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https://www.doubtnut.com/question-answer/let-the-diameter-of-a-subset-s-of-the-plane-be-defined-as-the-maximum-of-the-distance-between-arbitr-644089616 | HomeEnglishClass 12MathsChapterCircle
Let the diameter of a subset S...
# Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary pairs of points of S. <br> Q. Let S={(x,y):(sqrt(5)-1)x-sqrt(10+2sqrt(5))y ge 0, (sqrt(5)-1)x+sqrt(10+12sqrt(5)) y ge 0, x^(2)+y^(2) le 9} then the diameter of S is :
Updated On: 17-04-2022
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Text Solution
(3)/(2) (sqrt(5)-1) 3(sqrt(5)-1)3sqrt(2) 3
D
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Transcript
hello stunts welcome to doubtnut our question is let the diameter of a subject s of of the plane be defined as a maximum of the distance between arbitrary pair of contour S letters is equal to x y z root 5 minus 1 x minus root 3 + 2 root 5 is greater than equal to zero and root 5 minus 1 X + root 3 + 2 root 5 by greater than equal to zero and root and square + Y is far less than equal to 9 than the diameter of this version basically we have to part we have to tell the maximum distance between two points in the region as our plane that will be our diameter so let's see how we will solve this we will solve let's say our first thing is root 5 minus 1 X /
root 10 + 2 root 5 is greater than equal to y I have taken first condition from here so let's say this is the question number 1 and root 5 minus 1 upon root 10 + 2 root 5 x is greater than - why this is our second and third is X square + Y square less than equal to 9 which is equation of a circle so first all our first equation which is the equation of line first solve why less than
root 5 minus 1 upon root 3 + 2 root 5 so I can write it as another form like this I have taken root in the whole thing so it will equal to root 5 minus y square upon 10 + 2 root 5 time x so why will be less than equal to let open the bracket here and story so 5 + 1 equal to 6 minus 2 root 5 / 10 + 2 root 5 x so this is a form of line let the comparing Y equal to a so this is a line passing through a
M equal to root 6 minus 2 root 5 / 10 + 2 root 5 which is less than one because 10 + 2 root 5 is greater than 6 minus 2 root 5 so it will be less than 1 lakh let's go to the second line which is which year let's write it was greater than I am taking -2 interval will be changed so I am taking Root over so root 5 minus 1 square divided by 10 + 2 root 5 so let's square
and open the bracket it will equal to minus root 5 + 1 is 6 minus 2 root 5 upon 3 + root 5 is data then why you again comparing with Y equal to a so this is again a equal to minus root of x minus 2 root 5 upon 3 + root 5 is this thing is again less than one expect -2 when we will take my sorry plus one first sorry I should have right - year so this thing is a less
less than 1 and when we will - here it will look like this minor so our interval will be changed and -1 greater than -1 and 1 hour the equation of circle X square + Y square is less than equal to 9 vi we have find our three reasons so let's draw our region to we will for I will first Rock ordinate axis and make a circle of radius 3 radius \$3 draw circle this is this is like this is not have origin so this is a circle and first draw two line
Bhai call to X and Y equal to -1 sorry because our lines are less than 1 and -1 this is equal to X line and this is why equal to minus x line let se so this is what we call to wear which is no 1 and Y equal to -1 and our first line has its low less than 12 let's draw our line this is our first line which is it has a slow less than 12 it will look like this this is why quality to our first line and this is this has also Lo Plus
10 -12 it will also look like sorry I will ride by another colour so I can find it more easily so this is our second line this is our second line so just name it it is our first line first line and this is our second line and let se name the point so it will be comma zero cause its radius and it is 00803 so I have name in a point so we have to find is the reason between this so for this lets a03 point is UPS idea
so let's say that the value 0 commentary India was first line to what is our first line is here so when we we will put 032 we will put zero here and here he is not less than zero so our region will not be applied it will down side near the reason will be here on the first nine it will appear first division let take the second line so let's say this is again 037 2nd 9030 I will put zero here and here to minus 3 is less than zero which is correct so I will take region upside to region will be upside
will be this season will be this so reason will be this is our common reason will hear let's highlight of a reason this is our reason so we have to tell the maximum distance in this region between 2.2 and this is the first point zero and 200 first maximum distance we are getting his three first distance let's 80000 which is the distance is given equal to 3 let's check another point which is I will say and let this point and this point towards you mean to this point will be
from the intersection point it will require root 2 by root 2 from the intersection of Y equal to X and circle in this point will be read by root 2 minus 3 by root 2 so h Di to will be let se difference between two points distance between 6 by which is true to this is due to but this is the distance between Y equal to X and Y equal to minus x which is far greater than between these two lines to let check the option and as you from options so this is our first answer free and this is true to so it cannot be 302 is distance between Y equal to X and Y equal to minus
not take it or leave it to root 5 minus the value of root 5 is 2.3 and 2.3 - 1 is 1.3 and 1.3 X is 3.9 something let se 424 by 2 equal to 2 is less then restore it cannot be our maximum distance diameter and let sit 3 root 5 is 2.1.3 3.9.9 which is a comparatively 23 rupee Hero to it is not in our vision to our final answer will be free this is our correct answer thank you | 2022-05-20 00:33:12 | {"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": 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": 0.7111330628395081, "perplexity": 419.60992446781205}, "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/1652662530553.34/warc/CC-MAIN-20220519235259-20220520025259-00006.warc.gz"} |
https://socratic.org/questions/the-line-spectrum-for-element-x-includes-a-band-of-green-light-of-wavelength-522 | # The line spectrum for element X includes a band of green light of wavelength 522 nm. This band corresponds to what energy transition within an atom of element X? Let Planck’s constant h = 4.136 × 10−15 eV ∙ s, and the speed of light c = 3.00 × 108 m/s.
May 25, 2017
decrease of $2.38$ $\text{e V}$
#### Explanation:
Let's substitute the value of $h$ into the Planck equation $E = h f$:
Rightarrow E = (4.136 times 10^(- 15) $\text{e V}$ $\cdot$ "s") $\times f$
Then, let's substitute the definition of $f$, $f = \frac{c}{\lambda}$, into the above equation:
Rightarrow E = (4.136 times 10^(- 15) $\text{e V}$ $\cdot$ "s") $\times \frac{c}{\lambda}$
Let's substitute the values of $c$ and $\lambda$ into the equation:
Rightarrow E = (4.136 times 10^(- 15) $\text{e V}$ $\cdot$ "s") times frac(3.00 times 10^(8) " m s"^(- 1))(522 " nm")
Rightarrow E = (4.136 times 10^(- 15) $\text{e V}$ $\cdot$ "s") times frac(3.00 times 10^(8) " m s"^(- 1))(5.22 times 10^(- 7) " m")
Rightarrow E = frac(1.2408 times 10^(- 6) " e V" cdot "m")(5.22 times 10^(- 7) " m")
$R i g h t a r r o w E = 2.337$ $\text{e V}$
$\therefore E = 2.38$ $\text{e V}$
Therefore, this band corresponds to a decrease of $2.38$ $\text{e V}$. | 2022-11-27 09:35:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 35, "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": 0.8962472677230835, "perplexity": 6236.622561999519}, "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-2022-49/segments/1669446710218.49/warc/CC-MAIN-20221127073607-20221127103607-00412.warc.gz"} |
http://mathhelpforum.com/calculus/109598-directional-derivatives.html | # Math Help - directional derivatives
1. ## directional derivatives
Suppose that you are climbing a hill whose shape is given by , and that you are at the point (90, 40, 300).
1) In which direction (unit vector) should you proceed initially in order to reach the top of the hill fastest? Answer : $<-12.6/ \sqrt{174.76},-4/ \sqrt{174.76}>$
2) If you climb in that direction, at what angle above the horizontal will you be climbing initially (radian measure)?
How to do 2) ?
$f_{x}=-.14x, \;\ f_{y}=-.10y$
$\nabla f(90,40)=-12.6i-4j$
The max value of the directional derivative is $||{\nabla}f(90,40)||=\sqrt{(\frac{-63}{5})^{2}+(-4)^{2}}=\frac{\sqrt{4369}}{5}\approx 13.22$
A unit vector in this direction is:
$\frac{-12.6}{13.22}i-\frac{4}{13.22}j$
$\sqrt{174.76}=13.22$ | 2016-05-02 20:53:41 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "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": 0.3909037113189697, "perplexity": 885.0184228126068}, "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-2016-18/segments/1461860117405.91/warc/CC-MAIN-20160428161517-00007-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://planet.emacsen.org/ | ## Irreal: Remote Sudo
If your workflow involves administering remote systems or something similar, you probably have the following burned into your muscle memory. If, on the other hand, you’re like me and don’t have occasion to need root access on a remote machine very often, you can probably use a reminder. If you’re in the second category, here, courtesy of abo-abo, you go:
If you click on the tweet, you’ll see the next tweet, which explains that “cloud” is the name of a remote system defined in abo-abo’s ~/.ssh/config. Defining often used remote systems like this is something I do too and it saves me a bunch of time and mental cycles trying to remember domains or, worse yet, IP addresses.
-1:-- Remote Sudo (Post jcs)--L0--C0--October 17, 2017 05:16 PM
## Marcin Borkowski: emacs-reveal
Some time ago, I learned from the Org-mode mailing list about a very interesting extension to the well-known org-reveal package. The emacs-reveal allows to embed audio files in reveal.js presentations. I find this quite fascinating, especially that I actually did prepare quite a few educational presentations.
-1:-- emacs-reveal (Post)--L0--C0--October 16, 2017 04:19 PM
## Wilfred Hughes: These Weeks in Remacs III
Time for another Remacs update: lots of contributions, a wide range of features, and even a logo!
## Contributing
Since the last update, we’ve seen contributions from lots of new people. We’ve added @brotzeit and @shanavas786, bringing us to seven wonderful people who can approve your PRs.
Speaking of PRs, we’ve merged an amazing 64 pull requests since the last update!
If you’re looking for a good feature for your first contribution, @brotzeit has been regularly adding new suggestions under the ‘good first issue’ label.
## Features
Many Emacs features have now been ported to Rust, with new Rust APIs for accessing elisp datastructures.
Here’s an overview of the features that have landed.
Arithmetic: arithmetic, floating point, random number generation (using a Rust RNG!), and comparisons.
Checksums: MD5sum (using a Rust MD5 crate!).
Processes: accessing, type check, data structures and names.
Buffers: for the current thread, accessing, file names, size and modification.
Hash tables: copying and accessing.
Characters: multibyte conversions, character tables, category tables
Fonts: type checks.
Miscellaneous: prefix arguments and identity.
We’re also periodically pulling GNU Emacs features into Remacs, so all the features available GNU Emacs trunk are included in Remacs.
## Idiomatic Rust in Remacs
Remacs has gradually developed a set of conventions for elisp data types. For each type Foo, we define a LispObject::as_foo, LispObject::as_foo_or_error and a FooRef when you know your elisp datatype is actually a Foo.
For example, here’s how overlay-start was implemented in C:
The C codebase makes heavy use of macros for checking types (CHECK_OVERLAY) and for accessing struct attributes (OVERLAY_START).
Here’s the Rust equivalent:
We use procedural macros to simplify defining an elisp primitive function, and type checking is much more explicit.
(This example is from PR #298.)
Other exciting Rusty features include variadic macros to replace call1, call2 in C with just call! in Rust, and the ability to mock extern C functions so we can write unit tests.
## Hash Maps
We’re not always able to leverage the Rust libraries available. @DavidDeSimone showed some amazing Rust-fu exploring using Rust’s FnvHashMap inside Remacs.
Sadly, we weren’t able to use the Rust hash map implementation. The C layer assumes that it can mutate hash table keys in place, and unexec does not play nicely with mmap. See the PR for the full details.
Finally, we’re discussing a logo for Remacs. We’ve had some great submissions:
You can join the logo discussion at PR #360.
As always, if you fancy writing some Rust in support of the world’s lispiest text editor, you can join us on GitHub!
-1:-- These Weeks in Remacs III (Post Wilfred Hughes (me@wilfred.me.uk))--L0--C0--October 16, 2017 12:00 AM
## Pragmatic Emacs: Using a visible bell in Emacs
Here’s a tiny and basic tip. If you want you Emacs to flash at you instead of beeping for an error, add the following to your emacs config file
;; turn on visible bell
(setq visible-bell t)
-1:-- Using a visible bell in Emacs (Post Ben Maughan)--L0--C0--October 15, 2017 11:35 PM
## Irreal: Start an Engineering Notebook
Camilla over at Winterflower argues that software engineers should keep an engineering notebook. That’s advice that everybody knows they should follow but that too many of us don’t. We’re busy and we think, “I’ll remember what I just did, I don’t need to write it down.” Of course, a little later we don’t remember and have to go through the pain of figuring things out all over again.
I keep a journal in which I record everything I do and discover but I’ve only been doing this for about 3 years. I really wish I’d started earlier. A good way of making it easier to get started and keep with it is to have a good system for recording things.
Of course, as an Emacs and Org mode user that means I have a built-in infrastructure for such things. One of the things that Emacs and Org mode provide is an easy way of retrieving information from your notebook. Org mode tags provide an excellent way of finding things. For example months ago I revised the way I compile Emacs (which can be a bit finicky on macOS), put it in my journal, and added the tags emacs and compiling. When I want to compile Emacs, I merely type Ctrl+c a m emacs:comiling to find all my journal entries with those tags. Even better, I have the commands in a code block so I can run them automatically by just typing Ctrl+c Ctrl+c in the block. That’s a real win over trying to figure out everything each time I compile a new Emacs.
There’s lots of examples like this, of course, and the more you put into your notebook, the more you can get out and the easier it will make things for you. Of you aren’t already keeping an engineering notebook, you should start. Or at least make a New Years resolution to start. I promise you, you’ll be glad you did.
-1:-- Start an Engineering Notebook (Post jcs)--L0--C0--October 15, 2017 05:55 PM
## Endless Parentheses: Mold Slack entirely to your liking with Emacs
Although fine-tuning your slack notifications is already reason enough to run slack in Emacs, that’s only the beginning. Once everything is up and running you get to decide what you want out of your slack. Some of the snippets below simply make up for missing functionality, other customize the package beyond what you can do on the Slack Webapp.
Priorities first. The most important improvement you can implement is install emojify-mode and turn it on for slack chats.
Secondly, make sure you costumize the chat faces to your liking. Just open a chat buffer, place your cursor on a piece of text whose face you want to customize, and call customize-face.
In order to keep track of new messages in the mode-line, slack.el uses a package called tracking, which is the same one circe uses for IRC chats. The command tracking-next-buffer is a fantastic way to cycle through your pending messages, bind it to something short.
I’ll never know who thought user statuses were a good idea for Slack. But, thanks to a tip by _asummers on HackerNews, I can live in a world where they don’t exist.
I like notifications with minimal titles, and the package is kind enough to make these configurable.
Slack.el uses lui for the chat buffers. If you, like me, are a heavy user of abbrevs in Emacs, you’ll find it annoying that the final word of each message won’t get expanded unless you explicitly hit SPC before RET. That’s easy to remedy with an advice.
Finally, the biggest missing feature from this package is that it displays the author on every message output, even when the same user sends several messages in a row. The snippet below adds a hook to omit the author name for a message whenever it’s the same author as the previous message.
You don’t have to stop here, of course. Want to fine-tune which buffers get tracked on the mode-line? Hack into tracking.el. Want to change the face used for your own messages, or even align them to the right? Redefine slack-buffer-insert. Your workflow is yours to build.
Comment on this.
-1:-- Mold Slack entirely to your liking with Emacs (Post)--L0--C0--October 09, 2017 11:43 PM
## Pragmatic Emacs: Tree-style directory views in dired with dired-subtree
By default, Emacs’ file browser/manager dired usually presents you with a flat list of files in a given directory. Entering a subdirectory then opens a new buffer with the listing of the subdirectory. Sometimes you might want to be able to see the contents of the subdirectory and the current directory in the same view. Many GUI file browsers visualise this with a tree structure with nodes that can be expanded or collapsed. In Emacs there is a built-in function dired-insert-subdir that inserts a listing of the subdirectory under the cursor, at the bottom of the current buffer instead of in a new buffer, but I’ve never found that very helpful.
The dired-subtree package (part of the magnificent dired hacks) improves on this by allowing you to expand subdirectories in place, like a tree structure. To install the package, use the following code:
(use-package dired-subtree
:config
(bind-keys :map dired-mode-map
("i" . dired-subtree-insert)
(";" . dired-subtree-remove)))
This sets up the keybinds so that in dired, hitting i on a subdirectory expands it in place with an indented listing. You can expand sub-subdirectories in the same way, and so on. Hitting ; inside an expanded subdirectory collapses it.
Happily, some of my other favourite tools from dired hacks like dynamically narrowing the directory listing or copying and pasting files work as you would want in these expanded subdirectories.
-1:-- Tree-style directory views in dired with dired-subtree (Post Ben Maughan)--L0--C0--October 08, 2017 11:29 PM
## Manuel Uberti: To shell or not to shell
As much as most of my daily workflow revolves around Emacs, I always have GNOME terminal ready to fly with Fish shell and tmux. I keep EShell next to me for quick tasks, but I have never relied on shell-mode or ansi-term for other CLI-intensive work.
I don’t know what happened to the lovely “Emacs Chat” series from Sacha Chua, but more than three years ago she interviewed Mickey Petersen. Mickey talked with great enthusiasm about shell-mode and at the time I admittedly made a mental note about giving it a try. Regretfully, I have only recently come back to that note.
My first M-x shell didn’t look that great. I haven’t debugged the compatibility issues with Fish, probably something related to my heavily customised config.fish. Anyway, falling back to Bash is enough.
(validate-setq explicit-shell-file-name "/bin/bash")
Note that I am using validate-setq as explained here.
Another thing I have noticed is the lack of colours for the output of ls. Fortunately, Emacs StackExchange has an answer for that, so I have added this line to my .bash_aliases file:
alias ls="TERM=ansi ls --color=always"
The input echoing is easily turned off following the instructions on the manual. Also, the history is much cleaner and easier to navigate with counsel-shell-history.
(unbind-key "C-c C-l" shell-mode-map)
(bind-key "C-c C-l" #'counsel-shell-history shell-mode-map)
Note that unbind-key and bind-key are macros from bind-key.el, which is part of the fantastic use-package.
Last but not least, I like to have my shell buffer filling the whole window in the current frame. Thus, display-buffer-alist to the rescue.
(validate-setq
display-buffer-alist
(
;; … other stuff …
(,(rx bos "*shell")
(display-buffer-same-window)
(reusable-frames . nil))
;; … other stuff …
))
-1:-- To shell or not to shell (Post)--L0--C0--October 07, 2017 12:00 AM
## Modern Emacs: Solving ligature spacing in Emacs - proof of concept
Ligatures are single-character replacements of strings. Examples of ligatures: replacing "alpha" with the alpha symbol and "!=" with the a slashed equal sign. See Coding with Mathematical Notation for details and pictures.
There is a serious flaw with ligatures - either the indentation you see with ligatures or without ligatures is correct, not both. So if someone that does not use ligatures works on your code, your indentation's will not match. An example:
;; True indentation, what you want others to see
(alpha b
c)
;; Emacs indentation, what you want to see when working
(a b
c)
This problem significantly hampers ligature adoption.
I do not believe any editor implements a solution to ligatures such that you see the indentation you want to see, while the true indentation remains correct.
I present a proof-of-concept solution to ligature spacing,
# How Emacs displays text
Emacs associates text-properties with strings. A property can be anything. Some property names are special and tell Emacs to handle the text in a particular way, like face for how a text is highlighted.
An overlay has associated text-properties but is buffer-local. So when we move that text to another buffer, if that overlay had a face, then that face would not be carried over.
Properties to be aware of:
• display : How Emacs displays that region, can be any string.
• invisible : Whether the text should be displayed.
• modification-hooks : When text in the overlay is edited, run these hooks.
• evaporate (overlays) : Once the overlay is "done-with", delete the overlay.
## Compose region
Additionally, compose-region is similar to display in that the composed region is displayed as (possibly many) characters. Current implementations of ligatures all leverage compose-region by searching the buffer for say alpha and composing from alphas beginning to end point the Unicode symbol for alpha.
There are several important distinctions between compose-region and put-text-property 'display
1. Indentation uses the composed character for indenting while the text-property
2. display indents with the true, original string.
3. Composition cannot be set for overlays. The internal composition text property,
4. unlike all other properties, cannot be put manually.
5. Editing within a composed region will undo the composition while one must
6. delete the whole region with the display property to undo the display.
# Working through a solution
## To compose or display the ligature?
Because composition adjusts the underlying indentation, it cannot be used for a ligature spacing solution. Indentation cannot be adjusted in a major-mode agnostic manner. Indentation always considers the true number of characters preceding the text on the line, so dynamically adding invisible spaces will not work.
## But how to make editing a display behave like a composition?
It is a serious issue to have to delete the whole text for the ligature to disappear.
The solution is the modification-hooks text-property.
(defun lig-mod-hook (overlay post-mod? start end &optional _)
(when post-mod?
(overlay-put overlay 'display nil)
(overlay-put overlay 'modification-hooks nil))) ; force evaporation
(overlay-put lig-overlay 'modification-hooks '(lig-mod-hook))
Now editing text with the display property will behave as desired.
## So how to visually collapse the indentation?
We could set invisible on the first 5 spaces of the line to collapse the visual indentation by 5. But the invisible property will modify subsequent line's indentation by 5 fewer (if necessary), an issue that cannot be resolved as we cannot determine in general the "if necessary" part.
The trick is to make the 5 first spaces display as one space. Because display doesn't modify indentation, subsequent lines will be indented properly.
(overlay-put space-overlay 'display " ")
## How do we determine the indentation we want to see then?
We let Emacs do the work - we create a mirror buffer where the ligatures are actually composed and compare the differences in indentation.
Overlays are not just buffer-local, they also do not transfer to indirect buffers. Ideally we would have a hidden indirect buffer where we keep ligatures composed instead. Unfortunately, since the composition text property is special, it can only be set with compose-region which does not work for overlays.
Further, calculating indentation always adjusts the indentation. The significance is that whenever we indent the indirect buffer, all the text will move back-and-forth. So indirect buffers are out.
Instead we create temporary buffers for the composition and retrieve an alist of lines and their composed indentations.
# A working example
The current ligature snippets floating around hack font-locks to perform the ligature substitutions. I recently became familiar with context-sensitive syntax highlighting via the syntax-propertize-function in my work on hy-mode.
I develop a minimal major-mode lig-mode that uses the syntax function to implement ligatures.
## Setup
First we setup a basic major-mode for testing.
(provide 'lig-mode)
(define-derived-mode lig-mode fundamental-mode "Lig"
(setq-local indent-line-function 'lisp-indent-line)
(setq-local syntax-propertize-function 'lig-syntax-propertize-function))
This is a proof-of-concept - we implement spacing for a single ligature for now. Lets replace "hello" with a smiley face.
(defun lig--match-lig (limit)
(re-search-forward (rx word-start "hello" word-end) limit t))
(setq lig-char #x263a)
(setq lig-str "☺")
## Determining the indents we want to see
We copy the buffer contents to a temporary buffer, search and compose the symbols, indent the buffer, and copy the indentation for each line.
(defvar lig-diff-indents nil)
(defun lig-get-diff-indents ()
(setq lig-diff-indents nil)
(save-excursion
;; Compose the ligatures
(goto-char (point-min))
(while (re-search-forward (rx word-start "hello" word-end) nil t)
(compose-region (match-beginning 0) (match-end 0) lig-char))
;; Change indent to match the composed symbol
(indent-region (point-min) (point-max))
;; Build an alist of line and indention column
(goto-char (point-min))
(setq line 1)
(while (< (point) (point-max))
(push (cons line (current-indentation))
lig-diff-indents)
(forward-line)
(setq line (1+ line)))))
(defun run-lig-get-diff-indents ()
(let ((true-buffer (current-buffer)))
(with-temp-buffer
(fundamental-mode)
(setq-local indent-line-function 'lisp-indent-line)
(insert-buffer-substring-no-properties true-buffer)
(lig-get-diff-indents))))
## Bringing it together
For details on how syntax-propertize-function works, check this post.
Whenever we edit the buffer this hook will run, recalculating and visually collapsing all the leading spaces as needed.
(defun lig-syntax-propertize-function (start-limit end-limit)
;; Make sure visual indentations are current
(run-lig-get-diff-indents)
(save-excursion
(goto-char (point-min))
(while (lig--match-lig end-limit)
(let ((start (match-beginning 0))
(end (match-end 0)))
(unless (-contains? (overlays-at start) lig-overlay)
;; Create and set the lig overlays if not already set
(setq lig-overlay (make-overlay start end))
(overlay-put lig-overlay 'display lig-str)
(overlay-put lig-overlay 'evaporate t)
(overlay-put lig-overlay 'modification-hooks '(lig-mod-hook)))))
;; Remove all spacing overlays from buffer
(remove-overlays nil nil 'invis-spaces t)
;; Recalcualte and add all spacing overlays
(goto-char (point-min))
(setq line 1)
(while (< (point) (point-max))
;; Don't add the spacing overlay until we indent
(unless (> (+ (current-indentation) (point))
(point-max))
(let* ((vis-indent (alist-get line lig-diff-indents))
(num-spaces (- (current-indentation) vis-indent))
(start (point))
(end (+ num-spaces (point))))
;; only add invisible spaces if the indentations differ
(unless (<= num-spaces 1)
(setq space-overlay (make-overlay start end))
(overlay-put space-overlay 'invis-spaces t)
(overlay-put space-overlay 'display " ")
(overlay-put space-overlay 'evaporate t))
(setq line (1+ line))
(forward-line))))))
# The result
Enable lig-mode to see:
;; The true text
(hello how
are
you (hello hi
again))
;; What we see
(☺ how
are
you (☺ hi
again))
The indentation we see is not the true indentation anymore!
The full and current code is hosted here.
The missing space on the second hello is a bug. There are many issues with this implementation - this is a proof of concept. I suspect a completely correct solution to be still some time and effort away, if only because this approach is incredibly inefficient.
This post shows that we maybe can have our cake and eat it too in regards to ligatures.
-1:-- Solving ligature spacing in Emacs - proof of concept (Post)--L0--C0--October 05, 2017 12:00 AM
## Intro
When creating documents, context aware completion is a powerful mechanism that can help you improve the speed, correctness and discoverability.
Emacs provides context aware completion via the complete-symbol command, bound to C-M-i by default. In order for it to do something useful, completion-at-point-functions has to be set up.
Documentation:
Special hook to find the completion table for the thing at point.
Each function on this hook is called in turn without any argument and should
return either nil to mean that it is not applicable at point,
or a list of the form (START END COLLECTION) where
START and END delimit the entity to complete and should include
point, COLLECTION is the completion table to use to complete it.
For each major-mode, a different value of completion-at-point-functions can (and probably should) apply. One of the modes that's set up nicely by default is emacs-lisp-mode: press C-M-i to get completion for Elisp variable and function names. Org-mode, on the other hand, is quite lacking in this regard: nothing useful happens with C-M-i.
Here's my current setting for Org-mode:
(setq completion-at-point-functions
'(org-completion-symbols
ora-cap-filesystem
org-completion-refs))
## org-completion-symbols
When I write about code in Org-mode, I quote items like this:
=/home/oleh/=, =HammerFactoryFactory=, etc.
• It looks nice, since it's in a different face,
• flyspell doesn't need to check it, which makes sense since it would fail on most variable and class names,
• Prevents Org from confusing directory names for italics mark up.
Completion has one more advantage on top of that: if I refer to a symbol name multiple times within a document, completion helps me to enter it quickly and correctly. Here's the corresponding completion source:
(defun org-completion-symbols ()
(when (looking-back "=[a-zA-Z]+")
(let (cands)
(save-match-data
(save-excursion
(goto-char (point-min))
(while (re-search-forward "=\$$[a-zA-Z]+\$$=" nil t)
(cl-pushnew
(match-string-no-properties 0) cands :test 'equal))
cands))
(when cands
(list (match-beginning 0) (match-end 0) cands)))))
1. First of all, it checks if the point is e.g. after =A, i.e. we are in fact entering a new quoted symbol. If that's not the case, return nil and let the other completion sources have a go.
2. Next, it looks through the current buffer for each =foo= and =bar=, accumulating them into a list.
3. Finally, it returns the bounds of what we've got so far, plus the found candidates. It's important that the bounds are passed to the completion engine, so that it can delete everything inside the bounds before inserting the whole selected symbol.
## org-cap-filesystem
This source is for completing file names:
(defun ora-cap-filesystem ()
(let (path)
(when (setq path (ffap-string-at-point))
(let ((compl
(when compl
(let ((offset (ivy-completion-common-length (car compl))))
(list (- (point) offset) (point) compl)))))))
I usually enter ~, so that ffap-string-at-point recognizes it as a path. Then complete each part of the path with C-M-i. It's very similar to counsel-find-file. In fact, I could just use counsel-find-file for this, with M-o i to insert the file name instead of opening the selected file.
## org-completion-refs
org-completion-refs is very similar to org-completion-symbols: it will collect all instances of e.g. \label{foo}, and offer them for completion when you enter \ref{. If you want to look at the code, it's available in my config.
## Outro
I hope I convinced you about the usefulness of completion at point. It's especially cool since it's a universal interface for major-mode-specific completion. So any IDE-like package for any language could provide its own completion using the familiar interface. That could go a long way towards providing a "just works" experience, particularly when dealing with a new language.
-1:-- Extending completion-at-point for Org-mode (Post)--L0--C0--October 03, 2017 10:00 PM
## Modern Emacs: Deep diving into a major mode - Part 1
I've taken up maintaining hy-mode - a major mode for lispy python.
I narrate working through specific problems in auto-completion, indentation, shell integration, and so on.
This post touches on: syntax, indentation, font-locking, and context-sensitive syntax.
All code snippets require the Emacs packages dash and s.
# Syntax Tables
The first step in a major mode is the syntax table.
In any major mode run describe-syntax to see its syntax table. As we are working with a lisp, we copy its syntax-table to start with.
(defconst hy-mode-syntax-table
(-let [table
(copy-syntax-table lisp-mode-syntax-table)]
;; syntax modifications...
table)
"Hy modes syntax table.")
The syntax table isn't set explicitly, its name identifies and sets it for hy-mode.
Configuration is performed with modify-syntax-entry, its docstring provides all the syntactic constructs we can pick from.
A subset to be familiar with:
• ( ) : open/close parenthesis. These are for all bracket-like constructs such
• as [ ] or { }. The first character should be the syntactic construct, namely "(" or ")", and the second character should be the closing delimiter.
(modify-syntax-entry ?\{ "(}" table)
(modify-syntax-entry ?\} "){" table)
(modify-syntax-entry ?$"(]" table) (modify-syntax-entry ?$ ")[" table)
• ' : prefix character. Prefixes a symbol/word.
;; Quote characters are prefixes
(modify-syntax-entry ?\~ "'" table)
(modify-syntax-entry ?\@ "'" table)
• _ and w : symbol and word constituent respectively.
;; "," is a symbol in Hy, namely the tuple constructor
(modify-syntax-entry ?\, "_ p" table)
;; "|" is a symbol in hy, naming the or operator
(modify-syntax-entry ?\| "_ p" table)
;; "#" is a tag macro, we include # in the symbol
(modify-syntax-entry ?\# "_ p" table)
• | : generic string fence. A more general string quote syntactic construct.
• Used for delimiting multi-line strings like with triple quotes in Python. I go into depth on this construct in the "context-sensitive syntax" section.
# Indentation
Look through calculate-lisp-indent, the indentation workhorse of lisp-mode derivatives, and it is quickly seen that indentation is hard.
Indentation is set with indent-line-function.
In the case of a lisp, we actually do:
(setq-local indent-line-function 'lisp-indent-line)
(setq-local lisp-indent-function 'hy-indent-function)
Where the real work is performed by calculate-lisp-indent that makes calls to lisp-indent-function, accepting an indent-point and state.
The function at heart is parse-partial-sexp, taking limiting points and retrieving a 10 element list describing the syntax at the point.
As this is a (necessarily) excessive amount of information, I recommend as many other modes have done - define some aliases. I have:
(defun hy--sexp-inermost-char (state) (nth 1 state))
(defun hy--start-of-last-sexp (state) (nth 2 state))
(defun hy--in-string? (state) (nth 3 state))
(defun hy--start-of-string (state) (nth 8 state))
Observe you can also omit state and call syntax-ppss to get state which runs parse-partial-sexp from point-min to current point, with the caveat that the 2nd and 6th state aren't reliable. I prefer to pass the state manually.
These are the building blocks for indentation - we can then write utilities to better get our head around indentation like:
(defun hy--prior-sexp? (state)
(number-or-marker-p (hy--start-of-last-sexp state)))
## The indent function
The three cases:
;; Normal Indent
(normal b
c)
(normal
b c)
;; Special Forms
(special b
c)
;; List-likes
[a b
c]
Hy's current indent function:
(defun hy-indent-function (indent-point state)
"Indent at INDENT-POINT where STATE is parse-partial-sexp' for INDENT-POINT."
(goto-char (hy--sexp-inermost-char state))
(if (hy--not-function-form-p)
(1+ (current-column)) ; Indent after [, {, ... is always 1
(forward-char 1) ; Move to start of sexp
(cond ((hy--check-non-symbol-sexp (point)) ; Comma tuple constructor
(+ 2 (current-column)))
((hy--find-indent-spec state) ; Special form uses fixed indendation
(1+ (current-column)))
(t
(hy--normal-indent calculate-lisp-indent-last-sexp)))))
When we indent we jump to the sexp's innermost char, ie. "(", "[", "{", etc..
If that character is a list-like, then we 1+ it and are done.
Otherwise we move to the start of the sexp and investigate if (thing-at-point 'symbol). If it is, then we check a list of special forms like when, do, defn for a match. If we found a (possibly fuzzy) match, then regardless of whether the first line contains args or not, we indent the same.
(defun hy--normal-indent (last-sexp)
"Determine normal indentation column of LAST-SEXP.
Example:
(a (b c d
e
f))
1. Indent e => start at d -> c -> b.
Observe 'a' need not be on the same line as the ( will cause a match.
Then we determine indentation based on whether there is an arg or not.
2. Indenting f will go to e.
Now since there is a prior sexp d but we have no sexps-before on same line,
the loop will terminate without error and the prior lines indentation is it."
(goto-char last-sexp)
(-let [last-sexp-start nil]
(if (ignore-errors
(while (hy--anything-before? (point))
(setq last-sexp-start (prog1
;; Indentation should ignore quote chars
(if (-contains? '(?\' ?\ ?\~)
(char-before))
(1- (point))
(point))
(backward-sexp))))
t)
(current-column)
(if (not (hy--anything-after? last-sexp-start))
(1+ (current-column))
(goto-char last-sexp-start) ; Align with function argument
(current-column)))))
Normal indent does the most work. To notice, if we are on the next line without a function arg above, then last-sexp-start will be nil as backward-sexp will throw an error and the setq won't go off.
If there is a function call above, then the current-column of the innermost, non-opening sexp, will end up as the indent point.
If we indent the line of the funcall, it will jump to the containing sexp and calculate its indent.
Other indentation functions are a bit more advanced in that they track the number of prior sexps in the indent-function to distinguish between eg. the then and else clause of an if statement. Those cases use the same fundamentals that are seen here.
Developing indentation from scratch can be challenging. The approach I took was to look at clojure's indentation and trim it down until it fit this language. I've removed most of the extraneous details that it adds to handle special rules for eg. clojure.spec but it is still possible that I could trim this further.
# Font Locks and Highlighting
Two symbols are the entry points to be aware of into font locking: hy-font-lock-kwds and hy-font-lock-syntactic-face-function.
(setq font-lock-defaults
'(hy-font-lock-kwds
nil nil
(("+-*/.<>=!?\$%_&~^:@" . "w")) ; syntax alist
nil
(font-lock-mark-block-function . mark-defun)
(font-lock-syntactic-face-function ; Differentiates (doc)strings
. hy-font-lock-syntactic-face-function)))
## Font lock keywords
There exists many posts on modifying the variable font-lock-keywords.
The approach taken in hy-mode is to separate out the language by category:
(defconst hy--kwds-constants
'("True" "False" "None" "Ellipsis" "NotImplemented")
"Hy constant keywords.")
(defconst hy--kwds-defs
'("defn" "defun"
"defmacro" "defmacro/g!" "defmacro!"
"Hy definition keywords.")
(defconst hy--kwds-operators
'("!=" "%" "%=" "&" "&=" "*" "**" "**=" "*=" "+" "+=" "," "-"
"-=" "/" "//" "//=" "/=" "<" "<<" "<<=" "<=" "=" ">" ">=" ">>" ">>="
"^" "^=" "|" "|=" "~")
"Hy operator keywords.")
;; and so on
And then use the amazing rx macro for constructing the regexes.
Now due to rx being a macro and its internals, in order to use variable definitions in the regex construction we have to call rx-to-string instead.
The simplest definition:
(defconst hy--font-lock-kwds-constants
(list
(rx-to-string
(: (or ,@hy--kwds-constants)))
'(0 font-lock-constant-face))
"Hy constant keywords.")
A more complex example with multiple groups taking different faces:
(defconst hy--font-lock-kwds-defs
(list
(rx-to-string
(: (group-n 1 (or ,@hy--kwds-defs))
(1+ space)
(group-n 2 (1+ word))))
'(1 font-lock-keyword-face)
'(2 font-lock-function-name-face nil t))
"Hy definition keywords.")
Of course not all highlighting constructs are determined by symbol name. We can highlight the shebang line for instance as:
(defconst hy--font-lock-kwds-shebang
(list
(rx buffer-start "#!" (0+ not-newline) eol)
'(0 font-lock-comment-face))
"Hy shebang line.")
We then collect all our nice and modular font locks as hy-font-lock-kwds that we set earlier:
(defconst hy-font-lock-kwds
(list hy--font-lock-kwds-constants
hy--font-lock-kwds-defs
;; lots more ...
hy--font-lock-kwds-shebang)
"All Hy font lock keywords.")
## Syntactic face function
This function is typically used for distinguishing between string, docstrings, and comments. It does not need to be set unless you want to distinguish docstrings.
(defun hy--string-in-doc-position? (state)
"Is STATE within a docstring?"
(if (= 1 (hy--start-of-string state)) ; Identify module docstring
t
(-when-let* ((first-sexp (hy--sexp-inermost-char state))
(function (save-excursion
(goto-char (1+ first-sexp))
(thing-at-point 'symbol))))
(s-matches? (rx "def" (not blank)) function)))) ; "def"=="setv"
(defun hy-font-lock-syntactic-face-function (state)
"Return syntactic face function for the position represented by STATE.
STATE is a parse-partial-sexp' state, and the returned function is the
Lisp font lock syntactic face function. String is shorthand for either
a string or comment."
(if (hy--in-string? state)
(if (hy--string-in-doc-position? state)
font-lock-doc-face
font-lock-string-face)
font-lock-comment-face))
It is rather straightforward - we start out within either a string or comment. If needed, we jump to the first sexp and see if it is a "def-like" symbol, in which case we know its a doc.
This implementation isn't perfect as any string with a parent def-sexp will use the doc-face, so if your function returns a raw string, then it will be highlighted as if its a doc.
# Context sensitive syntax
An advanced feature Emacs enables is context-sensitive syntax. Some examples are multi-line python strings, where there must be three single quotes together, or haskell's multiline comments.
Hy implements multiline string literals for automatically escaping quote characters. The syntax is #[optional-delim[the-string]optional-delim] where the string can span lines.
In order to identify and treat the bracket as a string, we look to setting the syntax-propertize-function.
It takes two arguments, the start and end points with which to search through. syntax.el handles the internals of limiting and passing the start and end and applying/removing the text properties as the construct changes.
(defun hy--match-bracket-string (limit)
"Search forward for a bracket string literal."
(re-search-forward
(rx "#["
(0+ not-newline)
"["
(group (1+ (not (any "]"))))
"]"
(0+ not-newline)
"]")
limit
t))
(defun hy-syntax-propertize-function (start end)
"Implements context sensitive syntax."
(save-excursion
(goto-char start)
;; Start goes to current line, need to go to char-before the #[ block
(when (nth 1 (syntax-ppss))
(goto-char (- (hy--sexp-inermost-char (syntax-ppss)) 2)))
(while (hy--match-bracket-string end)
(put-text-property (1- (match-beginning 1)) (match-beginning 1)
'syntax-table (string-to-syntax "|"))
(put-text-property (match-end 1) (1+ (match-end 1))
'syntax-table (string-to-syntax "|")))))
We go to the start and jump before its innermost containing sexp begins minus two for the hash sign and bracket characters.
If the regex matches a bracket string, we then set the innermost brackets on both sides to have the string-fence syntax.
When the syntax is set - parse-partial-sexp and in particular font lock mode and indent-line will now recognize that block as a string - so proper indentation and highlighting follow immediately. And when we modify the brackets, the string-fence syntax is removed and behaves as expected.
This function can handle any kind of difficult syntactic constructs. For instance, I could modify it to only work if the delimiters on both side of the bracket string are the same. I could also associate some arbitrary, custom text property that other parts of hy-mode interact with.
Note that there is the macro syntax-propertize-rules for automating the searching and put-text-property portions. I prefer to do the searching and application manually to 1. have more flexibility and 2. step through the trace easier.
# Closing
Building a major mode teaches a lot about how Emacs works. I'm sure I've made errors, but so far this has been enough to get hy-mode up and running. The difference in productivity in Hy I've enjoyed since taking maintainer-ship has made the exercise more than worth it.
I also have auto-completion and shell/process integration working which I'll touch on in future posts.
-1:-- Deep diving into a major mode - Part 1 (Post)--L0--C0--October 03, 2017 12:00 AM
## Endless Parentheses: Turbo up your Ruby console in Emacs
Keeping a REPL (or a console) always by your side is never a bad habit, and if you use an IDE-package (like Robe for Ruby, or Cider for Clojure) it’s nigh unavoidable. Being an essential part of your environment, it would be ridiculous not to invest some time optimizing it.
One obvious optimization is to bind a key to your “start console” command, but that’s just the start. You pretty much never need two running consoles for the same project, so why not have the same key switch to it if it’s already running?
But we can go a bit farther with very little work. I have a file where I define a lot of small helper methods for my Ruby console, so let’s require it automatically whenever a new console is started.
If you use Projectile and want to go even faster, check out the j key on my post about Projectile.
Comment on this.
-1:-- Turbo up your Ruby console in Emacs (Post)--L0--C0--October 02, 2017 08:57 PM
## Marcin Borkowski: Converting TeX sequences to Unicode characters
I quite often deal with LaTeX files using stuff like \'a or \"e, and I really prefer having those encoded in UTF-8. So the natural question arises: how to convert one into another? The problem is especially frustrating because Emacs can do this – either via C-x 8 prefix, or with the TeX input method. It is not trivial, however, to find out how it does these things, and to get hold of the data used to actually perform the conversion. (At least, I didn’t find a way to do it.) After a bit of searching, however, I came up with another solution. I’m hesitant to call it “clever”; it’s rather hackish, but hey, it works, so who cares.
-1:-- Converting TeX sequences to Unicode characters (Post)--L0--C0--October 02, 2017 06:14 PM
## Jonas Bernoulli: Borg 2.0 and Epkg 3.0 released
I am excited to announce the release of Borg v2.0, Epkg v3.0, Closql v0.4 and Emir v2.0.
-1:-- Borg 2.0 and Epkg 3.0 released (Post)--L0--C0--September 20, 2017 03:00 PM
## Manuel Uberti: Taming closing delimiters in my s-expressions
As I explained when I wrote about my daily Clojure workflow, I rely heavily on Smartparens for my editing. With Lisp-like languages in particular, I enable smartparens-strict-mode to keep my s-expressions balanced even when I happen to use delete-char or kill-word dangerously near a closing parenthesis.
I have sp-kill-sexp bound to C-M-k, however out of habit I often use C-k to kill a line, which in my configuration is set up as Artur Malabarba explained in his Kill Entire Line with Prefix Argument. Doing that in the middle of an s-expression creates unnerving chaos.
Smartparens comes with a handy binding to temporarily disable the enforced balancing and let me insert a closing delimiter. Just pressing C-q followed by the desired matching parenthesis brings the order back.
Unfortunately, it’s not always that easy. Take this snippet which appears at the end of a ClojureScript function:
(when-not (empty? @data)
[:div
[graph]])]]))))
Carelessly hitting C-k near [graph] disrupts an otherwise elegant s-expression. I could undo, of course, but what if after C-k I do other kill-and-yank edits?
This is exactly why I have come to love syntactic-close.
(use-package syntactic-close ; Automatically insert closing delimiter
:ensure t
:bind ("C-c x c" . syntactic-close))
As soon as I discover an unbalanced s-expression, I can use C-c x c as many times as needed to add back the right closing delimiters.
-1:-- Taming closing delimiters in my s-expressions (Post)--L0--C0--September 17, 2017 12:00 AM
## Bryan Murdock: Not Leaky, Just Wrong
Intel recently announced new tools for FPGA design. I should probably try to understand OpenCL better before bagging on it, but when I read, "[OpenCL] allows users to abstract away hardware-specific development and use a higher-level software development flow." I cringe. I don't think that's how we get to a productive, higher-level of abstraction in FPGA design. When you look at the progress of software from low-level detailed design to high-level abstract design you see assembly to C to Java to Python (to pick one line of progression among many). The thing that happened every time a new higher-level language gained traction is people recognized patterns that developers were using over and over in one language and made language features in a new language that made those patterns one-liners to implement.
Examples of design patterns turning into language features are, in assembly people developed the patterns of function calls: push arguments onto the stack, save the program counter, jump to the code the implements the function, the function code pops arguments off the stack, does it's thing, then jumps back to the the code that called it. In C the tedium of all that was abstracted away by the language providing you with syntax to define a function, pass it arguments, and just call return at the end. In C people then started developing patterns of structs containing data and function pointers for operating on that data which turned into classes and objects in Java. Java also abstracted away memory management with a garbage collector. Patterns in Java (Visitor, State, etc.) are no longer needed in Python because of features in that language (related discussion here).
This is the path that makes most sense to me for logic design as well. Right now in RTL Verilog people use patterns like registers (always block that activates on posedge clk, has reset, inputs, outputs, etc.), state machines (case statement and state registers, next_state logic...), interfaces (SV actually attempted to add syntax for this), and so on. It seems like the next step in raising the abstraction level is to have a language with those sorts of constructs built-in. Then let people use that for a while and see what new patterns develop and encapsulate those patterns in new language features. Maybe OpenCL does this? I kind of doubt it if it's a "software development flow." It's probably still abstracting away CPU instructions.
-1:-- Not Leaky, Just Wrong (Post Bryan (noreply@blogger.com))--L0--C0--September 15, 2017 03:04 PM
## Timo Geusch: Emacs 25.3 released
Emacs 25.3 has been released on Monday. Given that it’s a security fix I’m downloading the source as I write this. If you’re using the latest Emacs I’d recommend you update your Emacs. The vulnerability as been around since Emacs Read More
The post Emacs 25.3 released appeared first on The Lone C++ Coder's Blog.
-1:-- Emacs 25.3 released (Post Timo Geusch)--L0--C0--September 15, 2017 04:20 AM
## punchagan: Emacs frame as a pop-up input
I wanted to try using a dialog box/pop-up window as a prompt to remind me to periodically make journal entries. I had the following requirements:
• Simple, light-weight dialog box that allows text of arbitrary length
• Ability to launch the dialog from the shell
• Ability to have some placeholder or template text, each time the dialog is shown
• Save the input text to a specific org-mode file
• Write as little code of my own, as possible, to do this
I had initially thought about using a tool like zenity, or write a simple dialog box in Python using Qt, wx or even tk, and then yank the input text at the desired location. This probably wouldn’t have turned out to be too hard, but getting things to look and work exactly the way I wanted would have required more code than I was willing to write or maintain.
After avoiding doing this for a while, I finally realized that I could simply use Emacs with a new frame with the appropriate dimensions, and with the correct file/buffer open to the desired location. This would
• eliminate the need for me to write the UI myself
• eliminate the need to do text manipulation in code, to yank it at the right place, in the right form. By directly opening up the editor at the required location, the onus is on me (as a text inputting user) to put it in, the way I want it.
• additionally provide me the comfort of being able to write with the full power of Emacs - keybindings and all that jazz.
• let me leverage elisp to do essentially whatever I want with the buffer being displayed as the dialog box.
I ended up with a command that looks something like this
emacsclient -c -n\
-F '((title . "Title") (left . (+ 550)) (top . (+ 400)) (width . 110) (height . 12))'\
-e '(pc/open-journal-buffer)'
This worked pretty nicely, except for the fact that with gnome-shell, the pop-up frame doesn’t always appear raised. It often gets hidden in the Emacs windows group, and the whole idea of the pop-up acting as a reminder goes for a toss! But, thanks to this Ask Ubuntu post, I could fix this pretty easily.
emacsclient -c -n\
-F '((title . "Title") (left . (+ 550)) (top . (+ 400)) (width . 110) (height . 12))'\
-e '(progn (pc/open-journal-buffer) (raise-frame) (x-focus-frame (selected-frame)))'
-1:-- Emacs frame as a pop-up input (Post)--L0--C0--September 14, 2017 04:56 PM
## Jonas Bernoulli: Magit 2.11 released
I am excited to announce the release of Magit version 2.11, consisting of 303 commits since the last feature release six months ago.
-1:-- Magit 2.11 released (Post)--L0--C0--September 13, 2017 11:00 AM
## Steven Pigeon: Undo that mess
During last marking season (at the end of the semester), I had, of course, to grade a lot of assignments. For some reason, every semester, I have a good number of students that write code like they just don’t care. I get code that looks like this:
int fonction (int random_spacing)^M{ ^M
int niaiseuses;
for (int i=0;i<random_spacing; i++){
{
{
std::cout
<< bleh
<< std::endl;
}}
}
}
There’s a bit of everything. Random spacing. Traces of conversions from one OS to another, braces at the end of line. Of course, they lose points, but that doesn’t make the code any easier to read. In a previous installment, I proposed something to rebuild the whitespaces only. Now, let’s see how we can repair as many defects as possible with an Emacs function.
Let’s start at the beginning: a list of the things to repair:
• OS-related conversion. Linux/*nixes end lines in \n, Windows in \r\n. Other platforms may use something else. Let’s not concern ourselves with the ZX80.
• Replace longs series of (white)spaces by only one space.
• Deal with braces at the end of lines.
• Reindent everything else using the defined style.
The first two items can be combined. Since transforming \r\n into \n only requires to remove \r, we can bundle series of (white)spaces and \r for replacement. I’m not a regex ninja: I came up with this:
; replaces multiple spaces and stray ^M
(while (re-search-forward "[[:space:]\|?\r]+" nil t)
(replace-match " " nil nil))
Trailing braces are a bit more complicated. They may, or mayn’t, be preceded by spaces and followedby spaces. This time, the regex is a bit more complicated:
; remove fiendish { at end of (non-empty) line
(while (re-search-forward
"\$$[^[:space:]{?\n]+\$$\$$[[:space:]]*\$$\$${\$$\$$[[:space:]]*\$$" nil t)
(replace-match "\\1\n{" nil nil))
It matches three parts. Something that is not whitespaces, followed by something that is whitespaces, the brace {, then whitespaces to the end of line. OK, that makes four. The only one we’re interested in not replacing is the first (the \\1 argument in replace). Everything else, most of it whitespaces, is replaced by newline, { , newline.
Now, the buffer should be in a rather messy state, possibly with trailing whitespaces and destroyed indentation. Calls to whitespace-cleanup and indent-region should finish the job.
Putting all that together:
(defun cleanup-whole-buffer()
"Removes ^M, tabs, and reindent whole buffer"
(interactive)
(save-excursion
(undo-boundary)
(beginning-of-buffer)
; replaces multiple spaces and stray ^M
(while (re-search-forward "[[:space:]\|?\r]+" nil t)
(replace-match " " nil nil))
(beginning-of-buffer)
; remove fiendish { at end of (non-empty) line
(while (re-search-forward
"\$$[^[:space:]{?\n]+\$$\$$[[:space:]]*\$$\$${\$$\$$[[:space:]]*\$$" nil t)
(replace-match "\\1\n{" nil nil))
(beginning-of-buffer)
(whitespace-cleanup)
(indent-region (point-min) (point-max) nil)
)
)
A few explanations on the other stuff we haven’t discussed yet. The save-excursion primitive saves cursor position so that when the function ends, we are still where we called it from. The undo-boundary makes sure that we won’t need a series of undos to undo the cleanup. beginning-of-buffer moves the cursor… at the beginning of the buffer.
Applying it to the above code snippet, we end up with:
int fonction (int random_spacing)
{
int niaiseuses;
for (int i=0;i<random_spacing; i++)
{
{
{
std::cout
<< bleh
<< std::endl;
}}
}
}
There are still a number of issues. For example, i++ has still an extraneous space before it, and we still have two closing braces on the same line. Maybe we should fix that sometime.
Filed under: emacs, hacks Tagged: braces, elisp, n!, newline, whitespace, \r, \r\n
-1:-- Undo that mess (Post Steven Pigeon)--L0--C0--September 12, 2017 03:33 PM
## Chris Wellons: Gap Buffers Are Not Optimized for Multiple Cursors
Gap buffers are a common data structure for representing a text buffer in a text editor. Emacs famously uses gap buffers — long-standing proof that gap buffers are a perfectly sufficient way to represent a text buffer.
• Gap buffers are very easy to implement. A bare minimum implementation is about 60 lines of C.
• Gap buffers are especially efficient for the majority of typical editing commands, which tend to be clustered in a small area.
• Except for the gap, the content of the buffer is contiguous, making the search and display implementations simpler and more efficient. There’s also the potential for most of the gap buffer to be memory-mapped to the original file, though typical encoding and decoding operations prevent this from being realized.
• Due to having contiguous content, saving a gap buffer is basically just two write(2) system calls. (Plus fsync(2), etc.)
A gap buffer is really a pair of buffers where one buffer holds all of the content before the cursor (or point for Emacs), and the other buffer holds the content after the cursor. When the cursor is moved through the buffer, characters are copied from one buffer to the other. Inserts and deletes close to the gap are very efficient.
Typically it’s implemented as a single large buffer, with the pre-cursor content at the beginning, the post-cursor content at the end, and the gap spanning the middle. Here’s an illustration:
The top of the animation is the display of the text content and cursor as the user would see it. The bottom is the gap buffer state, where each character is represented as a gray block, and a literal gap for the cursor.
Ignoring for a moment more complicated concerns such as undo and Unicode, a gap buffer could be represented by something as simple as the following:
struct gapbuf {
char *buf;
size_t total; /* total size of buf */
size_t front; /* size of content before cursor */
size_t gap; /* size of the gap */
};
This is close to how Emacs represents it. In the structure above, the size of the content after the cursor isn’t tracked directly, but can be computed on the fly from the other three quantities. That is to say, this data structure is normalized.
As an optimization, the cursor could be tracked separately from the gap such that non-destructive cursor movement is essentially free. The difference between cursor and gap would only need to be reconciled for a destructive change — an insert or delete.
A gap buffer certainly isn’t the only way to do it. For example, the original vi used an array of lines, which sort of explains some of its quirky line-oriented idioms. The BSD clone of vi, nvi, uses an entire database to represent buffers. Vim uses a fairly complex rope-like data structure with page-oriented blocks, which may be stored out-of-order in its swap file.
### Multiple cursors
Multiple cursors is fairly recent text editor invention that has gained a lot of popularity recent years. It seems every major editor either has the feature built in or a readily-available extension. I myself used Magnar Sveen’s well-polished package for several years. Though obviously the concept didn’t originate in Emacs or else it would have been called multiple points, which doesn’t quite roll off the tongue quite the same way.
The concept is simple: If the same operation needs to done in many different places in a buffer, you place a cursor at each position, then drive them all in parallel using the same commands. It’s super flashy and great for impressing all your friends.
However, as a result of improving my typing skills, I’ve come to the conclusion that multiple cursors is all hat and no cattle. It doesn’t compose well with other editing commands, it doesn’t scale up to large operations, and it’s got all sorts of flaky edge cases (off-screen cursors). Nearly anything you can do with multiple cursors, you can do better with old, well-established editing paradigms.
Somewhere around 99% of my multiple cursors usage was adding a common prefix to a contiguous serious of lines. As similar brute force options, Emacs already has rectangular editing, and Vim already has visual block mode.
The most sophisticated, flexible, and robust alternative is a good old macro. You can play it back anywhere it’s needed. You can zip it across a huge buffer. The only downside is that it’s less flashy and so you’ll get invited to a slightly smaller number of parties.
But if you don’t buy my arguments about multiple cursors being tasteless, there’s still a good technical argument: Gap buffers are not designed to work well in the face of multiple cursors!
For example, suppose we have a series of function calls and we’d like to add the same set of arguments to each. It’s a classic situation for a macro or for multiple cursors. Here’s the original code:
foo();
bar();
baz();
The example is tiny so that it will fit in the animations to come. Here’s the desired code:
foo(x, y);
bar(x, y);
baz(x, y);
With multiple cursors you would place a cursor inside each set of parenthesis, then type x, y. Visually it looks something like this:
Text is magically inserted in parallel in multiple places at a time. However, if this is a text editor that uses a gap buffer, the situation underneath isn’t quite so magical. The entire edit doesn’t happen at once. First the x is inserted in each location, then the comma, and so on. The edits are not clustered so nicely.
From the gap buffer’s point of view, here’s what it looks like:
For every individual character insertion the buffer has to visit each cursor in turn, performing lots of copying back and forth. The more cursors there are, the worse it gets. For an edit of length n with m cursors, that’s O(n * m) calls to memmove(3). Multiple cursors scales badly.
Compare that to the old school hacker who can’t be bothered with something as tacky and modern (eww!) as multiple cursors, instead choosing to record a macro, then play it back:
The entire edit is done locally before moving on to the next location. It’s perfectly in tune with the gap buffer’s expectations, only needing O(m) calls to memmove(3). Most of the work flows neatly into the gap.
So, don’t waste your time with multiple cursors, especially if you’re using a gap buffer text editor. Instead get more comfortable with your editor’s macro feature. If your editor doesn’t have a good macro feature, get a new editor.
If you want to make your own gap buffer animations, here’s the source code. It includes a tiny gap buffer implementation:
-1:-- Gap Buffers Are Not Optimized for Multiple Cursors (Post)--L0--C0--September 07, 2017 01:34 AM
## Chen Bin (redguardtoo): Split Emacs window with certain ratio
Emacs window with certain ratio :en:emacs:
The idea comes from yangdaweihit. Here is the implementation.
(defvar my-ratio-dict
'((1 . 1.61803398875)
(2 . 2)
(3 . 3)
(4 . 4)
(5 . 0.61803398875))
"The ratio dictionary.")
(defun my-split-window-horizontally (&optional ratio)
"Split window horizontally and resize the new window.
Always focus bigger window."
(interactive "P")
(let* (ratio-val)
(cond
(ratio
(setq ratio-val (cdr (assoc ratio my-ratio-dict)))
(split-window-horizontally (floor (/ (window-body-width)
(1+ ratio-val)))))
(t
(split-window-horizontally)))
(set-window-buffer (next-window) (other-buffer))
(if (or (not ratio-val)
(>= ratio-val 1))
(windmove-right))))
(defun my-split-window-vertically (&optional ratio)
"Split window vertically and resize the new window.
Always focus bigger window."
(interactive "P")
(let* (ratio-val)
(cond
(ratio
(setq ratio-val (cdr (assoc ratio my-ratio-dict)))
(split-window-vertically (floor (/ (window-body-height)
(1+ ratio-val)))))
(t
(split-window-vertically)))
;; open another window with other-buffer
(set-window-buffer (next-window) (other-buffer))
;; move focus if new window bigger than current one
(if (or (not ratio-val)
(>= ratio-val 1))
(windmove-down))))
(global-set-key (kbd "C-x 2") 'my-split-window-vertically)
(global-set-key (kbd "C-x 3") 'my-split-window-horizontally)
Usage is simple. For example, C-x 2 is similar to original split-winddow-vertically while C-u 1 C-x 2 split the window in golden ratio.
-1:-- Split Emacs window with certain ratio (Post Chen Bin)--L0--C0--September 05, 2017 01:26 PM
## Raimon Grau: Everyone welcome Wilfred to the emacs hall of fame.
The recent emacs' hall of fame: Magnars, Malabarba, abo-abo..... and now, we have Wilfred Hughes.
Thank you all for inspiring us, each one with different styles, influences and strategies. Kudos!
-1:-- Everyone welcome Wilfred to the emacs hall of fame. (Post Raimon Grau (noreply@blogger.com))--L0--C0--August 31, 2017 06:38 PM
I’ve just released Helpful, a new way of getting help in Emacs!
The *Help* built-in to Emacs is already pretty good. Helpful goes a step further and includes lots of contextual info. Let’s take a look.
Have you ever wondered which major modes have a keybinding for a function? Helpful reports keybindings in all keymaps!
When you’re hacking on some new code, you might end up with old function aliases after renaming a function. Helpful provides discoverable debug buttons, so you don’t need to remember fmakunbound.
Helpful also has strong opinions on viewing docstrings. Summaries are given focus, and text is fontified. We solve the text-quoting-style debate by removing superfluous puncuation entirely.
Helpful will even show all the references to the symbol you’re looking at, using elisp-refs. This is great for understanding how and where a function is used.
Finally, Helpful will rifle through your Emacs instance to find source code to functions:
• If you’ve defined a function interactively, Helpful will use edebug properties to find the source code.
• If Emacs can only find the raw closure, helpful will convert it back to an equivalent defun.
• If Emacs can only find the byte-compiled files, helpful will just pretty-print that.
I’ve just released v0.1, so there will be bugs. Please give it a try, and let me know what you think, or how we can make it even more, well, helpful!
-1:-- Helpful: Adding Contextual Help to Emacs (Post Wilfred Hughes (me@wilfred.me.uk))--L0--C0--August 30, 2017 12:00 AM
## emacsninja: Parsing the Hard Way
Hello again and sorry for the long downtime! My current Emacs project is an EPUB reader, something that requires parsing XML and validating/extracting data from it. The former can be done just fine in Emacs Lisp with the help of libxml2 (or alternatively, xml.el), for the latter there is no good solution. Typically people go for one of the following approaches:
• Don’t parse at all and just use regular expressions on the raw XML. This works somewhat okayish if your input is predictable and doesn’t change much.
• Parse and walk manually through the parse tree with car, cdr and assoc. Rather tedious and requires writing your own tree traversal functions for anything less than static XML.
• Invent your own library and use a selector DSL for DOM traversal. I’ve seen a few of those, like xml+.el, enlive.el and xml-query.el, however they support relatively little features in their selectors, use their own language instead of a well-established one (such as CSS selectors or XPath) and are usually not available from a package archive for easy installation.
As I’m a big fan of APIs like Python’s lxml with the cssselect module and used the esxml package before, I decided to implement CSS selectors for it. The general strategy was to take parse a CSS selector into a suitable form, do tree traversal by interpreting the parse tree and return the nodes satisfying the selector. Surprisingly enough, the hardest part of this were the parsing bits, so I’ll go into a bit more of detail on how you’d do it properly without any dependencies.
The approach taken in esxml-query.el is recursive-descent parsing, as seen in software like GCC. Generally speaking, a language can be described by a set of rules where the left side refers to its name and the right side explains what it expands to. Expansions are sequences of other rules or constants (which naturally cannot be expanded) and may contain syntactic sugar, such as the Kleene star (as seen in regular expressions). Given an input string described by the grammar, a parser breaks it down according to its rules until it has found a valid combination. The easiest way to turn a grammar into code is by expressing it with a function for each rule, with each function being free to call others. Success and failure can be expressed by returning a piece of the parse tree, a special sentinel value (I’ve chosen to return nil if the rule wasn’t completely matched) or throwing an error, thereby terminating the computation. If all recursively called rule functions returned a bit of the parse tree, the top-level call returns the complete parse tree and the parsing attempt has been successful.
Traditionally there is an extra step before parsing the string, as it’s a bit tedious to express the terminating rules as a sequence of characters, the string is typically preprocessed by a so-called lexer into a list of tagged tokens. This is relatively simple to do in Emacs Lisp by treating the string like a buffer, finding a token that matches the current position, adding it to the list of found tokens and advancing the position until the input has been exhausted. There is one non-trivial problem though, depending on the token definitions it can happen that there are two different kinds of tokens for a given position in the input string. A simple solution here is picking the longer match, this is why the tokenization in esxml--tokenize-css-selector finds all possible matches and picks the longest one.
The syntactical sugar used for the official CSS grammars consists of alternation (|), grouping ([...]), optionals (?) and greedy repetition (* and +). Given the basic token operations (peek) (return first token in the stream) and (next) (pop first token in the stream), it’s straight-forward to translate them to working code by using conditionals and loops. For example, the rule whitespace: SPACE* is consumed by calling (next) while (pop) returns a whitespace. To make things easier, I’ve also introduced an (accept TYPE) helper that uses (peek) to check whether the following token matches TYPE and either consumes it and returns the value or returns nil without consuming. With it the preceding example can be shortened to (while (accept 'space)). Similarly, alternation is expressed with cond and grouping with a while where the body checks whether the grouped content could be matched.
This parsing strategy allows for highly flexible error reporting going beyond “Invalid selector” errors I’ve seen previously in a browser console as you immediately know at which place the parser fails and are free to insert code dealing with the error as you see fit. Be warned though that you must understand the grammar well enough to transform it into a more suitable form, yet equivalent form if you run into rules that are hard or even impossible to express as code. Debugging isn’t too bad either, you can observe the junctions taken by your code and quickly spot at which it goes wrong.
I’m looking forward to venture into parser combinators and PEGs next as they follow the same approach, but involve less code to achieve similar results.
-1:-- Parsing the Hard Way (Post Vasilij Schneidermann)--L0--C0--August 26, 2017 09:52 PM
## Chris Wellons: Vim vs. Emacs: The Working Directory
Vim and Emacs have different internals models for the current working directory, and these models influence the overall workflow for each editor. They decide how files are opened, how shell commands are executed, and how the build system is operated. These effects even reach outside the editor to influence the overall structure of the project being edited.
In the traditional unix model, which was eventually adopted everywhere else, each process has a particular working directory tracked by the operating system. When a process makes a request to the operating system using a relative path — a path that doesn’t begin with a slash — the operating system uses the process’ working directory to convert the path into an absolute path. When a process forks, its child starts in the same directory. A process can change its working directory at any time using chdir(2), though most programs never need to do it. The most obvious way this system call is exposed to regular users is through the shell’s built-in cd command.
Vim’s spiritual heritage is obviously rooted in vi, one of the classic unix text editors, and the most elaborate text editor standardized by POSIX. Like vi, Vim closely follows the unix model for working directories. At any given time Vim has exactly one working directory. Shell commands that are run within Vim will start in Vim’s working directory. Like a shell, the cd ex command changes and queries Vim’s working directory.
Emacs eschews this model and instead each buffer has its own working directory tracked using a buffer-local variable, default-directory. Emacs internally simulates working directories for its buffers like an operating system, resolving absolute paths itself, giving credence to the idea that Emacs is an operating system (“lacking only a decent editor”). Perhaps this model comes from ye olde lisp machines?
In contrast, Emacs’ M-x cd command manipulates the local variable and has no effect on the Emacs process’ working directory. In fact, Emacs completely hides its operating system working directory from Emacs Lisp. This can cause some trouble if that hidden working directory happens to be sitting on filesystem you’d like to unmount.
Vim can be configured to simulate Emacs’ model with its autochdir option. When set, Vim will literally chdir(2) each time the user changes buffers, switches windows, etc. To the user, this feels just like Emacs’ model, but this is just a convenience, and the core working directory model is still the same.
### Single instance editors
For most of my Emacs career, I’ve stuck to running a single, long-lived Emacs instance no matter how many different tasks I’m touching simultaneously. I start the Emacs daemon shortly after logging in, and it continues running until I log out — typically only when the machine is shut down. It’s common to have multiple Emacs windows (frames) for different tasks, but they’re all bound to the same daemon process.
While with care it’s possible to have a complex, rich Emacs configuration that doesn’t significantly impact Emacs’ startup time, the general consensus is that Emacs is slow to start. But since it has a really solid daemon, this doesn’t matter: hardcore Emacs users only ever start Emacs occasionally. The rest of the time they’re launching emacsclient and connecting to the daemon. Outside of system administration, it’s the most natural way to use Emacs.
The case isn’t so clear for Vim. Vim is so fast that many users fire it up on demand and exit when they’ve finished the immediate task. At the other end of the spectrum, others advocate using a single instance of Vim like running a single Emacs daemon. In my initial dive into Vim, I tried the single-instance, Emacs way of doing things. I set autochdir out of necessity and pretended each buffer had its own working directory.
At least for me, this isn’t the right way to use Vim, and it all comes down to working directories. I want Vim to be anchored at the project root with one Vim instance per project. Everything is smoother when it happens in the context of the project’s root directory, from opening files, to running shell commands (ctags in particular), to invoking the build system. With autochdir, these actions are difficult to do correctly, particularly the last two.
### Invoking the build
I suspect the Emacs’ model of per-buffer working directories has, in a Sapir-Whorf sort of way, been responsible for leading developers towards poorly-designed, recursive Makefiles. Without a global concept of working directory, it’s inconvenient to invoke the build system (M-x compile) in some particular grandparent directory that is the root of the project. If each directory has its own Makefile, it usually makes sense to invoke make in the same directory as the file being edited.
Over the years I’ve been reinventing the same solution to this problem, and it wasn’t until I spent time with Vim and its alternate working directory model that I truly understood the problem. Emacs itself has long had a solution lurking deep in its bowels, unseen by daylight: dominating files. The function I’m talking about is locate-dominating-file:
(locate-dominating-file FILE NAME)
Look up the directory hierarchy from FILE for a directory containing NAME. Stop at the first parent directory containing a file NAME, and return the directory. Return nil if not found. Instead of a string, NAME can also be a predicate taking one argument (a directory) and returning a non-nil value if that directory is the one for which we’re looking.
The trouble of invoking the build system at the project root is that Emacs doesn’t really have a concept of a project root. It doesn’t know where it is or how to find it. The vi model inherited by Vim is to leave the working directory at the project root. While Vim can simulate Emacs’ working directory model, Emacs cannot (currently) simulate Vim’s model.
Instead, by identifying a file name unique to the project’s root (i.e. a “dominating” file) such as Makefile or build.xml, then locate-dominating-file can discover the project root. All that’s left is wrapping M-x compile so that default-directory is temporarily adjusted to the project’s root.
That looks very roughly like this (and needs more work):
(defun my-compile ()
(interactive)
(let ((default-directory (locate-dominating-file "." "Makefile")))
(compile "make")))
It’s a pattern I’ve used again and again and again, working against the same old friction. By running one Vim instance per project at the project’s root, I get the correct behavior for free.
-1:-- Vim vs. Emacs: The Working Directory (Post)--L0--C0--August 22, 2017 04:51 AM
# Emacs Start-Up: Speeding It Up
## 1 TL;DR:
Describes my Emacs start-up file, and what I did to speed it up from
12 seconds to under 4 seconds.
## 2 Overview Of Steps
• Byte-compile start-up files.
• Temporarily increase gc-cons-threshold during startup.
• Lexically bind file-name-handler-alist to nil if start-up
is split across many files.
• Used memoization to avoid network lookup of current location during startup.
I have a large number of elpa/melpa packages installed:
(length load-path)
400
With the above, my emacs (Emacs 26 built from Git) startup time is on
average 4 seconds. This includes starting up emacspeak (including
speech servers), as well as launching a number of project-specific
shell buffers. Given that I rarely restart Emacs, the startup time is
academic — but speeding up Emacs startup did get me to clean-up my
Emacs setup.
## 3 Introduction
I have now used Emacs for more than 25 years, and my Emacs start-up
file
has followed the same structure through this time.
1. The init file defines a start-up-emacs function that does the
bulk of the work.
2. Package-specific configuration is split up into
<package>-prepare.el files.
3. All of these files are byte-compiled.
of various modules.
## 4 Load Byte-Compiled Start-Up File
I keep my emacs-startup.el checked into GitHub.
My Emacs init-file is a symlink to the byte-compiled version of the
above — this is something that goes back to my time as a
grad-student at Cornell (when GitHub of course did not exist).
That is also when I originally learnt the trick of temporarily setting
gc-cons-threshold to 8MB — Emacs' default is 800K.
## 5 Package Autoloads And eval-after-load
Over time, some of the package-specific setup files had come to
package-specific setup at the time. As part of the cleanup, I updated
setup code in eval-after-load — this is effectively the same as
using use-package.
means you can load compressed, encrypted or remote files without
having to worry about it. That flexibility comes at a cost — if you
are sure you dont need this flexibility during start-up, then locally
binding file-name-handler-alist to nil is a big win — in my
case, it sped things up by 50%.
## 7 Avoid Network Calls During Start-Up
In my case, I set calendar-latitude and calendar-longitude by
geocoding my address — geocoding is done by calling the Google Maps
API. The geocoding API is plenty fast that you normally dont notice
it — but it was adding anywhere from 1–3 seconds during
startup. Since my address doesn't change that often, I updated module
gmaps to use a memoized version. My address is set via Customize,
and the geocoded lat/long is saved to disk automatically.
## 8 References
1. Emacs Speed What got it all started.
2. file-name-handler-alist The article that gave me the most useful
tip of them all.
Net
-1:-- Emacs Start-Up: Speeding It Up (Post T. V. Raman (noreply@blogger.com))--L0--C0--August 21, 2017 08:01 PM
## sachachua: 2017-08-21 Emacs news
Links from reddit.com/r/emacs, /r/orgmode, /r/spacemacs, Hacker News, planet.emacsen.org, YouTube, the changes to the Emacs NEWS file, and emacs-devel.
Past Emacs News round-ups
-1:-- 2017-08-21 Emacs news (Post Sacha Chua)--L0--C0--August 21, 2017 07:16 AM
## sachachua: 2017-08-14 Emacs news
Links from reddit.com/r/emacs, /r/orgmode, /r/spacemacs, Hacker News, planet.emacsen.org, YouTube, the changes to the Emacs NEWS file, and emacs-devel.
Past Emacs News round-ups
-1:-- 2017-08-14 Emacs news (Post Sacha Chua)--L0--C0--August 14, 2017 06:33 AM
## emacshorrors: make-temp-name
Update: Reddit points out that this has been fixed on master by replacing most of the code with a call to gnulib’s gen_tempname.
For someone not terribly experienced in writing safe programs, one can only hope that building blocks like make-temp-file are doing the right thing and cannot be subverted by a malicious third party. The general advice here is that it’s preferable to use the primitive for creating the temporary file instead of the primitive to generate its name. Now, does Emacs reuse mkstemp(3) for this? Or at least tmpnam(3)? Of course not! Where we go, we can just invent our own source of randomness:
make-temp-file looks as follows:
static const char make_temp_name_tbl[64] =
{
'A','B','C','D','E','F','G','H',
'I','J','K','L','M','N','O','P',
'Q','R','S','T','U','V','W','X',
'Y','Z','a','b','c','d','e','f',
'g','h','i','j','k','l','m','n',
'o','p','q','r','s','t','u','v',
'w','x','y','z','0','1','2','3',
'4','5','6','7','8','9','-','_'
};
static unsigned make_temp_name_count, make_temp_name_count_initialized_p;
/* Value is a temporary file name starting with PREFIX, a string.
The Emacs process number forms part of the result, so there is
no danger of generating a name being used by another process.
In addition, this function makes an attempt to choose a name
which has no existing file. To make this work, PREFIX should be
an absolute file name.
BASE64_P means add the pid as 3 characters in base64
encoding. In this case, 6 characters will be added to PREFIX to
form the file name. Otherwise, if Emacs is running on a system
with long file names, add the pid as a decimal number.
This function signals an error if no unique file name could be
generated. */
Lisp_Object
make_temp_name (Lisp_Object prefix, bool base64_p)
{
Lisp_Object val, encoded_prefix;
ptrdiff_t len;
printmax_t pid;
char *p, *data;
char pidbuf[INT_BUFSIZE_BOUND (printmax_t)];
int pidlen;
CHECK_STRING (prefix);
/* VAL is created by adding 6 characters to PREFIX. The first
three are the PID of this process, in base 64, and the second
three are incremented if the file already exists. This ensures
262144 unique file names per PID per PREFIX. */
pid = getpid ();
if (base64_p)
{
pidbuf[0] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidbuf[1] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidbuf[2] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidlen = 3;
}
else
{
#ifdef HAVE_LONG_FILE_NAMES
pidlen = sprintf (pidbuf, "%"pMd, pid);
#else
pidbuf[0] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidbuf[1] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidbuf[2] = make_temp_name_tbl[pid & 63], pid >>= 6;
pidlen = 3;
#endif
}
encoded_prefix = ENCODE_FILE (prefix);
len = SBYTES (encoded_prefix);
val = make_uninit_string (len + 3 + pidlen);
data = SSDATA (val);
memcpy (data, SSDATA (encoded_prefix), len);
p = data + len;
memcpy (p, pidbuf, pidlen);
p += pidlen;
/* Here we try to minimize useless stat'ing when this function is
invoked many times successively with the same PREFIX. We achieve
this by initializing count to a random value, and incrementing it
afterwards.
We don't want make-temp-name to be called while dumping,
because then make_temp_name_count_initialized_p would get set
and then make_temp_name_count would not be set when Emacs starts. */
if (!make_temp_name_count_initialized_p)
{
make_temp_name_count = time (NULL);
make_temp_name_count_initialized_p = 1;
}
while (1)
{
unsigned num = make_temp_name_count;
p[0] = make_temp_name_tbl[num & 63], num >>= 6;
p[1] = make_temp_name_tbl[num & 63], num >>= 6;
p[2] = make_temp_name_tbl[num & 63], num >>= 6;
/* Poor man's congruential RN generator. Replace with
++make_temp_name_count for debugging. */
make_temp_name_count += 25229;
make_temp_name_count %= 225307;
if (!check_existing (data))
{
/* We want to return only if errno is ENOENT. */
if (errno == ENOENT)
return DECODE_FILE (val);
else
/* The error here is dubious, but there is little else we
can do. The alternatives are to return nil, which is
as bad as (and in many cases worse than) throwing the
error, or to ignore the error, which will likely result
in looping through 225307 stat's, which is not only
dog-slow, but also useless since eventually nil would
have to be returned anyway. */
report_file_error ("Cannot create temporary name for prefix",
prefix);
/* not reached */
}
}
}
DEFUN ("make-temp-name", Fmake_temp_name, Smake_temp_name, 1, 1, 0,
doc: /* Generate temporary file name (string) starting with PREFIX (a string).
The Emacs process number forms part of the result, so there is no
danger of generating a name being used by another Emacs process
\(so long as only a single host can access the containing directory...).
This function tries to choose a name that has no existing file.
For this to work, PREFIX should be an absolute file name.
There is a race condition between calling make-temp-name' and creating the
file, which opens all kinds of security holes. For that reason, you should
normally use make-temp-file' instead. */)
(Lisp_Object prefix)
{
return make_temp_name (prefix, 0);
}
The generated file name is therefore a combination of the prefix, the Emacs PID and three characters from the above table. This makes about 200.000 possible temporary files that can be generated with a given prefix in an Emacs session. This range can be traversed in a negligible amount of time to recreate the state of the RNG and accurately predict the next temporary file name.
(defun make-temp-file (prefix &optional dir-flag suffix)
"Create a temporary file.
The returned file name (created by appending some random characters at the end
of PREFIX, and expanding against temporary-file-directory' if necessary),
is guaranteed to point to a newly created empty file.
You can then use write-region' to write new data into the file.
If DIR-FLAG is non-nil, create a new empty directory instead of a file.
If SUFFIX is non-nil, add that at the end of the file name."
;; Create temp files with strict access rights. It's easy to
;; loosen them later, whereas it's impossible to close the
;; time-window of loose permissions otherwise.
(with-file-modes ?\700
(let (file)
(while (condition-case ()
(progn
(setq file
(make-temp-name
(if (zerop (length prefix))
(file-name-as-directory
temporary-file-directory)
(expand-file-name prefix
temporary-file-directory))))
(if suffix
(setq file (concat file suffix)))
(if dir-flag
(make-directory file)
(write-region "" nil file nil 'silent nil 'excl))
nil)
;; make-temp-name' and write-region', let's try again.
-1:-- make-temp-name (Post Vasilij Schneidermann)--L0--C0--August 13, 2017 06:37 PM | 2017-10-18 07:09:28 | {"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": 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": 0.48115086555480957, "perplexity": 5828.640233523607}, "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-2017-43/segments/1508187822822.66/warc/CC-MAIN-20171018070528-20171018090528-00004.warc.gz"} |
https://quicklore.com/aptitude/problems-on-age | ## Aptitude question and answer on Problems on age
1 Santosh is four times as old as his son. four years later the sum of their ages will be 43 years. The present age of son is A. 5 years B. 7 years C. 8 years D. 10 years Answer : B. 7 years Explanation: Let son's age = x then, Santosh's age =4x $\dot{..}$ (x+4)+(4x+4)=43 x=7 $\dot{..}$Present age of the son = 7 years. Discuss 2 Ten years ago, P was half of Q's age. If the ratio of their present ages is 3:4, what will be the total of their present ages? A. 30 B. 35 C. 40 D. 45 Answer : B. 35 Explanation: Let the present age of P and Q be 3x and 4x respectively. Ten years ago, P was half of Q's age => (3x – 10) = $1\over2$(4x – 10) => 6x – 20 = 4x – 10 => 2x = 10 => x = 5 total of their present ages = 3x + 4x = 7x = 7 × 5 = 35 Discuss 3 A man's age is 125% of what it was 10 years ago, but $83\dfrac{1}{3}\%$ of what it will be after 10 years. What is his present age? A. 40 B. 50 C. 60 D. 70 Answer : B. 50 Explanation: Let the age before 10 years = x Then $125x \over100$ = x + 10 =>125x = 100x + 1000 => x = $1000 \over 25$ = 40 Present age = x + 10 = 40 +10 = 50 Discuss 4 Present ages of Kiran and Syam are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Syam's present age in years ? A. 24 B. 25 C. 26 D. 27 Answer : A. 24 Explanation: Ratio of the present age of Kiran and Syam = 5 : 4 Let the present age of Kiran = 5x Present age of Syam = 4x After 3 years, ratio of their ages = 11:9 => (5x + 3) : (4x + 3) = 11 : 9 => 9(5x + 3) = 11(4x + 3) => 45x + 27 = 44x + 33 => x = 33-27 =6 Syam's present age = 4x = 4×6 = 24 Discuss 5 Five years ago, the total of the ages of a father and his son was 40 years. The ratio of their present ages is 4 : 1. What is the present age of the father ? A. 30 years B. 20 years C. 25 years D. None of these Answer : D. None of these Explanation: Let son's age = x. Then father's age = 4x $(x-5)+(4x-5)=40$ $⇒x = 10$ Present age of father = 40 years Discuss 6 One year ago Jaya was four times as old as her daughter Nikitha. Six years hence, Mrs.Jaya's age will exceed her daughter's age by 9 years. The ratio of the present ages of Jaya and her daughter is : A. 9 : 2 B. 11: 3 C. 12: 5 D. 13: 4 Answer : D. 13: 4 Explanation: Let Nikitha's age 1 year ago = x Then Jaya's age 1 year ago = 4x After 6 years their ages are $4x+7, x+7$ $(4x+7) - (x+7) = 9$ or $x = 3$ Present age of Jaya = (12+1) years = 13 years Present age of Nikitha = (3+1) years = 4 years Ratio of their ages = 13 : 4 Discuss 7 The sum of the ages of a mother and daughter is 50 years. Also, 5 years ago, the mother's age was 7 times the age of the daughter. The present ages of the mother and daughter respectively are : A. 35 years, 15 years B. 38 years, 12 years C. 40 years, 10 years D. 42 years, 8 years Answer : C. 40 years, 10 years Explanation: Let the daughter's present age be x years. Then, mother's present age = (50-x) years. Now, $7(x-5)=(50-x-5)$ or $x = 10$ So, their present ages are 40 years and 10 years. Discuss 8 The sum of the ages of a son and father is 56 years. After 4 years, the age of the father will be three times that of the son. Their ages respectively are: A. 12 years, 44 years B. 16 years, 42 years C. 16 years, 48 years D. 18 years, 36 years Answer : A. 12 years, 44 years Explanation: Let the present ages of son and father be x years and $(56-x)$ years respectively. Then $(56-x+4)=3(x+4)$ or 4x =48 or x = 12 So their ages are 12 years, 44 years respectively. Discuss | 2020-02-25 14:31:34 | {"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": 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": 0.5099106431007385, "perplexity": 3121.35474341237}, "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-2020-10/segments/1581875146123.78/warc/CC-MAIN-20200225141345-20200225171345-00186.warc.gz"} |
https://www.cs.uaf.edu/2013/spring/cs321/lectures/lec-2013_03_01-mem_virt.html | CS 321 Spring 2013 > Lecture Notes for Friday, March 1, 2013
# CS 321 Spring 2013 Lecture Notes for Friday, March 1, 2013
## Dynamic Allocation (cont’d)
See the February 27 lecture notes.
## Virtual Memory
### Ideas
In modern virtual memory, logical memory is divided into equal-sized pages. Traditionally a page is 4K, but this may vary. Memory is allotted to a process, swapped out to disk, marked as copy-on-write, etc., individually for each page.
Pages are allotted to processes using memory mapping, which means establishing a correspondence between three things:
• A logical memory location
• A physical memory location
• A location in storage (e.g., in a file)
Memory mapping is a very versatile facility. In particular, once we are able to establish the above correspondence, all we need to do is give a name to the file, and we can do memory-mapped I/O: treating the contents of a file as an array of bytes, so that read and writing the file can be done simply by reading and writing memory in the usual way.
In modern virtual memory, address translation is done by means of a page table. This stores information on each page. For example, it might store the page’s permissions, whether it is present in memory, does it correspond to a named file, what is its copy-on-write status, etc.
A page table could be simply an array with one item for each page. However, as memory sizes grow, this does not scale well. A multi-level page table stores the table as an array of pointers to arrays. If one of the second-level arrays is not used, then it does not need to be allocated. An inverted page table stores only entries for allocated pages.
All of these methods reduce the space required by the page table, but increase the time required to access it. To speed this up, some entries in the page table are cached for fast access, in the translation lookaside buffer (TLB).
When the processor accesses a memory location, we would like the page’s entry to be in the TLB. If it is not, but is in memory, then we have a soft miss. If the page is not in memory at all, then we have a hard miss, also called a page fault; the page must then be loaded from disk.
### Using mmap
The *ix system call that does memory mapping is mmap. This call, along with its companion munmap and associated constants, is declared in the header <sys/mman.h>.
An mmap call is always considered to establish a correspondence with a file. When we want some plain old memory—i.e., when we are not doing memory-mapped I/O—we map to an “unnamed file”. (The documentation says this unnamed file is /dev/zero. That strikes me as a meaningless statement; I have no idea why the documentation says that.)
Function mmap has 6 parameters.
A (void *) giving the logical address to use. This should be a multiple of the page size. By default this is allowed to be ignored. Make it zero if you do not care (and you almost always do not care). See the file & sharing flags for more information.
Size
A size_t telling how much memory to map.
Access flags
The permissions requested for the mapped memory. This should be the bitwise-OR of one or more of the following: PROT_READ, PROT_WRITE, PROT_EXEC.
File & sharing flags
The bitwise-OR of various flags. You must include exactly one of the following two: MAP_SHARED, MAP_PRIVATE. These are strangely named; MAP_SHARED is normal operation, while MAP_PRIVATE establishes a copy-on-write mapping. Include MAP_ANONYMOUS if you are not mapping a named file. Include MAP_FIXED if you wish to specify the logical address for the mapping, using the first parameter (this is not recommended).
File descriptor
If you are mapping a named file, then this should be the file descriptor for an open file. Otherwise, make this (-1).
Offset
The byte offset in the file at which you want the mapping to start. This should be a multiple of the page size. Make it zero when using an unnamed file.
The return value of mmap is a (void *) giving the logical address of the mapped memory, or the special value MAP_FAILED if the call failed.
So a “normal” call to mmap, using an unnamed file, might look something like this.
[C++]
const size_t MEMSIZE = 1000;
char * p = (char *)mmap(0, MEMSIZE,
MAP_SHARED | MAP_ANONYMOUS,
-1, 0);
if (p == MAP_FAILED)
{
...
Function munmap deletes a mapping created by mmap. It has two parameters: address (the value returned by mmap) and size (the second parameter of mmap). It returns zero on success, nonzero on failure.
[C++]
int failure = munmap(p, MEMSIZE);
if (failure)
{
cout << "munmap failed and I have no idea what to do";
...
We can do memory-mapped I/O by mapping a named file. We must first open the file (with the open system call). On success this call returns a file descriptor, which we pass to mmap. We also leave MAP_ANONYMOUS out of the file & sharing flags.
The call to open should request the permissions (read, write) needed to do the mmap call. For example, if the call to mmap requests write access to the mapped memory, then the call to open should request write access to the file.
If the open and the map both succeed, then we must break the map (with munmap) and then close the file (with close). Thus, if all goes well, we do the following sequence of operations.
• Call open.
• Call mmap.
• Read or write the mapped memory as appropriate.
• Call munmap.
• Call close.
The result is a read or write to the file, without ever using a read or write system call.
[C++]
int fd = open("myfile.txt", O_RDONLY);
if (fd == -1)
{
// ERROR
...
return;
}
size_t MEMSIZE = 10000;
off_t offset = 4096;
char * p = (char *)mmap(0, MEMSIZE,
MAP_SHARED,
fd, OFFSET);
if (p == MAP_FAILED)
{
// ERROR
...
return;
}
// Do some I/O by treating p as an array of 10,000 bytes.
// And then eventually:
int failure = munmap(p, MEMSIZE);
...
close(fd);
CS 321 Spring 2013: Lecture Notes for Friday, March 1, 2013 / Updated: 6 May 2013 / Glenn G. Chappell / ggchappell@alaska.edu | 2019-02-19 20:40:43 | {"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": 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": 0.31713518500328064, "perplexity": 2947.3715552003473}, "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-2019-09/segments/1550247492825.22/warc/CC-MAIN-20190219203410-20190219225410-00132.warc.gz"} |
https://math.stackexchange.com/questions/linked/237779 | 11k views
### $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate]
assume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance
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### Proof If $AB-I$ Invertible then $BA-I$ invertible. [duplicate]
I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
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### Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ [duplicate]
I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
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### $I_m -AB$ ivertible if and only if $I_n-BA$ invertible
Let $A$ and $B$ be $m\times n$ and $n\times m$ matrices respectively. Prove that if $\lambda$ is a non-zero eigenvalue of $AB$ then it is also an eigenvalue of $BA$ Prove that $I_m-AB$ is ... | 2019-07-21 02:38:35 | {"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": 0.9668839573860168, "perplexity": 166.9921659829024}, "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-2019-30/segments/1563195526818.17/warc/CC-MAIN-20190721020230-20190721042230-00464.warc.gz"} |
http://nrich.maths.org/880/solution | ### Cutting a Cube
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
# Covering Cups
##### Stage: 3 Challenge Level:
Many thanks to Andrei of School 205 Bucharest for the inspiration for this solution. Well done Andrei. One might have expected the triangular solution to be best. I wonder what happens if you chop off the corners?
Using the following notation:
$h$ - for the height of the cup and
$d$ - for the diameter of the cup.
Test each of the two possible forms of the box, one with a rectangular base and one with a triangle as its base:
Cuboidal box
As there are six cups they could be ordered in two ways:
$1 \times 6$ cups
and
$2 \times 3$ cups
For any cuboid box, the surface area is: 2 times the top area + 2 times the side area + 2 times the front area:
The total area of a $1 \times 6$ cups box is:
$2 \times(h \times 6 \times d + d \times 6 \times d + d \times h) = 185470$ mm$^2$
The total area of a $2 \times3$ cups box is:
$2 \times(h \times2 \times d + h \times3 \times d + 2 \times d \times3 \times h) = 157250$ mm$^2$
For a triangular prism box
This case is illustrated in the figure of the problem.
The base is an equilateral triangle. The surface area is 2 times the area of the equilateral triangle + 3 times the area of each of the faces.
First calculate the side of the equilateral triangle $ABC$.
This is $2 \times BM + 2 \times85$ mm.
$BM$ is a tangent to the circumference of a corner cup (centre $O$, radius $d/2 = 85/2$ mm).
Also the angle $MOB = 60^{\circ}$
Therefore angle $MBO = 30^{\circ}$
\eqalign{ BM &= \frac{85}{2}\tan60^{\circ} \\ \; &= \frac{85}{2}\sqrt{3} \\ \; &= \frac{85\sqrt{3}}{2}\\ \mbox{The side of triangle ABC is} &= 2 \times d + 2 \times BM \\ \; &= 2 \times85 + 2 \times\frac{85\sqrt{3}}{2} \\ \; &= 85(2 + \sqrt{3}) \\ \mbox{The altitude of the equilateral triangle ABC is} &= AB \times\sin60^{\circ} \\ \; &= 85(2 + \sqrt{3}) \times\frac{\sqrt{3}}{2} \\ \; &= \frac{85\sqrt{3}(2 + \sqrt{3})}{2} \\ \mbox{Area of the two equilateral triangles} &= 85(2 + \sqrt{3}) \times\frac{85\sqrt{3}(2 + \sqrt{3})}{2} \\ \; &= 87149 \; \mbox{mm}^2 \\ \mbox{Area of the three faces} &= 3 \times83 \times85(2 + \sqrt{3}) \\ \; &= 78989 \; \mbox{mm}^2 \\ \mbox{The total surface area is } &= 166138 \; \mbox{mm}^2}
So the best box is a box of 2 x 3 cups. | 2015-05-29 08:29:01 | {"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": 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": 1.0000100135803223, "perplexity": 1482.2020791996965}, "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-2015-22/segments/1432207929956.54/warc/CC-MAIN-20150521113209-00065-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://escholarship.org/uc/item/64c311g4 | Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$
Open Access Publications from the University of California
## Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$
• Author(s): Golenishcheva-Kutuzova, T
• Gorodetski, A
• Kleptsyn, V
• Volk, D
• et al.
Abstract
We consider a minimal action of a finitely generated semigroup by homeomorphisms of a circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov's theorem and its corollaries.
Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you. | 2019-07-24 00:02:19 | {"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": 0.5565201044082642, "perplexity": 1657.6795548570876}, "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-2019-30/segments/1563195530246.91/warc/CC-MAIN-20190723235815-20190724021815-00370.warc.gz"} |
https://www.festadoavante.pcp.pt/2019/desporto | # Desporto
Warning: Missing argument 7 for lista(), called in /home/festa/public_html/2019/theme/templates/tag.php on line 74 and defined in /home/festa/public_html/2019/theme/templates/lista.php on line 15 | 2019-05-21 03:05:01 | {"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": 0.8714316487312317, "perplexity": 9041.261263925211}, "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-2019-22/segments/1558232256215.47/warc/CC-MAIN-20190521022141-20190521044141-00033.warc.gz"} |
https://tutorial.math.lamar.edu/Classes/DE/VibratingString.aspx | • Go To
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• Practice and Assignment problems are not yet written. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here.
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### Section 9-8 : Vibrating String
This will be the final partial differential equation that we’ll be solving in this chapter. In this section we’ll be solving the 1-D wave equation to determine the displacement of a vibrating string. There really isn’t much in the way of introduction to do here so let’s just jump straight into the example.
Example 1 Find a solution to the following partial differential equation.
\begin{align*}& \frac{{{\partial ^2}u}}{{\partial {t^2}}} = {c^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\\ & u\left( {x,0} \right) = f\left( x \right)\hspace{0.25in}\frac{{\partial u}}{{\partial t}}\left( {x,0} \right) = g\left( x \right)\\ & u\left( {0,t} \right) = 0\hspace{0.25in}u\left( {L,t} \right) = 0\end{align*}
Show Solution
One of the main differences here that we’re going to have to deal with is the fact that we’ve now got two initial conditions. That is not something we’ve seen to this point but will not be all that difficult to deal with when the time rolls around.
We’ve already done the separation of variables for this problem, but let’s go ahead and redo it here so we can say we’ve got another problem almost completely worked out.
So, let’s start off with the product solution.
$u\left( {x,t} \right) = \varphi \left( x \right)h\left( t \right)$
Plugging this into the two boundary conditions gives,
$\varphi \left( 0 \right) = 0\hspace{0.25in}\varphi \left( L \right) = 0$
Plugging the product solution into the differential equation, separating and introducing a separation constant gives,
\begin{align*}\frac{{{\partial ^2}}}{{\partial {t^2}}}\left( {\varphi \left( x \right)h\left( t \right)} \right) & = {c^2}\frac{{{\partial ^2}}}{{\partial {x^2}}}\left( {\varphi \left( x \right)h\left( t \right)} \right)\\ \varphi \left( x \right)\frac{{{d^2}h}}{{d{t^2}}} & = {c^2}h\left( t \right)\frac{{{d^2}\varphi }}{{d{x^2}}}\\ \frac{1}{{{c^2}h}}\frac{{{d^2}h}}{{d{t^2}}} & = \frac{1}{\varphi }\frac{{{d^2}\varphi }}{{d{x^2}}} = - \lambda \end{align*}
We moved the $${c^2}$$ to the left side for convenience and chose $$- \lambda$$ for the separation constant so the differential equation for $$\varphi$$ would match a known (and solved) case.
The two ordinary differential equations we get from separation of variables are then,
\begin{align*}\frac{{{d^2}h}}{{d{t^2}}} + {c^2}\lambda h = 0\hspace{0.25in} & \frac{{{d^2}\varphi }}{{d{x^2}}} + \lambda \varphi=0 \\ & \varphi \left( 0 \right) = 0\hspace{0.25in}\varphi \left( L \right) = 0\end{align*}
We solved the boundary value problem above in Example 1 of the Solving the Heat Equation section of this chapter and so the eigenvalues and eigenfunctions for this problem are,
${\lambda _{\,n}} = {\left( {\frac{{n\pi }}{L}} \right)^2}\hspace{0.25in}{\varphi _n}\left( x \right) = \sin \left( {\frac{{n\,\pi \,x}}{L}} \right)\hspace{0.25in}n = 1,2,3, \ldots$
The first ordinary differential equation is now,
$\frac{{{d^2}h}}{{d{t^2}}} + {\left( {\frac{{n\pi c}}{L}} \right)^2}h = 0$
and because the coefficient of the $$h$$ is clearly positive the solution to this is,
$h\left( t \right) = {c_1}\cos \left( {\frac{{n\pi c\,t}}{L}} \right) + {c_2}\sin \left( {\frac{{n\pi c\,t}}{L}} \right)$
Because there is no reason to think that either of the coefficients above are zero we then get two product solutions,
$\begin{array}{*{20}{c}}{{u_n}\left( {x,t} \right) = {A_n}\cos \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)}\\{{u_n}\left( {x,t} \right) = {B_n}\sin \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)}\end{array}\hspace{0.25in}n = 1,2,3, \ldots$
The solution is then,
$u\left( {x,t} \right) = \sum\limits_{n = 1}^\infty {\left[ {{A_n}\cos \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right) + {B_n}\sin \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)} \right]}$
Now, in order to apply the second initial condition we’ll need to differentiate this with respect to $$t$$ so,
$\frac{{\partial u}}{{\partial t}} = \sum\limits_{n = 1}^\infty {\left[ { - \frac{{n\pi c}}{L}{A_n}\sin \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right) + \frac{{n\pi c}}{L}{B_n}\cos \left( {\frac{{n\pi c\,t}}{L}} \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)} \right]}$
If we now apply the initial conditions we get,
\begin{align*}& u\left( {x,0} \right) = f\left( x \right) = \sum\limits_{n = 1}^\infty {\left[ {{A_n}\cos \left( 0 \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right) + {B_n}\sin \left( 0 \right)\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)} \right]} = \sum\limits_{n = 1}^\infty {{A_n}\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)} \\ & \frac{{\partial u}}{{\partial t}}\left( {x,0} \right) = g\left( x \right) = \sum\limits_{n = 1}^\infty {\frac{{n\pi c}}{L}{B_n}\sin \left( {\frac{{n\,\pi \,x}}{L}} \right)} \end{align*}
Both of these are Fourier sine series. The first is for $$f\left( x \right)$$ on $$0 \le x \le L$$ while the second is for $$g\left( x \right)$$ on $$0 \le x \le L$$ with a slightly messy coefficient. As in the last few sections we’re faced with the choice of either using the orthogonality of the sines to derive formulas for $${A_n}$$ and $${B_n}$$ or we could reuse formula from previous work.
It’s easier to reuse formulas so using the formulas form the Fourier sine series section we get,
\begin{align*}{A_{\,n}} & = \frac{2}{L}\int_{{\,0}}^{{\,L}}{{f\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}}\,\,\,\,\,\,\,n = 1,2,3, \ldots \\ \frac{{n\pi c}}{L}{B_{\,n}} & = \frac{2}{L}\int_{{\,0}}^{{\,L}}{{g\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}}\,\,\,\,\,\,\,n = 1,2,3, \ldots \end{align*}
Upon solving the second one we get,
\begin{align*}{A_{\,n}} & = \frac{2}{L}\int_{{\,0}}^{{\,L}}{{f\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}}\,\,\,\,\,\,\,n = 1,2,3, \ldots \\ {B_{\,n}} & = \frac{2}{{n\pi c}}\int_{{\,0}}^{{\,L}}{{g\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}}\,\,\,\,\,\,\,n = 1,2,3, \ldots \end{align*}
So, there is the solution to the 1-D wave equation and with that we’ve solved the final partial differential equation in this chapter. | 2020-08-11 16:46:10 | {"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": 0.9976648092269897, "perplexity": 1036.500290975291}, "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/1596439738816.7/warc/CC-MAIN-20200811150134-20200811180134-00316.warc.gz"} |
http://math.stackexchange.com/questions/270757/a-multiple-choice-question-on-uniformly-continuous-function | # a multiple choice question on uniformly continuous function
Let $f :(0, 1) \to \mathbb{R}$ be continuous. Pick out the statements which imply that $f$ is uniformly continuous.
a. $|f (x) − f (y)| ≤ \sqrt{|x − y|}$, for all $x, y ∈ (0, 1)$.
b. $f\left(\frac{1}{n}\right) \to \frac{1}{2}$ and $f\left(\frac{1}{n^2}\right) \to \frac{1}{4}$.
c. $f(x) = x ^{\frac{1}{2}}\sin\left(\frac{1}{x^3}\right)$.
My thoughts:
(c) is uniformly continuous as limits at $x=0$ and $x=1$ are exists.
(a) and (b) I am not sure.
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Do you know the difference between continuous and uniformly continuous? – Erick Wong Jan 5 '13 at 7:27
(b) Consider $\left\vert f\left(\frac1{n^2-1}\right)-f\left(\frac1{n^2}\right)\right\vert$. Note that $\left\vert \frac1{n^2-1}-\frac1{n^2}\right\vert$ will be smaller than any prespecified $\delta>0$ when $n$ is large enough.
Given that $f(1/n^2)$ is a subsequence of $f(1/n)$, I'm tempted to say the contradiction in (b) vacuously implies that $f$ is uniformly continuous. – Erick Wong Jan 5 '13 at 18:12
@ErickWong Absolutely agree! Yet I think part (b) was intened to mean that $f(1/n)\to1/2$ for the sequence of nonperfect squares. – user1551 Jan 5 '13 at 18:43 | 2014-04-24 03:56:50 | {"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": 0.881435215473175, "perplexity": 536.9750691928675}, "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-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/891575/3-ants-going-at-different-speeds-when-they-will-be-at-the-same-place-motion-pro | # 3 Ants going at different speeds, when they will be at the same place Motion Problem
The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per second, ant V 5 centimeters, and and C 10 centimeters. How long does it take for the three ants to arrive at the same spot for the first time?
I tried making a list and writing down the numbers, but they seem to never be the same. I know the distance formula is d=rt, but I don't know how to use it to solve this problem. Any help? Thanks!
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Hint. Suppose that the ants meet after $t$ seconds, and measure distance around the circle from where C starts. Then A has travelled $3t$ metres, but had a $60$ metre start for a total of $60+3t$ from the initial point. Likewise B will be a distance 30+5t from the initial point. However B may have travelled a number of times around the circle, say $x$ times more than A, and therefore has travelled $90x$ metres further than A. So we have the equation $$60+3t+90x=30+5t\ .$$ See if you can explain by using similar ideas why $$60+3t+90y=10t\ ,$$ where $y$ is the number of times C has "lapped" A.
Now eliminate $t$ from these two equations; then find the smallest possible values of $x$ and $y$, remembering that while $t$ could be any positive number, $x$ and $y$ must be positive integers.
Good luck!
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At the start, C is at location 0, B is at location 30 and A is at location 60. It is easy to see that C will catch up to B in 6 seconds at location 60. And then they will meet again at location 60 every 18 seconds after that. When will A be at location 60? Well, he starts out there and he gets back there every 30 seconds.
So, B and C will be at the right place in 6, 24, 42, 60, 78 etc. seconds while A will be at the right place in 30, 60, 90 etc. seconds. We now see that they meet for the first time after 60 seconds.
To help verify that this is correct you can see that 0 + 10 * 60 = 600; 30 + 5 * 60 = 330; 60 + 3 * 60 = 240. It is easy to see that 600, 330 and 240 all refer to the same spot on the circle. Formally expressed 600 mod 90 = 330 mod 90 = 240 mod 90 = 60.
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First focus on A and B. Since B goes 2 cm/sec faster than A, B first catches A after 15 seconds; and then every 45 seconds thereafter.
Now look at B and C. Since C goes 5 cm/sec faster than B, C first catches B after 6 seconds, and every 18 seconds there after.
So A and B coincide at times $15 + 45n$ seconds for $n = 0, 1, ...$; and B and C coincide at times $6 + 18k$ seconds for $k = 0, 1, 2,...$.
So you need $15 + 45n=6 + 18k$ for nonnegative integer $n,k$.
The smallest solution is $n=1, k=3$. Giving a time of $60$ seconds.
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Suppose, without affecting the result of the problem, that ant $A$ starts at the polar coordinates of angle $0$ on the unit circle, the other ants will be at position $2\frac{\pi}{3}$ and $4\frac{\pi }{3}$. Let's try to find three functions, $f$, $g$, $h$, that takes $t$ (time) as argument, and yields the angle at which ant $A$, $B$, and $C$ will be at the given $t$ moment, respectively. Since the speed is constant, the three functions will be linear.
We know that:
$f(0)=0$
Since the ant $A$ will walk $3$($\frac{90}{30}$) centimeters in $1$ second, it will walk $\frac{1}{30}$ of the circle in $1$ second, so the new angle will decrease by $2\frac{ \pi}{30}$.
$f(1)=-\frac{\pi}{15}$
$f(t)=at+b$
$f(0)=b=0$
$f(1)=a=-\frac{\pi}{15}=>a=-\frac{\pi}{15}$
Similarly, we can compute for $g$ and $h$.
$f(t)=-\frac{\pi}{15}t$
$g(t)=-\frac{\pi}{9}t+2\frac{\pi}{3}$
$h(t)=-2\frac{\pi}{9}t+4\frac{\pi}{3}$
Now, it only remains to solve the equation(for any $n$, $m$).
$f(t)=g(t)+2n\pi =h(t)+2m\pi$
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Expressing the position by the angle doesn't make anything simpler and fills your calculation with fractions of $\pi$, making it more likely that you'll make mistakes. – David Richerby Aug 9 '14 at 10:55
Another way of simplifying the problem:
A is at 4 o'clock, B is at 12 o'clock, and C is at 8 o'clock.
Let's subtract $5$ cm/s from each ant's speed. So now C is going clockwise at only $5$ cm/s, B is stationary, and A is going counter-clockwise at $2$ cm/s. And they're all still $30$ cm apart. Subtracting the same amount from each ant's speed doesn't change the solution to the problem: in physics terms, it's equivalent to working in ant B's frame of reference, instead of the circle's.
C will reach B after $\frac{30}{5}$ sec, and thereafter every $\frac{90}{5}$ sec, or after $6, 24, 42, 60, 78$, ... seconds
A will reach B after $\frac{30}{2}$ sec, and thereafter every $\frac{90}{2}$ sec, or after $15, 60, 105$, ... seconds
The first double hit is after 60 seconds...
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I suggest you use the angle as the ant's position indicator. Note that $$\theta = \frac{s}{r} = \frac{s}{{\frac{{90}}{{2\pi }}}} = \frac{{2\pi s}}{{90}}$$Now:$$\begin{array}{l}{\theta _{\rm{A}}}(t) = {\omega _{\rm{A}}}t + {\theta _{{{\rm{A}}_0}}}\\{\theta _{\rm{B}}}(t) = {\omega _{\rm{B}}}t + {\theta _{{{\rm{B}}_0}}}\\{\theta _{\rm{C}}}(t) = {\omega _{\rm{C}}}t + {\theta _{{{\rm{C}}_0}}}\end{array}$$and$$\begin{array}{l}{\theta _{{{\rm{A}}_0}}} = - \frac{{4\pi }}{3}\\{\theta _{{{\rm{B}}_0}}} = - \frac{{2\pi }}{3}\\{\theta _{{{\rm{C}}_0}}} = 0\end{array}$$where$$\begin{array}{l}{\omega _{\rm{A}}} = \frac{{6\pi }}{{90}}\\{\omega _{\rm{B}}} = \frac{{10\pi }}{{90}}\\{\omega _{\rm{C}}} = \frac{{20\pi }}{{90}}\end{array}$$Your question could be re-written as find a $t$ such that$$\begin{array}{l}{\theta _{\rm{C}}}(t) - {\theta _{\rm{B}}}(t)\mathop \equiv \limits^{2\pi } 0\\{\theta _{\rm{C}}}(t) - {\theta _{\rm{A}}}(t)\mathop \equiv \limits^{2\pi } 0\end{array}$$Replacing the data and simplifying yields$$\begin{array}{l}\frac{{\pi t + 6\pi }}{9}\mathop \equiv \limits^{2\pi } 0\\\frac{{7\pi t + 60\pi}}{{45}}\mathop \equiv \limits^{2\pi } 0\end{array}$$ which is equivalent to $$\begin{array}{l}t + 6\mathop \equiv \limits^{18} 0\\7t + 60\mathop \equiv \limits^{90} 0\end{array}$$Combining the second equation with the first, yields $$\begin{array}{l}t\mathop \equiv \limits^{18} 12\\t\mathop \equiv \limits^{30} 0\end{array}$$The first solution to these is $t=30$.
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That makes it seems harder though...what's the the greek alphavet for? I still gave an upvote for answering though. – Anonymous Aug 9 '14 at 4:04
Expressing the position by the angle doesn't make anything simpler and fills your calculation with fractions of $\pi$, making it more likely that you'll make mistakes. – David Richerby Aug 9 '14 at 10:55
@Anonymous Using $\theta$ for an angle and $\omega$ for an angular velocity is as standard as using $t$ for time or $r$ for radius. – David Richerby Aug 9 '14 at 12:56 | 2015-01-26 18:44:27 | {"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": 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": 0.7938206195831299, "perplexity": 398.4263833131683}, "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-2015-06/segments/1422115862015.5/warc/CC-MAIN-20150124161102-00168-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/80492/a-technical-problem-on-the-contragredient-representation-in-the-context-of-locall | # A technical problem on the contragredient representation in the context of locally compact totally disconnected groups
Let $\pi$ be an admissible representation of a locally compact totally disconnected group. I have a technical problem about the proof of
$\pi$ is irreducible if and only if its contragredient is so
given in 2.15(c) of the '76 article of Bernstein and Zelevinsky. There $\pi$ is assumed to have a nontrivial proper subrepresentation $E_1$, and it is asserted that the orthogonal complement of $E_1$ be a nontrivial proper subrepresentation of the contragredient, whence the result. What I cannot figure out is the nontriviality of this orthogonal complement. We simply have to find a nonzero smooth functional which vanishes on $E_1$; this shall follow from $E_1\neq E$ (as properness of the orthogonal complement follows from $E_1\neq 0$), but how?
-
The paper seems to be: numdam.org/item?id=ASENS_1977_4_10_4_441_0 But I cannot find 2.15(c)?? – Matthew Daws Nov 9 '11 at 19:56
That's the wrong B-Z paper. The right one is math.tau.ac.il/~bernstei/Publication_list/publication_texts/… – Faisal Nov 9 '11 at 20:05
@Faisal: Thanks! The spelling of Zelevinsky (or ..ii) defeated MathSciNet... – Matthew Daws Nov 9 '11 at 20:18
No problem. When I started learning about the rep theory of p-adic groups I was told that a good basic reference is "the paper by Bernstein and Zelevinsky". Like you, I thought that meant archive.numdam.org/ARCHIVE/ASENS/ASENS_1977_4_10_4/… because it was the first thing that showed up in my search. I was really confused until I realized that there was another (more appropriate) B-Z paper. – Faisal Nov 9 '11 at 20:27
This follows from two facts:
1. The complement $E_1^\perp$ of $E_1$ in $\tilde{E}$ is isomorphic to the contragredient of $E/E_1$.
2. If $V$ is admissible and nonzero then $\tilde{V}$ is nonzero (and admissible). For if $\tilde{V}=0$ then $V = \tilde{\tilde{V}} = 0$.
-
If I understand the question, the worry is about algebraic (=smooth) vectors-- here $x\in E$ is smooth if the stabilizer of $x$ for the $\pi$ action is an open subgroup of $G$. Could you say some words about this?? – Matthew Daws Nov 9 '11 at 19:57
The contragredient is by definition the "smooth dual" representation (i.e. the subrep of the dual rep on the subspace of smooth vectors). – Faisal Nov 9 '11 at 20:04
In symbols, $\tilde{V} = (V^\ast)^\infty$. – Faisal Nov 9 '11 at 20:14
@Faisel: Sure! But I think the original question wanted to know why there is an algebraic vector in the perp of $E_1$. That is, your answer seemed a bit brief, given what the original question asked (so I think maybe it won't be easy to understand). However, on a close reading of the original question, I see that $\pi$ is assumed "admissible". I personally don't understand what implications this has, but it seems to imply lots of powerful things; so maybe one needs to fully understand this definition...?? – Matthew Daws Nov 9 '11 at 20:21
As far as this question is concerned, admissibility is important because it gives us the equivalence $\tilde{\tilde{\pi}}=\pi$ (which I used in (2)). This equivalence doesn't hold for general inadmissible $\pi$. – Faisal Nov 9 '11 at 20:33
If you believe that the monomorphism $\pi \to \widetilde{\widetilde{\pi}}$ is an isomorphism for admissible $\pi$, then you can see that $\pi \mapsto \widetilde{\pi}$ is a contravariant autoequivalence on the abelian category of admissible (smooth) representations. Thus it carries simple objects to simple objects.
- | 2014-08-28 15:31:59 | {"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": 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": 0.9485996961593628, "perplexity": 604.0517993072983}, "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-2014-35/segments/1408500830903.34/warc/CC-MAIN-20140820021350-00308-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Definition:Main_Connective_(Propositional_Logic) | # Definition:Main Connective/Propositional Logic
## Definition
### Definition 1
Let $\mathbf C$ be a WFF of propositional logic.
Let $\circ$ be a binary connective.
Then $\circ$ is the main connective iff the scope of $\circ$ is $\mathbf C$.
### Definition 2
Let $\mathbf C$ be a WFF of propositional logic such that:
$\mathbf C = \left({\mathbf A \circ \mathbf B}\right)$
where both $\mathbf A$ and $\mathbf B$ are both WFFs and $\circ$ is a binary connective.
Then $\circ$ is the main connective of $\mathbf C$.
Alternatively, let $\mathbf A$ be a WFF of propositional logic such that:
$\mathbf A = \neg \mathbf B$
where $\mathbf B$ is a WFF.
Then $\neg$ is the main connective of $\mathbf A$.
### Definition 3
Let $T$ be a WFF of propositional logic in the labeled tree specification.
Suppose $T$ has more than one node.
Then the label of the root of $T$ is called the main connective of $T$.
## Also known as
The main connective is sometimes also called the principal operator. | 2020-01-29 01:16:13 | {"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": 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": 0.9004908204078674, "perplexity": 427.2605836942844}, "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-2020-05/segments/1579251783621.89/warc/CC-MAIN-20200129010251-20200129040251-00009.warc.gz"} |
https://dannypsnl.me/2020-03-08-note-stlc | # NOTE: simply typed lambda calculus
Last time I introduce lambda calculus. Lambda calculus is powerful enough for computation. But it's not good enough for people, compare with below Church Numerals
$add := \lambda m. \lambda n. \lambda s. \lambda z. m\;s\;(n\;s\;z)$
people prefer just + more.
But once we introduce such fundamental operations into the system, validation would be a thing. This is the main reason to have a $\lambda \to$ system(a.k.a. simply typed lambda calculus). It gets name $\lambda \to$ is because it introduces one new type: Arrow type, represent as $T_1 \to T_2$ for any abstraction $\lambda x.M$ where $x$ has a type is $T_1$ and $M$ has a type is $T_2$. Therefore we can limit the input to a specified type, without considering how to add two Car together!
To represent this, syntax needs a little change:
term ::= terms
x variable
λx: T.term abstraction
term term application
Abstraction now can describe it's parameter type. Then we have typing rules:
$\frac{ x:T \in \Gamma }{ \Gamma \vdash x:T } \;\;\;\; T-Variable \\ \frac{ \Gamma, x:T_1 \vdash t_2: T_2 }{ \Gamma \vdash \lambda x:T_1.t_2 : T_1 \to T_2 } \;\;\;\; T-Abstraction \\ \frac{ \Gamma, t_1:T_1 \to T_2 \; \Gamma \vdash t_2: T_1 }{ \Gamma \vdash t_1 \; t_2 : T_2 } \;\;\;\; T-Application$
Here is explaination:
• T-Variable: with the premise, term $x$ binds to type $T$ in context $\Gamma$ is truth. We can make a conclusion, in context $\Gamma$, we can judge the type of $x$ is $T$.
• T-Abstraction: with the premise, with context $\Gamma$ and term $x$ binds to type $T_1$ we can judge term $t_2$ has type $T_2$. We can make a conclusion, in context $\Gamma$, we can judge the type of $\lambda x:T_1.t_2$ is $T_1 \to T_2$.
• T-Application: with the premise, with context $\Gamma$ and term $t_1$ binds to type $T_1 \to T_2$ and with context $\Gamma$ we can judge term $t_2$ has type $T_1$. We can make a conclusion, in context $\Gamma$, we can judge the type of $t_1 \; t_2$ is $T_2$.
Date: 2020-03-08 Sun 00:00 | 2023-04-02 00:05:54 | {"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": 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": 0.8648489117622375, "perplexity": 970.1489261430737}, "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/1679296950363.89/warc/CC-MAIN-20230401221921-20230402011921-00765.warc.gz"} |
https://www.hpmuseum.org/forum/thread-9195-post-80930.html | 50g Mini-Challenge: Number of positive divisors of x!
10-02-2017, 08:42 AM
Post: #21
Joe Horn Senior Member Posts: 1,506 Joined: Dec 2013
RE: 50g Mini-Challenge: Number of positive divisors of x!
(10-02-2017 06:43 AM)Gerald H Wrote:
(10-02-2017 05:30 AM)Joe Horn Wrote: I keep << -40 CF MEM DROP 0.5 WAIT TEVAL -40 SF >> assigned to a key for accurate timings.
Why do you have MEM in the programme? TEVAL does garbage collection.
Wow, so it does! I never knew that. Good to know. I've removed MEM DROP from my assignment. Thank you!
(10-02-2017 06:43 AM)Gerald H Wrote: What's the point of 0.5 WAIT?
Short answer: Because I got annoyed having to tap the assigned key very rapidly, to prevent TEVAL's result from being polluted by the system slowdown caused by the keydown event.
Longer answer: Including 0.5 WAIT is only useful if TEVAL is executed via a key assignment. Try assigning just TEVAL by itself to a key, and you'll notice that you'll get the longest timings if you press and hold down the key during the entire TEVAL, and the fastest timings if you very rapidly tap the key (releasing the key before TEVAL even has a chance to begin), and various timings in between (when the key is released after TEVAL starts but before it finishes). Since my goal was reliably similar timings, I include 0.5 WAIT in the key assignment, which gives plenty of time for the key to be released before TEVAL begins to actually begin timing.
-40 CF turns off the "ticking" clock display, which causes a system interrupt once every second, which messes up timings. -40 SF is included because having the ticking clock display is the only known workaround for the Busy Bug.
<0|ɸ|0>
-Joe-
10-02-2017, 09:00 AM
Post: #22
Gerald H Senior Member Posts: 1,414 Joined: May 2014
RE: 50g Mini-Challenge: Number of positive divisors of x!
Thank you, Joe, have modified my key assignment accordingly.
10-02-2017, 09:13 AM (This post was last modified: 10-02-2017 09:13 AM by pier4r.)
Post: #23
pier4r Senior Member Posts: 2,016 Joined: Nov 2014
RE: 50g Mini-Challenge: Number of positive divisors of x!
(10-02-2017 05:30 AM)Joe Horn Wrote: Note that you can determine such timings yourself using the 50g's TEVAL command.
Yes I know, but if someone else did already, the result is already computed. Thanks nevertheless for the input! (that lead Gerald to inform us that TEVAL activates the GC)
Wikis are great, Contribute :)
10-02-2017, 12:44 PM
Post: #24
John Keith Senior Member Posts: 451 Joined: Dec 2013
RE: 50g Mini-Challenge: Number of positive divisors of x!
(10-01-2017 03:47 PM)pier4r Wrote: John it seems that you have some knowledge of speed of some commands (considering also the other thread where you today wrote that CEIL is faster than 1 + IP ). By chance do you have some measurements in time for each command tested by you?
Well, I have been benchmarking HP calculator commands and functions since the HP-71B but everything is scribbled in various notebooks and folders. Turning my notes into machine-readable form would be a monumental task which is far from the top of my To-do list. :-)
Hopefully someone has posted this information before but I don't know where. If enough people are interested I would be happy to contribute some basic stats but a full spreadsheet or article would be beyond the time I have available for such things.
John
10-02-2017, 01:24 PM
Post: #25
Gerald H Senior Member Posts: 1,414 Joined: May 2014
RE: 50g Mini-Challenge: Number of positive divisors of x!
There is this list
http://www.hpmuseum.org/cgi-sys/cgiwrap/...i?read=700
It may be best to start a new thread in General Forum.
10-04-2017, 03:40 PM
Post: #26
Gerald H Senior Member Posts: 1,414 Joined: May 2014
RE: 50g Mini-Challenge: Number of positive divisors of x!
A more efficient version of the programme:
Size: 124.
CkSum: # 6E3Ch
Code:
:: CK1&Dispatch # FF :: ZINT 1 SWAP ZINT 0 BEGIN FPTR2 ^Prime+ 2DUP Z>= WHILE :: 2DUP ZINT 0 SWAPROT BEGIN OVER FPTR2 ^ZQUOText ROTOVER FPTR2 ^RADDext 3UNROLL FPTR2 ^DupQIsZero? UNTIL 2DROP ZINT 1 FPTR2 ^RADDext 4ROLL FPTR2 ^RMULText 3UNROLL ; REPEAT 2DROP ; ;
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User(s) browsing this thread: 1 Guest(s) | 2019-11-19 04:57:53 | {"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": 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": 0.28562772274017334, "perplexity": 10275.587349469291}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-47/segments/1573496670006.89/warc/CC-MAIN-20191119042928-20191119070928-00044.warc.gz"} |
http://math.stackexchange.com/questions/82591/is-a-connected-first-countable-space-necessarily-hausdorff/82604 | # Is a connected, first countable space necessarily Hausdorff?
I've been trying for forever to come up with a counterexample but haven't had any luck.
-
We do not remove questions just because they have been answered. To show that you are satisfied with an answer, click the checkmark next to the answer you are accepting. – Austin Mohr Nov 16 '11 at 6:21
Trivial Topology – Amitesh Datta Nov 16 '11 at 6:43
The Sierpinski Two-Point Space is connected and first countable, but not Hausdorff (indeed, not even $T_1$).
Another example: the cofinite topology on the integers is $T_1$, second countable (so first countable), connected and very non-Hausdorff for the same reason: all non-empty open sets intersect. | 2014-03-12 19:15:46 | {"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": 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": 0.8370290994644165, "perplexity": 1015.6641112371801}, "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-2014-10/segments/1394023864559/warc/CC-MAIN-20140305125104-00069-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://en.turkcewiki.org/wiki/%C2%B1 | # Plus–minus sign
(Redirected from ±)
±
Plus–minus sign
In UnicodeU+00B1 ± PLUS-MINUS SIGN (HTML ± · ±, ±, ±)
Related
See alsoU+2213 MINUS-OR-PLUS SIGN (HTML ∓ · ∓, ∓, ∓)
The plus–minus sign (also, plus or minus sign), ± is a mathematical symbol with multiple meanings.
## History
A version of the sign, including also the French word ou ("or") was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as William Oughtred's Clavis Mathematicae (1631).[4]
## Usage
### In mathematics
In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the Plus and minus signs, + or , allowing the formula to represent two values or two equations.
For example, given the equation x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations x = +3 or x = −3. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula
${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$
describing the two solutions to the quadratic equation ax2 + bx + c = 0.
Similarly, the trigonometric identity
${\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B).}$
can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with on both sides. The two copies of the ± sign in this identity must both be replaced in the same way: it is not valid to replace one of them with + and the other of them with . In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.
The minus–plus sign (also minus-or-plus sign), is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z, which can be interpreted as meaning x + y − z and/or x − y + z, but not x + y + z nor x − y − z. The upper in is considered to be associated to the + of ± (and similarly for the two lower symbols) even though there is no visual indication of the dependency. (However, the ± sign is generally preferred over the sign, so if they both appear in an equation it is safe to assume that they are linked. On the other hand, if there are two instances of the ± sign in an expression, without a , it is impossible to tell from notation alone whether the intended interpretation is as two or four distinct expressions.) The original expression can be rewritten as x ± (y − z) to avoid confusion, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
${\displaystyle \cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)}$
which represents the two equations:
{\displaystyle {\begin{aligned}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\\cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{aligned}}}
Another example is
${\displaystyle x^{3}\pm 1=(x\pm 1)\left(x^{2}\mp x+1\right)}$
A third related usage is found in this presentation of the formula for the Taylor series of the sine function:
${\displaystyle \sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \pm {\frac {1}{(2n+1)!}}x^{2n+1}+\cdots .}$
Here, the plus-or-minus sign indicates that the term may be added or subtracted, in this case depending on whether n is odd or even, the rule can be deduced from the first few terms. A more rigorous presentation of the same formula would multiply each term by a factor of (−1)n, which gives +1 when n is even and −1 when n is odd.
### In statistics
The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity together with its tolerance or its statistical margin of error.[1] For example, "5.7±0.2" may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2 standard deviations (a probability of 68.3% or 95.4% in a normal distribution).
Operations involving uncertain values should always try to preserve the uncertainty in order to avoid propagation of error. If n = a ± b, any operation of the form m = f(n) must return a value of the form m = c ± d, where c is f(n) and d is range updated using interval arithmetic.
A percentage may also be used to indicate the error margin. For example, 230 ± 10% V refers to a voltage within 10% of either side of 230 V (from 207 V to 253 V inclusive).[citation needed] Separate values for the upper and lower bounds may also be used. For example, to indicate that a value is most likely 5.7 but may be as high as 5.9 or as low as 5.6, one may write 5.7+0.2
−0.1
.
### In chess
The symbols ± and are used in chess notation to denote an advantage for white and black respectively. However, the more common chess notation would be only + and .[3] If a difference is made, the symbols + and denote a larger advantage than ± and .
## Encodings
• In Unicode: U+00B1 ± PLUS-MINUS SIGN
• In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1hex. This location was copied to Unicode.
• The symbol also has a HTML entity representations of ± and ±.
• The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+2213 MINUS-OR-PLUS SIGN so can be used in HTML using ∓ or ∓.
• In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
• Although these characters may also be produced using underlining or overlining + symbol ( + or + ), this is deprecated because the formatting may stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users with screen readers.
### Typing
• Windows: Alt+241 or Alt+0177 (numbers typed on the numeric keypad).
• Macintosh: ⌥ Option+⇧ Shift+= (equal sign on the non-numeric keypad).
• Unix-like systems: Compose,+,- or ⇧ Shift+Ctrl+u B1space (second works on Chromebook)
• AutoCAD shortcut string: %%d
## Similar characters
The plus–minus sign resembles the Chinese characters (Radical 32) and (Radical 33), whereas the minus–plus sign resembles (Radical 51).
## References
1. ^ a b Brown, George W. (1982), "Standard Deviation, Standard Error: Which 'Standard' Should We Use?", American Journal of Diseases of Children, 136 (10): 937–941, doi:10.1001/archpedi.1982.03970460067015, PMID 7124681.
2. ^ Engineering tolerance
3. ^ a b Eade, James (2005), Chess For Dummies (2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334.
4. ^ Cajori, Florian (1928), A History of Mathematical Notations, Volumes 1-2, Dover, p. 245, ISBN 9780486677668. | 2020-08-08 03:50:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.8401452898979187, "perplexity": 1060.1945269398116}, "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/1596439737238.53/warc/CC-MAIN-20200808021257-20200808051257-00310.warc.gz"} |
http://papers.nips.cc/paper/6964-multiresolution-kernel-approximation-for-gaussian-process-regression | # NIPS Proceedingsβ
## Multiresolution Kernel Approximation for Gaussian Process Regression
[PDF] [BibTeX] [Supplemental] [Reviews]
### Abstract
Gaussian process regression generally does not scale to beyond a few thousands data points without applying some sort of kernel approximation method. Most approximations focus on the high eigenvalue part of the spectrum of the kernel matrix, $K$, which leads to bad performance when the length scale of the kernel is small. In this paper we introduce Multiresolution Kernel Approximation (MKA), the first true broad bandwidth kernel approximation algorithm. Important points about MKA are that it is memory efficient, and it is a direct method, which means that it also makes it easy to approximate $K^{-1}$ and $\mathop{\textrm{det}}(K)$. | 2019-03-24 11:53:46 | {"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": 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": 0.6961347460746765, "perplexity": 860.9229629525641}, "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-2019-13/segments/1552912203438.69/warc/CC-MAIN-20190324103739-20190324125739-00323.warc.gz"} |
https://www.sierrachart.com/index.php?page=doc/StudiesReference.php&ID=159&Name=Color_Bar_Open_Close | # Technical Studies Reference
### Color Bar Open/Close
This study colors the chart bars based on whether the bar's Open Price is higher or lower than its Close Price.
Let $$O_t$$ and $$C_t$$ be the values of the Open and Close Prices, respectively, at Index $$t$$. This study colors the chart bar at Index $$t$$ according to the following rules.
• If $$C_t > O_t$$, then the entire main price graph bar is colored with the Primary Color of the study (the first color button of the Color Bar Subgraph).
• If $$C_t < O_t$$, then the entire main price graph bar is colored with the Secondary Color of the study (the second color button of the Color Bar Subgraph).
• If $$C_t = O_t$$ and $$C_{t - 1} \geq O_{t - 1}$$, then the entire main price graph bar is colored with the Primary Color of the study.
• If $$C_t = O_t$$ and $$C_{t - 1} < O_{t - 1}$$, then the entire main price graph bar is colored with the Secondary Color of the study.
For this study to work correctly, the Color Bar Subgraph Draw Style must be set to Color Bar.
The output value is $$1$$ for all $$t$$.
#### Inputs
This study has no Inputs. | 2021-12-08 10:10:13 | {"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": 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": 0.43313607573509216, "perplexity": 1096.4466922066545}, "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/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00087.warc.gz"} |
https://plainmath.net/12781/projectile-cliff-above-ground-level-initial-speed-degrees-horizontal | Question
# A projectile is shot from the edge of a cliff 125 m above ground level with an initial speed of 65.0 m/s at an angle of 37 degrees with the horizontal
Other
A projectile is shot from the edge of a cliff 125 m above ground level with an initial speed of 65.0 m/s at an angle of 37 degrees with the horizontal. (a) determine the time taken by the projectile to hitthe point P at ground level. (b) determine the range X of the projectile as measured from the base of the cliff at the instant just before the projectile hits point P. Find (x) the horizontal and vertical components of its velocity and (d) the magnitude of the velocity and (e) the angle made by the velocity vector with the horizontal (f) Find the maximum height above the cliff top reached by the projectile.
2020-12-22
a)Find the time until projectile hits the ground
Use Y components
$$\displaystyle{Y}_{{f}}={Y}_{{i}}+{V}_{{i}}{T}+{\frac{{{1}}}{{{2}}}}{g}{T}^{{2}}$$
$$\displaystyle{Y}_{{f}}=-{125}{m}$$
$$\displaystyle{Y}_{{i}}={0}$$
$$\displaystyle{V}_{{i}}={65.0}{\sin{{37}}}$$ make sure your calculator is set to degrees
$$\displaystyle{g}=-{9.8}\ \frac{{m}}{{s}^{{2}}}$$
Solve for T
b)To find range use X coordinates
$$\displaystyle{X}_{{f}}={X}_{{i}}+{V}_{{i}}{T}+{\frac{{{1}}}{{{2}}}}{a}{T}^{{2}}$$
$$\displaystyle{X}_{{i}}={0}$$
$$\displaystyle{V}_{{i}}={65.0}{\cos{{37}}}$$
a=0 this is always the case in projectile motion
Plug T from a) and find $$\displaystyle{X}_{{f}}$$
c)Remember there is no air friction on this problem, so accelaration is zero. That means velocity stays the same on the X coordinate.
As for the Y coordinate $$\displaystyle{V}_{{f}}={V}_{{i}}+{a}{T}$$
You know T, a=-g, and $$\displaystyle{V}_{{i}}$$
The direction is down or course.
d)The magnitude is found by $$\displaystyle\sqrt{{{y}^{{2}}+{x}^{{2}}}}$$ you know the drill
e) I think this is $$\displaystyle\theta={\arccos{}}$$ (x/magnitude found in d)
f) $$\displaystyle{V}_{{f}}={V}_{{i}}+{a}{t}$$ use Y components
$$\displaystyle{V}_{{f}}={0}$$ at the top
we know $$\displaystyle{V}_{{i}}$$ and a=-g
find this T and use in the position formula to get $$\displaystyle{Y}_{{f}}$$ | 2021-09-23 14:23:35 | {"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": 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": 0.6226818561553955, "perplexity": 389.69072920687876}, "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-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00293.warc.gz"} |
https://homework.cpm.org/category/CC/textbook/cca/chapter/3/lesson/3.2.1/problem/3-40 | ### Home > CCA > Chapter 3 > Lesson 3.2.1 > Problem3-40
3-40.
Translate the Equation Mat at right into an equation. Do not simplify your equation. Remember that the double line represents “equals.”
Count up the number of each type of tile in each section, keeping track of the tiles' signs and whether they're being added or subtracted.
$2+(−2x)−(−x+4)=−3+2−(x−2)$ | 2021-10-23 04:33:36 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.37973347306251526, "perplexity": 1973.1223982430076}, "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-2021-43/segments/1634323585561.4/warc/CC-MAIN-20211023033857-20211023063857-00546.warc.gz"} |
https://search.r-project.org/CRAN/refmans/ARTool/html/anova.art.html | anova.art {ARTool} R Documentation
## Aligned Rank Transform Analysis of Variance
### Description
Conduct analyses of variance on aligned rank transformed data.
### Usage
## S3 method for class 'art'
anova(
object,
response = c("art", "aligned"),
type = c("III", "II", "I", 3, 2, 1),
factor.contrasts = "contr.sum",
test = c("F", "Chisq"),
all.rows = FALSE,
...
)
## S3 method for class 'anova.art'
print(x, verbose = FALSE, digits = 5, ...)
### Arguments
object An object of class art. response Which response to run the ANOVA on: the aligned responses ("aligned") or the aligned and ranked responses ("art"). This argument is passed to artlm. See 'Details'. type Type of ANOVAs to conduct. If type is 1 or "I", then conducts Type I ANOVAs using anova. Otherwise, conducts Type II or Type III ANOVAs using Anova. The default is Type III if the underlying model supports it. Models fit with Error terms are fit using aov, which only supports Type I ANOVAs. factor.contrasts The name of the contrast-generating function to be applied by default to fixed effect factors. See the first element of options("contrasts"). The default is to use "contr.sum", i.e. sum-to-zero contrasts, which is appropriate for Type III ANOVAs (also the default). This argument is passed to artlm. test Test statistic to use. Default "F". Note that some models and ANOVA types may not support "Chisq". all.rows Show all rows of the resulting ANOVA tables? By default (FALSE), shows only the rows that are relevant depending on the type of response. ... Additional arguments passed to Anova or anova by anova.art or to print by print.anova.art. x An object of class art. verbose When TRUE, sums of squares and residual sum of squares in addition to degrees of freedom are printed in some ANOVA types (e.g. repeated measures ANOVAs). Default FALSE, for brevity. digits Digits of output in printed table; see print.
### Details
This function runs several ANOVAs: one for each fixed effect term in the model object. In each ANOVA, the independent variables are the same, but the response is aligned by a different fixed effect term (if response is "aligned") or aligned and ranked by that fixed effect term (if response is "art"). These models are generated using artlm.
From each model, only the relevant output rows are kept (unless all.rows is TRUE, in which case all rows are kept).
When response is "art" (the default), only one row is kept from each ANOVA: the row corresponding to fixed effect term the response was aligned and ranked by. These results represent nonparametric tests of significance for the effect of each term on the original response variable.
When response is "aligned", all rows except the row corresponding to the fixed effect term the response was aligned by are kept. If the ART procedure is appropriate for this data, these tests should have all effects "stripped out", and have an F value of ~0. If that is not the case, another analysis should be considered. This diagnostic is tested by summary.art and a warning generated if the F values are not all approximately 0.
### Value
An object of class "anova", which usually is printed.
Matthew Kay
### References
Wobbrock, J. O., Findlater, L., Gergle, D., and Higgins, J. J. (2011). The aligned rank transform for nonparametric factorial analyses using only ANOVA procedures. Proceedings of the ACM Conference on Human Factors in Computing Systems (CHI '11). Vancouver, British Columbia (May 7–12, 2011). New York: ACM Press, pp. 143–146. doi: 10.1145/1978942.1978963
See art for an example. See also summary.art, artlm. | 2022-05-26 21:04:42 | {"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": 0.5538902878761292, "perplexity": 4722.012140652076}, "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/1652662625600.87/warc/CC-MAIN-20220526193923-20220526223923-00314.warc.gz"} |
https://en.wikibooks.org/wiki/Cell_Biology/Print_version | # Cell Biology/Print version
Note: current version of this book can be found at http://en.wikibooks.org/wiki/Cell_Biology Remember to click "refresh" to view this version.
Introduction
• Size of cell
• What is a cell?
• What is the difference between elements?
• What is living?
• What is interesting about cell biology?
• What is a tertiary protein?
Types of cells
• Prokaryotes
• Bacteria
• Eukaryotes
• Unique Properties of Plant Cells
Parts of the cell
• Membranes
• Organelles
• Genetic material
• Energy supply (chloroplasts and mitochondria)
Cell division
• Cell cycle
• Meiosis
• Mitosis
Genes
• Expression
• Translation
# Size of cells
## Size of Cells
Cells are so small that even a cluster of these cells from a mouse only measures 50 microns
Although it is generally the case that biological cells are too small to be seen at all without a microscope, there are exceptions as well as considerable range in the sizes of various cell types. Eukaryotic cells are typically 10 times the size of prokaryotic cells (these cell types are discussed in the next Chapter). Plant cells are on average some of the largest cells, probably because in many plant cells the inside is mostly a water filled vacuole.
So, you ask, what are the relative sizes of biological molecules and cells? The following are all approximations:
```0.1 nm (nanometer) diameter of a hydrogen atom
0.8 nm Amino Acid
2 nm Diameter of a DNA Alpha helix
4 nm Globular Protein
6 nm microfilaments
7 nm thickness cell membranes
20 nm Ribosome
25 nm Microtubule
30 nm Small virus (Picornaviruses)
30 nm Rhinoviruses
50 nm Nuclear pore
100 nm HIV
120 nm Large virus (Orthomyxoviruses, includes influenza virus)
150-250 nm Very large virus (Rhabdoviruses, Paramyxoviruses)
150-250 nm small bacteria such as Mycoplasma
200 nm Centriole
200 nm (200 to 500 nm) Lysosomes
200 nm (200 to 500 nm) Peroxisomes
800 nm giant virus Mimivirus
1 µm (micrometer)
(1 - 10 µm) the general sizes for Prokaryotes
1 µm Diameter of human nerve cell process
2 µm E.coli - a bacterium
3 µm Mitochondrion
5 µm length of chloroplast
6 µm (3 - 10 micrometers) the Nucleus
9 µm Human red blood cell
10 µm
(10 - 30 µm) Most Eukaryotic animal cells
(10 - 100 µm) Most Eukaryotic plant cells
90 µm small Amoeba
120 µm Human Egg
up to 160 µm Megakaryocyte
up to 500 µm giant bacterium Thiomargarita
up to 800 µm large Amoeba
1 mm (1 millimeter, 1/10th cm)
1 mm Diameter of the squid giant nerve cell
up to 40mm Diameter of giant amoeba Gromia Sphaerica
120 mm Diameter of an ostrich egg (a dinosaur egg was much larger)
3 meters Length of a nerve cell of giraffe's neck
```
What limits cell sizes?
• Prokaryotes — Limited by efficient metabolism
• Animal cells (eukaryotic) — Limited by surface area to volume ratio
• Plant cells (eukaryotic) — Have large sizes due to large central vacuole, which is responsible for their growth
• Some early history related to the development of an understanding of the existence and importance of cells. The importance of microscopy.
# What is a cell?
Cells are structural units that make up plants and animals; also, there are many single celled organisms. What all living cells have in common is that they are small 'sacks' composed mostly of water. The 'sacks' are made from a phospholipid bilayer membrane. This membrane is semi-permeable (allowing some things to pass in or out of the cell while blocking others). There exist other methods of transport across this membrane that we will get into later.
So what is in a cell? Cells are 90% fluid (called cytoplasm) which consists of free amino acids, proteins, carbohydrates, fats, and numerous other molecules. The cell environment (i.e., the contents of the cytoplasm and the nucleus, as well as the way the DNA is packed) affect gene expression/regulation, and thus are VERY important aspects of inheritance. Below are approximations of other components (each component will be discussed in more detail later):
### Elements
• 59% Hydrogen (H)
• 24% Oxygen (O)
• 11% Carbon (C)
• 4% Nitrogen (N)
• 2% Others - Phosphorus (P), Sulphur (S), etc.
### Molecules
• 50% protein
• 15% nucleic acid
• 15% carbohydrates
• 10% lipids
• 10% Other
### Components of cytoplasm
The following is optional reading, as all cell components will be discussed in subsequent chapters.
• Cytosol - contains mainly water and numerous molecules floating in it- all except the organelles.
• Organelles (which also have membranes) in 'higher' eukaryote organisms:
• Nucleus (in eukaryotes) - where genetic material (DNA) is located, RNA is transcribed.
• Endoplasmic Reticulum (ER) - Important for protein synthesis. It is a transport network for molecules destined for specific modifications and locations. There are two types:
• Rough ER - Has ribosomes, and tends to be more in 'sheets'.
• Smooth ER - Does not have ribosomes and tends to be more of a tubular network.
• Ribosomes - half are on the Endoplasmic Reticulum, the other half are 'free' in the cytosol, this is where the RNA goes for translation into proteins.
• Golgi Apparatus - important for glycosylation, secretion. The Golgi Apparatus is the "UPS" of the cell. Here, proteins and other molecules are prepared for shipping outside of the cell.
• Lysosomes - Digestive sacks found only in animal cells; the main point of digestion.
• Peroxisomes - Use oxygen to carry out catabolic reactions, in both plant and animals. In this organelle, an enzyme called catalase is used to break down hydrogen peroxide into water and oxygen gas.
• Microtubules - made from tubulin, and make up centrioles,cilia,etc.
• Cytoskeleton - Microtubules, actin and intermediate filaments.
• Mitochondria - convert foods into usable energy. (ATP production) A mitochondrion does this through aerobic respiration. They have 2 membranes, the inner membranes shapes differ between different types of cells, but they form projections called cristae. The mitochondrion is about the size of a bacteria, and it carries its own genetic material and ribosomes.
• Vacuoles - More commonly associated with plants. Plants commonly have large vacuoles.
• Organelles found in plant cells and not in animal cells:
• Plastids - membrane bound organelles used in storage and food production. These are similar to entire prokaryotic cells - for example, like mitochondria they contain their own DNA and self-replicate. They include:
• Chloroplasts - convert light/food into usable energy. (ATP production)
• Leucoplasts - store starch, proteins and lipids.
• Chromoplasts - contain pigments. (E.g. providing colors to flowers)
• Cell Wall - found in prokaryotic and plant cells; provides structural support and protection.
# What is the difference between elements?
The various elements that make up the cell are:
• 59% Hydrogen (H)
• 24% Oxygen (O)
• 11% Carbon (C)
• 4% Nitrogen (N)
• 2% Others - Phosphorus (P), Sulphur (S), etc.
The difference between these elements is their respective atomic weights, electrons, and in general their chemical properties. A given element can only have so many other atoms attached. For instance carbon (C) has 4 electrons in its outer shell and thus can only bind to 4 atoms; Hydrogen only has 1 electron and thus can only bind to one other atom. An example would be Methane which is CH4. Oxygen only has 2 free electrons, and will sometimes form a double bond with a single atom, which is an 'ester' in organic chemistry (and is typically scented).
Methane Water Methanol (Methyl Alcohol)
``` H
|
H-C-H
|
H
```
```H H
\ /
O
```
``` H
|
H-C-O-H
|
H
```
As for the organic molecules that make up a typical cell:
• 50% protein
• 15% nucleic acid
• 15% carbohydrates
• 10% lipids
• 10% Other
Here is a list of Elements, symbols, weights and biological roles.
Element Symbol Atomic Weight Biological Role
Bromine Br 79.9 Defense and pigmentation in certain marine organisms, esp. algae.
Calcium Ca 40.1 Bone; muscle contraction, second messenger
Carbon C 12.0 Constituent (backbone) of organic molecules
Chlorine Cl 35.5 Digestion and photosynthesis
Chromium Cr 52.0 Metabolism of sugars and lipids in humans.
Copper Cu 63.5 Part of Oxygen—carrying pigment of mollusk blood.
Fluorine F 19.0 For normal tooth enamel development
Hydrogen H 1.0 Part of water and all organic molecules
Iodine I 126.9 Part of thyroxine (a hormone)
Iron Fe 55.8 Hemoglobin, oxygen caring pigment of many animals
Magnesium Mg 24.3 Part of chlorophyll; essential cofactor for many enzymes of energy metabolism.
Manganese Mn 54.9 Essential to some enzyme actions.
Nitrogen N 14.0 Constituent of all proteins and nucleic acids.
Oxygen O 16.0 Respiration; part of water; and in nearly all organic molecules.
Phosphorus P 31.0 Constituent of DNA and RNA backbones; high energy bond in ATP.
Potassium K 39.1 Generation of nerve impulses.
Selenium Se 79.0 For the working of many enzymes.
Silicon Si 28.1 Diatom shells; grass leaves.
Sodium Na 23.0 Part of Salt; nerve conduction
Sulfur S 32.1 Constituent of most proteins. Important in protein structure: Sulfide bonds are strong.
Zinc Zn 65.4 Essential to alcohol oxidizing enzyme.
# What is living?
The question, "What is life?" has been one of many long discussions and the answer may depend upon your initial definitions.
Life is cells. Cell theory consists of three basic points.
1. All living things are made of cells.
2. The cell is the smallest living thing that can perform all the functions of life.
3. All cells must come from preexisting cells.
Some definitions of life are:
1. The quality that distinguishes a vital and functional being from a non-living or dead body or purely chemical matter.
2. The state of a material complex or individual characterized by the capacity to perform certain functional activities including metabolism, growth, and reproduction.
3. The sequence of physical and mental experiences that make up the existence of an individual.
## Seven Criteria
A cell undergoing mitosis. Reproduction is one of the seven criteria for life.
In biology, whether life is present is determined based on the following seven criteria:
1. It should maintain some balanced conditions in its inner structure. This is called Homeostasis
2. Its structure is highly organized.
3. It should be able to break down or build up nutrients to release or store energy based on need. This is called Metabolism
4. It should grow, which means its structure changes as time goes by in an advantageous manner.
5. It should show adaptation to the environment.
6. It should be able to respond to environmental stimuli on demand (as opposed to adaptation, which occurs over time).
7. It should be able to reproduce itself.
Another way of remembering the seven life processes for children is:-
```Movement
Respiration
Sensitivity
Growth
Reproduction
Excretion
Nutrition
```
Note the beginning letter of all the seven life processes, it spells out MRS GREN.
## Virus Controversy
Are these bacteriophages technically alive?
This definition of life has got some problems to it though: As an example, let's take viruses. Just by your intuition, what would you say: Are viruses alive or dead? Most people's intuitive answer is: Viruses are alive. When we suffer from any viral infection, we have the feeling that these viruses that cause or infection are alive. According to the seven principles as shown above, viruses are dead, as dead as a piece of plastic: They can't reproduce themselves. To understand that, we want to make a quick excursion to the replication mechanism of viruses: Viruses are really strange in their reproduction technique. Humans and other animals reproduce by the means of sexual intercourse, bacteria do something called binary fission: They divide. One cell divides itself into two, the two daughter cells divide again an so on. The point here is that both bacteria and animals or humans reproduce actively without any help from outside. Keep this point in mind as we move on to the viruses. Viruses need other cells to reproduce. They "drill" their way into another cell, called the host cell. Here, they release the genetic material they carry and, by a complex mechanism that shouldn't be explained further at this point, force their host cell to produce exact copies of the virus. After some time, the host cell is full of viruses and bursts, releasing the new viruses into the environment. Thus viruses need help to reproduce. They can't reproduce at all without a host cell and therefore do not fulfill the requirement "It should be able to reproduce itself". Looking at the other parts of the definition we find that viruses maintain some degree of homeostasis (1), being able to keep its protenatious and nucleic machinery separated from the outside world. Viruses also show adaptation(5), with their ability to mutate in order to affect new organisms. In addition to the reproduction problem, they also fail to meet the other requirements, showing no cellular organization (2) (or indeed cells at all), metabolism (3), or growth (4).
This example is just to illustrate the problems that arise using this definition. Life is not something one can define as any other technical term in science. Life arose from dead matter around 4 billion years ago. When life can arise from dead matter, there can't be a precise border line between these two.
## The cell is alive, what about parts of it?
Organelles are parts of eukaryotic cells (ones having a nucleus). They help the cell carry out its task. But, are they alive? Do they meet 7 criteria?
When a cell divides into two, organelles also 'reproduce'. They also age from young to old and then die. Some of them carry out the task of taking food, converting it to nutrients and energy. They can also react to stimuli, and surely they can evolve. Of course one can argue that all the above are coordinated by the nucleus. But it seems there are some signs of life there.
Yes, there are! Scientists have proven that some bacteria, in its evolutionary way, had found a home in other cells. They felt comfortable when living there, and gradually, they have become a part of that cell. Chloroplasts, for example, used to be bacteria. At some point in their evolutionary history these cyanobacteria formed a mutual symbiosis with the proto-eukaryote ancestors of algae. Since that time, chloroplasts have been helping plant cells photosynthesize.
Another example is mitochondria, organelles that produce energy for eukaryotes. Very likely a parasitic organism originally, the ancestor of the mitochondria we see today colonized the larger proto-eukarotes. It is unknown if the mitochondrial ancestor originally had a metabolic role in its life cycle or if it adapted to the changing conditions after it was engulfed.
# What is interesting about cell biology?
What makes Cell Biology particularly interesting is that there is so much that is not fully understood. A cell is a complex system with thousands of molecular components working together in a coordinated way to produce the phenomenon we call "life". During the 20th century these molecular components were identified (for example, see Human Genome Project), but research continues on the details of cellular processes like the control of cell division and cell differentiation. Disruption of the normal control of cell division can cause abnormal cell behavior such as rapid tumor cell growth.
Cells have complex interactions with the surrounding environment. Whether it is the external world of a single celled organism or the other cells of a multicellular organism, a complex web of interactions is present. Study of the mechanisms by which cells respond appropriately to their environments is a major part of cell biology research and often such studies involve what is called signal transduction. For example, a hormone such as insulin interacting with the surface of a cell can result in the altered behavior of hundreds of molecular components inside the cells. This sort of complex and finely tuned cell response to an external signal is required for normal metabolism and to prevent metabolic disorders like Type II diabetes.
Most of the cells of a multi-cellular organism have the same genetic material in every cell; yet, there may be hundreds of different types of cells that make up the organism's body each with its own distinctive shape, size, and function. In any case, all of these cells were developed from one special cell, a zygote. The study of how the many cell types develop during embryonic development (Developmental Biology) is a branch of Biology that is heavily dependent on the use of microscopy. Much of the control of cell differentiation is at the level of the control of gene transcription, the control of which mRNAs are made. Muscle cells make muscle proteins and nerve cells make brain proteins. Geneticists, molecular biologists and cell biologists are working to discover the details of how cells specialize to accomplish hundreds of functions from muscle contraction to memory storage.
## Summary
• Complexity in:
• inter-relations between cells
• signal transduction pathways inside cells
• control of cell death and cell reproduction
• control of cell differentiation
• control of cell metabolism.
# Prokaryotes
The structures of two prokaryotic cells. The bacterium (shown at the top) is a heterotroph, an organism that eats other organisms. Cyanophytes are autotrophs, organisms that make their food without eating other organisms.
Most of these prokaryotic cells are small, ranging from 1 to 10 microns with a diameter no greater than 1 micron. The major differences between Prokaryotic and Eukaryotic cells are that prokaryotes do not have a nucleus as a distinct organelle and rarely have any membrane bound organelles [mitochondria, chloroplasts, endoplasmic reticulum, golgi apparatus, a cytoskeleton of microtubules and microfilaments] (the only exception may be a bacterium discovered to have vacuoles). Both types contain DNA as genetic material, have a surrounding cell membrane, have ribosomes[70 s], accomplish similar functions, and are very diverse. For instance, there are over 200 types of cells in the human body, that vary greatly in size, shape, and function.
Prokaryotes are cells without a distinct nucleus.They have genetic material but that material is not enclosed within a membrane. Prokaryotes include bacteria and cyanophytes. The genetic material is a single circular DNA strand and is located within the cytoplasm. Recombination happens through transfers of plasmids (short circles of DNA that pass from one bacterium to another). Prokaryoytes do not engulf solids, nor do they have centrioles or asters. Prokaryotes have a cell wall made up of peptidoglycin.
In majority of prokaryotes, the genome consists of a circular chromosome whose structure includes fewer proteins that found in the linear chromosomes of eukaryotes. Their chromosome is located in the nucleoid, a region of cytoplasm that appears lighter than surrounding cytoplasm in electron micrographs. Also, a single chromosome have much smaller rings of separately replication DNA called plasmids.
## Cell Surface
Prokaryotic cell walls maintain cell shape, provide physical protection, and prevents the cell from bursting in a hypotonic environment. In hypertonic environment, most prokaryotes lose water and shrink away from their wall (plasmolyze). The cell walls of prokaryotes differ in molecular composition and construction from those of eukaryotes. The bacterial cell walls contain peptidoglycan, a network of modified-sugar polymers cross linked by short polypeptides. This molecular fabric encloses the entire bacterium and anchors other molecules that extend from its surface. Archaeal cell walls contain a variety of polysaccharides and proteins but lack peptidoglycan.
Gram-positive bacteria have simpler walls with a relatively large amount of peptidoglycan. It has a thick cell wall that traps the crystal violet in the cytoplasm. The alcohol rinse does not remove the crystal violet which masks the added red safanin dye.
Gram-negative bacteria have less peptidoglycan and are structurally more complex, with an outer membrane that contains lipopolysaccharides. It has a thinner layer of peptidoglycan, and it is located in a layer between the plasma membrane and an outer membrane. The crystal violet is easily rinsed from the cytoplasm, and the cell appears pink or red.
The cell wall of many prokaryotes is covered by a capsule, a sticky layer of polysaccharide or protein. The capsule enables prokaryotes to adhere to their substrate or to other individuals in a colony. Some capsules protect against dehydration, and some shield pathogenic prokaryotes from attack by their host's immune system. Some prokaryotes stick to their substrate or to one another by means of hair like protein appendages called fimbriae. They are also known as attachment pili. Fimbriae are usually shorter extension of the plasma membrane.
In uniform environment, flagellated prokaryotes move randomly, but in heterogeneous environment, many prokaryotes exhibit taxis, movement toward or away from a stimulus. For example, prokaryotes that exhibit chemotaxis change their movement pattern in response to chemicals. They move toward nutrients or oxygen (positive chemotaxis) or away from a toxic substance (negative chemotaxis).
Prokaryotes reproduce quickly in a favorable environment. By binary fission, a single prokaryotic cell divid into 2 cells, which then divide into 4, 8, 16, and on. Under optimal conditions, many prokaryotes can divide every 1-3 hours. However the cells eventually exhaust their nutrient supply, poison themselves with metabolic wastes, face competition from other microorganisms, or are consumed by other organisms. The prokaryotes are small, they reproduces by binary fission, and they have short generation times. The ability of some prokaryotes to withstand harsh conditions also contributes to their success. Certain bacteria develop resistant cell called endospores when an essential nutrient is laking. The original cell produces a copy of its chromosome and surrounds it with a tough wall, forming the endospore. Water is removed from the endospore, and its metabolism halt. The rest of the original cell then disintegrates, leaving the endospore behind. Most endospore are so durable that they can survive in boiling water. In less hostile environments, endospore can remain dormant but viable for centuries, able to rehydrate and resume metabolism when their environment improves.
Due to their short generation times, prokaryotic populations can evolve substantially in short periods of time. The ability of prokaryotes to adapt rapidly to new conditions highlights the fact that although the structure of their cells is simpler than that of eukaryotic cells, prokaryotes are not "primitive" or "inferior" in an evolutionary sense. They are highly evolved, and their population have responded successfully to many different types of environmental challenges.
Rapid reproduction and mutation In sexually reproducing species, the generation of a novel allele by a new mutation is rare at any particular gene. Instead, most of the genetic variation in sexual populations results from the way existing alleles are arranged in new combinations during meiosis and fertilization. Prokaryotes do not reproduce sexually, so at first glance their extensive genetic variation may seem puzzling.
After repeated rounds of division, most of the offspring cells are genetically identical to the original parent cell; however owing to insertions, deletions, and base-pair substitutions in their DNA, some of the offspring cells may differ genetically. The new mutations, though individually rare, can greatly increase genetic diversity in specie that has short generation times and large population sizes. This diversity, in turn, can lead to rapid evolution: individuals that are genetically better equipped for their local environment tend to survive and reproduce more prolifically than less fit individuals.
## Transformation and Transduction
In transformation, the genotype and possible phenotype of a prokaryotic cell are altered by the uptake of foreign DNA from its surroundings. For example, bacteria from a harmless strain of Streptococcus pneumoniae can be transformed to pneumonia causing cells if they are placed into a medium containing dead, broken-open cells of the pathogenic strain. This transformation occurs when a live nonpathogenic cell takes up a piece of DNA carry the allele for pathogenicity. The foreign allele is then incorporated into the cell's chromosome, replacing the existing nonpathogenic allele- an exchange of homologous DNA segments. The cell is now a recombinant: Its chromosome contains DNA derived from two different cells.
In transduction, bacteriophage carry bacterial genes from one hose cell to another; transduction is a type of horizontal gene transfer. For most phages, transduction results from accidents that occur during the phage reproductive cycle. A virus that carries bacterial DNA may not be able to reproduce because it lacks its own genetic material. However, the virus may be able to attach to another bacterium (a recipient) and inject the piece of bacterial DNA acquired from the first cell (the donor). Some of this DNA may subsequently replace the homologous region of the recipient cell's chromosome by DNA recombination. In such case, the recipient cell's chromosome becomes a combination of NA derived from two cells; genetic recombination has occurred.
Conjugation and Plasmids In a process called conjugation, genetic material is transferred between two bacterial cells ( of same or different species) that are temporarily joined. The DNA transfer is one way: One cell donates the DNA,a nd the other receives ti. The donor uses sex pili to attach to the recipient. After contacting a recipient cell, each sex pilus retracts, pulling the two cells together, much like a grappling hook. A temporary "mating bridge" then forms between the two cells, providing an avenue for DNA transfer. In most cases, the ability to form sex pili and donate DNA during conjugation results from the presence of a particular piece of DNA called F factor. The F factor consists about 25 genes, most required for the production of sex pili. The F factor can exist either as a plasmid or as a segment of DNA within the bacterial chromosome
The F factor in its plasmid form is called F plasmid. Cells containing the F plasmid, designated F+ cells, function as DNA donors during conjugation. Cells lacking the F factor, designated F-, function as DNA recipients during conjugation. The F+ condition is transferable in the sense that an F+ cell converts and F- cell to F+ is a copy of the entire F+ plasmid is transferred.
Chromosomal genes can be transferred during conjugation when the donor cell's F factor is integrated into the chromosome. A cell with the F factor built into its chromosome is called an Hfr cell. Like an F+ cell, an Hfr cell functions as a donor during conjugation with an F- cell. When chromosomal DNA from an Hfr cell enters and F- cell, homologous regions of the HFr and F- chromosomes may align, allowing segments of their DNA to be exchanged. This results in the production of a recombinant bacterium that has genes derived from two different cells- a new genetic cariant on which evolution can act. Though theses processes of horizontal gene transfer have so far been studied almost exclusively in bacteria, it is assumed that they are similarly important in archaea.
## Diverse nutritional and metabolic adaptations
The mechanisms discussed in the previous section- rapid reproduction, mutation, and genetic recombination- underlie that extensive genetic variation found in prokaryotic populations. This variation is reflected in the nutritional adaptations found in prokaryotes. Like all organisms, prokaryotes can be categorized by their nutrition; how they obtain every and the carbon used in building the organic molecules that make up cells. Nutritional diversity is greater among prokaryotes than among eukaryotes: Every type of nutrition observed in eukaryotes is represented among the prokaryotes, along with some nutritional modes unique to prokaryotes. Phototrophs are the organisms that obtain energy from light. Chemotrophs are the organisms that obtain energy from chemicals. Organisms that need only an inorganic compound are called autotrophs. Heterotrophs require at least one organic nutrient to make other organic compounds. Combining these possibilities for energy sources and carbon sources results in four major modes of nutrition.
Photoautotrophs: photosynthetic organisms that capture light energy and use it to drive the synthesis of organic compounds and other inorganic carbon compounds. Cyanobacteria and many other groups of prokaryotes are photoautotrophs, as are plants and algae.
Chemoautotrophs: also need only an inorganic compound; however, instead of using light as an energy source, they oxidize inorganic substance, such as hydrogen sulfide, ammonia, or ferrous ions. This mode of nutrition is unique to certain prokaryotes.
Photoheterotrophs: Harness energy from light but must obtain carbon in organic form. This mode is unique to certain marine and halophilic (salt-loving) prokaryotes.
Chemoheterotrphs: must consume organic molecules to obtain both energy and carbon. This nutritional mode is widespread among prokaryotes. Fungi, animals, most protists, and even some parasitic plants are also chemoheterotrophs.
## The Role of Oxygen In Metabolism
Prokaryotic metabolism also varies with respect to oxygen. Obligate aerobes use oxygen for cellular respiration and cannnot grow without it. Obligate anaerobes, however, are posioned by oxygen. Some obligate anaerobes live exclusively by fermentation; other extract chemical energy by anaerobic respiration, in which substance other than oxygen such as nitrate ions or sulfate ions accept electrons at the "downhill" end of electron transport chains. Facultative anaerobes use oxygen if it is present but can also carry out anaerobic respiration or fermentation in an anaerobic environment.
Nitrogen Metabolism Nitrogen is essential for the production of amino acids and nucleic acids in all organisms. Whereas eukaryotes can obtain nitrogen from only a limited group of nitrogen compounds, prokaryotes can metabolize nitrogen in a wide variety of forms. For example, some cyanobacteria and some methanogens covert atmospheric nitrogen to ammonia, a process called nitrogen fixation. The cells can then incorporate this "fixed" nitrogen into amino acids and other organic molecules. In terms of their nutrition, nitrogen-fixing cyanobacteria are some of the most self-sufficient organisms, since they need only light, carbon dioxide, nitrogen, water and some minerals to grow. Nitrogen fixation by prokaryotes has a large impact on other organisms. For example, nitrogen -fixing prokaryotes can increase the nitrogen but can use the nitrogen compounds that the prokaryotes produce from ammonia.
Metabolic Cooperation Cooperation between prokaryotes allows them to use environmental resource they could not use as individual cells. In some cases, this cooperation takes place between specialized cells of a colony. For instance, the cyanobacterium Anabaena has genes than encode proteins for photosynthesis and for nitrogen fixation, but a single cell cannot carry out both processes at the same time. The reason is that photosynthesis produces oxygen which inactivates the enzymes involved in nitrogen fixation. Instead of living as isolated cells, anabaena forms filamentous colonies synthesis while a few specialized cells called heterocytes carry out only nitrogen fixation. Each heterocyte is surrounded by a thickened cell wall that restricts entry of oxygen produced by neighboring photosynthetic cells. Intercellular connections allow heterocytes to transport fixed nitrogen to neighboring cells and to receive carbohydrates.
Metabolic cooperation between different prokaryotic species often occurs in surface-coating colonies known as biofilms. Cells in a biofilm secrete signaling molecules that recruit nearby cells, causing the colonies to grow. The cells also produce proteins that stick the cells to the substrate and on to another. Channels in the biofilm allow nutrients to reach cells in the interior and wastes to be expelled. Biofilm damage industrial and medical equipment, contaminate products, and contribute to tooth decay and more serious health problems. In another example of cooperation between prokaryotes, sulfate-consuming bacteria coexist with methane-consuming archaea in ball- shaped aggregates on the ocean floor. The bacteria appear to use the archaea's waste products, such as organic compounds and hydrogen. In turn, the bacteria produce compounds that facilitate methane consumption by the archaea. This partnership has global ramifications.
## Reference
Berg, Jeremy M., John L. Tymoczko, and Lubert Stryer. Biochemistry. 7th ed. New York: W.H. Freeman, 2012. Print.
Reece, Campbell, Lisa A. Urry, Michael L. Cain, Steven A. Wasserman, Peter V. Minosky, and Robert B. Jackson. Biology. 8th ed. San Francisco: Cummings, 2010. Print.
Eukaryotes
# Bacteria
Bacteria are prokaryotic, unicellular organisms. Bacteria are very small; so much so that 1 billion could fit on 1 square centimeter of space on the human gums, and 1 gram of digested food has 10 billion bacteria. Bacteria are the simplest living organisms. Previously they fell under the Kingdom Moneran, but now they fall into two different Domains: Archaebacteria and Eubacteria. There are several differences between the two. Typically, microbiologists in the 21st century call these groups "Archaea" and "Bacteria." One of the co-discoverers of the three Domains has argued that the term "prokaryote" should be removed from classrooms because it reflects an evolutionary hypothesis that has been disproved, given that the Archaea are more closely related to the Eukarya than they are to the Bacteria (Pace 2006, Nature 441 p. 289).
It is incorrect to think of bacteria as particularly "simplistic" for all that they do not have internal organelles that can be visualized using a light microscope. The bacterial nucleoid, for example, is a highly organized structure even though it typically contains just one or two circular chromosomes with a total of millions of basepairs of DNA (Thanbichier et el., 2005, J Cell. Biochem. 96:506-521; see also http://www.microbelibrary.org/Laboratory%20Diagnostics/details.asp?id=782&Lang=English). Bacteria have complex cell walls exterior to the cell membrane. Some bacteria contain plasmids, which are typically circular DNA with replication that is uncoupled from binary fission (cellular division). Plasmids in nature often encode traits of significant interest to humans, such as the ability to be resistant to clinically important antibiotics, or the ability to degrade "odd" carbon sources such as TNT or human-made pollutants. Plasmids from Escherichia coli have been "domesticated" and have long been in use for genetic engineering, as it is easier to isolate and modify plasmid DNA, and introduce it into a new cells, than it is to modify a bacterial chromosome (see "Techniques" at dnai.org; see also http://www.microbelibrary.org/Laboratory%20Diagnostics/details.asp?id=707&Lang=English). A few phylogenetic groups of bacteria can make endospores, which are metabolically inert but are able to resist high temperatures, radiation, and desiccation (see http://www.microbelibrary.org/Laboratory%20Diagnostics/details.asp?id=2511&Lang=English).
Bacterial reproduction is always asexual and usually occurs through binary fusion, once thought to be a simple process of growing and dividing. Microbiologists now know, however, that binary fission is complex in that it requires dozens of proteins cooperating to build the septum (new cell wall between 'daughter' cells) and to actively separate the two daughter chromosomes. Furthermore, there are other forms of reproduction in bacteria, all of them "asexual" in that they do not use gametes or involve genetic exchange (Angert 2005, Nature Reviews Microbiology 3:214-224). Genetic exchange in bacteria is instead called horizontal (or lateral) gene transfer, because bacteria can obtain genetic information from organisms that are not their parents (Amabile-Cuevas 2003, American Scientist 91:138-149). (Vertical genetic transmission is the inheritance of DNA down through the generations.) Horizontal gene transfer can happen any time and has nothing to do with cell division. There are three main types, conjugation which is the sharing of plasmid DNA; transduction, where a bacterial virus accidentally transfers bacterial DNA from one bacterium to another, and transformation, where bacteria bind to DNA in the environment, internalize it, and can use that DNA as genetic material.
## Archaea
Archaea are microbes that are more closely related to Eukaryotic cells than they are to the Bacteria (http://tolweb.org/tree/home.pages/aboutoverview.html). Under a light microscope, they visually resemble Bacteria, so that it wasn't until the advent of the use of molecular methods in evolutionary biology that they were recognized as belonging to their own Domain (a phylogenetic grouping above the level of Kingdom). Archaea have ultrastructural features that are superficially similar to those in Bacteria but are usually comprised of distinctive molecules. They do, for example, have a cell wall, yet that cell wall never contains peptidoglycan. Instead, peptidoglycan is a unique molecular signature of the Bacteria. Archaea also have odd lipids in their cell membranes. They were originally discovered living in extreme environments thought to resemble conditions on early earth, but now that microbiologists have become more adept at detecting them, it is clear that the Archaea are not confined to extreme habitats and can instead be found everywhere. It is true that some Archaea are "extremophiles," found in extremely salty or hot environments, but there are also extremophile Bacteria and even some very unusual extremophile Eukarya. The best-understood groups of Archaea are:
1. Methanogens use Carbon dioxide and Hydrogen to make Methane. They are found in sewage, cows, and swamps, and they do not take in oxygen.
2. Extreme Halophiles live in extremely salty places (i.e.: the dead sea and great salt lake).
3. Thermoacidophiles prefer extremely hot, acidic areas (i.e.: hot springs and volcanos).
## Bacteria
(sometimes called "eubacteria")
Bacteria have peptidoglycan in their cell walls, and they have no unusual phospholipids. Bacteria have four shapes:
• bacilli (rod shaped)
• vibrios (curved shaped)
• coccus (round shaped)
• spirilli (spiral shaped).
Bacteria can also have prefixes before their names: strepto, indicating chains of the shaped bacteria, and staphylo, indicating clusters of the shaped bacteria. A 19th century microbiologist invented the Gram stain, still used today to differentiate bacteria into two types, Gram negative and Gram positive (http://en.wikipedia.org/wiki/Hans_Christian_Gram). These types are not useful in determining phylogeny but can be very useful in a clinical setting, because Gram negative and Gram positive bacteria can exhibit differential sensitivity to some classes of antibiotics. There are probably dozens of "Kingdoms" within the Domain Bacteria, but the phylogeny of Bacteria is still disputed as microbiologists continue to study the evolution of bacteria using molecular methods. Some of the major types of Bacteria are:
1. Cyanobacteria are photoautotrophs that strip electrons from water and use them to fix carbon dioxide; they are a major source of organic carbon in marine ecosystems.
2. Spirochetes are Gram negative bacteria that have flexible cells and internal flagella in an unusual form of a more typical Gram negative cell wall.
3. Proteobacteria (E-coli)
Some bacteria produce virulence factors that can cause sickness. Some examples of these are serotoxins, which are given off by the Gram positive bacteria, and endotoxins, which are given off by Gram negative bacteria as they die. There are many other examples, however, and specific pathogens make a unique suite of virulence factors that lead to the particular disease caused by that pathogen.
# Eukaryotes
An animal Cell
Eukaryotes house a distinct nucleus, a structure in which the genetic material (DNA) is contained, surrounded by a membrane much like the outer cell membrane. Eukaryotic cells are found in most algae, protozoa, all multicellular organisms (plants and animals) including humans. The genetic material in the nucleus forms multiple chromosomes that are linear and complexed with proteins that help the DNA 'pack' and are involved in regulation of gene expression.
The cells of higher plants differ from animal cells in that they have large vacuoles, a cell wall, chloroplasts, and a lack of lysosomes, centrioles, pseudopods, and flagella or cilia. Animal cells do not have the chloroplasts, and may or may not have cilia, pseudopods or flagella, depending on the type of cell.
## Comparing Prokaryotic and Eukaryotic Cells
All cells have several basic features in common: they are all bounded by a selective barrier, plasma membrane. Cytosol is a jellylike substance that is semifluid. All cells contain chromosomes which carry genes in the form of DNA, and ribosomes that make proteins according to instructions from the gene. The major difference between prokaryotic and eukaryotic cells is the location of their DNA. In eukaryotic cell, DNA is found at the nucleus, which is bounded by a double membrane. (the word eukaryotic is from the Greek eu, true, and karyon, kernel, here referring to the nucleus).
Eukaryotic cells are much larger than prokaryotic cells; size is general aspect of cell structure that relates to function. The logistics of carrying out cellular metabolism sets limits on cell size. At the lower limit, the smallest cells, known are bacteria called mycoplasmas have diameters between 0.1 and 1.0mm. These are the smallest packages with enough DNA to program metabolism and enough enzymes and other cellular equipment to carry out the activities necessary for a cell to sustain itself and reproduce.
Metabolic requirements also impose theoretical upper limits on the size that is practical for a singel cell. Plasma membrane functions as a selective barrier that allows sufficient passage of oxygen, nutrients, and wastes to service the entire cell. For each square micrometer of membrane, only a limited amount of a particular substance can cross per second, so the ratio of surface area to volume is critical. As a cell increases in size, its volume grows proportionately more than its surface area. Area is proportional to a linear dimension squared, whereas volume is proportional to the linear dimension cubed. Therefore a smaller object has a greater ration of surface area to volume.
The need for a surface area sufficiently large to accommodate the volume helps explain the microscopic size of most cells, and the narrow, elongated shapes of others, such as nerve cells. Larger organisms has more cells compare to smaller cells. High ratio of surface area to volume is especially important in cells that exchange a lot of material with their surroundings such as intestinal cells. Such cells may have many long, thin projections from their surface called microvilli, which increase surface area without an appreciable increase in volume.
## Animal Cells
Flagellum: locomotion organelle present in some animal cells; composed of a cluster of microtubules within an extension of the plasma membrane.
Centrosome: region where the cell's microtubules are initiated contains a pair of centrioles which function is unknown.
Cytoskeleton: reinforces cell's shape, functions in cell movement components are made of protein. It includes microfilaments, intermediate filaments, and microtubules.
Microvilli: projections that increase the cell's surface area.
Peroxisome: organelle with carious specialized metabolic functions; produces hydrogen peroxide as a by-product, then converts it to water.
Mitochondrion: organelle where cellular respiration occurs and most ATP is generated.
Lysosome: digestive organelle where macromolecules are hydrolyzed.
Golgi apparatus: organelle active in synthesis, modification, sorting, and secretion of cell products.
Ribosomes: complexes (small brown dots) that make proteins; free in cytosol or bound to rough ER or nuclear envelope.
Plasma membrane: membrane enclosing the cell
Endoplasmic Reticulum (ER): network of membraneous sacs and tube; active in membrane synthesis and other synthetic and metabolic processes; has rough (ribosome-studded) and smooth regions. (Rough ER, and Smooth ER)
Nucleus: nucleus contains:
``` Nuclear envelope: double membrane enclosing the nucleus; perforated by pores; continuous with ER
Nucleolus: structure involved in production of ribosomes; a nucleus has one or more nucleoli
Chromatin: material consisting of DNA and proteins; visible as individual chromosomes in a dividing cell
```
In animal cells, lysosomes, centrosomes with centrioles, and flagella are present but not in plant cells.
## Plant Cell
Cell Wall: outer layer that maintains cell's shape and protects cell from mechanical damage; made of cellulose, other polysaccharide, and protein.
Plasmodesmata: channels through cell walls that connect the cytoplasms of adjacent cells.
Chloroplast: photosynthetic organelle; converts energy of sunlight to chemical energy stored in sugar molecules.
Central vacuole: prominent organelle in older plant cells; functions include storage, breakdown of waste products, hydrolysis of macromolecules; enlargement of vacuole is a major mechanism of plant growth.
Nucleus: nucleus contains:
``` Nuclear envelope: double membrane enclosing the nucleus; perforated by pores; continuous with ER
Nucleolus: structure involved in production of ribosomes; a nucleus has one or more nucleoli
Chromatin: material consisting of DNA and proteins; visible as individual chromosomes in a dividing cell
```
Golgi apparatus: organelle active in synthesis, modification, sorting, and secretion of cell products.
Endoplasmic Reticulum (ER): network of membraneous sacs and tube; active in membrane synthesis and other synthetic and metabolic processes; has rough (ribosome-studded) and smooth regions. (Rough ER, and Smooth ER)
Ribosomes: complexes (small brown dots) that make proteins; free in cytosol or bound to rough ER or nuclear envelope.
Cytoskeleton: reinforces cell's shape, functions in cell movement components are made of protein. It includes microfilaments, intermediate filaments, and microtubules.
In plant cell, chloroplasts, central vacuole, cell wall, and plasmodesmata are present but not in animal cells. Chromatin in the plant cell is a primary protein
## Nucleus
The nucleus contains most of the genes in the eukaryotic cell; some genes are located in mitochondria and chloroplast. It is generally the most conspicuous organelle in a eukaryotic cell. The nuclear envelope encloses the nucleus, sparating its contents from the cytoplasm. The nuclear envelope is a double membrane, each a lipid bilayer with associated proteins. The envelope is perforated by pore structure that are about 100nm in diameter. At the lip of each pore, the inner and outer membranes of the nuclear envelope are continuous. Pore complex lines each pore and regulates the entry and exit of most proteins and RNAs, as well as large complexes of macromolecules. Except at the pores, the nuclear side of the envelope is lined by the nuclear lamina, a netlike array of protein filaments that maintains the shape of the nucleus by mechanically supporting the nuclear envelope. Also nuclear matrix, a framework of fibers extending throughout the nuclear interior, present.
Chromosomes are organized DNA units that carry the genetic information. Each chromosome is made up of material called chromatin, a complex of proteins and DNA. Stained chromatic usually appears as a diffuse mass, byt as a cell prepares to divide, the thin chromatin fibers coil up and condense thick enough to be distinguished as chromosomes. Each eukaryotic species has a characteristic number of chromosomes. For example human has 46 chromosomes.
Nucleolus is a prominent structure within the nondividing nucleus. Ribosomal RNA (rRNA) is synthesized from instructions in the DNA; in nucleolus, proteins imported from the cytoplasm are assembled with rRNA into large and small ribosomal subunits. Theses subunits then exit the nucleus through the nuclear pores to the cytoplasm, where a large and a small subunit can assemble into a ribosome. the number depends on the species and the stage in the cell's reproductive cycle.
The Nucleus directs protein synthesis by synthesizing messenger RNA (mRNA) according to instructions provided by the DNA. The mRNA is then transported to the cytoplasm via the nuclear pores. Once an mRNA molecule reaches the cytoplasm, ribosomes translate the mRNA's genetic message into the primary structure of a specific poly peptide.
## Ribosomes
Ribosomes are complexes made of ribosomal RNA and protein; ribosomes are the cellular components that carry out proteins synthesis, also known as protein factories. Cells that have high rates of protein synthesis have particularly large number of ribosomes. Cells active in protein synthesis also have prominent nucleoli. Ribosomes build proteins in two cytoplasmic locales. Free ribosomes are suspended int he cytosol, while bound ribosomes are attached to the outside of the endoplasmic reticulum or nuclear envelope. Bound and free ribosomes are structurally identical, and ribosomes can alternate between the two roles. Most of proteins are made on free ribosomes function within the cytosol. Bound ribosomes generally make proteins that are destined for insertion into membranes, for packaging within certain organelles such as lysosomes, or for export from the cell (secretion).
## The Endomembrane System
Endomembrane system carries out a variety of tasks in the cell. These tasks include synthesis of proteins and their transport into membranes and organelles or out of the cell, metabolism and movement of lipids, and detoxification of poisons. The membrane of this system are related either through direct physical continuity or by the transfer of membrane segments as tiny vesicles. The various membranes are not identical in structure and function; the thickness, molecular composition, and types of chemical reactions carried out in a given membrane are not fixed but modified several times during the membrane's life. The endomembrane system includes the nuclear envelope, the endoplasmic reticulum, the Golgi apparatus, lysosomes, various kinds of vacuoles, and the plasma membrane.
## Endoplasmic Reticulum (ER)
Endoplasmic reticulum (ER) is an extensive network of membrane that it accounts for more than half the total membrane in many eukaryotic cells. The word endoplasmic means "within the cytoplasm", and reticulum is Latine for "little net". The ER consists of a network of membranous tubules and sacs called cisternae. The ER membrane separates the internal compartment of the ER, ER lumen (cavity) or cisternal space, from the cytosol. Since ER membrane is continuous with the nuclear envelope, the space between the two membranes of the envelope is continuous with the lumen of the ER. Smooth ER lacks ribosomes on its outer surface, and Rough ER has ribosomes on the outer surface of the membrane. Ribosomes are also attached to the cytoplasmic side of the nuclear envelope's outer membrane.
Smooth ER- The smooth ER functions in diverse metabolic processes, which vary with cell type. Theses processes include synthesis of lipids, metabolism of carbohydrates, and detoxification of drugs and poisons. Enzymes of the smooth ER are important in the synthesis of lipids, including oils, phospholipids, and steroids. Sex hormones of vertebrates and the various steroid hormones are produced by the smooth ER in animal cells. Other enzymes of the smooth ER help detoxify drugs and poisons in liver cells. Detoxification involves adding hydroxyl groups to drug molecules, making them more soluble and easier to flush from the body. For example, sedative phenobarbital and other barbiturates are the drugs that metabolized in this manner by smooth ER in liver cells. Barbiturates, alcohol, and many other drugs induce the proliferation of smooth ER and its associated detoxification enzymes, therefore, increasing tolerance to the drugs; in other words, higher doses are required to achieve a particular effect. Also, because some of the detoxification enzymes have relatively broad action, the proliferation of smooth ER in response to one drug can increase tolerance to other drugs as well. The smooth ER also stores calcium ions; in muscle cells, a specialized smooth ER membrane pumps calcium ions from the cytosol into the ER lumen. When a muscle cell is stimulated by a nerve impulse, calcium ions rush back across the ER membrane into the cytosol and trigger contraction of the muscle cell.
Rough ER- Many times of cells secrete proteins produced by ribosomes attached to rough ER. As a polypeptide chain grows from a bound ribosomes, it is threaded into the ER lumen through a pore formed by a protein complex in the ER membrane. As the new protein enters the ER lumen, it folds into its native shape. Most secretory proteins are glycoproteins, which have carbohydrates covalently bonded to them. After secretory proteins are formed, the ER membrane keeps them separate from proteins that are produced by free ribosomes and will remain in the cytosol. Secretory proteins depart from the ER wrapped in the membranes of vesicles that bud like bubbles from a specialized region called transitional ER. Transport vesicles are the vesicles in transit from one part of the cell to another. Rough ER is also a membrane factory for the cell; it grows in place by adding membrane proteins and phospholipids to its own membrane. As polypeptide destined to be membrane proteins grow from the ribosomes, they are inserted into the ER membrane and are anchored there by their hydrophobic portions. The rough ER makes its own membrane phospholipids; enzymes build into the ER membrane assemble phospholipids from precursors in the cytosol. The ER membrane expands and is transferred in the form of transport vesicles to other components of the endomembrane system.
## Golgi Apparatus
Golgi is a center of manufacturing, warehousing, sorting, and shipping. The products of the ER are modified and stored and then sent to other destinations. Golgi apparatus is extensive in cells specialized for secretion. The Golgi apparatus consists of flattened membranous sac, cisternae. The membrane of each cisterna in a stack separates ints internal space from the cytosol. Besicles concentrated in the vicinity of the Golgi apparatus are engaged in the transfer of material between parts of the Golgi and other structures. Golgi stack has a distinct structural polarity with the membrane of cisternae on opposite side of the stack different in thickness and molecular composition. The two poles of a Golgi stack are referred to as the cis face and the trans face; cis is the receiving and trans is shipping departments of the Golgi apparatus. The cis face is usually located near ER. Transport vesicles move material from the ER to the Golgi apparatus. A vesicle that buds from the ER can add its membrane and the contents of its lumen to the cis face by fusing with a Golgi membrane. The trans face give rise to vesicles, which pinch off and travel to other sites. The products of ER are usually modified during their transit from the cis region to the trans region of the Golgi. Various Golgi enzymes modify the carbohydrate portions of glycoproteins; carbohydrates are first added to proteins int he rough ER during the process of polypeptide synthesis. The carbohydrate on the resulting glycoprotein is then modified as it passes through the rest of the ER and the Golgi. The Golgi removes some sugar monomers and substitutes other, producing a large variety of carbohydrates. In addition, the Golgi apparatus manufactures certain macromolecules by itself. Many polysaccharides secreted by cells are Golgi products, including pectins and certain other non-cellulose polysaccharides made by plant cells and incorporated along with cellulose into their cell walls. Similar to secretory proteins, non-protein Golgi products will be secreted depart from the trans face of the Golgi inside transport vesicles that eventually fuse with the plasma membrane.
The Golgi manufactures and refines its products in stages, with different cisternae containing unique teams of enzymes. Recent research has give rise to a new model of the Golgi as a more dynamic structure; According to the cisternal maturation model, the cisternae of the Golgi actually progress forward from the cis to the tras face of the Golgi, carrying and modifying their cargo as they move. Before a Golgi stack dispatches its products by budding vesicles fromt he trans face, it sorts these products and targets them for various parts of the cell. Molecular identification tags, such as phosphate groups added to the Golgi products, aid in sorting. Transport vesicles budded fromt he Golgi may have external molecules on their membranes that recognize "docking site" on the surface of specific organelles or on the plasma membrane, therefore, targeting the vesicle appropriately.
## Lysosomes
Lysosome is a membranous sac of hydrolytic enzymes that an animal cell uses to digest macromolecules. Lysosomal enzymes work best in the acidic environment found in lysosomes. If a lysosome breaks open or leaks its contents, the released enzymes are not very active because the cytosol has a neutral pH. However, excessive leakage from a large number of lysosomes can destroy a cell by autodigestion. Hydrolytic enzymes and lysosomal membrane are made by rough ER and then transferred to the Golgi apparatus for further processing. Proteins of the inner surface of the lysosomal membrane and the digestive enzymes are spared from destruction by having three dimensional shapes that protect vulnerable bonds from enzymatic attack.
Phagocytosis is a process that amoebas and many other protists eat by engulfing smaller organisms or other food particles. The food vacuole formed , and then fuses with a lysosome and digests the food. Digestion products pass into cytosol and become nutrients for the cell. In human body, white blood cell helps defend the body by engulfing and destroying bacteria and other invaders.
Lysosome use their hydrolytic enzymes to recycle the cell's own organic material; this is called autophagy. During autophagy, damaged organelle or small amount of cytosol become surrounded by a double membranes, and lysosome fuses with the outer membrane of their vesicle. The lysosomal enzymes dismantle the enclosed material, and the organic monomers are returned to the cyotosol for reuse. The lysosomes become engorged with indigestible substrates, which begin to interfere with other cellular activities.
## Vacuoles
Vacuoles are membrane-bounded vesicles whose functions vary in different kinds of cells. Food vacuoles are formed by phagocytosis. Many freshwater protists have contractile vacuoles that pump excess water out of the cell, thereby maintaining a suitable concentration of ions and molecules inside the cell. In plants and fungi, which lacks lysosomes, vacuoles carry out hydrolysis;
The central vacuole develops by the coalescence of smaller vacuoles, themselves derived from the endoplasmic reticulum and Golgi apparatus. The vacuolar membrane is selective in transportin solutes. As result, the solution inside the central vacuole is called cell sap, is different in composition from the cytosol. It can hold reserves of important organic compounds such as proteins stockpiled in the vacuoles of storage cells in seeds. Also it is the plant cell's main repository of inorganic ions, such as potassium and chloride. Many plant cells use their vacuoles contain pigments that color the cells. Vacuoles may also help protect the plant against predators by containing compounds that are poisonous or unpalatable to animals. The vacuole has a major role in the growth of plant cells, which enlarge as their vacuoles absorb water, enabling the cell to become larger with a minimal investment in new cytoplasm.
## Mitochondria and Chloroplasts
Mitochondria and chloroplasts are the organelles that convert energy to forms that cells can use for work. Mitochondria are the site of cellular respiration, the metabolic process that generates ATP by extracting energy from sugars, fats, and other fuels with the help of oxygen. Chloroplasts, are found in plants and algae, and they are the sites of photosynthesis. They convert solar energy to chemical energy by absorbing sunlight and using it to drive the synthesis of organic compounds such as sugar from carbon dioxide and water. Both of them are not part of endomembrane system. Mitochondria have two membrane separating their innermost space from the cytosol, and chloroplasts have three. The membrane proteins of mitochondria and chloroplasts are made not by ribosomes bound to the ER, but by free ribosomes in the cyotosol and by ribosomes contained within these organelles themselves. They also contain small amount of DNA that programs the synthesis of the proteins made on the organelle's ribosomes. Mitochondria and chloroplasts are semiautonomous organelles that grow and reproduce within the cell.
Mitochondria Mitochondria are found in all eukaryotic cells; Some cells have a single large mitochondrion, but more often a cell has hundreds or thousands of mitochondria. The number correlates with the cell's level of metabolic activity. The mitochondrion is enclosed by two membranes, each a phospholipid bilayer with a unique collection of embedded proteins. The outer membrane is smooth, but the inner membrane is convoluted, with infolding called cristae. The inner membrane divides the mitochondrion into two internal compartments : the first is the inter-membrane space, the narrow region between the inner and outer membranes and the second compartment, the mitochondrial matrix, is enclosed by the inner membrane. the matrix contains many different enzymes as well as the mitochondrial DNA and ribosomes. Enzymes in the matrix catalyze some steps of cellular respiration. Other proteins that function in respiration, including the enzyme that makes ATP are built into the inner membrane, As highly folded surface, the cristae give the inner mitochondrial membrane a large surface ares, thus enhancing the productivity of cellular respiration.
Chloroplasts The chloroplast is a specialized member of related plant organelles called plastids. Chloroplasts contain the green pigment chlorophyll, along with enzymes and other molecules that function in the photosynthetic production of sugar. Its shape is lens-shaped and found in leaves. The contents of a chloroplast are partitioned from the cytosol by an envelope consisting of two membranes separated by a very narrow intermembrane space. Inside the chloroplast is another membranous system in the form of flattened, inter-connected sacs called thylakoids. Thylakoids are stacked like poker ships, and each stack is called granum. The fluid outside the thylakoids is the stroma which contains the chloroplast DNA and ribosomes as well as many enzymes. The membranes of the chloroplast divide the chloroplast space into three compartments: the intermembrane space, the stroma, and the thylakoid space.
## Cytoskeleton
Cytoskeleton is a network of fibers extending throughout the cytoplasm. It plays a major role in organizing the structure and activities of the cell. It is composed of three types of molecular structure: microtubules, microfilaments, and intermediate filaments. The main function of the cytoskeleton is to give mechanical support to the cell and maintain its shape. Cytoskeleton is stabilized by balance between opposing forces exerted by its elements. The cytoskeleton is more dynamic than an animal skeleton; it can be quickly dismantled in one part of the cell and reassembled in a new location, changing the shape of the cell. Also several types of cell motility involve the cytoskeleton. the cell motility encompasses both changes in cell location and more limited movements of parts of the cell. Cell motility require the interaction of the cytoskeleton with motor proteins. Cytoskeletal elements and motor proteins work together with plasma membrane molecules to allow whole cells to move along fibers outside the cell. The cytoskeleton is also involved in regulating biochemical activities in the cell in response to mechanical stimulation forces exerted by extracellular molecules via cell-surface proteins are apparently transmitted into the cell by cytoskeletal elements, and the forces may even reach the nucleus.
Microtubules- thickest
All eukaryotic cells have microtubules; the wall of the hollow tube is constructed from a globular protein called tubulin. Each tubulin protein is a dimer, a molecule made up of two subunits. A tubulin dimer consists of two slightly different polypeptides, alpha-tublin, and beta-tubulin. Microtubules grow in length by adding tubulin dimers. Due to the architecture of a microtubules, its two ends are slightly different; one end can accumulate or release tubulin dimers at a much higher rate than the other, therefore, growing and shrinking significantly during cellular activities. This is called the "plus end", not because it can only add tubulin proteins but because it's the end where both "on" and "off" rates are much higher. Microtubules shape and support the cell and also serve as tracks along which organelles equipped with other proteins can move.
In animal cells, microtubules grow out from a centrosomes, a region that is often located near the nucleus and considered a "microtubule-organizing center". Theses microtubules function as compression-resisting girders of the cytoskeleton. Within the centrosome are a pair of centrioles, each composed of nine sets of triplet microtubules arrange in a ring. Before division, the centrioles replicate; although centrosomes with centrioles may help organize microtubule assembly in animal cell,s they are not essential for this function in all eukaryotes.
Also specialized arrangement of microtubules is responsible for the beating of flagella and cilia. Thees are microtubule containing extensions that project from some cells. When cilia or flagella extend from cells that are held in place as part of a tissue layer, they can move fluid over the surface of the tissue. Flagella and cilia different in their beating patterns. A flagellum has an undulating motion that generates force in the same direction as the flagellum's axis. However cilia work more like oars, with alternating power and recovery strokes generating force in a direction perpendicular to the cilium's axis. A cillium may also act as a signal-receiving "antenna" for the cell. Cilia that have this function are nonmotile, and there is only one per cell. Membrane proteins on this kind of cilium transmit molecular signals from the cell's movement to its interior, triggering signaling pathways that may lead to changes int he cell's activities. Cillia-based signaling appears to be crucial to brain function and to embryonic development. Motile cilia and flagella share a common ultrastructure; each has a core of microtubules sheathed in an extension of the plasma membrane. Nine doublets of microtubules, the members of each parit sharing part of their walls, are arranged in a ring. This arrangement, referred to as the "9+2" pattern, is found in all eukaryotic flagella and motile cilia. Non-motile primary cilia have "9+0" pattern, lacking the central pari of microtubules. The microtubule assembly of a cilium or flagellum is anchored in the cell by a basal body, which is structurally very similar to a centriole.
In flagella and motile cilia, flexible cross-linking proteins, evenly spaced along the length of the cilium or falgellum, connect the outer doublets to each other and to the two central microtubules Each outer doublet also has paris of protruding proteins spaced along its length and reaching toward the neighboring doublet; These are large motor proteins called dyneins, composed of several polypeptides. Dyneins are responsible for the bending movements of the organelle. A dynein molecule performs a complex cycle of movements cause by changes in these shape of the proteins, with ATP providing the energy for these changes. The mechanics of dynein-based bending involve a process that resembles walking. A typical dynein protein has two "feet" that "walk" along the microtubule of the adjacent doublet, one foot maintaining contact while the other releases and reattaches one step further along the microtubules. Without any restraints ont he movement of the microtubules doublets, one doublet would continue to "walk" along and slide past the surface of the other, elongating the cilium or flagellum rather than bending it.
Microfillaments- thinnest
Microfilaments are solid rods about 7nm in diameter. They are also called as actin filaments because they are build from molecules of actin, a globular protein. A microfilament si a twisted double chain of actin subunits. Microfilaments can form structural networks, due to the presence of proteins that bind along the side of an actin filament and allow a new filament to extend as a branch. The structure role of microfilaments in the cytoskeleton is to bear tension. A cortical microfilaments, a three-dimensional network formed by microfilaments just inside the plasma membrane, helps support the cell's shape. This network give the outer cytoplasmic layer of a cell called the cortex.
In animal cells specialized for transporting materials across the plasma membrane, such as intestinal cells, bundles of microfilaments make up the core of microvilli. Microfillaments are well known for their role in cell motility, particularly as part of the contractile apparatus of muscle cells (myosin). Localized contraction brought about by actin and myosin also plays a role in amoeboid movement, which a cell such as an amoeba crawls along a surface by extending and flowing into cellular extension called pseudopodia. pseudopodia extend and contract through the reversible assembly of actin subunits nto microfilaments and of microfillaments into networks that convert cytoplasm fro a sol to a gel. The pseudopodium extends until the actin reassembles into a network.
In plant cells, both actin-myosin interactions and sol-gel transformations brought about by actin may be involved in cytoplasmic streaming, a circular flow of cytoplasm within cells.
intermediate filaments- middle range intermediate filaments are larger than the diameter of microfilaments but smaller than that of microtubules. Specialized for bearing tension (like microfilaments), intermediate filaments are a diverse class of cytoskeletal elements. Each type is constructed from a different molecular subunit such as keratins. Intermediate filaments are more permanent fixture of cells than are microfilaments and microtubules. Even after the death of the cell, intermediate filament networks often persist. Intermediate filaments are important in reinforcing the shape of a cell and fixing the position of certain organelles. For instance, the nucleus commonly sits within a cage made of intermediate filaments, fixed in location by braches of the filaments that extend into the cytoplasm. Other intermediate filaments make p the nuclear lamina that lines the interior of the nuclear envelope. In case where the shape of the entire cell is correlated with function, intermediate filaments support that shape.
## Cell Wall
Cell wall is an extracellular structure of plant cell that distinguishes them from animal cells. The wall protects the plant cell, maintains its shape, and prevents excessive uptake of water. The strong walls of specialized cells hold the plant up against the force of gravity. Plant cell walls are much thicker than the plasma membrane, and the exact chemical composition of the wall varies from species to species and even from one cell type to another in the same plant, but basic design of the wall is consistent. Microfibrils made of the polysaccharide cellulose are synthesized by an enzyme called cellulose synthase and secreted to the extracellular space, where they become embedded in a matrix of other polysaccharides and proteins. This combination of materials, strong fibers in a "ground substance" (matrix), is the same basic architectural design found in steel-reinforced concrete and in fiberglass.
A young plant cell first secrets a thins and flexible wall called the primary cell wall; as the cell grows, the cellulose fibrils are oriented at right angels to the direction of cell expansion, possibly affecting the growth pattern. Between primary walls of adjacent cell is the middle lamella, which is a thin layer rich in sticky polysaccharides pectins. The middle lamella flues adjacent cells together. When the cell mature and stops growing, it strengthens its wall. Some plant cells do this simply by secreting hardening substances into the primary wall, but other cells add a secondary cell wall between the plasma membrane and the primary wall. Then secondary wall, often deposited in several laminated layer, has a strong and durable matrix that afford the cell protection and support.
## References
Berg, Jeremy M., John L. Tymoczko, and Lubert Stryer. Biochemistry. 7th ed. New York: W.H. Freeman, 2012. Print.
Reece, Campbell, Lisa A. Urry, Michael L. Cain, Steven A. Wasserman, Peter V. Minosky, and Robert B. Jackson. Biology. 8th ed. San Francisco: Cummings, 2010. Print.
Prokaryotes
# Unique Properties of Plant Cells
Cell_biology | Types of cells
<< Eukaryotes | Unique Properties of Plant Cells
Plant Cells have a number of important differences compared to their animal counterparts. The major ones are the Chloroplasts, Cell walls and Vacuoles. Unlike animal cells, plant cells do not have centrioles.
## Chloroplasts
The chloroplasts are an organelle similar to the mitochondria in that they are self reproducing and they are the energy factories of the cell.they are near the large center vacuole. There most of the similarities ends. Chloroplasts capture light energy from the sun and convert it into ATP and sugar. In this way the cell can support itself without food.
## Vacuoles
Plants often have large structures containing water surrounded by a membrane in the center of their cells. These are vacuoles and act as a store of water and food (in seeds), a place to dump wastes and a structural support for the cell to maintain turgor. When the plant loses water the vacuoles quickly lose their water, and when plants have a lot of water the vacuoles fill up. In mature plants there is usually one large vacuole in the centre of the cell.
## Cell walls
Plant cells are not flaccid like animal cells and have a rigid cell wall around them made of fibrils of cellulose embedded in a matrix of several other kinds of polymers such as pectin and lignin. The cellulose molecules are linear and provide the perfect shape for intermolecular hydrogen bonding to produce long, stiff fibrils. It is the cell wall that is primarily responsible for ensuring the cell does not burst in hypotonic surroundings.
# Membranes
Parts of the cell The cell membrane is very important, because it works as a selective filter that allows only certain things to come inside or go outside the cell, it act as a body guard for our body.It can maintain a stable and healthy environment for cell in order to keep people healthy.
plant cell membranes are rigid walls, and animal cell membranes are lipid bilayers.
Plasma membrane bilayer
The phospholipid bilayer which the cell membrane is an example of, is composed of various cholesterol, phospholipids, glycolipids, blagoscony and proteins. Below is an example of a simple phospholipid bilayer.
The smaller molecules shown between the phospholipids are Cholesterol molecules. They help to provide rigidity or stability to the membrane. The two main components of phospholipids are shown in these figures by blue circles representing the hydrophilic head groups and by long thin lines representing the hydrophobic fatty acid tails.
Both the interior of the cell and the area surrounding the cell is made up of water or similar aqueous solution. Consequently, phospholipids orient themselves with respect to the water and with each other so that the hydrophilic ("water loving") head groups are grouped together and face the water, and the hydrophobic ("water fearing") tails turn away from the water and toward each other. This self-organization of phospholipids results in one of just a few easily recognizable structures. Cell membranes are constructed of a phospholipid bilayer as shown above.
Smaller structures can also form, known as 'micelles' in which there is no inner layer of phospholipid. Instead, the interior of a micell is wholly hydrophobic, filled with the fatty acid chains of the phospholipids and any other hydrophobic molecule they enclose. Micelles are not so important for the understanding of cellular structure, but are useful for demonstrating the principles of hydrophilicity and hydrophobicity, and for contrasting with lipid bilayers.
At least 10 different types of lipids are commonly found in cell membranes. Each type of cell or organelle will have a different percentage of each lipid, protein and carbohydrate. The main types of lipids are:
• Cholesterol
• Glycolipids
• Phosphatidylcholine
• Sphingomyelin
• Phosphatidylethnolamine
• Phosphatydilinositol
• Phosphatidylserine
• Phosphatidylglycerol
• Diphosphatidylglycerol (Cardiolipin)
• Phosphatidic acid
## The Cell Membrane is Asymmetric
The cell membrane tends to have different composition on one side of the membrane than on the other side of the membrane. The differences can be caused by the different ratios or types of amphipathic lipid-based molecules, the different positioning of the proteins (facing in or facing out), or the fixed orientations of proteins spanning the membrane. Additionally, there are different enzymatic activities in the outer and inner membrane surfaces.
The reason the cell membrane is asymmetric is because when the proteins are synthesized by the preexisting membranes, they are inserted into the membrane in an asymmetric manner. The asymmetry of the cell membrane allows the membrane to be rigid and allows the cell to have a different intracellular environment from the existing extracellular environment. Additionally, the cell membrane's phospholipids are distributed asymmetrically across the lipid bilayer, in a phenomenon called membrane phospholipid asymmetry. There are three mechanisms for transmembrane movement of phospholipids: 1) spontaneous diffusion, 2) facilitated diffusion, 3) ATP-dependent active translocation.
The spontaneous diffusion is a form of passive transport. Because passive transport does not require energy to transport non-polar substances through the membrane, this can happen spontaneously. Facilitated diffusion, like spontaneous diffusion, is a form of passive transport. The molecules or ions in this diffusion pass through the membrane by using specific transmembrane transport proteins.
Membrane transport of small molecules Because animal membrane proteins are lipid bilayer which are inner hydrophobic, this character prohibits polar molecules. Transport proteins can provide help for this situation. It can transport polar molecules across the membrane. There are several types of membrane transport proteins. They are uniports and cotransport. Uniports can move solutes from one side to another, change the position of the proteins. Cotransport systems can simultaneously sending two solutes across the lipid bilayer. Solutes are sent in the same direction or opposite directions Transport proteins does not need to be acts natural direction.
Membrane Transport of Macromolecules Membrane transport of Macromolecules can divide into two parts, they are exocytosis and endocytosis. In exocytosis, the contents of vesicles are released when the vesicle fuses with the cell membrane. There are five steps involved, which are vesicle trafficking, vesicke tethering, vesicle docking, vesicle priming and vesicle fusion. In endocytosis the membrane depresses and pinches off, enclosing the molecule. In receptor-mediated endocytosis, coated pits and vesicles bind to specific receptors on the cell surface, allowing the cell to select what molecules to take and what to reject.
# Organelles
Parts of the cell
Schematic of typical animal cell, showing subcellular components.
Organelles: (1) nucleolus (2) nucleus (3) ribosome (4) vesicle (5) rough endoplasmic reticulum (ER) (6) Golgi apparatus (7) Cytoskeleton (8) smooth ER (9) mitochondrion (10) vacuole (11) cytoplasm (12) lysosome (13) centrioles
## Nucleus
The nucleus contains genetic material or DNA in the form of chromatin, or, during mitosis or late interphase, chromosomes. All transcription and replication of genetic material takes place within the nucleus, as does RNA processing. The nucleolus also resides within the nucleus, and is responsible for RNA transcription and folding. Translation of RNA transcripts takes place outside of the nucleus.
## Mitochondria
A mitochondrian is the organelle responsible for a cell's metabolism. It synthetizes ATP through a protein called ATP synthase. Mitochondria have a double membrane. An outer membrane and a folded inner membrane. The internal membrane, called the cristae is invaginated (folded or creased), to maximize surface area enabling it to hold more ATP synthases. It is called as "the powerhouse of the cell" which is present in the eukaryotic organisms. It has matrix inside the inner membrane.
## Ribosomes
Ribosomes are responsible for protein synthesis. They are comprised of interacting protein and nucleic acid chains. Broadly, ribosomes are comprised of a large and a small subunit. The small subunit functions to attach to the mRNA strand and hold it in place during translation, while the large subunit holds and manufactures the growing polypeptide chain. The large subunit is further subdivided into the A (aminoacyl), P (peptidyl), and E (exit) binding sites.
Aminoacyl Binding Site The aminoacyl binding site binds a charged tRNA whose anticodon matches the codon in the A site.
Peptidyl Binding Site The peptidyl binding site contains the molecular machinery that transfers the bound polypeptide from the tRNA to the polypeptide chain, and holds the growing chain in place.
Exit Site The exit site is the terminal binding site for tRNA, where discharged tRNA's are released from the translation complex.
## Endoplasmic Reticulum
The Endoplasmic Reticulum (ER) acts as a transport from the nucleus and ribosomes to the Golgi apparatus. There are two types of endoplasmic reticulum:
### Smooth ER
Smooth ER act as transport for various things, mainly the RNA from the nucleus to the ribosomes (RNA is a small piece of the DNA code specifically designed to tell the ribosomes what to make). Smooth ER appears smooth in texture, hence the name. Smooth ER plays an important role in lipid emulsification and digestion in the cell.
### Rough ER
Rough ER are "rough" because of the ribosomes embedded in them. The rough ER takes the protein to the Golgi apparatus to be packaged into vacuoles
## Golgi Complex
The Golgi Complex basically functions as a "packaging center" for the cell, attaching "address labels" (functional groups) to various cell products to direct them to their respective locations, and "packaging" the products into vacuoles to ensure delivery. Anatomically, the Golgi Complex consists of layers of lipid membrane stacked one one top of another, with a cis face and a trans face. As the molecular product being packaged moves through the complex, various enzymes act upon it to induce vacuole formation and functional group attachment.
## Vacuole
Paramecium, with contractile vacuoles indicated using arrows.
Vacuoles are cellular storage places. Like the cell membrane, they are comprised of a lipid bilayer that functions as a selectively permeable barrier to regulate movement of materials into and out of the compartment. They can serve a variety of purposes, storing food, water, or waste products, or immune functions such as containing dangerous materials or maintaining turgor pressure (in plants). Vacuoles serve very different purposes in plant cells than they do in animal cells.
Plant Cells In plants, vacuoles comprise a significant portion of the cell's total volume and often contribute significantly to the function of a differentiated cell. For example, vacuoles in stomata cells contain large numbers of potassium ions, which can be pumped in or out to open or close the stomata.
Animal Cells In animal cells, vacuoles serve more subordinate roles, such as assisting in endo- and exocytosis or basic storage of food and waste.
Central Vacuole The central vacuole is found only in plant cells. It is filled with water and is pressurised, like a balloon. This forces all the other organelles within the cell out toward the cell wall. This pressure is called turgor pressure and is what gives plants their "crisp" and firm structure.
## Peroxisomes
Peroxisomes perform a variety of metabolic processes and as a by-product, produce hydrogen peroxide. Peroxisomes use peroxase enzyme to break down this hydrogen peroxide into water and oxygen.
## Lysosomes
Lysosomes are vacuoles containing digestive and destructive membranes. In white blood cells, these are used to kill the bacteria or virus, while in tadpole-tail cells they kill the cell by separating the tail from the main body.
They also do much of the cellular digestion involved in apoptosis, the process of programmed cell death.
# Energy supply (chloroplasts and mitochondria)
Parts of the cell
Chloroplasts are the organelles used for photosynthesis (a process that incorporates light energy into storage as chemical energy) whereas mitochondria used in respiration (a process that releases stored chemical energy). It assumed that you already know the information about these organelles explained in the organelles section. If you have not read the entries on chloroplasts and mitochondria from there yet, please go back and read them now.
# Cell cycle
The normal cell cycle consists of 2 major stages. The first is interphase, during which the cell lives and grows larger. The second is Mitotic Phase. Interphase is composed of three subphases. G1 phase (first gap), S phase (synthesis), and G2 phase (second gap). The interphase is the growth of the cell. The normal cell functions of creating proteins and organelles. The Mitotic Phase is composed of Mitosis and Cytokinesis. Mitosis, when the cell divides. Mitosis can be further divided into multiple phases. Cytokinesis, which is when the two daughter cells complete their separation. Mitosis is the division of the nucleus and cytokinesis is the division of the cytoplasm. There is some overlap between there two sub phases. Reproductive cell division is called meiosis, which yields a nonidentical daughter cells that have only one set of chromosomes. In other words, they have half as many chromosomes as the parent cell. Meiosis occurs in gonads, ovaries or testes. Therefore combining two gametes together produce 46 chromosomes.
## From Wikipedia
The cell cycle is the cycle of a biological cell, starting from the time it is first formed from a dividing parent cell until its own division into two cells, consisting of repeated mitotic cell division and interphase (the growth phase). A cell spends the overwhelming majority of its time in the interphase(about 90% of time).
## Background Information
DNA, deoxyribonucleic acid, consists of four nucleic acids, A, T, C, and G. In a cell, the DNA provides the directions for creating all of the proteins necessary for cell viability, health, growth, function, and replication. The unique DNA sequence that encodes each protein is called a gene, and the complete set of genes for an organism or cell is referred to as it's genome. Prokaryotic genomes are often a single long DNA molecule, and Eukaryotic genomes consist of number of DNA molecules. A typical human cell has about 2 m of DNA, which is 250,000 times greater than the cell's diameter. Before a cell divides the DNA is first copied then separated so that each daughter cell ends up with a complete genome. Chromosomes are the packaged DNA molecules. Because of chromosomes, the replication and distribution of so much DNA is manageable. Every eukaryotic species has a characteristic number of chromosomes in each cell nucleus. They contain two sets of each chromosome: one set inherited from each parent. For example human somatic cells (all body cells except the reproductive cells) each contain 46 chromosomes; the reproductive cells, gametes, have half as many chromosomes as somatic cells. The number of chromosomes in somatic cells varies widely among species. Eukaryotic chromosomes are made of chromatin that is a complex of DNA and associated protein molecules. Each single chromosome contains one very long, linear DNA molecule that carries several hundred to a few thousand genes; the associated proteins maintain the structure of the chromosome and help control the gene activity. When a cell is not dividing, each chromosome is a long thins chromatic fiber; however after DNA duplication chromosomes condense. Each chromatin fiber coils and folds. Each duplicated chromosome has two sister chromatids, containing an identical DNA molecule, initially attached along adhesive protein complex; such attachment is called sister chromatid cohesion. In condensed form of chromosome, a center narrow part is called centromere, a specialized region where the two chromatids are closely attached. The other part of a chromatid on either side of the centromere is referred as arm. Once the sister chromatids separate, they are considered individual chromosomes.
## Overview
Schematic of the cell cycle. I=Interphase, M=Mitosis. The duration of mitosis in relation to the other phases has been exaggerated in this diagram.
The mitotic phase includes both mitosis and cytokinesis which is usually the shortest part of the cell cycle. Interphase accounts about 90%of the cycle; during interphase the cell grows and copies its chromosomes in preparation for cell division. Interphase is divided into sub-phases: G1 phase ("first gap"), the S phase ("synthesis"), and G2 phase ("second gap"). The chromosomes are duplicated only during the S phase. During G1 phase cell grows until S phase where the cell prepares for the cell division during G2 phase. Based on human cell, M phase only takes about 1 hour while the S phase occupies about 10-12 hours.
The cell cycle consists of
• G1 phase, the first growth phase
• S phase, during which the DNA is replicated, where S stands for the Synthesis of DNA.
• G2 phase is the second growth phase, also the preparation phase for the next stage
• M phase or mitosis and cytokinesis, the actual division of the cell into two daughter cells
The cell cycle stops at several checkpoints and can only proceed if certain conditions are met, for example, if the cell has reached a certain diameter. Some cells, such as neurons, never divide once they become locked in a G0 phase.
## Mitosis
Mitosis has five stages: prophase, prometaphase, metaphase, anaphase, and telophase. Mitotic spindle starts to form in the cytoplasm during prophase. it is made of microtubules and other associated proteins. while the mitotic spindle assembles, the microtubules of the cytoskeleton disassemble, providing the material used to construct the spindle. In animal cells, the assembly of spindle microtubules starts at the centrosome, the microtubule-organizing center. In plant cells, the centrioles are not present.
During interphase in animal cells, the single centrosome replicates; the two centrosomes remain together near the nucleus and they move apart during prophase and prometaphase of mitosis as spindle microtubules grows. The two centrosomes are located at the opposite end of the cell. Then aster, a radial array of short microtubules, extends from each centrosome. Kinetochore is a structure of proteins associated with specific sections of chromosomal DNA at the centromere. Each of the two sister chromatids of a replicated chromosome contains kinetochore as it face in opposite direction. During prometaphase, kinetochore microtubules form as come of the spindle microtubules attach to the kinetochores. After the microtubuels are attached to chromosome's kinetochores, the chromosome begins to move towards the pole from which those microtubules extend. the chromosomes moves in a motion like a tug-of-war. Metaphase plate is the imaginary plane that formed during metaphase the centromeres of all the duplicated chromosomes are on the plane midway between the spindle's two poles. The other microtubules that did not attach to kinetochores overlap and interact with other nonkinetochore microtubules from the opposite pole. The nonkinetochore microtubules are responsible for elongating the whole cell during anaphase. During anaphase, the cohesins holding the sister chromatids of each chromosome are cleaved by enzymes. Then the chromatids separated, and they move towards the opposite ends of the cell. The region of overlap is reduced as motor proteins attached to the microtubules move away from one another, using ATP. As the microtubules push apart from each other, their spindle poles are pushed apart, elongating the cell. As the duplicate groups of chromosomes arrive at the opposite ends of the elongated parent cell, the telophase begins; during telophase nuclei reforms and cytokinesis begins.
• G2 of Interphase:During G2 phase, a nuclear envelope bounds the nucleus, and two centrosomes forms by replication of a single centrosome. In animal cells, each centrosome contains two centrioles. The chromosomes are duplicated during S phase but cannot be seen since they are not condensed yet.
• Prophase: the chromatin fibers coils and dense into chromosomes and the nucleoli disappear. Each duplicated chromosome has tow identical sister chromatids joined at their centromeres along with their arms by cohesins, then the mitotic spindle form. The asters are the radial arrays of shorter microtubules that extend from the centrosomes. Propelled by the lengthening microtubules, the centrosomes move away from each other.
• Prometaphase: As the nuclear envelope fragments, the microtubules extending from each centrosome invade the nuclear area. the chromosome become more condensed as each of the two chromatids of each chromosome has a kinetochore. Some of the microtubules attach to the kinetochores ("kinetochore microtubules" and other nonkinetochore microtubules interact with each from fromt he opposite pole of the spindle.
• Metaphase: Metaphse is the longest stage of mitosis. The centrosomes are placed at the opposite poles of the cell. The chromosomes' centromeres lie ont he metaphse plate as the chromosome convene on the metaphase plate. Each kinetochores of the sister chromatids are attacged to kinetochore microtubules coming from opposite poles.
• Anaphase: Anaphase is the shortest stage of mitosis, and begins when the cohesin proteins are cleaved, allowing the two sister chromatids of each pair to part suddenly. The two liberated daughter chromosomes moves towards oppostie ends of the cell as the kinetochore microtubules shorten. The cell starts to elongate and nonkinetochore microtubules lengthen. By the end of anaphase, the two ends of the cell have equivalent collections of chromosome.
• Telophase: Two daughter nuclei form in the cell, and nuclear envelopes arise from the fragments of the parent cell's nuclear envelope. As nucleoli reappear, the chromosome become less condensed, and completes the division of the one nucleus into two genetically identical nuclei.
• Cytokinesis: In animal cells, cytokinesis involves formation of cleavage furrow; in plante cell the cleavage furrow does not exist. The formation of cell wall in the middle of cell (cell plate) divides the cell into two daughter cells.
## Details of mitosis
Schematic of interphase (brown) and mitosis (yellow).
## Cytokinesis
The cytokinesis process begins with cleavage. Cleavage furrow, a shallow groove in the cell surface near the old metaphase plate, is the first sign of cleavage. As it process, contractile ring of actin microfilaments form on the cytoplasmic side. The actin microfilaments interact with the myosin molecules, and cause the ring to contract. As the cleavage furrow deepens, the cell is separated into two with its own nucleus. For plant cells, there is no cleavage furrow because they have the cell walls. Instead of forming cleavages, vesicles derived from the Golgi apparatus move along microtubules to the middle of the cells, and forms cell plate. As the cell plate enlarges, and surrounding membrane fuses with the plasma membrane along the perimeter of the cell and from two daughter cells.
## Binary Fission
Binary fission is a method of asexual reproduction by "division in half". In prokaryotes, binary fission does not involve mitosis, but in single celled eukaryotes that undergo binary fission. In bacteria, motst genes are carried on a single bacterial chromosome that consists of a circular DNA molecule and associated proteins. The chromosome of the bacterium Escherichia coli, is 500 times as long as the cell when it is sctreched out. At the origin of replication, DNA of the bacterial chromosome begins to replicate. As the chromosome continues to replicate, one origin moves rapidly toward the opposite end of the cell, and the cell elongates. When the replication is complete the bacterium is about twice its initial size, and its plasma membrane grows inward, dividing the parent E. coli cell into two daughter cells. Bacteria don’t have mitotic spindles; the two origins of replication end up at opposite ends of the cell or in some other very specific location.
## The Evolution of Mitosis
Since the prokaryotes were on Earth more than a billion years than eukaryotes that mitosis had its origins in simpler prokaryotic mechanism of the cell reproduction can be assumed. Some of the proteins involved in bacterial binary fission are related to eukaryotic proteins that function in mitosis. Possible hypothesis of evolution of mitosis is that prokaryotic cell's reproduction gave rise to mitosis.
## The Cell Cycle Control System
Based from mammalian cell grow experiment, possible hypothesis was supported: the cell cycle is driven by specific signaling molecules present in the cytoplasm. In this experiment two cells in different phase of the cell cycle were fused to form a single cell with two nuclei. One cell was in the S phase and the other was in G1, and G1 nucleus immediately entered the S phase, as though stimulated by chemicals present in the cytoplasm of the first cell. Therefore, if a cell undergoing mitosis (M phase) was fused with another cell in any stage of its cell cycle, the second nucleus enteres mitosis. Other experiments on animal cells and yeasts demonstrates the sequential events of the cell cycle control system; the cell cycle control system operates set of molecules in the cell that both triggers and coordinates key events in the cell cycles. The cell cycle control system proceeds on its own, but it is regulated at certain checkpoints by internal and external signals. Animal cells have built-in stop signals that halt the cell cycle at checkpoints until they get go-ahead signals. The signals report whether crucial cellular processes that should have occurred by that point have in fact been completed correctly and thus whether or not the cell cycle should proceed. The three check points are in G1, G2, and M phase. For mammalian cells, G1 check points are the most important. When a cell receives a go-ahead signal at the G1 checkpoint, the cell complete the G1, S, G2 and M phases and divide; however when a cell does not get a go-ahead signal, it will exit the cycle and enter non dividing state, G0 phase. Most of human cells are in G0 phase, such as mature nerve cells and muscle cells. However the liver cells can re-enter the cycle by external signals such as growth factor released during injury. Rhythmic fluctuations in the abundance and activity of cell cycle control molecules pase the sequential events of the cell cycle. The regulatory molecules are portins of two types: protein kinases and cyclins. Portin kinases are enzymes that activate or inactivate other proteins by phosphorylating. The protein kinases give the go-ahead signals at the G1 and G2 checkpoints. The kinases that drive the cell cycle are present at a constant concentration in the growing cell, but they are in an inactive form. In order to activate them, kinase must be attached to a cyclin, a protein that cyclically fluctuating concentration in the cell. Because of such requirement, these are called cyclin-dependent kinases or Cdks. The activity of cdks rises and falls with changes in the concentration of its cyclin partner. The cylclin level rises during the S and G2 phases and then falls rapidly during M phase. MPF, the maturation -promoting factor, or M-phase -promoting factor, activity corresponds to the peaks of cyclin concentration. MPF triggers the cell's passage past the G2 checkpoint into M phase. MPF acts both directly as a kinase and indirectly by activating other kinases. During anaphase, MPF hels switch itself off by initiating a process that leads to the destruction of its own cyclin. The Cdk, noncyclin part of MPF, persists in the cell in inactive form until it associates with new cyclin molecules synthesized during the S and G2 phase of the next round of the cycle. Density-dependent inhibition is a phenomenon in which crowded cells stop dividing. It is caused by external physical factor. Also most animal cells exhibit anchorage dependence; in order to divide, the cells must be attached to a substratum; like a cell density, anchorage is signaled to the cell cycle control system via pathways involving plasma membrane proteins and elements of cytoskeleton linked to them. The loss of cell cycle controls leads to cancer cells, which exhibit neither density-dependent inhibition nor anchorage dependence.
## Reference
Berg, Jeremy M., John L. Tymoczko, and Lubert Stryer. Biochemistry. 7th ed. New York: W.H. Freeman, 2012. Print.
Reece, Campbell, Lisa A. Urry, Michael L. Cain, Steven A. Wasserman, Peter V. Minosky, and Robert B. Jackson. Biology. 8th ed. San Francisco: Cummings, 2010. Print.
# Meiosis
Meiosis is a special type of cell division that is designed to produce gametes. Before meiosis occurs, the cell will be double diploid and have a pair of each chromosome, the same as before mitosis.
Meiosis consists of 2 cell divisions, and results in four cells. The first division is when genetic crossover occurs and the traits on the chromosomes are shuffled. The cell will perform a normal prophase, then enter metaphase during which it begins the crossover, then proceed normally through anaphase and telophase.
The first division produces two normal diploid cells, however the process is not complete. The cell will prepare for another division and enter a second prophase. During the second metaphase, the chromosome pairs are separated so that each new cell will get half the normal genes. The cell division will continue thorough anaphase and telophase, and the nuclei will reassemble. The result of the divisions will be 4 haploid gamete cells.
## Crossover
Crossover is the process by which two chromosomes paired up during prophase I of meiosis exchange a distal portion of their DNA. Crossover occurs when two chromosomes, normally two homologous instances of the same chromosome, break and connect to each other's ends. If they break at the same locus, this merely results in an exchange of genes. This is the normal way in which crossover occurs. If they break at different loci, the result is a duplication of genes on one chromosome and a deletion on the other. If they break on opposite sides of the centromere, this results in one chromosome being lost during cell division.
Any pair of homologous chromosomes may be expected to cross over three or four times during meiosis. This aids evolution by increasing independent assortment, and reducing the genetic linkage between genes on the same chromosome.
# Mitosis
Mitosis is the normal type of cell division. Before the cells can divide, the chromosomes will have duplicated and the cell will have twice the normal set of genes.
The first step of cell division is prophase, during which the nucleus dissolves and the chromosomes begin migration to the midline of the cell. (Some biology textbooks insert a phase called "prometaphase" at this point.)The second step, known as metaphase, occurs when all the chromosomes are aligned in pairs along the midline of the cell. As the cell enters anaphase, the chromatids, which form the chromosomes, will separate and drift toward opposite poles of the cell. As the separated chromatids, now termed chromosomes, reach the poles, the cell will enter telophase and nuclei will start to reform. The process of mitosis ends after the nuclei have reformed and the cell membrane begins to separate the cell into two daughter cells, during cytokinesis.
Mitosis divides genetic information during cell division.
The mitotic phase which includes both mitosis and cytokinesis is the shortest part of the cell cycle. The interphase cycle accounts for about 90% of the cell cycle. This phase is where the cell grows and copies its chromosomes in preparation for cell division. In the G1 phase which is also called the “first gap” the cell grows as it copies its chromosomes. In S phase, the cell starts to synthesize the DNA and completes preparation for cell division. In G2 it starts to divide.
In biology, Mitosis is the process of chromosome segregation and nuclear division that follows replication of the genetic material in eukaryotic cells. This process assures that each daughter nucleus receives a complete copy of the organism's genetic material. In most eukaryotes, mitosis is accompanied with cell division or cytokinesis, but there are many exceptions, for instance among fungi. There is another process called meiosis, in which the daughter nuclei receive half the chromosomes of the parent, which is involved in gamete formation and other similar processes, which makes the parent cell still active.
Mitosis is divided into several stages, with the remainder of the cell's growth cycle considered interphase. Properly speaking, a typical cell cycle involves a series of stages: G1, the first growth phase; S, where the genetic material is duplicated; G2, the second growth phase; and M, where the nucleus divides through mitosis. Mitosis is divided into prophase, prometaphase, metaphase, anaphase and telophase.
The whole procedure is very similar among most eukaryotes, with only minor variations. As prokaryotes lack a nucleus and only have a single chromosome with no centromere, they cannot be properly said to undergo mitosis.
## Prophase
The genetic material (DNA), which normally exists in the form of chromatin condenses into a highly ordered structure called a chromosome. Since the genetic material has been duplicated, there are two identical copies of each chromosome in the cell. Identical chromosomes (called sister chromosomes) are attached to each other at a DNA element present on every chromosome called the centromere. When chromosomes are paired up and attached, each individual chromosome in the pair is called a chromatid, while the whole unit (confusingly) is called a chromosome. Just to be even more confusing, when the chromatids separate, they are no longer called chromatids, but are called chromosomes again. The task of mitosis is to assure that one copy of each sister chromatid - and only one copy - goes to each daughter cell after cell division.
The other important piece of hardware in mitosis is the centriole, which serves as a sort of anchor. During prophase, the two centrioles - which replicate independently of mitosis - begin recruiting microtubules (which may be thought of as cellular ropes or poles) and forming a mitotic spindle between them. By increasing the length of the spindle (growing the microtubules), the centrioles push apart to opposite ends of the cell nucleus. It should be noted that many eukaryotes, for instance plants, lack centrioles although the basic process is still similar.
## Prometaphase
Some biology texts do not include this phase, considering it a part of prophase. In this phase, the nuclear membrane dissolves in some eukaryotes, reforming later once mitosis is complete. This is called open mitosis, found in most multicellular forms. Many protists undergo closed mitosis, in which the nuclear membrane persists throughout.
Now kinetochores begin to form at the centromeres. This is a complex structure that may be thought of as an 'eyelet' for the microtubule 'rope' - it is the attaching point by which chromosomes may be secured. The kinetochore is an enormously complex structure that is not yet fully understood. Two kinetochores form on each chromosome - one for each chromatid.
When the spindle grows to sufficient length, the microtubules begin searching for kinetochores to attach to.
## Metaphase
As microtubules find and attach to kinetochores, they begin to line up in the middle of the cell. Proper segragation requires that every kinetochore be attached to a microtubule before separation begins. It is thought that unattached kinetochores control this process by generating a signal - the mitotic spindle checkpoint - that tells the cell to wait before proceeding to anaphase. There are many theories as to how this is accomplished, some of them involving the generation of tension when both microtubules are attached to the kinetochore.
When chromosomes are bivalently attached - when both kinetochores are attached to microtubules emanating from each centriole - they line up in the middle of the spindle, forming what is called the metaphase plate. This does not occur in every organism - in some cases chromosomes move back and forth between the centrioles randomly, only roughly lining up along the midline.
## Anaphase
Anaphase is the stage of meiosis or mitosis when chromosomes separate and move to opposite poles of the cell (opposite ends of the nuclear spindle). Centromeres are broken and chromatids rip apart.
When every kinetochore is attached to a microtubule and the chromosomes have lined up along the middle of the spindle, the cell proceeds to anaphase. This is divided into two phases. First, the proteins that bind the sister chromatids together are cloven, allowing them to separate. They are pulled apart by the microtubules, towards the respective centrioles to which they are attached. Next, the spindle axis elongates, driving the centrioles (and the set of chromosomes to which they are attached) apart to opposite ends of the cell. These two stages are sometimes called 'early' and 'late' anaphase.
At the end of anaphase, the cell has succeeded in separating identical copies of the genetic material into two distinct populations.
## Telophase
The nonkinetochore microtubules elongate the cell and try to cut the cell in two. The nuclear envelopes start to become created by fragments of the parents cell’s nuclear envelope. Then, the chromatids start to become less tightly coiled together. By this point, cytokinesis is fully under way.
## Cytokinesis
Cytokinesis refers to the physical division of one eukaryotic cell. Cytokinesis generally follows the replication of the cell's chromosomes, usually mitotically, but sometimes meiotically. Except for some special cases, the amount of cytoplasm in each daughter cell is the same. In animal cells, the cell membrane forms a cleavage furrow and pinches apart like a balloon. In plant cells, a cell plate forms, which becomes the new cell wall separating the daughters. Various patterns occur in other groups.
In plant cells, cytokinesis is followed through by the usage of contracting ring of microfilaments that pull the cleavage furrow within itself, cutting the cell in two. In plant cells, vesicles from the Golgi apparatus start to form a cell plate within the center of the cell. When this cell plate solidifies and connects the two ends of the cell, a new cell wall is created and two daughter cells are produced.
## Regulation of Cell Cycle
Protein kinases are enzymes that activate or inactivate other proteins by phosphorylating them. These give out the signals for the G1 and G2 checkpoints to occur. However, to be active, the kinase must be attached to a cyclin. This is why it is called a CDK or a cyclin-dependent kinase.
Internal kinetochores exhibit a wait function. Not until all kinetochores are attached to a spindle microtubule does the cell process starts. This helps prevent some chromosomes from being left behind. Density dependent inhibition is when cells have a cue to multiply until a certain level of density is fulfilled. This means that a cell keeps multiplying until there is a full layer or until a certain level of pressure is built upon each other.
One possible explanation of why cancer cells do not follow normal signals is because they have an abnormality in the signaling pathway that conveys the growth factor’s signal to the cell-cycle control system. Usually, a cell will follow normal checkpoints due to the release of CDK in the system that regulate the cell process. However, in a cancer cell, the checkpoints are random. This means that because the cell does not follow density-dependent inhibition or follow the growth signals, the cell replicates at random points.
# Expression
Gene expression is the first stage of a process that decodes what the DNA holds in a cell. It is the expression of a gene that gives rise to a protein.
How does gene expression occur?
Genetic expression is a complex process. It is regulated by a series of mechanisms.
Gene expression begins with transcription of DNA, giving rise to messenger RNA (mRNA). This is performed by the enzyme RNA polymerase, which produces the mRNA. The mRNA in prokaryotes is coupled with several ribosomes which are responsible for translating proteins.
In eukaryotes, mRNA that is made from DNA is immature, and is called pre-mRNA. Pre-mRNA loses non-coding sections (called exons), maturing to mRNA. mRNA is coupled to ribosomes on Rough Endoplasmatic Reticle (RER) where translation happens. Translation is made when a new polypeptide is formed. The genetic code indeed says the order of pe polypeptides, but it doesn't give us a clue about it's tridimensional structure. Tridimensional structure is given by post-translational processes.
Translation occurs following transcription wherein the protein synthesis machinery gets into action and uses its tools to read out the message that the RNA holds.
There are some genes known to be without coding proteins. Yet, they work as regulation sequences in a cell. In this case, the sequences can enhance coding (called "enhancers") or they can inhibit (called "repressors"). When a protein is coupled with these genes, a substrate or hormone, they join together.
In multicellular organisms only certain cells will produce a certain type of protein; e.g.: Haemoglobin is encoded in every cell of a mammal organism (it includes humans), but only precursors of red blood cells are allowed to express it (red blood cells are not allowed to express it, because they lose their nucleus). However, the enhancers and repressors are present in every cell of a mammal.
Genetic Information
In nature, there is information found in all living cells. Different cultures have often studied this information and used various forms of recording techniques to display it. Ancient Egyptians, in particular, referred to this information and its records as "provider of attributes" and determined it ||| to mean several, and that was earlier in human history of recording something that was known about nature.
There were often other signs as well that accompanied Egyptian writings on the source of this "information key of life". Among them were double, water and wick of twisted flax. But the most central one, for modern science, of course, was the snake like determinative that meant a worm or serpent in the limit of life. This limit, water, was "N" meaning that something or someone is, the essence which would be referred to by the Greeks as "esse" or "ens", and in today's English terms, the "essence".
In anthropology, the language of gene expression is rooted in the sources of knowledge that Odhiambo Siangla of Kenya has called "rieko" and Jeremy Narby of Switzeland has termed the "cosmic serpent". Both Siangla and Narby are not only experts in cultures but are trained in communication and expression. And from both the key has been the "three letter word".
In the alphabet of the three letter word found in cell biology are the organic bases, which are adenine (A), guanine (G), cytosine (C) and thymine (T). It is the triplet recipe of these bases that make up the ‘dictionary’ we call in molecular biology genetic code.
The codal system enables the transmission of genetic information to be codified, which at the molecular level, is conveyed through genes.
What is gene ? A gene is a region of DNA that produces a functional RNA molecule. If a region of DNA is not functional, that region is not a transmissible form of information for protein synthesis. And because the information is not transmissible, it is not readily functional. There are various sizes of gene. The first recorded attempts to imagine the very small was the Horus Eye, which is also a pristine idea of limit. Today we talk about bases. The insulin gene, for example, has 1.7 x ${\displaystyle 10^{3}}$, about 1700 nucleotides. There exists a receptor gene known as low-density lipoprotein (LDL). This protein has 4.5 x${\displaystyle 10^{4}}$ nucleotides. In terms of nucleotides this (LDL) approximates to 45,000 nucleotides. Now, with the dystrophic gene as another example, we find the nucleotides to be around 2.0 x${\displaystyle 10^{8}}$, approximately 200,000,000 nucleotides in number.
Now, the introns. It is the noncoding regions of DNA that are called introns meaning the “intervening sequences”. Introns make up a greater part of the nucleotide sequences of a gene. The coding regions are called exons to mean “expression sequences”. They constitute a minority of the nucleotide progression of a DNA and they instruct cellular workshops for the formation of proteins via amino acids.
Through proteins, the expression of genetic information is achieved. In particular are the enzymes. Even during the ancient time the enzymes were understood and utilized well. The enzymes catalyze the chemical reactions of anabolic kind, that is, the building of cellular food and those of catabolic type, the braking down of food. The two processes are collectively termed metabolism. What, further, can we add about proteins?
We can further say that proteins are concentration of heteropolymers manufactured from amino acids. There are 20 amino acids used in synthesizing natural proteins. It is clear that a protein may consist of many, in fact, several hundred amino acid sediments. It is essentially unlimited in number to speak about how many different proteins we can make from combinations of amino acids. Mathematics explains it well. There is therefore a diverse set of proteins whose forms and functions can be achieved by means of a coding system explained below.
Genetic information flows unidirectional, from DNA to protein and with messenger RNA (mRNA) as intermediate. First, DNA encodes genetic information into an RNA molecule. This is called transcription (TC) of the information. Then the information gets converted into proteins, being named here translation (TL). It is this concept of information current that is called the Central Dogma of molecular biology. The Central Dogma is the fundamental theme in our exploration on gene articulation.
In order to complete the picture, we can add two further aspects of information flow. We can add duplication of the genetic material, which occurs prior to cell division. And that a DNA, in this case, represents duplication process, —DNA transfer. Wherefore in this case it is known as DNA replication. But where some viruses have RNA instead of DNA as their genetic material, we speak about reverse transcription (RT). With this transcription, we get a DNA molecule as a copy of the viral RNA genome.
In other words, genetic information, whether historically traced world wide (Narby,1998) or particularly assigned to ancient Africa (Siangla,1997) involves gene expression. Both DNA and RNA are polynucleotides, where nucleotides are the monomer—building units, which are composed of three basic subunits called nitrogenous base, sugar, and phosphoric acid. Genetic information is contained in DNA. The genetic code in DNA expresses the connection between the polynucleotide alphabet of four bases and 20 amino acids. In one strand of the parental DNA molecule, there is a dictated amino acid sequence strictly for protein production.
We will discuss in the next few postings, a relatively detailed understanding of the polymerization of amino acids sequence as directed by base sequences of messenger RNA.
At the moment, though, let us note that protein synthesis is an expression of genetic information. Protein synthesis is the cellular procedure, as we have said, of making proteins and involves two main processes: Transcription and Translation. The two processes mean that the direction of the synthesis is from DNA to RNA and then from RNA to protein respectively. Is this true to all organisms?
Yes. With a few exceptions, which are in mitochondria, and as stated above, some viruses become exceptions to this order because in their genetic material, they have RNA instead of DNA as their initial information source. However it is true that in all organisms, methods that relate the nucleotide sequence in messenger RNA to the amino acid sequence in proteins (genetic code proper) are the same. For in the given exceptions there occurs reverse transcription (RT). With that viral example of transcription noted, we get DNA molecular information being copied from the genome of viral RNA.
Building on this clue that is provided by transcription processes, we can readily see that a three-nucleotide sense codon denotes each amino acid. For example, UUU specifies phenylalanine, UCU specifies serine and GCA specifies alanine. But UAC and UAU both specify tyrosine. We will speak more about this tyrosine when expanding cell biology in the study of melanin.
Here now are other ways to see the remaining three properties of the genetic code. One is the contiguous property. With this property the codons do not overlap and at the same time they do not separated by spacers. The other is degenerate property in which there is more than one codon for some amino acids as exemplified by tyrosine in the above paragraph. And finally, there is the unambiguous property. With this genetic code of unambiguity, each codon specifies only one amino acid.
# Translation
The Translation Phase of Genetic Expression is divided into 2 Steps Transcription and Translation. During Transcription RNA Polymerase unzips the two halfs of the DNA where it needs to transcript. Then free RNA bases Attach to the DNA bases with the Polymerase starting at the promoter and ending at the Termination signal. From this the RNA can become mRNA, rRNA, or tRNA. The mRNA is a ribbon like strand that takes the genetic information from the nucleus of the cell to the ribosome. rRNA forms a globular ball that attaches to the rough E.R. to help make ribosomes. finally the tRNA forms a hair shaped landing base that reads the genetic information to make proteins. Translation happens when mRNA is pulled through a ribosome and tRNA reads the RNA bases on the mRNA to make anti-codons of 3 bases and brings amino-acids to form the protein. This starts with the condon AUG and ends at UAG. When done the protein forms the correct shape and does the task it was created for. This brings the genetic code from the nucleus, which it never leaves, to the cytoplasm of the cell where proteins are produced to upkeep the body. | 2017-06-24 09:01:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.376592218875885, "perplexity": 4237.3217973127585}, "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/1498128320243.11/warc/CC-MAIN-20170624082900-20170624102900-00178.warc.gz"} |
https://brilliant.org/discussions/thread/calculator-usage/ | ×
Calculator Usage?
Hi Everyone,
Actually I wanted to ask a general question. When you do Brilliant problems, do you typically use calculators?
The reason why I ask is sometimes I find problems that are made incredibly easy by the use of calculators, and I'm not quite sure that that's what their purpose is. At least in my point of view I like the idea of problems that are difficult even with the use of calculators, and to be fair a lot of the problems on Brilliant are as well. But do you have any personal opinions on the use of calculators, whether on Brilliant or elsewhere?
Thank you!
Note by Matt Wang
4 years, 9 months ago
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I'd say that it is fine to use a calculator if you already know how to manually arrive at the answer and the calculator is simply speeding up your progress. I don't think it is right to use it if you are trying to evade something that you may need to know how to do.
- 4 years, 9 months ago
Matt, I think the less you reach for your calculator while solving problems here, the more you'll get out of them.
I know if you want to compete in math competition, most will not allow calculators. I've been out of the realm of HS level math for a bit now and it's been a rude awakening trying to get back in without my trusty calculator (but that's kinda the point!). I can say that in my experience in college and professionally I've always gone straight to my calculator and have possibly lost a step as a result.
For this reason, I choose not to use a calculator while here. Obviously, I know how to cube numbers and do long division but my philosophy is that I'm solving these problems to get back into math fighting shape and taking an extra 30 seconds to evaluate 8! seems worth it to me.
- 4 years, 9 months ago
Well, I only use calculators when i there is a high risk of miscalculation , especially if not all the involve numbers are integers (e.g adding up fractions)
- 4 years, 9 months ago
I always try to avoid calculator while doing problems in brilliant . Rarely for some problems which need manual calculation with more time , i use calculator
- 4 years, 9 months ago
if the calculation is getting too long (however dat generally doesn't happens) and we know how to proceed manually we can use calculator.....[i think so]
- 4 years, 9 months ago
While I agree that it's useful to be proficient at arithmetic (useful for remembering certain numbers' properties and occasionally mildly entertaining), there's nothing particularly high or noble about it. I enjoy (and think I get most out of) the necessity of creativity (that is, being confused and trying to fix that), not the drudgery, in solving a problem. If I've worked out exactly what algorithm to apply to a problem and know I'm capable of carrying it out, it becomes boring*, and the rest is checking whether I'm correct.
However, improving your own arithmetical speed will increase the speed of getting trivia out of the way for minor calculations. *a few solutions are appealing by their very nature and are nice to ponder, though
- 4 years, 9 months ago
it's fine to use a calculator sometimes!
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https://www.aimsciences.org/article/doi/10.3934/jimo.2015.11.265 | American Institute of Mathematical Sciences
January 2015, 11(1): 265-289. doi: 10.3934/jimo.2015.11.265
A survey on models and algorithms for discrete evacuation planning network problems
1 Central Departments of Mathematics/CSIT, IOST, Tribhuvan University, Kathmandu, Nepal
Received May 2013 Revised January 2014 Published May 2014
With an increasing number of large-scale natural and man-created disasters over the last decade, there is growing focus on the application of operations research techniques for humanitarian relief in the emerging field of emergency evacuation. Even though a large diversity of models have been developed, many rely on solving network-flow problems on appropriate graphs. In this survey, we give a systematic collection of network flow models used in emergency evacuation and their applications. We especially focus on results interrelating these models. Considered models include max flows and min cost flows, lexicographic flows, quickest flows, and earliest arrival flows, as well as contraflows and time-dependent problems.
Citation: Tanka Nath Dhamala. A survey on models and algorithms for discrete evacuation planning network problems. Journal of Industrial and Management Optimization, 2015, 11 (1) : 265-289. doi: 10.3934/jimo.2015.11.265
References:
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https://beta.mxnet.io/api/gluon/_autogen/mxnet.gluon.loss.L2Loss.html | # L2Loss¶
class mxnet.gluon.loss.L2Loss(weight=1.0, batch_axis=0, **kwargs)[source]
Calculates the mean squared error between pred and label.
$L = \frac{1}{2} \sum_i \vert {pred}_i - {label}_i \vert^2.$
pred and label can have arbitrary shape as long as they have the same number of elements.
Parameters: weight (float or None) – Global scalar weight for loss. batch_axis (int, default 0) – The axis that represents mini-batch.
Inputs:
• pred: prediction tensor with arbitrary shape
• label: target tensor with the same size as pred.
• sample_weight: element-wise weighting tensor. Must be broadcastable to the same shape as pred. For example, if pred has shape (64, 10) and you want to weigh each sample in the batch separately, sample_weight should have shape (64, 1).
Outputs:
• loss: loss tensor with shape (batch_size,). Dimenions other than batch_axis are averaged out.
__init__(weight=1.0, batch_axis=0, **kwargs)[source]
Initialize self. See help(type(self)) for accurate signature.
Methods
__init__([weight, batch_axis]) Initialize self. apply(fn) Applies fn recursively to every child block as well as self. cast(dtype) Cast this Block to use another data type. collect_params([select]) Returns a ParameterDict containing this Block and all of its children’s Parameters(default), also can returns the select ParameterDict which match some given regular expressions. export(path[, epoch]) Export HybridBlock to json format that can be loaded by SymbolBlock.imports, mxnet.mod.Module or the C++ interface. forward(x, *args) Defines the forward computation. hybrid_forward(F, pred, label[, sample_weight]) Overrides to construct symbolic graph for this Block. hybridize([active]) Activates or deactivates HybridBlock s recursively. infer_shape(*args) Infers shape of Parameters from inputs. infer_type(*args) Infers data type of Parameters from inputs. initialize([init, ctx, verbose, force_reinit]) Initializes Parameter s of this Block and its children. load_parameters(filename[, ctx, …]) Load parameters from file previously saved by save_parameters. load_params(filename[, ctx, allow_missing, …]) [Deprecated] Please use load_parameters. name_scope() Returns a name space object managing a child Block and parameter names. register_child(block[, name]) Registers block as a child of self. register_forward_hook(hook) Registers a forward hook on the block. register_forward_pre_hook(hook) Registers a forward pre-hook on the block. save_parameters(filename) Save parameters to file. save_params(filename) [Deprecated] Please use save_parameters. summary(*inputs) Print the summary of the model’s output and parameters.
Attributes
name Name of this Block, without ‘_’ in the end. params Returns this Block’s parameter dictionary (does not include its children’s parameters). prefix Prefix of this Block. | 2018-12-12 11:43:57 | {"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": 0.17543189227581024, "perplexity": 9656.867474726918}, "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/1544376823872.13/warc/CC-MAIN-20181212112626-20181212134126-00441.warc.gz"} |
https://www.qb365.in/materials/stateboard/12th-standard-maths-english-medium-applications-of-differential-calculus-reduced-syllabus-important-questions-with-answer-key-2021-4409.html | #### 12th Standard Maths English Medium Applications Of Differential Calculus Reduced Syllabus Important Questions With Answer Key 2021
12th Standard
Reg.No. :
•
•
•
•
•
•
Maths
Time : 01:00:00 Hrs
Total Marks : 100
Multiple Choice Questions
15 x 1 = 15
1. A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t2. The stone reaches the maximum height in time t seconds is given by
(a)
2
(b)
2.5
(c)
3
(d)
3.5
2. The tangent to the curve y2 - xy + 9 = 0 is vertical when
(a)
y = 0
(b)
$\\ \\ y=\pm \sqrt { 3 }$
(c)
$y=\cfrac { 1 }{ 2 }$
(d)
$y=\pm 3$
3. The value of the limit $\\ \\ \\ \underset { x\rightarrow 0 }{ lim } \left( cotx-\cfrac { 1 }{ x } \right)$
(a)
0
(b)
1
(c)
2
(d)
4. The function sin4 x + cos4X is increasing in the interval
(a)
$\left[ \cfrac { 5\pi }{ 8 } ,\cfrac { 3\pi }{ 4 } \right]$
(b)
$\left[ \cfrac { \pi }{ 2 } ,\cfrac { 5\pi }{ 8 } \right]$
(c)
$\left[ \cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 2 } \right]$
(d)
$\left[ 0,\cfrac { \pi }{ 4 } \right]$
5. The number given by the Mean value theorem for the function $\cfrac { 1 }{ x }$,x∈[1,9] is
(a)
2
(b)
2.5
(c)
3
(d)
3.5
6. One of the closest points on the curve x2 - y2.= 4 to the point (6, 0) is
(a)
(2,0)
(b)
$\left( \sqrt { 5 } ,1 \right)$
(c)
$\left( 3,\sqrt { 5 } \right)$
(d)
$\left( \sqrt { 13 } ,-\sqrt { 3 } \right)$
7. The point of inflection of the curve y = (x - 1)3 is
(a)
(0,0)
(b)
(0,1)
(c)
(1,0)
(d)
(1,1)
8. The point on the curve y=x2 is the tangent parallel to X-axis is
(a)
(1,1)
(b)
(2,2)
(c)
(4,4)
(d)
(0,0)
9. Equation of the normal to the curve y=2x2+3 sin x at x=0 is
(a)
x + y = 0
(b)
3y = 0
(c)
x + 3y = 7
(d)
x + 3y = 0
10. The critical points of the function f(x) = $(x-2)^{ \frac { 2 }{ 3 } }(2x+1)$ are
(a)
-1, 2
(b)
1, $\frac { 1 }{ 2 }$
(c)
1, 2
(d)
none
11. The equation of the tangent to the curve x = t cost, y = t sin t at the origin is
(a)
x = 0
(b)
y = 0
(c)
x +y = 0
(d)
x + y = 7
12. In LMV theorem, we have f'(x1) =$\frac { f(b)-f(a) }{ b-a }$ then a < x1 _________
(a)
<b
(b)
≤b
(c)
=b
(d)
≠b
13. If the curves y = 2ex and y =ae-x intersect orthogonally, then a = _________
(a)
$\frac { 1 }{ 2 }$
(b)
-$\frac { 1 }{ 2 }$
(c)
2
(d)
2e2
14. $\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx }$ is _________
(a)
1
(b)
-1
(c)
0
(d)
15. The statement " If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________
(a)
the extreme value theorem
(b)
Fermats' theorem
(c)
Law of mean
(d)
Rolle's theorem
1. 2 Marks
10 x 2 = 20
16. A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) =100×(1− 0.1t)2, 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?
17. If the volume of a cube of side length x is v = x3. Find the rate of change of the volume with respect to x when x = 5 units.
18. Find the slope of the tangent to the curves at the respective given points.
y = x4 + 2x2 − x at x =1
19. Find the point on the curve y = x2 − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.
20. Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
$f(x)=x-2logx, x\in [2,7]$
21. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
f(x) = x2 − x, x ∈ [0, 1]
22. At what point on the curve y=x2 on [-2,2] is the tangent parallel to X-axis?
23. Find the point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa.
24. Using Rolle’s theorem find the value of c for f(x) = sin x in[0,2π]
25. Evaluate the following limits, if necessary using L’Hopitalrule
(i) $\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x }$
(ii) $\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 }$
(iii) $\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } }$
(iv) $\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } }$
1. 3 Marks
10 x 3 = 30
26. A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres
Find the average velocity of the points between t = 3 and t = 6 seconds.
27. A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres
Find the instantaneous velocities at t = 3 and t = 6 seconds.
28. A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?
29. Find the points on the curve y2 - 4xy = x2 + 5 for which the tangent is horizontal.
30. Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}$ there is a zero of the polynomial $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}$
31. Find the absolute extrema of the following functions on the given closed interval.
$f(x)=2cosx+sin2x;\left[ 0,\cfrac { \pi }{ 2 } \right]$
32. Suppose that for a function f(x), f'(x) ≤ 1for all 1 ≤ x ≤ 4. Show that f(4) - f(1) ≤ 3.
33. Verify LMV theorem for f (x) = x3 - 2x2 - x + 3 in [0, 1].
34. A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t2 where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?
35. The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.
1. 5 Marks
7 x 5 = 35
36. A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t2 in t seconds.
How long does the camera fall before it hits the ground?
37. A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?
38. A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
How fast is the top of the ladder moving down the wall?
39. A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?
40. Write the Maclaurin series expansion of the following function
log(1 - x); -1 ≤ x < 1
41. missle fired from ground level rises x metres vertically upwards in t seconds and $x=100t-\cfrac { 25 }{ 2 } { t }^{ 2 }$. Find the
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.
42. A manufacturer can sell x items at a price of rupees $\left( 5-\cfrac { x }{ 100 } \right)$ each. The cost price of x items is Rs.$\left( \cfrac { x }{ 5 } +500 \right)$ .Find the numbers of items he should sell to earn maximum profit. | 2021-05-12 12:27:40 | {"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": 0.559468150138855, "perplexity": 582.4511792405134}, "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-2021-21/segments/1620243989693.19/warc/CC-MAIN-20210512100748-20210512130748-00541.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=160754 | # Repeated Integrals
by Bucky
Tags: integrals, repeated
P: 82 1. The problem statement, all variables and given/known data Evaluate the following repeated integral $$\int^2_1 \int^4_2 \sqrt{x-y} dx dy$$ 2. Relevant equations 3. The attempt at a solution $$\int^2_1 \int^4_2 x^{1/2} - y^{1/2} dx dy$$ $$\int^2_1 [ \frac {2x^{3/2}}{3} - xy^{1/2} ]^4_2 dy$$ $$\int^2_1 [ \frac {2(4)^{3/2}}{3} - (4)y^{1/2} ] - [ \frac {2(2)^{3/2}}{3} - (2)y^{1/2} ] dy$$ $$\int^2_1 [ \frac {2\sqrt{64}}{3} - (4)y^{1/2} ] - [ \frac {2\sqrt{8}}{3} - (2)\sqrt{y} ] dy$$ $$\int^2_1 [ \frac {16}{3} - (4)\sqrt{y}] - [ \frac {2(2)^{3/2}}{3} - (4)\sqrt{2} ] dy$$ $$\int^2_1 [ \frac {16 - 4 \sqrt{2}}{3} - 2 y^{1/2}] dy$$ $$\frac {16 - 4 \sqrt{2}}{3} \int^2_1 [ - 2 y^{1/2}] dy$$ (can i take it out since it's a constant?) $$\frac {16 - 4 \sqrt{2}}{3} \int^2_1 [ - 2 y^{3/2}]^2_1$$ $$\frac {16 - 4 \sqrt{2}}{3} \int^2_1 [ \frac{- 2 y^{3/2}}{\frac{3}{2}}]^2_1$$ $$\frac {16 - 4 \sqrt{2}}{3} \int^2_1 [ \frac{- 4 y^{3/2}}{3}]^2_1$$ ...and so on. basically it doesn't work out. answer i'm meant to get is $$\frac{4(9\sqrt{3} - 4\sqrt{2} - 1)}{15}$$ can anyone see what i've done wrong? Invairably it's sloppy algebra -.-
P: 1,123 edit... Woops, you made a huge mistake on the first step! It is not always true that $\sqrt{x-y} = \sqrt{x}-\sqrt{y}$ Consider x = 2, y =1. Then $\sqrt{2-1} = \sqrt{1} = 1, \ \text{but} \ \sqrt{2}-\sqrt{1} = \sqrt{2} - 1 \neq 1$ So you cannot just switch things out like that. ----- Assuming that was OK, you made a mistake when you pulled out the "constant". It is not really a constant in the way you used it. Remember the properties of integrals: $$\int (f-g) = \int f - \int g$$ $$\int cf = c\int f$$ So you should have had: \begin{align*} \int_1^2 \left(\frac {16 - 4 \sqrt{2}}{3} - 2 y^{1/2}\right)dy & = \int_1^2 \frac{\left(16- 4\sqrt{2}}{3}\right)dy - \int_1^2 2y^{1/2}dy \\ & = \left(\frac{16- 4\sqrt{2}}{3}\right) \int_1^2 dy - 2\int_1^2 y^{1/2}dy \end{align*} Note that I posted this last part before I noticed the first mistake, so the idea behind my post is correct, but you should not get the same thing (since the first step was wrong). edit.. Hmm, my code is not being updated, and what is cmyk0000 lol?
P: 82 ok so i looked at this again and it may in fact be a standard integral? for the sake of my hands, i won't type the whole thing out in latex...but can someone confirm if i've started this properly? $$\int^2_1 \int^4_2 \sqrt{x-y} dx dy$$ using standard integral $$\int (ax+b)^n = \frac {(ax+b)^{n+1}}{a (n+1)}$$ $$\frac {(x-y)^{3/2}}{\frac{3}{2}}$$ $$\frac {2(x-y)^{3/2}}{3}$$
P: 82 ok so i've mucked it up... $$\int^2_1 [\frac {2(x-y)^{3/2}}{3} ]^4_2 dy$$ $$[\frac {2((4)-y)^{3/2}}{3} ] - [\frac {2((2)-y)^{3/2}}{3} ]$$ i get to this point. my first thought was "oh it's a letter and a number in a bracket to a power ( (4-y)^3/2 )...can't i just solve it out?", but that seems very akward and messy... $$2\frac {\sqrt{-2y^3 - 8y^2 - 96y + 128}}{3} - 2\frac {\sqrt{y^3 + 6y^2 + 12y + 8}}{3}$$
P: 1,123 What you wrote first is good, but you left out the integral in the second line, so I am not sure if you are just not writing it out, or if you are misunderstanding something. You should have: \begin{align*} \int^2_1 \frac {2(x-y)^{3/2}}{3} \Big |_2^4 dy & = \int_2^1 \left(\frac {2((4)-y)^{3/2}}{3} - \frac {2((2)-y)^{3/2}}{3} \right)dy \\ &= \int^2_1 \frac {2((4)-y)^{3/2}}{3} dy - \int_1^2 \frac {2((2)-y)^{3/2}}{3} dy \end{align*} From here $y$ is now a variable, so you have a new integral to evaluate. Try to evaluate this integral now.
P: 82 ok this is all mathcrafting, may be lies and methods that'd make no sense... $$\int^2_1 \frac {2((4)-y)^{3/2}}{3} dy - \int_1^2 \frac {2((2)-y)^{3/2}}{3} dy$$ if i take 2/3 out as a fraction in both equations, i am left with a bracket to integrate. $$\int^2_1 \frac {2}{3} * ((4)-y)^{3/2} dy - \int_1^2 \frac {2}{3} * ((2)-y)^{3/2} dy$$ from earlier, the rule for integrating the bracket is.. $$\int (ax+b)^n = \frac {(ax+b)^{n+1}}{a (n+1)}$$ giving $$\frac{2}{3} x \frac{(4-y)^{5/2}}{-1(\frac{5}{2})} - \frac{2}{3} x \frac{(2-y)^{5/2}}{-1(\frac{5}{2})}$$ $$[\frac{4(4-y)^{5/2}}{25}]^2_1 - [\frac{4(2-y)^{5/2}}{25}^2_1]$$ somethings wrong i'm sure, but that may just be complete lack of confidence in my ability to do maths.
P: 1,123 Everything looks good up to the very last step, I think you just made a silly mistake with the fractions. $$\frac{2}{3} \times \frac{(4-y)^{5/2}}{-1(\frac{5}{2})} - \frac{2}{3} \times \frac{(2-y)^{5/2}}{-1(\frac{5}{2})} \Big |_1^2$$ Note: If you want to use x to means times then you should use \times in your LaTeX code. From there you get (factor out the 2/3 from both parts, and "flip" the fraction in the denominator): $$\frac{2}{3}\left( -\frac{2}{5}(4-y)^{5/2} - -\frac{2}{5}(2-y)^{5/2} \Big |_1^2 \right)$$ From there we can factor the 2/5 out and we get: $$\frac{2}{3}\times \frac{2}{5}\left( -(4-y)^{5/2} + (2-y)^{5/2} \Big |_1^2 \right)$$ Now you just "plug and chug," and you should get the correct answer. | 2014-07-26 17:13:27 | {"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": 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": 0.9688379168510437, "perplexity": 550.2135989681018}, "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-2014-23/segments/1405997903265.4/warc/CC-MAIN-20140722025823-00055-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/1685285/prove-x-is-not-congruent | # Prove x is not congruent
If $\,x \not\equiv15\pmod{17},\text{ then }x^5 \not\equiv2\pmod{17}.$
I tried to take the contrapositive:
If $\,x^5 \equiv2\pmod{17},\text{ then }x \equiv15\pmod{17}$ and
then I assume that $x^5=17y+2$ for some integer $y$
But I am not what to do after this step.
How do I continue?
-
Proving the contrapositive is a good idea.
The key observation is that if $x$ is not divisible by $17$, then $x^{16} \equiv 1(mod\ 17)$ by Euler's theorem. If we assume $x^5 \equiv 2(mod\ 17)$, then it follows that $$15 \equiv -2 \equiv -2 \cdot x^{16} \equiv -2\cdot(x^5)^3\cdot x \equiv -2 \cdot 2^3 \cdot x \equiv -16x \equiv x\ (mod\ 17).$$
-
Will Fermat's Little Theorem do? If $x^5\equiv2\pmod{17}$, then $x^{20}\equiv-1\pmod{17}$ and $x^{40}\equiv1\pmod{17}$. We then have
$$x^{65}\equiv x\equiv2(-1)(1)\equiv-2\equiv15\pmod{17}$$
-
HINT:
$$a\equiv\pm1, a^5\equiv\pm1$$
$$a\equiv\pm2, a^5\equiv\pm32\equiv\mp2$$ and so on | 2016-07-01 02:52:57 | {"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": 0.9532763957977295, "perplexity": 176.26859458348807}, "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-2016-26/segments/1466783399522.99/warc/CC-MAIN-20160624154959-00134-ip-10-164-35-72.ec2.internal.warc.gz"} |
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We study a general credence goods model with $N$ problem types and $N$ treatments. Communication between the expert seller and the client is modeled as cheap talk. We find that the expert's equilibrium payoffs admit a geometric characterization, described by the quasiconcave envelope of his belief-based profits function under discriminatory pricing. We establish the existence of client-worst equilibria, apply the geometric characterization to previous research on credence goods, and provide a necessary and sufficient condition for when communication benefits the expert. For the binary case, we solve for all equilibria and analyze their welfare properties.
### Why Are Immigrants Always Accused of Stealing People's Jobs?
Immigrants are always accused of stealing people's jobs. Yet, in a neoclassical model of the labor market, there are jobs for everybody and no jobs to steal. (There is no unemployment, so anybody who wants to work can work.) In standard matching models, there is some unemployment, but labor demand is perfectly elastic so new entrants into the labor force are absorbed without affecting jobseekers' prospects. Once again, no jobs are stolen when immigrants arrive. This paper shows that in a matching model with job rationing, in contrast, the entry of immigrants reduces the employment rate of native workers. Moreover, the reduction in employment rate is sharper when the labor market is depressed -- because jobs are more scarce then. Because immigration reduces labor-market tightness, it makes it easier for firms to recruit and improves firm profits. The overall effect of immigration on native welfare depends on the state of the labor market. It is always negative when the labor market is inefficiently slack, but some immigration improves welfare when the labor market is inefficiently tight.
### Sequential Cauchy Combination Test for Multiple Testing Problems with Financial Applications
We introduce a simple tool to control for false discoveries and identify individual signals when there are many tests, the test statistics are correlated, and the signals are potentially sparse. The tool applies the Cauchy combination test recursively on a sequence of expanding subsets of $p$-values and is referred to as the sequential Cauchy combination test. While the original Cauchy combination test aims for a global statement over a set of null hypotheses by summing transformed $p$-values, the sequential version determines which $p$-values trigger the rejection of the global null. The test achieves strong familywise error rate control and is less conservative than existing controlling procedures when the test statistics are dependent, leading to higher global powers and successful detection rates. As illustrations, we consider two popular financial econometric applications for which the test statistics have either serial dependence or cross-sectional dependence: monitoring drift bursts in asset prices and searching for assets with a nonzero alpha. The sequential Cauchy combination test is a preferable alternative in both cases in simulation settings and leads to higher detection rates than benchmark procedures in empirics. | 2023-03-26 03:42:39 | {"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": 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": 0.6307899951934814, "perplexity": 1644.4662798184854}, "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-2023-14/segments/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00129.warc.gz"} |
http://www.joemariglio.com/blog/forcing-function-estimation/ | Forcing Function Estimation in Modal Analysis
In an earlier post, I described a technique for modal analysis which can estimate damped modes of vibration in an object, as measured at a variable number of points on its surface. I have applied this technique to field data from an array of laser microphones, measuring waves on the surface of the water. Now I will develop the mathematics and practical considerations behind estimating the forcing function in such recordings.
Estimating the forcing function using such an analysis should be possible using the following steps:
Estimate the eigenvector matrix $S$.
Perform a linear regression on the data using the eigenvector matrix, ie
Where $c_2$ is a vector of coefficients describing how the new timeseries vector $u_2$ maps to into the modal basis. Then, we can leverage the similarity transform $S^{-1} A S = \Lambda$ to our advantage as follows:
Where $c_1$ is the coefficient vector of the previous values in the time series.
This expresses $c_2$ as a sum of $\Lambda c_1$ and an error signal, but the result is still in modal space. To rotate back into the standard basis, we multiply by the eigenvector matrix:
Depending on the normalization of $S$, this rotation can display the error signal in numerous ways, some more useful than others. If we normalize $S$ such that each eigenvector's last term is 1, this rotation gives the error signal $F$ as a single vector whose number of components matches that of the measured time series.
To describe the entire error time series, we must predict the homogeneous response by rotating the initial conditions $c_1$ by $\Lambda$ every time step:
This equation gives the error function as a time series of points whose dimensions match the measured data in the standard basis. It begs the question: what is this error function? Is it useful?
I hypothesize that this error function is a combination of sensor noise, analytical error (including those caused by precision and quantization), nonlinear responses of the medium, and the forcing function. If the distributions of these terms can be estimated, their effect could, in theory, be minimized, and we would be able to estimate the forcing function using this regression model. Nonlinearities in the medium could be locally linearized by updating the eigendecomposition at a regular frequency. However, if the update frequency were too high, the system might mistake transients for changes in the steady state, so this parameter must be adjusted to fit known qualities of the dataset.
To assess the validity of the forcing function estimation, an ideal model was constructed to simulate modal phenomena, using simple allpole filters, with white noise as a forcing function. Because the forcing function and eigenmodes are known, the algorithm's accuracy can be tested.
This process of forcing function estimation was attempted using scripts written in GNU Octave. A test script can be found here, on my github. When this is run, a new model of a resonant object is synthesized, and the estimated versus actual forcing function are compared. The model was simulated with the following parameters:
mics = 1;
order = 4;
winsize = 10000;
mscale = 0.001;
mshift = 0.999;
Where 'mscale' is the range, and 'mshift' the offset, of randomly selected eigenvalue magnitudes. The angles were randomly selected with an even distribution across the entire bandwidth.
In one example, a correlation of -0.65030 was found between estimated and ideal forcing function time series.
In addition to caluclating the correlation, the first 50 points may be plotted against one another. Here, the ideal timeseries is in blue, and the estimated is in green (and inverted to show the correlation as positive).
Furthermore, the eigenvalues may be plotted against one another. Here the ideal eigenvalues are in red, and the estimated are in blue.
Previous research has demonstrated that the algorithm's accuracy in estimating modal behavior is highly sensitive to amount of damping, number of channels, and order of the analysis. The next set of experiments will seek to find the mean absolute value correlation between measured and estimated forcing function, as a function of damping. | 2017-11-20 16:59:18 | {"extraction_info": {"found_math": true, "script_math_tex": 12, "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": 16, "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": 0.8468542098999023, "perplexity": 507.8273665114991}, "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-2017-47/segments/1510934806086.13/warc/CC-MAIN-20171120164823-20171120184823-00051.warc.gz"} |
https://www.x-mol.com/paper/1285641477006475264 | Georgian Mathematical Journal ( IF 0.500 ) Pub Date : 2020-07-16 , DOI: 10.1515/gmj-2020-2066
Let $ℛ$ be a prime ring, $𝒬r$ the right Martindale quotient ring of $ℛ$ and $𝒞$ the extended centroid of $ℛ$. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e., | 2020-08-10 05:15:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.5692575573921204, "perplexity": 475.41323912243564}, "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/1596439738609.73/warc/CC-MAIN-20200810042140-20200810072140-00516.warc.gz"} |
http://tex.stackexchange.com/questions/147165/how-to-calculate-intersections-of-a-line-defined-by-a-segment-and-an-ellipse-in | # How to calculate intersections of a line defined by a segment and an ellipse in tikz
EDIT Here's what I intended to ask, but didn't clearly formulate initially. The Tikz package intersections allows one to calculate and manipulate the intersection point of two non-parallel lines defined by (non intersecting) line segments. The link in my original question shows how to do this, it is how the point (F) is defined. In the example below, the point (0) is defined as the intersection of the lines extending the line segments vertline and horline. What I want to do is calculate and manipulate intersection points of an ellipse with the line extending a given line segment, without having to rely on extending the line segment ''by hand'' as it were, to ensure it actually intersects the ellipse. When I copy the syntax used in the code from the link, and naively adapt it,
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\usetikzlibrary{intersections}
\path[name path=ellipse] (0,0) ellipse (4 and 3);
\path[name path=horline] (1,0) -- (2,0);
\path[name path=vertline] (0,1) -- (0,2);
\path[name intersections={of = horline and vertline, by={origin}}, draw]
(origin) node [above right] {$0$} circle (1.5pt);
\path[name intersections={of = ellipse and vertline}]
\coordinate (I) at (intersection-1);
\coordinate (J) at (intersection-2);
\draw (I) circle (1.5pt);
\draw (J) circle (1.5pt);
\end{tikzpicture}
\end{document}
it doesn't give me the desired result : the intersections (I) and (J) are of ellipse and vertline are totally misplaced. Moreover, the code is faulty, but I don't understand where the error lies. Please tell me if you'd code things differently, I don't want to develop bad habits.
Further Edit It appears (I) and(J) aren't misplaced, they simply aren't calculated at all, and what appeared on the pdf were points (I) and(J) defined in a previous drawing.
Oringinal Question - May be dicarded In this Example, the author gets tikz to calculate the coordinates of the two intersections (X and Y) of an ellipse with the line defined by a certain segment, and also the coordinates of the intersection (F) of two lines defined by two line segments.
On my machine, I have no problem with the second problem, finding the intersection point of two lines defined by (non intersecting line segments), but i get non-sense when I try to perform the intersection of an ellipse with the line defined by a line segment that doesn't intersect the ellipse.
-
Please add a minimal working example (MWE) that illustrates your problem. It will be much easier for us to reproduce your situation and find out what the issue is when we see compilable code, starting with \documentclass{...} and ending with \end{document}. – Kevin C Nov 27 '13 at 5:30
@KevinC The link provides this. – Olivier Bégassat Nov 27 '13 at 5:37
So you're saying that you can't get the points X and Y to show up? The linked example compiles okay with me (MikTeX 2.9), and the output is as expected. – Kevin C Nov 27 '13 at 5:44
@KevinC I'll get back at you tomorow. – Olivier Bégassat Nov 27 '13 at 5:48
@KevinC I've edited the question, I think it is clearer now what I'm asking. – Olivier Bégassat Nov 27 '13 at 15:31
You can calculate the intersection of two non-parallel lines using the coordinate specification (intersection of A--B and C--D), where A, B, C and D have to be named nodes. That's the approach used in the code you linked to. This, however, does not work with arbitrary named paths, which is what you're trying to do.
To calculate the intersections of a line with an arbitrary path, you'll have to make sure that the paths actually intersect. If you don't want to manually extend the line segment, you can use the approach from Intersection with rays in TikZ and create "infinitely long" (1 metre, say) line segments inside a interruptboundingbox environment:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\usetikzlibrary{intersections, calc}
\path[name path=ellipse] (0,0) ellipse (4 and 3);
\begin{pgfinterruptboundingbox}
\path[name path global=horline] ($(1,0)!-100cm!(2,0)$) -- ($(1,0)!100cm!(2,0)$);
\path[name path global=vertline] ($(0,1)!-100cm!(0,2)$) -- ($(0,1)!100cm!(0,2)$);
\end{pgfinterruptboundingbox}
\path[name intersections={of = horline and vertline, by={origin}}, draw]
(origin) node [above right] {$0$} circle (1.5pt);
\path[name intersections={of = ellipse and vertline}];
\coordinate (I) at (intersection-1);
\coordinate (J) at (intersection-2);
\draw (I) circle (1.5pt);
\draw (J) circle (1.5pt);
\end{tikzpicture}
\end{document}
-
I get an intersection, but it's not where it should be. – Olivier Bégassat Nov 27 '13 at 16:17
Check your .log file, there's probably the same error message in there and the node is just put at the origin. In the code you linked to, the point F is found using intersection of A--D and B--C, which is not the same as the intersection of two named paths. – Jake Nov 27 '13 at 16:18
In the .log file the error message is with respect to intersection-1, it says ! Package pgf Error: No shape named intersection-1 is known. – Olivier Bégassat Nov 27 '13 at 16:25
@OlivierBégassat: Yes, that's the same message that I got. On my system, that error halted the compilation (which it should, since everything after that is unreliable). I've edited my answer to show how your code can be fixed. – Jake Nov 27 '13 at 16:31
Your 'Intersection with rays' answer is very very neat. I think I'll use that. I still wish there was an intrinsic way to work with infinite lines or half lines. While we're talking, what manual do you recommend for learning Tikz? Also, do I need to learn pgftricks aswell? – Olivier Bégassat Nov 27 '13 at 16:43 | 2014-12-29 14:36:00 | {"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": 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": 0.8633020520210266, "perplexity": 1016.4930352700275}, "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/1419447563342.152/warc/CC-MAIN-20141224185923-00060-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://brilliant.org/discussions/thread/nice-functional-analysis-problem/ | ×
# Nice Functional Analysis Problem
If $$f: \mathbb R \rightarrow \mathbb R$$ is continuous everywhere and for every real $$x$$,
$f(x) = f(2x)$
Prove that it is indeed constant. The answer involves a bit of imagination.
Note by Romanos Molfesis
1 year, 9 months ago
Sort by:
I think I have seen this question before.
$f(x)=f(2x)$ $f(\frac12x)=f(x)$ and like this, we get, $f(\frac{x}{2^n})=f(x)$ Now as $$n\to\infty$$ $f(0)=f(x)$ and thus, we get that $$f(x)$$ is a constant function.
- 1 year, 9 months ago
Correct!
- 1 year, 9 months ago
We have $$f(x)=f(2^nx)$$ for all real $$x$$ and for all integers $$n$$. Now, by continuity, $$f(0)=\lim_{n\to -\infty}f(2^nx)=f(x)$$, showing that $$f(x)$$ is constant, taking the value $$f(0)$$ for all $$x$$.
- 1 year, 9 months ago
Correct!
- 1 year, 9 months ago
it is a continuous function therfore it has to be constant
- 8 months ago | 2017-10-17 13:35:12 | {"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": 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": 0.9622334241867065, "perplexity": 811.8482474845463}, "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-2017-43/segments/1508187821189.10/warc/CC-MAIN-20171017125144-20171017145144-00364.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=519&parent=2128 | ## Installation
### Solved? - was Re: Getting a course to see a Library
by Neal Caidin -
Number of replies: 0
I updated my global.conf file in the section "##### Directories used by PG" , underneath which one defines the macro search path in the subsection "# The macro file search path." The I restarted Apache and everything seemed to work fine.
I hope this is the correct approach. | 2021-12-03 17:03:15 | {"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": 0.849909245967865, "perplexity": 7563.534422920945}, "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-2021-49/segments/1637964362891.54/warc/CC-MAIN-20211203151849-20211203181849-00460.warc.gz"} |
http://www.solutioninn.com/the-us-energy-information-administration-us-eia-reported-that-the | # Question
The U.S. Energy Information Administration (US EIA) reported that the average price for a gallon of regular gasoline is \$3.94 (US EIA website, April 6, 2012). The US EIA updates its estimates of average gas prices on a weekly basis. Assume the standard deviation is \$.25 for the price of a gallon of regular gasoline and recommend the appropriate sample size for the US EIA to use if they wish to report each of the following margins of error at 95% confidence.
a. The desired margin of error is \$.10.
b. The desired margin of error is \$.07.
c. The desired margin of error is \$.05.
Sales4
Views192 | 2016-10-22 19:51:44 | {"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": 0.8623685240745544, "perplexity": 1680.4277760656096}, "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-2016-44/segments/1476988719041.14/warc/CC-MAIN-20161020183839-00415-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.nature.com/articles/sdata2017116?error=cookies_not_supported&code=6f59c3d0-9262-41f6-8aac-a71023c89998 | ## Background & Summary
Anthropogenic heat is wasted heat in the form of sensible and latent heat (http://www.ametsoc.org/amsedu/) that is released to the urban canopy mainly by means of heating and cooling, running appliances, transportation, and industrial processes, which convert energy into anthropogenic heat14. Nearly 70% of energy is consumed within cities occupying a mere 2% of the Earth’s surface area, and future scenarios indicate that global primary energy consumption will rise 1.6 times (864.7 quadrillion kJ) from 2010 to 2040 (http://www.worldenergyoutlook.org/). Although anthropogenic heat accounts for only 1% of the greenhouse gas forcing, it causes the majority of regional warming, such as urban heat islands1,2, urban boundary heights, and hourly intensity of precipitation at the city level25, especially at night. Anthropogenic heat reduces the intensity of precursor species (NOx) in the urban atmosphere due to the dilution, increases concentrations of near surface ozone6,7, and raises risks for morbidity and mortality of the residents through increasing the air temperatures in urban areas8. Meanwhile, many studies indicate that anthropogenic heat increases the temperature, especially in urban regions and cold seasons, which affects the Asian winter monsoon circulation9 and disrupts the normal global atmospheric circulation patterns10. Considering the remarkable growth of energy use expected in the future11, anthropogenic heat will play an increasingly important role in the radiative forcing of the atmosphere, global climate changes, and urban ecosystem impacts.
Therefore, calculating a high spatial resolution global scale anthropogenic heat flux (AHF) dataset is necessary for studying climate impacts. According to the characteristics of anthropogenic heat, three main approaches have been developed: observations, detailed statistical models (building/traffic energy modelling), and inventory approaches. As a classical method, the inventory approach is used globally to estimate global-scale anthropogenic heat12,13, using the assumption that any heat emitted by energy consumption is completely converted to anthropogenic heat. The observation approach includes an energy budget residual approach1416 and in situ eddy covariance observations13. Statistical detailed models, such as the town energy budget (TEB) or urban canopy model17, have been developed according to building materials, heights, equipment, as well as transportation equipment1820. Each approach has its own advantage; however, the inventory approach is more efficient for large-scale studies than the other two, which are limited by their demand categories and modelling resolutions. Some attempts have been made at developing a global AHF dataset. In the first global anthropogenic heat dataset (released in 2009, referred to as the Flanner study hereafter), Flanner proportionally downscales the total country-level non-renewable energy consumption to a 2.5-minute resolution according to the spatial population density21. However, this global dataset has unreliable information for regions where the population density does not have a clear statistical relationship with economic development and energy consumption, such as in China22. To overcome the limitations of Flanner’s study, in Chen’s study, a simple model was built based on the relationship of nighttime light data from the US Air Force Defense Meteorological Satellite Program/Operational LineScan System (DMSP/OLS) and energy consumption23. However, information in downtown areas has lower reliability due to the saturation and temporal fluctuation characteristics of nighttime light data24. In this study, we attempted to develop an updated method of generating an improved precision and high spatial resolution global anthropogenic heat dataset. Normalized difference vegetation index (NDVI) was used to calibrate DMSP/OLS data, and then used as a proxy to downscale the country-level inventory data to a 1×1 km spatial resolution. Using statistical methods, we validated the feasibility of the method and the reliability of the AH-DMSP dataset.
AH-DMSP shows that the majority of anthropogenic heat is mainly produced in the developed regions-the eastern part of North America, Western Europe, the eastern and southern parts of Asia as a result of the high intensity and magnitude of human activities in these regions. All of the annual maximum values appear in urban regions. Changes in anthropogenic heat between 1990 and 2010 were significant in the metropolis regions of both the developing and developed countries.
## Methods
### Data
#### DMSP/OLS nighttime light data
Cloud-free composite DMSP/OLS data from 1992 to 2010 were acquired from the National Oceanic and Atmospheric Administration’s (NOAA) National Geophysical Data Center25. Three of four satellites from the DMSP carry the OLS in low-altitude polar orbits to record nighttime data. The DMSP/OLS has the unique capability to detect low levels of visible near infrared (VNIR) radiance at night. With the OLS ‘VIS’ band data, it is possible to detect clouds illuminated by moonlight, plus lights from cities, towns, industrial sites, gas flares, and ephemeral events such as fires and lightning-illuminated clouds. Data values range from 1–63. Areas with no cloud-free observations are represented by the value 255. The stable average nighttime lights detected by DMSP/OLS contain the lights from cities, towns, and other sites with persistent lighting, including gas flares. Ephemeral events, such as fires, were discarded26,27. Background noise was identified and replaced with zero values. In this work, the DMSP/OLS data is regarded as a proxy to the energy consumption intensity in cities and towns, as well as in other persistent lighting sites, including gas flares (Data Citation 1).
#### Normalized difference vegetation index (NDVI) Data
Gap-filled, snow-free, Nadir Bidirectional Reflectance Distribution Function (BRFD) Adjusted Reflectance (NBAR) data derived from the Moderate Resolution Imaging Spectro radiometer (MODIS) MCD43D products were used to generate NBAR-NDVI time series. For each year 2001–2010, the annual mean NDVI was calculated for each pixel27,28. To calibrate DMSP/OLS data, we prefer the annual mean NDVI to the annual maximum NDVI, because the former is more stable and less sensitive to seasonal and intra-annual fluctuations. To keep temporal consistency with the DMSP/OLS data, we extended the annual mean NDVI from year 2001 as constant values back to 1992, to complement the period 1992–2001. This decision is supported by the observation that there is not significant annual variation in the annual mean NDVI for urban regions during 2001–2010 (Data Citation 2).
#### Gridded population data
We obtained gridded population density data for the year 2000 which are 2.5 arc-minute grid cells of the Gridded Population of the World v3 (GPWv3), provided by the Centre for International Earth Science Information Network (Data Citation 3). The population density grids were derived by dividing population count grids by a land area grid, and represent persons per square kilometre29,30.
#### Statistical energy consumption data
Statistics of national primary energy consumption for 224 countries were obtained from the U.S. Energy Information Administration (EIA)11. The statistics include energy from the four main primary energy sources: coal, petroleum, natural gas, and renewable energy. Chinese urban level energy consumption was obtained from the Urban Statistical Yearbook for China (http://data.stats.gov.cn/easyquery.htm?cn=E0103). Gross national income (GNI) per capita for each of 224 countries was obtained from the World Bank (http://data.worldbank.org/income-level/OEC). A portion of the primary energy sources are used to produce secondary energy (for example, the coal used by a power station to generate electricity) in non-urban areas that is then consumed by urban residents at the end of the energy flow. In this case, the wasting heat in the processing procedure was not taken into consideration since it has little influence on total heat emitted. We therefore use only the four primary energy source classifications in this study.
### Methodology
The algorithm for calculating the global anthropogenic heat product (AH-DMSP) developed in this study is presented in a flowchart (Fig. 1). First, we extracted the urban areas from calibrated DMSP-OLS and MODIS NDVI from 1992 to 2010 (details in step 1 and 3). Second, for each country, we separated the primary energy consumption (coal, petroleum, natural gas, and renewable energy) into urban and non-urban consumption using the urban consumption ratio from the International Energy Agency (IEA) (details in step 2). Third, a strong exponential relationship between urban area and energy consumption (R2=0.9, P=0.039) was found from the available statistical data (Fig. 2). Based on this relationship, the energy consumption in urban areas obtained from Step 2 was inserted in the urban map obtained from Step 3. Most large factories or refineries are located in urban areas. Therefore, the energy consumed in an urban area not only includes the energy consumption for domestic use but also industrial usage. Human metabolism occupied a small share of the total heat released by human activities, and we therefore excluded it. We note that statistical errors in urban energy consumption are a function of urban area and energy consumption. In order to alleviate their effects, we can allocate the total urban energy into patches based on percent-normalized urban energy from Fig. 2. Finally, we downscaled the total urban energy consumption to the pixel-level using the corresponding spatial proxy obtained in Step 1. The rural energy consumption at the country level was allocated directly into each grid. Finally, the energy consumption value was divided by elapsed time, and the resulting AH-DMSP of each grid cell was converted to anthropogenic heat flux (AHF, W.m−2).
The calculation is simplified by the following balance equation:
${A}_{n}=\sum _{i=1}^{4}{k}_{i}.{E}_{ni}$
where An is the total anthropogenic heat of country n (1, 224); ki is the conversion rate of type i energy to heat, in which type i refers to the four sub-types of energy consumption: coal, petroleum, natural gas, or renewable energy, and Eni is the energy consumption of type i for country n. The values for k were obtained from the EIA (http://www.eia.gov/beta/international/data).
### Step 1: Calibrating DMSP/OLS: Saturation and temporal fluctuation
Noise caused by less important persistent lighting sites such as gas flares, and ephemeral events such as fires in DMSP/OLS, occur mainly in the desert regions far from human habitants, but may affect anthropogenic heat precision. Here we masked out areas with a gridded population density less than 1 per 2.5 arc-minutes with the assumption that they would have no release of anthropogenic heat.
There are two drawbacks in the DMSP/OLS data: inter-annual variation and digital saturation in downtown areas3134. Due to the auto-calibration system on the remote sensors, DMSP/OLS composites from different years cannot be inter-compared31. To address this problem, we employed the method developed by Elvidge32, using a second-order polynomial function to calibrate the annual fluctuation of DMSP/OLS data. Saturation values in DMSP/OLS data largely restrict the capacity of characterizing variations of inner urban region. We used a new spectral index to correct the saturated pixels, the Vegetation Adjusted Normalized Urban Index(VANUI) developed by Zhang35, based on the assumption that vegetation abundance is closely and inversely correlated with the intensity of energy consumption in urban areas.
### Step 2: Estimating ratios of urban energy consumption to total national consumption
The International Energy Agency (IEA) only calculates the ratios of urban energy consumption to total national consumption for four regions: United States, European Union, Austral-Asia, and China (Data Citation 4). In addition to these four regions, we introduce a simple method to estimate the urban energy consumption ratios for other countries or regions. First, 224 countries were grouped into two classes: high-income and low-income countries (including middle-income countries) based on gross national income (GNI) per capita. Second, the average urban energy consumption ratios from the United States, European Union, and Australia were applied to the high-income countries, while the ratio from China was applied to the other countries. The urban and rural energy consumption for each country could then be separated.
### Step 3: Extracting urban boundaries by support vector machines (SVM)
The existing global urban datasets have been applied for many studies of climate change and ecology (e.g., the Global Rural Urban Mapping Project or MODIS 500 m urban extent maps)36, but their temporal resolutions do not meet our requirements. Many previous studies indicate that the SVM algorithm31,34,37, which is widely used for classification and regression analysis, performs well at monitoring urban patches. It has the advantage of improving the accuracy of classification by encompassing substantial information, such as economic activities and urban vegetables38. Moreover, this method is not sensitive to the initial training samples, and it has the flexibility of multi-temporal and multi-spatial analysis39.
Here, we use the annual mean NDVI and calibrated DMSP/OLS as input information to the SVM-based algorithm, and extracted the urban boundaries for every year. We set the threshold (excluding water pixels inside the urban boundary) at DMSP/OLS >30 and NDVI <0.3 for developing countries39, and at DMSP/OLS >53 and NDVI <0.3 for developed countries. The Gaussian radial basis function (RBF, exp(−γ|μν|^2)) was used as the kernel function, with the coefficient of cost (−c=2) set to 2 and gamma was 0.02 (−g=0.02). Noise in DMSP/OLS can be caused by a variety of reasons, such as variability in the atmosphere interference, and may cause errors in the boundary extraction output. Due to the stability of urban patches, we assumed that any urban patch detected in an earlier year by DMSP/OLS should be maintained in the results of later urban boundaries. In the end, we generated urban boundary data for each year during the period 2001–2012.
### Step 4: Estimating energy consumption for all urban patches
When a city starts expansion, it normally means its urban population grows and energy consumption grows accordingly. However, when a city reaches a mature stage of economic development and infrastructure construction, energy consumption will grow relatively slower than its expansion due to increase of energy efficiency. City energy data are difficulty to find and the difficult of acquiring urban energy data lies in consideration of city boundary issues (Data Citation 4). Therefore, we try to interpret energy consumption with city size in this study. The correlation parameters are developed with statistical data in China, which is provided in Supplementary File 1.
There was a strong exponential relationship between the urban area and energy consumption (y=−0.001x2+8.30x−24.25, P=0.039, where y is the urban energy consumption (10 million kg) and x is the urban magnitude (km2) (Fig. 2). This relationship was then applied to extract energy consumption for each urban patch from the total national urban energy consumption calculated in 2.2.2 and 2.2.3.
It is common practice to validate the accuracy of global-scale AHF estimation with anthropogenic heat data using a detailed statistical model, which consists of heat released from three components: the building sector, transportation, and human metabolism. The building sector is always divided into heat released from electricity consumption and from heating fuels, such as natural gas and fuel oil40. The detailed model focuses on the end-user of energy, while the inventory method focuses on primary energy. These methods can improve model accuracy by incorporating better information about individual cities20. For this study, the annual mean AHF estimated using our method was validated by AHF data estimated using the detailed model at multiple scales, from the city to province level. These data were estimated using the detailed statistical model and were compared with zone averaged AHF from the AH-DMSP for specific cities or provinces. Meanwhile, inter-validations of spatial patterns and single points for specific cities were also performed with previously created global gridded datasets. These annual mean global datasets were calculated by the inventory approach, and single points for specific AHF data were zone averaged.
### Code availability
The algorithm is coded in Matlab and C. AHF.m is the main procedure to calculate AH-DMSP. The global urban dataset is produced by Global_urban.m and its core parts are opened source code which are provided by Chang and Lin (http://www.csie.ntu.edu.tw/~cjlin/libsvm). Areas of global urban patches are calculated by Recursion.c and only binary files can be inputted and outputted in the procedure (write_binary.m and write_tif.m are designed for converting file format between binary and tiff). ArcGIS and Matplot or Python are exploited to draw figures. This code is available alongside the dataset at figshare (Data Citation 4).
## Data Records
### AH-DMSP dataset
AH-DMSP is a global gridded dataset of annual mean anthropogenic heat and its spatial resolution for the entire planet is 1×1 km2. Tagged image file format (TIFF) and network common data form (NetCDF) of AH-DSMP are provided in the repository (Data Citation 4). For the TIFF formats of AH-DMSP, its tfw format file is about spatial information of longitude and latitude for corresponding file. These data can be processed by GIS software, Matlab, Ncl or R etc. The uploaded data includes AH-DMSP at 1992,2001 and 2010 and is stored in uint16, which can be converted to annual mean anthropogenic flux (W.m−2) by a factor 0.1 (Data Citation 4).
## Technical Validation
### Spatial and temporal patterns of AH-DMSP
As shown in Fig. 3, spatial distributions of annual mean AHF are mainly in the developed regions, i.e., the eastern part of North America, Western Europe, and the eastern and southern parts of Asia. Obviously, the northern hemisphere contributes significantly more annual mean AHF than the southern hemisphere, which is consistent with the distribution of intensity and magnitude of human activities. All of the highest annual values appear in urban regions. Pixels with annual mean AHF greater than 1.59 W.m−2 occur in densely populated regions, and their numbers increase gradually with increasing global urbanization. The differences in spatial distribution of annual mean AHF between 2010 and 1992 are significant for the urban regions in rapidly developing countries, as well as in regions where there has been ongoing economic depression (for example, in Russia, Poland, and Ukraine).
Figure 4 shows a comparison between the annual mean AHF from AH-DMSP and AHF data estimated with a detailed statistical model (AHF-stat) from the city to province scales. The statistical tests at the city and province scales between AH-DMSP and AHF-stat were P=0.008 and P=0.65, respectively. At the city level, AH-DMSP has good estimates of anthropogenic heat for cities whose AHF was less than 30 W.m−2. However, with urban developments, the differences between AH-DMSP and AHF-stat became increasingly significant. Root mean square error (RMSE: W.m−2) between the AH-DMSP and AHF-stat both the city and province levels were 12.01 and 1.07, respectively. At the province scale, due to limited data from AHF-stat, the statistical test was insignificant.
To assess the performance of AH-DMSP at the global scale, we made a comparison with two other global scale datasets from the studies by Chen and Flanner studies (Fig. 5)20,21,41. There are significant differences between the AH-DMSP, Chen, and Flanner data, both in the value ranges and the spatial patterns. Spatial patterns are quite similar between AH-DMSP and Chen’s data, since both studies use DMSP/OLS nighttime data, but the two have quite different value ranges. Most areas had values lower than 0.44 W.m−2 or higher than 1.31 W.m−2 in AH-DMSP, but the Chen data had values in the range from 1.02 to 1.31 W.m−2. This implies a significant influence from saturation, and intra- and inter-annual fluctuations of the DMSP/OLS data. Conversely, the Flanner dataset, which is currently used by the climate modelling community, had a larger area with AHF values, most of which are lower than 0.44 W.m−2.
We make comparisons of the annual means from previous datasets of AHF and AH-DMSP using the AHF-stat results as the real values. Since it is difficult to access annual mean AHF-stat data corresponding to the study period, only the Chen and AH-DMSP datasets were parallel compared in this section. From Fig. 6, with a growth in city magnitude, there is a trend in biases between the global datasets and AHF-stat data, which shift from positive to negative values and become more significant (city names in Table 1). For small cities, although the AHF values from global datasets is underestimated, its values are similar to the AHF-stat (real) values. For mega cities, values are significantly overestimated. From Fig. 6, AH-DMSP is closer to the real value than the Chen data.
## Usage Notes
Although anthropogenic heat is only about 0.3% of the total energy transported to the extra-tropics by atmospheric and oceanic circulations, it could disrupt normal atmospheric circulation patterns and warm surface temperatures at the local and global scales10. Therefore, incorporating anthropogenic heat into climate models could improve the performance of simulations of surface climate warming6,10 and are beneficial for studying the impacts of increased urban heat on the concentration of precursor species (e.g., NOx and CO) and resident health, such as morbidity and mortality risk, in cities7,31.
However, detailed evaluation of the impact on climate is deterred by the limited precision and spatial and temporal resolutions of anthropogenic heat data. The DMSP/OLS data has provided an opportunity to produce an improved precision and resolution AHF dataset. The applied methodology has been validated by a detailed anthropogenic heat model at multiple scales, but there are still some needed improvements. The underlying assumption that all energy is converted into anthropogenic heat is unreasonable, since some energy consumed will be stored by buildings and converted to other forms of energy. This assumption could therefore lead to overestimation of anthropogenic heat. Due to difficulties in observation; it is difficult to establish ratios for converting energy consumption to anthropogenic heat.
The observation approach includes in situ eddy covariance observations13 and the energy budget residual approach1416. In situ eddy covariance observations of anthropogenic heat are seriously constrained by site-specific challenges, particularly in urban regions (e.g., permissions to install towers and equipment, access to measurement sites, and dense human activities)13. These challenges make it even more complex to find a site suitable for instrument observations and distinguish human activity signals from background noise in the highly heterogeneous urban environment. Previous studies have exploited the energy budget residual to calculate anthropogenic heat19; however, some uncertainties are introduced by latent heat and storage heat. A statistical detailed model, such as the town energy budget (TEB) or urban canopy model17, were developed by accounting for the building materials and heights, as well as the equipment used in buildings or transportation18,19. Conversely, for global scale anthropogenic heat estimations, available data about building, transportation, and urban energy consumption limits the scope of application. The inventory approach is more efficient for large-scale studies than the other two, which are limited by their demand categories and resolutions for modelling. The inventory approach is widely used to estimate global-scale anthropogenic heat12,13.
In order to keep temporal consistency with DMSP/OLS data, the annual mean NDVI for year 2001 was extended back as constant values to 1992, in order to complement the period 1992–2001. The simplified method produces less influence on the method performance, because the spatial resolution of MODIS instruments (1×1 km) has a limited ability for detecting variations in urban vegetation and cannot detect decade-scale variations35,39. Gridded population density data for the year 2000 were used to mask out noises in DMSP/OLS during the study periods. We assumed that areas with a gridded population density value lower than 1 have no release of anthropogenic heat and that the boundary of human habitations had no significant changes during the study period.
Validation revealed that the values of AH-DMSP were underestimated for certain mega-cities. The underestimation might relate to the selected ratio of urban energy consumption to the national, and an allocating function that distributes the total urban energy proportionally to each urban patch. Due to limitations in available statistics for urban sizes and energy use around the world, only four regions have ratios of urban energy consumption—United States, European Union, Austral-Asia, and China. The ratios for other countries were calculated based on gross national income (GNI) per capita, which could have negative influences on the accuracy of estimated anthropogenic heat for mega cities in these countries. Previous studies have argued that the SVM-based algorithm is not sensitive to initial training samples. Therefore, threshold values of NDVI and DMSP/OLS were used for extracting global urban patches42, and uniform training samples and thresholds of NDVI and DMSP/OLS were utilized around the world during the study period. This is another reason for underestimation in mega cities. However, separating total energy consumption into two types, urban and non-urban, then further downscaling urban energy to each city in each country is a step forward for calculating a robust global AHF product. | 2023-02-06 09:42:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.40456777811050415, "perplexity": 2648.6296502946843}, "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/1674764500334.35/warc/CC-MAIN-20230206082428-20230206112428-00808.warc.gz"} |
https://collegemathteaching.wordpress.com/category/student-learning/ | # College Math Teaching
## February 9, 2016
### An economist talks about graphs
Filed under: academia, economics, editorial, pedagogy, student learning — Tags: , — collegemathteaching @ 7:49 pm
Paul Krugman is a Nobel Laureate caliber economist (he won whatever they call the economics prize).
Here he discusses the utility of using a graph to understand an economic situation:
Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101. I say production possibilities yes, Edgeworth box no — which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box — and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days.
There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.
OK, this is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.
Beyond that, pictures are often the best way to convey global insights about the economy — global in the sense of thinking about all possibilities as opposed to small changes, not as in theworldisflat. […]
And it probably doesn’t hurt to remind ourselves that our students are, in general, NOT like us. What comes to us naturally probably does not come to them naturally.
## February 8, 2016
### Where these posts are (often) coming from
Filed under: academia, linear albegra, student learning — Tags: , — collegemathteaching @ 9:57 pm
Yes, my office is messy. Deal with it. 🙂 And yes, some of my professional friends (an accountant and a lawyer) just HAD to send me their office shots…pristine condition, of course.
(all in good fun!)
Note: this semester I teach 3 classes in a row: second semester “business/life science” calculus, second semester “engineering/physical science” calculus and linear algebra. Yes, I love the topics, but there is just enough overlap that I have to really clear my head between lessons. Example: we covered numerical integration in both of my calculus classes, as well as improper integrals. I have to be careful not to throw in $\int^{\infty}_{-\infty} \frac{dx}{1+x^2}$ as an example during my “life science calculus” class. I do the “head clearing” by going up the stairs to my office between classes.
Linear algebra is a bit tricky; we are so accustomed to taking things like “linear independence” for granted that it is easy to forget that this is the first time the students are seeing it. Also, the line between rigor and “computational usefulness” is tricky; for example, how rigorously do we explain “the determinant” of a matrix?
Oh well…back to some admin nonsense.
## February 16, 2015
### Topologist’s Sine Curve: connected but not path connected.
Filed under: student learning, topology — Tags: , , — collegemathteaching @ 1:01 am
I wrote the following notes for elementary topology class here. Note: they know about metric spaces but not about general topological spaces; we just covered “connected sets”.
I’d like to make one concession to practicality (relatively speaking). When it comes to showing that a space is path connected, we need only show that, given any points $x,y \in X$ there exists $f: [a,b] \rightarrow X$ where $f$ is continuous and $f(a) = x, f(b) = y$. Here is why: $s: [0,1] \rightarrow [a,b]$ by $s(t) = a + (b-a)t$ maps $[0,1]$ to $[a,b]$ homeomorphically provided $b \neq a$ and so $f \circ s$ provides the required continuous function from $[0,1]$ into $X$.
Now let us discuss the topologist’s sine curve. As usual, we use the standard metric in $R^2$ and the subspace topology.
Let $S = \{(t, sin(\frac{1}{t}) | t \in (0, \frac{1}{\pi} \}$. See the above figure for an illustration. $S$ is path connected as, given any two points $(x_1, sin(\frac{1}{x_1}), (x_2, sin(\frac{1}{x_2})$ in $S$, then $f(x) = (x, sin(\frac{1}{x})$ is the required continuous function $[x_1, x_2] \rightarrow S$. Therefore $S$ is connected as well.
Note that $(0,0)$ is a limit point for $S$ though $(0,0) \notin S$.
Exercise: what other limit points does $S$ that are disjoint from $S$?
Now let $T = S \cup \{ (0,0) \}$, that is, we add in the point at the origin.
Fact: $T$ is connected. This follows from a result that we proved earlier but here is how a “from scratch” proof goes: if there were open sets $U, V$ in $R^2$ that separated $T$ in the subspace topology, every point of $S$ would have to lie in one of these, say $U$ because $S$ is connected. So the only point of $T$ that could lie in $V$ would be $(0,0)$ which is impossible, as every open set containing $(0,0)$ hits a point (actually, uncountably many) of $S$.
Now we show that $T$ is NOT path connected. To do this, we show that there can be no continuous function $f: [0, \frac{1}{\pi}] \rightarrow T$ where $f(0) = (0,0), f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0 )$.
One should be patient with this proof. It will go in the following stages: first we show that any such function $f$ must include EVERY point of $S$ in its image and then we show that such a function cannot be extended to be continuous at $(0,0)$.
First step: for every $(z, sin(\frac{1}{z})),$ there exists $x \in (0,\frac{1}{\pi} ]$ where $f(x) = (z, sin(\frac{1}{z}) )$ Suppose one point was missed; let $z_0$ denote the least upper bound of all $x$ coordinates of points that are not in the image of $f$. By design $z_0 \neq \frac{1}{\pi}$ (why: continuity and the fact that $f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0)$ ) So $(z_0, sin(\frac{1}{z_0})$ cuts the image of TS into two disjoint open sets $U_1, V_1$ (in the subspace topology): that part with x-coordinate less than and that part with x-coordinate greater than $x = z_0$. So $f^{-1}(U_1)$ and $f^{-1}(V_1)$ form separating open sets for $[0,\frac{1}{\pi}]$ which is impossible.
Note: if you don’t see the second open set in the picture, note that for all $(w, sin(\frac{1}{w})), w > z_0$ one can find and open disk that misses the part of the graph that occurs “before” the $x$ coordinate $z_0$. The union of these open disks (an uncountable union) plus an open disk around $(0,0)$ forms $V_1$; remember that an arbitrary union of open sets is open.
Second step: Now we know that every point of $S$ is hit by $f$. Now we can find the sequence $a_n \in f^{-1}(\frac{1}{n \pi}, 0))$ and note that $a_n \rightarrow 0$ in $[0, \frac{1}{\pi}]$. But we can also find $b_n \in f^{-1}(\frac{2}{1 + 4n \pi}, 1)$ where $b_n \rightarrow 0$ in $[0, \frac{1}{\pi}]$. So we have two sequences in the domain converging to the same number but going to different values after applying $f$. That is impossible if $f$ is continuous.
This gives us another classification result: $T$ and $[0,1]$ are not topologically equivalent as $T$ is not path connected.
## August 25, 2014
### How to succeed at calculus, and why it is worth it!
Filed under: calculus, student learning — Tags: , — collegemathteaching @ 2:06 pm
This post is intended to help the student who is willing to put time and effort into succeeding in a college calculus class.
Part One: How to Study
The first thing to remember is that most students will have to study outside of class in order to learn the material. There are those who pick things up right away, but these students tend to be the rare exception.
Think of it this way: suppose you want to learn to play the piano. A teacher can help show you how to play it and provide a practice schedule. But you won’t be any good if you don’t practice.
Suppose you want to run a marathon. A coach can help you with running form, provide workout schedules and provide feedback. But if you don’t run those workouts, you won’t build up the necessary speed and endurance for success.
The same principle applies for college mathematics classes; you really learn the material when you study it and do the homework exercises.
Here are some specific tips on how to study:
1. It is optimal if you can spend a few minutes scanning the text for the upcoming lesson. If you do this, you’ll be alert for the new concepts as they are presented and the concepts might sink in quicker.
2. There is some research that indicates:
a. It is better to have several shorter study sessions rather than one long one and
b. There is an optimal time delay between study sessions and the associated lecture.
Look at it this way: if you wait too long after the lesson to study it, you would have forgotten much of what was presented. If you study right away, then you really have, in essence, a longer class room session. It is probably best to hit the material right when the initial memory starts to fade; this time interval will vary from individual to individual. For more on this and for more on learning for long term recall, see this article.
3. Learn the basic derivative formulas inside and out; that is, know what the derivatives of functions like $sin(x), cos(x), tan(x), sec(x), arctan(x), arcsin(x), exp(x), ln(x)$ are on sight; you shouldn’t have to think about them. The same goes for the basic trig identities such as $\sin ^{2}(x)+\cos ^{2}(x)=1$ and $\tan^{2}(x)+1 = \sec^{2}(x)$
Why is this? The reason is that much of calculus (though not all!) boils down to pattern recognition.
For example, suppose you need to calculate:
$\int \dfrac{(\arctan (x))^{5}}{1+x^{2}}dx=$
If you don’t know your differentiation formulas, this problem is all but impossible. On the other hand, if you do know your differentiation formulas, then you’ll immediately recognize the $arctan(x)$ and it’s derivative $\dfrac{1}{1+x^{2}}$ and you’ll see that this problem is really the very easy problem $\int u^{5}du$.
But this all starts with having “automatic” knowledge of the derivative formulas.
Note: this learning is something your professor or TA cannot do for you!
4. Be sure to do some study problems with your notes and your book closed. If you keep flipping to your notes and book to do the homework problems, you won’t be ready for the exams. You have to kick up the training wheels.
Try this; the difference will surprise you. There is also evidence that forcing yourself to recall the material FROM YOUR OWN BRAIN helps you learn the material! Give yourself frequent quizzes on what you are learning.
5. When reviewing for an exam, study the problems in mixed fashion. For example, get some note cards and write problems from the various sections on them (say, some from 3.1, some from 3.2, some from 3.3, and so on), mix the cards, then try the problems. If you just review section by section, you’ll go into each problem knowing what technique to use each time right from the start. Many times, half of the battle is knowing which technique to use with each problem; that is part of the course! Do the problems in mixed order.
If you find yourself whining complaining “I don’t know where to start” it means that you don’t know the material well enough. Remember that a trained monkey can repeat specific actions; you have to be a bit better than that!
6. Read the book, S L O W L Y, with pen and paper nearby. Make sure that you work through the examples in the text and that you understand the reasons for each step.
7. For the “more theoretical” topics, know some specific examples for specific theorems. Here is what I am talking about:
a. Intermediate value theorem: recall that if $f(x)=\frac{1}{x}$, then $f(-1)=-1,f(1)=1$ but there is no $x$ such that $f(x) = 0$. Why does this not violate the intermediate value theorem?
b. Mean value theorem: note also that there is no $c$ such that $f'(c) = \frac{f(1)-f(-1)}{2} = 0$. Why does this NOT violate the Mean Value Theorem?
c. Series: it is useful to know basic series such as those for $exp(x), sin(x), cos(x)$. It is also good to know some basic examples such as the geometric series, the divergent harmonic series $\sum \frac{1}{k}$ and the conditionally convergent series $\sum (-1)^{k}\frac{1}{k}$.
d. Limit definition of derivative: be able to work a few basic examples of the derivative via the limit definition: $f(x) = x^{n}, f(x) = \frac{1}{x}, f(x)=\sqrt{x}$ and know why the derivative of $f(x) = |x|$ and $f(x) = x^{1/3}$ do not exist at $x = 0$.
Part II: Attitude
Your attitude will be very important.
1. Remember that your effort will be essential! Again, you can’t learn to run a marathon without getting off of the couch and making your muscles sore. Learning mathematics involves some frustration and, yes, at times, some tedium. Learning is fun OVERALL but it isn’t always fun at all times. You will encounter discomfort and unpleasantness at times.
2. Remember that winners look for ways to succeed; losers and whiners look for excuses for failure. You can always find those who will be willing to enable your underachievement. Instead, seek out those who bring out your best.
3. Success is NOT guaranteed; that is what makes success rewarding! Think of how good you’ll feel about yourself if you mastered something that seemed impossible to master at first. And yes, anyone who has achieved anything that is remotely difficult has taken some lumps and bruises along the way. You will NOT be spared these.
Remember that if you duck the calculus challenge, you are, in essence, slamming many doors of opportunity shut right from the get-go.
4. On the other hand, remember that Calculus (the first two semesters anyway) is a Freshman level class; exceptional mathematical talent is not a prerequisite for success. True, calculus is easy for some but that isn’t the point. Most reasonably intelligent people can have success, if they are willing to put forth the proper effort in the proper manner.
Just think of how good it will feel to succeed in an area that isn’t your strong suit!
## February 25, 2014
### The potential harm in using outliers as examples…..
Filed under: academia, editorial, student learning — Tags: — blueollie @ 9:52 pm
I think that this is common in this day and age: I have some students who are struggling in our “elementary conceptual calculus” course. They come to class, but work a large number of hours at a job in order to make ends meet. So…they are often left with very little time to study.
And yes, IN THIS COURSE, most of the students need to study quite a bit in order to have a chance at even a “C”.
In short: most students need to have a certain number of hours in order to sleep and to study..in addition to making the classes and their part time jobs.
Now, some might say that this is nonsense.
I remember a professor I had at the Naval Academy. He said that when he was an undergraduate he studied very little for his math classes as he paid his own way through school by waiting tables. He made up for it by PAYING ATTENTION IN CLASS.
That is well and good…..but then remember that he had an earned Ph.D. in mathematics from MIT.
Most of us don’t have that type of natural ability.
Yes, Mohammed Ali could break the conventional rules of boxing (dangle his arms, lean away from punches):
But most, including most other professional boxers, don’t have that kind of ability.
Following the 1976 trials he trained by running 35 miles per week and ran “a 2:14:37 for second place at the Nike-Oregon Track Club Marathon in Eugene in 1978. After that, he ran 2:15:23 for 15th place in the Boston Marathon in 1979.”
But most of us aren’t that gifted (this was Tony Sandoval, cowinner of the 1980 US Olympic Trials Marathon)
Yes, some can make a successful film while being stoned on marijuana, but most of us aren’t as talented as the Beatles.
The list can go on and on. The bottom line: you can gain inspiration from the incredibly successful, but you won’t be able to get away with taking the short cuts that many of them got away with. Neither you nor I are outliers.
## December 21, 2013
### Rant: please stop with the teaching of “gimmicks for calculation”
Filed under: calculus, editorial, integrals, student learning, Uncategorized — Tags: — collegemathteaching @ 1:05 pm
I finished teaching calculus II (our course: techniques for integration, applications of integrals and infinite sequences/series) and noticed that some of our freshmen students came in knowing how to do many of the calculations…did well on the first exam…then didn’t do so well in the rest of the course.
Evidently, they were well versed in calculation tricks learned in high school; give them $\int x^3 sin(x) dx$ and they could whip out a table.
So here is my rant: we teach integration by parts not so much to calculate integrals like $\int x^3 sin(x) dx$ (which can be rapidly done with a calculator) but rather so they can understand the technique of integration by parts.
Why? Well, there are many uses of integration by parts and I’ll just display a few uses of them:
1. Taylor Polynomials. How do we get these? If we assume that $f$ has enough derivatives, we proceed in the following manner: calculate $\int ^x_0 f'(t) dt$ in two different ways: use the Fundamental Theorem of calculus on one side (to obtain $f(x) - f(0)$ and use integration by parts on the other side: $u = f'(t), dv = dt, du = f''(t), v = t-x$ (yes, we are being choosy about which anti derivative of $dv$ to use).
This means: $-\int^x_0 f''(t)(t-x)dt +f'(t)(t-x)|^x_0 = f(x)-f(0)$ so $f(x) = f(0) + f'(0)x -\int^x_0 f''(t)(t-x)dt =f(x)$ and one proceeds from there.
2. Differential equations: given $y' + p(x)y = f(x)$ one seeks to find an integrating factor (which is $e^{\int p(x)}$ so as to get:
$e^{\int p(x)}y' + p(x)e^{\int p(x)}y = f(x)e^{\int p(x)}$ which can be written as $\frac{d}{dx}(e^{\int p(x)}y) = f(x)e^{\int p(x)}$. That is, the left hand side is just the product rule for derivatives, which, as you know (if you are a calculus teacher), is really all integration by parts is!
Sure, one can jazz it up (as we subtly did in the Taylor Polynomial calculation); the integration by parts formula is really $\frac{d}{dx} (f(x)g(x)) = \frac{d}{dx}(f(x)+ C) g(x) + f(x)\frac{d}{dx}(g(x) + D)$ where $C, D$ are arbitrary constants. But, my main point is that integration by parts should be UNDERSTOOD; short cuts to do tedious calculations are relatively unimportant, IMHO.
Now if you want to ask students “why does tabular integration work”, then….GREAT!
## October 15, 2013
### What do you mean “you don’t know”???
Filed under: calculus, integrals, student learning — Tags: — collegemathteaching @ 6:38 pm
I am grading calculus II exams and the above photo looks a bit like me. I’ll calm down before I hand the exams back to the students.
But I swear: I had one student DURING THE EXAM say “hey, you can’t do $\int^1_{\frac{1}{2}} \frac{1}{x} dx$ because $\frac{1}{x} = x^{-1}$ and $x^{-1+1} \frac{1}{-1+1}$ is undefined. Yes, this person passed calculus I and yes, we did that integral in calculus I (some schools hold off and develop $ln(x) = \int^x_1 \frac{1}{t} dt$ using the Fundamental Theorem of Calculus). And yes, previously THIS SEMESTER we did integrals like $\int \frac{1}{(x-1)(x+1)} dx$.
(facepalm).
## October 1, 2013
### I’ve lost any reason to live…
Filed under: student learning — Tags: , — collegemathteaching @ 7:45 pm
Today, I was grading differential equation papers. Some were really good, others were not in that category.
A calculus 2 student came in (we teach techniques of integration, applications of integration, and infinite series).
She, in a very polite manner, complained that the course was. too. easy.
Too. Easy.
I wish that I drank.
But there is no denying it: we have strong freshmen and some very weak, “take the class multiple times” students in the same class, and heaven help you if your flunk out rate is too high.
I might encourage her to take her complaint to the dean and put it in writing.
#\$#@!!!
## September 19, 2013
### What we mean about poor algebra skills…
Filed under: basic algebra, calculus, student learning — collegemathteaching @ 4:47 pm
Yes, mathematics professors have been complaining about their students lack of algebra skills as long as there have been calculus courses.
No, we aren’t talking about a student who, in a moment of panic, decided to write $\int \sqrt{x^2+1} dx = \int \sqrt{x^2} + \sqrt{1} dx$ because they were stuck on an exam. And yes, I once saw a professor walk into an analysis class, write $\sqrt{x^2+1} dx \ne \sqrt{x^2} + \sqrt{1}$ on the board (while grinding the chalk into the board) while saying “the next person who makes this mistake will get an F for this class, ON THE SPOT! 🙂
But the weakness is more of the following: in class today, I wrote
$\int (sec^2(x) - 1)tan(x) dx = \int (sec^2(x)tan(x) -tan(x)) dx$ $= \frac{1}{2}tan^2(x) - ln(|sec(x)|+C$
The student actually understood the integration, but didn’t understand where the first equality came from! I said “it is just algebra” and he STILL didn’t get it.
I have a hard time believing that this student doesn’t understand the distributive axiom of algebra; what I think is going on is that they don’t have the concept as a regular working part of their math/science/engineering mind.
## September 14, 2013
### Reality of modern college teaching: students with Asperger’s syndrome
Filed under: academia, mathematics education, student learning — Tags: — collegemathteaching @ 4:43 pm
One of the major changes I’ve encountered since I started college teaching (first as a teaching assistant in 1986; then as a new professor in 1991) is that students with Asperger’s syndrome have been showing up.
Most of the time, it isn’t a big deal; the worst I’ve had is one of these students became completely disoriented when he got to class and someone was sitting in “his” seat (no, I don’t make seat assignments; this is college).
This semester, I have a transfer student (not sure why he transferred); in spots he is “disruptive to a minor degree”: you have to remind him that there are 34 OTHER students in the class; this isn’t a one-on-one dialogue just for him.
Also, I sometimes make side remarks (to explain a point to another student) and use analogies; that just confuses the heck out of him. But I am not going to stop being effective with the other 34 students just for him; I just tell him “see me in office hours” or “don’t worry about this”.
On the other hand, he is relatively easy to work with in office hours; the one-on-one exchanges are usually reasonable and pleasant.
Hence, when I see he is getting confused, I tell him “for this point, see me for office hours.”
I’ve searched the internet to see what is out there; most of it is what I already know and much of it is a series of tired cliches, finger wagging, etc. I haven’t found much of the following: “I had these issues in my calculus class; here is how they were resolved” or “these issues COULDN’T be resolved.” Sometimes they aren’t up to the task of being in college.
But, overall, it seems to be this way: we are told to be “more productive” which means more students per semester (105 students in 2 sections of calculus and 1 of differential equations). So no, one cannot tailor lessons and work to the learning style of a specific student, especially if that student is an outlier. One has to teach to a type of average or to the class as a whole; one can adjust for a class full of, say biology students, or one full of engineers or one full of computer science majors.
These students require time, more attention and resources and these COST MONEY. This is where some of the increased educational expense is coming from (some from technology as well). At times, it appears as if colleges and universities are being tugged in different directions.
Older Posts » | 2017-06-24 00:12:50 | {"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": 121, "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": 0.6428434252738953, "perplexity": 739.3242206637984}, "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/1498128320206.44/warc/CC-MAIN-20170623235306-20170624015306-00643.warc.gz"} |
https://math.stackexchange.com/questions/2860736/number-of-knight-cycle-free-n-tours-on-an-n-times-n-modular-chessboard | Number of knight cycle-free $n$-tours on an $n \times n$ modular chessboard?
Given an $n \times n$ modular (ie: the line resp. column after the last one is identified with the first) chessboard, I'd like to count the number of cycle-free $n$-paths a knight can do, starting from any square (ie: the knight will walk only $n$ squares instead of your usual $n^2$).
The solution with the knight graph adjacency matrix won't work here because I don't know $n$ in advance... I'd like to obtain the answer as a formula depending on $n$.
I tried thinking like this: there are $n^2$ squares to put the first knight. Since the board is modular that square threatens exactly other 8, so there are 8 possibilities for the 2nd. Since I want paths, for the third one there are only 7 possible squares, and so on until the last, giving the number $8n^27^{n-1}$ paths.
However I realized this number is merely an upper bound, not only because I'm counting some paths more than once but because that way some paths might also contain cycles...
So... How can I properly count the number of such paths? | 2019-07-24 06:34:20 | {"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": 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": 0.8327315449714661, "perplexity": 433.8035291279483}, "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-2019-30/segments/1563195531106.93/warc/CC-MAIN-20190724061728-20190724083728-00331.warc.gz"} |
https://www.semanticscholar.org/paper/The-Golomb-topology-of-polynomial-rings%2C-II-Spirito/acbc20a62f437307ff596e616aa8127bb4aabc88 | • Corpus ID: 252531950
# The Golomb topology of polynomial rings, II
@inproceedings{Spirito2022TheGT,
title={The Golomb topology of polynomial rings, II},
author={Dario Spirito},
year={2022}
}
. We study the interplay of the Golomb topology and the algebraic structure in polynomial rings K [ X ] over a field K . In particular, we focus on infinite fields K of positive characteristic such that the set of irreducible polynomials of K [ X ] is dense in the Golomb space G ( K [ X ]). We show that, in this case, the characteristic of K is a topological invariant, and that any self-homeomorphism of G ( K [ X ]) is the composition of multiplication by a unit and a ring automorphism of K [ X ].
## References
SHOWING 1-10 OF 13 REFERENCES
Abstract We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if K
We prove the following form of Dirichlet's theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F. For all relatively prime polynomials a(X), b(X) E F[X] and for
• Mathematics
• 2019
The $Golomb$ $space$ $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with
• Mathematics
• 2019
Abstract In 1959 Golomb defined a connected topology on ℤ. An analogous Golomb topology on an arbitrary integral domain was defined first by Knopfmacher-Porubský [KP97] and then again in a recent
• Mathematics
Commentationes Mathematicae Universitatis Carolinae
• 2019
The Golomb space $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with
### Golomb . Arithmetica topologica
• General Topology and its Relations to Modern Analysis and Algebra ( Proc . Sympos . , Prague , | 2023-01-28 19:18:09 | {"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": 0.8671781420707703, "perplexity": 613.1103903413132}, "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/1674764499654.54/warc/CC-MAIN-20230128184907-20230128214907-00402.warc.gz"} |
https://rstudio-pubs-static.s3.amazonaws.com/140060_976371289b864ec1a632199e762add68.html | ## Three-Dimensional Charting in R
Creating 3D charts is a powerful capability of R. These charts are particularly interesting because they are interactive. The viewing angle can be manipulated.
This is not as challenging as it sounds. Here is one way to accomplish this.
First, the basic code to open packages and set things up.
data(iris3, package = 'datasets')
iris3 <- as.data.frame(iris3)
names(iris3)<- make.names(names(iris3))
library(nlme)
library(mgcv)
## This is mgcv 1.8-9. For overview type 'help("mgcv-package")'.
library(rgl, pos = 4)
library(mgcv, pos = 4)
library(car)
Now for the creation of the charts. Note the code calls for the chart to be opened in a separate window. Also note the code creates charts one at at time.
The first chart plots petal and sepal characteristics of the Setosa variety:
scatter3d(iris3$Petal.W..Setosa, iris3$Petal.L..Setosa, iris3$Sepal.L..Setosa, fit = 'linear', residuals = TRUE, bg = 'white', axis.scales = TRUE, grid = TRUE, ellipsoid = FALSE, xlab = 'Petal.W..Setosa', ylab = 'Petal.L..Setosa', zlab = 'Sepal.L..Setosa') The second chart plots petal lengths for all three varieties. scatter3d(iris3$Petal.L..Versicolor, iris3$Petal.L..Setosa, iris3$Petal.L..Virginica, fit = 'linear', residuals = TRUE, bg = 'white', axis.scales = TRUE, grid = TRUE, ellipsoid = FALSE, xlab = 'Petal.L..Versicolor', ylab = 'Petal.L..Setosa', zlab = 'Petal.L..Verginica')
This last code chunk stores the chart as a PNG file on your computer.
rgl.snapshot("C:/Documents and Settings/abc/Desktop/RGLGraph.png")
## Warning in rgl.snapshot("C:/Documents and Settings/abc/Desktop/
## RGLGraph.png"): RGL: Pixmap save: unable to open file 'C:\Documents and
## Settings\abc\Desktop\RGLGraph.png' for writing
## Warning in rgl.snapshot("C:/Documents and Settings/abc/Desktop/
## RGLGraph.png"): 'rgl.snapshot' failed | 2020-06-04 14:51:44 | {"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": 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": 0.2665654420852661, "perplexity": 13683.171708880012}, "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-2020-24/segments/1590347441088.63/warc/CC-MAIN-20200604125947-20200604155947-00177.warc.gz"} |
https://codereview.stackexchange.com/questions/192546/incremental-count-and-its-position-in-python | # Incremental Count and it's position in Python [closed]
I have a requirement to get incremental counts. I am new to Python so please help me.
I have tried below:
Testword = 'foo boo'
letters = 'abcdefghijklmnopqrstuvwxyz'
for key in Testword:
print key, Testword.count(key)
Results :
f 1
o 4
o 4
1
b 1
o 4
o 4
I would like to get incremented count output in the same input order like
f=1
o=1
o=2
b=1
o=3
o=4
## closed as off-topic by Peilonrayz, t3chb0t, Toby Speight, alecxe, GraipherApr 20 '18 at 12:47
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "Code not implemented or not working as intended: Code Review is a community where programmers peer-review your working code to address issues such as security, maintainability, performance, and scalability. We require that the code be working correctly, to the best of the author's knowledge, before proceeding with a review." – Peilonrayz, t3chb0t, Toby Speight, alecxe, Graipher
If this question can be reworded to fit the rules in the help center, please edit the question.
What you are looking for is the number of occurences until the position, you are currently looking at. Using Testword[:pos+1], you can take the substring until the current position. Hence:
testword = 'foo boo'
letters = 'abcdefghijklmnopqrstuvwxyz'
for pos, key in enumerate(testword):
print(key, testword[:pos+1].count(key))
Gives
f 1
o 1
o 2
1
b 1
o 3
o 4
Btw, please use python3, not python2.
• Please note, questions involving code that's not working as intended are off-topic on this site and should not be answered. If OP edits their post to address the problem and makes their post on-topic, they may make your answer moot. Anyways, on-topic questions get more views and will get you more rep in the long run ;) See this meta discussion for reference: codereview.meta.stackexchange.com/q/6604/98493 – Graipher Apr 20 '18 at 12:49
• This code could do with a Code Review, to remove the $n^2$ complexity. – Peilonrayz Apr 20 '18 at 15:51 | 2019-08-22 10:13:52 | {"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": 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": 0.19088317453861237, "perplexity": 3501.400335104087}, "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/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00370.warc.gz"} |
https://kerodon.net/tag/008Z | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Example 2.2.5.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We let $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the strict functor which carries every object, $1$-morphism, and $2$-morphism of $\operatorname{\mathcal{C}}$ to itself. We will refer to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as the identity functor on $\operatorname{\mathcal{C}}$. Note that it is both a left and right unit for the composition of lax functors given in Construction 2.2.5.1. | 2021-09-24 23:41:11 | {"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": 0.9852918982505798, "perplexity": 198.13240138695667}, "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-2021-39/segments/1631780057584.91/warc/CC-MAIN-20210924231621-20210925021621-00379.warc.gz"} |
http://www.contrib.andrew.cmu.edu/~ryanod/?tag=learning-theory | [...]
## §6.4: Applications in learning and testing
In this section we describe some applications of our study of pseudorandomness.
[...]
## §3.4: Learning theory
Computational learning theory is an area of algorithms research devoted to the following task: given a source of “examples” $(x, f(x))$ from an unknown function $f$, compute a “hypothesis” function $h$ which is good at predicting $f(y)$ on future inputs $y$. We will focus on just one possible formulation of the task:
[...] | 2019-04-20 21:05:39 | {"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": 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": 0.6995217800140381, "perplexity": 797.1774791150482}, "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-2019-18/segments/1555578530040.33/warc/CC-MAIN-20190420200802-20190420222802-00154.warc.gz"} |
http://www.cs.bham.ac.uk/~sjv/papersfull.php | # Steven Vickers: Bibliography
This is a complete list of my academic papers, with abstracts and dowloadable preprint versions where available.
Elsewhere, you can also find -
#### Note on .ps papers
Some of the older .ps papers (PreFrPrePre, GeoTh+DBs, Infosys and QuProc) have been prepared from original Microsoft Word documents. They will differ from the published versions, certainly in format and perhaps also in details of content. Also, some of those .ps files have not been tested on a printer, so please report any problems with them by emailing me (s.j.vickers) at cs.bham.ac.uk.
## Contents (most recent first)
### ABSTRACTS
#### "A coherent account of geometricity"
(Steven Vickers and Christopher Townsend)
Unpublished.
Preprint version April 2014: pdf.
A "bundle endofunctor preserving proneness" for a cartesian category $\mathcal{C}$ is an endofunctor $\mathfrak{T}_\bullet$ of the arrow category $\mathcal{C}^{\downarrow}$, the identity on codomains, that preserves the property of morphisms of being pullback squares. Such a $\mathfrak{T}_\bullet$ is "slicewise strong", restricting to a strong endofunctor $\mathfrak{T}_\bullet$ on every slice $\mathcal{C}/B$, and with that structure preserved by pullback between slices.
In the other direction, a strong endofunctor $T$ on $\mathcal{C}$ extends to a bundle endofunctor $T_\bullet$ on $\mathcal{C}$, which preserves proneness if $T$ preserves coreflexive equalizers.
If a bundle endofunctor $\mathfrak{T}_\bullet$ preserving proneness also preserves coreflexive equalizers, then it is naturally isomorphic to the $T_\bullet$ arising from $T = \mathfrak{T}_1$.
Combining these, bundle endofunctors for $\mathcal{C}$ preserving proneness and coreflexive equalizers are equivalent to strong endofunctors on $\mathcal{C}$ preserving coreflexive equalizers; this latter structure is inherited by all slices and preserved by pullback between them.
The results extend to situations where $\mathfrak{T}_\bullet$ and $T$ are monads.
We propose that the structure of bundle endofunctor preserving proneness is a satisfactory categorical abstraction of the notion of geometric construction when $\mathcal{C}$ is the category $\mathbf{Loc}$ of locales. The powerlocales give rise to bundle monads on $\mathbf{Loc}$ preserving proneness and coreflexive equalizers; likewise for the covariant powerobject monad on any topos.
#### "Geometric constructions preserve fibrations"
(Bertfried Fauser and Steven Vickers)
Unpublished.
Archived at arXiv:1411.2457.
Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod} \colon \mathcal{C}^\downarrow \rightarrow \mathcal{C}$ and (ii) $\mathfrak{T}_\bullet$ preserves proneness (cartesianness) of morphisms in $\mathcal{C}^\downarrow$. Then $\mathfrak{T}_\bullet$ preserves fibrations and opfibrations in $\mathcal{C}$. The proof takes Street's characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads $\mathfrak{L}_B$ on slice categories $\mathcal{C}/B$ and develops it by defining a 2-monad $\mathfrak{L}_\bullet$ on $\mathcal{C}^\downarrow$ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.
#### "Positivity relations on a locale"
(Francesco Ciraulo and Steven Vickers)
Unpublished.
Preprint version March 2014: pdf.
This paper analyses the notion of a positivity relation of Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad).
#### "Domain theory in topical form"
In Coecke, Ong, Panangaden (ed.) "Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky"
Springer LNCS 7860 (2013), pages 348--349
ISSN 0302-9743, ISBN 978-3-642-38163-8,
doi:10.1007/978-3-642-38164-5_24
Preprint version: pdf.
Describes how "Topical categories of domains" developed out of Abramsky's "Domain theory in logical form".
#### "Gelfand spectra in Grothendieck toposes using geometric mathematics"
(Bas Spitters, Steven Vickers and Sander Wolters)
In Proceedings of the 9th International Workshop on Quantum Physics and Logic (QPL 2012), Brussels 2012 (ed. Duncan, Panangaden).
Electronic Proceedings in Theoretical Computer Science 158 (2014), pp. 77 - 107.
ISSN 2075-2180, doi:10.4204/EPTCS.158.7
Preprint version: pdf, or arXiv 1310.0705.
In the topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to any unital C*-algebra A, a topos T(A) of sheaves on a locale and a commutative C*-algebra A within that topos. The Gelfand spectrum of A is a locale Σ in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale Σ, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise.
As a by-product of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too.
#### "The Born rule as structure of spectral bundles"
(Extended abstract)
(Bertfried Fauser, Guillaume Raynaud and Steven Vickers)
In Proceedings of the 8th International Workshop on Quantum Physics and Logic (QPL 2011), Nijmegen 2011 (ed. Jacobs, Selinger, Spitters).
Electronic Proceedings in Theoretical Computer Science 95 (2012), pp. 81 - 90.
ISSN 2075-2180, doi:10.4204/EPTCS.95.8
Preprint version: pdf
Topos approaches to quantum foundations are described in a unified way by means of spectral bundles, where the base space is a space of contexts and each fibre is its spectrum. Differences in variance are due to the the bundle being a fibration or opfibration. Relative to this structure, the probabilistic predictions of the Born rule in finite dimensional settings are then described as a section of a bundle of valuations. The construction uses in an essential way the geometric nature of the valuation locale monad.
#### "A monad of valuation locales"
Preprint version: pdf
If X is a locale then its valuation locale has for its points the valuations on X. This construction is the functor part of a strong monad on the category of locales, a localic analogue of the Giry monad. It is commutative, i.e. product valuations exist and a Fubini Theorem holds. An analogue of the Riesz Representation theorem holds. Concrete representations are given for the tensor product of lattices and for the modular monoid. The work conforms with the constructive constraints of geometric logic.
#### "Continuity and geometric logic"
Journal of Applied Logic 12 (1) (2014), pages 14-27
ISSN: 1570-8683; doi: 10.1016/j.jal.2013.07.004.
Preprint version: pdf
Reports my Bordeaux talk "Aspects of geometric logic".
This paper is largely a review of known results about various aspects of geometric logic. Following Grothendieck's view of toposes as generalized spaces, one can take geometric morphisms as generalized continuous maps. The constructivist constraints of geometric logic guarantee the continuity of maps constructed, and can do so from two different points of view: for maps as point transformers and maps as bundles.
#### "Generalized powerlocales via relation lifting"
(Yde Venema, Steven Vickers and Jacob Vosmaer)
Mathematical Structures in Computer Science 23 (1) (2013), pages 142-199.
ISSN: 0960-1295; doi: 10.1017/S0960129512000229.
Preprint version: pdf
This paper introduces an endofunctor VT on the category of frames, parametrized by an endofunctor T on the category Set that satisfies certain constraints. This generalizes Johnstone's construction of the Vietoris powerlocale, in the sense that his construction is obtained by taking for T the finite covariant power set funtor. Our construction of the T-powerlocale VT L out of a frame L is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss's coalgebraic cover modality.
We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality, and the combination of zerodimensionality and compactness.
#### "An induction principle for consequence in arithmetic universes"
(Maria Emilia Maietti and Steven Vickers)
Journal of Pure and Applied Algebra 216 (8-9) (2012), pages 2049-2067.
ISSN: 0022-4049; doi: 10.1016/j.jpaa.2012.02.040.
Preprint version: pdf
Presentations: Amsterdam and Genova.
Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0)→ψ(0) and an induction step that, for generic n, the hypothesis φ(n)→ ψ(n) allows one to deduce φ(n+1)→ ψ(n+1). Then it is already true in that arithmetic universe that (∀ n)(φ(n)→ ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential φ(n)→ ψ(n).
The principle is applied to the question of locatedness of Dedekind sections.
The development analyses in some detail a notion of "subspace" of an arithmetic universe, including open or closed subspaces and a boolean algebra generated by them. There is a lattice of subspaces generated by the opens and the closeds, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.
#### "Presenting dcpos and dcpo algebras"
(Achim Jung, M. Andrew Moshier and Steven Vickers)
In Proceedings of the 24th Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIV) (ed. Bauer, Mislove), Electronic Notes in Theoretical Computer Science 218 (2008), pp. 209 - 229.
ISSN 1571-0661; doi:10.1016/j.entcs.2008.10.013
Preprint version: pdf
Dcpos can be presented by a preorder of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are "stable" for the operations. The resulting dcpo algebra has a natural universal characterization and satisfies all the inequational laws satisfied by the generating algebra.
Applications include known "coverage theorems" from locale theory.
#### "Issues of logic, algebra and topology in ontology"
Chapter 22 in "Theory and Applications of Ontology: Computer Applications" (ed. Poli, Healy, Kameas), vol. 2 of "Theory and Applications of Ontology", Springer, 2010.
ISBN 978-90-481-8846-8
Preprint version: pdf
When one uses a particular logical formalism, one makes an ontological commitment to being able to interpret the symbols involved. We discuss this in a case study of geometric logic, being aided by a presentation of the logic as a sequent calculus. We also discuss the connections of geometric logic with topology and algebra.
#### "Fuzzy sets and geometric logic"
Fuzzy Sets and Systems 161 (2010), pp. 1175 - 1204.
ISSN 0165-0114; doi: 10.1016/j.fss.2009.06.013.
Preprint version: pdf
Hoehle has identified fuzzy sets, valued in a frame (complete Heyting algebra) Ω, with certain sheaves over Ω: the subsheaves of constant sheaves More general sheaves can be got as quotients of the fuzzy sets. His principal approach to sheaves over Ω, and topos-theoretic constructions on them, is via complete Ω-valued sets.
In this paper we show how the geometric fragment of those constructions can be described in a natural "stalkwise" manner, provided one works also with incomplete Ω-valued sets.
Our exposition examines in detail the interactions between different technical expressions of the notion of sheaf, and highlights a conceptual view of sheaf as "continuous set-valued map".
You can also view my slides for the invited talk I gave on this topic at the 29th Linz Seminar on Fuzzy Set Theory, February 2008.
#### "Cosheaves and connectedness in formal topology"
Annals of Pure and Applied Logic 163 (2012), pp. 157-174.
ISSN 0168-0072; doi:10.1016/j.apal.2011.06.024
Preprint version: pdf
The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π0 together with certain additional structure and properties that constrain π0 to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. A cosheaf is the connected components cosheaf for a locally connected formal topology iff its complete spread is a homeomorphism, and in this case it is a terminal cosheaf.
A new, geometric proof is given of the topos-theoretic result that a cosheaf is a connected components cosheaf iff it is a "strongly terminal" point of the symmetric topos, in the sense that it is terminal amongst all the generalized points of the symmetric topos. It is conjectured that a study of sites as "formal toposes" would allow such geometric proofs to be incorporated into predicative mathematics.
Key words: Formal topology, predicative, locally connected, cosheaf, symmetric topos, complete spread
2008 MSC: Primary 03F60; Secondary 54D05, 54B20, 18F10
#### "A localic theory of lower and upper integrals"
Mathematical Logic Quarterly 54 (1) (2008), pp. 109 - 123.
ISSN 0942-5616 (print), 1521-3870 (online); doi:10.1002/malq.200710028
Preprint version: pdf
An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.
Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals.
#### "The connected Vietoris powerlocale"
Topology and its Applications 156 (2009), pp. 1886-1910.
ISSN: 0166-8641; doi:10.1016/j.topol.2009.03.043
Preprint version: pdf
The Vietoris powerlocale VX is a point-free analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale VcX whose points are those points of VX that (considered as sublocales of X) satisfy a constructively strong connectedness property. VcX is a strong monad on the category of locales. The strength gives rise to a product map
x: VcX x VcY -> Vc(X x Y),
showing that the product of two of these connected sublocales is again connected. If X is locally connected then VcX is overt. In the case where X is the localic completion of a generalized metric space Y, the points of VcX are characterized as certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric.
The results are applied to the particular case of the point-free real line, giving a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem.
The work is constructive in the sense of topos-validity with natural numbers object. Its geometric aspects (preserved under inverse image functors) are stressed, and exploited to give a pointwise development of the point-free locale theory. The connected Vietoris powerlocale itself is a geometric construction.
#### "Locales and toposes as spaces"
Chapter 8 in "Handbook of Spatial Logics" (ed. Aiello, Pratt-Hartman, van Bentham), Springer, 2007, pp. 429-496.
ISBN 978-1-4020-5586-7
Preprint version: pdf
This chapter was written as an introduction for logicians to the spatial content of geometric logic.
It first outlines the established ideas of Lindenbaum algebras of logical theories, and how they combine with Stone duality (between Boolean algebras and Stone topological spaces) to give a correspondence between theories in classical propositional logic and Stone spaces. This enables one to treat logical theories as topological spaces.
The aim of the chapter is to show how the same insights can be applied to geometric logic, whose connectives mirror the axioms of topology, even though its incompleteness means there is no exact Stone duality.
An important part of this is the role of constructive reasoning. This is paradoxical, since without classical principles (such as choice) it becomes even harder to find the models needed to show completeness. However, constructive reasoning can be applied to models in non-classical mathematics, and in particular the internal mathematics of sheaves. This has the effect of allowing access to a sufficiency of models.
The chapter also describes how the same principles apply to theories in predicate geometric logic. However, the spaces are now generalized in Grothendieck's sense as toposes. The chapter describes how the categorical technicalities of topos theory (e.g. in Mac Lane and Moerdijk) connects to the spatial intuitions.
#### "Partial Horn logic and cartesian categories"
(Erik Palmgren and Steven Vickers)
Annals of Pure and Applied Logic 145 (3) (2007), pp. 314 - 353.
ISSN: 0168-0072; doi:10.1016/j.apal.2006.10.001
Preprint version: pdf
A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as "partial Horn logic".
Various kinds of logical theory are equivalent:
• partial Horn theories
• "quasi-equational" theories, partial Horn theories without predicate symbols
• cartesian theories
• essentially algebraic theories
The logic is sound and complete with respect to models in $\mathbf{Set}$, and sound with respect to models in any cartesian (finite limit) category.
The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint.
Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory $\mathbb{T}$ another, $\mathrm{Cart}\varpi\mathbb{T}$, is constructed, whose models are cartesian categories equipped with models of $\mathbb{T}$. Its initial model, the "classifying category" for $\mathbb{T}$, has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors.
MSC: Primary 18C10. Secondary 03C05, 03G30, 08A55, 08B05, 08B20, 18C05.
Keywords: Cartesian theory, essentially algebraic, free algebra, classifying category, syntactic category, partial algebra.
#### "Sublocales in formal topology"
Journal of Symbolic Logic 72 (2) (2007) 463-482
ISSN: 0022-4812; doi:10.2178/jsl/1185803619
Preprint version: pdf
The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure.
Overt sublocales and weakly closed sublocales are described, and related via a new notion of "rest closed" sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with "lower powerpoints", arising from the impredicative theory of the lower powerlocale.
Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with "upper powerpoints", arising from the impredicative theory of the upper powerlocale.
#### "Some constructive roads to Tychonoff"
In: Laura Crosilla and Peter Schuster (eds.) From Sets and Types to Topology and Analysis: Practicable Foundations for Constructive Mathematics, pp. 223 - 238. Oxford Logic Guides 48, Oxford University Press (2005).
Preprint version: pdf
The Tychonoff Theorem is discussed with respect to point-free topology, from the point of view of both topos-valid and predicative mathematics.
A new proof is given of the infinitary Tychonoff Theorem using predicative, choice-free methods for possibly undecidable index set. It yields a complete description of the finite basic covers of the product.
#### "A language for configuring multi-level specifications"
(Gillian Hill and Steven Vickers)
Final journal version appeared as:
Theoretical Computer Science 351 (2006) 146 - 166
ISSN: 0304-3975; doi:10.1016/j.tcs.2005.09.065
The preprint version is closer to the original conference version, in:
Rattray, C., Maharaj, S. and Shankland, C. (eds) Algebraic Methodology and Software Technology, 10th International Conference, AMAST 2004, Stirling, Scotland, pp. 196-210. Springer Lecture Notes in Computer Science 3116 (2004).
ISBN 3-540-22381-9
Preprint version: ps
This paper shows how systems can be built from their component parts with specified sharing. Its principle contribution is a modular language for configuring systems. A configuration is a description in the new language of how a system is constructed hierarchically from specifications of its component parts. Category theory has been used to represent the composition of specifications that share a component part by constructing colimits of diagrams. We reformulated this application of category theory to view both configured specifications and their diagrams as algebraic presentations of presheaves. The framework of presheaves leads naturally to a configuration lanaguage that expresses structuring from instances of specifications, and also incorporates a new notion of instance reduction from a particular configuration . The language now expresses the hierarchical structuring of multi-level configured specifications. The syntax is simple because it is independent of any specification language; structuring a diagram to represent a configuration is simple because there is no need to calculate a colimit; and combining specifications is simple because structuring is by configuration morphisms with no need to flatten either specifications or their diagrams to calculate colimits.
#### "Localic completion of generalized metric spaces II: Powerlocales"
(Steven Vickers)
Journal of Logic and Analysis 1(11) (2009) pp. 1-48.
ISSN: 1759-9008; doi:10.4115/jla.2009.1.11
Preprint version: pdf
(This paper is a greatly revised version of part of "Localic completion of quasimetric spaces".)
The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann.
As applications, a localic completion is always open, and is compact iff its generalized metric space is totally bounded.
The representation is used to discuss closed intervals of the reals, with the localic Heine-Borel Theorem as a consequence.
The work is constructive in the topos-valid sense.
Keywords: locale, constructive, topos, metric, hyperspace, powerlocale.
AMS Subject Code Classifications: Primary -- 54B20. Secondary -- 06D22, 03G30, 54E50, 03F60.
#### "Localic completion of generalized metric spaces I"
(Steven Vickers)
Theory and Applications of Categories 14 (2005), pp. 328-356.
ISSN: 1201-561X.
Free on-line at TAC: pdf
(This paper is a greatly revised version of part of "Localic completion of quasimetric spaces".)
We give a constructive localic account of the completion of generalized metric spaces (gms's) in a sense derived from that of Lawvere: a set X equipped with a metric map from X2 to the interval of upper reals (approximated from above but not from below) from 0 to infinity inclusive, and satisfying the zero self-distance law and the triangle inequality. The completion is then a locale whose points can be viewed either as Cauchy filters of formal balls or (following Lawvere's approach using enriched categories) as flat left modules.
Non-expansive functions between the gms's lift to continuous maps between the completions.
Various examples and constructions are given, including finite products.
The points of this completion are classically equivalent to those of the "Yoneda completion" of Bonsangue et al., modified to use Cauchy nets instead of sequences.
The exposition is constructive in the topos-valid sense and exploits the use of geometric reasoning to approach locales and toposes as "topology-free spaces".
#### "Entailment systems for stably locally compact locales"
(S.J. Vickers)
Theoretical Computer Science 316 (2004) (special issue on Domain Theory) pp. 259-296.
ISSN: 0304-3975; doi:10.1016/j.tcs.2004.01.033
Preprint version: pdf, ps
The category $\mathbf{SCFr}_U$ of stably continuous frames and preframe homomorphisms (preserving finite meets and directed joins) is dual to the Karoubi envelope of a category $\mathbf{Ent}$ whose objects are sets and whose morphisms $X \rightarrow Y$ are upper closed relations between the finite powersets $\mathcal{F}X$ and $\mathcal{F}Y$. Composition of these morphisms is the "cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent calculus. Thus stably locally compact locales are represented by "entailment systems" $(X,\vdash)$ in which $\vdash$, a generalization of entailment relations, is idempotent for cut composition.
Some constructions on stably locally compact locales are represented in terms of entailment systems: products, duality and powerlocales.
Relational converse provides $\mathbf{Ent}$ with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If $A$ and $B$ are stably continuous frames, then the internal preframe hom $A \pitchfork B$ is isomorphic to $\hat{A} \otimes B$ where $\hat{A}$ is the Hofmann-Lawson dual.
For a stably locally compact locale $X$, the lower powerlocale of $X$ is shown to be the dual of the upper powerlocale of the dual of $X$.
#### "A universal characterization of the double powerlocale"
(S.J. Vickers + C.F. Townsend)
Theoretical Computer Science 316 (2004) (special issue on Domain Theory) pp. 297-321.
ISSN: 0304-3975; doi:10.1016/j.tcs.2004.01.034
Preprint version: pdf
The double powerlocale $\mathbb{P}(X)$ (found by composing, in either order, the upper and lower powerlocale constructions $P_U$ and $P_L$) is shown to be isomorphic in $[\mathbf{Loc}^{op}, \mathbf{Set}]$ to the double exponential $\mathbb{S}^{\mathbb{S}^X}$ where $\mathbb{S}$ is the Sierpinski locale. Further, $P_U(X)$ and $P_L(X)$ are shown to be the subobjects of $\mathbb{P}(X)$ comprising, respectively, the meet semilattice and join semilattice homomorphisms. A key lemma shows that, for any locales $X$ and $Y$, natural transformations from $\mathbb{S}^X$ (the presheaf $\mathbf{Loc}(- \times X, \mathbb{S})$) to $\mathbb{S}^Y$ (i.e. $\mathbf{Loc}(- \times Y, \mathbb{S})$) are equivalent to dcpo morphisms from the frame $\Omega(X)$ of $X$ to that of $Y$. It is also shown that $\mathbb{S}^X$ has a localic reflection in $[\mathbf{Loc}^{op}, \mathbf{Set}]$ whose frame is $\mathbf{dcpo}(\Omega(X), \Omega(Y))$.
The reasoning is constructive in the sense of topos validity.
#### "Compactness in locales and in formal topology"
(S.J. Vickers)
Annals of Pure and Applied Logic 137 (2006), pp. 413-438. (Special issue for the Proceedings of the 2nd Workshop on Formal Topology, Venice 2002.)
ISSN: 0168-0072; doi:10.1016/j.apal.2005.05.028
Preprint version: pdf
If a locale is presented by a "flat site", it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale (and also for its openness). Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies.
A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The discussion is then related to the double powerlocale.
#### "The double powerlocale and exponentiation: a case study in geometric logic"
(S.J. Vickers)
Theory and Applications of Categories 12 (2004) pp. 372-422
Free on-line at TAC: pdf
If $X$ is a locale, then its double powerlocale $\mathbb{P}(X)$ is defined to be $P_U(P_L(X))$ where $P_U$ and $P_L$ are the upper and lower powerlocale constructions. We prove various results relating it to exponentiation of locales, including the following. First, if $X$ is a locale for which the exponential $\mathbb{S}^X$ exists (where $\mathbb{S}$ is the Sierpinski locale), then $\mathbb{P}(X)$ is an exponential $\mathbb{S}^{\mathbb{S}^X}$. Second, if in addition $W$ is a locale for which $\mathbb{P}(W)$ is homeomorphic to $\mathbb{S}^X$, then $X$ is an exponential $\mathbb{S}^W$.
The work uses geometric reasoning, i.e. reasoning stable under pullback along geometric morphisms, and this enables the locales to be discussed using their points as though they were spaces. It relies on a number of geometricity results including those for locale presentations and for powerlocales.
#### "Localic sup-lattices and tropological systems"
(P. Resende + S.J. Vickers)
Theoretical Computer Science 305 (2003) (special issue on Topology in Computer Science) pp. 311-346.
ISSN: 0304-3975; doi:10.1016/S0304-3975(02)00702-8
Preprint version: ps, pdf
The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic sup-lattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from sup-lattices to localic sup-latices and, for each quantale Q, from left Q-modules to localic right Q-modules. A proof technique for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given.
#### "Presheaves as Configured Specifications"
(S.J.Vickers + G.A.Hill)
Formal Aspects of Computing 13 (2001) pp. 32-49.
ISSN: 0934-5043 (printed), 1433-299X (electronic); doi:10.1007/PL00003938
Preprint version: ps
The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is known that these can be expressed by presheaves and the present paper applies this idea to configuration.
We interpret the category theory informally as follows. Suppose C is a category whose objects are interpreted as specifications, and for which each morphism u: X -> Y is interpreted as contravariant "instance reduction", reducing instances of specification Y to instances of X. Then a presheaf P over C represents a collection of instances that is closed under reduction. We develop an algebraic account of presheaves in which we present configurations by generators (for components) and relations (for shared reducts), and we outline a proposed configuration language based on the techniques. Oriat uses diagrams to express colimits of specifications, and we show that Oriat's category Diag(C) of finite diagrams is equivalent to the category of finitely presented presheaves over C.
#### "Strongly Algebraic = SFP (Topically)"
Mathematical Structures in Computer Science 11 (2001) pp. 717-742.
ISSN 0960-1295; doi:10.1017/S0960129501003437
Preprint version: ps, pdf
Certain "Finite Structure Conditions" on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos. The conditions assert that every homomorphism from a finite structure of the theory to a model factors via a finite model, and they hold in cases where the finitely presentable models are all finite.
The conditions are shown to hold for the theory of strongly algebraic (or SFP) information systems and some variants, as well as for some other theories already known to be classified by presheaf toposes.
The work adheres to geometric constructivism throughout and in consequence provides "topical" categories of domains (internal in the category of toposes and geometric morphisms) with an analogue of Plotkin's double characterization of strongly algebraic domains, by sets of minimal upper bounds and by sequences of finite posets.
#### "Localic Completion of Quasimetric Spaces"
Research Report DoC 97/2, Department of Computing, Imperial College (1998).
(This has now been superseded by "Localic completion of generalized metric spaces I" and "Localic completion of generalized metric spaces II: Powerlocales".)
Preprint version: ps.gz
We give a constructive localic account of the completion of quasimetric spaces. In the context of Lawvere's approach, using enriched categories, the points of the completion are flat left modules over the quasimetric space. The completion is a triquotient surjective image of a space of Cauchy sequences and can also be embedded in a continuous dcpo, the "ball domain". Various examples and constructions are given, including the upper, lower and Vietoris powerlocales, which are completions of finite powerspaces. The exposition uses the language of locales as "topology-free spaces".
#### "Topical Categories of Domains"
Mathematical Structures in Computer Science 9 (1999) pp.569-616.
ISSN 0960-1295; doi:10.1017/S0960129599002741
Preprint version: ps, pdf
It is shown how many techniques of categorical domain theory can be expressed in the general context of topical categories (where "topical" means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as "generalized topological spaces" to be treated in a transparently spatial way, and also shows the constructivity of the arguments. The theory of strongly algebraic (SFP) domains is given as a case study in which the topical category is Cartesian closed.
Properties of local toposes and of lifting of toposes (sconing) are summarized, and it is shown that the category of toposes has a fixpoint object in the sense of Crole and Pitts. This is used to show that for a local topos, all endomaps have initial algebras, and this provides a general context in which to describe fixpoint constructions including the solution of domain equations involving constructors of mixed variance. Covariance with respect to embedding-projection pairs or adjunctions arises in a natural way.
The paper also provides a summary of constructive results concerning Kuratowski finite sets, including a novel strong induction principle; and shows that the topical categories of sets, finite sets and decidable sets are not Cartesian closed (unlike the cases of finite decidable sets and strongly algebraic domains).
#### "Topology via Constructive Logic"
Moss, Ginzburg and de Rijke (eds) "Logic, Language and Computation Vol II" (Proceedings of conference on Information-Theoretic Approaches to Logic, Language, and Computation, 1996). CSLI Publications, Stanford (1999) pp. 336-345.
ISBN 1575861801; 157586181X
Preprint version: ps, pdf
By working constructively in the sense of geometric logic, topology can be hidden. This applies also to toposes as generalized topological spaces.
#### "Toposes pour les vraiment nuls"
In : Edalat, Jourdan and McCusker (eds) "Theory and Formal Methods 1996", Third Imperial College Department of Computing Workshop on Theory and Formal Methods, April 1996, pp. 1-12. Imperial College Press, London, 1996.
ISBN 1-86094-031-5
Postprint version: ps, pdf
Restriction to geometric logic can enable one to define topological structures and continuous maps without explicit reference to topologies. This idea is illustrated with some examples and used to explain toposes as generalized topological spaces.
#### "Constructive points of powerlocales"
Mathematical Proceedings of the Cambridge Philosophical Society 122 (1997), 207-222.
ISSN 0305-0041; doi:10.1017/S0305004196001636
Preprint version: ps, pdf
Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. We then also describe the points of the Vietoris powerlocale.
In each of two special cases, an exponential $^D ($ being the Sierpinski locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular.
#### "Toposes pour les nuls"
In: "Semantics Society Newsletter, Issue 4", 1995.Also available as Research Report DoC 96/4, Department of Computing, Imperial College (1996).
Preprint version: ps, pdf
An introduction to Grothendieck's idea of toposes as generalized topological spaces.
Sheaves are described as continuous set-valued functions, geometric morphisms are described as continuous maps, and continuity itself is explained as "genericity + geometricity".
"Reasoned Programming"
(K. Broda + S. Eisenbach + H. Khoshnevisan + S. Vickers)
Prentice Hall International Series in Computer Science (1994).
ISBN 0-13-098831-6
Aimed at first or second year undergraduate students, Reasoned Programming shows how to apply mathematical reasoning to the development of the computer programs that users need, using logical specifications to express these and then writing program code to achieve the objectives set out in the specifications.
The book starts with functional programming (including tutorial material written for Miranda) for its simplicity, but also shows how it can be used as a reasoning tool for imperative programming.
#### "Locales are not pointless"
In: Hankin, Mackie and Nagarajan (eds) "Theory and Formal Methods of Computing 1994", 199-216, Imperial College Press.
ISBN 1-86094-003-X
Preprint version: ps, ps.gz, pdf
The Kripke-Joyal semantics is used to interpret the fragment of intuitionistic logic containing conjunction, implication, equality and universal quantification in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems.
Openness and properness of maps between locales are characterized by descriptions using the logic, and it is proved that a locale is open iff its lower powerlocale has a greatest point. The entire account is constructive and holds for locales over any topos.
#### "Geometric logic as a specification language"
In: Hankin, Mackie and Nagarajan (eds) "Theory and Formal Methods of Computing 1994", 321-340, Imperial College Press.
ISBN 1-86094-003-X
Preprint version: ps.gz, pdf
The "observational content" of geometric logic is discussed and it is proposed that geometric logic is an appropriate basis for a Z-like specification language in which schemas are used as geometric theory presentations.
A descriptional mechanism of "schema entailment", generalizing type constructions and logical entailment, is defined and investigated in some examples, and is also used in defining schema morphisms which are discussed briefly in connection with schema connectives, and with specifying and implementing operations.
#### "Towards a GeoZ toolkit"
(M. Dawson + S.J. Vickers)
In: Hankin, Mackie and Nagarajan (eds) "Theory and Formal Methods of Computing 1994", Imperial College Press.
ISBN 1-86094-003-X
Preprint version: ps.gz, pdf
The use of Geometric Logic as the foundation of a specification language called GeoZ is proposed elsewhere. In this note we explore GeoZ from the perspective of practitioners, already familiar with the existing Z notation, by explaining the issues that arise and the role of schema entailment in the GeoZ reformulation of Z's mathematical toolkit.
#### "Geometric Logic in Computer Science"
pp. 37-54 in G.L. Burn, S.J. Gay and M.D. Ryan (eds) "Theory and Formal Methods 1993", Proceedings of the first Imperial College Department of Computing workshop on Theory and Formal Methods, Springer Workshops in Computer Science, 1993.
ISBN 3-540-19842-3; 0-387-19842-3
Preprint version: ps.gz, ps, pdf
We present an introduction to geometric logic and the mathematical structures associated with it, such as categorical logic and toposes. We also describe some of its applications in computer science including its potential as a logic for specification languages.
#### "Quantales, Observational Logic and Process Semantics"
(S. Abramsky + S.J. Vickers)
pp.161-227 in Mathematical Structures in Computer Science vol. 3 (1993).
doi:10.1017/S0960129500000189
Preprint version: ps.gz, pdf
Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications.
NB - An earlier version of this paper was issued as a Departmental Report. The new version is substantially revised in its discussion and its mathematical techniques, though the overall approach is unchanged.
#### "Information Systems for Continuous Posets"
pp. 201-229 in "Theoretical Computer Science B" vol. 114, (1993)
ISSN 0304-3975; doi:10.1016/0304-3975(93)90072-2
Preprint version: ps.gz, pdf
NB - This paper was previously entitled "Continuous Information Systems". It has been renamed to avoid confusion with Raymond Hoofman's paper.
The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's "R-structures". Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe Hoffmann-Lawson duality (which could not be done easily with R-structures) and Vietoris power locales. We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets, Vietoris algebras are the same as localic semilattices.
#### "Geometric Theories and Databases"
pp. 288-314 in M.P. Fourman, P.T. Johnstone and A.M. Pitts (eds) "Applications of Categories in Computer Science" (Proceedings of the LMS Symposium, Durham 1991), London Mathematical Society Lecture Note Series 177, Cambridge University Press, 1992.
ISBN 0-521-42726-6
Preprint version: pdf
Domain theoretic understanding of databases as elements of powerdomains is modified to allow multisets of records instead of sets. This is related to geometric theories and classifying toposes, and it is shown that algebraic base domains lead to algebraic categories of models in two cases analogous to the lower (Hoare) powerdomain and Gunter's mixed powerdomain.
#### "Preframe Presentations Present"
(P. Johnstone + S.J. Vickers)
pp. 193-212 in A. Carboni, M.C. Pedicchio and G. Rosolini (eds.) Category Theory - Proceedings, Como 1990 (Springer Lecture Notes in Mathematics 1488,1991).
ISBN 3-540-54706-1; 0-387-54706-1; doi:10.1007/BFb0084221
Preprint version: pdf
Preframes (directed complete posets with finite meets that distribute over the directed joins) are the algebras for an infinitary essentially algebraic theory, and can be presented by generators and relations. This result is combined with a general argument concerning categories of commutative monoids to give a very short proof of the localic Tychonoff Theorem. It is also shown how frames can be presented as preframes, a result analogous to Johnstone's construction of frames from sites, and an application is given.
#### "Formal Implementation"
Chapter = 25 in J.A. McDermid (ed.) The Software Engineer's Reference Book, (Butterworth Scientific, London, 1991).
ISBN 0-750-61040-9
An account of the formal logic of pre- and post-conditions.
#### "Topology via Logic"
Cambridge Tracts in Theoretical Computer Science 5 (Cambridge University Press 1988).
ISBN 0-521-36062-5; 0-521-57651-2
An advanced textbook on topology for computer scientists with three unusual features. First, the introduction is from a localic viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the methods of locale theory are freely exploited. Third, there is a substantial discussion of some computer science applications. Although books on topology aimed at mathematics exist, not book has been written specifically for computer scientists, and as computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap.
#### "A fixpoint construction of the p-adic domain"
pp. 270-289 in D.H. Pitt, A. Poigne and D.E. Rydeheard, Category Theory and Computer Science (Proceedings of Edinburgh Workshop, 1986) (Springer Lecture Notes in Computer Science 283, 1987).
ISBN 3-540-18508-9; 0-387-18508-9
The Kahn domain on p symbols can be given an arithmetic structure so that its maximal elements are isomorphic to the p-adic integers. This is described as a fixpoint of a functor in a category of sheaves of rings.
#### "An algorithmic approach to the p-adic integers"
pp. 599-616 in M. Main, A. Melton, M. Mislove and D. Schmidt Mathematical Foundations of Programming Language Semantics (Proceedings of Tulane Workshop, 1987) (Springer Lecture Notes in Computer Science 298, 1988).
ISBN 3-540-19020-1; 0-387-19020-1
The ring of p-adic integers can be embedded as the maximal elements in a Scott domain with algebraic structure. We show how definitions and proofs in the mathematical theory of p-adics can be replaced by algorithms on the partial elements and formal programming methods working on the algorithms. Certain types of argumetns translate naturally into non-deterministic algorithms using the Smyth power domain.
#### "Theories as Categories"
(M.P. Fourman + S.J. Vickers)
pp. 434-448 in D. Pitt, S. Abramsky, A. Poigné and D. Rydeheard (eds.) Category Theory and Computer Programming (Springer Lecture Notes in Computer Science 240, 1986).
ISBN 3-540-17162-2; 0-387-17162-2
This paper is not, and is not intended to be, original. Its purpose is to present a couple of examples from the folklore of topos theory, the theory of classifying topoi in particular. This theory and its applications spread initially without the benefit of widespread publication. Many ideas were spread among a relatively small group, largely by word of mouth. The result of this is that the literature does not provide an accessible introduction to the subject. Computer scientists studying the logic of computing have recently become interested in this area. They form our intended audience. In the space (and time) available we can only hope to provide a small selection of the many ideas missing from, or buried in, the literature. We attempt to give a perspective of the structure of the subject. Our selection is, of necessity, idiosyncratic, and our treatment brief. We hope that missing technical details may be reconstructed from the literature. This may require some diligence. [Yes, it did - SJV]
To apportion credit for the ideas presented here is difficult so long after the event. Lawvere and Joyal have a special position in htis subject. Their intuitions have shaped it. Many others, who participated in the Peripatetic Seminars in Europe, the New York Topos Theory Seminar (which also wandered) and the Category Theory meetings at Oberwohlfach, contributed also. Their contributions are, in general, better reflected in their published works. Finally, to apportion blame, this paper derives from notes taken by SJV of a talk by MPF. Any misrepresentations are the responsibility of the latter. | 2014-11-22 23:37:54 | {"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": 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": 0.7136195302009583, "perplexity": 1700.3283687212793}, "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-2014-49/segments/1416400378815.18/warc/CC-MAIN-20141119123258-00071-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.tutorialspoint.com/how-to-calculate-average-cells-from-different-sheets-in-excel | How to calculate average cells from different sheets in Excel?
How can we do the calculations needed to determine the average of all the numbers on the various worksheets if we want to accomplish that? This post will show you an easy approach to compute the average for all numbers across several worksheets. Currently, we may know how to calculate the average for a selected range on a single worksheet. However, this article will show you how to do it for all numbers.
In terms of the reality that the data range on other worksheets could be the same or different, we have prepared two instances to use as an illustration.
Let’s understand step by step with an example.
Step 1
To begin, we will need to prepare two worksheets by populating them with numbers from the same range or different range. As shown in the below screenshots.
If, for instance, you want to calculate the average of Range A1:A10 across Sheets 1 through 3, you may easily solve the problem by doing so as follows.
Step 2
Now, choose a cell that is empty, such as cell D4, and after that enter formula into it. Please refer to the below screenshot for the same.
Formula
=AVERAGE(Sheet1:Sheet3!B2:B10)
Step 3
Then press the Enter key. Now, you will receive the overall average in the cell D3. Please check out below screenshot for the same.
Important Note
• Sheet1 through Sheet3 are the many adjacent sheets from which you will compute the average using the formula "=AVERAGE(Sheet1:Sheet3!B1:B10)".
• The range B1 through B10 refers to the cells or range from which you will calculate the average in each of the multiple sheets. You are allowed to modify them to suit your requirements.
• If you want to calculate the average of multiple cells or ranges from multiple worksheets in Excel, for example, if you want to calculate the average of Range B1:B5 in Sheet1, Range B3:B6 in Sheet2, and Range B7:B9 in Sheet3, you should apply this formula: "=AVERAGE(B1:B5,Sheet2!B3:B6,Sheet3!7:B9)".
Conclusion
In this tutorial, we used a simple example to demonstrate how you can calculate average cells from different sheets by using formulas in Excel. | 2023-03-30 12:13:56 | {"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": 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": 0.5163430571556091, "perplexity": 588.2459268906199}, "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-2023-14/segments/1679296949181.44/warc/CC-MAIN-20230330101355-20230330131355-00607.warc.gz"} |
http://openstudy.com/updates/55a826d6e4b071e6530c0217 | ## anonymous one year ago Whats the sum of the first 50 terms? $$\huge { 12 - -\frac{7}{10}n }$$ I am getting \ ( \huge -\frac{445}{2} \) is that right because it is stating it is wrong. They stat it is $$\huge -\frac{585}{2}$$ I am using $$\huge \frac{n}{2}[2*a_1 + (n-1)d) ] to solve$$
1. anonymous
Whats the sum of the first 50 terms? $$\huge { 12 - -\frac{7}{10}n }$$ I am getting \ ( \huge -\frac{445}{2} \) is that right because it is stating it is wrong. They stat it is $$\huge -\frac{585}{2}$$ I am using $$\huge \frac{n}{2}[2*a_1 + (n-1)d) ]$$ to solve
2. anonymous
$$\huge \frac{50}{2}[2*\frac{127}{10} + (50-1)*-\frac{7}{10} ]$$
3. anonymous
This $$\huge \huge { 12 - -\frac{7}{10}n }$$ SHOULD BE $$\huge\huge { 12 - \frac{7}{10}n }$$
4. anonymous
Wait, so is the sequence $$a_n=12-\frac{7}{10}n$$, $$n>0$$?
5. anonymous
Yes
6. anonymous
Ok, if so, I think what you did wrong is that your first term is wrong.$a_1=12-\frac{7}{10}=\frac{113}{10}$
7. anonymous
Ah and I got 127/10. Now how did I get 127...... That was the problem.... I hate mis calculating problems lol Thank you. It came up to be correct.
8. anonymous
lol np. Everyone makes these kinds of mistakes from time to time. :) | 2016-10-26 17:20:25 | {"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": 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": 0.8065179586410522, "perplexity": 1339.5975538505786}, "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-2016-44/segments/1476988720967.29/warc/CC-MAIN-20161020183840-00501-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/exponential-map.639625/ | # Exponential map
I'm having trouble understanding the exponential map for nonlinear vector fields.
If dσ/dt=X(σ)
for vector field X, then how does one interpret the solution:
σ(t)=exp[tX]σ(0) ?
If X is nonlinear, then X is not a matrix, so this expression wouldn't make sense.
If X is a matrix that maps:
X: point on manifold → vector (direction of flow) on manifold
then this expression makes sense.
## Answers and Replies
quasar987
Homework Helper
Gold Member
what's a nonlinear vector field? Plz define it.
what's a nonlinear vector field? Plz define it.
A linear vector field has the property that X(p1+p2)=X(p1) +X(p2), where p are points in your space. In 2 dim, they look like
X=(ax+by,cx+dy)
for constant a b c d.
A nonlinear field is
X=X(f(x,y),g(x,y))
quasar987
Homework Helper
Gold Member
I think I know what is confusing you. You are basically asking "what does exp(tX) means when X is not a matrix ?!?". The answer is that in the context of flows, exp(tX) is just a notation for the solution of the ODE dσ/dt=X(σ). The reason for this strange notation is tha the stolution of this equation has the "exponential property": exp([s+t]X) = exp(sX)exp(tX).
I think I know what is confusing you. You are basically asking "what does exp(tX) means when X is not a matrix ?!?". The answer is that in the context of flows, exp(tX) is just a notation for the solution of the ODE dσ/dt=X(σ). The reason for this strange notation is tha the stolution of this equation has the "exponential property": exp([s+t]X) = exp(sX)exp(tX).
I was looking at some online notes, and they explained it in terms of linear vector fields so that it's not just notationally true, but literally true (there are a few typos, but the first two pages has it):
http://mysite.science.uottawa.ca/rossmann//Lie_book_files/Section 1-1.pdf
But in textbooks the exponential map is applied to any flow, not just linear ones.
So it seems you can define an exponential map for a lot of things...things that obey the additive group for example, or just a Lie group in general if you expand the "exponential property" via Baker-Campbell Hausdorff.
quasar987 | 2021-05-13 15:19:06 | {"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": 0.842805802822113, "perplexity": 940.2022653530141}, "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-2021-21/segments/1620243989814.35/warc/CC-MAIN-20210513142421-20210513172421-00392.warc.gz"} |
https://web2.0calc.com/questions/please-help-thank-you_8 | +0
# Please help! Thank you!
0
336
2
+156
Please help! I'm confused! Show work please! Thank you.
Apr 27, 2018
### 2+0 Answers
#1
+5258
+2
$$y = {-7 \pm \sqrt{7^2-4(1)(-6)} \over 2}$$
Convenient that the first thing from the LaTeX button is the quadratic formula. (Before this, subtract 6 from both sides. It's now y2+7y-6=0)
Evaluate what is under the root.
-4*-6=24
72=49
49+24=73
We now have $$y= {-7 \pm \sqrt{73} \over 2}$$
Nothing past this can be done. Your answer is A .
Apr 27, 2018
#2
+156
+2
Thank you so much!! I appreciate your help.
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http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Peter+May | # nLab Peter May
J. Peter May is a homotopy theorist at the University of Chicago, inventor of operads as a technique for studying infinite loop spaces and spectra.
Peter May’s work makes extensive use of enriched- and model-category theory as power tools in algebraic topology/homotopy theory, notably in discussion of highly structured spectra in MMSS00‘s Model categories of diagram spectra (for exposition see Introduction to Stable homotopy theory – 1-2), or in the discussion of genuine equivariant spectra or K-theory of permutative categories, etc.. While he has co-edited a book collection on higher category theory (Baez-May 10) and eventually had high praise (May 16) for 2-category theory as a tool in algebraic topology/higher algebra, he has vocally warned against seeing abstract (∞,1)-category theory as a replacement for concrete realizations in model category-theory (P. May, MO comment Dec 2013).
## Selected writings
• Peter May, The geometry of iterated loop spaces, 1972 (pdf)
• Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (Euclid)
Infinite loop space theory revisited (pdf)
On higher algebra (brave new algebra) in stable homotopy theory, i.e. on ring spectra, module spectra etc.:
• Peter May, Equivariant and non-equivariant module spectra, Journal of Pure and Applied Algebra Volume 127, Issue 1, 1 May 1998, Pages 83–97 (pdf) | 2022-01-18 19:40:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.4072546362876892, "perplexity": 2044.2886842912958}, "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/1642320300997.67/warc/CC-MAIN-20220118182855-20220118212855-00221.warc.gz"} |
https://zbmath.org/authors/?q=ai%3Alyons.terence-j | ## Lyons, Terence John
Compute Distance To:
Author ID: lyons.terence-j Published as: Lyons, Terry; Lyons, T. J.; Lyons, Terry J.; Lyons, T.; Lyons, Terence J. more...less External Links: MGP · Wikidata · Math-Net.Ru · dblp · GND · IdRef
Documents Indexed: 108 Publications since 1980, including 2 Books 1 Contribution as Editor Biographic References: 1 Publication Co-Authors: 55 Co-Authors with 77 Joint Publications 1,408 Co-Co-Authors
all top 5
### Co-Authors
23 single-authored 8 Crisan, Dan O. 8 Qian, Zhongmin M. 6 Litterer, Christian 6 Yang, Danyu 5 Zhang, Tusheng S. 4 Cass, Thomas Richard 4 Hambly, Ben M. 3 Gyurkó, Lajos Gergely 3 Ni, Hao 3 Perez Arribas, Imanol 3 Xu, Weijun 3 Zheng, Weian 2 Boedihardjo, Horatio 2 Chang, Jiawei 2 Del Moral, Pierre 2 Friz, Peter Karl 2 Gaines, J. G. 2 Hara, Keisuke 2 Hayman, Walter Kurt 2 Nejad, Sina 2 Röckner, Michael 2 Stoica, Lucreţiu 2 Victoir, Nicolas B. 1 Albeverio, Sergio A. 1 Barnett, Chris 1 Bass, Richard F. 1 Beliaev, Dmitri B. 1 Boutaib, Youness 1 Caruana, Michael 1 Chevyrev, Ilya 1 Duplantier, Bertrand 1 Eremenko, Alexandre Émmanuilovich 1 Flint, Guy H. 1 Gaines, Jessica 1 Gamelin, Theodore W. 1 Gassiat, Paul 1 Geng, Xi 1 Kalsi, Jasdeep 1 Kershaw, Donald 1 Lawler, Gregory Francis 1 Le Gall, Jean-François 1 Ledoux, Michel 1 Lee, Wonjung 1 Lejay, Antoine 1 Levin, Daniel 1 Lévy, Thierry 1 Li, Xiang-Dong 1 Liang, Gechun 1 Lunt, John 1 MacGibbon, Brenda 1 Margarint, Vlad 1 McKean, Henry P. jun. 1 Oberhauser, Harald 1 Reuter, G. E. H. 1 Rozanov, Yuriĭ Anatol’evich 1 Salisbury, Thomas S. 1 Salvi, Cristopher 1 Sidorova, Nadia 1 Stroock, Daniel W. 1 Sullivan, Dennis Parnell 1 Taylor, John C. 1 Yam, Phillip S. C. 1 Yang, Weixin 1 Zeitouni, Ofer
all top 5
### Serials
10 The Annals of Probability 9 Journal of Functional Analysis 6 Bulletin of the London Mathematical Society 5 Probability Theory and Related Fields 3 Journal of Differential Geometry 3 Revista Matemática Iberoamericana 3 SIAM Journal on Applied Mathematics 3 Applied Mathematical Finance 2 Advances in Applied Probability 2 Advances in Mathematics 2 Journal of the London Mathematical Society. Second Series 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 Stochastic Processes and their Applications 2 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 2 Electronic Communications in Probability 2 Journal of the European Mathematical Society (JEMS) 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Revue Roumaine de Mathématiques Pures et Appliquées 1 Annales de l’Institut Fourier 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Bulletin des Sciences Mathématiques. Deuxième Série 1 Illinois Journal of Mathematics 1 Journal of the Mathematical Society of Japan 1 Mathematische Annalen 1 Proceedings of the London Mathematical Society. Third Series 1 SIAM Journal on Numerical Analysis 1 Transactions of the American Mathematical Society 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 The Annals of Applied Probability 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 1 Stochastics and Stochastics Reports 1 Mathematical Research Letters 1 Bulletin des Sciences Mathématiques 1 Monte Carlo Methods and Applications 1 Sbornik: Mathematics 1 Markov Processes and Related Fields 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Annals of Mathematics. Second Series 1 Communications in Mathematical Sciences 1 Fields Institute Communications 1 Lecture Notes in Mathematics 1 Publications in Sciences. University of Joensuu 1 SIAM Journal on Financial Mathematics 1 Oxford Mathematical Monographs 1 SIAM Journal on Mathematics of Data Science
all top 5
### Fields
82 Probability theory and stochastic processes (60-XX) 19 Potential theory (31-XX) 19 Systems theory; control (93-XX) 16 Numerical analysis (65-XX) 13 Ordinary differential equations (34-XX) 13 Global analysis, analysis on manifolds (58-XX) 8 Functions of a complex variable (30-XX) 8 Functional analysis (46-XX) 7 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 6 Partial differential equations (35-XX) 4 Real functions (26-XX) 4 Differential geometry (53-XX) 2 Measure and integration (28-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Approximations and expansions (41-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Abstract harmonic analysis (43-XX) 2 Operator theory (47-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX)
### Citations contained in zbMATH Open
96 Publications have been cited 1,892 times in 992 Documents Cited by Year
Differential equations driven by rough signals. Zbl 0923.34056
Lyons, Terry J.
1998
System control and rough paths. Zbl 1029.93001
Lyons, Terry; Qian, Zhongmin
2002
Differential equations driven by rough signals. I: An extension of an inequality of L. C. Young. Zbl 0835.34004
Lyons, Terry
1994
Differential equations driven by rough paths. Ecole d’Eté de Probabilités de Saint-Flour XXXIV – 2004. Lectures given at the 34th probability summer school, July 6–24, 2004. Zbl 1176.60002
Lyons, Terry J.; Caruana, Michael; Lévy, Thierry
2007
Cubature on Wiener space. Zbl 1055.60049
Lyons, Terry; Victoir, Nicolas
2004
Function theory, random paths and covering spaces. Zbl 0554.58022
Lyons, Terry; Sullivan, Dennis
1984
Variable step size control in the numerical solution of stochastic differential equations. Zbl 0888.60046
Gaines, J. G.; Lyons, T. J.
1997
A simple criterion for transience of a reversible Markov chain. Zbl 0509.60067
Lyons, Terry
1983
Integrability and tail estimates for Gaussian rough differential equations. Zbl 1278.60091
Cass, Thomas; Litterer, Christian; Lyons, Terry
2013
Decomposition of Dirichlet processes and its applications. Zbl 0804.60044
Lyons, T. J.; Zhang, T. S.
1994
Uniqueness for the signature of a path of bounded variation and the reduced path group. Zbl 1276.58012
Hambly, Ben; Lyons, Terry
2010
Random generation of stochastic area integrals. Zbl 0805.60052
Gaines, J. G.; Lyons, T. J.
1994
An extension theorem to rough paths. Zbl 1134.60047
Lyons, Terry; Victoir, Nicolas
2007
Nonlinear filtering and measure-valued processes. Zbl 0888.93056
Crisan, Dan; Lyons, Terry
1997
A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Zbl 0654.60059
Lyons, Terence J.; Zheng, Weian
1988
Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. Zbl 0914.60045
Lunt, John; Lyons, T. J.; Zhang, T. S.
1998
Lévy area of Wiener processes in Banach spaces. Zbl 1016.60071
Ledoux, M.; Lyons, T.; Qian, Z.
2002
On conditional diffusion processes. Zbl 0715.60097
Lyons, T. J.; Zheng, W. A.
1990
Discrete filtering using branching and interacting particle systems. Zbl 0967.93088
Crisan, D.; Del Moral, P.; Lyons, T.
1999
Winding of the plane Brownian motion. Zbl 0541.60075
Lyons, T. J.; McKean, H. P.
1984
Convergence of a branching particle method to the solution of the Zakai equation. Zbl 0915.93060
Crisan, Dan; Gaines, Jessica; Lyons, Terry
1998
Finely holomorphic functions. Zbl 0459.46038
Lyons, T. J.
1980
A particle approximation of the solution of the Kushner-Stratonovitch equation. Zbl 0951.93068
Crisan, D.; Lyons, T.
1999
Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. Zbl 0599.60011
Lyons, Terry
1987
Efficient and practical implementations of cubature on Wiener space. Zbl 1221.65009
Gyurkó, Lajos Gergely; Lyons, Terry J.
2011
Extending the Wong-Zakai theorem to reversible Markov processes. Zbl 1010.60070
Bass, R. F.; Hambly, B. M.; Lyons, T. J.
2002
Physical Brownian motion in a magnetic field as a rough path. Zbl 1390.60257
Friz, Peter; Gassiat, Paul; Lyons, Terry
2015
Stochastic area for Brownian motion on the Sierpiński gasket. Zbl 0936.60073
Hambly, B. M.; Lyons, T. J.
1998
High order recombination and an application to cubature on Wiener space. Zbl 1261.65011
Litterer, Christian; Lyons, Terence J.
2012
The signature of a rough path: uniqueness. Zbl 1347.60094
Boedihardjo, Horatio; Geng, Xi; Lyons, Terry; Yang, Danyu
2016
The limits of stochastic integrals of differential forms. Zbl 0969.60078
Lyons, Terry; Stoica, Lucreţiu
1999
Characteristic functions of measures on geometric rough paths. Zbl 1393.60008
Chevyrev, Ilya; Lyons, Terry
2016
Backward stochastic dynamics on a filtered probability space. Zbl 1238.60064
Liang, Gechun; Lyons, Terry; Qian, Zhongmin
2011
Flow of diffeomorphisms induced by a geometric multiplicative functional. Zbl 0918.60009
Lyons, Terry; Qian, Zhongmin
1998
Rough paths, signatures and the modelling of functions on streams. Zbl 1373.93158
Lyons, Terry
2014
Minimal entropy approximations and optimal algorithms. Zbl 1018.65014
Crisan, Dan; Lyons, Terry
2002
Evolving communities with individual preferences. Zbl 1321.60204
Cass, Thomas; Lyons, Terry
2015
On the non-existence of path integrals. Zbl 0745.60051
Lyons, Terry
1991
Smoothness of Itô maps and diffusion processes on path spaces. I. Zbl 1127.60033
Li, Xiang-Dong; Lyons, Terry J.
2006
Lévy’s area under conditioning. Zbl 1099.60054
Friz, P.; Lyons, T.; Stroock, D.
2006
Bases for positive continuous functions. Zbl 0675.30040
Hayman, W. K.; Lyons, T. J.
1990
On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization. Zbl 1199.60292
Lejay, Antoine; Lyons, Terry
2005
The critical dimension at which quasi-every Brownian path is self- avoiding. Zbl 0609.60087
Lyons, Terry J.
1986
Discretely sampled signals and the rough Hoff process. Zbl 1348.60058
Flint, Guy; Hambly, Ben; Lyons, Terry
2016
A note on tightness of capacities associated with Dirichlet forms. Zbl 0781.60063
Lyons, Terry; Röckner, Michael
1992
Flow equations on spaces of rough paths. Zbl 0890.58090
Lyons, Terry; Qian, Zhongmin
1997
A class of vector fields on path spaces. Zbl 0877.58059
Lyons, T. J.; Qian, Z. M.
1997
Stopping non-commutative processes. Zbl 0585.60058
Barnett, Chris; Lyons, Terry
1986
Convergence of non-symmetric Dirichlet processes. Zbl 0888.31005
Lyons, Terry; Zhang, Tusheng
1996
System control and rough paths. Zbl 1044.93009
Lyons, T. J.
2002
Inverting the signature of a path. Zbl 1429.70012
Lyons, Terry J.; Xu, Weijun
2018
Stochastic Jacobi fields and vector fields induced by varying area on path spaces. Zbl 0903.60008
Lyons, Terry; Qian, Zhongmin
1997
Hyperbolic development and inversion of signature. Zbl 1362.53009
Lyons, Terry J.; Xu, Weijun
2017
A signed measure on rough paths associated to a PDE of high order: results and conjectures. Zbl 1193.60071
Levin, Daniel; Lyons, Terry
2009
Instability of the conservative property under quasi-isometries. Zbl 0747.53034
Lyons, Terry
1991
Dimension-free Euler estimates of rough differential equations. Zbl 1399.60115
Boutaib, Youness; Gyurkó, Lajos Gergely; Lyons, Terry; Yang, Danyu
2014
Interacting particle systems approximations of the Kushner-Stratonovitch equation. Zbl 0947.60040
Crişan, D.; Del Moral, P.; Lyons, T. J.
1999
A synthetic proof of Makarov’s law of the iterated logarithm. Zbl 0708.30033
Lyons, Terry
1990
Diffusion processes with non-smooth diffusion coefficients and their density functions. Zbl 0715.60096
Lyons, T. J.; Zheng, W. A.
1990
Random thoughts on reversible potential theory. Zbl 0757.31007
Lyons, Terry
1992
The geometry of the Brownian curve. Zbl 0778.60058
Duplantier, B.; Lawler, G. F.; Le Gall, J.-F.; Lyons, T. J.
1993
Uncertain volatility and the risk-free synthesis of derivatives. Zbl 1466.91347
Lyons, T. J.
1995
Rough paths based numerical algorithms in computational finance. Zbl 1198.91229
Gyurkó, Lajos Gergely; Lyons, Terry
2010
The interpretation and solution of ordinary differential equations driven by rough signals. Zbl 0827.34049
Lyons, Terry J.
1995
Note on convergence of Dirichlet processes. Zbl 0793.31006
Lyons, T. J.; Zhang, T. S.
1993
On the radius of convergence of the logarithmic signature. Zbl 1103.60060
2006
The best harmonic approximant to a continuous function. Zbl 0541.41024
Hayman, Walter K.; Kershaw, Donald; Lyons, Terry J.
1984
A theorem in fine potential theory and applications to finely holomorphic functions. Zbl 0459.46039
Lyons, T. J.
1980
Cones of lower semicontinuous functions and a characterisation of finely hyperharmonic functions. Zbl 0492.31005
Lyons, Terry J.
1982
Conditional exponential moments for iterated Wiener integrals. Zbl 0961.60053
Lyons, Terry; Zeitouni, Ofer
1999
Systems controlled by rough paths. Zbl 1125.93026
Lyons, Terry
2005
Calculus for multiplicative functionals, Itô’s formula and differential equations. Zbl 0862.60043
Lyons, T. J.; Qian, Z. M.
1996
On Gauss-Green theorem and boundaries of a class of Hölder domains. Zbl 1103.28007
Lyons, Terry J.; Yam, Phillip S. C.
2006
Projection theorems for hitting probabilities and a theorem of Littlewood. Zbl 0566.58036
Lyons, T. J.; MacGibbon, K. B.; Taylor, J. C.
1984
Expected signature of Brownian motion up to the first exit time from a bounded domain. Zbl 1350.60086
Lyons, Terry; Ni, Hao
2015
Kusuoka-Stroock gradient bounds for the solution of the filtering equation. Zbl 1333.60147
Crisan, Dan; Litterer, Christian; Lyons, Terry
2015
Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length. (Supermultiplicativité et une borne inférieure pour la décroissance de la signature d’un chemin de longueur finie.) Zbl 1391.60163
Chang, Jiawei; Lyons, Terry; Ni, Hao
2018
The adaptive patched cubature filter and its implementation. Zbl 1343.60042
Lee, Wonjung; Lyons, Terry
2016
Rough paths on manifolds. Zbl 1280.58023
Cass, Thomas; Litterer, Christian; Lyons, Terry
2012
Introducing cubature to filtering. Zbl 1237.93168
Litterer, C.; Lyons, T.
2011
Cubature on Wiener space continued. Zbl 1187.35299
Litterer, Christian; Lyons, Terry
2007
Jensen measures for R(K). Zbl 0509.31002
Gamelin, T. W.; Lyons, T. J.
1983
Finely harmonic functions need not be quasi-analytic. Zbl 0541.31002
Lyons, Terry
1984
Uniform factorial decay estimates for controlled differential equations. Zbl 1333.60117
Boedihardjo, Horatio; Lyons, Terry; Yang, Danyu
2015
Numerical method for model-free pricing of exotic derivatives in discrete time using rough path signatures. Zbl 1437.91431
Lyons, Terry; Nejad, Sina; Perez Arribas, Imanol
2019
An optimal polynomial approximation of Brownian motion. Zbl 1434.60226
Foster, James; Lyons, Terry; Oberhauser, Harald
2020
Optimal execution with rough path signatures. Zbl 1443.91263
Kalsi, Jasdeep; Lyons, Terry; Arribas, Imanol Perez
2020
Recovering the pathwise Itô solution from averaged Stratonovich solutions. Zbl 1338.60148
Lyons, Terry; Yang, Danyu
2016
A uniform estimate for rough paths. Zbl 1296.60155
Lyons, Terry J.; Xu, Weijun
2013
Kiyoshi Itô (1915–2008). Zbl 1201.60005
Lyons, Terence J.
2010
The partial sum process of orthogonal expansions as geometric rough process with Fourier series as an example – an improvement of Menshov-Rademacher theorem. Zbl 1284.42082
Lyons, Terry J.; Yang, Danyu
2013
Smooth rough paths and applications to Fourier analysis. Zbl 1146.26004
Hara, Keisuke; Lyons, Terry
2007
On the limit of stochastic integrals of differential forms. Zbl 0899.60046
Lyons, Terry; Stoica, Lucreţiu
1996
On boundary conditions for stochastic evolution equations with an extremally chaotic source. Zbl 0869.60054
Albeverio, S.; Lyons, T.; Rozanov, Yu.
1995
Continuity in $$\kappa$$ in $$\mathrm{SLE}_\kappa$$ theory using a constructive method and rough path theory. Zbl 07374667
Beliaev, Dmitry; Lyons, Terry J.; Margarint, Vlad
2021
Corrigendum to: “Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length”. Zbl 1400.60089
Chang, Jiawei; Lyons, Terry; Ni, Hao
2018
Continuity in $$\kappa$$ in $$\mathrm{SLE}_\kappa$$ theory using a constructive method and rough path theory. Zbl 07374667
Beliaev, Dmitry; Lyons, Terry J.; Margarint, Vlad
2021
An optimal polynomial approximation of Brownian motion. Zbl 1434.60226
Foster, James; Lyons, Terry; Oberhauser, Harald
2020
Optimal execution with rough path signatures. Zbl 1443.91263
Kalsi, Jasdeep; Lyons, Terry; Arribas, Imanol Perez
2020
Numerical method for model-free pricing of exotic derivatives in discrete time using rough path signatures. Zbl 1437.91431
Lyons, Terry; Nejad, Sina; Perez Arribas, Imanol
2019
Inverting the signature of a path. Zbl 1429.70012
Lyons, Terry J.; Xu, Weijun
2018
Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length. (Supermultiplicativité et une borne inférieure pour la décroissance de la signature d’un chemin de longueur finie.) Zbl 1391.60163
Chang, Jiawei; Lyons, Terry; Ni, Hao
2018
Corrigendum to: “Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length”. Zbl 1400.60089
Chang, Jiawei; Lyons, Terry; Ni, Hao
2018
Hyperbolic development and inversion of signature. Zbl 1362.53009
Lyons, Terry J.; Xu, Weijun
2017
The signature of a rough path: uniqueness. Zbl 1347.60094
Boedihardjo, Horatio; Geng, Xi; Lyons, Terry; Yang, Danyu
2016
Characteristic functions of measures on geometric rough paths. Zbl 1393.60008
Chevyrev, Ilya; Lyons, Terry
2016
Discretely sampled signals and the rough Hoff process. Zbl 1348.60058
Flint, Guy; Hambly, Ben; Lyons, Terry
2016
The adaptive patched cubature filter and its implementation. Zbl 1343.60042
Lee, Wonjung; Lyons, Terry
2016
Recovering the pathwise Itô solution from averaged Stratonovich solutions. Zbl 1338.60148
Lyons, Terry; Yang, Danyu
2016
Physical Brownian motion in a magnetic field as a rough path. Zbl 1390.60257
Friz, Peter; Gassiat, Paul; Lyons, Terry
2015
Evolving communities with individual preferences. Zbl 1321.60204
Cass, Thomas; Lyons, Terry
2015
Expected signature of Brownian motion up to the first exit time from a bounded domain. Zbl 1350.60086
Lyons, Terry; Ni, Hao
2015
Kusuoka-Stroock gradient bounds for the solution of the filtering equation. Zbl 1333.60147
Crisan, Dan; Litterer, Christian; Lyons, Terry
2015
Uniform factorial decay estimates for controlled differential equations. Zbl 1333.60117
Boedihardjo, Horatio; Lyons, Terry; Yang, Danyu
2015
Rough paths, signatures and the modelling of functions on streams. Zbl 1373.93158
Lyons, Terry
2014
Dimension-free Euler estimates of rough differential equations. Zbl 1399.60115
Boutaib, Youness; Gyurkó, Lajos Gergely; Lyons, Terry; Yang, Danyu
2014
Integrability and tail estimates for Gaussian rough differential equations. Zbl 1278.60091
Cass, Thomas; Litterer, Christian; Lyons, Terry
2013
A uniform estimate for rough paths. Zbl 1296.60155
Lyons, Terry J.; Xu, Weijun
2013
The partial sum process of orthogonal expansions as geometric rough process with Fourier series as an example – an improvement of Menshov-Rademacher theorem. Zbl 1284.42082
Lyons, Terry J.; Yang, Danyu
2013
High order recombination and an application to cubature on Wiener space. Zbl 1261.65011
Litterer, Christian; Lyons, Terence J.
2012
Rough paths on manifolds. Zbl 1280.58023
Cass, Thomas; Litterer, Christian; Lyons, Terry
2012
Efficient and practical implementations of cubature on Wiener space. Zbl 1221.65009
Gyurkó, Lajos Gergely; Lyons, Terry J.
2011
Backward stochastic dynamics on a filtered probability space. Zbl 1238.60064
Liang, Gechun; Lyons, Terry; Qian, Zhongmin
2011
Introducing cubature to filtering. Zbl 1237.93168
Litterer, C.; Lyons, T.
2011
Uniqueness for the signature of a path of bounded variation and the reduced path group. Zbl 1276.58012
Hambly, Ben; Lyons, Terry
2010
Rough paths based numerical algorithms in computational finance. Zbl 1198.91229
Gyurkó, Lajos Gergely; Lyons, Terry
2010
Kiyoshi Itô (1915–2008). Zbl 1201.60005
Lyons, Terence J.
2010
A signed measure on rough paths associated to a PDE of high order: results and conjectures. Zbl 1193.60071
Levin, Daniel; Lyons, Terry
2009
Differential equations driven by rough paths. Ecole d’Eté de Probabilités de Saint-Flour XXXIV – 2004. Lectures given at the 34th probability summer school, July 6–24, 2004. Zbl 1176.60002
Lyons, Terry J.; Caruana, Michael; Lévy, Thierry
2007
An extension theorem to rough paths. Zbl 1134.60047
Lyons, Terry; Victoir, Nicolas
2007
Cubature on Wiener space continued. Zbl 1187.35299
Litterer, Christian; Lyons, Terry
2007
Smooth rough paths and applications to Fourier analysis. Zbl 1146.26004
Hara, Keisuke; Lyons, Terry
2007
Smoothness of Itô maps and diffusion processes on path spaces. I. Zbl 1127.60033
Li, Xiang-Dong; Lyons, Terry J.
2006
Lévy’s area under conditioning. Zbl 1099.60054
Friz, P.; Lyons, T.; Stroock, D.
2006
On the radius of convergence of the logarithmic signature. Zbl 1103.60060
2006
On Gauss-Green theorem and boundaries of a class of Hölder domains. Zbl 1103.28007
Lyons, Terry J.; Yam, Phillip S. C.
2006
On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization. Zbl 1199.60292
Lejay, Antoine; Lyons, Terry
2005
Systems controlled by rough paths. Zbl 1125.93026
Lyons, Terry
2005
Cubature on Wiener space. Zbl 1055.60049
Lyons, Terry; Victoir, Nicolas
2004
System control and rough paths. Zbl 1029.93001
Lyons, Terry; Qian, Zhongmin
2002
Lévy area of Wiener processes in Banach spaces. Zbl 1016.60071
Ledoux, M.; Lyons, T.; Qian, Z.
2002
Extending the Wong-Zakai theorem to reversible Markov processes. Zbl 1010.60070
Bass, R. F.; Hambly, B. M.; Lyons, T. J.
2002
Minimal entropy approximations and optimal algorithms. Zbl 1018.65014
Crisan, Dan; Lyons, Terry
2002
System control and rough paths. Zbl 1044.93009
Lyons, T. J.
2002
Discrete filtering using branching and interacting particle systems. Zbl 0967.93088
Crisan, D.; Del Moral, P.; Lyons, T.
1999
A particle approximation of the solution of the Kushner-Stratonovitch equation. Zbl 0951.93068
Crisan, D.; Lyons, T.
1999
The limits of stochastic integrals of differential forms. Zbl 0969.60078
Lyons, Terry; Stoica, Lucreţiu
1999
Interacting particle systems approximations of the Kushner-Stratonovitch equation. Zbl 0947.60040
Crişan, D.; Del Moral, P.; Lyons, T. J.
1999
Conditional exponential moments for iterated Wiener integrals. Zbl 0961.60053
Lyons, Terry; Zeitouni, Ofer
1999
Differential equations driven by rough signals. Zbl 0923.34056
Lyons, Terry J.
1998
Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. Zbl 0914.60045
Lunt, John; Lyons, T. J.; Zhang, T. S.
1998
Convergence of a branching particle method to the solution of the Zakai equation. Zbl 0915.93060
Crisan, Dan; Gaines, Jessica; Lyons, Terry
1998
Stochastic area for Brownian motion on the Sierpiński gasket. Zbl 0936.60073
Hambly, B. M.; Lyons, T. J.
1998
Flow of diffeomorphisms induced by a geometric multiplicative functional. Zbl 0918.60009
Lyons, Terry; Qian, Zhongmin
1998
Variable step size control in the numerical solution of stochastic differential equations. Zbl 0888.60046
Gaines, J. G.; Lyons, T. J.
1997
Nonlinear filtering and measure-valued processes. Zbl 0888.93056
Crisan, Dan; Lyons, Terry
1997
Flow equations on spaces of rough paths. Zbl 0890.58090
Lyons, Terry; Qian, Zhongmin
1997
A class of vector fields on path spaces. Zbl 0877.58059
Lyons, T. J.; Qian, Z. M.
1997
Stochastic Jacobi fields and vector fields induced by varying area on path spaces. Zbl 0903.60008
Lyons, Terry; Qian, Zhongmin
1997
Convergence of non-symmetric Dirichlet processes. Zbl 0888.31005
Lyons, Terry; Zhang, Tusheng
1996
Calculus for multiplicative functionals, Itô’s formula and differential equations. Zbl 0862.60043
Lyons, T. J.; Qian, Z. M.
1996
On the limit of stochastic integrals of differential forms. Zbl 0899.60046
Lyons, Terry; Stoica, Lucreţiu
1996
Uncertain volatility and the risk-free synthesis of derivatives. Zbl 1466.91347
Lyons, T. J.
1995
The interpretation and solution of ordinary differential equations driven by rough signals. Zbl 0827.34049
Lyons, Terry J.
1995
On boundary conditions for stochastic evolution equations with an extremally chaotic source. Zbl 0869.60054
Albeverio, S.; Lyons, T.; Rozanov, Yu.
1995
Differential equations driven by rough signals. I: An extension of an inequality of L. C. Young. Zbl 0835.34004
Lyons, Terry
1994
Decomposition of Dirichlet processes and its applications. Zbl 0804.60044
Lyons, T. J.; Zhang, T. S.
1994
Random generation of stochastic area integrals. Zbl 0805.60052
Gaines, J. G.; Lyons, T. J.
1994
The geometry of the Brownian curve. Zbl 0778.60058
Duplantier, B.; Lawler, G. F.; Le Gall, J.-F.; Lyons, T. J.
1993
Note on convergence of Dirichlet processes. Zbl 0793.31006
Lyons, T. J.; Zhang, T. S.
1993
A note on tightness of capacities associated with Dirichlet forms. Zbl 0781.60063
Lyons, Terry; Röckner, Michael
1992
Random thoughts on reversible potential theory. Zbl 0757.31007
Lyons, Terry
1992
On the non-existence of path integrals. Zbl 0745.60051
Lyons, Terry
1991
Instability of the conservative property under quasi-isometries. Zbl 0747.53034
Lyons, Terry
1991
On conditional diffusion processes. Zbl 0715.60097
Lyons, T. J.; Zheng, W. A.
1990
Bases for positive continuous functions. Zbl 0675.30040
Hayman, W. K.; Lyons, T. J.
1990
A synthetic proof of Makarov’s law of the iterated logarithm. Zbl 0708.30033
Lyons, Terry
1990
Diffusion processes with non-smooth diffusion coefficients and their density functions. Zbl 0715.60096
Lyons, T. J.; Zheng, W. A.
1990
A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Zbl 0654.60059
Lyons, Terence J.; Zheng, Weian
1988
Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. Zbl 0599.60011
Lyons, Terry
1987
The critical dimension at which quasi-every Brownian path is self- avoiding. Zbl 0609.60087
Lyons, Terry J.
1986
Stopping non-commutative processes. Zbl 0585.60058
Barnett, Chris; Lyons, Terry
1986
Function theory, random paths and covering spaces. Zbl 0554.58022
Lyons, Terry; Sullivan, Dennis
1984
Winding of the plane Brownian motion. Zbl 0541.60075
Lyons, T. J.; McKean, H. P.
1984
The best harmonic approximant to a continuous function. Zbl 0541.41024
Hayman, Walter K.; Kershaw, Donald; Lyons, Terry J.
1984
Projection theorems for hitting probabilities and a theorem of Littlewood. Zbl 0566.58036
Lyons, T. J.; MacGibbon, K. B.; Taylor, J. C.
1984
Finely harmonic functions need not be quasi-analytic. Zbl 0541.31002
Lyons, Terry
1984
A simple criterion for transience of a reversible Markov chain. Zbl 0509.60067
Lyons, Terry
1983
Jensen measures for R(K). Zbl 0509.31002
Gamelin, T. W.; Lyons, T. J.
1983
Cones of lower semicontinuous functions and a characterisation of finely hyperharmonic functions. Zbl 0492.31005
Lyons, Terry J.
1982
Finely holomorphic functions. Zbl 0459.46038
Lyons, T. J.
1980
A theorem in fine potential theory and applications to finely holomorphic functions. Zbl 0459.46039
Lyons, T. J.
1980
all top 5
### Cited by 993 Authors
50 Friz, Peter Karl 32 Lyons, Terence John 29 Tindel, Samy 23 Hairer, Martin 20 Gubinelli, Massimiliano 19 Nualart, David 19 Qian, Zhongmin M. 19 Zhang, Tusheng S. 17 Crisan, Dan O. 16 Bailleul, Ismaël F. 15 Inahama, Yuzuru 15 Lejay, Antoine 14 Cass, Thomas Richard 14 Hu, Yaozhong 13 Oberhauser, Harald 13 Russo, Francesco 12 Geng, Xi 12 Riedel, Sebastian 12 Victoir, Nicolas B. 11 Boedihardjo, Horatio 11 Deya, Aurélien 11 Diehl, Joscha 11 Röckner, Michael 10 Müller-Gronbach, Thomas 10 Ouyang, Cheng 10 Teichmann, Josef 10 Yamada, Toshihiro 9 Baudoin, Fabrice 9 Coutin, Laure 9 Prömel, David J. 8 Aida, Shigeki 8 Bayer, Christian 8 Chevyrev, Ilya 8 Fan, Xiliang 8 Litterer, Christian 8 Neuenkirch, Andreas 8 Unterberger, Jérémie M. 8 Zheng, Weian 7 Chen, Zhen-Qing 7 Garrido-Atienza, María José 7 Melbourne, Ian 7 Perkowski, Nicolas 7 Rovira, Carles 7 Xiong, Jie 7 Yaroslavtseva, Larisa 6 Catellier, Rémi 6 Del Moral, Pierre 6 Duc, Luu Hoang 6 Fuglede, Bent 6 Hofmanová, Martina 6 Mazzucchi, Sonia 6 Nilssen, Torstein K. 6 Nourdin, Ivan 6 Ritter, Klaus 6 Zhu, RongChan 5 Albeverio, Sergio A. 5 Besalú, Mireia 5 Beznea, Lucian 5 Brault, Antoine 5 Bruned, Yvain 5 Ebrahimi-Fard, Kurusch 5 Fitzsimmons, Patrick J. 5 Gardiner, Stephen J. 5 Gess, Benjamin 5 Harang, Fabian Andsem 5 Hinz, Michael 5 Ito, Yu 5 Kubilius, Kȩstutis 5 Kuwae, Kazuhiro 5 Liu, Chong 5 Rößler, Andreas 5 Schmalfuß, Björn 5 Tudor, Ciprian A. 5 Weber, Hendrik 5 Yang, Xue 4 Burrage, Kevin 4 da Silva, José Luís 4 Delarue, François 4 Franchi, Jacques 4 Gassiat, Paul 4 Hong, Phan Thanh 4 Kouritzin, Michael A. 4 Kuznetsov, Dmitriĭ Feliksovich 4 Le Jan, Yves 4 León, Jorge A. 4 Liu, Yanghui 4 Manolarakis, Konstantinos 4 Marie, Nicolas 4 Ni, Hao 4 Papavasiliou, Anastasia 4 Peres, Yuval 4 Tapia, Nikolas 4 Vas’kovskiĭ, Maksim Mikhaĭlovich 4 Yang, Danyu 4 Zähle, Martina 4 Zhang, Huilin 4 Zhu, Xiang-Chan 3 Amaba, Takafumi 3 Ancona, Alano 3 Bock, Wolfgang ...and 893 more Authors
all top 5
### Cited in 229 Serials
87 Stochastic Processes and their Applications 62 The Annals of Probability 53 Journal of Functional Analysis 31 The Annals of Applied Probability 29 Probability Theory and Related Fields 27 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 26 Journal of Differential Equations 21 Potential Analysis 17 Stochastic Analysis and Applications 16 Transactions of the American Mathematical Society 15 Journal of Theoretical Probability 15 Electronic Journal of Probability 15 Bernoulli 14 Journal of Mathematical Analysis and Applications 14 Stochastics and Dynamics 13 Journal of Computational and Applied Mathematics 12 Proceedings of the American Mathematical Society 12 Stochastics 11 Stochastic and Partial Differential Equations. Analysis and Computations 10 Bulletin des Sciences Mathématiques 9 Numerical Algorithms 8 Statistics & Probability Letters 8 European Series in Applied and Industrial Mathematics (ESAIM): Probability and Statistics 7 Lithuanian Mathematical Journal 7 Annales de l’Institut Fourier 7 Journal of the Mathematical Society of Japan 7 Journal of Evolution Equations 7 Quantitative Finance 7 Comptes Rendus. Mathématique. Académie des Sciences, Paris 6 Communications in Mathematical Physics 6 The Annals of Statistics 6 Journal of Complexity 6 Revista Matemática Iberoamericana 6 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 6 Discrete and Continuous Dynamical Systems. Series B 5 Communications on Pure and Applied Mathematics 5 Advances in Mathematics 5 Applied Mathematics and Optimization 5 Journal of Applied Probability 5 Proceedings of the Edinburgh Mathematical Society. Series II 5 SIAM Journal on Numerical Analysis 5 Acta Applicandae Mathematicae 5 Applied Numerical Mathematics 5 Acta Mathematicae Applicatae Sinica. English Series 5 Journal of Dynamics and Differential Equations 5 Finance and Stochastics 5 Infinite Dimensional Analysis, Quantum Probability and Related Topics 5 SIAM Journal on Financial Mathematics 4 Journal of Statistical Physics 4 Mathematics of Computation 4 BIT 4 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 4 Physica D 4 Journal de Mathématiques Pures et Appliquées. Neuvième Série 4 Bulletin of the American Mathematical Society. New Series 4 Monte Carlo Methods and Applications 4 Applied Mathematical Finance 4 Discrete and Continuous Dynamical Systems 4 Acta Mathematica Sinica. English Series 4 Annales Henri Poincaré 4 Differential Equations 4 Differentsial’nye Uravneniya i Protsessy Upravleniya 4 ALEA. Latin American Journal of Probability and Mathematical Statistics 4 Science China. Mathematics 4 International Journal of Stochastic Analysis 4 Forum of Mathematics, Sigma 3 Archive for Rational Mechanics and Analysis 3 International Journal of Control 3 Journal of Mathematical Physics 3 Nonlinearity 3 Arkiv för Matematik 3 Inventiones Mathematicae 3 Osaka Journal of Mathematics 3 SIAM Journal on Control and Optimization 3 Journal of Scientific Computing 3 Forum Mathematicum 3 Communications in Partial Differential Equations 3 SIAM Journal on Mathematical Analysis 3 Annals of Mathematics. Second Series 3 Journal of Dynamical and Control Systems 3 Oberwolfach Reports 3 Forum of Mathematics, Pi 2 Advances in Applied Probability 2 Israel Journal of Mathematics 2 Journal of Computational Physics 2 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 2 Collectanea Mathematica 2 Duke Mathematical Journal 2 Mathematische Annalen 2 Mathematics and Computers in Simulation 2 Mathematische Nachrichten 2 Nagoya Mathematical Journal 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Proceedings of the London Mathematical Society. Third Series 2 Tôhoku Mathematical Journal. Second Series 2 Statistical Science 2 Science in China. Series A 2 Japan Journal of Industrial and Applied Mathematics 2 Aequationes Mathematicae 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics ...and 129 more Serials
all top 5
### Cited in 54 Fields
809 Probability theory and stochastic processes (60-XX) 188 Numerical analysis (65-XX) 134 Partial differential equations (35-XX) 106 Ordinary differential equations (34-XX) 68 Potential theory (31-XX) 64 Systems theory; control (93-XX) 59 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 54 Statistics (62-XX) 47 Global analysis, analysis on manifolds (58-XX) 44 Dynamical systems and ergodic theory (37-XX) 42 Real functions (26-XX) 42 Operator theory (47-XX) 40 Statistical mechanics, structure of matter (82-XX) 37 Functional analysis (46-XX) 23 Functions of a complex variable (30-XX) 22 Quantum theory (81-XX) 19 Measure and integration (28-XX) 17 Calculus of variations and optimal control; optimization (49-XX) 15 Combinatorics (05-XX) 15 Differential geometry (53-XX) 14 Approximations and expansions (41-XX) 13 Associative rings and algebras (16-XX) 13 Harmonic analysis on Euclidean spaces (42-XX) 11 Fluid mechanics (76-XX) 10 Computer science (68-XX) 9 Nonassociative rings and algebras (17-XX) 8 Integral equations (45-XX) 6 Topological groups, Lie groups (22-XX) 6 Biology and other natural sciences (92-XX) 5 General and overarching topics; collections (00-XX) 5 Algebraic geometry (14-XX) 5 Information and communication theory, circuits (94-XX) 4 Algebraic topology (55-XX) 4 Operations research, mathematical programming (90-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Group theory and generalizations (20-XX) 3 Several complex variables and analytic spaces (32-XX) 3 Geometry (51-XX) 3 General topology (54-XX) 2 Number theory (11-XX) 2 Commutative algebra (13-XX) 2 Special functions (33-XX) 2 Mechanics of particles and systems (70-XX) 2 Relativity and gravitational theory (83-XX) 1 History and biography (01-XX) 1 Category theory; homological algebra (18-XX) 1 Difference and functional equations (39-XX) 1 Abstract harmonic analysis (43-XX) 1 Integral transforms, operational calculus (44-XX) 1 Convex and discrete geometry (52-XX) 1 Mechanics of deformable solids (74-XX) 1 Optics, electromagnetic theory (78-XX) 1 Astronomy and astrophysics (85-XX) 1 Geophysics (86-XX)
### Wikidata Timeline
The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata. | 2022-08-09 17:47:29 | {"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": 0.6909084320068359, "perplexity": 6867.836673838881}, "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-2022-33/segments/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00066.warc.gz"} |
http://coefficientlee.blogspot.com/ | ## Wednesday, April 17, 2013
### Base Ten and Place Value Flashcards
So I'm shifting to include some elementary level math activities on this page. My son is in 1st grade and may or may not have inherited a math brain. Anyways there are days he shows signs of math brilliance and there are some days he brings home a not so great score on a math assignment. He told me the other day that he liked doing fractions...let's hope this continues! Here's the flashcards I made for him. I was inspired by these cute cards found here and I used a free border found here to spice them up.
April is the month of birthdays for my family. You know age in years doesn't look so bad in base ten blocks...just a thought.
You may download and use these flashcards for your own purposes, especially to help a child learn math. Do not use or sell these cards for profit. All I ask in exchange is for you to become a follower and/or to share my blog with your friends. Thanks!
## Friday, March 15, 2013
### Midpoint and Distance Formula Project
This is a small project I assigned students. It reinforces application of the midpoint and distance formulas in addition to providing an opportunity for the students to problem solve and be creative. I specifically designed the rubric for extremely quick grading. You will notice that every student has the same problem, which eases checking for accuracy. I also did not require students to use the midpoint and/or distance formulas...as long as they could explain in writing their methodology in solving the problem that was good enough for me. The end product was a variety of maps that were great to display.
## Tuesday, March 12, 2013
### Percent of Change Worksheet
Before I taught students how to compute any percent of change I would also present the following competitors and ask, "Who's the Biggest Loser?" This would lead to laughs, but of course it intrigued the students and opened up a great discussion. Do you know who the Biggest Loser is?
Here's a short activity I gave the students to complete:
Standards:
CCSS Math Content 7.RP.A.3
Minnesota 7.2.2.2
Texas TEKS 111.24 (b) 3 (B)
Virginia SOL 7.4, SOL 8.3
## Saturday, March 9, 2013
### Pi Day Resources
Pi Day is approaching! Here are some resources to help you out...even though I think we should be celebrating the ratio of a circle's circumference to its radius or tau. Why tau? Well most people will tell you that a circle's circumference is 2πr and that the area of a circle is πr2. Notice how both formulas refer to the circle's radius? Also if we celebrated tau, then we could eat twice the pie! I rest my case.
## Friday, March 8, 2013
### Multiplying Two Binomials
For those of you that don't know...a binomial is a two-term polynomial like these:
Probably the most common strategy of teaching students how to multiply two binomials is using the FOIL method. FOIL is an acronym for "First, Outer, Inner, Last." That is, multiply the first terms, multiply the outer terms, multiply the inner terms, and finally multiply the last terms. It's easy to remember, so that's why I think it's popular.
Then there's the visual strategy of drawing a face. I like to think I have a good imagination, but this visual just doesn't do it for me. There's also the bird's beak. If these strategies don't get the students excited about multiplying binomials then try this beak...it should do the trick.
Standards:
Common Core Standard A-APR
Minnesota 9.2.3.2
Texas TEKS 111.32 (b) (4)
Virginia Math SOL A.2
## Thursday, March 7, 2013
### Keeping Track of the Assignments
I penalized students for turning in homework or daily practice assignments late, which was after a short one class safety buffer. If you didn't get a chance, you can read more by clicking here.
How did I keep track of what was turned in and when? I would of loved to have one of those time stamp machine things, but alas there are two words to describe the problem...Teacher's Salary.
I kept track of all the incoming assignments by creating my own time stamp machine. Here's what you need: A calendar, stamps, and ink pads. I found a free printable calendar here at Hello, Cuteness!; stamps I purchased at Michael's (don't forget to use the weekly %-off coupons and your teacher discount); and the ink pads I picked up from a "free" bin from an outgoing teacher (these felt ink pads, found here, last longer, make less of a mess than the sponge kind, and are scented!).
Instructions:
1. Each day select a stamp and color and record it on the calendar. Place only the stamp and color of the day out...or else students will get sneaky! I have 23 different stamps and 7 different color inks...that's 23 x 7 = 161 possible combinations!
2. On the day an assignment is due (and you want to give a numerical grade for it) go around the room with a clipboard/tablet to record a grade for each student. Any paper that is given full-credit is stamped with the corresponding stamp/color for the day. These papers are not collected as the students were responsible for keeping them in their notebook/binder. Incomplete papers are not stamped and also not collected (with the hopes that the student will make the attempt to complete it by the next class for full-credit).
3. If a student has an assignment to turn in that is not on time, then he or she should stamp the paper with the corresponding stamp/color for the day and physically place it in the proper basket.
4. If any sheet of paper in the basket is missing a stamp or written detail of the assignment (i.e. Section 3.2 - p.342 #1-21odd), then it is returned to the student immediately without being recorded. (Do this consistently and they won't make the same mistake again).
5. If any sheet of paper in the basket is missing a name, it should be placed immediately on a "No Name" board. (My "No Name" board consisted of a string with clothes pins).
Pros:
• Student's can't easily lie about when they turned in an assignment.
• You can fall behind with your grading and recording and still know when an assignment was turned in.
• Doesn't cost a ton.
Cons:
• Some students are slobs with ink.
• A desperate student can possibly acquire and use the same stamp/ink you used on a specific day and claim they turned in the assignment on time.
Tips:
• Keep a container of wet wipes in your room to clean the stamps.
• Put the stamp/ink out for the next day just before leaving, that way it's ready to go if you have an unexpected absence.
• Give a mini lesson to the students on how to ink and stamp without making a mess.
• Display the stamped calendar for the students.
• Make sure you record the due date for each assignment in your gradebook.
## Wednesday, March 6, 2013
### Requests?
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https://codereview.stackexchange.com/questions/86041/calculator-in-ruby/86042#86042 | Calculator in Ruby
I have created a calculator in Ruby. I am using Ruby 2.1.0. I'm fairly sure that someone will be able to improve this, as I am quite new to Ruby.
puts "Welcome to Calc"
puts ""
puts "Please enter the first number"
n1 = gets.to_i()
puts ""
puts "Please enter the second number"
n2 = gets.to_i()
puts ""
subtract = n1 - n2
multiply = n1 * n2
divide = n1 / n2
power = n1 ** n2
sqrt1 = Math.sqrt(n1)
sqrt2 = Math.sqrt(n2)
puts "#{n1} + #{n2} = #{add}"
puts "#{n1} - #{n2} = #{subtract}"
puts "#{n1} * #{n2} = #{multiply}"
puts "#{n1} / #{n2} = #{divide}"
puts "#{n1} ** #{n2} = #{power}"
puts "#{n1} √ = #{sqrt1}"
puts "#{n2} √ = #{sqrt2}"
gets()
One thing you forgot to check was if n2 is 0 and n1 is non zero, in which case the answer is either undefined or a signed infinity.
• In addition: divide on integers is a modulo division. If it is intented to use the modulo division, then I would expect both values using divmod. Or you have to convert one value to float value. | 2022-09-24 22:53:35 | {"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": 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": 0.4169900119304657, "perplexity": 6696.868720309085}, "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-2022-40/segments/1664030333541.98/warc/CC-MAIN-20220924213650-20220925003650-00497.warc.gz"} |
http://nrich.maths.org/685/solution | ### Three Way Split
Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.
### Pareq Calc
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.
### Pareq Exists
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
# Farhan's Poor Square
##### Stage: 4 Challenge Level:
Splendid solutions to Farhan's problem came in from Shabbir Telani, age 13, Jack Hunt School, Peterborough, Prav Idaikkadar,age 13, and Megan Mitchell, age 14, North London Collegiate School Maths Club, Richard Mason, Madras College, and Rachel Evans, the Mount School York. This is Shabbir's solution:
Using trigonometry:
$$x = \frac{35}{\cos 42^o} = 47.09714 = 47.10$$ (to 2 decimal places).
To find the area of the square not covered by the circle work out the area of the square minus the area of the circle.
The formula for the area is
$A = x^2 - \pi \left(\frac{x}{2}\right)^2$
So the area of the square not covered by the circle and triangle is 2218.14 - 1742.12 = 476.02 cm 2 (to 2 decimal places). | 2014-04-21 07:23:23 | {"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": 0.6701022982597351, "perplexity": 1045.7853396500643}, "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-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00632-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://iwaponline.com/view-large/2484515 | Table 5
Developed mathematical equations using the WGEP model
Area(z)Equations
UL 1
2
3
1
2
3
1
2
3
SR 1
2
3
KM 1
2
3
HJM 1
2
3
Area(z)Equations
UL 1
2
3
1
2
3
1
2
3
SR 1
2
3
KM 1
2
3
HJM 1
2
3
st: denotes spatial and time for s and t, respectively.
Close Modal | 2021-09-17 23:38:12 | {"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": 0.9952903389930725, "perplexity": 450.03051678096546}, "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-2021-39/segments/1631780055808.78/warc/CC-MAIN-20210917212307-20210918002307-00427.warc.gz"} |
https://thephilosophyforum.com/discussion/2928/donald-trump-all-general-trump-conversations-here/p592 | Donald Trump (All General Trump Conversations Here)
• 17.4k
‘Donald Trump has attacked Rupert Murdoch in a blistering statement, accusing him of betraying his Fox News television hosts by admitting that he doubted their conclusions about the 2020 election.
“Why is Rupert Murdoch throwing his anchors under the table,” the former US president posted to his platform Truth Social.’
https://www.smh.com.au/world/north-america/trump-attacks-murdoch-for-throwing-his-anchors-under-the-table-20230302-p5cor5.html
Trial hasn’t even started yet!
• 5.7k
I think Trumpy got his idioms mixed up. He wants the anchors to be under the table (with their secrets).
• 6.8k
Remember when Rupert Murdoch was supposed to be some puppet master? He cannot even control his own employees. Another conspiracy theory turned nothingburger.
• 17.4k
Even the mighty Rupert abases himself for the holy $. • 4.2k Murdoch saying he doubted their conclusions does not mean he cannot control his own employees. The news hosts themselves doubt the truth of what they reported. It is not about the truth. It is about pandering to their viewers. It may be that this puppet master story is like a version of Pinocchio. • 6.8k Who cares? After years of Russia collusion, Covid propaganda, Ukraine warmongering, January 6th handwringings, and all the deep-state dinner theater news outlets have spoon-fed us these past few years, I’m now supposed to give a hoot over Murdoch disagreeing with Fox News anchors about the results of an election? • 4.2k I’m now supposed to give a hoot over Murdoch disagreeing with Fox News anchors about the results of an election? He did not disagree with them. Fox News knowingly peddles lies about the election. Is there any evidence that he attempted to stop them? Now you may not give a hoot that a major "news" network did this, but it is a serious matter. It is not simply that what the claimed turned out to be wrong, they were well aware that it was not the truth. • 6.8k I honestly don’t care because everything to the anti-Trump brigade is a serious matter until one looks closely. Every conspiracy theory regarding Trump, whether it was Russia collusion or his tax returns, have been massively and comically overstated, and as a result has turned justice into nonsense, journalism into a joke, politics into circuses, and the US into clown world. It’s gotten so bad that one can adopt a contrary belief without any evidence to do so and he’ll be right most of the time. • 4.2k Spoken like a true Trumpster. Dear Leader would be pleased by your loyalty. Despite Trump's claim that there was no collusion and Barr's attempt to sweep it under the rug, the Mueller investigation did not exonerate him. Whether you call it collusion or something else, the Senate Intelligence Committee found clear evidence of cooperation between the Trump campaign and the Kremlin. Here is a summary of the findings. But you can safely ignore it because by your logic Trump and his campaign did no wrong. As to tax evasion, the Trump Organization was found guilty of fraud and fined 1.6 million dollars. There are several ongoing cases. I won't go into any of it because by your logic, despite whatever the facts reveal, you are right to conclude he did nothing wrong and you are likely to be right. • 6.8k Again I could care less about any of your propaganda. Fact is, all these investigations and conspiracy theories over the years and he has yet to be found guilty of anything, despite your assumptions of guilt. But, like a true fanatic, you double your efforts long after you have forgotten your aim. • 4.2k Again I could care less about any of your propaganda. The Senate Intelligence Committee findings, led by eight Republicans and seven Democrats. are not my "propaganda". The fact that the Trump Organization was found guilty of fraud is not my "propaganda". The grand jury's indictment recommendations in the Georgia investigation into election interference are not my "propaganda". • 5.7k In new campaign promises Trump wants to take government owned land and build 10 new “freedom cities” on it. If I remember right they do things like that in communist China and they sometimes end up “ghost cities”. Maybe it won’t seem as communistic if he also promises that Mexico will pay for the cities. • 12.4k Real Estate Developer in Chief • 11.6k Again I could care less about any of your propaganda. — NOS4A2 The Senate Intelligence Committee findings, led by eight Republicans and seven Democrats. are not my "propaganda". The fact that the Trump Organization was found guilty of fraud is not my "propaganda". The grand jury's indictment recommendations in the Georgia investigation into election interference are not my "propaganda". :up: :up: :up: Just a guess, but I've always had a strong suspicion, based soley on his/her posts, that @NOS4A2 is someone who desperately needs to be lied to by FOX Noise, OANN, Newsmax, Alex Jones, Steve Bannon, Pravda (RT) and other wingnut media. :mask: Here's some more "propaganda" for NOS: Making Attorneys Get Attorneys :victory: :smirk: • 11.6k @NOS4A2 & others MAGAs who love to be lied to We are very, very close to being able to ignore Trump most nights. I truly can't wait. I hate him passionately. That's the last four years. We're all pretending we've got a lot to show for it, because admitting what a disaster it's been is too tough to digest. But come on. There really isn't an upside to Trump. — Tucker Carlson, the FOX Noise paid actor, Jan. 4, 2021, from Dominion defamation lawsuit • 6.8k I think you guys are just mad at the guy because he exposed the extent of the propaganda you’ve been fed for years in one single segment. So you have to sift through one or two out of context texts for gossip. No matter; I’ll listen to the worst propagandist before I consider a peep from any busybody. • 11.6k :lol: Denial is a helluva drug! It’s easier to fool people than to convince them that they have been fooled. — Mark Twain • 12k Trump says he expects to be arrested on Tuesday Former US President Donald Trump says he is expecting to be arrested on Tuesday in a case about alleged hush money paid to an ex-porn star. Mr Trump called on his supporters to protest against such a move in a post on his Truth Social platform. One of Mr Trump's lawyers said his claim was based on media reports that he could be indicted next week. • 4.2k What form does he want and expect these protests to take? Given what has happened in the past, it does not seem likely that they would be peaceful. But this is typical of Trump. It is clear that he does not want his day in court. Above all else is the court of public opinion. But in a selective rather than general sense, that is, limited to the opinion of his followers. • 6.8k It’s typical of his critics to erect some sort of show-trial for political purposes. Impeachments, J6 hearings (complete with a television producer from ABC), raiding his home, multiple investigations and civil suits, and now thinks he might be arrested by a Manhattan district attorney (of course). This is the deep-state dinner theater that has made Trump a folk devil in the eyes of the establishment base. • 4.2k It’s typical of his critics It is typical of apologists such as yourself to jump to Trump's defense by making vague accusations that portray him as an innocent treated unfairly by the media, the courts, politicians, and anyone else who, because they dare to question the legality of Trump's actions, are part of a deep state conspiracy. the establishment base ? Empty rhetoric. Trump, his Republican supporters, Fox News, the Federalist society, big money supporters are all entrenched part of "the establishment". • 14.3k FAKE NEWS ACCUSATIONS ARE FAKE NEWS! I DID NOT DO IT BECAUSE NO TRUE ME DID IT! DO NOT BELIEVE FAKE NEWS PROSECOTURS OF NEW YORK! EAT MORE ICE CREAM! WHEN I AM PRESIDENT ICE CREAM WILL BE FREE AND SO WILL I! PROTEST NOW OR THEY WILL JAIL ICE CREEM FOR LIFE! • 11.6k FOX Noise just got hit with a second defamation suit for$2.7 billion by Smartmatic (adding to Dominion Voting Machine's \$1.6 billion defamation lawsuit) and "Individual-1" was notified that he's imminently becoming "Defendent-1" :clap: . Belated Happy St. Paddy's! :party:
:up:
:lol:
:rofl:
• 14.3k
:cool: :party:
• 12.4k
Trump, his Republican supporters... are all entrenched part of "the establishment".
No, I don't think so.
• 7.4k
Trump, his Republican supporters... are all entrenched part of "the establishment".
— Fooloso4
No, I don't think so.
I think so. Rather in the same way that a corrupt policeman is still a member of the police.
• 12.4k
think so. Rather in the same way that a corrupt policeman is still a member of the police.
Broadly speaking, he's an anti-establishment figure. He was a perfect rendition of the demagogue who leads the fight against the establishment. In this case, the establishment is neoliberal, so MAGA was about taking the country back to somewhere around 1965 when embedded liberalism assured the average white guy a good job with benefits.
It wasn't a good look when Obama and Clinton tried to explain that we can't go back. That assured that they'd be taken as exactly what they are: representatives of the establishment.
Another hint is that establishment figures don't try to arrange coups.
• 7.4k
Another hint is that establishment figures don't try to arrange coups.
That's like saying that policemen do not commit crimes. Dangerous falsehood. Hitler came to power democratically and then established his dictatorship. Likewise any revolutionary government comes to power in a coup and immediately becomes established or is overthrown by a counter coup by the disestablished establishment. Trump is a corrupt member of the establishment, seeking to exploit the establishment and resentment of the establishment in equal measure and with no loyalty to either.
• 12.4k
That's like saying that policemen do not commit crimes.
I guess we're thinking of different meanings of "establishment.". Today's establishment is those college kids who wrestled with the police at Kent State. Clinton was very progressive in her youth and then became part of the new establishment. It's not about authority or policing anything. If anything, our establishment is "let the markets do what they want and defund aid to the inner cities.". That's what both Clinton and her husband ultimately stood for.
I know it's hard to think of Trump as the leader of the young rabble rousers, but he actually is. He was elected in part by refugees from the Democratic party. It didn't turn out well. He didn't strike a particularly heroic pose, but he did reveal a well of frustration with, and anger toward...
the establishment.
Likewise any revolutionary government comes to power in a coup and immediately becomes established or is overthrown by a counter coup by the disestablished establishment.
I'm sure Trump would have liked to establish his own ment. He wanted to be a dictator.
• 4.2k
He was a perfect rendition of the demagogue who leads the fight against the establishment.
His anti-establishment rhetoric helped to get him elected, but he is no "man of the people". He is every bit the kind of elite he rails against.
the establishment is neoliberal
Trump is neoliberal.
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reveal | 2023-03-30 13:50:25 | {"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": 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": 0.22902998328208923, "perplexity": 8972.396871860656}, "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-2023-14/segments/1679296949331.26/warc/CC-MAIN-20230330132508-20230330162508-00378.warc.gz"} |
http://mathhelpforum.com/trigonometry/7086-several-problems-trig.html | 1. Several Problems Trig
1. y=cot(x-(pi/6))
What is the first negative asymptote?
2. Originally Posted by qbkr21
1. y=cot(x-(pi/6))
What is the first negative asymptote?
$\displaystyle x=\pi/6$?
RonL
3. Re:
4. Originally Posted by qbkr21
Opps negative - missed that
$\displaystyle -5\pi/6$
5. Originally Posted by qbkr21
1. y=cot(x-(pi/6))
What is the first negative asymptote?
hello,
the cot-function is not defined for $\displaystyle x \in \{ ...-2\pi, -\pi, 0, \pi, 2\pi...\}$
Then calculate x if
$\displaystyle x-\frac{\pi}{6} = k \cdot \pi, \ k \in \mathbb{Z}$
I've got the the first negative asymptote at $\displaystyle x=-\frac{5}{6} \pi$
EB
6. Re:
Ok yea that right, great thanks
But what formula did you use. Keep in mind that I am in Pre-Calc
7. Originally Posted by qbkr21
Ok yea that right, great thanks
But what formula did you use. Keep in mind that I am in Pre-Calc
The vertical asymtotes occur where tan(x-pi/6) is zero (and so the cot goes
to infinity), which are the points where x-pi/6=n pi, n=0, +/-1, +/-2, ....
RonL
(and checking by ploting the graph of tan(x-pi/6) and or cot(x-pi/6)) | 2018-04-24 09:14:52 | {"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": 0.935656726360321, "perplexity": 8738.03348580179}, "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/1524125946578.68/warc/CC-MAIN-20180424080851-20180424100851-00320.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=260557 | # Integral of full-wave rectified sinewave, find equivalent sine waveform
by gfgray
Tags: equivalent, fullwave, integral, rectified, sine, sinewave, waveform
P: 1
1. The problem statement, all variables and given/known data
I am given two waveforms (a square wave and DC) that define the maximum allowable operating parameters for an LED. I wish to derive the maximum allowable fully-rectified sine waveform (120hz).
Square wave: 442mW peak, 0.1ms pulse width, 10% duty cycle
DC: 87mW
See attached PDF. Use the power on the right axis instead of current on the left. Also ignore the temperature stuff. I was trying to figure out why the energy for each waveform is different.
Here is a link to the actual spec sheet:
http://marktechopto.com/pdfs/Cree/LO...%2003DEC07.pdf
This is a two-part question. First, how do I write the integral for the full-wave rectified sine? Note that the sinewave's x-axis intersection is advanced/delayed by ~1.12ms. Integrate from 0 to pi?
Second, how do I deal with the fact that the total energy for the given waveforms is different? Or is this unsolvable?
2. Relevant equations
3. The attempt at a solution
See PDF.
Attached Files
Pulse Width Calculations 09-30-2008.pdf (659.5 KB, 11 views)
Related Discussions Engineering, Comp Sci, & Technology Homework 3 Calculus & Beyond Homework 3 Introductory Physics Homework 7 Introductory Physics Homework 1 General Discussion 5 | 2014-03-09 11:05:56 | {"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": 0.8097921013832092, "perplexity": 2905.887418031227}, "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-2014-10/segments/1393999677441/warc/CC-MAIN-20140305060757-00067-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/319854/square-matrices-with-one-constant-on-the-main-diagonal-and-a-second-constant-eve | # Square matrices with one constant on the main diagonal and a second constant everywhere else.
Suppose that A_n is the n×n matrix which has 2’s on the diagonal, and 1’s everywhere else:
A_2 matrix : Vector (2,1) and (1,2) A_3 matrix : Vector (2,1,1), (1,2,1), (1,1,2) A_4 matrix : Vector (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2) A_n matrix : vector (2,1,1…), (1,2,1,1…), (…)
and suppose that B_n is the n × n matrix which is just filled with minus-ones:
B_2 matrix : Vector (-1,-1), (-1,-1) B_3 matrix : Vector (-1,-1,-1), (-1,-1,-1), (-1,-1,-1) B_n matrix : Vector (-1,-1,-1,…), (…)
Explain why det(An) = det(In − Bn), where In is the n × n identity matrix.
If Pn(λ) is the characteristic polynomial of Bn, explain why det(An) = Pn(1).
Since Bn has rank 1, explain why this means that λn−1 has to divide Pn(λ).
Either by looking at the trace of Bn, or by seeing what happens to the vector ⃗v = (1,1,...,1) of all 1’s when you put it through Bn, find the value of a.
What is det(An)?
What is the determinant of the n × n matrix Cn which has 5’s on the diagonal, and 1’s everywhere else?
• This isn't really on topic, but what do all of you think of the alternative "diagonable matrix"? It's in Horn and Johnson and I had a friend who said it instead just because it sounds funny. – Graphth Mar 3 '13 at 21:44
• @Anna-Banana: What is the connection to diagonalizability? – Dennis Gulko Mar 3 '13 at 21:44
• I think that $\det(A_n) = \det(I_n -B_n)$ because $A_n = I_n -B_n$. – Ben Mar 3 '13 at 21:45
• Oh, I please: write your mathematics with LaTeX...! You can find directyions in the FAQ section. – DonAntonio Mar 3 '13 at 22:03
There is no mystery to this. The square matrix consisting of all 1's has the eigenvalue $n$ with multiplicity one and $0$ with multiplicity $(n-1).$ A basis of orthogonal eigenvectors, but of varying length, is: $$n: \; \; (1,1,1, \ldots,1),$$ $$0: \; \; (1,-1,0,0,\ldots,0),$$ $$0: \; \; (1,1,-2,0,\ldots,0),$$ $$0: \; \; (1,1,1,-3,0,\ldots,0),$$ $$0: \; \; (1,1,1,1,-4,\ldots,0),$$ $$0: \; \; (1,1,1,1,1,\ldots,1,1-n).$$
To get $w$ on the main diagonal you add $(w-1)I,$ which does not alter the eigenvectors but adds $(w-1)$ to each eigenvalue. The determinant is the product of the eigenvalues, giving $$(n+w-1) (w-1)^{n-1}.$$
I see. I missed the evident homework questions interspersed. Sigh. Call the matrix of all ones $T,$ there is no difficulty dealing with $\alpha T + \beta I.$ Please do those yourself. An eigenvector of $T$ with eigenvalue $\lambda$ is also an eigenvector of $\alpha T + \beta I$ with eigenvalue $\alpha \lambda + \beta .$ | 2019-07-18 17:28:05 | {"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": 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": 0.8145739436149597, "perplexity": 781.32850920236}, "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-2019-30/segments/1563195525699.51/warc/CC-MAIN-20190718170249-20190718192249-00492.warc.gz"} |
https://www.gradesaver.com/textbooks/engineering/mechanical-engineering/engineering-mechanics-statics-and-dynamics-14th-edition/chapter-2-force-vectors-section-2-3-vector-addition-of-forces-problems-page-29/5 | ## Engineering Mechanics: Statics & Dynamics (14th Edition)
Published by Pearson
# Chapter 2 - Force Vectors - Section 2.3 - Vector Addition of Forces - Problems - Page 29: 5
#### Answer
$F_{AB}=314lb$ $F_{AC}=256lb$
#### Work Step by Step
We can use the law of sines to find the magnitudes of the components of $F$. $\sin75^{\circ}/350lb=\sin60^{\circ}/F_{AB}$ $F_{AB}=\sin60^{\circ} *350lb/\sin75^{\circ}$ $F_{AB}=314lb$ $\sin75^{\circ}/350lb=\sin45^{\circ}/F_{AC}$ $F_{AC}=\sin45^{\circ} *350lb/\sin75^{\circ}$ $F_{AC}=256lb$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-10-22 01:00:41 | {"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": 0.6284717321395874, "perplexity": 1104.051520450153}, "config": {"markdown_headings": true, "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-2018-43/segments/1539583514443.85/warc/CC-MAIN-20181022005000-20181022030500-00104.warc.gz"} |
http://spot.pcc.edu/math/orcca/section-square-root-properties.html | ## Section8.2Square Root Properties
###### ObjectivesPCC Course Content and Outcome Guide
In this chapter, we will learn how to both simplify square roots and to do operations with square roots.
###### Definition8.2.2The Definition of the Square Root of a Number
If $y^2=x$ for a positive number $y\text{,}$ then $y$ is called the square root of $x\text{,}$ and we write $y=\sqrt{x}\text{,}$ where the $\sqrt{\phantom{x}}$ symbol is called the radical or the root. We call expressions with a root symbol radical expressions. The number inside the radical is called the radicand.
For example, since $4^2=16\text{,}$ then $\sqrt{16}=4\text{.}$ Both $\sqrt{2}$ and $3\sqrt{2}$ are radical expressions. In both expressions, the number $2$ is the radicand. You can review the square root basics in Section 1.3.
The word “radical” means something like “on the fringes” when used in politics, sports, and other places. It actually has that same meaning in math, when you consider a square with area $A$ as in Figure 8.2.3.
### Subsection8.2.1Estimating Square Roots
When the radicand is a perfect square, its square root is a rational number. If the radicand is not a perfect square, the square root is irrational. We want to be able to estimate square roots without using a calculator.
To estimate $\sqrt{10}\text{,}$ we can find the nearest perfect squares that are whole numbers on either side of $10\text{.}$ Recall that the perfect squares are $1, 4, 9, 16, 25, 36, 49, 64,\dots$ The perfect square that is just below $10$ is $9$ and the perfect square just above $10$ is $16\text{.}$ This tells us that $\sqrt{10}$ is between $\sqrt{9}$ and $\sqrt{16}\text{,}$ or between $3$ and $4\text{.}$ We can also say that $\sqrt{10}$ is much closer to $3$ than $4$ because $10$ is closer to $9\text{,}$ so we think $3.1$ or $3.2$ would be a good estimate.
To check our estimate, let's find $\sqrt{10}$ with a calculator:
\begin{equation*} \sqrt{10}\approx3.162 \end{equation*}
The actual value is just above $3$ as we estimated, and between $3.1$ and $3.2\text{.}$ Let's look at some more examples.
###### Example8.2.5
Estimate $\sqrt{3.2}$ without a calculator.
Explanation
The radicand $3.2$ is between $1$ and $4\text{,}$ so $\sqrt{3.2}$ is between $\sqrt{1}$ and $\sqrt{4}\text{,}$ or between $1$ and $2\text{.}$
To be more precise, we notice that $3.2$ is much closer to $4$ than $1\text{.}$ We estimate $\sqrt{3.2}$ to be about $1.8\text{.}$
We will check our estimate with a calculator:
\begin{equation*} \sqrt{3.2}\approx1.788 \end{equation*}
### Subsection8.2.2Multiplication and Division Properties of Square Roots
Here is an example using perfect squares and the rules of exponents to show a relationship between the product of two square roots:
\begin{gather*} \sqrt{9\cdot16}=\sqrt{3^2\cdot 4^2}=\sqrt{(3\cdot4)^2}=3\cdot4=12\\ \end{gather*}
and
\begin{gather*} \sqrt{9}\cdot\sqrt{16}=\sqrt{3^2}\cdot\sqrt{4^2}=3\cdot4=12 \end{gather*}
Whether we multiply the radicands first or take the square roots first, we get the same result. This tells us that in multiplication with radicals, we can combine factors into a single radical or separate them as needed.
Now let's look at division. When we learned how to find the square root of a fraction in Section 1.3, we saw that the numerators and denominators could be simplified separately. We multiply the numerators and denominators independently. Here is an example of two different ways to simplify a fraction in a square root:
\begin{gather*} \sqrt{\frac{25}{9}}=\sqrt{\left(\frac{5}{3}\right)^2}=\frac{5}{3}\\ \end{gather*}
and
\begin{gather*} \frac{\sqrt{25}}{\sqrt{9}}=\frac{\sqrt{5^2}}{\sqrt{3^2}}=\frac{5}{3} \end{gather*}
Just like with multiplication, we can separate the numerators and denominators in a radical expression or combine them as needed. Note that we worked with expressions that were perfect squares, but these properties will work regardless of the number inside the radical. Let's summarize these properties.
### Subsection8.2.3Simplifying Square Roots
We can use Multiplication and Division Properties of Square Roots to simplify a radicand that is not a perfect square. Simplifying radicals is similar to simplifying fractions because we want the radicand to be as small as possible.
To understand why we can simplify radicals, let's use a calculator to compare $\sqrt{12}$ and $2\sqrt{3}\text{.}$
\begin{align*} \sqrt{12}\amp=3.4641\ldots \amp\amp\text{and}\amp2\sqrt{3}\amp=3.4641\ldots \end{align*}
These are equivalent expressions so let's see how we can simplify $\sqrt{12}$ to $2\sqrt{3}\text{.}$
First, we will make a table of factor pairs for the number $12\text{,}$ as we did in Section 7.3.
\begin{align*} 1\amp\cdot12\\ 2\amp\cdot6\\ 3\amp\cdot4 \end{align*}
The factor pair with the largest perfect square is $3\cdot4\text{.}$ We will use the property of multiplying radicals to separate the perfect square from the other factor. We write the perfect square first because it will end up in front of the radical.
\begin{align*} \sqrt{12}\amp=\sqrt{4}\cdot\sqrt{3}\\ \amp=2\cdot\sqrt{3}\\ \amp=2\sqrt{3} \end{align*}
This process can be used to simplify any square root, or to determine that it is fully simplified. Let's look at a few more examples.
###### Example8.2.7
Simplify $\sqrt{72}\text{.}$
Explanation
Here is a table of factor pairs for the number $72\text{.}$
\begin{align*} 1\amp\cdot72\amp4\amp\cdot18\\ 2\amp\cdot36\amp6\amp\cdot12\\ 3\amp\cdot24\amp8\amp\cdot9 \end{align*}
The largest perfect square is $36$ so we will rewrite $72$ as $36\cdot2\text{.}$
\begin{align*} \sqrt{72}\amp=\sqrt{36\cdot2}\\ \amp=\sqrt{36}\cdot\sqrt{2}\\ \amp=6\sqrt{2} \end{align*}
Notice that if we had chosen $4\cdot18$ we could simplify the radical partially but we would need to continue and find the perfect square of $9$ in $18\text{.}$
###### Example8.2.9
Simplify $\sqrt{30}\text{.}$
Explanation
Here is a table of factor pairs for the number $30\text{.}$
\begin{align*} 1\amp\cdot30\amp3\amp\cdot10\\ 2\amp\cdot15\amp5\amp\cdot6 \end{align*}
The number $30$ does not have any factors that are perfect squares so it cannot be simplified further.
We can also use Division Property of Square Roots to simplify expressions.
###### Example8.2.10
1. Simplify $\sqrt{\frac{9}{16}}\text{.}$
2. Simplify $\frac{\sqrt{50}}{\sqrt{2}}\text{.}$
Explanation
1. For the first expression, we will use the Division Property of Square Roots:
\begin{align*} \sqrt{\frac{9}{16}}\amp=\frac{\sqrt{9}}{\sqrt{16}}\\ \amp=\frac{3}{4} \end{align*}
2. For the second expression, we use the same property in reverse: $\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}\text{:}$
\begin{align*} \frac{\sqrt{50}}{\sqrt{2}}\amp=\sqrt{\frac{50}{2}}\\ \amp=\sqrt{25}\\ \amp=5 \end{align*}
### Subsection8.2.4Multiplying Square Root Expressions
If we use the Multiplication Property of Square Roots and the Division Property of Square Roots in the reverse order as
\begin{align*} \sqrt{x}\cdot\sqrt{y}\amp=\sqrt{xy}\amp\amp \text{and}\amp \frac{\sqrt{x}}{\sqrt{y}}\amp=\sqrt{\frac{x}{y}}\text{,} \end{align*}
we can use these properties to multiply and divide square root expressions. We want to simplify each radical first to keep the radicands as small as possible. Let's look at a few examples.
###### Example8.2.11
Multiply $\sqrt{8}\cdot\sqrt{54}\text{.}$
Explanation
We will simplify each radical first, and then multiply them together. We do not want to multiply $8\cdot54$ because we will end up with a larger number that is harder to factor.
\begin{align*} \sqrt{8}\cdot\sqrt{54}\amp=\sqrt{4\cdot2}\cdot\sqrt{9\cdot6}\\ \amp=2\sqrt{2}\cdot3\sqrt{6}\\ \amp=2\cdot3\sqrt{2\cdot6}\\ \amp=2\cdot3\sqrt{2\cdot2\cdot3}\\ \amp=6\cdot2\sqrt{3}\\ \amp=12\sqrt{3} \end{align*}
We could have multiplied $2\cdot6$ inside the radical to get $12$ and then factored $12$ into $4\cdot3\text{.}$ Whenever you find a pair of identical factors, this is a perfect square.
###### Example8.2.13
Multiply $\sqrt{\frac{6}{5}}\cdot\sqrt{\frac{3}{5}}\text{.}$
Explanation
\begin{align*} \sqrt{\frac{6}{5}}\cdot\sqrt{\frac{3}{5}}\amp=\sqrt{\frac{6}{5}\cdot\frac{3}{5}}\\ \amp=\sqrt{\frac{18}{25}}\\ \amp=\frac{\sqrt{18}}{\sqrt{25}}\\ \amp=\frac{\sqrt{9\cdot2}}{5}\\ \amp=\frac{3\sqrt{2}}{5} \end{align*}
### Subsection8.2.5Adding and Subtracting Square Root Expressions
We learned the Multiplication Property of Square Roots previously and applied this to multiplication of square roots, but we cannot apply this property to the operations of addition or subtraction. Here are two examples to demonstrate this:
\begin{align*} \sqrt{9+16}\amp\stackrel{?}{=}\sqrt{9}+\sqrt{16}\amp\sqrt{169-25}\amp\stackrel{?}{=}\sqrt{169}-\sqrt{25}\\ \sqrt{25}\amp\stackrel{?}{=}3+4\amp\sqrt{144}\amp\stackrel{?}{=}13-5\\ 5\amp\stackrel{\text{no}}{=}7\amp12\amp\stackrel{\text{no}}{=}8 \end{align*}
We do not get the same result if we separate the radicals, so we must complete any additions and subtractions inside the radical first.
To add and subtract radical expressions, we will need to recognize that we can only add and subtract like terms. In this case, we will call them like radicals. In fact, adding like radicals will work just like adding like terms
\begin{align*} x+x\amp=2x\\ \end{align*}
and
\begin{align*} \sqrt{5}+\sqrt{5}\amp=2\sqrt{5} \end{align*}
We can verify that the second equation is true by replacing $x$ with $\sqrt{5}$ in the second equation. Let's look at a few more examples.
###### Example8.2.14
Simplify $\sqrt{2}+\sqrt{8}\text{.}$
Explanation
\begin{align*} \sqrt{2}+\sqrt{8}\amp=\sqrt{2}+\sqrt{4\cdot2}\\ \amp=\sqrt{2}+2\sqrt{2}\\ \amp=3\sqrt{2} \end{align*}
To help understand $\sqrt{2}+2\sqrt{2}=3\sqrt{2}\text{,}$ think of $x+2x=3x$ or “a taco plus two tacos is three tacos.”
###### Example8.2.16
Simplify $\sqrt{2}+\sqrt{27}\text{.}$
Explanation
\begin{align*} \sqrt{2}+\sqrt{27}\amp=\sqrt{2}+\sqrt{9\cdot3}\\ \amp=\sqrt{2}+3\sqrt{3} \end{align*}
We cannot simplify the expression further because $\sqrt{2}$ and $\sqrt{3}$ are not like radicals.
###### Example8.2.17
Simplify $\sqrt{6}-\sqrt{18}\cdot\sqrt{12}\text{.}$
Explanation
In this example, we should multiply the latter two square roots first and then see if we have like radicals.
\begin{align*} \sqrt{6}-\sqrt{18}\cdot\sqrt{12}\amp=\sqrt{6}-\sqrt{9\cdot2}\cdot\sqrt{4\cdot3}\\ \amp=\sqrt{6}-3\sqrt{2}\cdot2\sqrt{3}\\ \amp=\sqrt{6}-3\cdot2\cdot\sqrt{2}\cdot\sqrt{3}\\ \amp=\sqrt{6}-6\sqrt{6}\\ \amp=-5\sqrt{6} \end{align*}
### Subsection8.2.6Rationalizing the Denominator
When simplifying square root expressions, we have seen that we need to make the radicand as small as possible. Another rule is that we do not leave any irrational numbers, such as $\sqrt{3}$ or $2\sqrt{5}\text{,}$ in the denominator of a fraction. In other words, we want the denominator to be rational. The process of dealing with such numbers in the denominator is called rationalizing the denominator.
Let's see how we can remove the square root symbol from the denominator in $\frac{1}{\sqrt{5}}\text{.}$ If we multiply a radical by itself, the result is the radicand, by Definition 8.2.2. As an example:
\begin{equation*} \sqrt{5}\cdot\sqrt{5}=5 \end{equation*}
To write $\frac{1}{\sqrt{5}}$ as an equivalent fraction, we must multiply both the numerator and denominator by the same number. If we multiply the numerator and denominator by $\sqrt{5}\text{,}$ we have:
\begin{align*} \frac{1}{\sqrt{5}}\amp=\frac{1}{\sqrt{5}} \multiplyright{\frac{\sqrt{5}}{\sqrt{5}}}\\ \amp=\frac{1\cdot\sqrt{5}}{\sqrt{5}\cdot\sqrt{5}}\\ \amp=\frac{\sqrt{5}}{5} \end{align*}
We can use a calculator to verify that $\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\text{.}$ They both are $0.4472\ldots\text{.}$ Let's look at a few more examples.
###### Example8.2.18
Rationalize the denominator in $\frac{6}{\sqrt{3}}\text{.}$
Explanation
We will rationalize this denominator by multiplying the numerator and denominator by $\sqrt{3}\text{:}$
\begin{align*} \frac{6}{\sqrt{3}}\amp=\frac{6}{\sqrt{3}}\multiplyright{\frac{\sqrt{3}}{\sqrt{3}}}\\ \amp=\frac{6\cdot\sqrt{3}}{\sqrt{3}\cdot\sqrt{3}}\\ \amp=\frac{6\sqrt{3}}{3}\\ \amp=2\sqrt{3} \end{align*}
Note that we reduced any fractions that are outside the radical.
###### Example8.2.20
Rationalize the denominator in $\sqrt{\frac{2}{7}}\text{.}$
Explanation
\begin{align*} \sqrt{\frac{2}{7}}\amp=\frac{\sqrt{2}}{\sqrt{7}}\\ \amp=\frac{\sqrt{2}}{\sqrt{7}}\multiplyright{\frac{\sqrt{7}}{\sqrt{7}}}\\ \amp=\frac{\sqrt{2}\cdot\sqrt{7}}{\sqrt{7}\cdot\sqrt{7}}\\ \amp=\frac{\sqrt{14}}{7} \end{align*}
### Subsection8.2.7More Complicated Square Root Operations
In Section 6.4, we learned how to multiply polynomials like $2(x+3)$ and $(x+2)(x+3)\text{.}$ All the methods we learned apply when we multiply square root expressions. We will look at a few examples done with different methods.
###### Example8.2.21
Multiply $\sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)\text{.}$
Explanation
We will use the distributive property to do this problem:
\begin{align*} \sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)\amp=\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{2}\\ \amp=\sqrt{15}-\sqrt{10} \end{align*}
###### Example8.2.22
Multiply $\left(\sqrt{6}+\sqrt{12}\right)\left(\sqrt{3}-\sqrt{2}\right)\text{.}$
Explanation
We will use the FOIL Method to do this problem:
\begin{align*} \left(\sqrt{6}+\sqrt{12}\right)\left(\sqrt{3}-\sqrt{2}\right)\amp=\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+\sqrt{12}\sqrt{3}-\sqrt{12}\sqrt{2}\\ \amp=\sqrt{18}-\sqrt{12}+\sqrt{36}-\sqrt{24}\\ \amp=3\sqrt{2}-2\sqrt{3}+6-2\sqrt{6} \end{align*}
When simplifying radicals it is useful to keep in mind that for any $x\ge 0\text{,}$
\begin{equation*} \sqrt{x}\cdot\sqrt{x}=x\text{.} \end{equation*}
###### Example8.2.23
Expand $\left(\sqrt{3}-\sqrt{2}\right)^2\text{.}$
Explanation
We will use the FOIL method to expand this expression:
\begin{align*} \left(\sqrt{3}-\sqrt{2}\right)^2\amp=\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)\\ \amp=\left(\sqrt{3}\right)^2-\sqrt{3}\sqrt{2}-\sqrt{2}\sqrt{3}+\left(\sqrt{2}\right)^2\\ \amp=3-\sqrt{6}-\sqrt{6}+2\\ \amp=5-2\sqrt{6} \end{align*}
###### Example8.2.24
Multiply $\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}+\sqrt{7}\right)\text{.}$
Explanation
We will again use the FOIL method to expand this expression, but will note that it is a special form $(a-b)(a+b)$ and will simplify to $a^2-b^2\text{:}$
\begin{align*} \left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}+\sqrt{7}\right)\amp=\left(\sqrt{5}\right)^2+\sqrt{5}\sqrt{7}-\sqrt{7}\sqrt{5}-\left(\sqrt{7}\right)^2\\ \amp=5+\sqrt{35}-\sqrt{35}-7\\ \amp=-2 \end{align*}
### Subsection8.2.8Exercises
###### 1
Which of the following are square numbers? There may be more than one correct answer.
• $16$
• $86$
• $40$
• $9$
• $132$
• $64$
###### 2
Which of the following are square numbers? There may be more than one correct answer.
• $53$
• $121$
• $25$
• $84$
• $93$
• $1$
###### 3
Evaluate the following.
1. $\displaystyle{ \sqrt{49} }$ =
2. $\displaystyle{ \sqrt{36} }$ =
3. $\displaystyle{ \sqrt{25} }$ =
###### 4
Evaluate the following.
1. $\displaystyle{ \sqrt{64} }$ =
2. $\displaystyle{ \sqrt{4} }$ =
3. $\displaystyle{ \sqrt{121} }$ =
###### 5
Evaluate the following.
1. $\displaystyle{ \sqrt{{{\frac{81}{121}}}} }$ =
2. $\displaystyle{ \sqrt{{-{\frac{9}{100}}}} }$ =
###### 6
Evaluate the following.
1. $\displaystyle{ \sqrt{{{\frac{121}{36}}}} }$ =
2. $\displaystyle{ \sqrt{{-{\frac{64}{25}}}} }$ =
###### 7
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{144} }$ =
2. $\displaystyle{ \sqrt{1.44} }$ =
3. $\displaystyle{ \sqrt{14400} }$ =
###### 8
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{4} }$ =
2. $\displaystyle{ \sqrt{0.04} }$ =
3. $\displaystyle{ \sqrt{400} }$ =
###### 9
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{9} }$ =
2. $\displaystyle{ \sqrt{900} }$ =
3. $\displaystyle{ \sqrt{90000} }$ =
###### 10
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{16} }$ =
2. $\displaystyle{ \sqrt{1600} }$ =
3. $\displaystyle{ \sqrt{160000} }$ =
###### 11
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{25} }$ =
2. $\displaystyle{ \sqrt{0.25} }$ =
3. $\displaystyle{ \sqrt{0.0025} }$ =
###### 12
Evaluate the following.
Do not use a calculator.
1. $\displaystyle{ \sqrt{49} }$ =
2. $\displaystyle{ \sqrt{0.49} }$ =
3. $\displaystyle{ \sqrt{0.0049} }$ =
###### 13
Without using a calculator, estimate the value of $\sqrt{65}\text{:}$
• 7.94
• 8.94
• 8.06
• 7.06
###### 14
Without using a calculator, estimate the value of $\sqrt{78}\text{:}$
• 9.83
• 8.83
• 9.17
• 8.17
###### 15
Evaluate the following.
$\displaystyle{\sqrt{{{\frac{100}{121}}}}={}}$.
###### 16
Evaluate the following.
$\displaystyle{\sqrt{{{\frac{121}{144}}}}={}}$.
###### 17
Evaluate the following.
$-\sqrt{4}={}$.
###### 18
Evaluate the following.
$-\sqrt{9}={}$.
###### 19
Evaluate the following.
$\sqrt{-25}=$.
###### 20
Evaluate the following.
$\sqrt{-36}=$.
###### 21
Evaluate the following.
$\displaystyle{\sqrt{-{{\frac{36}{49}}}}={}}$.
###### 22
Evaluate the following.
$\displaystyle{\sqrt{-{{\frac{49}{144}}}}={}}$.
###### 23
Evaluate the following.
$\displaystyle{-\sqrt{{{\frac{81}{100}}}}={}}$.
###### 24
Evaluate the following.
$\displaystyle{-\sqrt{{{\frac{100}{121}}}}={}}$.
###### 25
Evaluate the following.
1. $\displaystyle{\sqrt{169}-\sqrt{144}=}$
2. $\displaystyle{\sqrt{169-144}=}$
###### 26
Evaluate the following.
1. $\displaystyle{\sqrt{25}-\sqrt{9}=}$
2. $\displaystyle{\sqrt{25-9}=}$
###### 27
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ \frac{{\sqrt{8}}}{{\sqrt{2}}} =}$
###### 28
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ \frac{{\sqrt{54}}}{{\sqrt{6}}} =}$
###### 29
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ \frac{{\sqrt{4}}}{{\sqrt{36}}} =}$
###### 30
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ \frac{{\sqrt{2}}}{{\sqrt{32}}} =}$
###### 31
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{343}} = }$
###### 32
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{90}} = }$
###### 33
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{360}} = }$
###### 34
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{2156}} = }$
###### 35
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{231}} = }$
###### 36
Simplify the radical expression or state that it is not a real number.
$\displaystyle{ {\sqrt{70}} = }$
###### 37
Simplify the expression.
$4\sqrt{7} \cdot 7\sqrt{5}=$
###### 38
Simplify the expression.
$5\sqrt{7} \cdot 4\sqrt{2}=$
###### 39
Simplify the expression.
$6\sqrt{13} \cdot 2\sqrt{{25}} =$
###### 40
Simplify the expression.
$6\sqrt{3} \cdot 7\sqrt{{121}} =$
###### 41
Simplify the expression.
$\displaystyle{4\sqrt{6} \cdot 3\sqrt{12}=}$
###### 42
Simplify the expression.
$\displaystyle{5\sqrt{3} \cdot 5\sqrt{150}=}$
###### 43
Simplify the expression.
$\displaystyle{ {\sqrt{2}} \cdot {4\sqrt{50}} = }$
###### 44
Simplify the expression.
$\displaystyle{ {\sqrt{2}} \cdot {6\sqrt{8}} = }$
###### 45
Simplify the expression.
$\displaystyle{ \sqrt{\frac{7}{3}} \cdot \sqrt{\frac{5}{3}} =}$
###### 46
Simplify the expression.
$\displaystyle{ \sqrt{\frac{3}{4}} \cdot \sqrt{\frac{1}{4}} =}$
###### 47
Simplify the expression.
$\displaystyle{ {\sqrt{\frac{6}{19}}} \cdot {\sqrt{\frac{3}{19}}} =}$
###### 48
Simplify the expression.
$\displaystyle{ {\sqrt{\frac{28}{11}}} \cdot {\sqrt{\frac{4}{11}}} =}$
###### 49
Simplify the expression.
$\displaystyle{{16\sqrt{10}} - {17\sqrt{10}} =}$
###### 50
Simplify the expression.
$\displaystyle{{17\sqrt{3}} - {18\sqrt{3}} =}$
###### 51
Simplify the expression.
$\displaystyle{{19\sqrt{2}} - {19\sqrt{2}} + {14\sqrt{2}} =}$
###### 52
Simplify the expression.
$\displaystyle{{20\sqrt{23}} - {15\sqrt{23}} + {19\sqrt{23}} =}$
###### 53
Simplify the expression.
$\displaystyle{{\sqrt{8}} + {\sqrt{18}} =}$
###### 54
Simplify the expression.
$\displaystyle{{\sqrt{50}} + {\sqrt{18}} =}$
###### 55
Simplify the expression.
$\displaystyle{{\sqrt{48}} - {\sqrt{12}} =}$
###### 56
Simplify the expression.
$\displaystyle{{\sqrt{12}} - {\sqrt{75}} =}$
###### 57
Simplify the expression.
$\displaystyle{{\sqrt{180}} + {\sqrt{125}} + {\sqrt{27}} + {\sqrt{75}} =}$
###### 58
Simplify the expression.
$\displaystyle{{\sqrt{54}} + {\sqrt{24}} + {\sqrt{108}} + {\sqrt{12}} =}$
###### 59
Simplify the expression.
$\displaystyle{{\sqrt{294}} - {\sqrt{54}} - {\sqrt{72}} - {\sqrt{50}} =}$
###### 60
Simplify the expression.
$\displaystyle{{\sqrt{175}} - {\sqrt{63}} - {\sqrt{180}} - {\sqrt{20}} =}$
###### 61
Evaluate the following.
$\displaystyle{\frac{9}{\sqrt{{25}}}}$ = .
###### 62
Evaluate the following.
$\displaystyle{\frac{1}{\sqrt{{4}}}}$ = .
###### 63
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{1}{\sqrt{2}} = }$
###### 64
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{1}{\sqrt{3}} = }$
###### 65
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{4}{\sqrt{5}} = }$
###### 66
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{7}{\sqrt{5}} = }$
###### 67
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{7}{8\sqrt{6}} = }$
###### 68
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{10}{3\sqrt{7}} = }$
###### 69
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{12}{\sqrt{30}} = }$
###### 70
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{15}{\sqrt{35}} = }$
###### 71
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{10}{\sqrt{2}} = }$
###### 72
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{6}{\sqrt{2}} = }$
###### 73
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{1}{{\sqrt{18}}} = }$
###### 74
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{1}{{\sqrt{63}}} = }$
###### 75
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{8}{{\sqrt{180}}} = }$
###### 76
Rationalize the denominator and simplify the expression.
$\displaystyle{ \frac{10}{{\sqrt{252}}} = }$
###### 77
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{13}{64}} = }$
###### 78
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{7}{81}} = }$
###### 79
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{9}{2}} = }$
###### 80
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{81}{2}} = }$
###### 81
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{3}{7}} = }$
###### 82
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{5}{2}} = }$
###### 83
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{54}{7}} = }$
###### 84
Rationalize the denominator and simplify the expression.
$\displaystyle{ \sqrt{\frac{48}{11}} = }$
###### 85
Expand and simplify the expression.
$\displaystyle{{\sqrt{7}} \left({\sqrt{3}} + {\sqrt{13}}\right) =}$
###### 86
Expand and simplify the expression.
$\displaystyle{{\sqrt{7}} \left({\sqrt{19}} + {\sqrt{5}}\right) =}$
###### 87
Expand and simplify the expression.
$\displaystyle{\left(5 + {\sqrt{13}}\right)\left(6 + {\sqrt{13}}\right) =}$
###### 88
Expand and simplify the expression.
$\displaystyle{\left(10 + {\sqrt{13}}\right)\left(9 + {\sqrt{13}}\right) =}$
###### 89
Expand and simplify the expression.
$\displaystyle{\left(8 - {\sqrt{5}}\right)\left(7 - 3 {\sqrt{5}}\right) =}$
###### 90
Expand and simplify the expression.
$\displaystyle{\left(5 - {\sqrt{5}}\right)\left(6 - 4 {\sqrt{5}}\right) =}$
###### 91
Expand and simplify the expression.
$\displaystyle{ \left(3+\sqrt{7}\right)^2 =}$
###### 92
Expand and simplify the expression.
$\displaystyle{ \left(4+\sqrt{5}\right)^2 =}$
###### 93
Expand and simplify the expression.
$\displaystyle{ \left(\sqrt{3}-5\right)^2 =}$
###### 94
Expand and simplify the expression.
$\displaystyle{ \left(\sqrt{7}-6\right)^2 =}$
###### 95
Expand and simplify the expression.
$\displaystyle{ \left(\sqrt{35} - \sqrt{5}\right)^2 =}$
###### 96
Expand and simplify the expression.
$\displaystyle{ \left(\sqrt{14} + \sqrt{7}\right)^2 =}$
###### 97
Expand and simplify the expression.
$\displaystyle{\left(9 - 5 {\sqrt{7}}\right)^2 =}$
###### 98
Expand and simplify the expression.
$\displaystyle{\left(6 - 3 {\sqrt{5}}\right)^2 =}$
###### 99
Expand and simplify the expression.
$\displaystyle{\left(4 - {\sqrt{5}}\right)\left(4 + {\sqrt{5}}\right) =}$
###### 100
Expand and simplify the expression.
$\displaystyle{\left(9 - {\sqrt{6}}\right)\left(9 + {\sqrt{6}}\right) =}$
###### 101
Expand and simplify the expression.
$\displaystyle{\left({\sqrt{6}} + {\sqrt{7}}\right)\left({\sqrt{6}} - {\sqrt{7}}\right) =}$
###### 102
Expand and simplify the expression.
$\displaystyle{\left({\sqrt{7}} + {\sqrt{5}}\right)\left({\sqrt{7}} - {\sqrt{5}}\right) =}$
###### 103
Expand and simplify the expression.
$\displaystyle{\left({5\sqrt{5}} + {6\sqrt{7}}\right)\left({5\sqrt{5}} - {6\sqrt{7}}\right) =}$
###### 104
Expand and simplify the expression.
$\displaystyle{\left({6\sqrt{6}} + {4\sqrt{5}}\right)\left({6\sqrt{6}} - {4\sqrt{5}}\right) =}$ | 2018-10-20 20:17:13 | {"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": 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": 0.9565126895904541, "perplexity": 1585.301721167286}, "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-2018-43/segments/1539583513384.95/warc/CC-MAIN-20181020184612-20181020210112-00301.warc.gz"} |
http://mathoverflow.net/feeds/question/58113 | Kronecker Approximation theorem and Fibonacci numbers - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:50:20Z http://mathoverflow.net/feeds/question/58113 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58113/kronecker-approximation-theorem-and-fibonacci-numbers Kronecker Approximation theorem and Fibonacci numbers Ostap Chervak 2011-03-10T22:53:24Z 2011-06-14T19:33:53Z <p>There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.</p> <p>Recently I found that the same result is true if we replace $\alpha n$ by $\alpha n^2$ or any polinomial p such that $p(0)=0$.</p> <p>Could this result be generalised to other functions? Particularly I'm curious about sequences $\alpha 2^n$ and $\alpha F_n$ where by $F_n$ I denote n-th Fibonacci number.</p> <p>Does anyone know anything about it?</p> http://mathoverflow.net/questions/58113/kronecker-approximation-theorem-and-fibonacci-numbers/58116#58116 Answer by Gerry Myerson for Kronecker Approximation theorem and Fibonacci numbers Gerry Myerson 2011-03-10T23:31:30Z 2011-03-10T23:31:30Z <p>$\alpha2^n$ is clearly not going to work, e.g., for $\alpha=1/3$. One place to look is the Kuipers and Niederreiter book on uniform distribution of sequences, although uniform distribution is a bit of overkill for the question you are asking about. </p> http://mathoverflow.net/questions/58113/kronecker-approximation-theorem-and-fibonacci-numbers/58119#58119 Answer by Nikita Sidorov for Kronecker Approximation theorem and Fibonacci numbers Nikita Sidorov 2011-03-11T00:35:57Z 2011-03-11T01:13:02Z <p>Well, as Gerry has pointed out, this is certainly not true for all $\alpha$. On the other hand, this is true for a.e. $\alpha$. More precisely, the sequence $2^n\alpha$ is equidistributed mod 1 for a.e. $\alpha$.</p> <p>I believe this result is due to H. Weyl and can be found in Cornfeld, Fomin and Sinai `Ergodic Theory'. (I don't have it with me.) </p> <p>The same must be true for the Fibonacci sequence, I'm sure. </p> <p>So, what you probably need is for this to be true for all $\alpha$, except some small (countable?) set. After all, $\|2^n\alpha\|<\varepsilon$ is indeed much weaker than equidistibution. </p> <p><strong>Update.</strong> Come to think about it, the answer is as follows: let $$\alpha = \sum_{k=1}^\infty a_k2^{-k}$$ be the binary expansion of $\alpha$. Then the sequence $2^n\alpha\bmod 1$ gets arbitrarily close to 0 if and only if the sequence $(a_k)$ has unbounded strings of 0s. In particular, any rational $\alpha$ is out of the picture, apart from the binary rationals, of course.</p> <p>All in all, your set of $\alpha$'s is indeed of full measure, but the exceptional set is of Hausdorff dimension 1, i.e., pretty big.</p> <p>For the Fibonacci sequence you'll need to replace binary expansion with the $\beta$-expansion, where $\beta=(\sqrt5-1/)2$, with the same conclusion. </p> http://mathoverflow.net/questions/58113/kronecker-approximation-theorem-and-fibonacci-numbers/67797#67797 Answer by Asaf for Kronecker Approximation theorem and Fibonacci numbers Asaf 2011-06-14T19:33:53Z 2011-06-14T19:33:53Z <p>You've received two good answers, but I'll elaborate a bit. Usually equidistribution on the torus (or more general, compact groups) wrt the Haar measure is achieved by computing the Weyl sums and showing that there are some cancellations.</p> <p>The question you are referring to is studied in the area of so-called "sparse equidistribution" (although it is more of sparse density if you would like).</p> <p>The problem with the harmonic-analytic approach is by summing (or integrating) over very sparse part of your period. It is usually not stright forward to bound such exponential sums. For example, Vinogradov proved for example that ${p_{n}x}$ is equidistributed mod 1 for all irrational x, to bound the Weyl sums, he used sieves with what is called now Vinogradov sums, and a result about the odd Goldbach conjecture.</p> <p>Now if you are interested in a metric result (i.e. a.e. x), then it is a very classical result that for every increasing unbounded sequence \${a_{n}}$ and for a.e. x, one have that ${a_{n}x}$ is equi. mod 1, this is done by taking the Weyl sum, computing its L^2 norm, and then sub. limit and integration by the DCT.</p> <p>Now the question if such a result follows for every x is very subtle, and not always amenable to harmonic-analytic approach, and the current state of the art actually lies in the ergodic approaches.</p> <p>If you have a sequence which is contained inside a geometric progression, then there exists x's for which ${a_{n}x}$ is not equi. more generally, for ${q^{n}}$ say, you can find x's whose orbit closure is with any Hausdorff dimension you want (the reason here that as a dynamical system, this is isomorphic to Bernoulli shift on $q$ letters). More generally, a result due to Boshernitzan says that if you have a lacunary sequence (the limit of the ratios of consecutive elements is larger than 1), then the Hausdorff dimension of the set of exceptional x's (such that ${a_{n}x}$ is not dense/equidistributed) is 1. On the contrary, Boshernitzan shown that if the sequence is non-lacunary (the ratio tends to 1, you should think about it as having sub-exp. growth), then the Hausdorff dimension of the set for which ${a_{n}x}$ is equi. is 1. There were even some old results due to Erdos from the 1950's about it (he talked about convolution of Bernoulli measures, which can be interpreted in this sense as well).</p> <p>A very peculiar discovery by Furstenberg (67) shown that if you have a non-lacunary semigroup, say ${2^{n}3^{m}}$ then for every irrational x you get that ${2^{n}3^{m}x}$ is dense mod 1 (certainly not equidistributed). This result is very interesting, because you have density for every x. Moreover, recently, Bourgain-Lindenstrauss-P. Michel and Venkatesh proved an effective version of that theorem (meaning that you fix some epsilon, you can estimate how far you need to go in-order to find an element which is epsilon-close to an integer). An even recent work (by myself, still preprint), generalizing the Bourgain-Lindenstrauss paper, and I shown that sets like ${2^{n}3^{3^{m}}3^{3^{k^2}}x}$ are dense for every x.</p> <p>About Fibonacci sequences, it follows from my work (based on other work of D. Meiri whith Yuval Peres and Elon Lindnstrauss), that you can prove density of sequences such as ${2^{n}3^{3^{m}}F_{k}x}$ for every irrational x. For the general Fibonachi sequences, you are basically in the lacunary case, which Boshernitzan already covered in certain sense.</p> | 2013-06-19 03:50:21 | {"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": 0.9455900192260742, "perplexity": 489.1002084091554}, "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-2013-20/segments/1368707440693/warc/CC-MAIN-20130516123040-00082-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.elibm.org/article/10011992 | ## Real Trace Expansions
##### Doc. Math. 24, 2159-2202 (2019)
DOI: 10.25537/dm.2019v24.2159-2202
### Summary
In this paper, we investigate trace expansions of operators of the form $A\eta(t\mathcal{L})$ where $\eta:\mathbb{R}\rightarrow\mathbb{C}$ is a Schwartz function, $A$ and $\mathcal L$ are classical pseudo-differential operators on a compact manifold $M$ with $\mathcal L$ elliptic. In particular, we show that, under certain hypotheses, this trace admits an expansion in powers of $t\rightarrow 0^+$. We also relate the constant coefficient to the non-commutative residue and the canonical trace of $A$. Our main tool is the continuous inclusion of the functional calculus of $\mathcal{L}$ into the pseudo-differential calculus whose proof relies on the Helffer-Sjöstrand formula.
58J40, 58J42
### Keywords/Phrases
pseudodifferential operators on manifolds, non-commutative residues, canonical trace
### References
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### Affiliation
Fischer, Véronique
Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, UK | 2020-08-13 12:12:00 | {"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": 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": 0.7201040983200073, "perplexity": 4104.000325330723}, "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-2020-34/segments/1596439738982.70/warc/CC-MAIN-20200813103121-20200813133121-00221.warc.gz"} |
https://studyqas.com/in-which-number-is-the-value-of-the-3-one-tenth-the-value/ | # In which number is the value of the 3 one tenth the value of the 3 in 712.83?a.28.13 b.157.038c.63.81 d.80.345
In which number is the value of the 3 one tenth the value of the 3 in 712.83? a.28.13
b.157.038
c.63.81
d.80.345
## This Post Has 4 Comments
1. Expert says:
Ummm ok. for informing us
$Got a second account cause my old one annarose61 was deleted$
2. Expert says:
the image is given below
step-by-step explanation:
since the options are not given so we draw it from the equation
the given equation is
y = (x-1) (x+4)
y = y² + 3x - 4
the above equation is a quadratic equation which will have two roots. if we draw the graph for it, it will look like this
3. Expert says:
the speed is $135$miles per hour > is false
step-by-step explanation:
we know that
a relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form $y/x=k$ or $y=kx$
in a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin
the graph shown in the figure represent a direct variation
the slope of the line represent the speed
$y/x=k$
for the point $(1.5,90)$
substitute the values of x and y
$k=90/1.5=60\ miles/hour$
the linear equation is equal to
$y=60x$
statements
case a) it takes $2$ hours to go $120$miles
the statement is true
because
substitute
for $x=2$
find the value of y in the linear equation
$y=60(2)=120\ miles$ > is correct
case b) the speed is $135$miles per hour
the statement is false
because the speed is $60\ miles/hour$
case c) the unit rate for the trip is $60\ miles/hour$
the statement is true
because the speed is equal to the slope of the linear equation
case d) it took $20$ minutes to go the first $20$miles
the statement is true
because
substitute
for $x=20\ minutes=(1/3)\ hour$
find the value of y in the linear equation
$y=60(1/3)=20\ miles$ > is correct
4. Expert says:
What's the denominator | 2023-02-05 19:52:22 | {"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": 0.7471466660499573, "perplexity": 913.7935991606743}, "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/1674764500288.69/warc/CC-MAIN-20230205193202-20230205223202-00204.warc.gz"} |
https://stats.stackexchange.com/questions/74627/categorical-response-variable-prediction/74657 | # Categorical response variable prediction
I have the following kind of data (coded in R):
v.a = c('cat', 'dog', 'dog', 'goat', 'cat', 'goat', 'dog', 'dog')
v.b = c(1, 2, 1, 2, 1, 2, 1, 2)
v.c = c('blue', 'red', 'blue', 'red', 'red', 'blue', 'yellow', 'yellow')
set.seed(12)
v.d = rnorm(8)
aov(v.a ~ v.b + v.c + v.d) # Error
I would like to know if the value of v.b or the value of v.c has any ability to predict the value of v.a. I would run an ANOVA (as shown above) but I think it does not make any sense since my response variable is not ordinal (it is categorical). What should I do?
• Learn about multinomial logit. Here are two, free online books by Kenneth Train (elsa.berkeley.edu/books/choice.html and elsa.berkeley.edu/books/choice2.html). I think those are graduate level books. Or just google around for "multinomial logit."
– Bill
Nov 5 '13 at 13:59
• @Bill this looks like a good start on an answer. Please consider expanding a little bit on what multinomial logit models are, and posting it as an answer. Nov 5 '13 at 15:03
You could use ANY classifier. Including Linear Discriminants, multinomial logit as Bill pointed out, Support Vector Machines, Neural Nets, CART, random forest, C5 trees, there are a world of different models that can help you predict $$v.a$$ using $$v.b$$ and $$v.c$$. Here is an example using the R implementation of random forest:
# packages
library(randomForest)
#variables
v.a= c('cat','dog','dog','goat','cat','goat','dog','dog')
v.b= c(1,2,1,2,1,2,1,2)
v.c= c('blue', 'red', 'blue', 'red', 'red', 'blue', 'yellow', 'yellow')
# model fit
# note that you must turn the ordinal variables into factor or R wont use
# them properly
model <- randomForest(y=as.factor(v.a),x=cbind(v.b,as.factor(v.c)),ntree=10)
#plot of model accuracy by class
plot(model)
# model confusion matrix
modelconfusion Clearly these variables don't show a strong relation. • @JEquihua Could you please tell me a bit more about what is a "tree" and what is the meaning of the output(confusion matrix and the plot). Thanks a lot! Nov 8 '13 at 8:58 • I will. I'm very busy, give me a little time. @Remi.b Nov 19 '13 at 15:59 This is a more a partial practical answer, but it works for me to do some exercises before getting deeply into theory. This ats.ucla.edu link is a reference that might help beggining to understand about multinomial logistic regression (as pointed out by Bill), in a more practical way. It presents reproducible code to understand function multinom from nmet package in R and also gives a briefing about outputs interpretation. Consider this code: va = c('cat','dog','dog','goat','cat','goat','dog','dog') # cat will be the outcome baseline vb = c(1,2,1,2,1,2,1,2) vc = c('blue','red','blue','red','red','blue','yellow','yellow') # blue will be the vc predictor baseline set.seed(12) vd = round(rnorm(8),2) data = data.frame(cbind(va,vb,vc,vd)) library(nnet) fit <- multinom(va ~ as.numeric(vb) + vc + as.numeric(vd), data=data) # weights: 18 (10 variable) initial value 8.788898 iter 10 value 0.213098 iter 20 value 0.000278 final value 0.000070 converged fit Call: multinom(formula = va ~ as.numeric(vb) + vc + as.numeric(vd), data = data) Coefficients: (Intercept) as.numeric(vb) vcred vcyellow as.numeric(vd) dog -1.044866 120.3495 -6.705314 77.41661 -21.97069 goat 47.493155 126.4840 49.856414 -41.46955 -47.72585 Residual Deviance: 0.0001656705 AIC: 20.00017 This is how you can interpret the log-linear fitted multinomial logistic model: \begin{align} \ln\left(\frac{P(va={\rm cat})}{P(va={\rm dog})}\right) &= b_{10} + b_{11}vb + b_{12}(vc={\rm red}) + b_{13}(vc={\rm yellow}) + b_{14}vd \\ &\ \\ \ln\left(\frac{P(va={\rm cat})}{P(va={\rm goat})}\right) &= b_{20} + b_{21}vb + b_{22}(vc={\rm red}) + b_{23}(vc={\rm yellow}) + b_{24}vd \end{align} Here is an excerpt about how the model parameters can be interpreted: • A one-unit increase in the variable vd is associated with the decrease in the log odds of being "dog" vs. "cat" in the amount of 21.97069 (b_{14}$). the same logic for the second line but, considering "goat" vs. "cat" with ($b_{24}$=-47.72585). • The log odds of being "dog" vs. "cat" will increase by 6.705314 if moving from vc="blue" to vc="red"($b_{12}\$).
.....
There is much more in the article, but I thought this part to be the core.
Reference:
R Data Analysis Examples: Multinomial Logistic Regression. UCLA: Statistical Consulting Group.
from http://www.ats.ucla.edu/stat/r/dae/mlogit.htm (accessed November 05, 2013). | 2021-09-26 23:08:26 | {"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": 3, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9002543091773987, "perplexity": 4726.79687879416}, "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-2021-39/segments/1631780057973.90/warc/CC-MAIN-20210926205414-20210926235414-00128.warc.gz"} |
https://docs.micetoolkit.com/nodes.image.mri.t2t2_map.html | # T2/T2* Map
Class: NodeT2Map
Calculates a T2 or T2* map from spin echo or gradient echo data with multiple echo times. At least two images are required, however, more images are advised if a larger range of T2/T2* values are expected. Echo times (TE) are detected automatically and it is required that other parameters affecting the contrast must be the same for all images. An optional mask can also be provided to limit the calculations to a region.
Produces a T2 / T2 * map[ms], and S0 which contains all weighting that is independent of T2/T2*. The parameter values are obtained by fitting the data to the signal equation
$$S = S_0\exp(-T_E/T_2)$$
in which T2 also can be T2*. Two algorithms can be applied. Either a nonlinear algorithm that performs a nonlinear least squares fit to the data, or a linear least squares fit which fits the linear model that results from taking the logarithm of the equation above.
## Inputs
#### Echo Series
Input image series with different echo times.
Type: Image4DFloat, Required, Multiple
Type: Image4DBool, Optional, Single
## Outputs
#### T2
Resulting T2 map.
Type: Image4DFloat
## Settings
#### Method Selection
Use the linearized or the nonlinear algorithm.
Values: Linear, NonLinear
## References
1. P.A. Boulby and F. Rugg-Gunn, “T2: the Transverse Relaxation Time,” in Quantitative MRI of the brain: Measuring changes caused by disease, P. S. Tofts, Ed. Wiley, 2003, pp. 143–201.
Keywords: T2, T2*, T2star, star, transverse, relaxation, parameter, map, linear, nonlinear | 2020-06-05 06:27:38 | {"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": 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": 0.4026956558227539, "perplexity": 4950.583295348365}, "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-2020-24/segments/1590348493151.92/warc/CC-MAIN-20200605045722-20200605075722-00383.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/115881-cyclic-group-generators-print.html | # Cyclic Group Generators
• November 21st 2009, 05:52 AM
tntcoda
Cyclic Group Generators
Hi,
Dealing with integer multiplicative groups and given this:
Say I take the group ${Z_{3^2}}^*$ so thats $p^n$ as mentioned above, which makes me think any element should be a generator of the group except the identity element.
So the group contains {1, 2, 4, 5, 7, 8}
Now say I try to generate the group using element 4 (modulo 9)
$4\cdot4 = 16 = 7$
$7\cdot4 = 28 = 1$
$1\cdot4 = 4$
Subgroup of order 3 (fits in with lagrange rule but not equal to G).
What am I doing wrong that gives me a non-cyclic group? (Im very new to group theory and not great with maths generally), am i even using the generator correctly?
I'm basically just trying to understand the p^n rule for odd-primes, and every element of the group being a generator.
Thanks for any help
• November 21st 2009, 08:10 AM
tonio
Quote:
Originally Posted by tntcoda
Hi,
Dealing with integer multiplicative groups and given this:
Say I take the group ${Z_{3^2}}^*$ so thats $p^n$ as mentioned above, which makes me think any element should be a generator of the group except the identity element.
So the group contains {1, 2, 4, 5, 7, 8}
Now say I try to generate the group using element 4 (modulo 9)
$4\cdot4 = 16 = 7$
$7\cdot4 = 28 = 1$
$1\cdot4 = 4$
Subgroup of order 3 (fits in with lagrange rule but not equal to G).
What am I doing wrong that gives me a non-cyclic group? (Im very new to group theory and not great with maths generally), am i even using the generator correctly?
I'm basically just trying to understand the p^n rule for odd-primes, and every element of the group being a generator.
Thanks for any help
First, what is that Shanks and Sloane quotes you give above? Second, what is $M_n$ ? Third, I think you're making a big mix of several very different things and I'm no sure how you're dealing with this stuff if, as you say, you're "new at group theory"...!?!
You take $\mathbb{Z}_{3^2}^{*}=$ the group of units in the ring $\mathbb{Z}_{32}$ of (integer) residues modulo 32 (and somehow I feel you'd rather have to deal with $\mathbb{F}_{3^2}^{*}=$ the multiplicative group of the FIELD $\mathbb{F}_{3^2}$ with $3^2=9$ elements...) , but whatever: we have a nice theorem in group theory that says that the units group in $\mathbb{Z}_{3^2}^{*}=\{1,2,4,5,7,8\}$ is cyclic of order 6, but NOT all its non-unit elements are generators since this is not a group of order a prime: we have $\phi(6)=2$ generators here: $2,5$ , and 4 cannot be a generator since $4=2^2$ and its power $2$ is not coprime with 6...For example, $5 = 2^5$ as 5 is coprime with 6 so 5 is a generator.
I think it'd be a good idea if you'd explain what is this about, explain the background and the symbols. Perhaps that way there's a chance somebody will be able to explain you something.
Tonio
• November 21st 2009, 08:29 AM
tntcoda
The context here is cryptography working in the multiplicative group of integers mod n. Im really just trying to get a basic grasp of group theory so I can understand the cryptography more.
I think it now makes sense why ${Z}_{3^2}^{*}$ isn't cyclic, because it doesn't have a prime order.
I was somewhat confused though because this page says that the group should be cyclic if the modulus is of the form $p^n$ where p is an odd prime, i still don't really understand why that fails to work with a modulus of 3^2?
If i can explain more on my context and where this is all coming from, part of a cryptosystem im looking at has this step:
Quote:
Alice generates an efficient description of a cyclic group G of order q with two distinct, random generators g1,g2.
So, I came to the conclusion that if a group is going to have multiple generators it a) needs to be cyclic and b) needs to have a prime order. Does that sound correct?
From here I went to trying the group ${Z}_{3^2}^{*}$ hoping the p^n (powers of odd primes) rule would work, but as you said it isnt of prime order, so every element wont be a generator.
Does that help explain where my mind is at? Thanks for any advice
• November 21st 2009, 08:56 AM
tonio
Quote:
Originally Posted by tntcoda
The context here is cryptography working in the multiplicative group of integers mod n. Im really just trying to get a basic grasp of group theory so I can understand the cryptography more.
I think it now makes sense why ${Z}_{3^2}^{*}$ isn't cyclic, because it doesn't have a prime order.
I was somewhat confused though because this page says that the group should be cyclic if the modulus is of the form $p^n$ where p is an odd prime, i still don't really understand why that fails to work with a modulus of 3^2?
As previously said, you're confusing stuff here: first, why did you add that * to $\mathbb{Z}_{3^2}$?? It doesn't appear at all in Wolfram's page and mathematicians understand that you're talking of the multiplicative group of units in the commutative finite ring $\mathbb{Z}_{3^2}$ of residues modulo $3^2=9$, which is of order 6. The additive group of this ring, which is what Wolfram's talking about and denoting by $M_n$ , the hell knows why, IS cyclic of order 9, but it is ADDITIVE, not multiplicative!
If i can explain more on my context and where this is all coming from, part of a cryptosystem im looking at has this step:
So, I came to the conclusion that if a group is going to have multiple generators it a) needs to be cyclic and b) needs to have a prime order. Does that sound correct?
Not even close, I'm afraid to say. Plenty of groups have multiple generators and are not cyclic and, thus, neither of prime order. Take notice that there are cyclic groups of ANY finite order and also one (up to isomorphism) infinite cyclic group.
From here I went to trying the group ${Z}_{3^2}^{*}$ hoping the p^n (powers of odd primes) rule would work, but as you said it isnt of prime order, so every element wont be a generator.
Does that help explain where my mind is at? Thanks for any advice
I must say that it is strange seeing someone trying to cope with cryptography without having a hefty background in abstract algebra (at least a very good first theory group course), some serious number theory and, if you're trying to reach far, some rather advanced cryptographic systems use elliptic functions stuff and etc.
I'm afraid that it will be a very tough task for you to succeed in this if you don't worry to fill up those gaps in your mathematical education, and worse: it may become pretty disappointing and frustrating to deal with such a thing without understanding much what's going on.
Tonio
• November 21st 2009, 08:59 AM
tntcoda
Ok thanks very much, I will do some serious work patching up my maths background. | 2014-08-27 10:14:05 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 32, "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": 0.7185255885124207, "perplexity": 414.70406435810315}, "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-2014-35/segments/1408500828050.28/warc/CC-MAIN-20140820021348-00187-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/jess/RABI_RANJAN_TRIPATHY | • RABI RANJAN TRIPATHY
Articles written in Journal of Earth System Science
• Evaluation of the impact of high-resolution winds on the coastal waves
This study discusses the impact of high-resolution winds on the coastal waves and analyses the effectiveness of the high-resolution winds in recreating the fine-scale features along the coastal regions during the pre-monsoon season (March–May). The influence of the diurnal variation of winds on waves is studied for the Tamil Nadu coastal region using wind fields from weather research and forecast (WRF) (3 km) and European Centre for Medium-Range Weather Forecasts (ECMWF) (27.5 km). The improvement in the coastal forecast is then quantified with wave rider buoy observations. The high-resolution wind fields simulated fine-scale features like land–sea breeze events and showed good agreement with observation results. The error in the wave height and period is reduced by 8% and 46%, respectively, with the use of high-resolution forcing winds WRF over ECMWF, although the overestimation of wave energy on high frequencies due to overestimated WRF winds remains as a challenge in forecasting. The analysis also shows the importance of accurate wave forecast during a short-duration sudden wind ($\sim$12 m/s) occurrence in southern Tamil Nadu near Rameswaram during the pre-monsoon period. Low pressure forms over Tamil Nadu due to the land surface heating, resulting in a sudden increase of winds. High winds and steep waves which cause damage to the property of the coastal community near Rameswaram also were well simulated in the high-resolution forecast system with WRF winds.
• Journal of Earth System Science
Volume 132, 2023
All articles
Continuous Article Publishing mode
• Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | 2023-03-20 11:48:53 | {"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": 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": 0.5239992737770081, "perplexity": 6157.389108535364}, "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-2023-14/segments/1679296943483.86/warc/CC-MAIN-20230320114206-20230320144206-00785.warc.gz"} |
https://tolstoy.newcastle.edu.au/R/help/05/03/0015.html | # Re: [R] Sweave and \input or \include LaTeX commands
From: Gregor GORJANC <gregor.gorjanc_at_bfro.uni-lj.si>
Date: Tue 01 Mar 2005 - 17:53:47 EST
Hi!
What is wrong if there would be the same command? Recall my example from previous posts and at the end of this mail. If I have a file a.Rnw and this one inputs file a1.Rnw. My idea was that Sweave would check \input{a1} or \include{a1} statements. If file a1 would have extension .Rnw it would parse it otherwise (i.e. having .tex) it would skip it. Sweave would just parse .Rnw files, while latex would put all of them in one file during typesetting.
Example:
- we have files
a.Rnw, which has \input{a1} or \include{a1} a1.Rnw
- run Sweave(a.Rnw) and you get
a.tex
a1.tex
- run LaTeX (i.e. texi2dvi --pdf a.tex) and you get
a.pdf
Having new command i.e. \SweaveInput{} would also do the job perfectly, however I don't se any new benefits of it. One should write more or less the same R code as for \input or \include.
[...]
> Gabor
> Since this might not be desirable in all instances,
> if Sweave were to have an include facility then it should
> not be implemented in such a way that the latex include facility
> can no longer be used. The point was just that it should be possible
> to do the include at the Sweave or at the latex level.
Friedrich
I agree that it should not be the same command. I have put an \SweaveInput{} on my 2do list, should be doable for R 2.1.0.
--
Lep pozdrav / With regards,
Gregor GORJANC
-----------------------------------------------------------------------
University of Ljubljana
Biotechnical Faculty URI: http://www.bfro.uni-lj.si/MR/ggorjan
Zootechnical Department mail: gregor.gorjanc <at> bfro.uni-lj.si
Groblje 3 tel: +386 (0)1 72 17 861
SI-1230 Domzale fax: +386 (0)1 72 17 888
Slovenia, Europe
-----------------------------------------------------------------------
: Imagine this situation:
:
: % --- a.Rnw start ---
: \documentclass{book}
: \usepackage{Sweave}
: \begin{document}
: % some toy example
: <<print=TRUE>>=
: x <- 1:10
: <at>
: % now we input additional file
: \input{a1}
: % and lets look again at x
: <<print=TRUE>>=
: x
: <at>
: \end{document}
: % --- a.Rnw end ---
:
: % --- a1.Rnw start ---
: %\usepackage{Sweave}
: % add 1 to x
: <<print=TRUE>>=
: x <- x + 1
: <at>
: % --- a1.Rnw end ---
______________________________________________
R-help@stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-help | 2020-07-11 06:20:45 | {"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": 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": 0.9043524265289307, "perplexity": 13786.209290654091}, "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-2020-29/segments/1593655921988.66/warc/CC-MAIN-20200711032932-20200711062932-00277.warc.gz"} |
https://en.wikipedia.org/wiki/Amplitude_versus_offset | # Amplitude versus offset
In geophysics and reflection seismology, amplitude versus offset (AVO) or amplitude variation with offset is the general term for referring to the dependency of the seismic attribute, amplitude, with the distance between the source and receiver (the offset). AVO analysis is a technique that geophysicists can execute on seismic data to determine a rock’s fluid content, porosity, density or seismic velocity, shear wave information, fluid indicators (hydrocarbon indications).[1]
The phenomenon is based on the relationship between the reflection coefficient and the angle of incidence and has been understood since the early 20th century when Karl Zoeppritz wrote down the Zoeppritz equations. Due to its physical origin, AVO can also be known as amplitude versus angle (AVA), but AVO is the more commonly used term because the offset is what a geophysicist can vary in order to change the angle of incidence. (See diagram)
Diagram showing how the layout of sources and receivers affects the angle of incidence
## Background and theory
Diagram showing the mode conversions that occur when a P-wave reflects off an interface at non-normal incidence
For a seismic wave reflecting off an interface between two media at normal incidence, the expression for the reflection coefficient is relatively simple:
${\displaystyle R={\frac {Z_{1}-Z_{0}}{Z_{1}+Z_{0}}}}$,
where ${\displaystyle Z_{0}}$ and ${\displaystyle Z_{1}}$ are the acoustic impedances of the first and second medium, respectively.
The situation becomes much more complicated in the case of non-normal incidence, due to mode conversion between P-waves and S-waves, and is described by the Zoeppritz equations.
### Zoeppritz equations
Main article: Zoeppritz equations
In 1919, Karl Bernhard Zoeppritz derived four equations that determine the amplitudes of reflected and refracted waves at a planar interface for an incident P-wave as a function of the angle of incidence and six independent elastic parameters.[2] These equations have 4 unknowns and can be solved but they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved.[3]
### Richards and Frasier (1976), Aki and Richards (1980)
P. Richards and C. Frasier[4] expanded the terms for the reflection and transmission coefficients for a P-wave incident upon a solid-solid interface and simplified the result by assuming only small changes in elastic properties across the interface. Therefore, the squares and differential products are small enough to tend to zero and be removed. This form of the equations allows one to see the effects of density and P- or S- wave velocity variations on the reflection amplitudes. This approximation was popularized in the 1980 book Quantitative Seismology by K. Aki and P. Richards and has since been commonly referred to as the Aki and Richards approximation.[5]
### Ostrander (1980)
Ostrander was the first to introduce a practical application of the AVO effect, showing that a gas sand underlying a shale exhibited amplitude variation with offset.[6]
### Shuey (1985)
Shuey further modified the equations by assuming – as Ostrander had – that Poisson's ratio was the elastic property most directly related to the angular dependence of the reflection coefficient.[3] This gives the 3-term Shuey Equation:[7]
${\displaystyle R(\theta )=R(0)+G\sin ^{2}\theta +F(\tan ^{2}\theta -\sin ^{2}\theta )}$
where
${\displaystyle R(0)={\frac {1}{2}}\left({\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}+{\frac {\Delta \rho }{\rho }}\right)}$
and
${\displaystyle G={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}-2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P} }^{2}}}\left({\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)}$ ; ${\displaystyle F={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}}$
where ${\displaystyle {\theta }}$=angle of incidence; ${\displaystyle {V_{p}}}$ = P-wave velocity in medium; ${\displaystyle {{\Delta }V_{p}}}$ = P-wave velocity contrast across interface;${\displaystyle {V_{s}}}$ = S-wave velocity in medium; ${\displaystyle {{\Delta }V_{s}}}$ = S-wave velocity contrast across interface; ${\displaystyle {\rho }}$ = density in medium; ${\displaystyle {{\Delta }{\rho }}}$ = density contrast across interface;
In the Shuey equation, R(0) is the reflection coefficient at normal incidence and is controlled by the contrast in acoustic impedances. G, often referred to as the AVO gradient, describes the variation of reflection amplitudes at intermediate offsets and the third term, F, describes the behaviour at large angles/far offsets that are close to the critical angle. This equation can be further simplified by assuming that the angle of incidence is less than 30 degrees (i.e. the offset is relatively small), so the third term will tend to zero. This is the case in most seismic surveys and gives the “Shuey Approximation”:
${\displaystyle R(\theta )=R(0)+G\sin ^{2}\theta }$
This was the final development needed before AVO analysis could become a commercial tool for the oil industry.[7]
## Use
Diagram showing how to construct an AVO crossplot
Modern seismic reflection surveys are designed and acquired in such a way that the same point on the subsurface is sampled multiple times, with each sample having a different source and receiver location. The seismic data is then carefully processed to preserve seismic amplitudes and accurately determine the spatial coordinates of each sample. This allows a geophysicist to construct a group of traces with a range of offsets that all sample the same subsurface location in order to perform AVO analysis. This is known as a Common Midpoint Gather[8] (a midpoint being the area of the subsurface that a seismic wave reflects off before returning to the receiver) and in a typical seismic reflection processing workflow, the average amplitude would be calculated along the time sample, in a process known as “stacking”. This process significantly reduces random noise but loses all information that could be used for AVO analysis.[9]
### AVO crossplots
A CMP gather is constructed, the traces are conditioned so that they reference the same two-way travel time, sorted in order of increasing offset and the amplitude of each trace at a specific time horizon is extracted. Remembering the 2-term Shuey Approximation, the amplitude of each trace is plotted against sin^2 of its offset and the relationship becomes linear, as seen in the diagram. Using linear regression, a line of best fit can now be calculated that describes how the reflection amplitude varies with offset using just 2 parameters: the intersect, P, and the gradient, G.
As per the Shuey approximation, the intersect P corresponds to R(0), the reflection amplitude at zero-offset, and the gradient G describes the behaviour at non-normal offset, a value known as the AVO gradient. Plotting P (or R(0)) against G for every time sample in every CMP gather produces an AVO crossplot and can be interpreted in a number of ways.
## Interpretation
An AVO anomaly is most commonly expressed as increasing (rising) AVO in a sedimentary section, often where the hydrocarbon reservoir is "softer" (lower acoustic impedance) than the surrounding shales. Typically amplitude decreases (falls) with offset due to geometrical spreading, attenuation and other factors. An AVO anomaly can also include examples where amplitude with offset falls at a lower rate than the surrounding reflective events.
## Applications in the oil and gas industry
The most important application of AVO is the detection of hydrocarbon reservoirs. Rising AVO is typically pronounced in oil-bearing sediments, and even more so in gas-bearing sediments. Particularly important examples are those seen in turbidite sands such as the Late Tertiary deltaic sediments in the Gulf of Mexico (especially during the 1980s–1990s), West Africa, and other major deltas around the world. Most major companies use AVO routinely as a tool to "de-risk" exploration targets and to better define the extent and the composition of existing hydrocarbon reservoirs.
## AVO is not fail-safe
An important caveat is that the existence of abnormally rising or falling amplitudes can sometimes be caused by other factors, such as alternative lithologies and residual hydrocarbons in a breached gas column. Not all oil and gas fields are associated with an obvious AVO anomaly (e.g. most of the oil found in the Gulf of Mexico in the last decade), and AVO analysis is by no means a panacea for gas and oil exploration.
## References
1. ^ http://www.glossary.oilfield.slb.com/Display.cfm?Term=amplitude%20variation%20with%20offset Schlumberger Oilfield Glossary
2. ^ Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press.
3. ^ a b Shuey, R. T. [1985] A simplification of the Zoeppritz equations. Geophysics, 50:609–614
4. ^ Richards, P. G., and Frasier, C. W., 1976, Scattering of elastic wave from depth-dependent inhomogeneities: Geophysics, 41, 441–458
5. ^ Aki, K. and Richards, P. G., 1980, Quantitative seismology: Theory and methods, v.1 : W.H. Freeman and Co.
6. ^ Ostrander, W.J., 1984, Plane wave reflection coefficients for gas sands at non normal angles of incidence: Geophysics, 49, 1637–1648.
7. ^ a b Avseth, P, T Mukerji and G Mavko (2005). Quantitative seismic interpretation. Cambridge University Press, Cambridge, UK
8. ^ http://www.glossary.oilfield.slb.com/Display.cfm?Term=CMP Schlumberger Oilfield Glossary
9. ^ Young, R. & LoPiccolo, R. 2005. AVO analysis demystified. E&P. http://www.e-seis.com/white_papers/AVO%20Analysis%20Demystified.pdf | 2017-03-27 12:18:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 15, "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": 0.6989942789077759, "perplexity": 2212.466136893768}, "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-2017-13/segments/1490218189471.55/warc/CC-MAIN-20170322212949-00498-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/121388/random-ink-blotches-from-tikz | Random ink blotches from tikz
How can I create this type of effect in tikz, random ink spots from a fountain pen, for a watermark on select feature pages.
See update.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\usetikzlibrary{calc}
\def\blob#1#2{\draw[fill,rounded corners=#1*3mm] (#2) +($(0:#1*2+#1*rnd)$)
\foreach \a in {20,40,...,350} { -- +($(\a: #1*2+#1*rnd)$) } -- cycle;}
\begin{tikzpicture}
\blob{0.4}{0,0}
\foreach \a in {0,20,...,350} {
\fill[black] let \p1 = (\a+20*rnd:3*rnd),
\n1 = {0.2*rnd}
in (\p1) circle(\n1);
}
\blob{0.2}{1,3}
\foreach \a in {0,20,...,350} {
\fill[black] let \p1 = ($(1,3)+(\a+20*rnd:2*rnd)$),
\n1 = {0.15*rnd}
in (\p1) circle(\n1);
}
\end{tikzpicture}
\end{document}
Explanations
The picture is composed by two kind of "objects". The irregular shaped blotches, which are drawn by the macro \blob, and lots of smaller black circles.
The \blob macro takes two arguments. The first one is a scale factor. The second one is the coordinates of its center. The macro uses polar coordinates to generate a series of points at regular angles, but random distances from its center, and connects all of them with a rounded corners polyline, which is filled in black.
The smaller circles are drawn with a similar technique, using polar coordinates but this time not only the distance to the center, but also the angles are random. The center of each drop is calculated as the point \p1. The radius of each drop is also random, and precomputed in \n1.
For some unknown reason, if I tried to avoid the let...in syntax and use directly rnd in the expressions, such as circle(0.2*rnd), for example, I got some non-circular drops (very eccentric ellipses), so I resorted to the let...in syntax to precompute the center and radii, and the problem disappeared.
Update
As requested in a comment, I added shadows and shines. It was more difficult than expected, because, since all the shapes and locations are inherently random, but the shadows and shines had to be drawn at the same places and with the same shape than the random drops, I had to store the points of the contour to draw it twice. Also, since some drops can be drawn one "on top" of the other (because of its random positioning), this could cause the shadow of a drop to be drawn on top of the ink of another drop, which is ugly. I had to add pgflayers to ensure that all shadows are in the background.
I tried to devise a better algorithm to decide the size of the random drops, basend on their distance to the center of the splat. I'm not entirely satisfied with the result, but it is not too bad.
Code
\documentclass{article}
\usepackage{tikz}
\begin{document}
\usetikzlibrary{calc}
% Initial setup
\pgfmathsetseed{1234}
\pgfdeclarelayer{ink}
\pgfdeclarelayer{lights}
% Macro to draw the big random shaped blobs
% Compute the points of the path
\foreach \a in {0,20,...,350} {
\path let
\p1 = (#1), % center
\n1 = {#2+(#3-#2)*rnd} % distance at center
in (\p1) node[coordinate] (P\a) at +(\a:\n1) {};
}
% Draw first the shadow (-.5mm, -1mm) is the offset
\fill let \n1 = {(#3-#2)*10/6} in
[fill=black!20,,rounded corners=\n1 mm] ($(P0)+(-.5mm,-1mm)$)
\foreach \a in {20,40,...,350} { -- ($(P\a)+(-.5mm,-1mm)$) } -- cycle;
\end{pgfonlayer}
% Then the ink
\begin{pgfonlayer}{ink}
\draw let \n1 = {(#3-#2)*10/6} in
[fill,rounded corners=\n1 mm] (P0)
\foreach \a in {20,40,...,350} { -- (P\a) } -- cycle;
\end{pgfonlayer}
% I tried to draw a light near to the upper border, using the same path, but
% the results were bad, and I deleted it
}
% Macro to draw a drop of ink, relative to one splat
\pgfmathsetmacro{\distance}{#3+(#4-#3)*rnd}
\pgfmathsetmacro{\distancetoborder}{abs(\distance-#3)}
\pgfmathsetmacro{\size}{#3/(15+10*rnd)/sqrt(\distancetoborder)}
\pgfmathsetmacro{\angle}{#2+20*rnd}
\fill[black!20] ($(#1)+(-\size/5,-\size/3)$) +(\angle:\distance) circle(\size);
\end{pgfonlayer}
\begin{pgfonlayer}{ink}
\fill[black] (#1) +(\angle:\distance) circle(\size);
\end{pgfonlayer}
\begin{pgfonlayer}{lights}
\fill[white] (#1) ++(\angle:\distance) ++(80:0.7*\size) circle(0.2*\size);
\end{pgfonlayer}
}
% Main drawing
\begin{tikzpicture}
\blob{0,0}{1}{3}
\foreach \a in {0,10,...,350} {
\drop{0,0}{\a}{2.5}{5}
}
\blob{2,4}{0.3}{1}
\foreach \a in {0,20,...,350} {
\drop{2,4}{\a}{1}{3}
}
\end{tikzpicture}
\end{document}
Result
• The problem with the rnd circles becoming ellipses is due to the fact that circle internally sets both x radius and y radius to the same value, rnd in this case, which gets evaluated separately and therefore leads to two different radii. Same problem as in tex.stackexchange.com/a/119209/2552 – Jake Jun 27 '13 at 16:46
• Could replicate the shading from the OP image? That would be very nice. – dustin Jun 28 '13 at 0:36
• +1: nice!! I think you can also try to use radial shadings to get better effects: perhaps How to create a ball shading and to customize 3D lighting manually? can help. – Claudio Fiandrino Jun 28 '13 at 12:03
With PSTricks. Too long as a comment.
\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot,pst-node}
\usepackage[nomessages]{fp}
\pstVerb{realtime srand}
\expandafter\FPseed\expandafter=\pdfuniformdeviate 1000000
\def\blotches#1{%
\psLoop{#1}{%
\def\points{}%
\rput(!Rand 12 mul 3 sub Rand 12 mul 3 sub){%
\FPrandom\NP
\FPeval\NP{round(NP*17+3:0)}%
\curvepnodes[plotpoints=\NP]{0}{360}{Rand t PtoC}{P}%
\multido{\i=0+1}{\Pnodecount}{\xdef\points{\points(P\i)}}%
\expandafter\psccurve\expandafter*\points}}%
\ignorespaces}
\def\bubbles#1{%
\psLoop{#1}{%
\qdisk(!Rand 12 mul 3 sub Rand 12 mul 3 sub){!Rand 4 div}}%
\ignorespaces}
\begin{document}
\begin{pspicture}(-3,-3)(3,3)
\bubbles{100}
\blotches{20}
\end{pspicture}
\end{document}
Animation
To ease choosing the most realistic output.
Ink
\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot,pst-node}
\usepackage[nomessages]{fp}
\pstVerb{realtime srand}
\expandafter\FPseed\expandafter=\pdfuniformdeviate 1000000
\def\blotches#1{%
\psLoop{#1}{%
\def\points{}%
\rput(!Rand 12 mul 3 sub Rand 12 mul 3 sub){%
\FPrandom\NP
\FPeval\NP{round(NP*13+7:0)}%
\curvepnodes[plotpoints=\NP]{0}{360}{Rand 2 mul t PtoC}{P}%
\multido{\i=0+1}{\Pnodecount}{\xdef\points{\points(P\i)}}%
\expandafter\psccurve\expandafter*\points}}%
\ignorespaces}
\def\bubbles#1{%
\psLoop{#1}{%
\qdisk(!Rand 12 mul 3 sub Rand 12 mul 3 sub){!Rand 4 div}}%
\ignorespaces}
\begin{document}
\psLoop{10}{%
\begin{pspicture}(-3,-3)(3,3)
\bubbles{200}
\blotches{10}
\end{pspicture}}
\end{document}
Blood
Hopefully blood blotches are more interesting.
\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot,pst-node}
\usepackage[nomessages]{fp}
\pstVerb{realtime srand}
\expandafter\FPseed\expandafter=\pdfuniformdeviate 1000000
\def\blotches#1{%
\psLoop{#1}{%
\def\points{}%
\rput(!Rand 12 mul 3 sub Rand 12 mul 3 sub){%
\FPrandom\NP
\FPeval\NP{round(NP*13+7:0)}%
\curvepnodes[plotpoints=\NP]{0}{360}{Rand 1.5 mul t PtoC}{P}%
\multido{\i=0+1}{\Pnodecount}{\xdef\points{\points(P\i)}}%
\expandafter\psccurve\expandafter*\points}}%
\ignorespaces}
\def\bubbles#1{%
\psLoop{#1}{%
\qdisk(!Rand 12 mul 3 sub Rand 12 mul 3 sub){!Rand 4 div}}%
\ignorespaces}
\psset{linecolor=red}
\begin{document}
\psLoop{10}{%
\begin{pspicture}(-3,-3)(3,3)
\bubbles{250}
\blotches{10}
\end{pspicture}}
\end{document}
Attention
Note that Rand no longer produces a random real number between 0 and 0.5 inclusive. Its definition had been tacitly changed. Now it produces a random real number between 0 and 1 inclusive. It is not documented, nor announced, but it is still fun!
The code given above has not been updated yet so it will produce different output. I have no time to update it right now. Sorry for this inconvenience. | 2019-12-06 05:07:20 | {"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": 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": 0.756790816783905, "perplexity": 3287.366441110576}, "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-2019-51/segments/1575540484815.34/warc/CC-MAIN-20191206050236-20191206074236-00507.warc.gz"} |
http://math.virginia.edu/calendar/ | # University of Virginia Math Seminars
## List of seminars
### Colloquium
Thursdays at 3:45 in Kerchof 317. Refreshments served at 3:15
### Algebra Seminar
Wednesdays and some Fridays, 3:30-4:30pm, Ker 317
### Geometry Seminar
Tuesdays at 2:00 in Ker 317
Description The Geometry Seminar talks usually focus on aspects of low-dimensional topology and geometry, including knot theory and categorification, Floer homology, 3- and 4-dimensional manifolds, and symplectic and contact topology. The lectures are often given by outside speakers, however UVa graduate students and faculty give talks as well.
### Harmonic Analysis and PDE Seminar
Tuesdays at 3:30
Description Harmonic Analysis and PDE seminar features a mix of local speakers (graduate students and faculty) and visitors. The ideal topics live on the interface between the two fields; luckily, the interface has been steadily expanding.
### Mathematical Physics Seminar
Wednesdays and some Fridays at 2:00 in Ker 326
Description The Mathematical Physics Seminar features talks on a wide variety of topics such as, for instance, Schrödinger operators, the mathematics of quantum systems, statistical mechanics, the renormalization group and quantum field theory. Lectures typically are of research level and are given by local as well as outside speakers. Graduate students in mathematical physics are encouraged to give presentations at this seminar about their ongoing research. Everyone is welcome to attend.
### Operator theory seminar
Tuesdays at 4:00 in Ker 326
Description The Seminar in Operator Theory and Operator Algebras covers a wide variety of topics in functional analysis, including $C^*$-algebras and von Neumann algebras, composition operators, Banach spaces, noncommutative convexity, and applications of complex function theory. Most lectures are research level, but we also feature expository talks.
### Probability Seminar
Wednesdays at 4:35
Description The Probability Seminar is the place to see talks on active research topics in probability theory, as well as informal discussions of basic notions of probability. We typically have invited speakers every 2-3 weeks presenting a wide array of research in probability. Most other weeks are informal discussions led by local participants, often graduate students discussing recently studied topics. The seminar is open to all. Feel free to attend regularly or occasionally.
### Topology Seminar
Thursdays at 2:00 in Ker 326
Description Topology Seminar talks are on recent developments in algebraic topology—including homotopy theory, ordinary and extraordinary homology and cohomology, cobordism theory, and K-theory—and related subjects like differential topology and homological algebra.
Fridays at 2:30 in Ker 317
Description The Graduate Seminar provides a friendly atmosphere for grad students to give talks about current interests, research, or teaching. This seminar is for grad students only and encourages audience participation while keeping the intensity level below that of other seminars.
The graduate student seminar was started in the spring of 1999 by several junior faculty members. They hoped it would serve as a meeting place for junior faculty members and graduate students to socialize and to talk about mathematics.
The seminar is designed with several goals in mind. The seminar gives everyone a chance to interact outside of class while providing exposure to some of the current interests of the department. As a result, graduate students in their early years have a chance to become more familiar with the potential areas of study. Perhaps the most important goal of the seminar is to provide graduate students with an open forum in which to practice giving mathematical talks in a supportive environment.
The seminar is intended for graduate students and junior faculty in an attempt to foster a less intimidating atmosphere for discussion. Such a friendly, informal environment not only makes it easier for the speakers, but promotes more audience participation. Most talks last about 45 minutes, which leaves sufficient time for comments and questions afterward. There have been a wide variety of topics covered. Many speakers have presented material related to their research while others have chosen to speak about topics that may not be directly related to their studies. Some people have even used the seminar to prepare for professional talks. Though topics vary, the goal is to keep the mathematical intensity at an appropriate level so that graduate students not specializing in that discipline can still follow the presentation and learn something.
Tuesdays at 5:00 in Ker 314 (Math lounge)
Description The Undergraduate Math Club at University of Virginia is a weekly seminar and a club (with an official CIO status) for students interested in mathematics and related areas. It meets on Tuesdays at 5pm for 60-80 minutes. The activities vary from talks by faculty members, graduate and undergraduate students to presentations by local industries, panel discussions (on REUs, careers for math majors, etc.), and outside activities such as a movie outing or a visit to a 3D printing lab.
The Math Club is a students' space, and the CIO structure provides the Club with a managing board which helps plan and organize events.
### Galois-Grothendieck seminar
Tuesdays at 3:30 in Monroe 114
Description The Galois-Grothendieck Seminar is an expository seminar about various aspects of Galois theory and arithmetic geometry. Each semester/year has a coherent program, with graduate students contributing many of the talks. | 2017-10-18 23:56:19 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.2843693494796753, "perplexity": 1686.0633041555197}, "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-2017-43/segments/1508187823168.74/warc/CC-MAIN-20171018233539-20171019013539-00296.warc.gz"} |
https://codereview.stackexchange.com/questions/33506/max-digits-of-digit-summing | # Max-digits of digit summing
As a challenge I decided to do the following problem:
How many different numbers with n digits are there whose digits add up to k?
As this is an embarrassingly parallel problem I decided to use a bit of GPGPU programming (partly for learning purposes) using C++AMP and came up with the following solution:
#include <iostream>
#include <vector>
#include <assert.h>
#include <amp.h>
#include <chrono>
#include <ctime>
#include <numeric>
inline unsigned int AddDigits( unsigned int n ) restrict(amp, cpu)
{
unsigned int sum = 0;
while( n > 0 )
{
sum += n % 10;
n /= 10;
}
return sum;
}
int main()
{
unsigned int iSize, iSumRequired;
std::cout << "Please enter the number of digits: ";
std::cin >> iSize;
std::cout << std::endl << "Please enter the sum required: ";
std::cin >> iSumRequired;
std::cout << std::endl;
unsigned long long iMaxNum = std::pow( 10, iSize );
assert( vecData.max_size() > iMaxNum );
std::vector<unsigned int> vecData( iMaxNum );
std::vector<int> vecNumValid( iMaxNum );
auto tpBegin = clock.now();
std::iota( vecData.begin(), vecData.end(), 1 );
concurrency::array_view<const unsigned int, 1> arrayView( iMaxNum, vecData );
concurrency::array_view<int, 1> numValid( iMaxNum, vecNumValid );
concurrency::parallel_for_each( numValid.extent, [=]( concurrency::index<1> idx ) restrict( amp ) {
numValid[idx] = (AddDigits( arrayView[idx] ) == iSumRequired ? 1 : 0);
} );
numValid.synchronize();
int iNumValid = concurrency::parallel_reduce( vecNumValid.begin(), vecNumValid.end(), 0 );
std::cout << "The number of valid numbers are: " << iNumValid << std::endl;
auto tpTimeTaken = clock.now() - tpBegin;
std::cout << "Time Taken: " << std::chrono::duration_cast<std::chrono::milliseconds>(tpTimeTaken).count() << std::endl;
return 0;
}
This offers a significant improvement over parallel brute-forcing on the CPU (2x speedup versus 8 threads on an 8-core CPU), however out of curiosity I was wondering whether any further speedup could be gained, perhaps through somehow running the parallel_reduce on the GPU rather than on the CPU; removing the necessity to synchronize the array_view?
Moreover, this only allows for up to 8-digits (because of the restriction of the maximum value being less than 231-1) because the GPU only supports types of float, double, int, unsigned int and bool; is there a way to circumvent this and improve the maximum number of digits allowed?
• Could you please clarify this: How many different numbers with n digits are there that add up to k Do you mean: How many different numbers with n digits are there whose digits add up to k? Sorry probably my fault for not understanding. – Christopher J Nov 6 '13 at 11:05
I agree with @Christopher. The brute force might be embarrassingly parallel, but its complexity is dreadful.
Calling f(n,k) the answer, there is a simple approach : the case n>1 boils down to sum (g(n-1,k-i) for i in [1..min(k,9)]), where g also allows i=0. The case n=1 is left as an exercise, and some other early stop shortcuts might be used.
With this algorithm, you don't even need C++. Instead, a language with lightweight threads (scala) will prove profitable, since up to 9^n threads may be instantiated.
I understand that you are asking from a parallel point of view, but I am unable to provide feedback on that, as I have never worked with this system before. But from my point of view, creating a parallel solution is has been done premature.
Prior to creating a parallel solution to speed up your solution, you should improve the efficiency of the algorithm. At present you consider all numbers in the range of [1, 10N), whereas you only need to evaluate numbers where the number of digits in the number is equal to N. In which case you only need to evaluate numbers in the range of [10N-1, 10N).
You could further improve this via considering permutations and combinations. This would allow you to calculate all the different combinations of the digits for the given range. Once you have these you can then check which of the combinations digits sum to the required total.
If X combinations sum to a given total then you can quickly calculate how many numbers within the require range have N digits and sum to the goal. By computing X * (N Perm N) where N is the number of digits and where permutations are calculated without replacement. Representing just the N digits in the solution.
At this point you would be using a parallel solution to calculate the combination hits, but you may find the bottle neck of data transfer to a GPU causes the parallel solution to perform slower than the non-GPU enabled solution. | 2020-09-24 01:46:02 | {"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": 0.4004964828491211, "perplexity": 2019.6118479051931}, "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/1600400213006.47/warc/CC-MAIN-20200924002749-20200924032749-00296.warc.gz"} |
https://math.meta.stackexchange.com/questions/27072/publishing-answer | It seems to me that my answer at
If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?
deserves to become a little paper; everyone I mention it to is surprised by the result, and I doubt that it's well known. The paper I have in mind would be a considerably cleaned up version of what appears there.
Q: What legalities, protocols, and bits of etiquette are involved in publishing an answer to a MSE question?
In particular, assuming that such a publication is not simply unacceptable: Presumably I'd cite MSE as the source of the conjecture. I imagine there's a standard way to give such a citation, that I can find when I get around to looking for it. In particular squared: Would I cite the author as user228168 or is there a way to find his/her real name for this purpose?
• To see how others dealt with similar situation, you might have a look at the papers mentioned in this post: Papers that originated on math.SE. Maybe some of other posts linked there might be of interest for you, too. – Martin Sleziak Sep 28 '17 at 15:35
• @MartinSleziak Thanks. Just noticed the "cite" link below posts - the existence of that link answers my main question, yes, people do cite MSE in papers. That thread should be very useful regarding various details... – David C. Ullrich Sep 28 '17 at 16:03
• Well, there is a separate tag called (citation) here on meta - currently 29 questions have this tag; as you can see people asked various things about citing posts from this site. – Martin Sleziak Sep 28 '17 at 16:13
Mostly what you say suffices, I think. That is, it seems apt to mention SE as the source of the problem.
There is no mechanism on the site to find out the identity of the user, especially not in the current case where it is a deleted account.
If the account were still active you could mention your intent in a comment and ask for their preferences regarding using their displayname or give their real name.
In the current case, one could say something like: The result of the paper first appeared as a post by the author on the site Mathamatics Stack Exchange, responding to the question of a now anonymous account (user228168) [Reference to the thread.].
Searching one can find the display name used at the time the question was asked. Arguably, you could also use that. But I am not sure it is a good idea.
Regarding legalities, it may be worth recalling that you licensed the content of your answer under a CC license to Stack Exchange. Especially, if you want to reuse the same write-up this could be an issue for some journals but the policy of others is such that it is no problem.
• Thanks. I'm not a lawyer, don't even play one on TV, but I tend to suspect the license should be no problem. The paper I have in mind is substantially different from the writeup in my answer (I fill in various details left fuzzy in the answer, and having done that it turns out that most of what I said in the answer doesn't need to be said...) – David C. Ullrich Sep 28 '17 at 16:57
• You are welcome. I think, but I am not a lawyer, if the write-up is different it should a non-issue; – quid Sep 28 '17 at 17:11
• Incidentally, how does the licence granted on this site compare to the standard arXiv licence? – Tobias Kildetoft Sep 29 '17 at 6:48
• @TobiasKildetoft the license here is BY-SA, that is it requires mention of SE yet allows to redistribute and to modify the content. On the arXiv.org one only gives permission to the arXiv to distribute the content. That mention of SE is required is irrelevant for the present context as OP as author can always relicense to the journal their content in a way that does not require this. An issue a journal might see is that the content is licensed in a way that everybody can distribute it, say, a competing journal could also print the SE content while this is not possible with content on arXiv. – quid Sep 29 '17 at 9:02
• Practically I doubt it is relevant for many journals and not few will even be fine with it. But strictly there could be issues with some author agreements of journals that have some explicit exceptions to allows arXiv and other preprint servers yet those exceptions might not cover SE. I mean things like "This work was not published before except [some list]" @TobiasKildetoft – quid Sep 29 '17 at 9:07 | 2020-02-19 22:42:22 | {"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": 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": 0.5010265111923218, "perplexity": 625.4862538036002}, "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-2020-10/segments/1581875144429.5/warc/CC-MAIN-20200219214816-20200220004816-00057.warc.gz"} |
http://www.math.psu.edu/calendars/ | # Math Calendar
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A live feed of seminars and special events in the upcoming week.
March 2nd, 2015 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Auction theory from an applied Math point of view
Speaker: Nir Gavish, Technion – Israel Institute of Technology (Host: C Liu)
Location: MB114
Auctions are typically related to electronic trading sites like Ebay or when auctions of art masterpieces hit the news. It is less commonly recognized that auctions are central to the backbone of economy with wide use in electricity markets, treasury auctions, foreign exchanges, mineral rights and more. For example, in 2014 the US Treasury used auctions to issue approximately $7 trillion in securities to finance the public dept of the US. Most of auction theory concerns the case where all bidders are symmetric (identical). This is not because bidders are believed to be symmetric, but rather because the analysis of asymmetric auctions is considerably harder. For example, in the case of the common first-price auction, the symmetric case is governed by a single ODE which is easy to solve explicitly. In contrast, the model for asymmetric first-price auction consists of n first-order nonlinearly coupled ODES with 2n boundary condition and an unknown location of the right boundary, where n is the number of bidders. This nonstandard boundary value problem is challenging to analyze, or even to solve numerically. Therefore, very little is known about its solutions. In this talk, I will review various approaches to this problem (perturbation analysis, dynamical systems, numerical methods), and in particular focus on the case of large asymmetric first-price auctions. Joint work with Gadi Fibich and Arieh Gavious March 2nd, 2015 (02:30pm - 03:30pm) Seminar: Computational and Applied Mathematics Colloquium Title: Generalized Poisson Boltzmann and Differential Capacitance data: an inverse problem Speaker: Nir Gavish, Technion – Israel Institute of Technology (Host: C Liu) Location: MB106 The contact between a charged object (metal surface, macromolecule, membrane, etc.) and an electrolyte solution results in the rearrangement of ion distribution near the interface and formation of the so-called electrical double layer. One of the important experimentally available quantities for characterising the structure of electrolyte solutions near such interfaces are differential capacitance measurements. From a mathematical point of view, the double layer structure is commonly modelled by the Poisson-Boltzmann equation and generalizations of it. In this work, we conduct a systematic study of the differential capacitance data. In particular, we focus on the inverse problem: Given differential capacitance data, we ask whether it is possible to derive a generalized Poisson-Boltzmann model which gives rise to the prescribed data. We show that such models do exist, characterise their variational action in terms of a PDE, and provide a method for solving the PDE and deriving the appropriate generalized Poisson-Boltzmann model. This method does not yield a unique model, and so we find that a wide class of models can give rise to the same differential capacitance data. Using our method, we derive generalized Poisson-Boltzmann models from differential capacitance data coming from either theoretical models or experimental measurements. In particular, derive novel models which accurately recover experimental data. This is a joint work with Keith Promislow. March 2nd, 2015 (03:35pm - 04:35pm) Seminar: Dynamical systems seminar Title: Examples of analytic non-standard realization of some irrational circle rotations the torus Speaker: Shilpak Banerjee, Penn State Location: MB114 I will present a brief survey on one of the applications of the "approximation by conjugation" scheme developed by Anosov and Katok. Namely, this scheme can be used to produce examples of smooth ergodic diffeomorphisms on various manifolds metrically isomorphic to an irrational circle rotation. Then I will talk about some modifications that can be done and extended this technique to the analytic set-up and produce similar examples on the torus March 3rd, 2015 (11:15am - 12:05pm) Seminar: Combinatorics/Partitions Seminar Title: Properties of a Restricted Binary Partition Function a la Andrews and Lewis Speaker: James Sellers, Penn State Location: MB106 In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. At the end of their paper, they define a family of functions$W_1(S_1, S_2;n)$to be the number of partitions of$n$into parts from$S_1 \cup S_2$which do not contain both$a_j$and$b_j$as parts (where$S_1 = \left\{ a_1, a_2, a_3, \dots\right\}$and$S_2 = \left\{ b_1, b_2, b_3, \dots\right\}$and$S_1 \cap S_2 = \phi$). This definition is motivated by the main results of their paper; in that case,$S_1$and$S_2$contain elements in arithmetic progression with the same skip value''$k$. Our goal in this work is to consider more general examples of such partition functions where$S_1$and$S_2$satisfy the requirements mentioned above but do not simply contain elements in an arithmetic progression. In particular, we consider the situation where$S_1$and$S_2$contain specific powers of$2.We then prove a number of arithmetic properties satisfied by this function using elementary generating function manipulations and classic results from the theory of partitions. This work was completed in collaboration with my undergraduate student Bin Lan. March 3rd, 2015 (01:00pm - 02:00pm) Seminar: Theoretical Biology Seminar Title: Stochastic modeling of carcinogenesis Speaker: Rafael Meza, University of Michigan (Host: Jessica Conway) Location: MB106 Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of (stochastic) genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carcinogenesis. In particular, two stochastic (mechanistic) models, the Armitage-Doll and the two-stage clonal expansion (TSCE) model, have been widely used in the past for cancer risk assessment and for the analysis of cancer population and experimental data. In this talk, I will introduce some of the biological and mathematical concepts behind the theory of multistage carcinogenesis, and discuss in detail the use of these models in cancer epidemiology. Recent applications of multistage models in lung and colon cancer will be reviewed. March 3rd, 2015 (02:30pm - 03:45pm) Seminar: Logic Seminar Title: Kolmogorov Random Graphs Speaker: John Pardo, Penn State Location: MB315 We will discuss several properties of Kolmogorov random graphs using deficiency functions, i.e. functions that bound how far away a graph is from maximum complexity, and relate these properties back to the usual notion of randomness for binary strings as well as connect them to the property of quasirandomness. March 3rd, 2015 (03:30pm - 06:00pm) Seminar: Working Seminar: Dynamics and its Working Tools Title: Introduction to KAM (Kolmogorov-Arnold-Moser) theory, IV Speaker: Alena Erchenko, Penn State Location: MB114 March 3rd, 2015 (04:00pm - 05:00pm) Seminar: Applied Analysis Seminar Title: ANTHROPOMORPHIC IMAGE RECONSTRUCTION VIA OPTIMAL CONTROL AND HYPOELLIPTIC DIFFUSION Speaker: Ugo Boscain, CNRS, CMAP, École Polytechnique, Paris Location: MB106 In this talk I will present a model of geometry of vision due to Petitot, Citti, Sarti, and our research group. One of the main features of this model is that the primary visual cortex V1 lifts an image from R^2 to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process intrinsically defines an hypoelliptic heat equation on the bundle of directions of the plane. The numerical integration of this equation is difficult and require techniques of non-commutative Fourier analysis. The purpose of this research is to validate the biological model and to obtain an algorithm of image inpainting going beyond the state of the art. [1] U. Boscain, J. Duplaix, J.P. Gauthier, F. Rossi, “Anthropomorphic image reconstruction via hypoelliptic diffusion”. SIAM J. CONTROL OPTIM.Vol. 50, No. 3, pp. 1309–1336, 2012. http://arxiv.org/abs/1006.3735 [2] U. Boscain, R. Chertovskih, J.P. Gauthier, A. Remizov. Hypoelliptic diffusion and human vision: a semi-discrete new twist. SIAM Journal on Imaging Sciences 2014, Vol. 7, No. 2, pp. 669-695. http://arxiv.org/abs/1304.2062 March 4th, 2015 (12:05pm - 01:20pm) Seminar: Geometry Luncheon Seminar Title: Geodesics on the convex surfaces. Speaker: Anton Petrunin, Penn State Location: MB114 We give a universal bound for the variation of turn of minimizing geodesics on convex surfaces. This is a joint work with Nina Lebedeva. March 4th, 2015 (03:30pm - 05:00pm) Seminar: Complex Fluids Seminar Title: The dynamic boundary condition and Dirichlet to Neumann map Speaker: Chun Liu, Penn State University Location: MB106 March 4th, 2015 (03:30pm - 05:30pm) Seminar: Applied Algebra and Network Theory Seminar Title: (RESCHEDULED due to university closure) Introduction to Reaction Network Theory Speaker: Jacob Biamonte, ISI Foundation Location: MB315 There is a widely used and successful theory of “chemical reaction networks”, which provides a framework describing any system governed by mass action kinetics. Computer science and population biology use the same ideas under a different name: “stochastic Petri nets”. But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas—yet in a context where probabilities replace amplitudes. We have recently been working to explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. Our general idea is about merging concepts from quantum physics and reaction network theory to provide a bidirectional bridge of relevant analysis tools to address networks in both disciplines. http://arxiv.org/abs/1209.3632 March 5th, 2015 (08:30am - 11:00am) Seminar: Ph.D. Thesis Defense Title: “Studies on the weak convergence of partial sums in Gibbs-Markov dynamical systems” Speaker: Xuan Zhang, Adviser: Manfred Denker, Penn State Location: MB114 We investigates distributional limit theorems of partial sums of the formf_{n,1}+f_{n,2}\circ T_n+\cdots+f_{n,n}\circ T_n^{n-1}$for Gibbs-Markov dynamical systems$(X_n, \mathscr B_n, T_n,\mu_n,\alpha_n)$and an array of functions$f_{n,m}: X_n\to \mathbb R$of certain classes. We show a Central Limit Theorem (CLT) for this array, a CLT of Lindeberg type (with uniformly bounded functions) and we also investigate the Poisson limit case. We relate the Poisson limit theorem to escape rates of sweep-out sets and the CLT is applied in various situations, in particular to some statistical functions. March 5th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Zeros of Dirichlet series Speaker: Robert Vaughan, Penn State University Location: MB106 We are concerned here with Dirichlet series f(s) = 1 +\sum_{n=2}^{\infty} \frac{c(n)}{n^s} which satisfy a function equation similar to that of the Riemann zeta function, typically of the form f(s) = \epsilon 2^s q^{1/2-s} \pi^{s-1} \Gamma(1-s) \big(\sin\textstyle\frac{\pi}{2}(s+\kappa)\big) f(1-s), but for which the Riemann hypothesis is false. March 5th, 2015 (12:30pm - 02:59pm) Seminar: Ph.D. Thesis Defense Title: " A Complete Set of Invariants for Density Operators Under Local Conjugation" Speaker: Jacob Turner, Adviser: Jason Morton, Penn State Location: MB114 A density operator of is a trace one, positive semi-definite matrix in the tensor product of the spaces End (V_i) for i=1,...,n. These are used in physics to represent a quantum system of n particles, the ith of which has dim (V_i) spins. One of the most important questions about a density operator is the entanglement of the state it represents. Almost every notion of entanglement is invariant under conjuagation by the affine cone over the Segre product of the unitary groups over each V_i. Using techniques from algebraic geometry and representation theory, we determine a finite set of invariant polynomials that completely seperate orbits of density operators. March 5th, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: Intermediate C*-norms Speaker: Matthew Wiersma, University of Waterloo Location: MB106 It is known that C*-algebras admit unique C*-norms, but this is not true in general for dense *-subalgebras of C*-algebras. For example, if G is a discrete group, then its group ring algebra may admit more than one C*-norm. Similarly, the algebraic tensor product of two C*-algebras may admit multiple C*-norms. Each of these examples admits two canonical C*-norms. During this talk, we will investigate C*-norms which fall between these canonical constructions. March 5th, 2015 (03:30pm - 04:20pm) Seminar: Department of Mathematics Colloquium Title: Techniques and concepts of amenability of discrete groups Speaker: Kate Juschenko (Nate Brown), Northwestern University Location: MB114 The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on different spaces. In particular the study of isometry-invariant measure led to the Banach-Tarski decomposition theorem in 1924. The class of amenable groups was introduced and studied by von Neumann in 1929 and he explained why the paradox appeared only in dimensions greater or equal to three. In 1940's and 1950's a major contribution was made by M. Day in his paper on amenable semigroups. We will give an introductory to amenability talk, and explain more recent developments in this field. March 5th, 2015 (06:30pm - 08:30pm) Title: Private Location: MB102 March 10th, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Spring break Speaker: Spring break Location: MB106 March 12th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: No seminar today Speaker: Spring Break, Somewhere sunny Location: MB106 March 12th, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: No seminar Speaker: Spring Break Location: MB106 March 12th, 2015 (03:30pm - 04:20pm) Seminar: Department of Mathematics Colloquium Title: SPRING BREAK Speaker: SPRING BREAK Location: MB114 March 12th, 2015 (06:30pm - 08:30pm) Title: Private Location: MB102 March 16th, 2015 (12:20pm - 01:30pm) Seminar: CCMA Luncheon Seminar Title: Fitness, Games, and Public Goods Speaker: Andrew Belmonte, Penn State University Location: MB114 March 16th, 2015 (02:30pm - 03:30pm) Seminar: Computational and Applied Mathematics Colloquium Title: Mathematical Modeling of Micromagnetic Complex Fluids Speaker: Johannes Forster, University of Wuerzburg (Host: C Liu) Location: MB106 Magnetic fluids (ferrofluids) have many technological applications. They can not only be found in medical applications, but also in loud speakers and shock absorbers. We investigate magnetic fluids with micromagnetic particles in the framework of complex fluids. From a continuum mechanical setting and an energetic ansatz for the material, we derive PDEs to describe their behavior. We outline the process of modeling and the energetic variational approach. Moreover, we highlight the mathematical problems that arise in the establishment of the PDEs. This is joint work with Carlos Garcia-Cervera (Mathematics Department, University of California, Santa Barbara, USA), Chun Liu (Department of Mathematics, Penn State University, University Park, USA), and Anja Schloemerkemper (Institute for Mathematics, University of Wuerzburg, Germany). March 17th, 2015 (10:00am - 11:00am) Seminar: Hyperbolic and Mixed Type PDEs Seminar Title: Generic singularities of solutions to a nonlinear wave equation. Speaker: Alberto Bressan, Penn State Location: MB216 The talk will be concerned with conservative solutions to the nonlinear wave equation u_{tt} - c(u)(c(u) u_x)_x = 0 For an open dense set of C^3 initial data, the conservative solution is piecewise smooth in the t - x plane, while the gradient u_x can blow up along finitely many characteristic curves. The analysis relies on a variable transformation which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem. A detailed description of the solution profile can be given, in a neighborhood of every singular point and every singular curve. Some results on structurally stable singularities have been obtained also for dissipative solutions. (This work is in collaboration with Geng Chen, Tao Huang, and Fang Yu). March 17th, 2015 (12:20pm - 01:10pm) Seminar: Teaching Mathematics Discussion Group Seminar Title: TBA Speaker: Atendees, Penn State Location: MB216 Abstract: http:// March 17th, 2015 (01:00pm - 02:00pm) Seminar: Mathematical Biology Colloquium Title: Ecological theory for the nonstationary world Speaker: Peter Chesson, University of Arizona (Host: Tim Reluga) Location: MB106 The concept of equilibrium has always been controversial and has always been central in ecological thought. It has been the basis of prediction in ecology, as in many sciences. The vicinity of equilibrium commonly defines the properties expected of a system. In conservation, equilibrium, as a formalization of the ancient concept of the balance of nature, has been imagined to define the essence of a system and to be treated with reverence.However, most natural populations fluctuate greatly, and may exhibit trends on observable time scales. Limit cycles, strange attractors, and stationary probability distributions are various replacements for the equilibrium concept, but they all suffer from the complaint that they are merely equilibria on different scales. None account for long-term climate fluctuations, which are nonstationary and preclude these alternative concepts because they all imply stable long-term frequencies of population states. I demonstrate a new concept, asymptotic environmentally-determined trajectories (AEDTS), able to replace the traditional equilibrium concept while retaining much of its predictive power even though the environment is realistically nonstationary incorporating the fact that the physical environment, including climate, changes on all time scales without stable repetition frequencies. An AEDT is a function of time and is determined by the time series of environmental states and the dynamical rules for the system but is independent of initial population sizes. It thus reflects the multiplicities of interactions between organisms and with their environment.It invokes the kinds of questions and predictions long familiar to ecologists but in a much more realistic context. Convergence of system dynamics on an AEDT involves consideration of feedback loops and stability properties that are generalizations of much traditional theoretical ecology while not requiring the usual stationarity assumption. Realistic consideration of environmental history and a future changing profoundly due to human influence becomes possible. March 17th, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Symplectic Mackey Theory Speaker: Francois Ziegler, Georgia Southern University Location: MB106 When a Lie group G has a closed normal subgroup N, the “Mackey Machine” breaks down the classification of its irreducible representations into two smaller problems: a) find the irreducible representations of N; b) find the irreducible projective representations of certain subgroups of G/N. The desired classification often follows inductively. Key parts of this machine are 1) the “inducing construction” (building representations of G out of those of its subgroups); 2) the “imprimitivity theorem” (characterizing the range of the inducing construction); 3) a “tensoring” construction (combining objects of types a) and b) above). Many years ago Kazhdan, Kostant and Sternberg defined the notion of inducing a hamiltonian action from a Lie subgroup, thus introducing a purely symplectic geometrical analog of 1); and the question arose whether analogs of 2) and 3) could be found and built into an effective “symplectic Mackey Machine”. In this talk I will describe a complete solution to this problem, obtained recently. March 17th, 2015 (02:30pm - 03:30pm) Seminar: Center for Dynamics and Geometry Colloquium Title: Entropy for generalized beta-transformations Speaker: Dan Thompson, Ohio State University Location: MB114 Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation x↦βx (mod1), where β > 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting. I will also describe a connection with some of the results of Thurston's fascinating final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps are shown to describe a beautiful fractal. The talk will be suitable for a general dynamics audience, and for graduate students. March 17th, 2015 (02:30pm - 03:45pm) Seminar: Logic Seminar Title: Strong treeability of planar groups Speaker: Clinton Conley, Carnegie Mellon University Location: MB315 An equivalence relation is called treeable if it can be realized as the connectedness relation of an acyclic Borel graph. We call a finitely generated group planar if there is some finite generating set such that the induced Cayley graph of the group is planar. Using techniques originally created to analyze measure-theoretic chromatic numbers of graphs, we show that any orbit equivalence relation of a free measure-preserving action of a planar group on a standard probability space is treeable on a conull set. This is joint work with Gaboriau, Marks, and Tucker-Drob. March 17th, 2015 (03:30pm - 06:00pm) Seminar: Working Seminar: Dynamics and its Working Tools Title: Introduction to KAM (Kolmogorov-Arnold-Moser) theory, V Speaker: Changguang Dong, Penn State Location: MB114 March 18th, 2015 (03:30pm - 05:00pm) Seminar: Complex Fluids Seminar Title: What is known in elasticity and Ball's open problem #1 Speaker: Barbora Benesova, Institute for Mathematics, University of Wurzburg, Germany Location: MB106 March 18th, 2015 (03:30pm - 05:30pm) Seminar: Applied Algebra and Network Theory Seminar Title: Operadic modularity in networks Speaker: David Spivak, MIT Location: MB315 An operad is a category-theoretic structure that encodes many-input, one-output mappings. In this talk, we will discuss how operads and their algebras can serve as a framework for thinking about modular systems of all kinds, including various kinds of networks. In this setup, an operad O lays out an abstract language of architecture---rules for how interfaces can be arranged to form "higher level" interfaces---and an O-algebra expresses an interpretation of this abstract language. I will also discuss some new connections between operad algebras and various flavors of monoidal categories. March 18th, 2015 (03:35pm - 04:35pm) Seminar: Geometry Working Seminar Title: Combinatorial systolic inequalities Speaker: Minemyer, Barry, Ohio State U. Location: MB114 In this talk (research is joint with Ryan Kowalick and J.F. Lafont) I will establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the$1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. Our main theorem is that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulations. Along the way, we show that any closed Riemannian manifold has a smooth triangulation which "remembers'' the geometry of the Riemannian metric, and conversely, that every smooth triangulation gives rise to Riemannian metrics which encode the combinatorics of the triangulation. I will also show how our main result can be used to "fill" triangulated surfaces via a triangulated 3-manifold with a bounded number of tetrahedra. March 19th, 2015 (10:00am - 11:00am) Seminar: Hyperbolic and Mixed Type PDEs Seminar Title: Piecewise smooth solutions to the Burgers-Hilbert equation. Speaker: Alberto Bressan, Penn State University Location: MB216 In 2009 J.Biello and J.Hunter derived a balance law for nonlinear waves with constant frequency, obtained from Burgers' equation by adding the Hilbert transform as a source term. Recent work has established the global existence of solutions, in the space L^2(R). This talk will describe the construction of piecewise smooth solutions, locally in time. The analysis provides a detailed description of the solution profile in a neighborhood of each shock. Various related open problems will be discussed. (This is a joint work with Tianyou Zhang). March 19th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Construction of abelian varieties with a given Weil number Speaker: Frans Oort, University of Utrecht, visiting University of Pennsylvania Location: MB106 In this talk we sketch methods of algebraic geometry to show once a Weil number is given how to construct an abelian variety with that number as Frobenius. This result was known before, but proofs were through analytic parametrizations. This is joint work with Ching-Li Chai. For a given prime power q a Weil q-number is an algebraic integer having root q as absolute value. We will see that these numbers are easily classified, and using elementary algebra we can construct many examples. Weil showed that the Frobenius of an abelian variety over a field with q elements is a Weil q-number (the first proven case of the Weil conjectures). We recall a (well-known) easy proof of this deep theorem. Honda and Tate showed that every Weil number appears in this way. Hence we have access to existence of abelian varieties just by choosing Weil numbers. We will present a proof that indeed every Weil number appears this way (the trickiest part of the Hoda-Tate theory). In my talk I will give explicit definitions of concepts used, and I will present proofs, that are understandable for a general audience. These deep and beautiful results are now easily understood! March 19th, 2015 (11:30am - 01:00pm) Seminar: Teaching Seminar Title: ALEKS Update Speaker: Jim Hager, Tanya Furman Location: MB114 Abstract: http:// March 19th, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: Higher signatures on Witt spaces Speaker: Zhizhang Xie, Texas A&M Location: MB106 The signature is a fundamental homotopy invariant for topological manifolds. However, for spaces with singularities, this usual notion of signature ceases to exist, since, in general, spaces with singularities fail the usual Poincaré duality. A generalized Poincaré duality theorem for spaces with singularities was proven by Goresky and MacPherson using intersection homology. The classical signature was then extended to Witt spaces by Siegel using this generalized Poincaré duality. Witt spaces are a natural class of spaces with singularities. For example, all complex algebraic varieties are Witt spaces. In this talk, I will describe a combinatorial approach to higher signatures of Witt spaces, using methods of noncommutative geometry. This is based on joint work with Nigel Higson. March 19th, 2015 (03:30pm - 04:20pm) Seminar: Department of Mathematics Colloquium Title: Noncommutative triangulations and the Laurent phenomenon Speaker: Vladimir Retakh (Host: Yuri Zarhin), Rutgers University Location: MB114 The celebrated Ptolemy relation plays an important role in various studies of triangulated surfaces including hyperbolic geometry, geometrical applications of cluster algebras and so on. We will discuss a noncommutative version of the relation which can be seen as a "categorification" of the classical one. This leads to new noncommutative invariants of the surfaces and provides several examples of the noncommutative Laurent phenomenon. (Joint work with Arkady Berenstein from University of Oregon) March 19th, 2015 (06:30pm - 08:30pm) Title: Private Location: MB102 March 20th, 2015 (03:35pm - 04:35pm) Seminar: Probability and Financial Mathematics Seminar Title: Local limit theorems in dynamics Speaker: Manfred Denker, PSU Location: MB106 I will review the results on local limit theorems for continuous maps satisfying some distortion property. Applications to skew product transformations provide proves for conservativity. Some open problems will be mentioned as well. March 23rd, 2015 (02:30pm - 03:30pm) Seminar: Computational and Applied Mathematics Colloquium Title: An introduction to the Vlasov-Poisson system Speaker: Daniel Han-Kwan, Ecole Polytechnique, France (Hosts: A Bressan and T Nguyen) Location: MB106 The Vlasov-Poisson system is a classical model of plasma physics, used to describe the dynamics in phase space of interacting charged particles. We shall review in this lecture some remarkable mathematical properties of this system. The topics reviewed should include (1) the existence of weak or strong solutions, (2) the stability and instability theory of certain equilibria, (3) the quasineutral limit, i.e. the regime when the Debye length is small compared to the typical observation length. March 23rd, 2015 (02:30pm - 03:20pm) Seminar: GAP Seminar Title: Complete integrability from Poisson Nijenhuis structures on compact hermitian symmetric spaces Speaker: Francesco Bonechi, Istituto Nazionale Fisica Nucleare - Firenze Location: MB216 The so called Bruhat-Poisson structure is compatible with the Kostant-Kirillov-Souriau bracket when considered on compact hermitian symmetric spaces. This property allows the definition of a Poisson Nijenhuis structure. We study the spectrum of the Nijenhuis tensor, which proves to be non degenerate and defines a completely integrable model. On the Grassmannians this is the well known Gelfand-Cetlin model. By construction these models have a bihamiltonian description, i.e. the hamiltonians are in involution with respect to both Poisson structures. In this talk I will review the basic facts needed for this analysis, namely the construction of integrable models from collective hamiltonians (Thimm method) and Poisson vector bundles. The point of view mainly focuses on the geometry of the Bruhat-Poisson structures, in particular on its symplectic groupoid. March 23rd, 2015 (03:35pm - 04:35pm) Seminar: Dynamical systems seminar Title: Dynamics from Complex Continued Fractions Speaker: Adam Zydney, Penn State Location: MB106 While complex continued fractions in number theory and dynamics from real continued fractions are well-studied areas, there is little work in dynamical systems coming from complex continued fractions. This talk will discuss the relevant maps and structures involved in continued fraction dynamics and will show an explicit description of the attractor region for a particular system. March 23rd, 2015 (08:00pm - 09:00pm) Seminar: Marker Lecture Series Title: Geometric analysis and topology Speaker: Gang Tian, (Host: Jinchao Xu), Princeton University and Peking University Location: MB114 There are strong ties between geometry and topology. For decades, geometric methods have been applied to attacking topological problems. One distinguished example is the solution of the famous Poincare conjecture by using Hamilton’s Ricci flow. The Poincare conjecture is a famous topological problem which gives a characterization of the simplest topological 3-space, while the Ricci flow had been studied in geometric analysis for many years before it was used for solving the conjecture. The other examples include application of the gauge theory to studying differentiable topology of 4-manifolds in 80s and the use of Cauchy-Riemann equation in constructing the Gromov-Witten invariants in symplectic topology in 90s. In this talk aiming at general audience, I will show how geometric methods can be applied to studying topological spaces. First I will recall some classical facts on surfaces. Secondly, I will give a brief tour on Perelman’s works on geometrization of 3-manifolds and discuss geometric aspects of 4-manifolds. Finally, I will show some geometric methods in symplectic topology, particularly, constructing the Gromov-Witten invariants related to the σ-model in physics. March 24th, 2015 (11:15am - 12:05pm) Seminar: Combinatorics/Partitions Seminar Title: Asymptotics of Multidimensional Partitions Speaker: Daniel Hirsbrunner, PSU Location: MB106 Although MacMahon’s conjecture about the generating function for multidimensional partitions was disproved by Atkin, et al. in 1967, there has been renewed interest in the asymptotic accuracy of this conjecture among physicists since the mid 1990s. Many of the resulting publications are computational in nature, providing very suggestive data. Others make headway in rigorously establishing the asymptotics of the number of multidimensional partitions. The best known result is that log p_d(n) is asymptotically equivalent to n^{d/(d+1)}. March 24th, 2015 (12:20pm - 01:10pm) Seminar: Teaching Mathematics Discussion Group Seminar Title: TBA Speaker: Atendees, Penn State Location: MB216 Abstract: http:// March 24th, 2015 (01:00pm - 02:00pm) Seminar: Theoretical Biology Seminar Title: Delayed action insecticides and their role in mosquito and malaria control Speaker: Rongsong Liu, University of Wyoming Location: MB106 There is considerable interest in the management of insecticide resistance in mosquitoes. One possible approach to slowing down the evolution of resistance is to use late-life-acting (LLA) insecticides that selectively kill only the old mosquitoes that transmit malaria, thereby reducing selection pressure favoring resistance. In this project we consider an age-structured compartmental model for malaria with two mosquito strains that differ in resistance to insecticide, using a compartmental model to describe malaria in the mosquitoes and thereby incorporating the parasite developmental times for the two strains. The human population is modeled using a susceptible-exposed-infected compartmental model. We consider both conventional insecticides that target all adult mosquitoes, and LLA insecticides that target only old mosquitoes. According to linearised theory the potency of the insecticide affects mainly the speed of evolution of resistance. Mutations that confer resistance can also affect other parameters such as mean adult life span and parasite developmental time. For both conventional and LLA insecticides the stability of the malaria-free equilibrium, with only the resistant mosquito strain present, depends mainly on these other parameters. This suggests that the main long term role of an insecticide could be to induce genetic changes that have a desirable effect on a vital parameter such as adult life span. However, when this equilibrium is unstable, numerical simulations suggest that a potent LLA insecticide can slow down the spread of malaria in humans but that the timing of its action is very important. March 24th, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Gravity in three dimensions: physical foundations Speaker: Marc Geiller, Penn State Location: MB106 The year 2015 marks the centennial anniversary of the birth of Einstein's theory of general relativity, which is so far the most successful and experimentally tested description of the gravitational interaction. Although the physical spacetime around us is four-dimensional, general relativity, because of its geometrical nature, can also be formulated in three spacetime dimensions. There it exhibits special mathematical structure, and can be studied in exactly soluble ways. The goal of this series of lectures is to present some aspects of the rich interplay that exists between mathematics and physics within the context of three-dimensional gravity. Part 1: Physical foundations Part 2: Mathematical formulations Part 3: Quantization via loop groups Part 4: Path integral quantization and topological invariants March 24th, 2015 (02:30pm - 03:45pm) Seminar: Logic Seminar Title: Precisely Constructed Taxa and Higher Order Dangers in Models of Randomness Speaker: Steven Pincus, Guilford, CT Location: MB315 The certification, explicit construction and delineation of individual, infinite length random sequences have been longstanding, yet incompletely resolved problems. We address this topic via the study of normal numbers, which have often been viewed as reasonable proxies for randomness, given their limiting equidistribution of subblocks of all lengths. However, limitations arise within this perspective. First, we develop several criteria motivated by classical theorems for symmetric random walks, which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums, including rates of sign changes, patterns of return times to 0, and the extent of fairness of the sequence. Such characteristics are generally unaddressed in most evaluations of randomness. Second, we explicitly construct a normal number that satisfies the Law of the Iterated Logarithm (LIL), yet exhibits pairwise bias towards repeated values, rendering it inappropriate for any collection of random numbers. Accordingly, we deduce that the evaluation of higher order block dynamics, even beyond limiting equidistribution and fluctuational typicality, is imperative in proper evaluation of sequential randomness. More broadly, we can now differentiate normal numbers both on the basis of multiple distinct qualitative attributes, as well as quantitatively via a spectrum of rates within each attribute. Furthermore, we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications, including profound biases. Overall, we elucidate the vast diversity within the category of normal sequences. March 24th, 2015 (03:30pm - 04:30pm) Seminar: Marker Lecture Series Title: Introduction to gauged Witten equation Speaker: Gang Tian, (Host: Jinchao Xu), Princeton University and Peking University Location: MB114 In this and subsequent two talks, I will discuss my recent program with Guangbo Xu on constructing a mathematical theory of the gauged linear - model. First, I will introduce the gauged Witten equation, which also generalizes the symplectic vortex equation studied in the gauged Gromov-Witten theory. I will discuss some of its analytical properties, including the asymptotical behavior of nite energy solutions and the linear Fredholm property. March 25th, 2015 (12:05pm - 01:20pm) Seminar: Geometry Luncheon Seminar Title: Geodesics on convex surfaces. Speaker: Anton Petrunin, Penn State Location: MB114 We give a universal bound for the variation of turn of minimizing geodesics on convex surfaces. This is a joint work with Nina Lebedeva. March 25th, 2015 (03:30pm - 05:30pm) Seminar: Applied Algebra and Network Theory Seminar Title: Mesoscale topological statistics of force chain networks Speaker: Chad Giusti, University of Pennsylvania Location: MB315 Densely packed granular media exhibit a rich internal network of interactions characterized by so-called "force chains" consisting of particles which exert above-average forces on one another. The structure of these chains plays a substantial role in the response of the media to perturbation, but the mechanisms by which this happens are not well understood. A vital first step toward prediction and design of material packings is the development of techniques for measuring physically salient properties of force chains. Here, we describe work in progress on a data-driven approach to the problem which combines community-detection techniques for extracting force chains with topological statistics of the resulting structures to provide a mesoscale description of the force chain network. March 25th, 2015 (03:35pm - 04:35pm) Seminar: Marker Lecture Series Title: Compactness theorem for gauged Witten equation Speaker: Gang Tian, (Host: Jinchao Xu), Princeton University and Peking University Location: MB114 In this talk, I will discuss compactness results for the gauged Witten equation and its perturbation. The key is to establish a uniform C0-bound for solutions. I will explain how this can be done. March 26th, 2015 (10:00am - 11:00am) Seminar: Marker Lecture Series Title: Correlation functions for gauged linear σ-model Speaker: Gang Tian, (Host: Jinchao Xu), Princeton University and Peking University Location: MB114 In this last talk, I will give the definition of the correlation function of the gauged linear σ-model for a fixed smooth r-spin curve. I will first discuss certain cohomology groups which are used as state spaces for the gauged linear σ-model and which generalize the state spaces in Landau-Ginzburg A-model. The correlation function is defined as a family of multi-linear maps on those generalized state spaces by using the moduli for solutions of the gauged Witten equation. March 26th, 2015 (10:00am - 11:00am) Seminar: Hyperbolic and Mixed Type PDEs Seminar Title: Blow up for the two-component Camassa--Holm system Speaker: Katrin Grunert, Norwegian University of Science and Technology Location: MB216 The two-component Camassa--Holm system u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x&=0, serves as a model for shallow water. Furthermore, it is a generalization of the famous Camassa--Holm equation, which has been studied intensively due to its rich mathematical structure. Thus a huge class of solutions enjoys wave breaking within finite time, but there is also a regularising effect which prevents many solutions form blow up. Hence the aim of this talk is twofolded. On the one hand we want to study this regularising effect in some detail and on the other hand we want to focus on how to predict if a solution enjoys wave breaking in the nearby future or not. This talk is based on joint work with H. Holden and X. Raynaud. March 26th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Arithmetic Combinatorics and Character Sums Speaker: Brandon Hanson, University of Toronto Location: MB106 In this talk I will present a few ideas as to how character sums may be useful in arithmetic combinatorics and vice versa. I will talk about how character sums can be used to make progress on problems coming from arithmetic combinatorics. On the other hand, arithmetic combinatorics can prove useful when going the other way. Indeed, many character sums are easy to estimate provided they have enough summands - this is sometimes called the square-root barrier and is a natural obstruction. I will show how the sum-product phenomenon can be leveraged to push past this barrier. March 26th, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: Gravity in three dimensions: discussion Speaker: Nigel Higson, Penn State Location: MB106 March 26th, 2015 (03:30pm - 04:20pm) Seminar: Department of Mathematics Colloquium Title: Fast-slow systems with chaotic noise Speaker: David Kelly (Host: John Harlim), Courant Institute Location: MB114 It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the chaotic noise is multidimensional and multiplicative. The tools from rough path theory prove useful in this difficult setting. March 26th, 2015 (06:30pm - 08:30pm) Title: Private Location: MB102 March 27th, 2015 (01:00pm - 02:30pm) Seminar: CCMA PDEs and Numerical Methods Seminar Series Title: Existence of a dynamic system of ionic electrodiffusion Speaker: Tao Huang, Penn State Location: MB315 We consider a dynamic system of ionic electrodiusion which can be considered as a special case of cardiac bidomain model. A global weak solution has been constructed by Galerkin argument and maximum principle. March 27th, 2015 (03:35pm - 04:35pm) Seminar: Probability and Financial Mathematics Seminar Title: G/G/N Queues with Service Interruptions in the Halfin-Whitt Regime Speaker: Guodong Pang, PSU, Dept. of Industrial and Manufacturing Engineering Location: MB106 We consider G/G/N queues with renewal alternating service interruptions. The arrival process is general and the service times forms a stationary and weakly dependent sequence satisfying some strong mixing condition. The system experiences up and down alternating periods. Both the arrival and service processes operate normally in the up periods. In the down periods, arrivals continue entering the system, but all servers break down and the amount of service a customer has received will be conserved and resumed when the next up period starts. We assume that the up times are of the same order as the service times but the down times are asymptotically negligible compared with the service times. In the QD and QED regimes, we prove FLLNs and FCLTs for the total count processes and the two-parameter queueing processes tracking elapsed or residual times. The limit processes in the FCLTs are characterized via stochastic integral equations with jumps, and the convergence requires Skorohod M_1 topology in the spaces D([0,T], R) and D([0, T], D([0, T], R)). (This is joint work with Yuhang Zhou and Hongyuan Lu.) March 30th, 2015 (12:20pm - 01:30pm) Seminar: CCMA Luncheon Seminar Title: Energy-Stable Open Boundary Conditions for Two-Phase Outflows Speaker: Suchuan Steven Dong, Purdue University (Host: J Xu) Location: MB114 This talk focuses on the motion of a mixture of two immiscible incompressible fluids in a domain with open boundaries. The domain boundary is open in the sense that the fluids can freely leave or even enter the domain through such boundaries. In particular, we concentrate on situations where the interface formed between the two fluids passes through the open portions of the domain boundary. The problem therefore involves truly two-phase outflow/open boundaries. The challenge facing the design of effective techniques for treating two-phase outflows in numerical simulations is manifold. Some of the primary issues are associated with the viscosity contrasts, density contrasts, surface tension, and the presence of fluid interface, backflows or strong vortices on the open boundaries. Large density ratios and large viscosity ratios of the two fluids make two-phase outflow simulations tremendously challenging. In this talk we present a set of boundary conditions, and an associated numerical algorithm, for two-phase outflow simulations within the phase field framework. These open boundary conditions have the characteristic that they all ensure the energy stability of the two-phase system, even in situations where strong vortices, backflows, large density contrast and large viscosity contrast are present at the open boundaries. We will show the physical accuracy of the method by comparing simulation results with the theory and experimental data. Numerical experiments will be presented to demonstrate the long-term stability of the method in situations where large density contrast, large viscosity contrast, and backflows are present at the outflow/open boundaries. March 30th, 2015 (02:30pm - 03:30pm) Seminar: Computational and Applied Mathematics Colloquium Title: Incompressible N-Phase Flows: Physical Formulation and Numerical Algorithm Speaker: Suchuan Steven Dong, Purdue University (Host: J Xu) Location: MB106 This talk focuses on simulating the motion of a mixture of N (N>=2) immiscible incompressible fluids with given densities, dynamic viscosities and pairwise surface tensions. We present an N-phase formulation within the phase field framework that is thermodynamically consistent, in the sense that the formulation satisfies the conservations of mass/momentum, the second law of thermodynamics and Galilean invariance. In addition, we also present an efficient algorithm for numerically simulating the N-phase system that has overcome the issues caused by the variable mixture density/viscosity and the couplings among the (N-1) phase field variables and the flow variables. We compare simulation results with the Langmuir-de Gennes theory to demonstrate that the presented method produces physically accurate results for multiple fluid phases. Numerical experiments will be presented for several problems involving multiple fluid phases, large density contrasts and large viscosity contrasts to demonstrate the capabilities of the method for studying the interactions among multiple types of fluid interfaces. March 30th, 2015 (03:35pm - 04:35pm) Seminar: Dynamical systems seminar Title: Kolmogorov and Bernoulli property for partially hyperbolic diffeomorphisms Speaker: Ali Tahzibi, ICMC-USP Sao Carlos-Brazil Location: MB114 There is a hierarchy among the order of unpredictability of dynamical systems. The Bernoulli property (conjugacy with a Bernoulli Shift) is the strongest form of unpredictability. Each Bernoulli system, in particular, is a Kolmogorov system. However, the inverse is not always true. In this talk we review some known results and prove the equivalence between Kolmogorov and Bernoullicity of volume measure for partially hyperbolic systems which are derived from Anosov and have one dimensional central bundle on Torus. In particular we announce some recent results on disintegration of measures along central foliation of partially hyperbolic dynamics. This is a joint work with J.R.Varão and G. Ponce. March 31st, 2015 (11:15am - 12:05pm) Seminar: Combinatorics/Partitions Seminar Title: Enumeration Through Partial Bell Polynomials Speaker: Mike Weiner, PSU Altoona Location: MB106 We give a brief introduction to partial Bell polynomials and discuss how they can be used to enumerate trees, paths and polygon partitions. In this talk we will focus on finding the total number of colored partitions of a convex polygon by non-intersecting diagonals into convex polygons with prescribed properties. We give explicit examples and discuss how this approach uni es several known results. March 31st, 2015 (12:20pm - 01:10pm) Seminar: Teaching Mathematics Discussion Group Seminar Title: TBA Speaker: Atendees, Penn State Location: MB216 Abstract: http:// March 31st, 2015 (01:00pm - 02:00pm) Seminar: Theoretical Biology Seminar Title: A game-theoretic dispersal mechanism in PDE models of interacting populations Speaker: Russ deForest, doctoral candidate, Department of Mathematics, PSU Location: MB106 We adapt a fitness from evolutionary game theory as a dispersal mechanism in spatial PDE models of interacting populations. Evolutionary games are used to model selection dynamics among competing traits or strategies. The relative frequencies of competing strategies evolve according to an ODE model governed by a replicator equation. We spatially extend these models by allowing populations to travel up a fitness gradient. We discuss results for some two-species models, including cross-diffusive instabilities and pattern formation in a spatial Lotka-Volterra model. Some background on PDE models for interacting populations and spatial games will be given with a focus on PDE systems that are normally parabolic, but in general non-coercive. March 31st, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Gravity in three dimensions: mathematical foundations Speaker: Marc Geiller, Penn State Location: MB106 The year 2015 marks the centennial anniversary of the birth of Einstein's theory of general relativity, which is so far the most successful and experimentally tested description of the gravitational interaction. Although the physical spacetime around us is four-dimensional, general relativity, because of its geometrical nature, can also be formulated in three spacetime dimensions. There it exhibits special mathematical structure, and can be studied in exactly soluble ways. The goal of this series of lectures is to present some aspects of the rich interplay that exists between mathematics and physics within the context of three-dimensional gravity. Part 1: Physical foundations Part 2: Mathematical formulations Part 3: Quantization via loop groups Part 4: Path integral quantization and topological invariants March 31st, 2015 (02:30pm - 03:30pm) Seminar: Center for Dynamics and Geometry Colloquium Title: Deformations of boundary distances and geodesic flows Speaker: Sergei Ivanov, Russian Academy of Sciences, St. Petersburg Location: MB114 This is a joint work with Dima Burago. We show that a simple Finsler metric on the n-disc can be deformed so as to induce an arbitrary perturbation of the boundary distance function, or an arbitrary symplectic perturbation of the geodesic scattering map. Among the applications is a construction of a metric on the 4-sphere arbitrarily close to the standard "round" metric and having positive metric entropy of its geodesic flow (which is regarded as a Hamiltonian flow). March 31st, 2015 (02:30pm - 03:45pm) Seminar: Logic Seminar Title: No Seminar this week Speaker: No Seminar this week Location: MB315 March 31st, 2015 (03:30pm - 06:00pm) Seminar: Working Seminar: Dynamics and its Working Tools Title: Dynamics in threshold-linear networks Speaker: Carina Curto, Penn State Location: MB114 Threshold-linear networks are simplified models of neural networks in the brain. I will first describe how these networks can operate as traditional attractor neural networks (similar to the Hopfield model), and what we can say about the set of stable fixed points. I will then show how we can construct networks without fixed points, and state a conjecture about the dynamics in this regime. April 1st, 2015 (12:05pm - 01:20pm) Seminar: Geometry Luncheon Seminar Title: A Fary-Milnor theorem for CAT(0) spaces Speaker: Stephan Stadler, University of Cologne Location: MB114 We prove the following CAT(0) version of Fary-Milnor's theorem on knots. If$\gamma$is a Jordan curve of total curvature$\leq4\pi$in a CAT(0) space, then either$\gamma$spans an embedded disc or else the total curvature of$\gamma$equals$4\pi$and$\gamma$bounds a star shaped subset intrinsically isometric to a Euclidean cone of cone angle equal to$4\pi$. April 1st, 2015 (03:30pm - 05:30pm) Seminar: Applied Algebra and Network Theory Seminar Title: A Reformulation of the CSSR Algorithm and Application to Optimal Deception Strategy Speaker: Elisabeth Paulson, Penn State Location: MB315 In this talk we explore a reformulation of the Casual State Splitting and Reconstruction (CSSR) algorithm and an application to optimal strategies for deception in two-player games. The CSSR algorithm is used to infer probabilistic finite-state machines from an input stream of data. We formulate an integer programming version of the CSSR algorithm which always results in minimal probabilistic finite-state machine. This reformulation is shown to be NP-hard by comparing it to the minimal clique covering problem in graph theory. We then apply this algorithm to optimal deception strategies in repeated two-player games. We find that this deception can be modeled by combining both linear optimization with a genetic algorithm. We present numerical examples of optimal deception as well as some theoretical results. April 1st, 2015 (03:35pm - 04:35pm) Seminar: Geometry Working Seminar Title: TBA Speaker: Stephan Stadler, University of Cologne Location: MB114 TBA April 2nd, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Homogeneous additive equations over p-adic fields Speaker: Mike Knapp, Loyola University Location: MB106 In this talk, we will study solutions of the equation a_1x_1^d + a_2x_2^d + ... + a_sx_s^d = 0 in p-adic integers. It has been known since the 1960s that if s>= d^2 + 1, then this equation will have nontrivial p-adic solutions for any prime p, regardless of the coefficients. This bound is sharp when$d+1$is prime, but can be reduced when$d+1$is composite. Given a degree d, we define \Gamma^*(d) to be the smallest number of variables which guarantees that the above equation has nontrivial p-adic solutions for all p. In the first half of the talk, we will evaluate the exact values of \Gamma^*(d) for some small degrees. After that, we will focus specifically on the 2-adic version of the problem and give an exact formula for the smallest number of variables which guarantees that the equation has nontrivial 2-adic solutions. April 2nd, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: Gravity in three dimensions: quantization via loop groups Speaker: Marc Geiller, Penn State Location: MB106 The year 2015 marks the centennial anniversary of the birth of Einstein's theory of general relativity, which is so far the most successful and experimentally tested description of the gravitational interaction. Although the physical spacetime around us is four-dimensional, general relativity, because of its geometrical nature, can also be formulated in three spacetime dimensions. There it exhibits special mathematical structure, and can be studied in exactly soluble ways. The goal of this series of lectures is to present some aspects of the rich interplay that exists between mathematics and physics within the context of three-dimensional gravity. Part 1: Physical foundations Part 2: Mathematical formulations Part 3: Quantization via loop groups Part 4: Path integral quantization and topological invariants April 2nd, 2015 (03:30pm - 04:20pm) Seminar: Department of Mathematics Colloquium Title: Algebraic operations in geometry, topology and physics Speaker: Ralph Kaufmann (Host: Ping Xu), Purdue University Location: MB114 Algebraic structures help in studying complex problems, by both organizing the data as well as providing finer invariants. Paradigmatic examples are groups of invariants, ring structures on them, but also generating functions and their properties, such as satisfying certain PDEs. I will start by giving examples of such structures and then proceed to give a common framework for these algebraic structures themselves. April 2nd, 2015 (06:30pm - 08:30pm) Title: Private Location: MB102 April 3rd, 2015 (03:35pm - 04:35pm) Seminar: Probability and Financial Mathematics Seminar Title: Ergodicity Results for Stochastic Boussinesq Equations Speaker: Geordie Richards, Department of Mathematics Rochester University Location: MB106 We will review some recent results on invariant measures for stochastic Boussinesq equations (model equations for Rayleigh-Benard convection perturbed by an additive noise). First we will discuss ergodicity and mixing results in the two-dimensional periodic domain with a spatially degenerate stochastic forcing. These results generalize recent progress of Hairer and Mattingly for the stochastic Navier-Stokes equations. Then, with a less degenerate forcing but more physical boundary conditions, we present a simplified proof of ergodicity, and discuss some singular parameter limits. This is a joint work with Nathan Glatt-Holtz (Virginia Tech), Juraj Foldes (Universite Libre de Bruxelles) and Enrique Thomann (Oregon State University). April 6th, 2015 (03:35pm - 04:35pm) Seminar: Dynamical systems seminar Title: Conservative Anosov transformations on the two torus without absolutely continuous Lebesgue invariant measures Speaker: Zemer Kosloff, University of Warwick Location: MB114 Abstract: http://www.personal.psu.edu/sxk37/Abstract_Conservative_Anosov.pdf April 7th, 2015 (12:20pm - 01:10pm) Seminar: Teaching Mathematics Discussion Group Seminar Title: TBA Speaker: Atendees, Penn State Location: MB216 Abstract: http:// April 7th, 2015 (01:00pm - 02:00pm) Seminar: Theoretical Biology Seminar Title: TBA Speaker: Raibatak Das, University of Colorado Denver (Host: Jessica Conway) Location: MB106 April 7th, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Gravity in three dimensions: path integral quantization and topological invariants Speaker: Marc Geiller, Penn State Location: MB106 The year 2015 marks the centennial anniversary of the birth of Einstein's theory of general relativity, which is so far the most successful and experimentally tested description of the gravitational interaction. Although the physical spacetime around us is four-dimensional, general relativity, because of its geometrical nature, can also be formulated in three spacetime dimensions. There it exhibits special mathematical structure, and can be studied in exactly soluble ways. The goal of this series of lectures is to present some aspects of the rich interplay that exists between mathematics and physics within the context of three-dimensional gravity. Part 1: Physical foundations Part 2: Mathematical formulations Part 3: Quantization via loop groups Part 4: Path integral quantization and topological invariants April 7th, 2015 (02:30pm - 03:30pm) Seminar: Center for Dynamics and Geometry Colloquium Title: Beyond the obstructions of KAM theory Speaker: Raphael Krikorian, University of Paris VI Location: MB114 KAM theory is a very powerful and useful tool in Analysis and Dynamical Systems. It is a far reaching generalization of the classical fixed point theorem. In Dynamical Systems it originates from the works of Kolmogorov, Arnold and Moser and in Analysis from the works of Nash and Moser. Though KAM theory is a very versatile paradigm, its scope of application is limited by three classical obstructions: it is perturbative, it requires to deal with small denominators and as a consequence to exclude some parameters from the analysis of the problem under study. I will try to describe some cases where these obstructions can be removed. April 7th, 2015 (03:30pm - 06:00pm) Seminar: Working Seminar: Dynamics and its Working Tools Title: Coexistence of absolutely continuous and pure point spectrum for 1D quasi-periodic Schrödinger operators Speaker: Raphael Krikorian, University of Paris, Jussieu Location: MB114 I will describe new examples of quasi-periodic potentials for which the spectrum of the corresponding Schrödinger operators has both absolutely continuous and pure point part in its spectrum. This is a joint work with K. Bjerklöv April 7th, 2015 (04:00pm - 05:00pm) Seminar: Applied Analysis Seminar Title: Stabilization by noise Speaker: Jonathan Mattingly, Duke University Location: MB106 Noise is usually though of as a destabilizing force. I will discuss a few examples where the noise has a stabilizing effect. I will begin with a simple class of planer ODEs which exhibit blow-up for some initial data. I will show how careful balancing of the dynamics near the unstable manifold with the noise will lead to stable longtime behavior. While system will exhibit intermittent behavior with a fat-tailed invariant distribution, it will converge exponentially fast to equilibrium. The proofs will turn on building a optimal Lyapunov function using associated Poisson equations. If time permits, I will discuss some other examples of stabilization by noise including possibly Hamiltonian dynamics with a singular potential and the example of selection of long term statistics by the addition of noise. The last example is a toy version of the selection of an unique invariant measure in the inviscid limit of and stochastic PDE. April 8th, 2015 (03:30pm - 05:30pm) Seminar: Applied Algebra and Network Theory Seminar Title: Matrix Completion for the Independence Model Speaker: Zvi Rosen, University of California, Berkeley Location: MB315 Suppose you are given some entries of a matrix of probabilities for a pair of discrete random variables. When is it possible that these entries come from the independence model? In other words, when can we complete a partial matrix to a rank-1 nonnegative matrix whose entries add up to one? We will approach this problem using combinatorics and algebraic geometry. This talk is based on joint work with Kaie Kubjas (Aalto). April 9th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Rational points near hypersurfaces: with applications to the Dimension Growth Conjecture and metric diophantine approximation Speaker: Jing-Jing Huang, University of Toronto Location: MB106 The distribution of rational points on algebraic varieties is a central problem in number theory. An even more general problem is to investigate rational points near manifolds, where the algebraic condition is replaced with the non-vanishing curvature condition. In this talk, we will establish a sharp bound for the number of rational points of a given height and within a given distance to a hypersurface. This has surprising applications to counting rational points lying on the manifold; indeed setting the distance to zero, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties) in this general setup. In the second half of the talk, we will focus on metric diophantine approximation on manifolds. A long standing conjecture in this area is the Generalized Baker-Schmidt Problem. As another consequence of the main counting result above, we settle this problem for all hypersurfaces with non-vanishing Gaussian curvatures. Finally, if time permits, we will briefly elaborate on the main ideas behind the proof. April 9th, 2015 (02:30pm - 03:30pm) Seminar: Noncommutative Geometry Seminar Title: Loop groups and Dirac operators on quasi-Hamiltonian G-spaces Speaker: Yanli Song, University of Toronto Location: MB106 A quasi-Hamiltonian G-space is a finite dimensional model, introduced by Alekseev-Malkin-Meinrenken, for a Hamiltonian loop group space. In this talk, I will discuss some basic properties of q-Hamiltonian G-spaces, and construct twisted spinor bundles and twisted prequantum bundles on them. Then I will define the Dirac operator on a q-Hamiltonian G-space, with index given by a positive energy representation of the loop group. This generalizes the quantization of Hamiltonian G-spaces to quasi-Hamiltonian G-spaces. April 10th, 2015 (03:35pm - 04:35pm) Seminar: Probability and Financial Mathematics Seminar Title: Kolmogorov complexity versions of the Slepian-Wolf Theorem (joint with logic seminar) Speaker: Marius Zimand, Towson University Location: MB106 By the Shannon noiseless coding theorem, two correlated random variables (X,Y) can be optimally compressed to length H(X,Y) (where H is Shannon entropy). The Slepian-Wolf theorem shows that the rate R=H(X,Y) can be achieved even for separate encoding of X and Y. We will discuss several Kolmogorov complexity versions of the Slepian-Wolf theorem. As one particular application of our results, consider the following situation: Alice has a string x which she wants to communicate to Bob, and Bob has a correlated string y. We will show that, under some reasonable assumptions, Alice can compute in polynomial time and send to Bob a string p of length approximately C(x | y) (where C is the Kolmogorov complexity) such that Bob can reconstruct x from p and y. April 13th, 2015 (12:20pm - 01:30pm) Seminar: CCMA Luncheon Seminar Title: Limit dynamics of a small solid in a perfect incompressible fluid. Speaker: Olivier Glass, Université Paris-Dauphine (Host: A. Bressan) Location: MB114 This is an introductory talk to the Computational and Applied Mathematics Colloquium April 13th, 2015 (02:30pm - 03:30pm) Seminar: Computational and Applied Mathematics Colloquium Title: Limit dynamics of a small solid in a perfect incompressible fluid. Speaker: Olivier Glass, Université Paris-Dauphine (Host: A Bressan) Location: MB106 We consider a solid in a two-dimensional perfect incompressible fluid. The fluid is driven by the classical Euler equation, and the solid evolves according to Newton's law under the influence of the pressure on its surface. We consider the limit of the system as the solid shrinks to a point. We obtain various different models in the limit. A first model is obtained when the mass of the solid and the circulation around it are fixed; in that case the system converges to a variant of Marchioro and Pulvirenti's vortex-wave system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force. A second one is obtained when the mass of the solid and its density are fixed; in that case, we recover in the limit the vortex-wave system itself. These results are obtained in collaboration with Christophe Lacave (Paris-Diderot), Alexandre Munnier (Nancy), and Franck Sueur (Bordeaux). April 13th, 2015 (03:35pm - 04:35pm) Seminar: Dynamical systems seminar Title: Classification of Thurston maps with parabolic orbifolds Speaker: Nikita Selinger, SUNY Stony Brook Location: MB114 In a joint work with M. Yampolsky, we give a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a partial solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps. April 14th, 2015 (12:20pm - 01:10pm) Seminar: Teaching Mathematics Discussion Group Seminar Title: TBA Speaker: Atendees, Penn State Location: MB216 Abstract: http:// April 14th, 2015 (01:00pm - 02:00pm) Seminar: Theoretical Biology Seminar Title: TBA Speaker: Garrett Mitchener, College of Charleston (Host: Andrew Belmonte) Location: MB106 April 14th, 2015 (02:30pm - 03:30pm) Seminar: GAP Seminar Title: Split octonions and the rolling ball Speaker: John Baez, UC Riverside Location: MB106 Understanding exceptional Lie groups as the symmetry groups of more familiar objects is a fascinating challenge. The compact form of the smallest exceptional Lie group, G2, is the symmetry group of an 8-dimensional nonassociative algebra called the octonions. However, another form of this group arises as symmetries of a simple problem in classical mechanics! The space of configurations of a ball rolling on another ball without slipping or twisting defines a manifold where the tangent space of each point is equipped with a 2-dimensional subspace describing the allowed infinitesimal motions. Under certain special conditions, the split real form of G2 acts as symmetries. We can understand this using the quaternions together with an 8-dimensional algebra called the 'split octonions'. The rolling ball picture makes the geometry associated to G2 quite vivid. This is joint work with James Dolan and John Huerta. April 14th, 2015 (02:30pm - 03:30pm) Seminar: Center for Dynamics and Geometry Colloquium Title: Stochastic Arnold diffusion. Speaker: Vadim Kaloshin, University of Maryland, College Park Location: MB114 In 1964 V. Arnold constructed an example of nearly integrable deterministic system exhibiting instabilities. In the 1970s physicist B. Chirikov coined the term for this phenomenon Arnold diffusion'', where diffusion refers to stochastic nature of instability. One of most famous example of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by astronomer J. Wisdom. During the talk we describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. This result is conceptually different from all known mathematical results, where existence of a diffusing orbit’' is shown. This work is based on three papers: one is joint with O. Castejon, another is joint with M. Guardia and J. Zhang, and the third one is joint with J. Zhang and K. Zhang. April 14th, 2015 (03:30pm - 06:00pm) Seminar: Working Seminar: Dynamics and its Working Tools Title: TBA Speaker: Raphael Krikorian, University of Paris, Jussieu Location: MB216 April 15th, 2015 (12:05pm - 01:20pm) Seminar: Geometry Luncheon Seminar Title: Exceptional Jordan algebras and the Leech lattice Speaker: John Baez, University of California (Host: John Roe) Location: MB114 When Jordan, Wigner and von Neumann classified algebras of observables in their work on the foundations of quantum mechanics, they found 4 infinite series and one exception. This 'exceptional Jordan algebra' is 27-dimensional and consists of 3x3 self-adjoint octonionic matrices. The Leech lattice is another exceptional structure: the unique 24-dimensional even unimodular lattice with no vectors of length squared 2. I'll explain these entities and describe some work with Greg Egan where we made the Leech lattice into a 'Jordan subring' of the exceptional Jordan algebra. April 16th, 2015 (11:15am - 12:05pm) Seminar: Algebra and Number Theory Seminar Title: Power Partitions Speaker: Ayla Gafni, Penn State University Location: MB106 In 1918, Hardy and Ramanujan published a seminal paper which included an asymptotic formula for the partition function. In their paper, they also state without proof an asymptotic equivalence for the number of partitions of a number into$k$-th powers. In 1934, E. Maitland Wright [Acta Mathematica, 63 (1934) 143--191] gives a very precise asymptotic formula for this restricted partition function, but his argument is quite long and difficult. In this talk, I will present an asymptotic formula for the number of partitions into$k\$-th powers using a relatively simple method, while maintaining a decent error term.
April 16th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Singular foliations and their C* algebras: calculations. 1.
Speaker: Iakovos Androulidakis, University of Athens
Location: MB106
Singular foliations are examples of dynamical systems. They are abundant in many branches of mathematics, for instance control theory and Poisson geometry. In fact singular foliations appear much more often than regular ones. In this series of talks we discuss how to deal with the leaf space of such foliations, including calculations of various examples. Information about this space is encapsulated in the holonomy groupoid of the foliation and the associated C*-algebra. A tentative program for these lectures is: (1) singular foliations and bisubmersions, with examples (foliation by the flow of a single vector field, by orbits of the SO(3) action, by orbits of the action of SL(2,R)), (b) calculation of the holonomy groupoid for the above examples, (c) construction of the foliation C*-algebra, and (d) K-theory calculation for the above examples (the right-hand side of the Baum-Connes assembly map).
April 16th, 2015 (03:30pm - 04:20pm)
Seminar: Department of Mathematics Colloquium
Title: Department Faculty Meeting
Speaker: Department Faculty Meeting, Penn State
Location: MB114
April 20th, 2015 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Computational Challenges in Reservoir Simulations
Speaker: Ilya Mishev, ExxonMobil Upstream Research Company (Host: L Zikatanov)
Location: MB114
Reservoir simulations are extensively used in the petroleum industry to predict the optimal way to produce oil and gas from the reservoirs. Understanding how oil, gas, and water flow in the subsurface is essential for the success of the simulations. Modeling of fluid flow in porous media requires solving a strongly couple system of nonlinear PDEs in highly heterogeneous and anisotropic media. The PDEs exhibit both hyperbolic and parabolic features. Following accurately the geologic layers leads to grids that challenge the approximation methods. We will sketch the approaches used in the petroleum industry and some of the academic research for the discretization of the equations and solving the linear systems and discuss the challenges.
April 20th, 2015 (02:30pm - 03:30pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Linear Solvers for Reservoir Simulation problems
Speaker: Ilya Mishev, ExxonMobil Upstream Research Company (Host: L Zikatanov)
Location: MB106
Linear solvers are usually the most time consuming part of reservoir simulations. A considerable amount of research has been devoted to reducing the time necessary for solving the linear system. One approach is to use an operator splitting scheme that replace the fully implicit formulation with IMPES (Implicit Pressure Explicit Saturation) or Sequential Implicit (Implicit Pressure followed by Implicit Saturation). The size of the linear systems is reduced considerably and there is no coupling of hyperbolic and parabolic features. Nevertheless, the linear systems from the discretization of the pressure equation are still difficult to solve because of the size and the properties of the mesh, the heterogeneity and the anisotropy of the porous media. I will share my experience in development of overlapping and non-overlapping Additive Schwarz Domain Decomposition preconditioners for solving the pressure linear systems and discuss the use of Algebraic MultiGrid.
April 21st, 2015 (12:20pm - 01:10pm)
Seminar: Teaching Mathematics Discussion Group Seminar
Title: TBA
Speaker: Atendees, Penn State
Location: MB216
Abstract: http://
April 21st, 2015 (01:00pm - 02:00pm)
Seminar: Theoretical Biology Seminar
Title: TBA
Speaker: Tomo Pisanski
(Host: Carina Curto)
Location: MB106
April 21st, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: TBA
Speaker: Francesco D'Andrea, University of Naples
Location: MB106
April 22nd, 2015 (03:30pm - 05:30pm)
Seminar: Applied Algebra and Network Theory Seminar
Title: A Compositional Approach to Networks
Speaker: Brendan Fong, Oxford University
Location: MB315
Throughout engineering and computer science, network-type diagrams are used to represent and reason about systems. The most familiar such framework is perhaps that of electrical circuit diagrams, while other examples include signal flow graphs, bond graphs, Petri nets, automata, and similar. In this talk we ask what common algebraic structures underlie such diagrams, finding suitable language to address this question in the mathematical field known as monoidal category theory. This allows us to formalise relationships between different diagrammatic languages, as well as discuss their semantics; we shall give examples arising from electrical circuits. This is joint work with John Baez.
April 23rd, 2015 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: To be announced
Speaker: Alina Cojocaru, University of Illinois at Chicago
Location: MB106
April 23rd, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Singular foliations and their C* algebras: calculations. 2.
Speaker: Iakovos Androulidakis, University of Athens
Location: MB106
Singular foliations are examples of dynamical systems. They are abundant in many branches of mathematics, for instance control theory and Poisson geometry. In fact singular foliations appear much more often than regular ones. In this series of talks we discuss how to deal with the leaf space of such foliations, including calculations of various examples. Information about this space is encapsulated in the holonomy groupoid of the foliation and the associated C*-algebra. A tentative program for these lectures is: (1) singular foliations and bisubmersions, with examples (foliation by the flow of a single vector field, by orbits of the SO(3) action, by orbits of the action of SL(2,R)), (b) calculation of the holonomy groupoid for the above examples, (c) construction of the foliation C*-algebra, and (d) K-theory calculation for the above examples (the right-hand side of the Baum-Connes assembly map).
April 24th, 2015 (03:35pm - 04:35pm)
Seminar: Probability and Financial Mathematics Seminar
Title: TBA
Speaker: Yuri Suhov, PSU
Location: MB106
April 27th, 2015 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Inverse Obstacle Scattering
Speaker: Rainer Kress, University of Goettingen (Host: L Zikatanov)
Location: MB114
This talk serves as an introduction to the afternoon's CAM Colloquium.
April 27th, 2015 (02:30pm - 03:30pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Inverse Obstacle Scattering
Speaker: Rainer Kress, University of Goettingen (Host: L Zikatanov)
Location: MB106
We consider the inverse problem to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic acoustic or electromagnetic waves, i.e., an inverse boundary value problem for the Helmholtz and Maxwell equations. For the sake of simplicity, we will concentrate on the case of scattering from a sound-soft obstacle or a perfect conductor. We will review some basics on uniqueness and ill-posedness for this inverse problem and discuss some more recently developed reconstruction algorithms with an emphasis on iterative methods. The luncheon seminar will include a short survey on the corresponding direct scattering problem. For a flavour of the topic see D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Springer, New York, 2013. 1
April 28th, 2015 (12:20pm - 01:10pm)
Seminar: Teaching Mathematics Discussion Group Seminar
Title: TBA
Speaker: Atendees, Penn State
Location: MB216
Abstract: http://
April 28th, 2015 (01:00pm - 02:00pm)
Seminar: Theoretical Biology Seminar
Title: TBA
Speaker: Abhyudai Singh, University of Delaware (Host: Jessica Conway)
Location: MB106
April 28th, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: TBA
Speaker: Stephane Korvers, University of Luxembourg
Location: MB106
April 30th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Singular foliations and their C* algebras: calculations. 3.
Speaker: Iakovos Androulidakis, University of Athens
Location: MB106
Singular foliations are examples of dynamical systems. They are abundant in many branches of mathematics, for instance control theory and Poisson geometry. In fact singular foliations appear much more often than regular ones. In this series of talks we discuss how to deal with the leaf space of such foliations, including calculations of various examples. Information about this space is encapsulated in the holonomy groupoid of the foliation and the associated C*-algebra. A tentative program for these lectures is: (1) singular foliations and bisubmersions, with examples (foliation by the flow of a single vector field, by orbits of the SO(3) action, by orbits of the action of SL(2,R)), (b) calculation of the holonomy groupoid for the above examples, (c) construction of the foliation C*-algebra, and (d) K-theory calculation for the above examples (the right-hand side of the Baum-Connes assembly map). | 2015-03-31 18:07:43 | {"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": 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": 0.4424721300601959, "perplexity": 2388.526408625796}, "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-2015-14/segments/1427131300905.36/warc/CC-MAIN-20150323172140-00133-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://mailman.ntg.nl/pipermail/ntg-context/2010/053789.html | # [NTG-context] Stop with an error signal
Khaled Hosny khaledhosny at eglug.org
Sun Oct 31 06:04:10 CET 2010
On Sat, Oct 30, 2010 at 07:27:02PM -0400, Aditya Mahajan wrote:
> Hi,
>
> I want to write a macro that checks for some settings and if the
> settings are wrong stop the current compilation and terminate with
> an error message. Right now I have
>
> \def\ERROR
> {\writeline
> \showmessage\??externalfilter??{forbidden}\getexternalfilterdirectory
> \batchmode
> \normalend}
Well, the first thought that came in my mind is using os.exit(1):
\def\ERROR{\directlua{os.exit(1)}}
But since this is pretty obvious, I'm sure I'm missing something.
Regards,
Khaled
--
Khaled Hosny
Arabic localiser and member of Arabeyes.org team
Free font developer | 2021-12-01 00:10:41 | {"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": 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": 0.8921582698822021, "perplexity": 12418.588578679593}, "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/1637964359082.76/warc/CC-MAIN-20211130232232-20211201022232-00134.warc.gz"} |
https://socratic.org/questions/how-do-you-graph-the-function-label-the-vertex-axis-of-symmetry-and-x-intercepts-6 | # How do you graph the function, label the vertex, axis of symmetry, and x-intercepts. y=x^2+3x-5?
##### 1 Answer
May 31, 2015
x of vertex:$\left(- \frac{b}{2} a\right) = - \frac{3}{2}$
y of vertex : $f \left(- \frac{3}{2}\right) = \frac{9}{4} - \frac{9}{2} - 5 = - \frac{9}{4} - \frac{20}{4} = - \frac{29}{4}$
x- intercepts;
$D = {d}^{2} = 9 + 20 = 29 \to d = \pm \sqrt{29}$
$x = - \frac{3}{2} \pm \frac{\sqrt{29}}{2}$ | 2021-09-17 04:04:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.3727148175239563, "perplexity": 10946.093886426}, "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-2021-39/segments/1631780054023.35/warc/CC-MAIN-20210917024943-20210917054943-00258.warc.gz"} |
https://math.stackexchange.com/questions/2052980/probability-and-graphs-from-a-math-contest | # Probability and graphs from a math contest [closed]
There are $n$ planets. Luke is on planet $A$. For each planet, except planet $B$ and $C$, there are two unidirectional roads that join other planets. From at most one of these two roads, Luke can find a sequence of roads that bring him back to the planet he had just left. Luke's journey ends when he reaches planet $B$ or $C$. When he leaves a planet, he chooses one of the two roads with equal probability. If the probability that he arrives on planet $B$ is $\frac{1}{2016}$, what is the minimum value of $n$?
How would you attack this problem? I have no idea :(.
Any help would be greatly appreciated.
## closed as off-topic by heropup, user223391, Leucippus, Jack, user91500Dec 11 '16 at 6:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Community, Leucippus, Jack, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.
We can solve the problem with $n=13$, by starting with $$A\to X_1\to X_2\to X_3\to X_4\to X_5\to X_6\to X_7\to X_8\to X_9\to X_{10}\to B$$ and letting all other arrows point to $C$, except that the second arrow of $X_5$ points back to $A$. Then the probability that we ever take the edge $X_5\to X_6$ is $\frac1{2^6}+\frac1{2^{12}}+\frac1{2^{18}}+\ldots = \frac1{63}$. Once we are at $X_6$, the probability to reach $B$ is $2^{-5}$, hence in total we have a probability of $\frac1{63\cdot 32}=\frac1{2016}$ to reach $B$.
On the other hand, there is no solution with $n\le 12$, for in that case the shortest path from $A$ to $B$ (which must exist) would consist of at most $10$ edges (as it cannot pass through $C$), leading to a lower bound of $2^{-10}>\frac1{2016}$ for the probability. | 2019-07-17 01:01:15 | {"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": 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": 0.7915829420089722, "perplexity": 318.99981432589294}, "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-2019-30/segments/1563195525004.24/warc/CC-MAIN-20190717001433-20190717023433-00374.warc.gz"} |
https://www.tutorialspoint.com/find-the-other-number-when-lcm-and-hcf-given-in-cplusplus | # Find the other number when LCM and HCF given in C++
C++Server Side ProgrammingProgramming
Suppose we have a number A, and LCM and GCD values, we have to find another number B. If A = 5, LCM is 25, HCF = 4, then another number will be 4. We know that −
$$𝐴∗𝐵=𝐿𝐶𝑀∗𝐻𝐶𝐹$$
$$𝐵= \frac{LCM*HCF}{A}$$
## Example
Live Demo
#include <iostream>
using namespace std;
int anotherNumber(int A, int LCM, int GCD) {
return (LCM * GCD) / A;
}
int main() {
int A = 5, LCM = 25, GCD = 4;
cout << "Another number is: " << anotherNumber(A, LCM, GCD);
}
## Output
Another number is: 20
Published on 21-Oct-2019 11:37:52 | 2021-04-12 22:31:34 | {"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": 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": 0.3835562765598297, "perplexity": 6578.688562108159}, "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/1618038069267.22/warc/CC-MAIN-20210412210312-20210413000312-00048.warc.gz"} |
https://indico.cern.ch/event/634426/contributions/3090505/ | We upgraded Indico to version 3.0. The new search is now available as well.
# Hard Probes 2018: International Conference on Hard & Electromagnetic Probes of High-Energy Nuclear Collisions
30 September 2018 to 5 October 2018
Aix-Les-Bains, Savoie, France
Europe/Zurich timezone
PROCEEDINGS OPEN UNTIL DECEMBER 15th 2018
## Charged-particle production in Pb+Pb and Xe+Xe collisions measured with the ATLAS detector
2 Oct 2018, 09:40
20m
Aix-Les-Bains, Savoie, France
#### Aix-Les-Bains, Savoie, France
Aix-Les-Bains, Congress Center Student Lectures Day: September 30 at CERN
2a) Jets and high-pT hadrons (TALK)
### Speaker
Petr Balek (Weizmann Institute of Science (IL))
### Description
Measurements of charged-particle production in heavy-ion collisions and their comparison to $pp$ data provide insight into the properties of the quark-gluon plasma. In 2015, the ATLAS detector at the LHC recorded 0.49 nb$^{-1}$ of Pb+Pb collisions and 25 pb$^{-1}$ of $pp$ collisions at a center-of-mass energy of $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. In addition, around 3 $\mu$b$^{-1}$ of Xe+Xe collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.44 TeV were recorded in 2017. These samples provide an opportunity to study the system size dependence of parton energy loss. The large acceptance of the ATLAS detector allows measurements of charged-particle spectra in a wide range of both pseudorapidity and transverse momentum, and differential in collision centrality. Charged-particle spectra measured in Pb+Pb and Xe+Xe collisions are compared to the analogous spectra measured in $pp$ collisions, and the resulting nuclear modification factors are scrutinized. In particular, the nuclear modification factors are found to scale approximately with the number of participating nucleons, which may be a key to predicting the behavior of even smaller collision systems.
### Primary author
ATLAS Collaboration | 2021-06-12 19:10:36 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7660092115402222, "perplexity": 5417.165788370846}, "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-2021-25/segments/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00332.warc.gz"} |
https://gis.stackexchange.com/questions/270749/qgis-calculate-radius-of-an-arc-to-attributes | # QGIS calculate radius of an arc to attributes
I have QGIS 2.18.16 installed. My problem is, how to calculate radius of an arc / curve feature (line)? The lines are originated from .dgn file, which I read with FME and wrote to PostGis DB. I have created primary keys etc, and the table is fully editable. I calculated lengths for the arcs successfully, but cant figure out how to calculate radius for these curves. This ( https://www.mathopenref.com/arcradius.html ) might give some idea on the math side, though I can't figure out how to use it in QGIS calculator. So I need to calculate the "R" for all the lines i have in the DB.
Bellow is an example. My data has Arcs / Curves on different table than "normal" polylines.
• When I click on an arc with the Identify tool, one of the derived attributes listed is the "closest vertex radius." So there's no need to calculate the radius, you only need to figure out how to access this property in the field calculator. Or simply copy it from the identify tool. – csk Feb 8 '18 at 18:26
• You can also view the properties of a vertex when a layer is in edit mode by selecting vertices with the Node Tool. Then the vertex coordinates (x,y,r) display in the Vertex Editor panel. – csk Feb 8 '18 at 18:30
• Nice to know it is at least possible to see the radius via identify tool. With only few arcs copy-paste would be an option, but I plan to use this to the whole data I have, which has too many arc features to be done manually. :/ – Sisuaski Feb 9 '18 at 8:01
• I googled around a bit, and it seems like not many people use circular string features in QGIS, so there's not a ready-made tool or function in the field calculator. You may need to define a custom Python function. If you want to go that route, add the pygis tag to your question. – csk Feb 9 '18 at 17:31
• How many vertexs do you have for feature? What do you see in the Vertex Editor when you click the Node Tool? – Marco Feb 12 '18 at 12:01
Lightly tested formulas follow, so proceed with caution. But following along with an example here: https://www.mathopenref.com/arcradius.html
If your circular arcs have a vertex at the middle point along the arc (which I am saying is x1, y1 in the figure), you could use it along with the start and end points to calculate the chord length "W" and the height "h" to get the radius "R" using the following formula:
and saying
W =
sqrt(
( $x_at(-1) -$x_at(0) )^2 +
( $y_at(-1) -$y_at(0) )^2
)
and
H =
sqrt(
( $x_at(1) - ($x_at(-1) + $x_at(0) )/2 )^2 + ($y_at(1) - ( $y_at(-1) +$y_at(0) )/2 )^2
)
in your Expression Dialog of the Field Calculator you'd have this long equation for the radius calculation:
R =
sqrt(
( $x_at(1) - ($x_at(-1) + $x_at(0) )/2 )^2 + ($y_at(1) - ($y_at(-1) +$y_at(0))/2 )^2
) /2
+
( ( $x_at(-1) -$x_at(0) )^2 ) +
( $y_at(-1) -$y_at(0) )^2 ) )
/ ( 8 * sqrt(
( ($x_at(1) - ($x_at(-1) + $x_at(0) )/2 )^2 + ($y_at(1) - ($y_at(-1) +$y_at(0) )/2 )^2
)
)
• In LaTe X $$W= \sqrt {(x_{-1} - x_0)^2 +(y_{-1}-y_0)^2}$$ $$H = \sqrt { (x_1- \frac {(x_{-1}+x_0)} 2 )^2 + (y_1- \frac {(y_{-1}+y_0)} 2 )^2}$$ and $$R = \frac {\sqrt { (x_1- \frac {(x_{-1}+x_0)} 2 )^2 + (y_1- \frac {(y_{-1}+y_0)} 2 )^2}} {2} + \frac {(x_{-1} - x_0)^2 +(y_{-1}-y_0)^2} {8 \sqrt { (x_1- \frac {(x_{-1}+x_0)} 2 )^2 + (y_1- \frac {(y_{-1}+y_0)} 2 )^2}}$$ – Marco Feb 17 '18 at 16:22
• Very good! I did not imagine how to access the vertexes. Is it possible in PostGIS? – Marco Feb 17 '18 at 16:26
• Marco, I believe it is possible to extract whatever vertice you are looking for. See the link here: postgis.net/docs/ST_PointN.html. Thank you for the most-excellent formula additions and edits. – cm1 Feb 18 '18 at 20:17
• While waiting for developers to add a button to "easy calculate radius", this is most helpful answer I could hope for. Now I (and possibly others too) have at least some way to extract radius. :) I have not yet tested this, but I will in a few days when I have enough time. Thank you a lot for your help! – Sisuaski Feb 19 '18 at 13:25
What about extracting the radius while extracting the data with FME? There you have the ArcPropertyExtractor transformer, which should give you the radius values in attributes ready to set in your PostGIS DB
• That would be one possible way to proceed, a good way to handle the whole data. My problem is, there is not enough FME users in my organization and I need to make the process as easy as possible. That means, the whole process, including arcs, areas, lines etc., should be able to be done with Qgis. Ability to update Arc feature radius to new objects is what I need to be done with Qgis. Thank you for your reply though, I'll be updating radius attributes with FME for now, but hope there will be a way to do it with QGIS. – Sisuaski Feb 12 '18 at 10:00 | 2019-07-22 08:42:28 | {"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": 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": 0.623813271522522, "perplexity": 1098.1352242036535}, "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-2019-30/segments/1563195527828.69/warc/CC-MAIN-20190722072309-20190722094309-00287.warc.gz"} |
https://wenjie-stat.me/splines2/reference/dbs | This function produces the derivative of given order of B-splines. It is an implementation of the close form derivative of B-spline basis based on recursion relation. At knots, the derivative is defined to be the right derivative.
dbs(x, derivs = 1L, df = NULL, knots = NULL, degree = 3L,
intercept = FALSE, Boundary.knots = range(x, na.rm = TRUE), ...)
## Arguments
x The predictor variable. Missing values are allowed and will be kept and returned as they were. A positive integer specifying the order of derivative. By default, it is 1L for the first derivative. Degrees of freedom of the B-spline basis to be differentiated. One can specify df rather than knots, then the function chooses "df - degree" (minus one if there is an intercept) knots at suitable quantiles of x (which will ignore missing values). The default, NULL, corresponds to no inner knots, i.e., "degree - intercept". The internal breakpoints that define the B-spline basis to be differentiated. The default is NULL, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots. See also Boundary.knots. Non-negative integer degree of the piecewise polynomial to be differentiated. The default value is 3 for the integral of cubic B-splines. If TRUE, an intercept is included in the basis; Default is FALSE. Boundary points at which to anchor the B-spline basis to be differentiated. By default, they are the range of the non-NA data. If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Optional arguments for future usage.
## Value
A matrix of dimension length(x) by df = degree + length(knots) (plus on if intercept is included). Attributes that correspond to the arguments specified are returned for usage of other functions in this package.
## Details
The function is similar with splineDesign. However, it provides a more user-friendly interface, a more considerate NA's handling. Internally, it calls bSpline and generates a basis matrix for representing the family of piecewise polynomials and their corresponding derivative with the specified interior knots and degree, evaluated at the values of x. The function splineDesign in splines package can also be used to calculate derivative of B-splines.
## References
De Boor, Carl. (1978). A practical guide to splines. Vol. 27. New York: Springer-Verlag.
predict.dbs for evaluation at given (new) values; deriv.dbs for derivative method; bSpline for B-splines; ibs for integral of B-splines.
## Examples
library(splines2)
x <- seq.int(0, 1, 0.01)
knots <- c(0.2, 0.4, 0.7)
## the second derivative of cubic B-splines with three internal knots
dMat <- dbs(x, derivs = 2L, knots = knots, intercept = TRUE)
## compare with the results from splineDesign
ord <- attr(dMat, "degree") + 1L
bKnots <- attr(dMat, "Boundary.knots")
aKnots <- c(rep(bKnots[1L], ord), knots, rep(bKnots[2L], ord))
res <- splines::splineDesign(aKnots, x = x, derivs = 2L)
stopifnot(all.equal(res, dMat, check.attributes = FALSE)) | 2020-01-21 22:28:53 | {"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": 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": 0.4556155800819397, "perplexity": 2779.192289640272}, "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/1579250606226.29/warc/CC-MAIN-20200121222429-20200122011429-00246.warc.gz"} |
http://math.ivanovo.ac.ru/dalgebra/Khashin/2016_usa/acmes/2Khashin/index.html | ### Sergey Khashin, Department of Mathematics, Ivanovo State University
Counterexamples for Frobenius primality test
The most powerful elementary probabilistic method for primality test is the Frobenius test. Frobenius pseudoprimes are the natural numbers for which this test fails. There are several slightly different definitions of Frobenius pseudoprimes (FPP), which are almost equivalent. We are using the following one.
Definition Frobenius pseudoprime (FPP) is a composite odd integer $n$ such that it is not a perfect square provided $z^n \equiv \overline{z} \mod n$, where $z=1+\sqrt{c}$ and $c$ is the smallest odd prime number with the Jacobi symbol $J(c/n)=-1$.
Theorem 1(multiple factors). Let $n=p^sq$ is FPP where $p$ is prime and $s>1$. Then ${z^p = \overline{z} \mod p^2}$.
Experiment 1. There are no exist such prime $p<2^{32}$.
Theorem 2. Let $n=pq$ be FPP where $J(c/p)=+1$. Then $z_1^q \equiv z_2$, $z_2^q \equiv z_1 \mod p$, where $z_{1,2} = 1 \pm d \mod p$ and $d^2 = c \mod p$.
Experiment 2. For each $c$ there exists a short list ($<20$) of such primes $p$.
Experiment 3. There exist no FPP less than $2^{60}$. | 2022-07-05 10:03:26 | {"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": 0.92649906873703, "perplexity": 686.8017235400282}, "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/1656104542759.82/warc/CC-MAIN-20220705083545-20220705113545-00467.warc.gz"} |
https://www.statistics-lab.com/category/math4530-topology/ |
## MATH4530 Topology课程简介
Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous transformations, such as stretching, bending, and twisting. It is often described as qualitative geometry because it focuses on the properties of spaces that are not dependent on specific measurements or coordinates.
This course starts with basic point-set topology, which is concerned with the properties of sets and the relationships between them. Some of the topics covered in this section may include connectedness, compactness, and metric spaces. Connectedness refers to the idea that a space is made up of one piece, while compactness relates to the idea that a space has no “holes” or “gaps”. Metric spaces involve using a distance function to measure the distance between points in a space.
The later topics of the course may include the classification of surfaces such as the Klein bottle and Möbius band. These are examples of non-orientable surfaces, which means they do not have a consistent “up” and “down” orientation. Elementary knot theory, which studies the properties of knots and their classifications, may also be covered. The fundamental group and covering spaces are important concepts in algebraic topology and relate to the ways in which spaces can be mapped onto each other.
Overall, this course will introduce students to the foundational concepts of topology and its applications in various fields of mathematics.
## PREREQUISITES
The course prerequisites indicate that students must have completed one of the following courses: MATH 2210, MATH 2230, MATH 2310, or MATH 2940. Additionally, students must have taken at least one mathematics course numbered 3000 or above, or they must obtain permission from the instructor.
Students are also expected to be comfortable with proofs, which means they should have experience in constructing and understanding mathematical proofs. This indicates that the course will likely involve advanced mathematical concepts that require a strong foundation in proof-based mathematics.
In summary, to enroll in this course, students must have completed the prerequisite courses or obtained permission from the instructor, and they must be comfortable with constructing and understanding mathematical proofs.
## MATH4530 Topology HELP（EXAM HELP， ONLINE TUTOR）
Throughout, a topological space $X$ is endowed with a topology $\tau$, even if not explicitly mentioned.
1. A collection $\left{A_\alpha\right}$ of subsets of $X$ satisfies the finite intersection property if $\bigcap_{i=1}^n A_{\alpha_i} \neq \emptyset$ for any finite subcollection.
(a) Prove Cantor’s “finite intersection lemma”: Suppose $\left{K_\alpha\right}$ is a collection of compact sets of a Hausdorff space $X$. If $\bigcap_{i=1}^n K_{\alpha_i} \neq \emptyset$ for any finite subcollection, then $\bigcap_\alpha K_\alpha \neq$ 0.
(b) Prove that $X$ is compact if and only if every collection of closed sets $\left{F_\alpha\right}$ satisfying
(a) To prove Cantor’s “finite intersection lemma”, suppose that ${K_\alpha}$ is a collection of compact sets in a Hausdorff space $X$, and that $\bigcap_{i=1}^n K_{\alpha_i} \neq \emptyset$ for any finite subcollection. We wish to show that $\bigcap_\alpha K_\alpha \neq \emptyset$.
Suppose, for the sake of contradiction, that $\bigcap_\alpha K_\alpha = \emptyset$. Then, for each $\alpha$, there exists an open set $U_\alpha$ such that $K_\alpha \subseteq U_\alpha$ and $\bigcap_\alpha U_\alpha = \emptyset$. Since $X$ is a Hausdorff space, for each pair of distinct points $x,y \in X$, there exist disjoint open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively. For each $\alpha$, let $x_\alpha \in K_\alpha$ be any point, and let $U_{x_\alpha}$ be an open set containing $x_\alpha$ such that $U_{x_\alpha} \subseteq U_\alpha$. Then, ${U_{x_\alpha}}$ is an open cover of $\bigcup_\alpha K_\alpha$.
Since each $K_\alpha$ is compact, there exists a finite subcollection ${U_{x_{\alpha_1}}, \dots, U_{x_{\alpha_n}}}$ that covers $\bigcup_{i=1}^n K_{\alpha_i}$. But then, by construction, $\bigcap_{i=1}^n U_{x_{\alpha_i}} \neq \emptyset$, which contradicts the fact that $\bigcap_\alpha U_\alpha = \emptyset$. Therefore, we must have $\bigcap_\alpha K_\alpha \neq \emptyset$, as desired.
1. On HW 2, you proved that if a function $f: X \rightarrow Y$ between metric spaces is continuous, then its graph
$$\Gamma_f:={(x, f(x)) \mid x \in X}$$
is a closed subset of $X \times Y$. Now, suppose $f: X \rightarrow Y$ is a map between topological spaces, and $Y$ is Hausdorff.
(a) Show that if $f$ is continuous, then the graph $\Gamma_f$ is closed in $X \times Y$.
(b) Show that the conclusion of Part (a) may fail of $Y$ is not Hausdorff.
(c) Show that if $X$ and $Y$ are both compact and Hausdorff, then the converse to Part (a) holds.
(a) Suppose $f$ is continuous and let $(x,y)\in \overline{\Gamma_f}$, i.e., $(x,y)$ is a limit point of $\Gamma_f$. We want to show that $(x,y)\in \Gamma_f$, i.e., $y=f(x)$. Since $(x,y)$ is a limit point of $\Gamma_f$, there exists a sequence $(x_n,y_n)\in \Gamma_f$ such that $(x_n,y_n)\to (x,y)$. Since $Y$ is Hausdorff, we have $y_n\to y$. Since $f$ is continuous, we have $f(x_n)\to f(x)$. But $(x_n,y_n)\in \Gamma_f$, so $y_n=f(x_n)$ for all $n$. Therefore, $y=f(x)$, and $(x,y)\in \Gamma_f$. Hence, $\overline{\Gamma_f}\subseteq \Gamma_f$, so $\Gamma_f$ is closed in $X\times Y$.
(b) Let $X=Y={0,1}$ with the indiscrete topology, i.e., the only open sets are $\emptyset$ and $X$. Let $f:X\to Y$ be the identity map. Then $f$ is continuous, but $\Gamma_f={(0,0),(1,1)}$ is not closed in $X\times Y$.
(c) Suppose $X$ and $Y$ are both compact and Hausdorff, and suppose $\Gamma_f$ is closed in $X\times Y$. We want to show that $f$ is continuous. Let $x_0\in X$ and let $\epsilon>0$. We want to find a neighborhood $U$ of $x_0$ such that $f(U)\subseteq B(f(x_0),\epsilon)$, where $B(f(x_0),\epsilon)$ is the open ball of radius $\epsilon$ centered at $f(x_0)$ in $Y$. Since $Y$ is Hausdorff, for each $x\in X\setminus{x_0}$, there exist disjoint open sets $U_x$ and $V_x$ in $Y$ such that $f(x_0)\in U_x$ and $f(x)\in V_x$. The sets $U_x$ cover $f(x_0)$, so by compactness, there exist finitely many $x_1,\dots,x_n$ such that $U_{x_1},\dots,U_{x_n}$ cover $f(x_0)$. Let $V=V_{x_1}\cap\dots\cap V_{x_n}$, which is an open neighborhood of $x_0$ in $X$. We claim that $f(V)\subseteq B(f(x_0),\epsilon)$. To see this, let $y\in f(V)$, so there exists $x\in V$ such that $f(x)=y$. Then $x\in U_{x_i}$ for some $i$, so $f(x_i)\in U_{x_i}$ and $f(x)=f(x_i)$. Therefore, $y=f(x_i)\in U_{x_i}$, so $U_{x_i}\cap B(f(x_0),\epsilon)\neq\emptyset$. Since $U_{x_i}$ and $V_{x_i}$ are disjoint, we have $f(x_0)\notin V_{x_i}$, so $y=f(x_i)\notin V_{x_i}$.
## Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
Statistics-lab™可以为您提供stanford.edu EE364A convex optimization凸优化课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。 | 2023-03-24 12:10:25 | {"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": 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": 0.912006139755249, "perplexity": 96.45239845279173}, "config": {"markdown_headings": true, "markdown_code": false, "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-2023-14/segments/1679296945282.33/warc/CC-MAIN-20230324113500-20230324143500-00504.warc.gz"} |
http://umj.imath.kiev.ua/article/?lang=en&article=10543 | 2019
Том 71
№ 5
# On Generalized Hardy Sums $s_5(h, k)$
Simsek Y.
Abstract
The aim of this paper is to study generalized Hardy sums $s_5(h, k)$. By using mediants and the adjacent difference of Farey fractions, we establish a relationship between $s_5(h, k)$ and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums $s_5(h, k)$. A relationship between $s_5(h, k)$ and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between $s_5(h, k)$ and Lambert series.
English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 10, pp 1712–1719.
Citation Example: Simsek Y. On Generalized Hardy Sums $s_5(h, k)$ // Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1434–1440.
Full text | 2019-05-22 04:33:12 | {"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": 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": 0.7421759366989136, "perplexity": 864.7159638586402}, "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-2019-22/segments/1558232256763.42/warc/CC-MAIN-20190522043027-20190522065027-00200.warc.gz"} |
https://www.esaral.com/q/the-oxide-that-shows-magnetic-property-is-46887 | # The oxide that shows magnetic property is :
Question:
The oxide that shows magnetic property is :
1. $\mathrm{SiO}_{2}$
2. $\mathrm{Mn}_{3} \mathrm{O}_{4}$
3. $\mathrm{Na}_{2} \mathrm{O}$
4. $\mathrm{MgO}$
Correct Option: , 2
Solution:
$\mathrm{Mn}_{3} \mathrm{O}_{4}$ shows magnetic properties | 2023-03-26 02:18:19 | {"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": 0.877765953540802, "perplexity": 9961.426461507273}, "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/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00173.warc.gz"} |
https://blog.aurorasolar.com/orient-panels-for-peak-tou-rates | Solar design tips, sales advice, and industry insights from the premier solar design software platform
Author
### Andrew Gong
Andrew Gong is a Research Engineer at Aurora. He previously worked on two Solar Decathlon projects, and he received his M.S. from Stanford and B.S. from Caltech.
# Should I Point My Solar Panels West to Optimize for Afternoon Peak TOU Rates?
Here at Aurora, we like to tackle hard questions. One question we see often is: “Given that electricity rates are higher from 4 - 9 pm in California, should I place my solar panels facing west instead of south to maximize savings?”
In this post we will dive into why west-facing panels might be more advantageous than south-facing ones, and provide recommendations based on your location and how your peak time-of-use (TOU) rate works.
## Context: Moving West in California
### Aggregate Demand and the Duck Curve
The three main investor-owned utilities in California have been utilizing peak TOU rates 4 - 9 pm since late 2017; customers are charged more for the electricity they use during the late afternoon and evening hours. Prior to this, peak hours in the middle of the day—between 11 am and 6 pm—were most common.
The peak hours shift was driven by aggregate demand on California’s grid. Prior to the early 2010s, homeowners and businesses used the most energy in the middle of the day and early evening. Midday peak-pricing provided the financial incentive to reduce energy usage or shift loads to non-peak hours. Since then, the falling cost of solar has drastically shifted the total demand on the grid as seen in the infamous duck curve, pushing the peak demand, and moving peak-pricing to early evenings.
Source: CAISO. California’s “Duck Curve” shows the low mid-day net demand on the California electricity grid after solar and wind generation is accounted for. The low mid-day trough and steep afternoon ramp are part of why time-of-use rates have peak pricing between 4pm and 9pm.
### Directional Variation
Taking a look at the energy production of a south-facing system in the Los Angeles area, about 87% of its summer energy is produced during off-peak hours and 13% is during peak hours.
Chart showing a PV production curve for a south-facing system; only 12.6% of energy is produced during peak TOU hours.
If we take the same system and rotate it to face a different direction, we estimate that you can get up to 20% of that system’s energy produced during peak hours, but the total production (yield) from the system is lower. The tradeoff is having more peak-hour energy at the expense of having less total energy; the peak yield of a system in this location at an azimuth of 190, just slightly west of south.
Chart showing energy yield (kWh produced per W per year) and the percent of production that’s during peak hours. Maximum energy yield is around an azimuth of 190 degrees. The most on-peak kWh production occurs facing west, but the overall yield is substantially lower.
### Net Metering Valuation
In standard net metering policies, energy production from the PV system can be assigned a dollar value based on the utility rate at that time*. For example, producing 2 kWh in an hour with a rate of $0.15/kWh provides a credit of$0.30, and producing 2 kWh in an hour with a rate of $0.35/kWh is valued at$0.70. It’s important to understand the time-dependent production profile of a PV system, not just daily total yields.
Taking our south-facing production curve from above, about 12.6% of energy is produced during peak hours, but that energy is valued at nearly double that of off-peak energy. As a result, 21.2% of the value of the energy from the PV system comes from on-peak production.
Chart showing how the on-peak energy produces a greater value to the homeowner than off-peak energy, thanks to the peak TOU rate.
Back when the evening-peak rates were newly implemented, we reviewed several hundred San Diego projects that had been designed in Aurora and found that the rate change would substantially increase bills for customers who were still large net-consumers after going solar, but that solar was still a solid investment even with the less favorable TOU periods.
## Is It Worth Pointing Panels to the West?
After simulating production profiles for systems with various orientations and combining it with residential TOU rates, we created an “energy value” of the PV systems. We calculated energy value by taking the kWh created by a system during each hour of the year and multiplying it by the current retail rate, and then tabulating the sum of all hours. The data is presented in gauge charts below. The wheel color indicates the system yield similar to the irradiance maps in Aurora. The sunshine icon indicates the optimal azimuth with the maximum energy production, and the needle showing which orientation achieves the most “energy value” throughout the course of the year.
The example below is for a customer in Southern California Edison territory near Los Angeles, using the TOU-D-4-9-PM rate. The ideal orientation for energy production is around 190 degrees azimuth, and because of the late peak hours, the ideal orientation for solar value is slightly further west at 200 degrees azimuth.
In other locations, this trend still holds true. The optimal orientation for system production value is slightly west of the optimal orientation for overall energy production, but not substantially so. Nearly all sites studied (see below) have an optimal orientation of 190 or 200 degrees azimuth, just slightly west of south. The rate that favors the furthest west-facing panels is San Diego Gas & Electric’s TOU-DR-SES for solar energy systems, which features an extraordinarily high peak price and lower off-peak rates.
## What Happens If…
In the previous scenarios, there aren’t strong arguments to face your panels west when you have a choice to point them south. However, west-facing panels might be the way to go in some situations, e.g., a return to 3 - 8 pm peak pricing or a higher on-peak price. We will look at a couple of these what-if cases next.
### Higher Peak Pricing
In some of the current TOU rates, the peak rate is roughly double that of the off-peak rate ($0.16 vs$0.31 per kWh in the SCE example). Adjusting this ratio from 2:1 to 3:1 or 4:1 by either increasing the peak rate or reducing the off-peak rate will provide more relative values to the on-peak energy production. We tested this out with the SCE rate, and got the following results:Increasing the difference between on-peak and off-peak prices to such an extreme level is not expected, but doing so would favor southwest-facing systems.
The energy production (color scale) remains the same, but the needle shifts west indicating that there’s an advantage for southwest-facing systems when there is an extreme difference between on-peak and off-peak pricing. We don’t expect rates like this to show up frequently, but an off-peak to on-peak price difference of $0.15 vs$0.45 would be enough to favor systems that face more west.
### Earlier Peak Pricing Hours
For most PV systems, whether they face south or west, there is more energy production between 3 - 4 pm than 4 - 5 pm. What if peak TOU hours began at 3 pm, would there be an advantage to face further west?
Yes, if the rate is E-TOU-A in PG&E territory, which has a 3 - 8 pm peak pricing period. E-TOU-B is similar to E-TOU-A, but features a 4 - 9 pm peak rate. Shown side-by-side here, there’s no change in the optimal orientation.
### Non-Standard NEM Scenarios
The above analysis applies to utilities that have standard NEM policies, or have export rules that are still very close to retail rate. In markets such as Nevada or Hawaii, in which the gap between the purchase rate and the credit for excess energy is substantial,, there is additional value in designing PV production to coincide with home loads (i.e., self-consumed energy offsets the bill at the retail rate while exported energy is credited at a lower amount, or not at all). To correctly model these scenarios, it’s important to have a good measure of the customer’s home energy usage (such as Green Button Data) and to make sure your modeling tools support advanced NEM rules.
## Conclusions
If you are planning a ground-mount system or have a flat roof surface, you might be able to boost the value of your solar PV system by pointing it slightly to the west. If your roof is south-facing, the actual difference in produced energy value between due south and the optimal azimuth was between 0.3% and 0.7% in all the cases we looked at. Other factors, such as shade on the site, which roof surfaces are available for solar, and even the efficiency of the inverter, can have a much larger impact on the value of the system than picking between south and slightly southwest.
A choice with greater consequence is picking between a southeast-facing (135 degrees azimuth) and a southwest-facing (215 degrees azimuth) roof face. The southwest-facing surfaces typically have about 99% of the maximum energy value while southeast surfaces were around 95% across the board. Both are great, but if all other aspects were equal, the southwest surface would be the better choice.
#### Note
Our data models here ignore non-bypassable charges, which are 1.5-3 cents per kWh depending on the utility company and rate. Models also simplify non-peak costs to be an average of off-peak and shoulder-peak prices, since these typically only differ by a few cents. We used 4pm-9pm hours for TOU, and used May, June, July, and August as the summer-pricing months. Production profiles were simulated using unshaded panels sloped at 20 degrees. Actual energy yield and optimal orientation will depend on site conditions including shading, and the actual optimal-value orientation may vary by tilt, shade conditions, and the customer’s load profile. You can complete a more in-depth analysis of system performance and utility bill savings using Aurora’s tools.
• pv installation
• solar policy
Author
### Andrew Gong
Andrew Gong is a Research Engineer at Aurora. He previously worked on two Solar Decathlon projects, and he received his M.S. from Stanford and B.S. from Caltech. | 2020-04-06 05:57:38 | {"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": 0.43116599321365356, "perplexity": 2300.4302057936975}, "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-2020-16/segments/1585371618784.58/warc/CC-MAIN-20200406035448-20200406065948-00419.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-2-section-2-5-an-introduction-to-problem-solving-exercise-set-page-165/1 | ## Introductory Algebra for College Students (7th Edition)
Let $x$ = the unknown number Then "a number increased by 60" means $x+60$. Thus, the equation that represents the situation is: $x+60=410$ Subtract 60 on both sides of the equation to obtain: $x= 350$ | 2018-06-19 13:00:07 | {"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": 0.6293540000915527, "perplexity": 557.0758943554514}, "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-2018-26/segments/1529267862929.10/warc/CC-MAIN-20180619115101-20180619135101-00437.warc.gz"} |
https://www.studysmarter.us/explanations/math/geometry/rectangle/ | ### Select your language
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# Rectangle
When you look at an object, some questions come to mind and right there you give quick answers to a number of them. One of such questions answered is the shape of the object.
In this article, we will explore the definition of a rectangle, its properties, the formulas for the perimeter and area of a rectangle, and examples of their application.
## Definition of rectangle
A rectangle is a quadrilateral with four sides and four angles, where all the interior angles are right angles (90 degrees).
A rectangle is a special case of a parallelogram. In other words, what makes a parallelogram become a rectangle is having its sides perpendicular to each other. This can be illustrated in the image below.
Rectangle - StudySmarter Original
We can notice that the opposite sides AB and CD are equal in size and parallel, the same for BC and AD. Moreover, the four sides are perpendicular to each other, thus, the quadrilateral is a rectangle.
## Rectangles Properties
A rectangle being a parallelogram has all the properties of a parallelogram, but being a special case of it, it has its own unique properties that make it the geometric shape it is.
To better understand the properties of a rectangle, let's consider the following rectangle ABCD in the image below.
Rectangle ABCD - StudySmarter Original
Property Example 1. Opposite sides of a rectangle are equal and parallel. AB = CD, and AB is parallel to CD.Likewise, AD = BC, and AD is parallel to BC. 2. All four angles in a rectangle are right angles. $\angle A=\angle B=\angle C=\angle D=90°$ 3. The sum of all interior angles of a rectangle is 360º. $\angle A+\angle B+\angle C+\angle D=360°$ 4. The diagonals of a rectangle are equal in length and bisect each other – they intersect each other in their middle. AC and BD are the diagonals of rectangle ABCD.AC = BDAC bisects BD and BD bisects AC.
## Construction of a rectangle
For the construction of a rectangle, follow these steps.
Step 1: Draw a straight line (R), then place 2 points A and B on the line.
Step 2: Draw 2 perpendicular lines (S) and (T), passing by the two points A and B.
Step 3: Locate two points C and D respectively on the two lines (S) and (T). However, C and D must be on the same level.
The three steps mentioned earlier can be illustrated in the drawing below:
Image resulting from steps 1, 2 and 3 - StudySmarter Original
Step 4: Draw a straight line joining the two points C and D, as shown in the image below:
Image resulting from step 4 - StudySmarter Original
After reaching step 4, you will notice that the 4 points A, B, C, and D will form a rectangular shape.
## Formula for the area of a rectangle
The area of a flat shape or the surface of an object can be defined in geometry as the space occupied by it.
The area of a shape is usually measured considering the number of unit squares that cover the surface of the shape. Square centimeters, square feet, square inches, and other similar units are used to measure area.
Given a rectangle with height h and base b, its area will be equal to:
$A=b×h$.
Find the area of the rectangular shape in the image below. Consider the square composed of 25 smaller squares the square of side 1 unit.
We can notice that the height of the rectangle is equal to 2 unit squares, so its length is 2 units. Similarly, the base of the rectangle is 5 units. So the area of this rectangle can be calculated by multiplying the height by the base:
A= 2 unit × 5 unit = 10 unit2
## Formula for the perimeter of a rectangle
The perimeter of a shape is the distance around its outside.
Consequently, the shape's perimeter is calculated by summing the lengths of all its sides. The same concept also applies to a rectangular shape. So, the total length of all the sides of a rectangle is known as the perimeter.
A rectangle has its opposite sides equal to each other (one of its properties). Thus, the rectangle's perimeter of a rectangle with sides of lengths a, b, a, b is P = a + b + a + b, or P = 2a + 2b, or even P = 2 (a + b).
So, we just need to calculate the lengths of two sides to find the perimeter of a rectangle since opposite sides of a rectangle are always equal.
Find the perimeter and the area of the shape illustrated in the image below:
Step 1: Try to identify the rectangle shapes. We can notice that 2 rectangles are present in the shape above. The rectangles identified are illustrated in the image below:
The following properties are checked to make sure that the shapes identified are rectangles:
1. The opposite sides are equal and parallel. For example, the first rectangle identified has the opposite sides parallel and equal to 8, and the other two opposite sites also parallel and equal to 3.
2. All the angles are right angles, or in other words all the sides are perpendicular to each others.
The Perimeter of the first rectangle PA can be calculated as follows:
${P}_{A}=\left(4+4+3+3\right)cm=14cm$
The Perimeter of the second rectangle PB can be calculated as follows:
${P}_{B}=\left(10+10+5+5\right)cm=30cm$
The Perimeter of the overall shape PAB:
${P}_{AB}={P}_{A}+{P}_{B}=\left(14+30\right)cm=\mathbf{44}\mathbit{c}\mathbit{m}\mathbf{}\mathbf{}$
The Area of the first rectangle AA can be calculated as follows:
${A}_{A}=height×base=4cm×3cm=12c{m}^{2}$
The Area of the second rectangle AB can be calculated as follows:
${A}_{B}=height×base=5cm×10cm=50c{m}^{2}$
The Area of the overall shape:
${A}_{AB}={A}_{A}+{A}_{B}=12c{m}^{2}+50c{m}^{2}=\mathbf{62}\mathbit{c}{\mathbit{m}}^{\mathbf{2}}$
## Square and rectangle
You can notice in the figure below that a square and a rectangle are both quadrilateral with four sides.
A rectangle and a square - StudySmarter Original
A square and a rectangle have similar properties as illustrated in the table below:
Properties Rectangle Square The four sides are equal X ✔ Opposite sides are equal ✔ ✔ Opposite sides are parallel ✔ ✔ Diagonals bisect each others ✔ ✔ Diagonals are perpendicular to each others X ✔ All angles are equal ✔ ✔ Opposite angles are equal ✔ ✔ Sum of two adjacent angles is 180 degrees ✔ ✔
### What Characterizes a Square as a Unique Rectangle?
As illustrated in the table above, a square is a special type of rectangle for the following reasons:
1. A square has all the properties of a rectangle.
2. The only two differences between the square and the rectangle are that a square has its diagonal perpendicular to each other, and all its sides are equal.
## Rectangle - Key takeaways
• A rectangle is also a quadrilateral with four sides and four angles.
• All four angles in a rectangle are right angles.
• Opposite sides in a rectangle are equal and parallel A rectangle's diagonals are equal and bisect each other's. They bisect each other means that they intersect each other in their middle.
• Consecutive angles in a rectangle are supplementary. Their summation is equal to 180 degrees.
• Given a rectangle with height equal to h and base equal to b, then its corresponding area will be equal to the multiplication of b by h.
• A rectangle has its opposite sides equal to each other. Thus, the specified rectangle's perimeter is 2(a + b).
• A square is a unique rectangle.
Any four-sided shape with all its interior angles being right angles are an example of rectangles.
To find the area of a rectangle, use the formula A = b × h, where b is the base and h is the height of the rectangle.
To find the perimeter of a rectangle, use the formula P = 2(a + b), where a and b are the lengths of the sides.
Yes, all squares are rectangles.
The formula for the area of the rectangle is A = b × h.
## Final Rectangle Quiz
Question
Can the diagonal of a rectangle be perpendicular?
No
Show question
Question
In a rectangle the opposite sides are equal and parallel. Is is true or false?
True
Show question
Question
In a rectangle at least 2 angles must be right angles. Is it true or false?
False
Show question
Question
The perimeter of a flat shape or the surface of an object can be defined in geometry as the space occupied by it. Is is true or False?
False
Show question
Question
All rectangles are squares. Is it true or false?
False
Show question
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Over 10 million students from across the world are already learning smarter. | 2023-02-01 12:04:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.6336477398872375, "perplexity": 510.6418790018217}, "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-2023-06/segments/1674764499934.48/warc/CC-MAIN-20230201112816-20230201142816-00244.warc.gz"} |
https://codereview.stackexchange.com/questions/8232/one-time-property-reading | # One time property reading
I have following code that is used of one time skipping some functionality(actually do something only one time until SkipSomeStuff is true):
private bool _skipSomeStuff;
public bool SkipSomeStuff
{
set
{
_skipSomeStuff = value;
}
get
{
if (_skipSomeStuff)
{
_skipSomeStuff = false;
return true;
}
return false;
}
}
Is such construction ok to use, or I should change it with something?
## 1 Answer
It is very confusing to set a property just to discover that it returns something else just after you set it. Properties are supposed to keep the value you give them. (At least logically)
And since your SkipSomeStuff property is public, you have no guarantee you won't access that property by accident before the intended recipient accesses it, and the stuff isn't skipped after all.
How about keeping the skipSomeStuff member, but set it using a method called SkipStuffNextTime() or something like that? Nobody will misunderstand the intent of your code that way, and only the code it's relevant for will access and reset it.
private bool skipSomeStuff = false;
public void SkipStuffNextTime() {
skipSomeStuff = true;
}
public void DoSomeStuff() {
if (skipSomeStuff) {
skipSomeStuff = false;
return;
}
// do the stuff
}
• +1 for the plug for maintainability. P.S. A certain property that emits a value different from the underlying field - but never updates the field itself! This damn thing has wreaked havoc in our application. And guess what you see in the debugger when you hover over the property? Yes, the wrong answer (i.e. wrong value). – radarbob Jan 24 '12 at 2:56
• @radarbob - I imagine that if you accidentally hover your mouse over that property at the wrong time while debugging, you might inadvertently "use up" the value, causing your application to run incorrectly. Nasty problem. – Dr. Wily's Apprentice Jan 24 '12 at 8:24
• @Dr.Wily'sApprentice interesting suggestion. I'll try) – anatoliiG Jan 24 '12 at 9:06
• @Lars-Erik thanks for answering. I'll refactor this crap) – anatoliiG Jan 24 '12 at 9:07
• :) If you add an accessor property (only getter), you can even write some unit-tests which confirms that everything works as expected. – Lars-Erik Jan 24 '12 at 11:32 | 2019-11-20 05:50:54 | {"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": 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": 0.268730491399765, "perplexity": 3759.0993980020125}, "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-2019-47/segments/1573496670448.67/warc/CC-MAIN-20191120033221-20191120061221-00211.warc.gz"} |
https://www.daniweb.com/hardware-and-software/information-security/threads/49038/automatic-updates-error | Hi, I'm having an error box pop up consistantly when I turn my lap top on. The error is due to an Automatic Updates error. I recently searched the forums and found a person who had a similar problem [Volta06] who posted his problem about a year ago. However, he posted a highjackthis report, which may have helped, yet I do not know how to access or retrieve that report. If anyone has had this problem or just anyone in general who knows how to correct this problem, please help.. Thanks
3
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Discussion Span
Last Post by DMR
Hi sdeguzman, welcome to DaniWeb :)
First of all, you need to give us the full and exact text of the error(s) you get, as well as any other details that might be related to the problem. The more information we have to go on, the faster we can help you get things sorted out.
If it seems that a HijackThis log would help, we'll give you instructions on just how to do that.
okay.. well i was able to attain the hijackthis which came out to:
Logfile of HijackThis v1.99.1
Scan saved at 6:58:45 PM, on 6/30/2006
Platform: Windows XP SP2 (WinNT 5.01.2600)
MSIE: Internet Explorer v6.00 SP2 (6.00.2900.2180)
Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\lsass.exe
C:\WINDOWS\system32\svchost.exe
C:\Program Files\Common Files\Microsoft Shared\Ink\KeyboardSurrogate.exe
C:\WINDOWS\system32\spoolsv.exe
C:\Acer\eManager\anbmServ.exe
C:\Program Files\WIDCOMM\Bluetooth Software\bin\btwdins.exe
C:\WINDOWS\system32\nvsvc32.exe
C:\Program Files\Sophos\AutoUpdate\ALsvc.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\SYSTEM32\WISPTIS.EXE
C:\WINDOWS\System32\tabbtnu.exe
C:\WINDOWS\Explorer.EXE
C:\WINDOWS\system32\ctfmon.exe
C:\Program Files\Common Files\Microsoft Shared\Ink\TCServer.exe
C:\WINDOWS\SOUNDMAN.EXE
C:\Program Files\Synaptics\SynTP\SynTPLpr.exe
C:\Program Files\Synaptics\SynTP\SynTPEnh.exe
C:\WINDOWS\system32\SetCrSr.exe
C:\WINDOWS\system32\rundll32.exe
C:\acer\epm\epm-dm.exe
C:\Program Files\Launch Manager\LaunchAp.exe
C:\Program Files\Launch Manager\PowerKey.exe
C:\Program Files\Launch Manager\HotkeyApp.exe
C:\Program Files\Common Files\Microsoft Shared\Ink\TabTip.exe
C:\Program Files\Launch Manager\OSDCtrl.exe
C:\Program Files\Launch Manager\Wbutton.exe
C:\Acer\Empowering Technology\eRecovery\Monitor.exe
C:\Program Files\Acer soft button\SB.exe
C:\Program Files\Common Files\Real\Update_OB\realsched.exe
C:\Program Files\iTunes\iTunesHelper.exe
C:\Program Files\HP\hpcoretech\hpcmpmgr.exe
C:\Program Files\HP\HP Software Update\HPWuSchd2.exe
C:\Program Files\Java\jre1.5.0_06\bin\jusched.exe
C:\Program Files\Viewpoint\Viewpoint Manager\ViewMgr.exe
C:\Program Files\AIM\aim.exe
C:\Program Files\iPod\bin\iPodService.exe
C:\Program Files\WIDCOMM\Bluetooth Software\BTTray.exe
C:\Program Files\HP\Digital Imaging\bin\hpqtra08.exe
C:\Program Files\Sophos\AutoUpdate\ALMon.exe
C:\Program Files\Internet Explorer\IEXPLORE.EXE
C:\Program Files\Mozilla Firefox\firefox.exe
C:\WINDOWS\System32\svchost.exe
C:\DOCUME~1\SHERWI~1\LOCALS~1\Temp\Temporary Directory 1 for hijackthis[1].zip\HijackThis.exe
R1 - HKLM\Software\Microsoft\Internet Explorer\Main,Default_Page_URL = http://global.acer.com/
O4 - HKLM\..\Run: [TabletWizard] C:\WINDOWS\help\SplshWrp.exe
O4 - HKLM\..\Run: [TabletTip] "C:\Program Files\Common Files\microsoft shared\ink\tabtip.exe" /resume
O4 - HKLM\..\Run: [SoundMan] SOUNDMAN.EXE
O4 - HKLM\..\Run: [SynTPLpr] C:\Program Files\Synaptics\SynTP\SynTPLpr.exe
O4 - HKLM\..\Run: [SynTPEnh] C:\Program Files\Synaptics\SynTP\SynTPEnh.exe
O4 - HKLM\..\Run: [CentralCrSr] C:\WINDOWS\system32\SetCrSr.exe
O4 - HKLM\..\Run: [RemoteControl] "C:\Program Files\CyberLink\PowerDVD\PDVDServ.exe"
O4 - HKLM\..\Run: [IMJPMIG8.1] "C:\WINDOWS\IME\imjp8_1\IMJPMIG.EXE" /Spoil /RemAdvDef /Migration32
O4 - HKLM\..\Run: [MSPY2002] C:\WINDOWS\system32\IME\PINTLGNT\ImScInst.exe /SYNC
O4 - HKLM\..\Run: [PHIME2002ASync] C:\WINDOWS\system32\IME\TINTLGNT\TINTSETP.EXE /SYNC
O4 - HKLM\..\Run: [PHIME2002A] C:\WINDOWS\system32\IME\TINTLGNT\TINTSETP.EXE /IMEName
O4 - HKLM\..\Run: [NvCplDaemon] RUNDLL32.EXE C:\WINDOWS\system32\NvCpl.dll,NvStartup
O4 - HKLM\..\Run: [nwiz] nwiz.exe /install
O4 - HKLM\..\Run: [BluetoothAuthenticationAgent] rundll32.exe bthprops.cpl,,BluetoothAuthenticationAgent
O4 - HKLM\..\Run: [EPM-DM] c:\acer\epm\epm-dm.exe
O4 - HKLM\..\Run: [ePowerManagement] C:\Acer\ePM\ePM.exe boot
O4 - HKLM\..\Run: [LaunchAp] C:\Program Files\Launch Manager\LaunchAp.exe
O4 - HKLM\..\Run: [PowerKey] "C:\Program Files\Launch Manager\PowerKey.exe"
O4 - HKLM\..\Run: [LManager] C:\Program Files\Launch Manager\HotkeyApp.exe
O4 - HKLM\..\Run: [CtrlVol] C:\Program Files\Launch Manager\CtrlVol.exe
O4 - HKLM\..\Run: [LMgrOSD] C:\Program Files\Launch Manager\OSDCtrl.exe
O4 - HKLM\..\Run: [Wbutton] "C:\Program Files\Launch Manager\Wbutton.exe"
O4 - HKLM\..\Run: [eRecoveryService] C:\Acer\Empowering Technology\eRecovery\Monitor.exe
O4 - HKLM\..\Run: [Software Button] "C:\Program Files\Acer soft button\SB.exe"
O4 - HKLM\..\Run: [TkBellExe] "C:\Program Files\Common Files\Real\Update_OB\realsched.exe" -osboot
O4 - HKLM\..\Run: [iTunesHelper] "C:\Program Files\iTunes\iTunesHelper.exe"
O4 - HKLM\..\Run: [HP Component Manager] "C:\Program Files\HP\hpcoretech\hpcmpmgr.exe"
O4 - HKLM\..\Run: [HP Software Update] C:\Program Files\HP\HP Software Update\HPWuSchd2.exe
O4 - HKLM\..\Run: [SunJavaUpdateSched] C:\Program Files\Java\jre1.5.0_06\bin\jusched.exe
O4 - HKLM\..\Run: [ViewMgr] C:\Program Files\Viewpoint\Viewpoint Manager\ViewMgr.exe
O4 - HKLM\..\Run: [SpySweeper] "C:\Program Files\Webroot\Spy Sweeper\SpySweeper.exe" /startintray
O4 - HKCU\..\Run: [ctfmon.exe] C:\WINDOWS\system32\ctfmon.exe
O4 - HKCU\..\Run: [AIM] C:\Program Files\AIM\aim.exe -cnetwait.odl
O4 - HKCU\..\Run: [MSMSGS] "C:\Program Files\Messenger\msmsgs.exe" /background
O4 - HKCU\..\Run: [Yahoo! Pager] C:\Program Files\Yahoo!\Messenger\ypager.exe -quiet
O4 - Global Startup: Bluetooth.lnk = ?
O4 - Global Startup: HP Digital Imaging Monitor.lnk = C:\Program Files\HP\Digital Imaging\bin\hpqtra08.exe
O4 - Global Startup: AutoUpdate Monitor.lnk = C:\Program Files\Sophos\AutoUpdate\ALMon.exe
O8 - Extra context menu item: E&xport to Microsoft Excel - res://C:\PROGRA~1\MI1933~1\OFFICE11\EXCEL.EXE/3000
O8 - Extra context menu item: Send To &Bluetooth - C:\Program Files\WIDCOMM\Bluetooth Software\btsendto_ie_ctx.htm
O9 - Extra button: (no name) - {08B0E5C0-4FCB-11CF-AAA5-00401C608501} - C:\Program Files\Java\jre1.5.0_06\bin\npjpi150_06.dll
O9 - Extra 'Tools' menuitem: Sun Java Console - {08B0E5C0-4FCB-11CF-AAA5-00401C608501} - C:\Program Files\Java\jre1.5.0_06\bin\npjpi150_06.dll
O9 - Extra button: Research - {92780B25-18CC-41C8-B9BE-3C9C571A8263} - C:\PROGRA~1\MI1933~1\OFFICE11\REFIEBAR.DLL
O9 - Extra button: AIM - {AC9E2541-2814-11d5-BC6D-00B0D0A1DE45} - C:\Program Files\AIM\aim.exe
O9 - Extra button: @btrez.dll,-4015 - {CCA281CA-C863-46ef-9331-5C8D4460577F} - C:\Program Files\WIDCOMM\Bluetooth Software\btsendto_ie.htm
O9 - Extra 'Tools' menuitem: @btrez.dll,-4017 - {CCA281CA-C863-46ef-9331-5C8D4460577F} - C:\Program Files\WIDCOMM\Bluetooth Software\btsendto_ie.htm
O9 - Extra button: Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O9 - Extra 'Tools' menuitem: Windows Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O16 - DPF: {9A9307A0-7DA4-4DAF-B042-5009F29E09E1} (ActiveScan Installer Class) - http://acs.pandasoftware.com/activescan/as5free/asinst.cab
O16 - DPF: {B8BE5E93-A60C-4D26-A2DC-220313175592} (ZoneIntro Class) - http://cdn2.zone.msn.com/binFramework/v10/ZIntro.cab34246.cab
O16 - DPF: {E5D419D6-A846-4514-9FAD-97E826C84822} (HeartbeatCtl Class) - http://fdl.msn.com/zone/datafiles/heartbeat.cab
O20 - Winlogon Notify: TabBtnWL - C:\WINDOWS\SYSTEM32\TabBtnWL.dll
O20 - Winlogon Notify: tpgwlnotify - C:\WINDOWS\SYSTEM32\tpgwlnot.dll
O20 - Winlogon Notify: WRNotifier - C:\WINDOWS\SYSTEM32\WRLogonNTF.dll
O23 - Service: Notebook Manager Service (anbmService) - OSA Technologies Inc. - C:\Acer\eManager\anbmServ.exe
O23 - Service: Bluetooth Service (btwdins) - Broadcom Corporation. - C:\Program Files\WIDCOMM\Bluetooth Software\bin\btwdins.exe
O23 - Service: InstallDriver Table Manager (IDriverT) - Macrovision Corporation - C:\Program Files\Common Files\InstallShield\Driver\11\Intel 32\IDriverT.exe
O23 - Service: iPodService - Apple Computer, Inc. - C:\Program Files\iPod\bin\iPodService.exe
O23 - Service: NVIDIA Display Driver Service (NVSvc) - NVIDIA Corporation - C:\WINDOWS\system32\nvsvc32.exe
O23 - Service: Pml Driver HPZ12 - HP - C:\WINDOWS\system32\HPZipm12.exe
O23 - Service: Sophos Anti-Virus status reporter (SAVAdminService) - Sophos plc - C:\Program Files\Sophos\Sophos Anti-Virus\SAVAdminService.exe
O23 - Service: Sophos Anti-Virus (SAVService) - Sophos plc - C:\Program Files\Sophos\Sophos Anti-Virus\SavService.exe
O23 - Service: Sophos AutoUpdate Service - Sophos plc - C:\Program Files\Sophos\AutoUpdate\ALsvc.exe
O23 - Service: Webroot Spy Sweeper Engine (svcWRSSSDK) - Webroot Software, Inc. - C:\Program Files\Webroot\Spy Sweeper\WRSSSDK.exe
the error box actually displays, "
Automatic Updates has encountered a problem and needs to close. We are sorry for the inconvenience."
szAppName : wuauclt.exe szAppVer : 5.8.0.2469
szModName : esent.dll
szModVer : 5.1.2600.2780 offset : 00057a11
if anymore information is needed let me know how to obtain it. thanks!
Your log shows no indications of infections, nor signs of anything (non-malicious) which might be causing the error.
1. Your log does show one thing you need to fix before we continue:
C:\DOCUME~1\SHERWI~1\LOCALS~1\Temp\Temporary Directory 1 for hijackthis[1].zip\HijackThis.exe
One of the normal steps in eliminating malicious programs is to entirely delete the contents of all Temp folders. Given that, if HijackThis (and other data that you care about) is living in those Temp folders, it will be erased along with everything else!
* Create a folder for HJT outside of any Temp/Temporary folders. A folder such such as C:\HijackThis or C:\Spyware Tools\HijackThis will do.
* Right-click on the HijackThis.zip folder and choose the "Extract all..." option from the resulting drop-down menu. This will start Windows' Folder Extraction Wizard. Click the "Next" button to start the wizard.
* In the next window, click on the "Browse" button. In the destination selection box, navigate to the new folder you created for HJT, hilight it, and click "OK".
* Click "Next", and then click "Finished"; a window dispaying the newly-extracted hijackthis.exe file should open.
* Double-click on the hijackthis.exe file to verify that the program works. If it does, just close hijackthis for now.
2. Open the Event Viewer utility in your Administrative Tools control panel and look through your System and Application logs for entries flagged with "Error" or "Warning", especially those related to esent, wuauclt, or Automatic Update. Double-clicking on such an entry will open a properties window with more detailed information on the error; post the details from a representative sample of some of the different error messages (please don't post duplicates of a given entry, or flood us with the entire contents of the logs).
To post the details:
In the Properties window of a given entry, click on the button with the graphic of two pieces of paper on it; the button is at the right of the window just below the up arrow/down arrow buttons. You won't see anything happen when you click the button, but it will copy all of the details to the Windows clipboard. You can then paste the details into your next post here.
Okay.. here are some of the misc. error reports that you have requested. These are just some errors that are part are from the application folder. Most of them seem to be the repeats of the first one posted of wuauclt.exe:
Event Type: Error
Event Source: Application Error
Event Category: (100)
Event ID: 1000
Date: 7/1/2006
Time: 6:22:52 PM
User: N/A
Computer: ACER-885E81581C
Description:
Faulting application wuauclt.exe, version 5.8.0.2469, faulting module esent.dll, version 5.1.2600.2780, fault address 0x0005362d.
Data:
0000: 41 70 70 6c 69 63 61 74 Applicat
0008: 69 6f 6e 20 46 61 69 6c ion Fail
0010: 75 72 65 20 20 77 75 61 ure wua
0018: 75 63 6c 74 2e 65 78 65 uclt.exe
0020: 20 35 2e 38 2e 30 2e 32 5.8.0.2
0028: 34 36 39 20 69 6e 20 65 469 in e
0030: 73 65 6e 74 2e 64 6c 6c sent.dll
0038: 20 35 2e 31 2e 32 36 30 5.1.260
0040: 30 2e 32 37 38 30 20 61 0.2780 a
0048: 74 20 6f 66 66 73 65 74 t offset
0050: 20 30 30 30 35 33 36 32 0005362
0058: 64 d
Event Type: Warning
Event Source: Userenv
Event Category: None
Event ID: 1517
Date: 7/1/2006
Time: 9:46:30 AM
User: NT AUTHORITY\SYSTEM
Computer: ACER-885E81581C
Description:
Windows saved user ACER-885E81581C\Sherwin De Guzman registry while an application or service was still using the registry during log off. The memory used by the user's registry has not been freed. The registry will be unloaded when it is no longer in use.
This is often caused by services running as a user account, try configuring the services to run in either the LocalService or NetworkService account.
\
Event Type: Error
Event Source: Application Hang
Event Category: (101)
Event ID: 1002
Date: 7/1/2006
Time: 1:38:54 AM
User: N/A
Computer: ACER-885E81581C
Description:
Hanging application YPager.exe, version 7.0.2.120, hang module hungapp, version 0.0.0.0, hang address 0x00000000.
Data:
0000: 41 70 70 6c 69 63 61 74 Applicat
0008: 69 6f 6e 20 48 61 6e 67 ion Hang
0010: 20 20 59 50 61 67 65 72 YPager
0018: 2e 65 78 65 20 37 2e 30 .exe 7.0
0020: 2e 32 2e 31 32 30 20 69 .2.120 i
0028: 6e 20 68 75 6e 67 61 70 n hungap
0030: 70 20 30 2e 30 2e 30 2e p 0.0.0.
0038: 30 20 61 74 20 6f 66 66 0 at off
0040: 73 65 74 20 30 30 30 30 set 0000
0048: 30 30 30 30 0000
Event Type: Error
Event Source: Application Error
Event Category: (100)
Event ID: 1000
Date: 6/30/2006
Time: 6:58:21 PM
User: N/A
Computer: ACER-885E81581C
Description:
Faulting application svchost.exe, version 5.1.2600.2180, faulting module esent.dll, version 5.1.2600.2780, fault address 0x00050c08.
Data:
0000: 41 70 70 6c 69 63 61 74 Applicat
0008: 69 6f 6e 20 46 61 69 6c ion Fail
0010: 75 72 65 20 20 73 76 63 ure svc
0018: 68 6f 73 74 2e 65 78 65 host.exe
0020: 20 35 2e 31 2e 32 36 30 5.1.260
0028: 30 2e 32 31 38 30 20 69 0.2180 i
0030: 6e 20 65 73 65 6e 74 2e n esent.
0038: 64 6c 6c 20 35 2e 31 2e dll 5.1.
0040: 32 36 30 30 2e 32 37 38 2600.278
0048: 30 20 61 74 20 6f 66 66 0 at off
0050: 73 65 74 20 30 30 30 35 set 0005
0058: 30 63 30 38 0c08
Event Type: Error
Event Source: Application Error
Event Category: None
Event ID: 1001
Date: 6/30/2006
Time: 6:49:29 PM
User: N/A
Computer: ACER-885E81581C
Description:
Fault bucket 254364936.
Data:
0000: 42 75 63 6b 65 74 3a 20 Bucket:
0008: 32 35 34 33 36 34 39 33 25436493
0010: 36 0d 0a 6..
Event Type: Error
Event Source: Disk
Event Category: None
Event ID: 7
Date: 6/25/2006
Time: 3:39:47 PM
User: N/A
Computer: ACER-885E81581C
Description:
The device, \Device\Harddisk0\D, has a bad block.
Data:
0000: 03 00 68 00 01 00 b6 00 ..h...¶.
0008: 00 00 00 00 07 00 04 c0 .......À
0010: 00 01 00 00 9c 00 00 c0 ......À
0018: 00 00 00 00 00 00 00 00 ........
0020: 00 40 46 65 06 00 00 00 [EMAIL=".@Fe"].@Fe[/EMAIL]....
0028: 07 46 d8 01 00 00 00 00 .FØ.....
0030: ff ff ff ff 01 00 00 00 ÿÿÿÿ....
0038: 40 00 00 84 02 00 00 00 @......
0040: 00 20 0a 12 40 03 20 40 . ..@. @
0048: 00 00 00 00 0a 00 00 00 ........
0050: 00 00 00 00 e0 dd ae 86 ....àÝ®
0058: 00 00 00 00 28 08 f2 84 ....(.ò
0060: 02 00 00 00 20 a3 32 03 .... £2.
0068: 28 00 03 32 a3 20 00 00 (..2£ ..
0070: 08 00 00 00 00 00 00 00 ........
0078: f0 00 03 00 00 00 00 0b ð.......
0080: 00 00 00 00 00 00 00 00 ........
0088: 00 00 00 00 00 00 00 00 ........
I don't know if you wanted me to post all of the variations of the errors. I think that the ones I have posted were what the majority of them were. Please let me know if you want me to search for all the different ones. Thanks for all of your help
DMR are you considering Viewpoint manager to be malicious, because its in his log just so you know.
DMR are you considering Viewpoint manager to be malicious, because its in his log just so you know.
Yeah- I space on that one a lot, but it should go. Thanks for the eyes.
sdeguzman,
This isn't related to your primary problem, but you should open your Add/Remove Programs control panel, hilight the Viewpoint package, and click the "Remove" button.
(I'm still looking for a solution that directly relates to your AU/esent crashes)
thanks for helping.. i found a post when i was trying to search for a solution. here's the link: http://www.daniweb.com/techtalkforums/thread20973.html, the person who posted this seemed to have the same problem, but it's probably unique for each user. I don't know if it may help
can anyone help me with my problem? I deleted everything from my TEMP folder and deleted the viewpoint files out of my computer. Is there anything I can do to stop the error boxes from appearing?
This topic has been dead for over six months. Start a new discussion instead.
Have something to contribute to this discussion? Please be thoughtful, detailed and courteous, and be sure to adhere to our posting rules. | 2018-03-18 17:53:47 | {"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": 0.8382518291473389, "perplexity": 10860.88093828644}, "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/1521257645830.10/warc/CC-MAIN-20180318165408-20180318185408-00388.warc.gz"} |
https://ashishkumarletslearn.com/quadratic-equations-class-10-maths/ | There will be obstacles. There will be doubters. There will be mistakes. But with hard work, there are no limits. – Michael Phelps
# Quadratic Equations Class 10 Maths
In this chapter, you have studied the following points:
1. A quadratic equation in the variable x is of the form $ax^2+bx+c=0$, where a, b, c are real numbers and a ≠ 0.
2. A real number α is said to be a root of the quadratic equation $ax^2 + bx + c = 0$, if $aα^2+ bα + c = 0$. The zeroes of the quadratic polynomial $ax^2 + bx + c = 0$ and the roots of the quadratic equation $ax^2+ bx + c$
3. If we can factorise $ax^2+bx+c$, $a \ne 0$ into a product of two linear factors, then the roots of the quadratic equation $ax^2+bx+c$ can be found by equating each factor to zero.
4. A quadratic equation can also be solved by the method of completing the square.
5. Quadratic formula: The roots of a quadratic equation $ax^2+bx+c=0$ are given by $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$, provided $b^2-4ac \ge 0$.
6. A quadratic equation $ax^2 + bx + c = 0$ has
(i) two distinct real roots, if $b^2 – 4ac > 0$,
(ii) two equal roots (i.e., coincident roots), if $b^2 – 4ac = 0$, and
(iii) no real roots, if $b^2– 4ac < 0$. | 2019-11-20 09:10:40 | {"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": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 15, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7719722390174866, "perplexity": 160.58644343790007}, "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/1573496670535.9/warc/CC-MAIN-20191120083921-20191120111921-00150.warc.gz"} |
https://academy.vertabelo.com/course/data-visualization-101/line-chart/work-with-your-chart/baseline | Only this week, get the SQL Complete Track of 9 courses in a special prize of $330$89!
Visualize your data - line chart
As you probably noticed, ggplot automatically chooses the limits of the vertical and horizontal axes. If you look carefully at the vertical axis, you will see that – contrary to a bar chart – it doesn't start at zero. Is this a mistake? | 2019-04-23 00:19:38 | {"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": 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": 0.5384520888328552, "perplexity": 555.3818885484577}, "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-2019-18/segments/1555578583000.29/warc/CC-MAIN-20190422235159-20190423021159-00216.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-y-8x-3-and-y-4x-3 | # How do you solve y= -8x-3 and y=4x-3?
May 24, 2018
$x = 0 \mathmr{and} y = - 3$
#### Explanation:
You might notice that both equations represent straight lines as they are both in the form $y = m x + c$.
By solving them you are finding the point of intersection of the two lines.
The easiest way to solve equation in this form is to equate the equations. They are both given as $y = \ldots$
$\textcolor{b l u e}{y = - 8 x - 3} \text{ " and " } \textcolor{red}{y = 4 x - 3}$
$\textcolor{w h i t e}{\times \times \times \times \times \times} \textcolor{b l u e}{y} = \textcolor{red}{y}$
$\textcolor{w h i t e}{\times \times x} \therefore \textcolor{b l u e}{- 8 x - 3} = \textcolor{red}{4 x - 3}$
$\textcolor{w h i t e}{\times \times x} \therefore - 3 + 3 = 4 x + 8 x$
$\textcolor{w h i t e}{\times \times \times \times \times} \therefore 0 = 12 x$
$\textcolor{w h i t e}{\times \times \times \times \times} \therefore 0 = x$
(We could have seen this at the beginning because the $3$ is subtracted from different terms in $x$, yet both give $y$.
$x = 0$ is the only possible value of $x$
$y = 4 \left(0\right) - 3 = - 3$ | 2019-09-23 17:45:33 | {"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": 0.830630898475647, "perplexity": 256.64499284586094}, "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-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00441.warc.gz"} |
http://scholarworks.umass.edu/climate_nuclearpower/2011/nov19/17/ | Day 2: Saturday, November 19
Event Title
Session F: Contributed Oral Papers - F1: Physics Education: Class Explorations in Space: From the Blackboard and History to the Outdoors and Future
Location
Concourse, Campus Center, University of Massachusetts - Amherst
Event Website
http://blogs.umass.edu/nes2011/
Start Date
19-11-2011 8:00 AM
End Date
19-11-2011 8:12 AM
Description
Our everyday activities occur so seamlessly in the space around us as to leave us unawares of space, its properties, and our use of it. What might we notice, wonder about and learn through interacting with space exploratively? My seminar class took on that question as an opening for personal and group experiences during this semester. In the process, they observe space locally and in the sky, read historical works of science involving space, and invent and construct forms in space. All these actions arise responsively, as we respond to: physical materials and space; historical resources; our seminar participants, and future learners. Checks, revisions and further developments -- on our findings, geometrical constructions, shared or personal inferences---come about observationally and collaboratively. I teach this seminar as an expression of the research pedagogy of critical exploration, developed by Eleanor Duckworth from the work of Jean Piaget, B\"{a}rbel Inhelder and the Elementary Science Study. This practice applies the quest for understanding of a researcher to spontaneous interactions evolving within a classroom. The teacher supports students in satisfying and developing their curiosities, which often results in exploring the subject matter by routes that are novel to both teacher and student. As my students mess about'' with geometry, string and chalk at the blackboard, in their notebooks, and in response to propositions in Euclid's \textit{Elements}, they continually imagine further novel venues for using geometry to explore space. Where might their explorations go in the future? I invite you to hear from them directly!
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Nov 19th, 8:00 AM Nov 19th, 8:12 AM
Session F: Contributed Oral Papers - F1: Physics Education: Class Explorations in Space: From the Blackboard and History to the Outdoors and Future
Concourse, Campus Center, University of Massachusetts - Amherst
Our everyday activities occur so seamlessly in the space around us as to leave us unawares of space, its properties, and our use of it. What might we notice, wonder about and learn through interacting with space exploratively? My seminar class took on that question as an opening for personal and group experiences during this semester. In the process, they observe space locally and in the sky, read historical works of science involving space, and invent and construct forms in space. All these actions arise responsively, as we respond to: physical materials and space; historical resources; our seminar participants, and future learners. Checks, revisions and further developments -- on our findings, geometrical constructions, shared or personal inferences---come about observationally and collaboratively. I teach this seminar as an expression of the research pedagogy of critical exploration, developed by Eleanor Duckworth from the work of Jean Piaget, B\"{a}rbel Inhelder and the Elementary Science Study. This practice applies the quest for understanding of a researcher to spontaneous interactions evolving within a classroom. The teacher supports students in satisfying and developing their curiosities, which often results in exploring the subject matter by routes that are novel to both teacher and student. As my students mess about'' with geometry, string and chalk at the blackboard, in their notebooks, and in response to propositions in Euclid's \textit{Elements}, they continually imagine further novel venues for using geometry to explore space. Where might their explorations go in the future? I invite you to hear from them directly!
http://scholarworks.umass.edu/climate_nuclearpower/2011/nov19/17 | 2014-03-11 18:00:42 | {"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": 0.2904971241950989, "perplexity": 5281.8326186919685}, "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-2014-10/segments/1394011239452/warc/CC-MAIN-20140305092039-00021-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathematica.stackexchange.com/questions/23644/quantity-and-trig-functions | # Quantity and Trig functions
I have a bunch of data to which I've assigned units using Quantity, including some angular measurements in degrees. I'd like to be able to use these measurements with trigonometric functions, but it seems that trig functions don't "know" about Quantity, so I'm forced to convert those measurements to radians and then apply the trig function to the magnitudes, like this:
x = Quantity[23.4, "AngularDegrees"] | 2015-01-28 12:18:17 | {"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": 0.8561981916427612, "perplexity": 701.5100249262765}, "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-2015-06/segments/1422122102237.39/warc/CC-MAIN-20150124175502-00045-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41598-017-03411-7?error=cookies_not_supported&code=32a15b6d-bfa5-40ef-ac95-a6c5bbf22ead | # Chronic treatment with a smart antioxidative nanoparticle for inhibition of amyloid plaque propagation in Tg2576 mouse model of Alzheimer’s disease
## Abstract
The present study aimed to assess whether our newly developed redox nanoparticle (RNPN) that has antioxidant potential decreases Aβ levels or prevents Aβ aggregation associated with oxidative stress. The transgenic Tg2576 Alzheimer’s disease (AD) mice were used to investigate the effect of chronic ad libitum drinking of RNPN solution for 6 months, including memory and learning functions, antioxidant activity, and amyloid plaque aggregation. The results showed that RNPN-treated mice had significantly attenuated cognitive deficits of both spatial and non-spatial memories, reduced oxidative stress of lipid peroxide, and DNA oxidation. RNPN treatment increased the percent inhibition of superoxide anion and glutathione peroxidase activity, neuronal densities in the cortex and hippocampus, decreased Aβ(1-40), Aβ(1-42) and gamma (γ)-secretase levels, and reduced Aβ plaque observed using immunohistochemistry analysis and thioflavin S staining. Our results suggest that RNPN may be a promising candidate for AD therapy because of its antioxidant properties and reduction in Aβ aggregation, thereby suppressing its adverse side effect.
## Introduction
The number of patients affected with Alzheimer’s disease (AD) is increasing1. AD is a slow progressing, neurodegenerative disorder, which exhibits the cardinal hallmarks of amyloid beta (Aβ) plaques accumulation and neurofibrillary tangles. In addition, the loss of synapses, atrophy of the cerebral cortex, neuroinflammation, and increase in free radicals in patients with AD is also reported2.
In the progression of AD, the conformation and misfolding of Aβ proteins causes them to clump together into small oligomers, protofibrils, and mature fibrils3. The soluble oligomers are structurally heterogeneous, which have been determined as the main toxic forms in AD4. The oligomer-induced toxicity of the cell is related to its ability to permeabilize cellular membranes and lipid bilayers5 to form distinct pores or single channels in the membranes6 and disturb intracellular calcium homeostasis, leading to oxidative stress including overproduction of reactive oxygen species (ROS), altered signaling pathways, mitochondrial dysfunction, and neuronal death7.
Currently, one of the most promising strategies for AD therapy is to decrease Aβ aggregation, especially Aβ(1-42) and/or destroy the oligomers and fibrils formed in the brain8, 9. Previous studies have proposed an effective inhibitor of Aβ, Quercetin, which blocks the formation of fibrils, including the inhibition of Aβ aggregation10, reduction in ROS production11, and alteration in the functions of immune and inflammatory cells12. Therefore, there are several targets such as inhibition of Aβ aggregation, and Aβ-induced oxidative stress, especially ROS, to improve therapeutic and preventive strategies for the development of disease-modifying drugs for AD.
Antioxidants are widely used to scavenge ROS13, 14. Although versatile antioxidants such as vitamin C and E, many phytochemicals, and synthetic drugs have been developed so far, none of them work effectively. One of the serious issues in the use of antioxidants is the dysfunction of normal redox reaction in healthy cells, including the electron transport chain due to the internalization of these low-molecular-weight (LMW) antioxidants because of their size. This undesired adverse side effect limits the effective dosage of these LMW antioxidants. Recently, we designed polymer antioxidants, where the antioxidant moieties are covalently bound to the amphiphilic block copolymer backbone. Due to the high molecular weight, the internalization of these polymer antioxidants is decreased in healthy cells and their mitochondria, which significantly reduce its adverse effects15, 16.
We designed the polymer poly(ethylene glycol)-b-poly[4-(2,2,6,6-tetramethylpiperidine-1-oxyl)aminomethylstyrene] (PEG-b-PMNT). The PEG segment is water-soluble part, while PMNT is the water-insoluble part. Thus, it forms a polymer micelle of several tens of nanometer size in transparent aqueous media (we abbreviate it as RNPN, as shown in Supplementary Figure 1). The 2,2,6,6-tetramethylpiperidine-1-oxyl (so called TEMPO) groups as a side chain of the PMNT segment is known to be a stable radical and does not react with each other. However, it catalytically reacts with ROS and is regarded as one of the strongest antioxidants17. Because the PMNT segment possesses repeating amino groups, it protonates in response to pH decrease and becomes hydrophilic to result in disintegration under acidic conditions15, 18, 19. After oral administration of RNPN solution, it disintegrates in the stomach, and the disintegrated antioxidant polymer is absorbed in the bloodstream from the small intestine because the total molecular weight of the polymer was limited to around 7–20 kDa19. We have previously confirmed that our redox nanoparticle exhibits an anti-apoptotic effect on Aβ-induced cell death in vitro 20. We have also confirmed that oral administration of RNPN showed an antioxidant effect on the brain of senescence-accelerated (SAMP8) mice and did not exhibit any detectable toxicity in the vital organs after continuous administration for 1 month19. The objective of this work was to confirm the effectiveness of RNPN in transgenic AD mice model (Tg2576) for an extended time by ad libitum drinking of RNPN for the prevention of Aβ accumulation in Tg2576 mice overexpressing a mutant form of amyloid precursor protein (APP).
## Results
### Delivery of redox polymer to the blood and brain by chronic ad libitum drinking of RNPN solution
To confirm the uptake of the redox polymer in the blood by oral administration, electron spin resonance (ESR) measurements of the blood and brain were carried out. The ESR signal of LMW TEMPO derivatives showed a triplet signal due to the coupling between the unpaired electron on oxygen and 14N nuclei (Fig. 1A). After ad libitum drinking of LMW TEMPO, the triplet signal was clearly observed in both the blood and brain samples as observed in the standard solution (Fig. 1B,C). We have previously reported that the ESR pattern of an aqueous solution of RNPN does not appear as a typical triplet signal but as broadened spectra as shown in Fig. 1D. When RNPN was mixed with untreated brain, and the homogenate was analyzed by ESR measurement, the broadened spectrum was observed, similar to RNPN in saline solution (Fig. 1E). After ad libitum drinking of RNPN solution, small but definite signal was observed in the homogenized brain samples, although the signal pattern changed from broad ESR spectra (Fig. 1E) to triplet signal in both the blood and brain samples (Fig. 1F and G), indicating the disintegration of RNPN in the stomach to result in the uptake of redox polymer in the bloodstream and brain tissues. These results were consistent with the quantitative analysis of ESR spectra intensities as displayed in Supplementary Table S1.
### Effect of redox polymer on learning and memory deficit in transgenic mice expressing mutant APP
We analyzed the recovery of learning and memory functions in transgenic mice expressing APP in the brain as shown in Fig. 2. In the object recognition and location memory tests, the time period was lower for AD mice aged around 10–11 months, compared with the wild type group while in the groups treated with LMW TEMPO and nRNP, which is a polymer micelle without TEMPO moiety, the exploration time and discrimination index did not increase. In contrast, the exploration time and discrimination index for the RNPN-treated group significantly increased compared with other treated groups for both the object recognition and location tests (Fig. 2A–D). Furthermore, the effect of RNPN on object recognition test after 24 h also displayed the same pattern as after 1 h of ad libitum drinking (Figure S4). The Morris water maze tests also showed a tendency similar to the object recognition tests; the RNPN-treatment significantly shortened the acquisition time compared with other groups (Fig. 2E). The aforementioned results confirmed that the RNPN-treatment significantly improved the learning and memory dysfunctions in AD transgenic mice (p < 0.05) when compare to the water-treated, LMW TEMPO-treated, and nRNP-treated groups. It is interesting to note that the LMW TEMPO-treated group did not exhibit a strong improvement effect on the learning and memory functions than those of RNPN-treated mice, although it was detected in the brain using ESR measurements.
### Suppression of oxidative stress in the brain by ad libitum drinking of RNPN solution
The lipid peroxide or malondialdehyde (MDA), 8-hydroxy-2′ -deoxyguanosine (8-OHdG) levels, the percent inhibition of superoxide anion (O2 ·−), and glutathione peroxidase (GPx) assays were used to determine the free radical scavenging capacity of RNPN in comparison with non-treated, water-treated, TEMPO-treated, and nRNP-treated groups. As shown in Fig. 3, RNPN treatment decreased MDA and 8-OHdG levels, while increasing percent inhibition of O2 ·− and GPx activity when compared to the other treatment groups (p < 0.05), indicating the effectiveness of RNPN treatment compared to LMW antioxidant treatment.
### Decreased Aβ(1-40), Aβ(1-42), and γ-secretase levels in the brain and plasma of transgenic mice treated with ad libitum drinking of RNPN solution
As mentioned above, the formation and aggregation of Aβ(1-42) is more toxic, widespread, and abundant than Aβ(1-40)21. In addition, γ-secretase is a protease with substrates that cleave APP and catalyze Aβ aggregation via the amyloidogenic pathway22. We observed that the levels of Aβ(1-40), (1-42), and γ-secretase in both the brain and plasma samples were significantly reduced after RNPN-treatment compared to AD mice treated with water, TEMPO, and nRNP (p < 0.05) as shown in Fig. 4.
### Reduction in amyloid plaque formation by chronic ad libitum drinking of RNPN solution
To confirm amyloid plaque formation and/or aggregation in mice brain, thioflavin S and immunohistochemistry staining of Aβ(1-40) and Aβ(1-42) were carried out. From the representative photomicrographs of thioflavin S staining in the cerebral cortex, as shown in Fig. 5A, no remarkable Aβ fibrils was observed in wild-type mice (Fig. 5A-1), while transgenic mice in the water-treated group (Fig. 5A-2) showed remarkable Aβ fibrils in the cortex area. Although the LMW TEMPO treatment attenuated the number Aβ fibrils to some extent, AD mice treated with RNPN showed much higher attenuation efficiency (Fig. 5A-4) (p < 0.05, Figs 5A-3,5), which was confirmed by the quantitative fluorescent intensity in Fig. 5B.
The immunostaining of Aβ(1-40) and Aβ(1-42) revealed no immunoreactive cells in the cerebral cortex of wild-type mice (Fig. 5C-1, E-1). On the contrary, in AD transgenic mice that were treated with water, numerous immunoreactive Aβ(1-40) and Aβ(1-42) positive cells were observed (Fig. 5C-2, E-2), similar to that after treatment with nRNP (Fig. 5C-5, E-5). The RNPN-treated AD mice group showed significant reduction in Aβ(1-40) and Aβ(1-42) positive cells (Fig. 5C-4, E-4). This reduction was higher than that in the LMW TEMPO group (Fig. 5C-3, E-3), which are quantitatively confirmed as shown in Fig. 5D,F. For quantitative evaluation of number of plaques as shown in Fig. 5D,F, we employed randomly captured 5 slices per group under magnification of 20X (See Figure S5).
## Discussion
Transgenic Tg2576 mice overexpressing human amyloid precursor protein (hAPP) are widely used as an AD mouse model to evaluate the treatment effects on Aβ pathology. Previous reports on the onset of cognitive deficits in Tg2576 have determined that abnormalities appear as early as 3 months and as late as 12 months24. In this study, the aged Tg2576 mice (approximate 7–12 months) were used for the evaluation of the learning and memory of AD mice; Morris water maze, novel object recognition and object location memory tests25,26,27,28. The transgenic mice we used in this study showed statistically significant reduction of the duration of both object recognition and location tests and elongation of acquisition time in Morris water maze test than that of wild-type mice, which is in accordance with the previous study where Tg2576 mice displayed impaired spontaneous alternation of Y-maze test at both 3 and 10 months of age, as well as advanced impairment in the acquisition time of Morris water maze test in 9–10 months of AD mice29. We have previously reported that oral RNPN treatment recovered both cognition and memory levels in 17-week-old SAMP8 mice19. Here, RNPN-treated group extended the time in the object recognition and location tests and decreased the acquisition time in Morris water maze test, indicating the effective attenuation of the learning and memory deficits in AD transgenic mice (Tg2576) (Fig. 2). Taking together these results and our previous data on SAMP8, it suggests that our antioxidative nanoparticle treatment is robust strategy for AD therapy.
Oxidative stress has been implicated in Aβ accumulation and progression via mitochondrial dysfunction, which caused by the generation of ROS30, 31 and reduction in the level of detoxifying enzymes including superoxide dismutase (SOD), GPx and catalase (CAT) in the early stages of the AD32, 33. ROS destroys the polyunsaturated fatty acids of cellular membranes to generate lipid peroxidation products such as MDA, which may serve as an indicator of the level of oxidative damage32, 34, and induce neuronal deterioration35, 36. In addition, 8-OHdG has been determined as a pivotal biomarker of oxidative DNA damage37. Additionally, the toxic effect of Aβ peptide in neuronal cells has been proposed via the interaction between the peptide and Cu2+ and Fe3+ ions because Aβ is a metalloprotein that displays high-affinity binding to these ions, leading to amyloid plaque formation. Aβ catalyzes the reduction of Cu2+ to Cu+ and Fe3+ to Fe2+, that produce hydrogen peroxide (H2O2)38,39,40. Furthermore, in AD transgenic mouse models of mutants of APP elevated production of H2O2 and nitric oxide increases protein and lipid peroxidation. These were associated with age-related Aβ accumulation, and Aβ further enhances oxidative stress41. As expected, the observed neuroprotection by RNPN was due to its antioxidant properties through enhancement of the activity of GPx and percent inhibition of O2 ·−, while reducing MDA and 8-OHdG levels, compared to other AD groups (p < 0.05) as shown in Fig. 3. This was consistent with our previous result, where we showed that RNPN eliminates superoxide anion and hydroxyl radicals, which cause lipid peroxidation and protein and DNA oxidation19, 20.
ROS are implicated in the formation of senile plaques in the brains of patients with AD, which may result in neuronal death30. As the result displayed that scavenging ROS by orally administered RNPN significantly attenuates the neuronal loss in both the cerebral cortex and hippocampus when compared to other AD-treated groups (Supplementary Figures S2-3, p < 0.05).
The development of γ-secretase inhibitors has been explored as drugs for AD52. Our in vivo result shows that RNPN-treatment significantly reduced γ-secretase levels, followed by a decrease in Aβ-level compared to AD mice treated with water, TEMPO, and nRNP respectively (p < 0.05), as shown in Fig. 4, which is in accordance with the previous in vitro and knockdown experiments.
Finally, we confirmed amyloid plaque formation and/or aggregation via thioflavin S and immunohistochemistry staining of Aβ(1-40) and Aβ(1-42). The results shown in Fig. 5 display that chronic oral administration of RNPN significantly reduced Aβ plaques and/or fibril aggregation when compared to other AD treatment groups (p < 0.05). It has been reported that the accumulation of Aβ is through the interaction of soluble Aβ with metal ions, mainly Zn2+, Cu2+, and Fe3+38. The prevention of soluble Aβ formation by RNPN treatment inhibits Aβ aggregation.
## Conclusion
Here, we confirmed that after ad libitum drinking of our pH-sensitive redox nanoparticle, RNPN, which is a self-assembling polymer antioxidant, RNPN is internalized in the brain and eliminates the increased oxidative stress by scavenging ROS and attenuates cognitive deficits. The increased levels of Aβ and γ-secretase and radical-scavenging activity were decreased after RNPN treatments. Finally, the number of the thioflavin S-stained neurons was significantly higher in the RNPN-treated group than in other groups. On the basis of these obtained results in addition to our previous results on the senescence-accelerated mice-treated with RNPN, we suggest that RNPN may be a promising candidate for the treatment of brain disorders including AD therapy.
## Methods
### Drugs and reagents
Amino-2,2,6,6-tetramethylpiperidine-N-oxyl (TEMPO), 4-hydroxy-2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPOL) were purchased from Sigma-Aldrich (MO, USA), Mouse Aβ(1-42), Aβ(1-40) and γ-secretase ELISA kits were bought from MyBioSource, Inc. (San Diego, USA), Oxidative DNA damage ELISA kit was purchased from Cell Biolab, Inc. (San Diego, USA), Liquid 3,3′-diaminobenzidine (DAB) substrate kit was bought from Life technologies Corp. (Waltham, USA) and primary anti-Aβ(1-42), Aβ(1-40) and secondary antibodies were purchased from Abcam Plc. (Tokyo, Japan). All other chemicals in this study, which are analytical-reagent grade, were purchased locally from Wako Pure Chemical Industries Ltd. Japan.
### Animals
Female APPSWE/hemi-rd1 Tg2576 and wild-type mice were used in this study (CLEA, Japan, Inc., Tokyo, Japan). They were housed in the experimental animal center of the University of Tsukuba under controlled temperature (23 ± 1 °C), humidity (50 ± 5%), and lighting (12 h light/dark cycles). The animals had unrestricted access to food and water. All the experiments were carried out in accordance with the guidelines for animal care and use of Faculty of Medicine, Tsukuba University and were approved by the animal ethics committee of the Institutional Animal Experiment Committee at the University of Tsukuba (Protocol number 13-407) and in accordance with the Regulation for Animal Experiments in our University and the Fundamental Guidelines for Proper Conduct of Animal Experiments and Related Activities in Academic Research Institutions under the jurisdiction of the Ministry of Education, Culture, Sports, Science, and Technology.
### Biodistribution of RNPN in the blood and brain after ad libitum drinking
Tg2576 mice age approximately 12 months old were anesthetized via an intraperitoneal injection of sodium pentobarbital (50 mg/kg) after 6-month ad libitum drinking of RNPN solution (5 mg/mL/day) and LMW TEMPO (0.6 mg/mL/day). The blood and brain tissue were collected immediately after perfusion. Whole blood samples were subjected immediately to ESR measurement to quantify the drug levels. The brain tissue was immediately placed on ice and homogenized.
The ESR signals from the blood were recorded at room temperature using a Bruker EMX-T ESR spectrometer operating at 9.8 GHz with a 100 kHz magnetic field modulation. Signals were collected with the following parameters: center field, 5000 G; sweep width, 7000 G; microwave power, 100.2 mW; receiver gain, 1 × 103; time constant, 81.92 ms; and conversion time, 160 ms and sweep time, 163.84 s. The blood samples were corrected at predetermined time points and subjected to the ESR.
The total amount of drug (nitroxide radicals + hydroxyamines) in the brain was estimated from X-band ESR spectrometer (JES-TE25X; JEOL, Tokyo, Japan) at room temperature after the oxidation of hydroxylamine by K3[Fe(CN)6], which was prepared at 200 mM as a stock solution. The ESR measurements were carried out under the following conditions: frequency, 9.41 GHz; power, 8.00 mW; field, 333.8 ± 5 mT; sweep time, 1.0 min; modulation, 0.1 mT; time constant, 0.1 s.
### Voluntary test fluids consumption
Fifty milliliters of drinking fluids were provided in 75 mL drinking bottles equipped with stainless steel spouts, placed on cage covers. During the first 5 days, mice were given water as their only drinking fluid from bottles replacement water tap system to habituate them. For the next 6 months (at 7–12 months of age), all mice were allowed drinking bottles containing RNPN (5 mg/mL), nRNP (5 mg/mL), LMW TEMPO (0.6 mg/mL), and water in each group. The consumption of the fluids and the body weights was recorded once a week, and the bottles were filled with fresh solutions.
### Experimental design
We used 7 month-old female APPSWE/hemi-rd1 Tg2576 and their non-transgenic littermates or wild-type (WT) mice (40 Tg2576 mice and 10 WT mice; weight, 25 ± 2.0 g and 30 ± 2.0 g, respectively) in this study. All Tg2576 mice were randomly assigned to various groups for RNPN (5 mg/mouse/day) or blank micelles (5 mg/mL/day) or LMW TEMPO (0.6 mg/mL/day) or vehicle (water) by unrestricted access from drinking bottles every day for a 6-month period. In the fourth and fifth month after ad libitum drinking, the animals were tested in object recognition and object location tests. In the last month of the experiment, we also tested the mice in the Morris water maze test. After 6 months of the experiment, 50 mice were sacrificed, and half of the brain tissues were used for the assays for soluble Aβ(1-40) and Aβ(1-42), γ-secretase activity, ROS production (MDA and DNA oxidation), ROS levels, scavenging enzyme activity (GPx activity), and the other halves were used for Aβ immunohistochemical, thioflavin S, and cresyl violet staining.
### Plasma and brain collection
After their behavioral tests were finished, the mice were anesthetized with sodium pentobarbital (50 mg/kg i.p.); blood samples were kept to separate plasma, and brains were quickly isolated. The tissues were prepared for biochemical, histological and immunohistochemical examinations and stored at −80 °C until determination.
### Antioxidant enzyme assay
The glutathione peroxidase (GPx) activity was determined by the previous method of Hussain et al.53 based on that the activity was measured indirectly by a coupled reaction with glutathione reductase. Oxidized glutathione, produced upon reduction of hydrogen peroxide by glutathione peroxidase, was recycled to its reduced state by glutathione reductase and NADPH. The oxidation of NADPH to NADP+ was accompanied by a decrease in absorbance at 340 nm. The rate of decrease in the A340 nm was directly proportional to the glutathione peroxidase activity. In the final 1 mL of the system mixture contained 48 mM sodium phosphate, 0.38 mM EDTA, 0.12 mN β-NADPH, 0.95 mM sodium azide, 3.2 units of glutathione reductase, 1 mM glutathione (GSH), 0.02 mM DL-dithiothreitol, 0.0007% H2O2, and the standard enzyme glutathione peroxidase solution or a homogenate brain sample. The glutathione peroxidase solution was used as a standard enzyme activity. The standard curve was plotted as the rate of A340 nm per minute against the GPx activity. One unit activity was defined as the amount of enzyme necessary to catalase the oxidation by H2O2 of 1 µmole of GSH to GSSG per minute at pH 7 at 25 °C. The data were reported in units of GPx per mg protein.
### Reactive oxygen species (ROS) products assays
Lipid peroxidation (LPO) was measured by determining malonyldialdehyde (MDA) which have been used as an indicator of lipid peroxidation according to Ohkawa et al.54, were measured by using a commercial assay kit (BIOMOL International, USA). Each sample was homogenized (Potter-Elvehjem) in a 10-fold volume of ice-cold 20 mM pH 7.4 of PBS containing 0.5 mM butylated hydroxytoluene to prevent sample oxidation. The homogenized sample was centrifuged at 3,000 g at 4 °C for 10 min, and a 200 µL aliquot of the supernatant was used to measure MDA plus HAE levels according to the instructions of the manufacturer. Values were standardized to micrograms of protein.
Each sample was homogenized (Potter-Elvehjem) in a 10-fold volume of ice-cold 20 mM pH 7.4 of PBS containing 1% streptomycin sulfate and incubated for 30 min at room temperature. The nucleic acid precipitates were removed by centrifuging at 6,000 g for 10 min at 4 °C to avoid erroneous contribution to a higher estimation of the carbonyl content from nucleic acid in the cells. The supernatant was used to measure protein levels according to the instructions of the manufacturer. The obtained values were standardized to milligrams of protein. Deoxyguanosine (dG) is one of the constituents of DNA and when it is oxidized, it is altered into 8-hydroxy-2′-deoxyguanosine (8-OHdG). 8-OHdG is useful as a general DNA oxidation marker in the body. A commercial assay kit (Cell Biolab, Inc., San Diego,USA) was used in this measurement. Tissue homogenate and supernatant were used to measure 8-OHdG levels according to the instructions of the manufacturer. The obtained values were standardized to milligrams of protein.
### Brain oxidative DNA damage determined by ELISA
The remained supernatant of mice’ brains were also analyzed of 8-hydroxydeoxyguanosine (8-OHdG) which is the common marker of DNA oxidative stress according to the protocol of product manual (Cell Biolab, Inc., San Diego, USA). First, prepared 8-OHdG coated plate, overnight at 4 °C. Washed, filled assay diluent to each well and incubated for 1 h at room temperature (RT). Removed the solution, added 50 µL of sample to the coated wells following with anti-8-OHdG antibody and incubated on an orbital shaker for 1 h at RT. Washed properly, filled 100 µL of secondary antibody. Added substrate solution and inhibit reaction by stop solution. Eventually, read the absorbance at 450 nm on a spectrophotometer to compare to the standard curve of 8-OHdG.
### Assay of superoxide anion level
The reaction mixture consisted of 10 mM phosphate buffer (pH 7.4) containing 0.1 mM xanthine, 0.1 mM EDTA, 0.1 mM nitroblue tetrazolium, and 0.1 unit xanthine oxidase (XO) at a final volume of 1 mL. The formation rate of formazan produced was determined from the slope of the absorbance curve during the initial 2 min of the reaction at 560 nm. In order to analyze the anti-oxidation activity, each sample of different groups was added to the reaction mixture. The change of absorbance was compared with that of the control in the same time reaction, and anti-oxidation activity was calculated according to the following Equation (1)
$$\mathrm{Anti}\mbox{-}\mathrm{oxidation}\,{\rm{activity}}\,( \% )=\mathrm{100}\times ({\rm{A}}-{\rm{B}})/{\rm{A}}$$
(1)
where A and B are the rate of formazan formation in the absence and presence of sample, respectively.
### Measurement soluble Aβ(1-40), soluble Aβ(1-42) and γ-secretase in brain tissue and plasma
The brain tissue from one brain hemisphere of each mouse was homogenized in PBS, pH 7.4 and centrifuged at 1500 g for 15 min. The anticoagulated blood was drawn and spinned at 1000 g for 10 min. The supernatants of both types of sample were collected and protein quantification was performed using the bicinchoninic acid (BCA) assay (Bio-Rad Laboratories, Hercules, CA). Samples were analyzed for soluble Aβ(1-40), Aβ(1-42) or γ-secretase (BioSource International, Inc., Camarillo, CA) according to the manufacturer’s instructions. Briefly, added 50–100 µL of samples to each well of pre-coated microtiter plate, mixed with 5–10 µL of balance solution, filled 50–100 µL of conjugate solution to each specimen and then incubated for 1 h at 37 °C on automated shaker. Washed properly, added substrate A and B respectively and then incubated for 10–15 min at 37 °C. Finally, the reaction was stopped by terminate solution and determine the optical density at 450 nm immediately to compare with the typical standard curve of mouse Aβ(1-40) or Aβ(1-42) or γ-secretase respectively.
### Histopathology
Brain was isolated for immunohistopathology. The corrected tissues were fixed in 10% formalin solution. Paraffin blocks were prepared after completing the tissue processing in different grades of alcohol and xylene. Brain sections (5 µm) were prepared from paraffin blocks using microtome, stained with 0.5% cresyl violet according to Paxions and Chorles55. Images were taken using OLYMPUS camera connected to the microscope to examine gross cellular damage and neuronal density determination using Image JTM (NIH, MD, USA) software56.
### Aβ plaque staining and quantification
The brain sections of 5 µm were also studied of immunohistochemistry analysis for Aβ(1-40) and Aβ(1-42) respectively. Pre-heated slices with microwave in acetic solution, cool at RT, blocked with methanol and hydrogen peroxide solution for 30 min. Washed properly, prepared moist chamber, blocked with BSA and added primary anti-Aβ(1-40) and Aβ(1-42) respectively (1:100, 1:100 and 1:50) of each slices for overnight at RT. Washed again and incubated with secondary antibody conjugated-horseradish peroxidase (HRP) for 45 min. Developed color by using DAB kit and slices were represented as brown positive staining cells, compared to WT group and used without primary antibody as negative control group under light microscope (Olympus model BZ-X710 Keyence, Tokyo, Japan).
### Thioflavin S staining and semi-quantification
This method is suitable for staining of Aβ fibrils following Schmidt et al.57. After tissue processing, sections were immersed into 0.25% potassium permanganate solution for 20 min, bleaching solution for 2 min, blocking solution for 20 min, 0.25% acetic acid for 5 s, dropped thioflavin S staining solution on slices for 3–5 min, washed by 50% ethanol and following distilled water. Mounted with glycerin and observed under fluorescent microscope (Olympus model BZ-X710 Keyence, Tokyo, Japan).
### The behavioural tests
In this experiment using 3 behavioural tests for measuring cognition status. The Morris water maze test was selected as a method for the evaluation of the spatial learning and memory according to the Morris’s method58. A circular water tank (120 cm in diameter and 50 cm in height) was filled with water to a depth of 30 cm inside the tank; an escape platform (11 cm in diameter) was placed, with the top of 1 cm below the water surface. The platform was in the middle of the target quadrant, and its position remained fixed during the experiment. Above the tank, a white floor-to-ceiling cloth curtain was drawn around the pool, and four kinds of black cardboard (circle, triangular, rhombus and square) were hung equidistantly on the interior of the curtain serving as spatial cues. Each mouse had daily sessions of one trial for 5 consecutive days. When they succeeded, mice were allowed to stay on the platform for 30 s. When the mice failed to find the platform within 60 s, they were assisted by the experimenter and allowed to stay the platform for the same time. Probe trials were performed after the last training session at 6 months after free drinking substance.
The object location test or spatial novelty was conducted according to Barker et al. with some modification59. One day before object location test, all mice were exposed for 30 min to the empty test box for habituation. The object location tests consisted of 2 trials which are sample phase (T1) and test phase (T2) with a 30 min interval between the 2 trials. In sample phase, mice were exposed to object O1 and O2, which were placed in the far corner of the area. The animal was allowed to explore both objects during a sample phase for 3 min and the amount of exploration of each object was recorded. After a delay of 30 min, the test phase was started. In the test phase, object O3 was placed in the same position as object O1 in the sample test while object O4 was placed in the corner adjacent to the original position of O2, so that object O3 and O4 were in diagonal corners. Thus, both objects in the test phase were equally familiar, but one was in a new location. The position of the moved object was counterbalanced between mice. All measures experiments were made with the experimenter blind to the treatment status of each animal. The basic measure was the total time spent by mice exploring each object during T1 and T2 trials. Exploratory behavior was defined as the animal directing the nose toward the object at a distance of <2 cm. Looking around while sitting or resting against the object was not considered as exploration. On object location task the amount of time exploring each object (object in the new location versus object in familiar position) is reported as an object discrimination ratio and calculated using the following Equation (2)
$$\begin{array}{c}({\rm{E}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}\,{\rm{i}}{\rm{n}}\,{\rm{t}}{\rm{h}}{\rm{e}}\,{\rm{n}}{\rm{e}}{\rm{w}}\,{\rm{l}}{\rm{o}}{\rm{c}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}-{\rm{E}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}\,{\rm{i}}{\rm{n}}\\ \quad {\rm{f}}{\rm{a}}{\rm{m}}{\rm{i}}{\rm{l}}{\rm{i}}{\rm{a}}{\rm{r}}\,{\rm{l}}{\rm{o}}{\rm{c}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}})/{\rm{T}}{\rm{o}}{\rm{t}}{\rm{a}}{\rm{l}}\,{\rm{e}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{b}}{\rm{o}}{\rm{t}}{\rm{h}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}{\rm{s}}\end{array}$$
(2)
The object recognition task was performed in a circle open-field apparatus (60 cm in diameter and 50 cm in height). The objects used in this task were different in shapes, colors, and textures according to Antunes and Biala60. The open field and the objects were cleaned between each trial using 70% ethanol to avoid odor trails. Before the experiment day, the animals were allowed to acclimatize to the experimental environment. During habituation, the animals were allowed to freely explore the apparatus without objects for 5 min, once a day for three consecutive days before testing. On the experimental day, animals were submitted to two trials spaced. During the first trial (T1), animals were placed in the area containing the same two identical objects for an amount of time necessary to spend 15 s exploring these two objects in a limit of 4 min. Any mice which did not explore the objects for 15 s within the 4 min period were excluded from experiments. 1 h after exposing to the first trial, the animals were exposed to the second trial (T2). According to this trial, one of the objects presented in the first trial was replaced by an unknown object (novel object). Animals were placed back in the arena for 3 min, the total times which the animals spent to explore or directed the nose within 2 cm of the object while looking at, sniffing, or touching the novel object were recorded and recognized as total exploration time upon novel object.
### Statistical analysis
All data are presented as mean ± SEM values. All data were analyzed by One-way analysis of variance (ANOVA), followed by Tukey’s post hoc test. Probability (p) value of less than 0.05 was considered significant. Statistical analysis was performed using SPSS 17.0 software package for Windows.
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## Acknowledgements
This study was partially supported by a Grant-in-Aid for Scientific Research (S) (25220203) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (MEXT). One of the authors (YN) appreciates 2012-grant in psychoneurosis field from Senshin Medical Research Foundation.
## Author information
Authors
### Contributions
Conceived and designed the experiments: Y.N. and P.C. Performed the experiments: P.B., P.C., L.B.V., and S.S. Analyzed the data: P.B., P.C., and Y.N. Contributed reagents/materials/analysis tools: Y.T., K.I., A.T., and Y.N. Wrote the paper: P.B., P.C., L.B.V., and Y.N.
### Corresponding author
Correspondence to Yukio Nagasaki.
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### Competing Interests
The authors declare that they have no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Boonruamkaew, P., Chonpathompikunlert, P., Vong, L.B. et al. Chronic treatment with a smart antioxidative nanoparticle for inhibition of amyloid plaque propagation in Tg2576 mouse model of Alzheimer’s disease. Sci Rep 7, 3785 (2017). https://doi.org/10.1038/s41598-017-03411-7
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Biomaterials (2020) | 2021-01-26 16:36:56 | {"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": 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": 0.4540814459323883, "perplexity": 14463.413936886769}, "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/1610704800238.80/warc/CC-MAIN-20210126135838-20210126165838-00673.warc.gz"} |
https://www.princeton.edu/~achaney/tmve/wiki100k/docs/L_H%C3%B4pital_s_rule.html | # L'Hôpital's rule
related topics {math, number, function} {work, book, publish}
In calculus, l'Hôpital's rule (also called Bernoulli's rule) uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published the rule in his book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (literal translation: Analysis of the Infinitely Small to Understand Curved Lines) (1696), the first textbook on differential calculus.[1] However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.[2]
The Stolz-Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.
In its simplest form, l'Hôpital's rule states that for functions f and g:
If $\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0 \,$ or $\pm\infty$ and $\lim_{x\to c}f'(x)/g'(x)$ exists,
then $\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}.$
The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily. | 2017-11-24 11:38:13 | {"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": 0, "img_math": 4, "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": 0.9770673513412476, "perplexity": 1176.3556632027078}, "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-2017-47/segments/1510934807650.44/warc/CC-MAIN-20171124104142-20171124124142-00085.warc.gz"} |