 ft/s) to each resulting velocity. Also, again ignoring air resistance, any and all objects, when dropped from the same height, will hit the ground at the same time.
According to Newton's 3rd Law, the Earth itself experiences a force equal in magnitude and opposite in direction to that which it exerts on a falling object. This means that the Earth also accelerates towards the object until they collide. Because the mass of the Earth is huge, however, the acceleration imparted to the Earth by this opposite force is negligible in comparison to the object's. If the object does not bounce after it has collided with the Earth, each of them then exerts a repulsive contact force on the other which effectively balances the attractive force of gravity and prevents further acceleration.
The force of gravity on Earth is the resultant (vector sum) of two forces: [53] (a) The gravitational attraction in accordance with Newton's universal law of gravitation, and (b) the centrifugal force, which results from the choice of an earthbound, rotating frame of reference. The force of gravity is weakest at the equator because of the centrifugal force caused by the Earth's rotation and because points on the equator are furthest from the center of the Earth. The force of gravity varies with latitude and increases from about 9.780 m/s2 at the Equator to about 9.832 m/s2 at the poles.
### Equations for a falling body near the surface of the Earth
Under an assumption of constant gravitational attraction, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9.81 m/s2 on Earth. This resulting force is the object's weight. The acceleration due to gravity is equal to this g. An initially stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. The image on the right, spanning half a second, was captured with a stroboscopic flash at 20 flashes per second. During the first 120 of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 220 it has dropped at total of 4 units; by 320, 9 units and so on.
Under the same constant gravity assumptions, the potential energy, Ep, of a body at height h is given by Ep = mgh (or Ep = Wh, with W meaning weight). This expression is valid only over small distances h from the surface of the Earth. Similarly, the expression ${\displaystyle h={\tfrac {v^{2}}{2g}}}$ for the maximum height reached by a vertically projected body with initial velocity v is useful for small heights and small initial velocities only.
### Gravity and astronomy
The application of Newton's law of gravity has enabled the acquisition of much of the detailed information we have about the planets in the Solar System, the mass of the Sun, and details of quasars; even the existence of dark matter is inferred using Newton's law of gravity. Although we have not traveled to all the planets nor to the Sun, we know their masses. These masses are obtained by applying the laws of gravity to the measured characteristics of the orbit. In space an object maintains its orbit because of the force of gravity acting upon it. Planets orbit stars, stars orbit galactic centers, galaxies orbit a center of mass in clusters, and clusters orbit in superclusters. The force of gravity exerted on one object by another is directly proportional to the product of those objects' masses and inversely proportional to the square of the distance between them.
The earliest gravity (possibly in the form of quantum gravity, supergravity or a gravitational singularity), along with ordinary space and time, developed during the Planck epoch (up to 10−43 seconds after the birth of the Universe), possibly from a primeval state (such as a false vacuum, quantum vacuum or virtual particle), in a currently unknown manner. [4]
General relativity predicts that energy can be transported out of a system through gravitational radiation. Any accelerating matter can create curvatures in the spacetime metric, which is how the gravitational radiation is transported away from the system. Co-orbiting objects can generate curvatures in spacetime such as the Earth-Sun system, pairs of neutron stars, and pairs of black holes. Another astrophysical system predicted to lose energy in the form of gravitational radiation are exploding supernovae.
The first indirect evidence for gravitational radiation was through measurements of the Hulse–Taylor binary in 1973. This system consists of a pulsar and neutron star in orbit around one another. Its orbital period has decreased since its initial discovery due to a loss of energy, which is consistent for the amount of energy loss due to gravitational radiation. This research was awarded the Nobel Prize in Physics in 1993.
The first direct evidence for gravitational radiation was measured on 14 September 2015 by the LIGO detectors. The gravitational waves emitted during the collision of two black holes 1.3 billion-light years from Earth were measured. [55] [56] This observation confirms the theoretical predictions of Einstein and others that such waves exist. It also opens the way for practical observation and understanding of the nature of gravity and events in the Universe including the Big Bang. [57] Neutron star and black hole formation also create detectable amounts of gravitational radiation. [58] This research was awarded the Nobel Prize in physics in 2017. [59]
As of 2020, the gravitational radiation emitted by the Solar System is far too small to measure with current technology.
### Speed of gravity
In December 2012, a research team in China announced that it had produced measurements of the phase lag of Earth tides during full and new moons which seem to prove that the speed of gravity is equal to the speed of light. [60] This means that if the Sun suddenly disappeared, the Earth would keep orbiting the vacant point normally for 8 minutes, which is the time light takes to travel that distance. The team's findings were released in the Chinese Science Bulletin in February 2013. [61]
In October 2017, the LIGO and Virgo detectors received gravitational wave signals within 2 seconds of gamma ray satellites and optical telescopes seeing signals from the same direction. This confirmed that the speed of gravitational waves was the same as the speed of light. [62]
## Anomalies and discrepancies
There are some observations that are not adequately accounted for, which may point to the need for better theories of gravity or perhaps be explained in other ways.
• Extra-fast stars: Stars in galaxies follow a distribution of velocities where stars on the outskirts are moving faster than they should according to the observed distributions of normal matter. Galaxies within galaxy clusters show a similar pattern. Dark matter, which would interact through gravitation but not electromagnetically, would account for the discrepancy. Various modifications to Newtonian dynamics have also been proposed.
• Flyby anomaly : Various spacecraft have experienced greater acceleration than expected during gravity assist maneuvers.
• Accelerating expansion: The metric expansion of space seems to be speeding up. Dark energy has been proposed to explain this. A recent alternative explanation is that the geometry of space is not homogeneous (due to clusters of galaxies) and that when the data are reinterpreted to take this into account, the expansion is not speeding up after all, [63] however this conclusion is disputed. [64]
• Anomalous increase of the astronomical unit : Recent measurements indicate that planetary orbits are widening faster than if this were solely through the Sun losing mass by radiating energy.
• Extra energetic photons: Photons travelling through galaxy clusters should gain energy and then lose it again on the way out. The accelerating expansion of the Universe should stop the photons returning all the energy, but even taking this into account photons from the cosmic microwave background radiation gain twice as much energy as expected. This may indicate that gravity falls off faster than inverse-squared at certain distance scales. [65]
• Extra massive hydrogen clouds: The spectral lines of the Lyman-alpha forest suggest that hydrogen clouds are more clumped together at certain scales than expected and, like dark flow, may indicate that gravity falls off slower than inverse-squared at certain distance scales. [65]
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The theory of general relativity required the adaptation of existing theories of physical, electromagnetic, and quantum effects to account for non-Euclidean geometries. These physical theories modified by general relativity are described below.
The following outline is provided as an overview of and topical guide to black holes:
Entropic gravity, also known as emergent gravity, is a theory in modern physics that describes gravity as an entropic force—a force with macro-scale homogeneity but which is subject to quantum-level disorder—and not a fundamental interaction. The theory, based on string theory, black hole physics, and quantum information theory, describes gravity as an emergent phenomenon that springs from the quantum entanglement of small bits of spacetime information. As such, entropic gravity is said to abide by the second law of thermodynamics under which the entropy of a physical system tends to increase over time. | 2022-05-25 16:33:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8030880689620972, "perplexity": 1031.6540970365754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662588661.65/warc/CC-MAIN-20220525151311-20220525181311-00299.warc.gz"} |
https://plainmath.net/90661/n-a | # How do you find the zeros of f(x)=(x^2−3x−4)/(x^2−x−12)?
How do you find the zeros of $f\left(x\right)=\frac{{x}^{2}-3x-4}{{x}^{2}-x-12}$?
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Factorising and simplifying f(x) as follows.
$f\left(x\right)=\frac{{x}^{2}-3x-4}{{x}^{2}-x-12}=\frac{\overline{)\left(x-4\right)}\left(x+1\right)}{\overline{)\left(x-4\right)}\left(x-3\right)}=\frac{x+1}{x-3}$
with exclusion x ≠ 4
The zeros of f(x) are the values of x which make f(x) equal zero. That is f(x)=0.
For $f\left(x\right)=\frac{x+1}{x+3}$
The denominator of f(x) cannot be zero as this would make f(x) undefined. The numerator is the only part of the rational function that can equal zero.
$⇒x+1=0⇒x=-1\phantom{\rule{1ex}{0ex}}\text{is the zero of f(x)}$ | 2022-09-30 20:18:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8242969512939453, "perplexity": 826.2475577662805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.22/warc/CC-MAIN-20220930181143-20220930211143-00623.warc.gz"} |
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In either case, the shutdown source must be cleared before the restart can take place. That is, either the Shutdown condition must be removed, or the corresponding ASyE bit must be cleared. | 2023-02-01 08:22:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8713761568069458, "perplexity": 4266.225510404622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499919.70/warc/CC-MAIN-20230201081311-20230201111311-00757.warc.gz"} |
https://www.physicsforums.com/threads/minimum-and-maximum-values.905359/ | Homework Help: Minimum and Maximum Values
Tags:
1. Feb 24, 2017
a1234
1. The problem statement, all variables and given/known data
The problem is in the attached file. The part I need a little help with is part b.
2. Relevant equations and attempt at a solution
For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2.
For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to get 49 (because 35 = 5*7). And f(5) = 0, so f(108) also equals 0. I got the tangent line y - 49 = 0(x - 35).
For part b, I think we're supposed to consider x = 0 and x = 3, since this is where the graph starts increasing from decreasing and vice versa. But I also think we have to evaluate the integral at these points. Can someone explain this?
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• Calculus.PNG
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2. Feb 24, 2017
haruspex
The period is not 5, so why is that interesting?
3. Feb 24, 2017
BvU
If I consider
as an attempt, I agree with 0 and 3 but I don't see it decrease before 0... I also see another interval where f is not bigger than at 0 ...
If you want help for part (a) -- and I think you need it -- , please post your working in detail.
strike because the question is about $h$ and I mistakenly read $f$ --- bedtime
4. Feb 24, 2017
haruspex
Did you make a similar mistake here? I agree with all the answers for a).
5. Feb 24, 2017
BvU
YES !! aaaarrghh
6. Feb 24, 2017
a1234
I thought the graph would keep repeating every 5 units, but that's not true.
It seems to repeat every 8 units, so could we do this?
integral from 0 to 8 = 1
8*4 = 32
1*4 = 4
integral from 0 to 3 = 3
4 + 3 = 7
y - 7 = 0(x - 35)?
7. Feb 24, 2017
haruspex
You are told it does.
Didn"t you calculate it as 2 in part a?
I do not understand that step. What is the gradient of h at x=35?
8. Feb 25, 2017
a1234
I didn't count the triangle from 0 to 1 when I first calculated the integral from 0 to 8. I think it should be 1 because the area of the trapezoid is -6, area of the large triangle is 8, and area of the small triangle is -1. So 8 + (-6) + (-1) = 1.
For the derivative at x = 35...
h(35) = integral from 0 to 35 = 7
h'(35) = f(35) = 4 because the graph repeats every 8 units...f(3) = 4 for the remaining 3 units after 8*4 = 32. So the derivative at x = 35 is 4.
If I put it into point-slope form, y - 7 = 4(x - 35) is the tangent line.
9. Feb 25, 2017
haruspex
The trapezoid extends to x=9.
10. Feb 25, 2017
a1234
Ugh. I'm making a lot of mistakes here.
You're right, we only need the area of the trapezoid up to x = 8. So then the area of the trapezoid is 5, and h(8) = 2. The integral from 0 to 35 is 2*4 + 3 = 11.
y - 11 = 4(x - 35)
11. Feb 25, 2017
haruspex
Looks right.
12. Feb 25, 2017
a1234
How do I go about part b?
13. Feb 25, 2017
haruspex
Do you understand about local extrema and absolute extrema?
How do you find a local extremum?
14. Feb 26, 2017
a1234
The absolute maximum is the highest point over the entire graph. The absolute minimum is the lowest point over the graph. The relative max/min is the maximum or minimum over a certain interval.
I think the first step is to find do h'(x), which is just f(x) and find the critical points and endpoints of the graph. The critical points are 1 and 5, and the endpoints are 0 and 7. And then we need to evaluate g at those values. I came up with the following:
h(0) = 0
h(1) = -1
h(5) = 7
h(7) = 4
The absolute min is -1 and max is 7. The relative min is 0 and max is 4.
15. Feb 26, 2017
haruspex
You are asked for the absolute min and max over the whole interval, so yes, -1 and 7.
16. Feb 26, 2017
a1234
Okay. Thank you very much for the help! | 2018-06-23 08:49:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6864871382713318, "perplexity": 435.7047633660034}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864953.36/warc/CC-MAIN-20180623074142-20180623094142-00028.warc.gz"} |
https://zbmath.org/?q=an:0864.03026 | # zbMATH — the first resource for mathematics
The Mordell-Lang conjecture for function fields. (English) Zbl 0864.03026
The Mordell-Lang conjecture is a hypothesis formulated by Lang and extending the Mordell conjecture on rational points of curves, as well as the Manin-Mumford conjecture on torsion points of Abelian varieties. In characteristic 0, the Mordell-Lang conjecture was proved in a series of papers by Raynaud, Faltings and Vojta. In positive characteristic, only partial cases had been solved before this paper. Hrushovski presents here a uniform proof valid in any characteristic and using a model theoretic approach. Here is the statement of the main theorem. One deals with an algebraically closed field $$k$$ and a field extension $$K/k$$. Let $$S$$ be a semi-Abelian variety over $$K$$ and $$X$$ be a subvariety of $$S$$. Take a subgroup $$\Gamma$$ of $$S$$ such that $${\mathbb Q}_p \otimes \Gamma$$ is finitely generated as a $${\mathbb Q}_p$$-module (here $${\mathbb Q}_p$$ denotes $${\mathbb Q}$$ if $$p=0$$, and $$\{ m/n \in {\mathbb Q} : n$$ prime to $$p \}$$ otherwise). Suppose that $$X \cap \Gamma$$ is Zariski dense in $$X$$. Then there are a semi-Abelian variety $$S_0$$ defined over $$k$$, a subvariety $$X_0$$ of $$S_0$$ also defined over $$k$$, and a rational homomorphism $$h$$ from a group subvariety of $$S$$ into $$S_0$$ such that $$X$$ is a translate of $$h^{-1}(X_0)$$. As a consequence, when $$S$$ has $$K/k$$ trace 0, then $$X \cap \Gamma$$ is a finite union of cosets of subgroups of $$\Gamma$$.
The core of the Hrushovski approach does not depend on the characteristic $$p$$. But, just to explain the idea, let us treat before the case $$p=0$$. The argument in this case uses differential algebra and was inspired by some recent papers of Buium. Without loss of generality, one can assume that $$K$$ is algebraically closed and has infinite transcendence degree over $$k$$. Accordingly, one can equip $$K$$ with a derivation $$D$$ making $$K$$ a differential field, and even a differentially closed field (so an $$\omega$$-stable structure) with constant field $$k$$. Model theory assigns an ordinal dimension to every non-empty set definable in $$(K, D)$$ (its Morley rank); in particular, the Morley rank agrees with the usual dimension for varieties. $$X$$ is definable and its Morley rank is an integer. By using the Manin homomorphism in $$(K, D)$$, one sees that, with no loss of generality, $$\Gamma$$ is definable, too, and its Morley rank is again an integer. Let us apply a dichotomy theorem of Hrushovski and Zil’ber to certain groups related to $$X$$ and $$\Gamma$$. It comes out that such a group is either a module over a suitable local ring, with no additional structure, or an algebraic group over an algebraically closed field. A theorem of Sokolovic says that the only algebraically closed field definable in $$(K, D)$$ is – up to definable isomorphism – the constant field $$k$$. So the previous dichotomy just leads to the subvarieties mentioned in the main theorem: group subvarieties, or subvarieties defined over $$k$$. And actually the analysis of the two cases accomplishes the proof.
What happens when $$p>0$$? Here the plan is similar, but needs some changes. Most notably, differential algebra can be avoided now. Separably closed fields $$F$$ of characteristic $$p$$ such that $$|F : F^p|$$ is finite and $$>1$$ are enough. Their theory is not $$\omega$$-stable, but only stable, so Morley rank cannot be used in this setting. However some suitable modifications let the previous machinery work; for instance, the role of the Manin homomorphism is played now by the projection modulo $$\bigcap_n p^n S$$ (which is not a definable set, but is $$\infty$$-definable).
The paper is well written and well worth reading for several reasons. As recalled before, it gives the first proof of the conjecture (in the general form stated above). Moreover it provides a very brilliant and ingenious application of Model Theory to a question of Diophantine Geometry, using some deep model theoretic tools (like differentially and separably closed fields, or the Hrushovski-Zil’ber Dichotomy Theorem) to obtain the positive solution. So the paper is a significant step within the increasing connection between Model Theory and Algebraic Geometry.
##### MSC:
03C60 Model-theoretic algebra 14G05 Rational points 11G10 Abelian varieties of dimension $$> 1$$ 03C45 Classification theory, stability, and related concepts in model theory
Full Text:
##### References:
[1] Dan Abramovich and José Felipe Voloch, Toward a proof of the Mordell-Lang conjecture in characteristic \?, Internat. Math. Res. Notices 5 (1992), 103 – 115. · Zbl 0787.14026 [2] A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math. (2) 136 (1992), no. 3, 557 – 567. · Zbl 0817.14021 [3] Alexandru Buium, Effective bound for the geometric Lang conjecture, Duke Math. J. 71 (1993), no. 2, 475 – 499. · Zbl 0812.14029 [4] Alexandru Buium and José Felipe Voloch, Integral points of abelian varieties over function fields of characteristic zero, Math. Ann. 297 (1993), no. 2, 303 – 307. · Zbl 0789.14017 [5] Françoise Delon, Idéaux et types sur les corps séparablement clos, Mém. Soc. Math. France (N.S.) 33 (1988), 76 (French, with English summary). · Zbl 0678.03016 [6] Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117 – 138. · Zbl 0713.03015 [7] U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233 – 244. [8] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, Trans. Amer. Math. Soc. (to appear). [9] E. Hrushovski and B. Zil’ber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), 1–56. CMP 95:06 [10] Ehud Hrushovski and Boris Zilber, Zariski geometries, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 315 – 323. · Zbl 0781.03023 [11] Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229 – 234. · Zbl 0151.27401 [12] Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. · Zbl 0744.14012 [13] Daniel Lascar, Ranks and definability in superstable theories, Israel J. Math. 23 (1976), no. 1, 53 – 87. · Zbl 0326.02038 [14] Ju. I. Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 737 – 756 (Russian). · Zbl 0151.27503 [15] Ju. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395 – 1440 (Russian). · Zbl 0166.16901 [16] M. Messmer, Groups and fields interpretable in separably closed fields, preprint. · Zbl 0803.03023 [17] A. Pillay, Model theory, stability theory, and stable groups, The Model Theory of Groups , Notre Dame Math. Lectures, no. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 1–22.CMP 21:09 · Zbl 0792.03019 [18] A. Robinson and P. Roquette, On the finiteness theorem of Siegel and Mahler concerning Diophantine equations, J. Number Theory 7 (1975), 121 – 176. · Zbl 0299.12107 [19] Gerald E. Sacks, Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. · Zbl 0242.02054 [20] Z. Sokolovic, Model theory of differential fields, Ph.D. Thesis, Notre Dame, July, 1992. [21] A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948. · Zbl 0037.16202 [22] Carol Wood, Notes on the stability of separably closed fields, J. Symbolic Logic 44 (1979), no. 3, 412 – 416. · Zbl 0424.03014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-10-18 16:55:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8256220817565918, "perplexity": 474.685509433171}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585204.68/warc/CC-MAIN-20211018155442-20211018185442-00212.warc.gz"} |
https://aimsciences.org/article/doi/10.3934/dcds.2015.35.5631 | # American Institute of Mathematical Sciences
December 2015, 35(12): 5631-5663. doi: 10.3934/dcds.2015.35.5631
## Density estimates for vector minimizers and applications
1 Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece 2 Università degli Studi dell'Aquila, Via Vetoio, 67010 Coppito, L'Aquila, Italy
Received April 2014 Revised September 2014 Published May 2015
We extend the Caffarelli--Córdoba estimates to the vector case (L. Caffarelli and A. Córdoba, Uniform Convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995)). In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of minimal solutions. We also give applications.
Citation: Nicholas D. Alikakos, Giorgio Fusco. Density estimates for vector minimizers and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5631-5663. doi: 10.3934/dcds.2015.35.5631
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Google Scholar [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, Arch. Rat. Mech. Analysis, 124 (1993), 355. doi: 10.1007/BF00375607. Google Scholar [13] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbbR^2$ for a variational problem with a symmetric three-well potential,, Comm. Pure. Appl. Math., 49 (1996), 677. doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. Google Scholar [14] L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem,, Comm. Pure Appl. Math., 48 (1995), 1. doi: 10.1002/cpa.3160480101. Google Scholar [15] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal Amer. Math. Society , 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar [16] A. Cesaroni, C. M. Muratov and M. Novaga, Front propagation and phase field models of stratified media,, Archive for Rational Mechanics and Analysis, 216 (2015), 153. doi: 10.1007/s00205-014-0804-3. Google Scholar [17] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). doi: 10.1090/gsm/019. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [19] A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions,, J. Funct. Anal., 214 (2004), 386. doi: 10.1016/j.jfa.2003.07.012. Google Scholar [20] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Part. Diff. Eqs., 33 (2008), 1. doi: 10.1007/s00526-007-0146-1. Google Scholar [21] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calculus of Variations and Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x. Google Scholar [22] G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, Calc. Var. Part. Diff. Eqs., 49 (2014), 963. doi: 10.1007/s00526-013-0607-7. Google Scholar [23] G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy,, Comm. Pure Appl. Anal., 13 (2014), 1045. doi: 10.3934/cpaa.2014.13.1045. Google Scholar [24] G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 363 (2011), 4285. doi: 10.1090/S0002-9947-2011-05356-0. Google Scholar [25] E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint,, Indiana Univ. Math. Journal, 32 (1983), 25. doi: 10.1512/iumj.1983.32.32003. Google Scholar [26] C. Gui and M. Schatzman, Symmetric quadruple phase transitions,, Ind. Univ. Math. J., 57 (2008), 781. doi: 10.1512/iumj.2008.57.3089. Google Scholar [27] J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116. doi: 10.1137/0149007. Google Scholar [28] J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps,, SIAM J. Appl. Math., 49 (1989), 1722. doi: 10.1137/0149104. Google Scholar [29] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [30] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Journal de Mathématiques Pures et Appliquées, 101 (2014), 1. doi: 10.1016/j.matpur.2013.05.001. Google Scholar [31] O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality,, SIAM J. Math. Anal., 43 (2011), 2675. doi: 10.1137/110831040. Google Scholar [32] Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace,, Interfaces And Free Boundaries, 14 (2012), 153. doi: 10.4171/IFB/277. Google Scholar [33] P. Smyrnelis, Personal, communication., (). Google Scholar [34] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems,, Rocky Mountain J. Math., 21 (1991), 799. doi: 10.1216/rmjm/1181072968. Google Scholar [35] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,, Ann. Math., 103 (1976), 489. doi: 10.2307/1970949. Google Scholar [36] E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals,, J. Reine Angew. Math., 574 (2004), 147. doi: 10.1515/crll.2004.068. Google Scholar [37] B. White, Topics in GMT,, Notes by O. Chodash., (2012). Google Scholar
show all references
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Mech. Analysis, 202 (2011), 567. doi: 10.1007/s00205-011-0441-z. Google Scholar [6] N. D. Alikakos and G. Fusco, In, preparation., (). Google Scholar [7] N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima,, Indiana Univ. Math. Journal, 57 (2008), 1871. doi: 10.1512/iumj.2008.57.3181. Google Scholar [8] N. D. Alikakos and G. Fusco, Asymptotic and rigidity results for symmetric solutions of the elliptic system $\Delta u=W_u(u)$,, Ann. Scuola Norm Sup. Pisa Cl. Sci., 9 (2009), 1. Google Scholar [9] N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves,, to appear in JEMS, (). Google Scholar [10] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,, Ann. Inst. Henri Poincare, 7 (1990), 67. Google Scholar [11] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices,, Birkhäuser, (1994). doi: 10.1007/978-1-4612-0287-5. Google Scholar [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, Arch. Rat. Mech. Analysis, 124 (1993), 355. doi: 10.1007/BF00375607. Google Scholar [13] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbbR^2$ for a variational problem with a symmetric three-well potential,, Comm. Pure. Appl. Math., 49 (1996), 677. doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. Google Scholar [14] L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem,, Comm. Pure Appl. Math., 48 (1995), 1. doi: 10.1002/cpa.3160480101. Google Scholar [15] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal Amer. Math. Society , 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar [16] A. Cesaroni, C. M. Muratov and M. Novaga, Front propagation and phase field models of stratified media,, Archive for Rational Mechanics and Analysis, 216 (2015), 153. doi: 10.1007/s00205-014-0804-3. Google Scholar [17] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). doi: 10.1090/gsm/019. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [19] A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions,, J. Funct. Anal., 214 (2004), 386. doi: 10.1016/j.jfa.2003.07.012. Google Scholar [20] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Part. Diff. Eqs., 33 (2008), 1. doi: 10.1007/s00526-007-0146-1. Google Scholar [21] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calculus of Variations and Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x. Google Scholar [22] G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, Calc. Var. Part. Diff. Eqs., 49 (2014), 963. doi: 10.1007/s00526-013-0607-7. Google Scholar [23] G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy,, Comm. Pure Appl. Anal., 13 (2014), 1045. doi: 10.3934/cpaa.2014.13.1045. Google Scholar [24] G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 363 (2011), 4285. doi: 10.1090/S0002-9947-2011-05356-0. Google Scholar [25] E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint,, Indiana Univ. Math. Journal, 32 (1983), 25. doi: 10.1512/iumj.1983.32.32003. Google Scholar [26] C. Gui and M. Schatzman, Symmetric quadruple phase transitions,, Ind. Univ. Math. J., 57 (2008), 781. doi: 10.1512/iumj.2008.57.3089. Google Scholar [27] J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116. doi: 10.1137/0149007. Google Scholar [28] J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps,, SIAM J. Appl. Math., 49 (1989), 1722. doi: 10.1137/0149104. Google Scholar [29] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [30] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Journal de Mathématiques Pures et Appliquées, 101 (2014), 1. doi: 10.1016/j.matpur.2013.05.001. Google Scholar [31] O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality,, SIAM J. Math. Anal., 43 (2011), 2675. doi: 10.1137/110831040. Google Scholar [32] Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace,, Interfaces And Free Boundaries, 14 (2012), 153. doi: 10.4171/IFB/277. Google Scholar [33] P. Smyrnelis, Personal, communication., (). Google Scholar [34] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems,, Rocky Mountain J. Math., 21 (1991), 799. doi: 10.1216/rmjm/1181072968. Google Scholar [35] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,, Ann. Math., 103 (1976), 489. doi: 10.2307/1970949. Google Scholar [36] E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals,, J. Reine Angew. Math., 574 (2004), 147. doi: 10.1515/crll.2004.068. Google Scholar [37] B. White, Topics in GMT,, Notes by O. Chodash., (2012). Google Scholar
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[Back to Top] | 2019-09-18 11:25:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6307887434959412, "perplexity": 4139.117692148868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573284.48/warc/CC-MAIN-20190918110932-20190918132932-00151.warc.gz"} |
https://essaygoaway.info/2021/04/06/solve-algebra-problems-online_aa/ | # Solve algebra problems online
Einstein, albert (1879-1955) do not worry about your difficulties in mathematics, i assure you that mine are greater. author math10 banners. use owl hat algebra calculator solve algebra problems online with steps. follow us on twitter facebook. thanks for making algebra easy! for example, enter 3x 2=14 into the text box to get a step-by-step essays on legalization of marijuana explanation of how to solve 3x 2=14 try this example now! humans are still better at solving word problems than the best artificial intelligence essays on technology science homework sheets available today. the following steps outline the basic usage of summary essay outline our online algebra calculator: simply enter your problem and click answer to find out if you worked the problem correctly.now, i hope you realize mit college essays that if you simply put your homework problems into the calculator and copy the college essay mentor. answer down, you’re cheating yourself in the long run because you haven’t really learned anything problem solved! الجبر, romanized: popular argumentative essay topics skip to main content. 'reunion of broken parts, bonesetting') is how to write a good thesis paper one of the solve algebra problems online broad areas do i have homework of solve algebra problems online mathematics, together with creative writing for class 1 number theory, geometry and analysis. $$\frac{a}{b}=\frac{x}{100}$$ \frac{a}{{\color{red} {b}}}\cdot. | 2021-04-18 17:24:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1708654910326004, "perplexity": 3527.8802465243493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038507477.62/warc/CC-MAIN-20210418163541-20210418193541-00591.warc.gz"} |
http://mathhelpforum.com/algebra/64050-factorial-notation-help-print.html | # factorial notation help
• Dec 8th 2008, 10:24 PM
Carrick
factorial notation help
(n+2)/n!=56
I'm new to this and having trouble trouble breaking this down, could someone please give me a hint to get this started?
• Dec 8th 2008, 10:29 PM
chiph588@
you are saying to divide $n+2$ by $n!$ and get 56.
This can never happen though...
Are you sure you have the right problem?
• Dec 8th 2008, 10:33 PM
Carrick
Sorry,
(n+2)!/n!=56
• Dec 8th 2008, 10:38 PM
chiph588@
Note that $(n+2)! = (n+2)(n+1)(n)(n-1)(n-2) \dots = (n+2)(n+1)n!$
So $\frac{(n+2)!}{n!} = \frac{(n+2)(n+1)n!}{n!}$
Do you see where to go now?
• Dec 8th 2008, 10:42 PM
winsome
Given: (n + 2)!/n! = 56
(n + 2)(n + 2 - 1)(n + 2 - 2)!/n! = 56
(n + 2)(n + 1) = 56
n + 2 = 56 or n + 1 = 56
Which gives n = 54, 55
May be it helps You
• Dec 8th 2008, 10:49 PM
chiph588@
You solved that wrong...
$(n+2)(n+1) = n^2+3n+2=56$
So $n^2+3n-54 = 0$
$(n+9)(n-6) = 0$
$n=-6,9$
• Dec 8th 2008, 10:56 PM
Carrick
thanks for the help chip, when down to
(n+9)(n-6) how do you know which to make positive and negative?
eg how do you know its not (n+6)(n-9) without substiting into the initial equation?
• Dec 8th 2008, 11:06 PM
chiph588@
its simple factoring
given that $(n-a)(n-b) = n^2+3n-54$
doing foil... $n^2+(-a-b)n+ab = n^2+3n-54$
So $a+b = -3$ and $ab = -54$
By inspection it can be seen $a=-9 , \; b=6$
• Dec 9th 2008, 02:16 AM
mr fantastic
Quote:
Originally Posted by chiph588@
You solved that wrong...
$(n+2)(n+1) = n^2+3n+2=56$
So $n^2+3n-54 = 0$
$(n+9)(n-6) = 0$
$n=-6,9$
Small correction: n = -9, 6.
But only n = 6 is a valid solution to the original equation (to the OP: why?). | 2018-01-19 18:17:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.841511607170105, "perplexity": 4032.272326655126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888077.41/warc/CC-MAIN-20180119164824-20180119184824-00145.warc.gz"} |
http://answers.wikia.com/wiki/How_do_you_convert_electrical_energy_to_heat_energy | How do you convert electrical energy to heat energy?
1,038,851questions on
Apply a potential difference (a voltage) across a resistance, current flows according to Ohm's law (V=IR) and energy is dissipated as heat according to the power rule P=$I^2R$ | 2017-01-21 21:53:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41121944785118103, "perplexity": 678.3478408741802}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281226.52/warc/CC-MAIN-20170116095121-00062-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://bora.uib.no/bora-xmlui/handle/1956/915/browse?rpp=20&offset=1513&etal=-1&sort_by=1&type=title&starts_with=V&order=ASC | Viser treff 1514-1533 av 1600
• #### Testing the presence of CP violation in the 2HDM
(Conference object; Peer reviewed; Journal article, 2015-04-27)
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https://www.math24.net/properties-applications-triple-integrals/ | Select Page
Calculus
# Properties and Applications of Triple Integrals
Functions of three variables: $$f\left( {x,y,z} \right),$$ $$g\left( {x,y,z} \right),$$ $$f\left( {u,v,w} \right)$$
Independent variables: $$x,$$ $$y,$$ $$z,$$ $$u,$$ $$v,$$ $$w$$
Small changes: $$\Delta {x_i},$$ $$\Delta {y_j},$$ $$\Delta {z_k}$$
Limits of integration: $$a,$$ $$b,$$ $$c,$$ $$d,$$ $$r,$$ $$s$$
Domains of integration: $$G,$$ $$T,$$ $$S$$
Cylindrical coordinates: $$r,$$ $$\theta,$$ $$z$$
Spherical coordinates: $$r,$$ $$\theta,$$ $$\varphi$$
Volume of a solid: $$V$$
Mass of a solid: $$m$$
Density: $$\mu \left( {x,y,z} \right)$$
Coordinates of the center of mass: $${x_c},$$ $${y_c},$$ $${z_c}$$
First moments: $${M_{xy}},$$ $${M_{yz}},$$ $${M_{xz}}$$
Moments of inertia: $${I_{xy}},$$ $${I_{yz}},$$ $${I_{xz}},$$ $${I_x},$$ $${I_y},$$ $${I_z},$$ $${I_0}$$
1. The triple integral of a function $$f\left( {x,y,z} \right)$$ over a parallelepiped $$\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {r,s} \right]$$ is defined to be the limit of the integral sum (Riemann sum):
$$\require{AMSmath.js}{\large\iiint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {r,s} \right]}\normalsize} {f\left( {x,y,z} \right)dV} =$$ $$\lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0\\ \text{max}\,\Delta {z_k} \to 0}} \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^p {f\left( {{u_i},{v_j},{w_k}} \right)}}}$$ $${{{ \Delta {x_i}\Delta {y_j}\Delta {z_k}} } },$$
where $${\left( {{u_i},{v_j},{w_k}} \right)}$$ is some point in the parallelepiped $$\left( {{x_{i – 1}},{x_i}} \right) \times \left( {{y_{j – 1}},{y_j}} \right)$$ $$\times\, \left( {{z_{k – 1}},{z_k}} \right),$$ and the corresponding increments of the variables are equal to $$\Delta {x_i} = {x_i} – {x_{i – 1}},$$ $$\Delta {y_j} = {y_j} – {y_{j – 1}},$$ $$\Delta {z_k} = {z_k} – {z_{k – 1}}.$$
2. The triple integral of the sum of two functions is equal to the sum of the integrals of these functions:
$${\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) + g\left( {x,y,z} \right)} \right]dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$+\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV}$$
3. The triple integral of the difference of two functions is equal to the difference of the corresponding integrals of these functions:
$${\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) – g\left( {x,y,z} \right)} \right]dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$-\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV}$$
4. A constant factor can be moved across the triple integral sign:
$${\large\iiint\limits_G\normalsize} {kf\left( {x,y,z} \right)dV} =$$ $$k{\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$
5. If $${f\left( {x,y,z} \right)} \ge 0$$ and $$G$$ and $$T$$ are non-overlapping regions, then
$${\large\iiint\limits_{G \cup T}\normalsize} {f\left( {x,y,z} \right)dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$+\;{\large\iiint\limits_T\normalsize} {f\left( {x,y,z} \right)dV}$$
Here $$G \cup T$$ is the union of the regions of integrations $$G$$ and $$T$$.
6. Expressing a triple integral as a double integral
If the integration domain $$G$$ consists of a set of points $${\left( {x,y,z} \right)}$$ satisfying the condition
$$\left( {x,y} \right) \in \mathbb{R},$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
then the triple integral is expressed as
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iint\limits_R\normalsize} {\left[ {{\large\int\limits_{{\lambda _1}\left( {x,y} \right)}^{{\lambda _2}\left( {x,y} \right)}\normalsize} {f\left( {x,y,z} \right)dz} } \right]dxdy} ,$$
where $$R$$ is the projection of $$G$$ onto the $$xy$$-plane.
7. Expressing a triple integral as an iterated integral
If the integration domain $$G$$ consists of a set of points $${\left( {x,y,z} \right)}$$ such that
$$a \le x \le b,$$ $${\varphi _1}\left( x \right) \le y \le {\varphi _2}\left( x \right),$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
then the triple integral is given by
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\int\limits_a^b\normalsize} {\left[ {{\large\int\limits_{{\varphi _1}\left( x \right)}^{{\varphi _2}\left( x \right)}\normalsize} }\right.}$$ $${\left.{ {\left( {{{\large\int\limits_{{\lambda _1}\left( {x,y} \right)}^{{\lambda _2}\left( {x,y} \right)}\normalsize}} {f\left( {x,y,z} \right)dz} } \right) dy} } \right]dx}$$
8. Triple integral over a parallelepiped
If the domain of integration $$G$$ is a parallelepiped $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ $$\times\, \left[ {r,s} \right]$$, then
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\int\limits_a^b\normalsize} {\left[ {{\large\int\limits_c^d\normalsize} {\left( {{\large\int\limits_r^s\normalsize} {f\left( {x,y,z} \right)dz} } \right)dy} } \right]dx}$$
In the special case when the integrand $${f\left( {x,y,z} \right)}$$ can be written as the product
$$g\left( x \right)h\left( y \right)k\left( z \right),$$ the triple integral is given by
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $$\left( {{\large\int\limits_a^b\normalsize} {g\left( x \right)dx} } \right)$$ $$\left( {{\large\int\limits_c^d\normalsize} {h\left( y \right)dy} } \right)$$ $$\left( {{\large\int\limits_r^s\normalsize} {k\left( z \right)dz} } \right)$$
9. Change of variables
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\big[ {x\left( {u,v,w} \right),y\left( {u,v,w} \right), }}$$ $${{ z\left( {u,v,w} \right)} \big]\left|{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,v,w} \right)}}\normalsize}\right| dxdydz},$$ where $$\left| {{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,v,w} \right)}}}\normalsize} \right| =$$ $$\left| {\begin{array}{*{20}{c}} {\large\frac{{\partial x}}{{\partial u}}\normalsize} & {\large\frac{{\partial x}}{{\partial v}}\normalsize} & {\large\frac{{\partial x}}{{\partial w}}\normalsize}\\ {\large\frac{{\partial y}}{{\partial u}}\normalsize} & {\large\frac{{\partial y}}{{\partial v}}\normalsize} & {\large\frac{{\partial y}}{{\partial w}}\normalsize}\\ {\large\frac{{\partial z}}{{\partial u}}\normalsize} & {\large\frac{{\partial z}}{{\partial v}}\normalsize} & {\large\frac{{\partial z}}{{\partial w}}\normalsize} \end{array}} \right|$$ $$\ne 0$$ is the jacobian of the transformation
$$\left( {x,y,z} \right) \to \left( {u,v,w} \right)$$ and $$S$$ is the pullback of the integration domain $$G,$$ which can be computed by substituting $$x = x\left( {u,v,w} \right),$$ $$y = y\left( {u,v,w} \right),$$ $$z = z\left( {u,v,w} \right)$$ into the definition of $$G.$$
10. Triple integral in cylindrical coordinates
The differential $$dxdydz$$ in cylindrical coordinates is defined by the expression
$$dxdydz = \left|{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {r,\theta ,z} \right)}}\normalsize}\right| drd\theta dz =$$ $$rdrd\theta dz.$$
Let the solid $$G$$ is determined by the inequalities
$$\left( {x,y} \right) \in \mathbb{R},$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
where $$R$$ is the projection of $$G$$ onto the $$xy$$-plane. Then
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\left( {r\cos \theta ,r\sin \theta ,z} \right)rdrd\theta dz} =$$ $${\large\iint\limits_{R\left( {r,\theta } \right)}\normalsize} {\Big[ {{\large\int\limits_{{\lambda _1}\left( {r\cos \theta ,r\sin \theta } \right)}^{{\lambda _2}\left( {r\cos \theta ,r\sin \theta } \right)}\normalsize} {f\big( {r\cos \theta , }}}}$$ $${{{{ r\sin \theta ,z} \big)dz} } \Big]rdrd\theta }.$$
Here $$S$$ is the pullback of $$G$$ in cylindrical coordinates.
11. Triple integral in spherical coordinates
The differential $$dxdydz$$ in spherical coordinates is expressed by the formula
$$dxdydz = \left| {{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {r,\theta ,\varphi } \right)}}}\normalsize} \right|drd\theta d\varphi =$$ $${r^2}\sin \theta drd\theta d\varphi$$
In spherical coordinates, the triple integral is written as
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\big( {r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi , }}$$ $${{ r\cos \theta } \big){r^2}\sin \theta drd\theta d\varphi },$$
where $$S$$ is the pullback of $$G$$ in spherical coordinates. The angle $$\theta$$ ranges from $$0$$ to $$2\pi$$, the angle $$\varphi$$ ranges from $$0$$ to $$\pi.$$
1. Volume of a solid
$$V = {\large\iiint\limits_G\normalsize} {dxdydz}$$
2. Volume of a solid in cylindrical coordinates
$$V = {\large\iiint\limits_{S\left( {r,\theta ,z} \right)}\normalsize} {rdrd\theta dz}$$
3. Volume of a solid in spherical coordinates
$$V = {\large\iiint\limits_{S\left( {r,\theta ,\varphi } \right)}\normalsize} {{r^2}\sin \theta drd\theta d\varphi }$$
4. Mass of a solid
$$m = {\large\iiint\limits_G\normalsize} {\mu \left( {x,y,z} \right)dV} ,$$
where the solid occupies the domain $$G$$ and its density at a point $${\left( {x,y,z} \right)}$$ is equal to $${\mu \left( {x,y,z} \right)}.$$
5. Center of mass of a solid
$${x_C} = {\large\frac{{{M_{yz}}}}{m}\normalsize},\;$$ $${y_C} = {\large\frac{{{M_{xz}}}}{m}\normalsize},\;$$ $${z_C} = {\large\frac{{{M_{xy}}}}{m}\normalsize}$$, where
$${M_{yz}} = {\large\iiint\limits_G\normalsize} {x\mu \left( {x,y,z} \right)dV},\;$$
$${M_{xz}} = {\large\iiint\limits_G\normalsize} {y\mu \left( {x,y,z} \right)dV},\;$$
$${M_{xy}} = {\large\iiint\limits_G\normalsize} {z\mu \left( {x,y,z} \right)dV}$$
are the first moments about the coordinate planes $$x = 0,$$ $$y = 0$$ and $$z = 0,$$ respectively, and the function $${\mu \left( {x,y,z} \right)}$$ describes the density of the solid.
6. Moments of inertia about the $$xy$$-plane (or $$z = 0\text{),}$$ $$yz$$-plane $$\left({x = 0}\right),$$ and $$xz$$-plane $$\left({y = 0}\right)$$
$${I_{xy}} = {\large\iiint\limits_G\normalsize} {{z^2}\mu \left( {x,y,z} \right)dV} ,$$
$${I_{yz}} = {\large\iiint\limits_G\normalsize} {{x^2}\mu \left( {x,y,z} \right)dV} ,$$
$${I_{xz}} = {\large\iiint\limits_G\normalsize} {{y^2}\mu \left( {x,y,z} \right)dV}$$
7. Moments of inertia about the $$x$$-axis, $$y$$-axis, and $$z$$-axis
$${I_x} = {I_{xy}} + {I_{xz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{z^2} + {y^2}} \right)\mu \left( {x,y,z} \right)dV},$$
$${I_y} = {I_{xy}} + {I_{yz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{z^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV},$$
$${I_z} = {I_{xz}} + {I_{yz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{y^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV}$$
8. Polar moment of inertia
$${I_0} = {I_{xy}} + {I_{yz}} + {I_{xz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{x^2} + {y^2} + {z^2}} \right) }$$ $${\mu \left( {x,y,z} \right)dV}$$ | 2018-10-16 22:44:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.963594913482666, "perplexity": 202.52883941613445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510893.26/warc/CC-MAIN-20181016221847-20181017003347-00312.warc.gz"} |
http://physics.stackexchange.com/questions/17509/what-is-the-physical-meaning-of-the-affine-parameter-for-null-geodesic | # What is the physical meaning of the affine parameter for null geodesic?
For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is
$$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma_{\rho\sigma}^{\mu}\frac{dx^{\rho}}{d\tau}\frac{dx^{\sigma}}{d\tau}=0.$$
But proper time $\Delta\tau=0$ for null paths, so what the physical meaning of is the affine parameter for null geodesic?
-
Spacetime is locally flat, and in any flat space you have parallelism. This structure of parallels is completely independent of whether you think of the points as representing points in relativistic spacetime, points in Newtonian spacetime, or points in Euclidean space. It could even be a space such as a graph of temperature versus time. Once you've got a notion of parallelism, you are automatically able to construct a system of measurement along any given line. You can see the construction worked out here: lightandmatter.com/html_books/genrel/ch02/ch02.html#Section2.1 – Ben Crowell Nov 27 '11 at 15:00
If you forget about the affine-ness for a moment: you can parametrize a null geodesic in any way you want. Actually, you can parametrize any geodesic (heck, even any curve) in any way you want; all you need is a monotonic function that maps points on the geodesic to unique values of the parameter. But for timelike geodesics, you almost always use the proper time because it's a nice, sensible physical quantity that also happens to work as a parameter.
With null geodesics, you don't have the proper time as an option because the proper time mapping assigns the same value to all points on the geodesic. So you have to pick some other parametrization. In principle, again, it can be any monotonic function that maps points on the geodesic to unique values of the parameter.
However, it's possible to pick a way to parametrize the null geodesic in a way that is "sensible" in the same way that proper time is "sensible" for a timelike geodesic. This is called an affine parameter. In particular, one way to define an affine parameter is that it satisfies the geodesic equation. (Note: the geodesic equation does not work for just any arbitrary parametrization of a geodesic. You have to use an affine parameter.) Another way is to say that iff the parametrization is affine, parallel transport preserves the tangent vector, as Wikipedia does. Another way is to say that the acceleration is perpendicular to the velocity given an affine parameter, as Ron did. All these definitions are equivalent.
It turns out, although I don't know the details of a proof, that there is a unique affine parameter for any geodesic, up to transformations of the form $t \to at+b$.
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https://members.loria.fr/VCortier/files/Publications/b2hd-Prouverap2.html | ## A Survey of Algebraic Properties Used in Cryptographic Protocols
A Survey of Algebraic Properties Used in Cryptographic Protocols. Véronique Cortier, Stéphanie Delaune, and Pascal Lafourcade. Technical Report 2, projet RNTL PROUVÉ, 2004. 19 pages.
### Abstract
Using the perfect encryption assumption, cryptographic primitives are often represented by free function symbols. However some attacks and even honest runs may use algebraic properties of the operators like the exclusive or, the modular exponentiation, the addition, etc.
We give here a survey of protocols and attacks using such algebraic properties.
### BibTeX
@techreport{Prouve:rap2,
author = {Cortier, V{\'e}ronique and Delaune, St{\'e}phanie and
institution = {projet RNTL PROUV{\'E}},
month = jun,
note = {19~pages},
number = {2},
type = {Technical Report},
title = {A Survey of Algebraic Properties Used in
Cryptographic Protocols},
year = {2004},
abstract = {Using the \emph{perfect encryption assumption},
cryptographic primitives are often represented by
free function symbols. However some attacks and even
honest runs may use algebraic properties of the
operators like the exclusive or, the modular
exponentiation, the addition, etc.\par We give here a
survey of protocols and attacks using such algebraic
properties.},
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https://www.askdifference.com/affix-vs-suffix/ | VS.
# Affix vs. Suffix
Published:
Views: 92
Affixnoun
That which is affixed; an appendage.
Suffixnoun
A morpheme added at the end of a word to modify the word's meaning.
‘The suffix "-able" changes "sing" into "singable".’;
Affixnoun
(linguistic morphology) A bound morpheme added to the word’s stem's end.
Suffixnoun
(mathematics) A subscript.
Affixnoun
A bound morpheme added to a word’s stem; a prefix, suffix, etc.
Suffixnoun
(computing) A final segment of a string of characters.
‘The string "abra" is both a prefix and a suffix of the string "abracadabra".’;
Affixnoun
(mathematics) The complex number $a+bi$ associated with the point in the Gauss plane with coordinates $\left(a,b\right)$.
Suffixverb
(transitive) To append (something) to the end of something else.
Affixnoun
(decorative art) Any small feature, as a figure, a flower, or the like, added for ornament to a vessel or other utensil, to an architectural feature.
Suffixnoun
A letter, letters, syllable, or syllables added or appended to the end of a word or a root to modify the meaning; a postfix.
Affixverb
(transitive) To attach.
‘to affix a stigma to a person’; ‘to affix ridicule or blame to somebody’;
Suffixnoun
A subscript mark, number, or letter. See Subscript, a.
Affixverb
(transitive) To subjoin, annex, or add at the close or end; to append to.
‘to affix a syllable to a word’; ‘to affix a seal to an instrument’; ‘to affix one's name to a writing’;
Suffixverb
To add or annex to the end, as a letter or syllable to a word; to append.
Affixverb
(transitive) To fix or fasten figuratively; with on or upon.
‘eyes affixed upon the ground’;
Suffixnoun
an affix that is added at the end of the word
Affixverb
To subjoin, annex, or add at the close or end; to append to; to fix to any part of; as, to affix a syllable to a word; to affix a seal to an instrument; to affix one's name to a writing.
Suffixverb
attach a suffix to;
‘suffix words’;
Affixverb
To fix or fasten in any way; to attach physically.
‘Should they [caterpillars] affix them to the leaves of a plant improper for their food.’;
Suffix
In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs.
Affixverb
To attach, unite, or connect with; as, names affixed to ideas, or ideas affixed to things; to affix a stigma to a person; to affix ridicule or blame to any one.
Affixverb
To fix or fasten figuratively; - with on or upon; as, eyes affixed upon the ground.
Affixnoun
That which is affixed; an appendage; esp. one or more letters or syllables added at the end of a word; a suffix; a postfix.
Affixnoun
a linguistic element added to a word to produce an inflected or derived form
Affixverb
attach to;
‘affix the seal here’;
Affixverb
‘He appended a glossary to his novel where he used an invented language’;
Affixverb
attach or become attached to a stem word;
‘grammatical morphemes afix to the stem’;
Affixverb
stick, attach, or fasten (something) to something else
‘panels to which he affixes copies of fine old prints’;
Affixverb
be able to be fixed
‘the strings affix to the back of the bridge’;
Affixnoun
an addition to the base form or stem of a word in order to modify its meaning or create a new word.
Affix
In linguistics, an affix is a morpheme that is attached to a word stem to form a new word or word form. Affixes may be derivational, like English -ness and pre-, or inflectional, like English plural -s and past tense -ed. | 2022-07-03 15:10:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3430229425430298, "perplexity": 10474.167863451137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104244535.68/warc/CC-MAIN-20220703134535-20220703164535-00152.warc.gz"} |
https://ckeditor.com/docs/ckeditor4/latest/api/CKEDITOR_ui_panel_block.html | CKEDITOR.ui.panel.block
class
class and all methods
Methods
• constructor() → blockCKEDITOR.ui.panel.block#constructor
Creates a block class instances.
Returns
block
• getItems() → nodeListCKEDITOR.ui.panel.block#getItems
Returns a CKEDITOR.dom.nodeList of block items.
Returns
nodeList
• markItem( index )CKEDITOR.ui.panel.block#markItem
Mark the item specified by the index as current activated.
Parameters
index : Object
• markFirstDisplayed( beforeMark )CKEDITOR.ui.panel.block#markFirstDisplayed
private
Marks the first visible item or the one whose aria-selected attribute is set to true. The latter has priority over the former.
Parameters
beforeMark : Object
function to be executed just before marking. Used in cases when any preparatory cleanup (like unmarking all items) would simultaneously destroy the information that is needed to determine the focused item. | 2018-09-20 16:18:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.356467604637146, "perplexity": 13301.396897058012}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156524.34/warc/CC-MAIN-20180920155933-20180920180333-00248.warc.gz"} |
https://www.wpm2011.org/yy58w5/47c6o.php?18f23e=quadratic-inequalities-grade-9 | To solve a quadratic inequality we must determine which part of the graph of a quadratic function lies above or below the $$x$$-axis. All answers are included. Negative 3 squared is positive 9, you have a negative out front, it becomes negative 9 plus 6, which is negative 3. Solving quadratic inequalities, quadratic equations and quadratic simultaneous equations. This is a quadratic inequality. • You … Welcome to the presentation on quadratic inequalities. Since the original inequality was less than or equal to, the boundary points are included. Sum and product of the roots of a quadratic equations Algebraic identities. Solving quadratic equations by quadratic formula. Grade 9 – Algebra Solving Quadratic Inequalities – This activity provides practice in using the graph of a quadratic function to solve quadratic inequalities. Solving quadratic equations by factoring. It then becomes a negative 9 plus 3, it becomes negative 3, and negative 3 will also become a negative 3. Solving quadratic equations by completing square. Since this statement is false, the region between -9 and 1 is not correct. YouMore Kwenturuan tungkol sa kung paano gawin ang solving quadratic … This entry was posted in MATH and tagged math , math solver , mathway . 1. Graphing Quadratic inequalities – Example 1: Sketch the graph of $$y>3x^2$$. GCSE Grade 9 Quadratic Inequalities, Quadratic Equations and Quadratic Simultaneous Equations. 0 times. Visual intuition of what a quadratic inequality means. jamespearce. I need to find where y = –x 2 + 6x – 9 is above the axis. Since this quadratic is not factorable using rational numbers, the quadratic formula will be used to solve it. • Answer the questions in the spaces provided – there may be more space than you need. To solve a quadratic inequality we must determine which part of the graph of a quadratic function lies above or below the $$x$$-axis. So it must be the region on either side of those points. Grade mathematics: Quadratic Inequalities 1. Use the zero product property to find the solutions to the equation x2 - 9 = 16. x = -5 or x = 5 Adam is using the equation (x)(x + 2) = 255 to find two consecutive odd integers with a product of 255. Edit. This is the same quadratic equation, but the inequality has been changed to $$\red . However, this inequality is an "or equal to" inequality, so the "equal" part counts as part of the solution. [TAGALOG] Grade 9 Math Lesson: HOW TO SOLVE QUADRATIC INEQUALITY? GCSE (1 – 9) Quadratic Inequalities Name: _____ Instructions • Use black ink or ball-point pen. The range of values that satisfy the inequality is between -8 and 8.This can be expressed as, Then fill in the region either inside or outside of it, depending on the inequality. Quadratic Inequalities Example: x2 + x – 6 = … Let's say I had f of x is equal to x squared plus x minus 6. The above is an equation (=) but sometimes we need to solve inequalities like these: These unique features make Virtual Nerd a viable alternative to private tutoring. You da real mvps! An inequality can therefore be solved graphically using a graph or algebraically using a table of signs to determine where the function is positive and negative. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. These are imaginary answers and cannot be graphed on a real number line. Thanks to all of you who support me on Patreon. Step by step guide to solve Solving Quadratic Inequalities . And that represents the graph of the inequality. In this case, we have drawn the graph of inequality using a pink color. 1 per month helps!! Each STEM Case level has an associated Handbook. :) https://www.patreon.com/patrickjmt !! 0% average accuracy. The same basic concepts apply to quadratic inequalities like$$ y x^2 -1 $$from digram 8. Save. OBJECTIVES : •Define the interval correctly •Express the Quadratic Inequalities •Solve the Quadratic Inequalities through graphs and interval notation 3. Before we get to quadratic inequalities, let's just start graphing some functions and interpret them and then we'll slowly move to the inequalities. These are interactive guides that focus on … So all values from -infinity to -9 inclusive, and from 1 inclusive to infinity, are solutions. an hour ago. Try (–7, 10). Solving absolute value equations Solving Absolute value inequalities Mathematics 9 Lesson 2: Quadratic Inequalities 1. Mathematics.$$.. More Lessons for Grade 9 Math Math Worksheets Videos, worksheets, games and activities to help Algebra and Grade 9 students learn how to solve quadratic inequalities. Consider the standard form of the quadratic equation \(ax^2 + bx + … It can be used in conjunction with Module 1 Lesson 7 of the DepEd Grade 9 Learning Modules. Displaying top 8 worksheets found for - Quadratic Inequalities. Quadratic Inequalities It is an inequality of the form of ax2 + bx + c (<, >, ≤, ≥) 0, where a, b and c are real numbers and a ≠ 0 3. 6. We need to find solutions. Add 4 to both sides of each inequality: 1 ≤ W ≤ 7. But I know (and can verify from the above graph) that this quadratic only touches the axis from below; it is never fully above the axis. MHR • Pre-Calculus 11 Solutions Chapter 9 Page 1 of 84 Chapter 9 Linear and Quadratic Inequalities Section 9.1 Linear Inequalities in Two Variables Section 9.1 Page 472 Question 1 a) y < x + 3 Try (7, 10). Quadratic Inequalities 2. ... 2014/11/03 آ 3.1-Quadratic Functions & Inequalities Quadratic. Multiple grade appropriate versions, or levels, exist for each STEM Case. Example: Solve the inequality x^2 \lt 64 When solving quadratic inequalities it is important to remember there are two roots. To solve a quadratic inequality, find the roots of its corresponding equality. If I were to graph it, Let's try it with 3, as well-- if we put a 3 there, 3 squared is 9. Solve the quadratic inequality: x2+11x+10≥0x^2+11x+10\ge0x2+11x+10≥0 Solving Quadratic Inequalities DRAFT. The software Geogebra is used as graphing tool. an hour ago. Yes we have two inequalities, because 3 2 = 9 AND (−3) 2 = 9. $< – 3 , – 2 > \cup < 1 , 2 ]$ Quadratic inequalities worksheets Quadratic inequalities (4.4 MiB, 620 hits) A Quadratic Equation in Standard Form ( a , b , and c can have any value, except that a can't be 0.) Worked example 16: Solving quadratic inequalities Worked example 16: Solving quadratic inequalities Left Side Right Side Left Side Right Side y x + 3 y x + 3 = 10 = 7 + 3 = 10 = –7 + 3 = 10 = –4 By using this website, you agree to our Cookie Policy. 0. quadratic inequalities, conduct a spin-o˛ Think-Pair-Share activity (Lyman, 1981). Quadratic Inequalities (Visual Explanation) How to solve a quadratic inequality. An inequality can therefore be solved graphically using a graph or algebraically using a table of signs to determine where the function is positive and negative. Choose a testing point and check the solution section. 9th - 11th grade. Check: Therefore, the inequality x 2 + 2 x + 5 < 0 has no real solutions. The method of completing the square provides a way to derive a formula that can be used to solve any quadratic equation. Some of the worksheets for this concept are Quadratic inequalities date period, Solving quadratic inequalities, Solving quadratic inequalities 1, Solving quadratic inequalities l3s1, Graphing quadratic, Graphing and solving quadratic inequalities, Solving quadratic inequalities algebraically work, Inequalitiesmep pupil text 16. Website uses cookies to ensure you get the best experience 2 = 9 and ( −3 ) 2 =.... Any quadratic equation will also become a negative 3, it becomes 3! X=\Pm a Inequalities 2 ) 2 = 9 and ( −3 ) 2 = and! Graph a quadratic inequality calculator - solve quadratic Inequalities 1 material best serves needs... Inequalities Example: solve the inequality x^2 \lt 64 When Solving quadratic Inequalities step-by-step this uses! Fill in the spaces provided – there may be more space than you need Grade. = … Grade Mathematics: quadratic FunctionsThis instructional material was collaboratively developed and reviewed byeducators from and! 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The boundary points are included a negative 9 plus 3, it becomes negative 3, it becomes quadratic inequalities grade 9,.
Creepy Crawly Bugs, Oss 117 Film Complet, Klopman Diamond Joke, Lightning Strike Meaning, Creepy Crawly Bugs, Renault Duster 2020, Blue Velvet Cast, | 2021-01-23 10:01:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44519126415252686, "perplexity": 1272.4255155718702}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703537796.45/warc/CC-MAIN-20210123094754-20210123124754-00356.warc.gz"} |
http://science-media.org/showCategoryPapers.php?c=1&s=1&nvid=75&npos=76&npres=205&nconferences=20 | The ScienceMedia Network
#### Inversions for Deep Solar Meridional Flow Using Spherical Born Kernels
Author: Vincent Böning
The solar meridional flow is a crucial ingredient in modern dynamo theory. Seismic estimates of this flow have, however, been contradictory in deeper layers below about $0.9\,R_\odot$. Results from time-distance helioseismology h ... more
Views: 375
#### Experimental results for infrared aberration tracking using a correlation algorithm on two star extended field
Author: Narsireddy Anugu
We report an experimental results of different correlation algorithms for an extended source wavefront sensor.
Views: 318
#### The Carina Nebula and Gum 31 molecular complex: I. Molecular gas distribution, column densities and dust temperatures
Author: Michael Burton
We report high resolution observations of the 12CO(1 → 0) and 13CO(1 → 0) molecular lines in the Carina Nebula and the Gum 31 region obtained with the 22-m Mopra telescope as part of the The Mopra Southern Galactic Plane CO Su ... more
Views: 307
#### The GREGOR Fabry‐Pérot Interferometer (GFPI) ‐ Technical Innovations and Results achieved in 2013
Author: Klaus Gerhard Puschmann
This paper shall provide a summary of not yet published technical innovations to the GREGOR Fabry-Pérot Interferometer (GFPI) at the 1.5m GREGOR Solar Telescope (Europe's largest solar telescope) that I implemented in 2013 as the ... more
Views: 939
#### The Importance of Long-term Synoptic Observations and Data Sets for Solar Physics and Helioseismology
Author: Markus Roth
A casual single glance at the Sun would not lead an observer to conclude that it varies. The discovery of the 11-year sunspot cycle was only made possible through systematic daily observations of the Sun over 150 years and even to ... more
Views: 363
#### Modelling solar and stellar brightness variabilities
Author: Alexander Shapiro
Total and spectral solar irradiance, TSI and SSI, have been measured from space since 1978. This is accompanied by the development of models aimed at replicating the observed variability by relating it to solar surface magnetism. ... more
Views: 302
#### Measurement of the VLT pupil motions using a 2 × 2 lenslet evaluated aberrations
Author: Narsireddy Anugu
We present the telescope pupil tracking strategy for the GRAVITY interferometer.
Views: 319
#### Workshop Report: A New Synoptic Solar Observing Network
Author: Markus Roth
Since all space weather originates in and on the Sun, forecasters need continuous, long-term, consistent, and reliable solar data as a foundation for useful predictions. An effective strategy to obtain nearly continuous solar data ... more
Views: 229
#### VARIATION OF ACOUSTIC CUTOFF PERIOD WITH HEIGHT IN THE SOLAR ATMOSPHERE: THEORY VERSUS OBSRVETIONS
Author: Aneta Wisniewska
Recently Wiśniewska et al. demonstrated observationally how the acoustic cutoff frequency varies with height in the solar atmosphere including the upper photosphere and the lower and middle chromosphere, and showed that the obser ... more
Views: 318
#### LORD OF THE RINGS: A KINEMATIC DISTANCE TO CIRCINUS X-1 FROM A GIANT X-RAY LIGHT ECHO
Author: Michael Burton
Circinus X-1 exhibited a bright X-ray flare in late 2013. Follow-up observations with Chandra and XMM-Newton from 40 to 80 days after the flare reveal a bright X-ray light echo in the form of four well-defined rings with radii fro ... more
Views: 384 | 2018-01-21 22:34:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44043442606925964, "perplexity": 10358.604159262306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890893.58/warc/CC-MAIN-20180121214857-20180121234857-00436.warc.gz"} |
https://www.riskresearch.org/papers/DanielssonValenzuelaZer2015/ | # Daníelsson, J., M. Valenzuela, and I. Zer (2018, January). Learning from history: Volatility and financial crises. Review of Financial Studies 31, 2774–2805.
We study the effects of stock market volatility on risk-taking and financial crises by constructing a cross-country database spanning up to 211 years and 60 countries. Prolonged periods of low volatility have strong in-sample and out-of-sample predictive power over the incidence of banking crises and can be used as a reliable crisis indicator, whereas volatility itself does not predict crises. Low volatility leads to excessive credit build-ups and balance sheet leverage in the financial system, indicating that agents take more risk in periods of low risk, supporting the dictum that stability is destabilizing.'' | 2018-12-19 02:02:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1756417155265808, "perplexity": 5332.753786868344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031231-00505.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/differential-equations-and-linear-algebra-4th-edition/chapter-6-linear-transformations-6-1-definition-of-a-linear-transformation-problems-page-389/3 | ## Differential Equations and Linear Algebra (4th Edition)
$T(a+b)=(a+b)''+a_1(a+b)'+a_2(a+b)=a''+b''+a_1a'+a_1b'+a_2a+a_2b=T(a)+T(b)$ $T(ca)=(ca)''+a_1(ca)'+a_2(ca)=c\cdot a''+c\cdot a_1a'+c\cdot a_2a=cT(a)$ Thus it is a linear transformation. | 2019-12-06 14:08:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9573651552200317, "perplexity": 478.57297566661623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540488620.24/warc/CC-MAIN-20191206122529-20191206150529-00194.warc.gz"} |