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https://validmeasures.org/bbw/ | ## Overview
The blocked weighted bootstrap (BBW) is an estimation technique for use with data from two-stage cluster sampled surveys in which either prior weighting (e.g. population-proportional sampling or PPS as used in Standardized Monitoring and Assessment of Relief and Transiations or SMART surveys) or posterior weighting (e.g. as used in Rapid Assessment Method or RAM and Simple Spatial Sampling Method or S3M surveys). The method was developed by Accion Contra la Faim, Brixton Health, Concern Worldwide, Global Alliance for Improved Nutrition, UNICEF Sierra Leone, UNICEF Sudan and Valid International. It has been tested by the Centers for Disease Control (CDC) using infant and young child feeding (IYCF) data.
## Installation
# Install bbw using the following code in R:
install.packages("bbw")
# Or install the development version from GitHub:
# install.packages("devtools")
devtools::install_github("validmeasures/bbw")
## Usage
The BBW used in RAM and S3M is a modification to the percentile bootstrap to include blocking and weighting to account for a complex sample design.
With RAM and S3M surveys, the sample is complex in the sense that it is an unweighted cluster sample. Data analysis procedures need to account for the sample design. A blocked weighted bootstrap (BBW) can be used:
Blocked: The block corresponds to the primary sampling unit (PSU = cluster). PSUs are resampled with replacement. Observations within the resampled PSUs are also sampled with replacement.
Weighted: RAM and S3M samples do not use population proportional sampling (PPS) to weight the sample prior to data collection (e.g. as is done with SMART surveys). This means that a posterior weighting procedure is required. BBW uses a “roulette wheel” algorithm (see illustration below) to weight (i.e. by population) the selection probability of PSUs in bootstrap replicates.
In the case of prior weighting by PPS all clusters are given the same weight. With posterior weighting (as in RAM or S3M) the weight is the population of each PSU. This procedure is very similar to the fitness proportional selection technique used in evolutionary computing.
A total of m PSUs are sampled with replacement for each bootstrap replicate (where m is the number of PSUs in the survey sample).
The required statistic is applied to each replicate. The reported estimate consists of the 0.025th (95% LCL), 0.5th (point estimate), and 0.975th (95% UCL) quantiles of the distribution of the statistic across all survey replicates.
The main reason to use BBW is that the bootstrap allows a wider range statistics to be calculated than model-based techniques without resort to grand assumptions about the sampling distribution of the required statistic. A good example for this is the confidence interval on the difference between two medians which might be used for many socio-economic variables. The BBW also allows for a wider range of hypothesis tests to be used with complex sample survey data. | 2019-07-15 21:01:06 | {"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": 0.5167617797851562, "perplexity": 3677.2725043087044}, "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-2019-30/segments/1563195524111.50/warc/CC-MAIN-20190715195204-20190715220440-00064.warc.gz"} |
https://mathoverflow.net/questions/280442/on-unstable-manifold-and-incidence-number-of-novikov-complex | # On unstable manifold and incidence number of Novikov complex
Novikov complex is an extension of Morse theory to (closed) Morse 1-form $\omega$, which is not necessarily exact. Suppose for simplicity, $\omega$ is in the integer cohomology class and the universal covering is infinite cyclic. Then $\omega$ can be lifted to the universal cover $\hat M$, i.e., $\pi^\star \omega=d\hat f$, for some Morse function $\hat f$ on $\hat M$. The Morse 1-form $\omega$ generalizes the Morse function and the zeros of $\omega$ take the place of the critical points of a Morse function. Indices of the zeros (or critical points) of $\omega$ can be defined similarly and we have the stable and unstable manifolds of a critical point. The incidence number between a pair of critical points respectively of index $k$ and $k-1$ of $\omega$ can be defined for a Morse-Smale pair $(\omega, g)$, where $g$ is a Riemann metric on $M$. One can lift the stable and unstable manifolds to $\hat M$ and define the corresponding incidence number in $\hat M$ similarly. One can also define the Novikov complex.
My question is: $\hat M$ is not compact and the unstable manifold (or the descending disc) of a critical point in $\hat M$ in general goes infinitely downwards. Also, for fixed $x$ in $\hat M$, it is possible that the incidence numbers between the critical points $x$ of index $k$ and $y$ of index $k-1$ in $\hat M$ are non-zero for infinitely many $y$. These are different from Morse theory on compact manifolds. Can someone give a (simple) example of an unstable manifold in $\hat M$ going infinitely downwards for some Morse 1-form? Maybe also an example of infinite number of non-zero incidence numbers? I am sorry that I pose the question rather imprecisely. | 2019-10-22 17:43:15 | {"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": 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": 0.8786854147911072, "perplexity": 130.31768613286644}, "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-2019-43/segments/1570987822458.91/warc/CC-MAIN-20191022155241-20191022182741-00269.warc.gz"} |
https://ai.stackexchange.com/questions/9491/any-papers-regarding-different-inconsistent-action-space-in-reinforcement-learni/9514#9514 | # Any papers regarding different/inconsistent action space in Reinforcement Learning? [closed]
This question is regarding Reinforcement Learning and different/inconsistent action space for every/some states.
## What do I mean by different/inconsistent action space?
Let say you have an MDP where the number of actions varies between states (for example like in Figure 1 or Figure 2). We can express "inconsistent action space" formally as $$\forall s \in S: \exists s' \in S: A(s) \neq A(s') \wedge s \neq s'$$
That is, for every state, there exists some other state which do not have the same action set. In the figures (1, 2) there's a relatively small amount of actions per state. Instead imagine states $$s \in S$$ with $$m_s$$ number of actions, where $$1 \leq m_s \leq n$$ and $$n$$ is a really large integer.
## Environment
To get a better grasp of the question, here's an environment example. Take Figure 1 and let it explode into a really large directed acyclic graph with a source node, huge action space and a target node. The goal is to traverse a path, starting at any start node, such that we'll maximize the reward which we'll only receive at the target node. At every state, we can call a function $$M : s \rightarrow A'$$ that takes a state as input and returns a valid number of actions.
## Approches
1. A naive approach to this problem (discussed here and here) is to define the action set equally for every state, return a negative reward whenever the performed action $$a \notin A(s)$$ and move the agent into the same state, thus letting the agent "learn" what actions are valid in each state. This approach has two obvious drawbacks:
• Learning $$A$$ takes time, especially when the Q-values are not updated until either termination or some statement is fulfilled (like in experience replay)
• We know $$A$$, why learn it?
2. Another approach (first answer here, also very much alike proposals from papers such as Deep Reinforcement Learning in Large Discrete Action Spaces and Discrete Sequential Prediction of continuous action for Deep RL) is to instead predict some scalar in continuous space and, by some method, map it into a valid action. The papers are discussing how to deal with large discrete action spaces and the proposed models seem to be a somewhat solution for this problem as well.
3. Another approach that came across was to, assuming the number of different action set $$n$$ is quite small, have functions $$f_{\theta_1}$$, $$f_{\theta_2}$$, ..., $$f_{\theta_n}$$ that returns the action regarding that perticular state with $$n$$ valid actions. In other words, the performed action of a state $$s$$ with 3 number of actions will be predicted by $$\underset{a}{\text{argmax}} \ f_{\theta_3}(s, a)$$.
None of the approaches (1, 2 or 3) are found in papers, just pure speculations. I've searched a lot, but I cannot find papers directly regarding this matter.
Does anyone know any papers, or having other approaches, regarding different/inconsistent action space?
• Have a similar issue, and my immediate thoughts are to perform some transformation of the problem into a domain where the action space is fixed. For instance, if I am working in active learning, where the action is to select an example from a unlabelled training dataset (without replacement), then perhaps a different formulation where the action is to select a class, or point in data space might work as well (this will have a fixed/static action space) Dec 14 '19 at 23:28
• I have the same question. The best answer I have found is the following paper: Learning Action Representation for Reinforcement Learning Jan 23 '20 at 9:17
• How about your master thesis? Have you figured out methods to solve your question or anything else you want to share? I found the most related works as listed below are large action spaces. May 11 '20 at 19:40
• @DongDongChen we used a modified version of the arxiv.org/pdf/1512.07679.pdf paper, along with an ILP solver, which worked ok for us in the thesis. However, I think our version was very specific to our instance of the problem and was hard to generalize. May 11 '20 at 22:08
1. Does anyone know any paper regarding this subject?
I'm not familiar with any off the top of my head... I do know that the vast majority of Reinforcement Learning literature focuses on settings with a fixed action space (like robotics where your actions determine how you attempt to move / rotate a particular part of the robot, or simple games where you always have the same set of actions to move and maybe ''shoot'' or ''use'' etc.). Another common class of settings is where the action space can easily be treated as if it always were the same (by enumerating all actions that every could be legal in some state), and filtering out illegal actions in some sort of post-processing steps (e.g. RL work in board games).
So... there might be something out there, but it's certainly not common. Most RL people like to involve as little domain knowledge as possible, and I suppose that a function that generates a legal set of actions given a particular state can very much be considered to be domain knowledge.
1. Is the terminology wrong? "Inconsistant", "Irregular", "Different"... ?
I wouldn't use inconsistent, because that word can be interpreted as implying that something would be "wrong" or "ill-defined". I would say that you have a variable action set (the action set varies per state). When I search for that, there aren't a lot of results either though... but I think that term would be more promising.
1. Anyone having another approach worth digging into?
The problem you describe is mostly a problem in Reinforcement Learning with function approximation, in particular function approximation using Neural Networks. If you can get away with using tabular RL algorithms, the problem instantly disappears. For example, a table of $$Q(s, a)$$ values as commonly used in the tabular, value-based algorithms does not need to contain entries for all possible $$(s, a)$$ pairs; it's fine if it only contains entries for $$(s, a)$$ pairs such that $$a$$ is legal in $$s$$.
Variable action spaces primarily turn into a problem in Deep RL approaches, because we normally work with a fixed neural network architecture. A DQN-style algorithm involves neural networks that take feature vectors describing states $$s$$ as inputs, and provide $$Q(s, a)$$ estimates as outputs. This immediately implies that we need one output node for every action, which means you have to enumerate all the actions... which is where your problem comes in. Similarly, policy gradient methods traditionally also require one output node per action, which again means you have to be able to enumerate all the actions in advance (when determining the network architecture).
If you still want to use Neural Networks (or other kinds of function approximators with similar kinds of inputs and outputs), the key to addressing your problem (if none of the suggestions you've already listed in the question are to your liking) is to realize that you'll have to find a different way to formulate your inputs and outputs, such that you are no longer required to enumerate all actions in advance.
The only way I can think of doing that really is if you are able to compute meaningful input features for complete state-action pairs $$(s, a)$$. If you can do that, then you could, for example, build neural networks which:
• Take a feature vector $$x(s, a)$$ as input, which describes (hopefully in some meaningful way) the full pair of the state $$s$$ and the action $$a$$
• Produce a single $$\hat{Q}(s, a)$$ estimate as output, for the specific pair of state + action given as input, rather than producing multiple outputs.
If you can do that, then in any given state $$s$$ you can simply loop through all the legal actions $$A(s)$$, compute $$\hat{Q}(s, a)$$ estimates for them all (note: we now require $$\lvert A(s) \rvert$$ passes through the network rather than just a single pass as would normally be required in DQN-style algorithms), and otherwise proceed similarly to standard DQN-style algorithms.
Obviously the requirement of having good input features for actions is not always going to be satisfied... but I doubt there are many good ways to get around that. It's very similar to the situation with states really. In tabular RL, we enumerate all states (and all actions). With function approximation, we usually still enumerate all actions, but avoid the enumeration of all states by replacing them with meaningful feature vectors (which enables generalization across states). If you want to avoid enumerating actions, you'll in a very similar way have to have some way of generalizing across actions, which again means you need features to describe actions.
• This is great feedback and interesting thoughts, thanks for that. I find many papers regarding "large action space"'s, which is in pretty much the same issue. I think the paper "Discrete Sequential Prediction of continuous action for Deep RL" is very interesting since it predicts a sequence of actions using Recurrent Neural Networks instead, which solves the fixed network issue. We will do our master thesis in this subject and I hope we'll collect more information in this matter. Dec 17 '18 at 7:41
• I am trying to do this state-action representation and create a DQN-like network that would output the Q-value of the x(s,a), like a regression problem. Having read the tf_agents documentation, I do not think that I can model it with this. Is there a way to model it with tf_agents or should I do it from scratch ? Jul 8 '20 at 16:45
• @ddaedalus Hmmm I'm not sure, I'm not super familiar with tf_agents. I must say that I don't believe I've ever seen anyone in the literature actually doing this thing with state-action pairs being the inputs, rather than states inputs and actions outputs. Almost everyone in all the literature always seems to assume they can just predetermine the number of actions. So, since this is so rarely done in the literature, it would indeed be very well possible that it's not easily supported by such frameworks and requires more work from scratch Jul 8 '20 at 17:32
• In my case, every action is different from the other. If you choose an action, then it cannot be appeared again. Also from each state, you always have different actions or actions that you have seen again but not taken yet. Do you think that I could create an environment that inherits from gym environment? I doubt that because of what you said. Jul 8 '20 at 17:48
• @ddaedalus Again not 100% sure... I'm thinking that might be possible though. If I remember correctly, you do have to predefine your action space in advance, but it does not have to be discrete; you can specify continuous action spaces, which is used for all these continuous control envs (the ones where you have to make all sorts of robots learn how to walk etc.). I suppose that in your situation you'll also have to find some action representation (probably with binary or numeric features). If you have upper/lower bounds on your feature values, you can define your action space with that? Jul 8 '20 at 18:14
(3) Another approach that came across was to, assuming the number of different action set $$n$$ is quite small, have functions $$f_{\theta_1}$$, $$f_{\theta_2}$$, ..., $$f_{\theta_n}$$ that returns the action regarding that perticular state with $$n$$ valid actions. E.i, the performed action of a state $$s$$ with 3 number of actions will be predicted by $$\underset{a}{\text{argmax}} \ f_{\theta_3}(s, a)$$.
That sounds pretty complicated and the number of different action sets is usually very high, even for simplest games. Imagine checkers, ignore promotions and jumping for simplicity and there are some $$7 \cdot 4 \cdot 2=56$$ possible actions (which is fine), but the number of different sets of these actions is much higher. It's actually difficult to compute how many such sets are possible in a real game - it's surely much less than $$2^{56}$$, but also surely far too big for being practical.
1. Anyone having another approach worth digging into?
Assuming the number of actions is not too big, you can simply ignore actions which don't apply in a given state. That's different from learning - you don't have to learn returning negative award for illegal actions, you simply don't care and select the legal action returning the best award.
$$\forall s \in S: \exists s' \in S: A(s) \neq A(s') \wedge s \neq s'$$
$$\forall s \in S: \exists s' \in S: A(s) \neq A(s')$$
$$|A(s)|_{s \in S} > 1$$ | 2021-10-17 18:56:26 | {"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": 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": 44, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7516316771507263, "perplexity": 492.1232516898296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585181.6/warc/CC-MAIN-20211017175237-20211017205237-00053.warc.gz"} |
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-9th-edition/chapter-3-stoichiometry-exercises-page-129/51 | ## Chemistry 9th Edition
a. $17.03$ grams b. $30.03$ grams
a. $NH_3$ Molar mass = $1*14.007+3*1.008=17.03$ grams b. $N_2H_4$ Molar mass = $2*14.007+2*1.008=30.03$ grams | 2018-09-26 12:19:39 | {"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": 0.3997398018836975, "perplexity": 13249.008119776778}, "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-2018-39/segments/1537267164925.98/warc/CC-MAIN-20180926121205-20180926141605-00460.warc.gz"} |
https://www.physicsforums.com/threads/limit-problem.842926/ | # Limit problem
1. Nov 13, 2015
### geoffrey159
1. The problem statement, all variables and given/known data
Let $f$ be piecewise continuous from $[0,+\infty[$ into $V = \mathbb{R}$ or $\mathbb{C}$, such that $f(x) \longrightarrow_{ x\rightarrow +\infty} \ell$.
Show that $\frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell$
2. Relevant equations
Integration of asymptotic comparisons
3. The attempt at a solution
Can you tell me if this is correct please ?
Since $f - \ell = o_{+\infty}(1)$, and since $u: x \rightarrow 1$ is non-negative, piecewise continuous, and non-integrable on $[0,+\infty[$, then
$\int_0^x f(t) - \ell \ dt = o_{+\infty}(\int_0^x u(t) \ dt)$
which is the same as saying that $\int_0^x f(t) \ dt - x \ell = o_{+\infty}(x)$.
Multiplying left and right by $\frac{1}{x}$, I get that $\frac{1}{x}\ \int_0^x f(t) \ dt - \ell = o_{+\infty}(1)$ which proves that
$\frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell$.
Is this OK ?
2. Nov 13, 2015
### geoffrey159
Never mind, I've had confirmation. Thanks !
3. Nov 13, 2015
### Staff: Mentor
I think this is nicer notation: $\lim_{x \to \infty} f(x) = \ell$
My LaTeX script is $\lim_{x \to \infty} f(x) = \ell$
4. Nov 13, 2015
### HallsofIvy
Surely you meant "integrable" not "non-integrable" here?
5. Nov 13, 2015
### geoffrey159
:-) Ok thanks, I'll try to follow that notation in the future
No, why do you say that? $u = 1$ is non-integrable on $[0,+\infty[$ since $\int_0^x u(t) \ dt$ does not have a finite limit as $x$ tends to infinity. | 2018-02-24 02:38:42 | {"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": 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": 0.8596652746200562, "perplexity": 1297.3145033363821}, "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-2018-09/segments/1518891815034.13/warc/CC-MAIN-20180224013638-20180224033638-00727.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-7-exponents-and-exponential-functions-7-3-multipying-powers-with-the-same-base-practice-and-problem-solving-exercises-page-430/43 | ## Algebra 1
$a^{-4}\times a^{4}=1$
$a^{?}\times a^{4}=1$ The zero as an exponent rule states that for every nonzero number $a$, $a^0=1$. Therefore, $a^{?}\times a^{4}=a^0$ To multiply powers with the same base, we add the exponents. When we add $4$ and the first exponent, we must get $0$. The only way this is true is if the first exponent is $-4$ because $4+(-4)=0$. Therefore, $a^{-4}\times a^{4}=a^0$ We rewrite the equation in its original form: $a^{-4}\times a^{4}=1$ | 2021-04-19 07:26:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9818719029426575, "perplexity": 146.01365794810062}, "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/1618038878326.67/warc/CC-MAIN-20210419045820-20210419075820-00065.warc.gz"} |
https://rpg.stackexchange.com/questions/62338/with-street-grimoire-out-are-there-more-ways-by-which-mystic-adepts-can-gain-pow | # With street grimoire out are there more ways by which mystic adepts can gain power points?
As mentioned here: How do mystic adepts gain power points in Shadowrun 5? mystic adepts can gain power points at character creation. Later on they can only gain them by not choosing a metamagic ability during an initiation.
With street grimoire out now are there additional ways / what are the ways now by which (mystic) adepts can gain power points after character creation?
• possible duplicate of How do mystic adepts gain power points in Shadowrun 5? – Thomas E. May 24 '15 at 8:43
• I just saw that there is a question that is essentially the same as mine (the question itself. The answer here though has a few more details to it): rpg.stackexchange.com/questions/27290/… – Thomas E. May 24 '15 at 8:44
• Updated the question so that it shouldn't be a complete duplicate any longer (saw that the other question was out before street grimoire came out so focused on that in the edit) – Thomas E. May 24 '15 at 9:49
• Following a way (Streetgrimoire p. 176ff) can reduce the cost of adept powers, which allows you to buy new ones; but the number of points you can reduce the costs sum up to at most MAG/4 | 2020-01-23 00:11:09 | {"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": 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": 0.26335278153419495, "perplexity": 2685.759781197894}, "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/1579250607596.34/warc/CC-MAIN-20200122221541-20200123010541-00178.warc.gz"} |
https://jeopardylabs.com/play/module-3-rational-numbers-2 | Integers
Opposites
Comparing Integers
Absolute Value
Coordinates
### 100
Write an integer to represent a loss of $50. What is -$50?
### 100
The opposite of 5 is _____.
What is -5?
### 100
Write an inequality comparing -5 and 0.
-5 < 0 or 0 > -5
### 100
What does absolute value mean?
Absolute value means how many units (or jumps) away a number is from zero.
### 100
How is an ordered pair written?
What is (x, y)?
### 200
Write an integer to represent a gain of $5,000. What is positive$5,000?
### 200
The opposite of a credit of $20 is ____. What is a debit/loss of$20?
### 200
Write an inequality statement comparing |-90| and 90.
90 = 90
(The absolute value of 90 is equal to 90.)
The |0| is ___.
What is zero?
### 200
Which quadrant does (-3, -8) lie in?
### 300
Write an integer to represent 14,000 feet above sea level.
What is positive 14,000 feet?
### 300
The opposite of 400 feet above sea level is _____.
What is 400 feet below sea level?
### 300
Order the following from least to greatest: -8, -19, -0, -12, 12
What is -19, -12, -8, 0, 12?
### 300
The |-14| is ___.
What is 14?
### 300
What is the y-coordinate in the ordered pair (-5, 17)?
What is 17?
### 400
Write an integer to represent a debit of $900. What is -$900?
### 400
What is the opposite of -9 and describe it's location on a number line?
What is positive 9? It's location is 9 units to the right of (or above) zero.
### 400
Write a statement comparing −10℉ and −20℉.
−10℉ is warmer than −20℉.
### 400
True or False: As you approach zero from the left on the number line, the integers increase, but the absolute values of those integers decrease . This means that the order of negative integers is opposite the order of their absolute values.
True
### 400
Find the length of the line segment with end points (7, 2) and (-4, 2).
What is 11 units?
### 500
Write an integer that represents a loss of 10 yards.
What is -10?
### 500
Mr. Cordova wrote −(−$800) on the board, and said, “The opposite of the opposite of$800 is \$800.” Is his reasoning correct? Explain.
Yes, he is correct because the opposite of 800 is -800, and the opposite of that is 800.
### 500
Karen, Dulce, and Sayuri are playing a card game. Their scores are as follows:
Karen: -1, Dulce: -2, and Sayuri: -4. Who won? Who came in last place? Explain.
Karen won because -1 is greater (closer to zero). Sayuri came in last place because -4 is smallest (furthest away from zero).
### 500
In math class, Eric and Jesus are debating about integers and absolute value. Eric said two integers can have the same absolute value, and Jesus said one integer can have two absolute values. Who is right? Explain.
Eric is correct because |-#| and |#| both have the same amount of jumps from zero.
### 500
Locate and label the ordered pairs: A (-2 1/2,0); B (-4,3.75); C (8,1); and D (5, -8.5)
Response determined by teacher. | 2018-03-23 11:05:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5294773578643799, "perplexity": 4420.6739063695595}, "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/1521257648207.96/warc/CC-MAIN-20180323102828-20180323122828-00532.warc.gz"} |
http://www.reference.com/browse/use+temporarily | Related Searches
Definitions
# Zu Chongzhi
Zu Chongzhi (429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.
## Life and works
Zu Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravages of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of "Minister of Great Works" (大匠卿) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuo (祖朔) also served the court and was greatly respected for his erudition.
Zu was born in Jiankang. His family had historically been involved in astronomy research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute. When Emperor Xiaowu of Liu Song heard of him, he was sent to an Academy, the Hualin Xuesheng (華林學省), and later at the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today Zhenjiang, Jiangsu), he was engaged in work at the office of the local governor.
## Zhui Shu
Zu Chongzhi, along with his son Zu Gengzhi, wrote a mathematical text entitled Zhui Shu (Method of Interpolation). There is a high possibility of astronomical calculation techniques due to the accuracy of his calendars. It is said that the treatise contains formulas for the volume of the sphere, cubic equations and the accurate value of pi. Sadly, this book didn't survive to the present day, since it has been lost since the Song Dynasty.
His mathematical achievements included:
• the Daming calendar (大明曆) introduced by him in 465.
• distinguishing the Sidereal Year and the Tropical Year, and he measured 45 years and 11 months per degree between those two, and today we know the difference is 70.7 years per degree.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today; using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving two approximations of pi, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355113 (密率, Milu, detailed approximation) and 227 (約率, Yuelu, rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 212 × 3) sided polygon. This was an impressive feat for the time, especially considering that the device Counting rods he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. Japanese mathematician Yoshio Mikami pointed out, "$tfrac\left\{22\right\}\left\{7\right\}$ was nothing more than the π value obtain several hundred years earlier by the Greek mathematician Archimedes,however Milu $pi=tfrac\left\{355\right\}\left\{113\right\}$ could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraodinary fraction over a whole millennium earlier than Europe". Hence Mikami strongly urged that the fraction $tfrac\left\{355\right\}\left\{113\right\}$ be named after Zu Chongzhi as Zu Chongzhi fraction. In Chinese literature, this fraction is known as "Zu rate". Zu rate is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16600.
• finding the volume of a sphere as πD3/6 where D is diameter (equivilent to 4πr3/3).
• discovering the Cavalieri's principle, 1000 years before Bonaventura Cavalieri in the West.
## Astronomy
Zu was an accomplished astronomer who calculated the values of time to almost pinpoint precision. His methods of interpolating and the usage of integration is far ahead of his time. Even the astronomer's Yi Xing isn't comparable to his value(whom was beginning to utilize foreign knowledge). The Sung dynasty calendar was backwards to the "Northern barbarians" because they were implementing their daily lives with the Da Ming Li. It is said that his methods of calculation was so advance, the scholars of the Sung dynasty and Indo influence astronomers of the Tang dynasty found it confusing.
## Mathematics
Most of Zu's great mathematical works, are recorded in his lost text Zhui Shu. Most scholars argue about his complexity. Since traditionally, the Chinese developed mathematics as algebraic, and equational. Logically, scholars assume that his work, Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describes the lengthy calculations. Zu used the method of exhaustion, inscribing a 12,288-gon. Interestingly, Zu's value of pi is precise to 8 decimal places. No mathematician since his time, computed a value this precise until another 1000 years. Zu also worked on deducing the formula for the volume of the sphere. Zu used the Cavalieri Method, another method of integral calculus.
## The South Pointing Chariot
The South Pointing Chariot device was first invented by the Chinese mechanical engineer Ma Jun (c. 200-265 AD). It was a wheeled vehicle that incorporated an early use of differential gears to operate a fixed figurine that would constantly point south, hence enabling one to accurately measure their directional bearings. This effect was achieved not by magnetics (like in a compass), but through intricate mechanics, the same design that allows equal amounts of torque applied to wheels rotating at different speeds for the modern automobile. After the Three Kingdoms period, the device fell out of use temporarily. However, it was Zu Chongzhi who successfully re-invented it in 478 AD, as described in the texts fo the Song Shu (c. 500 AD) and the Nan Chi Shu, with a passage from the latter below:
When Emperor Wu of Liu Song subdued Guanzhong he obtained the south-pointing carriage of Yao Xing, but it was only the shell with no machinery inside. Whenever it moved it had to have a man inside to turn (the figure). In the Sheng-Ming reign period, Gao Di commissioned Zi Zu Chongzhi to reconstruct it according to the ancient rules. He accordingly made new machinery of bronze, which would turn round about without a hitch and indicate the direction with uniformity. Since Ma Jun's time such a thing had not been.
## Named for him
• $pi=tfrac\left\{355\right\}\left\{113\right\}$ as Zu Chongzhi $pi$ rate.
• The lunar crater Tsu Chung-Chi
• 1888 Zu Chong-Zhi is the name of asteroid 1964 VO1.
## References
• Needham, Joseph (1986). Science and Civilization in China: Volume 4, Part 2. Taipei: Caves Books, Ltd.
• Du, Shiran and He, Shaogeng, "Zu Chongzhi" Encyclopedia of China (Mathematics Edition), 1st ed. | 2013-05-23 15:25:14 | {"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": 0.7001482248306274, "perplexity": 5405.577599151058}, "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/1368703489876/warc/CC-MAIN-20130516112449-00034-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://library.kiwix.org/politics.stackexchange.com_en_all_2021-04/A/question/31734.html | ## Non-subjective definition of "terrorist", or widely used equivalent term?
24
7
...We all know that one side's terrorists are another side's freedom fighters...
Suppose big Country X provides arms to little Country y's native "freedom fighters", who do certain scary things, ("for freedom!"), which goes on for a decade or so, but then Country y's freedom fighters wind up fighting against Country X, doing the same scary things, which Country X now labels "terrorism".
Meanwhile Country y's fighters still consider themselves freedom fighters, and let's suppose that for the average fighter from Country y the daily routine never changed, they still get up in the morning, report in, and follow orders to get the same-old same old scary things done.
That'd be a subjective usage of the term "terrorism".
Is there a non-subjective usage of the term, one that both sides could agree upon as unequivocally terrorism, irrespective of the cause or target?
If not, is there any non-subjective term, (which describes the job of doing the scary things Country y's fighters do), that both sides would always agree upon?
15How uncomfortable are you with soldiers being classified as terrorists? – origimbo – 2018-06-24T17:38:06.647
4
Possible duplicate of Why are partisan groups in Afghanistan called terrorists?
– user4012 – 2018-06-24T17:40:24.780
6"We all know that one side's terrorists are another side's freedom fighters" - this is all about justification. As user4012 pointed out, you can be both, or none, so both terms are virtually independent. The point is that "terrorism" is conceived as bad, so people supporting the political cause will try to avoid the term while people opposing the cause will try to impose it onto their opponents. This should not be confused with the meaning of terrorism. Who kills civilians with the intent to spread fear to further a political cause is a terrorist, it is not important if the cause is just. – Thern – 2018-06-25T08:27:44.097
2It's ok. You can say United States (Country X) and Afghanistan (County y); however, they didn't become 'ersatz terrorists' for 'doing the same scary things'. Their resistance to Soviet invasion and 9/11 were different things, and the daily Afghan soldier didn't become a terrorist; the leadership became guilty of harboring (mostly Saudi) terrorists. – lly – 2018-06-25T10:35:19.700
1Why would any individual vote to close this question? – guest271314 – 2018-06-26T15:41:58.673
57
# TL;DR: Yes, there is an objective term.No, there is no way to force people to use the term objectively in political contexts and they don't tend to.
The term "terrorism" isn't subjective.
Quoting Wikipedia:
Since 1994, the United Nations General Assembly has repeatedly condemned terrorist acts using the following political description of terrorism:
"Criminal acts intended or calculated to provoke a state of terror in the general public, a group of persons or particular persons for political purposes are in any circumstance unjustifiable, whatever the considerations of a political, philosophical, ideological, racial, ethnic, religious or any other nature that may be invoked to justify them."
And:
A definition proposed by Carsten Bockstette at the George C. Marshall Center for European Security Studies underlines the psychological and tactical aspects of terrorism:
Terrorism is defined as political violence in an asymmetrical conflict that is designed to induce terror and psychic fear (sometimes indiscriminate) through the violent victimization and destruction of noncombatant targets (sometimes iconic symbols). Such acts are meant to send a message from an illicit clandestine organization. The purpose of terrorism is to exploit the media in order to achieve maximum attainable publicity as an amplifying force multiplier in order to influence the targeted audience(s) in order to reach short- and midterm political goals and/or desired long-term end states."
Note that the three components are required, which makes this an objective definition:
1. Acts that are intended to instill fear/terror
2. Acts against general public (civilians/non-combatants). This is why for example attacks on the military during armed conflict generally aren't universally considered terrorism.
3. For a political purpose (note that just what the purpose is is 100% irrelevant to the definition, as long as it's politics and not, say, robbery)
Now, the confusion that birthed your question arises out of two things:
1. You (or whatever your sources are) are confusing the well-defined objective tactics (terrorism) with a wholly orthogonal point, the goal of the movement.
Yes. The oft-repeated "one man's terrorist is another man's freedom fighter" is basically a word game designed to confuse people.
Someone is a terrorist if and only if they engage in above-defined objectively defined acts of terrorism as a tactic.
Someone is a freedom fighter if they do something to advance freedom (whether they advance freedom or not is a bit more subjective and squishy, but let's pretend we can agree on that).
The two are wholly orthogonal - you can be a freedom fighter using a wide variety of tactics, only one of which - and often, the least effective - is terrorism. You can be a freedom fighter and not a terrorist (Mahatma Gandhi is the typical example) or you can be a terrorist and NOT a freedom fighter (Taliban seems to fit here - they don't by any stretch of imagination fight for anyone's freedom in any stretch of the word; they fight to oppress other inhabitants of Afghanistan into their version of Sharia) or you can be a freedom fighter who engages in acts of terrorism and become both (IRA, Jewish fighters attacking the British during Mandate times, Basque separatists).
1. Also very importantly, just because there is an objective definition, it does not at all mean that political bodies will not disingenuously ignore that definition when it suits their political/ideological purpose.
The USSR didn't recognize the IRA as terrorists for a variety of political and ideological reasons. Many people in the USA and Israel refuse to recognize the PKK (a Kurdish organization) as terrorists for the same reason.
This willful ignoring of the objective definition applies to both type 1 and type 2 errors. That is, not only people refuse to apply "terrorist" label to clearly objectively terrorist organizations (PKK, Hamas, IRA), but they also apply the label to things that don't fit that definition.
# TL;DR: Yes there is an objective term. No, there is no way to force people to use the term objectively in political contexts.
Comments are not for extended discussion; this conversation has been moved to chat.
– Philipp – 2018-06-25T13:45:00.553
4In Addition it should be noted that terrorism is a valid military tactic that has been used throughout human history to various levels of effect to achieve a military goal from Sherman's March to the Sea, to the Dresden Bombing, to Northern Ireland's IRA goals. – Frank Cedeno – 2018-06-25T14:47:52.820
2I think an important addition here is that the definition can be clear and crisp like this, but when you apply it to real life situations, "intent" is a much more complicated topic. One could argue all military operations intend to instill terror -- they also intend to damage the other nation's military capabilities. A good military leader understands that an act can have more than one intent, and acts on all of them. (I'd make this an answer myself, but your answer is so solid, I'd rather make it a comment beneath it) – Cort Ammon – 2018-06-25T15:52:31.237
@CortAmmon - this was in the chat that was deleted by moderator. No, you cannot argue that the intent was to instill terror by killing civilians, if the military operation explicitly chose an option that kills LESS civilians, at the expense of an option that kills more. In that case, the "terror via killing of civilians" is 100% obviously NOT the intent, as the demonstrated action clearly conflicts with that hypothetical intent. – user4012 – 2018-06-25T15:59:40.927
3@user4012 So the line in the sand is "as long as there's a way that kills more civilians, your approach is not terrorism?" Assigning intent to any individual other than yourself is known to be a hazardous process. Intent has never been a simple thing, as long as there have been humans. Just look at the political process and how hard it is to untangle intent there. – Cort Ammon – 2018-06-25T16:05:25.113
4Is it clear what's a "noncombatant target" is, e.g. during apartheid? I think they started with arson attacks against police targets, also state property such as telephone exchanges. – ChrisW – 2018-06-25T16:10:50.977
@ChrisW - police and military are a grey area as far as that definition is concerned, which is why I explicitly exclude them from the narrower definition (see #2) for the sake of making it as objective as possible. Personally I would say it depends - attack on policeman enforcing bad law is not terrorism, attack on police barracks is (basically, whether police are acting in combatant role or member of civil society role). – user4012 – 2018-06-25T16:54:56.380
@CortAmmon - no, that was not what I said. I said that, as long as there is a way to affect more civilians (ceterus parabus), the fact that you choose not to is clear evidence that your intent was NOT to attack civilians. Because if that was your intent, you would have assuredly chosen a target with more victims. – user4012 – 2018-06-25T16:57:26.070
1@CortAmmon Think of it this way. You are moving furniture in the room with a 70' fancy TV costing $2000, and a$100 vase 2 feet away from the TV. You do something while moving that shatters the vase. Any jury would clearly conclude that you did NOT shatter the vase out of intent to cause material damage to apartment owner - because, if that was your intent, you would have shattered the TV instead. Your choice of actions proves your intent. – user4012 – 2018-06-25T17:04:50.227
1@user4012 If I chose the path which drops the \$800 laptop, do I get to argue that I did not intend to cause material damage because, if I did, I would have gone after the TV instead? – Cort Ammon – 2018-06-25T17:12:33.997
@CortAmmon - I mean, in theory, someone could argue that you're just being whily and deliberately chose laptop JUST so you could mount such a defense. But the underlying issue comparisons aren't 2000 vs 800, it's 2000 vs. 100, which is a whole different comparison - 100 just isn't enough damage if your goal IS to cause damage. And that is the order of magnitude we are talking about in this topic - collateral damage being single to low double digits, vs high hundreds that is possible if you actually set out to deliberately attack civilians on purpose. – user4012 – 2018-06-25T17:49:43.943
6This definition of "Terrorists" makes the UK and US government in World War 2 terrorits, as they purposedly used terror against the unarmed civilian german population through bombing, and did so for a political purpose. – Bregalad – 2018-06-25T18:18:34.143
@user4012 So can you tell me the objective line in the sand between the two? Obviously its' trivial to define examples that are black and white. A Napoleonic era infantry charge is clearly military. Setting off a dirty bomb in the middle of manhatan without any military followthrough is clearly terrorism. But those are the easy cases. If the definition is objective, we should be able to draw an objective line through the more grey cases. – Cort Ammon – 2018-06-25T18:36:52.283
@CortAmmon - I suspect that the term "combatant" has a fairly non-subjective legal definition, probably in some treaty or another. As such, non-combatant is whatever doesn't fit that definition, making it objective as well. – user4012 – 2018-06-25T19:14:46.900
@user4012 So as long as you only ever affect one or the other in any way, it should be crystal clear, right? That way you'd never have to worry about whether killing your one combatant hiding in a city kills 5 civilians or 6 civilians turning your military action into a terrorist one. – Cort Ammon – 2018-06-25T19:16:42.620
1
@user4012, The 70" TV might also be broken by dangerously clumsy friend, which might lead to mistaken inferences of ill intent; the military equivalent of whom might be an army of friendly accidental terrorists, sturdy and persistent allies whose weapons-grade obliviousness is more dangerous to their friends than hostile terrorists.
– agc – 2018-06-25T20:03:11.673
@user4012 as per Hague convention, a combatant is a member of an armed force bearing weapons. Note that an armed force per that convention also can include non-combatants, for example, medical and religious personnel. – Danila Smirnov – 2018-06-26T04:33:16.703
1@CortAmmon Objective definition also requires full information on the object to be known. If you know the intent behind an action, this definition is precise, if not - then you cannot really apply it, so you can't tell a terrorist act from military action bar subjective judgement (which is exactly what is usually happening IRL). – Danila Smirnov – 2018-06-26T04:51:39.287
@Bregalad - Is that an issue? Nothing precludes a government from using terrorist tactics - ISIS is or was briefly a government of sorts. I don't know if I'd call an organization a "terrorist" organization, precisely, if they do lots of other things: governments enforce laws, levy taxes, and many other non-terror activities. But I'd have no problem labeling their activities "terrorism." – Obie 2.0 – 2018-06-26T07:21:41.760
2That is, "terrorism" is what was defined in the answer. A "terrorist group" is a group that uses terrorism as their primary strategy. A "terrorist" is someone who employs terrorism as their primary method (yes, this could include some soldiers or officers in a government military). – Obie 2.0 – 2018-06-26T07:25:30.670
4
@Bregalad: Actually, this definition could be seen to make all the major powers in WW II terrorists, because they all bombed civilians (ignoring other acts which would also qualify as terrorism). However, they all acted under the doctrine of total war, which considers almost everything a legitimate military target - so under that definition, bombing civilans is legitimate and thus not terrorism. So as usual, it depends on the assumptions you build on.
– sleske – 2018-06-26T08:02:29.317
3@sleske It's not major surprise nazi germany or the soviet union would enter in the definition of terrorists, so I didn't point it out, I just pointed out that this definition makes the western allies terroritsts, too. Kinda hironic when those same governments declared the "war on terrorism". – Bregalad – 2018-06-26T10:28:24.867
1@sleske Just because one side (or both) adopts a doctrine legitimizing terrorism doesn't change anything objectively : ) Which only reinforces the original premise "No, there is no way to force people to use the term objectively in political contexts." – Agent_L – 2018-06-26T10:48:21.303
@CortAmmon- as I said before, it's about the intent, not about numbers. Was the goal to kill 1 combatant? (and 5 civilians were an unintended, unfortunate side casualties that couldn't easily be avoided?) Or was the goal to kill random people, and an easy option to take out 1 combatant without affecting non-combatants was deliberately not chosen? (again, with a modern military capable of killing 10000 people with great ease in a couple of minutes, you have to have pretty convincing proof that 5 civilian casualties are actually deliberate and by design) – user4012 – 2018-06-26T12:01:59.503
@agc - I'm not sure what your point is. I never argued that mass casualty is proof of ill intent (though it is a solid evidence of one). I argued that LACK of mass casualty is a very solid evidence, if not proof, of LACK of ill intent. – user4012 – 2018-06-26T12:06:24.827
4
The plain definition of "terrorism" does apply to both sides
Merriam-Webster dictionary
terrorism: the systematic use of terror especially as a means of coercion
Wikipedia, citing Terrorism & Communication: A Critical Introduction Matusitz, Jonathan (2013)
• It is the use of violence or threat of violence in the pursuit of political, religious, ideological or social objectives.
• It can be committed by governments, non-state actors, or undercover personnel serving on the behalf of their respective governments.
• It reaches more than the immediate target victims and is also directed at targets consisting of a larger spectrum of society.
• It is both mala prohibita (i.e., crime that is made illegal by legislation) and mala in se (i.e., crime that is inherently immoral or wrong).
Two examples of state sponsored terrorism in the U.S. are the Trail of Tears (note, there were several historical trails of tears; see The Debate over Indian Removal in the 1830s) and the Tulsa Race Riot of 1921 where so-called "black Wall Street" was destroyed, in part using aircraft to drop incendiary devices onto homes and business.
M-W seems a bit vague, so some hair-splitting: it doesn't appear to distinguish between incidental terror, (suppose a murderer is terrified of going to jail, yet he is not a victim of terrorism), and terror as an end in of itself, (applies to parents telling children about boogeymen, or Krampus, etc., directors of horror movies, barking dogs, wild bears, sharks, skunks...), and coercion, (all of criminal Law, and much of the rest). The Matusitz definition means the accused then battles over whether some action is really an instance of mala in se. – agc – 2018-06-24T20:10:59.783
@agc Any non-subjective definition is going to be results-based, which precludes classifying coming as incidental or deliberate. – origimbo – 2018-06-24T20:20:02.430
2@agc Are you suggesting that the plain meaning of the term "terrorism" is ambiguous, equivocal, and is capable of being applied, used, misused and interpreted in an arbitrary and capricious manner? – guest271314 – 2018-06-24T20:21:06.277
@guest271314, Not exactly. Rather I'm looking for any "plain meaning". Nobody much disputes who is and isn't a Plumber, a Tailor, or a Cook, even if one of them is from an enemy land or has immoral goals. So what exactly does a terrorist make or do (or unmake and undo) that's distinct from who they're working for or why they work. – agc – 2018-06-24T20:40:07.627
@agc The first bullet point at the answer is the plainest meaning. The previous comment applies, for example, to individuals whom are not considered a "terrorist" if a church or school is attacked, others are citizens whom are groomed entirely by the individuals who later arrest and charge them for doing nothing but following the instructions of agents of the state. – guest271314 – 2018-06-24T21:24:43.727
@agc Just as important relevant to the meaning of the term is the inverse of the plain meaning; can you provide a concise definition of what is not "terrorism"? – guest271314 – 2018-06-24T22:32:02.853
1
Inverse? Maybe not, but we can devise a contrary from the M-W definition. Admirable-ism: the systematic use of beauty and good taste especially as a self-evident means of persuasion. Proper artistry, really. Or if not that, satyagraha maybe...
– agc – 2018-06-25T02:38:56.513
Dictionaries don't provide non-subjective definitions, they simply describe how a word was/is used. See the disclaimer on the definition of racism.
– Rob Rose – 2018-06-25T23:39:59.397
@RobRose Dictionaries are used within the process of the construction of words. If there is ambiguity as to the interpretation of the plain meaning of a word or term then legislative notes, technical documents or other materials can be reviewed to determine what the word was intended to mean by the first individual or entity which used the word or term. – guest271314 – 2018-06-25T23:43:15.390
Sorry, but how does the Trail of Tears fit the definition of terrorism? It was cruel and unjust, yes, but its main purpose (as far as I understand) was to clear land for white settlers, not specifically to instill fear in the victims of the relocation. – sleske – 2018-06-26T08:05:42.843
1@sleske "Sorry, but how does the Trail of Tears fit the definition of terrorism? It was cruel and unjust, yes, but its main purpose (as far as I understand) was to clear land for white settlers, not specifically to instill fear in the victims of the relocation." That is post-genocide apologist commentary. When you write "clear land" do you realize you are referring to the land of sovereign nations? The actions of the United States went far beyond the notion of "instill fear". The United States waged wars of terror and committed genocide against sovereign indigenous nations for material gain – guest271314 – 2018-06-26T14:14:47.840
@guest271314 Yes, but the question asked for a non-subjective definition, which cannot be found in a dictionary. – Rob Rose – 2018-06-26T18:11:51.810
1@RobRose Disagree with your assessment of the non-subjective definition. People apply their subjectivity to the objective definition for their own political interests. – guest271314 – 2018-06-26T18:16:13.600
4
There is no objective definition.
User 4012 provides the typical formal definition in their answer:
"Criminal acts intended or calculated to provoke a state of terror in the general public, a group of persons or particular persons for political purposes are in any circumstance unjustifiable, whatever the considerations of a political, philosophical, ideological, racial, ethnic, religious or any other nature that may be invoked to justify them."
However, there are some reasons this can never possibly be an objective definition:
• "... intended or calculated..." It is well known in philosophy and psychology that ascribing intent to any individual besides ourself is a hazardous process. You never know what someone else is thinking, according to the most accepted beliefs of philosophers over the last few thousand years. Hence why I use the phrase "ascribe intent." You declare "here is your intent for the action you just did." Such a concept can never be objective.
• "... are in any circumstance unjustifiable." This is an inherently subjective phrasing. "Unjustifiable" is a word which implicitly requires a justifier to make the judgement. The only way to divorce this concept from the individual doing the judging is to invoke an external judge, such as a deity. This, itself, is recognized as another one of those hazardous processes.
Now that does not mean we cannot use the term, we just have to be specific about our subject. The major Western powers all generally ascribe to the same philosophy, so it is easy to declare some actor's actions to be "intended to provoke terror" with respect to their shared viewpoint. However, that is only as objective as their shared viewpoint is.
4The first word, "criminal" is also difficult. – Stig Hemmer – 2018-06-26T09:06:30.147
1
@StigHemmer On the other hand, the word "criminal" somewhat helps nudge towards the convention that "terrorism" is usually by non-state actors, as states can amend domestic criminal definitions to exclude their own actions from mala prohibita criminality (and inversely, potentially explicitly include particular groups; this also is to say nothing of the norm of the state's "monopoly on violence"). This of course does not address international norms of mala prohibita actions, nor of mala in se actions in general.
– Myles – 2018-06-26T10:21:48.963
While I think this is a valuable consideration, Isn't this kind of an isolated demand for rigor? How many definitions of things in real life would clear this bar? – Jared Smith – 2018-06-26T12:35:25.950
1@JaredSmith I don't think it's isolated. I think this sort of thing appears all over. Consider the "jury of your peers." If everything that mattered in the criminal justice system was objective evidence, we'd have clerks taking care of verdicts and sentencing. However, in practice, nearly everything we deal with in the criminal system has an element of subjectivity in it. Even the "objective" parts typically include some subjectivity ("beyond reasonable doubt"). It's why jury selection is such a big deal. – Cort Ammon – 2018-06-26T15:17:48.073
1The purpose of a jury is to identify a subject for these statements: the subject is your peers. Those 12 jurors (in USA) are a proxy for "the people" and their collective opinion. If anything, I'd say not holding "terrorism" to this bar would be an isolated lack of demand for rigor. And, in my experience, holding the belief that a subjective position is, in fact, the objective one is an enormously common cause of conflict in the world today. One might even be able to argue it is the only cause. – Cort Ammon – 2018-06-26T15:18:58.200
1
I would use the word "combatant", although that term can be imprecise given the exact conditions of the scenario in question. Unlike the other answers, and like you, I consider "terrorist" is subject to a moral point of view.
1"Combatant" is a little less general than "soldier". That is, one might say that all terrorists are combatants, but not all combatants are terrorists. – agc – 2018-06-25T19:37:21.853
Sorry, but this is just plain wrong. The generally agreed-upon definiton of "combatant" is someone who engages in legal violence (usually as the member of an organized, legally recognized military force). A terrorist is by definition not a combatant. – sleske – 2018-06-26T08:07:40.883
"generally agreed-upon" you say. Yet Merriam-Webster: "Definition of combatant : one that is engaged in or ready to engage in combat." "Definition of combat 1 : a fight or contest between individuals or groups". So combatant is as generic as you'd like. – JD Gamboa – 2018-06-26T16:33:54.813
1
Geneva definitions are different as you say, but there is still a space for what they called "unlawful combatant" or also belligerent, which is yet another word as generic as you'd find pleasing. https://en.wikipedia.org/wiki/Unlawful_combatant
– JD Gamboa – 2018-06-26T16:35:29.990
0
Yes, let's start with the assumption that Group Z first operates inside country y without Country X (!) and the group wants either to overthrow the government of Country X or split apart from Country X to create a new Country K, then Group Z members are "insurgents" or "rebels".
This is a definition both sides can live with. It does not nullify the "freedom fighter" label (yeah, we are fighting against the government), but it also says nothing about the way the insurgents operate (fear and terror, "terrorism"). In fact, the area of warfare which is designed to counter rebel operations has the name counterinsurgency.
The whole things gets again a lot murkier once Country X comes into play. Once Group Z starts to make attacks against Country X to overthink their support or trying to convince people to join their fight against Country X, then you call the attackers non-subjectively and neutrally "infiltrators".
It is typical that Group Z is also marked as "terrorist" when in fact the government of Country y is a puppet government of Country X who only tries to give the impression that Country y is independent and actually suppress people of Country y.
1Note: In the OP Country X is a foreign sponsor, not Group y's native land.
Suggested edit: s/group Y/_Group **y**_/g; s/country X/_Country **y**_/g.
– agc – 2018-06-24T20:24:39.300
Possible typo: In the revised intro, first paragraph, maybe it should be that Z operates in y, and therefore wants to overthrow y, (not X), and change the name y to k? Note k is lower case, since it'd still be a little country. – agc – 2018-06-25T02:53:34.317
This answer ignores that terrorism actually is a concrete act. So things are not really getting murky, it is just that people might get confused when propaganda sets in, or that they believe that it can't be terrorism if it is a just cause. Yes of course it can. – Thern – 2018-06-25T08:50:22.017 | 2021-07-31 21:00:50 | {"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": 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": 0.3486500382423401, "perplexity": 3202.4793358403717}, "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-2021-31/segments/1627046154126.73/warc/CC-MAIN-20210731203400-20210731233400-00405.warc.gz"} |
https://www.w3spoint.com/projectile-motion | # Projectile Motion
Projectile motion is a type of motion in which an object called a projectile is thrown or projected. It is an example of a two-dimensional motion with constant acceleration.
The projectile is thrown with some initial velocity near the earth’s surface, and it moves along a curved path under the influence of gravity.
There are two simultaneous motion in mutually perpendicular directions which are completely independent of each other. The diagram is shown below.
Acceleration of the projectile: When a projectile is thrown in the air with some velocity, the only force acting on it during its time in the air is because of acceleration due to gravity (g). This acceleration acts vertically downward. There is no acceleration in the horizontal direction i.e. the horizontal direction remains constant.
Figure:8.a
Suppose a projectile is launched with velocity $u$making an angle $\theta$with the horizontal.
The point O is called the point of projection; θ is the angle of projection.
Figure:8.b
Let us find different parameters related to projectile motion using kinematic equations.
Component of velocity along X-axis, $u_{x}=u cos\theta$
Horizontal acceleration, $a_{x}=0$
Component of velocity along Y-axis, $u_{y}=u sin\theta$
Vertical acceleration, $a_{y}=-g$
### Time of maximum height:
Let us consider the motion along Y-direction:
Let us first consider the motion of the projectile from point O to A.
Initial velocity along Y-direction, $u_{y}=u sin\theta$and acceleration, $a_{y}=-g$
If B is the maximum height that the projectile has reached and after that it falls, so final velocity at B along Y-direction, $v_{y}=$0
If t is the time taken to travel from O to A, then using the equation of motion $v=u+gt$,
We have, $0=u sin\theta +(-g)t$
Or, $t=u sin\theta /g$ ………….(1)
Equation (1) is known as the time of maximum height.
### Time of flight:
Now as the projectile reaches the ground after some time, the displacement along Y-direction is zero. If T is the total time taken by the projectile to reach B, then, we have
$s_{y}=$0. Using the equation of motion $s=ut+\frac{1}{2}gt^{2}$,
$0=T u sin\theta +1/2 (-g)T^{2}$
Or, $T(u sin\theta -gT/2)=0$
as $T\ne 0$
We have $T=2u sin\theta /g$ …………….. (2)
Equation (2) is known as time of flight.
Now, if $H_{\max }$ is the maximum height of the projectile, then, using the equation $v^{2}$=$u^{2}+$$2gs We get, 0=u^{2}\sin ^{2}\theta -2gH_{\max } Or, H_{\max }=u^{2}\sin ^{2}\theta /2g ……………… (3) Equation (3) is known as maximum height of a projectile. ### Horizontal range: The horizontal distance travelled by the projectile from its initial position O to the position B is called the horizontal range, R. As the horizontal range is along X-direction, we will consider the motion along X. We have, u_{x}=u cos\theta , a_{x}=0and s_{x}=R=horizontal range Also, T=2u sin\theta /g Using the equation of motion s=ut+\frac{1}{2}gt^{2}, R=u cos\theta (2u sin\theta /g)+1/2\times 0 Or, R=u^{2} 2sin\theta cos\theta /g Or, R=u^{2} sin2\theta /g ……………… (4) Equation (4) is known as the horizontal range of the projectile. ### Equation of trajectory or path: The equation of the trajectory of the projectile establishes a relationship between motion along the X and Y directions . Now displacement (x) along the X direction is given by, x=t u cos\theta +1/2 \times 0 Or, t=x/ u cos\theta ……………(5) displacement (y) along the Y direction is given by, y=t u sin\theta -1/2 gt^{2} Or, y=(x/ u cos\theta ) u sin\theta -1/2 g(x/ u cos\theta )^{2}, using the value of t from equation (5) Or, y=x tan\theta -$$\frac{1}{2}\frac{gx^{2}}{u^{2}\cos ^{2}\theta}$ ……………….. (6)
This is the equation of a parabola. Therefore, the path of a projectile is a parabola.
Equation (6) is known as the equation of trajectory. | 2021-04-14 02:12:23 | {"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": 0.8134943842887878, "perplexity": 1165.123119169065}, "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/1618038076454.41/warc/CC-MAIN-20210414004149-20210414034149-00638.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/91829/ | # Amplitude analysis and branching fraction measurement of Bs->J/ψK+K-
LHCb Collaboration; et al; Bernet, R; Müller, K; Steinkamp, O; Straumann, U; Vollhardt, A (2013). Amplitude analysis and branching fraction measurement of Bs->J/ψK+K-. Physical Review D (Particles, Fields, Gravitation and Cosmology), 87(7):072004.
## Abstract
An amplitude analysis of the final state structure in the B¯¯¯0s→J/ψK+K− decay mode is performed using 1.0 fb−1 of data collected by the LHCb experiment in 7 TeV center-of-mass energy pp collisions produced by the LHC. A modified Dalitz plot analysis of the final state is performed using both the invariant mass spectra and the decay angular distributions. Resonant structures are observed in the K+K− mass spectrum as well as a significant nonresonant S-wave contribution over the entire K+K− mass range. The largest resonant component is the ϕ(1020), accompanied by f0(980), f′2(1525), and four additional resonances. The overall branching fraction is measured to be B(B¯¯¯0s→J/ψK+K−)=(7.70±0.08±0.39±0.60)×10−4, where the first uncertainty is statistical, the second systematic, and the third due to the ratio of the number of B¯¯¯0s to B− mesons produced. The mass and width of the f′2(1525) are measured to be 1522.2±2.8+5.3−2.0 MeV and 84±6+10−5 MeV, respectively. The final state fractions of the other resonant states are also reported.
## Abstract
An amplitude analysis of the final state structure in the B¯¯¯0s→J/ψK+K− decay mode is performed using 1.0 fb−1 of data collected by the LHCb experiment in 7 TeV center-of-mass energy pp collisions produced by the LHC. A modified Dalitz plot analysis of the final state is performed using both the invariant mass spectra and the decay angular distributions. Resonant structures are observed in the K+K− mass spectrum as well as a significant nonresonant S-wave contribution over the entire K+K− mass range. The largest resonant component is the ϕ(1020), accompanied by f0(980), f′2(1525), and four additional resonances. The overall branching fraction is measured to be B(B¯¯¯0s→J/ψK+K−)=(7.70±0.08±0.39±0.60)×10−4, where the first uncertainty is statistical, the second systematic, and the third due to the ratio of the number of B¯¯¯0s to B− mesons produced. The mass and width of the f′2(1525) are measured to be 1522.2±2.8+5.3−2.0 MeV and 84±6+10−5 MeV, respectively. The final state fractions of the other resonant states are also reported.
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38 citations in Web of Science®
57 citations in Scopus® | 2022-06-25 17:49:54 | {"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": 0.8255130052566528, "perplexity": 2591.0826129140696}, "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/1656103036077.8/warc/CC-MAIN-20220625160220-20220625190220-00311.warc.gz"} |
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|
https://tex.stackexchange.com/questions/394096/image-not-found-in-command-with-etoolbox | I'm having problems compiling the following code:
\documentclass{article}
\usepackage{etoolbox}
\usepackage[pdftex]{graphicx}
\newcommand{\newInfo}[3][]{%
\edef\@creatingInfo{1}%
\edef\@printingInfo{0}%
\edef\currentname{#2}%
\ifstrempty{#1}{}{\csedef{info#2Img}{#1}}%
\csedef{info#2Cnt}{#3}%
\nullfont#3\normalfont%
\edef\@creatingInfo{0}
}
\newcommand{\printInfo}[1]{%
\edef\currentname{#1}%
\edef\@printingInfo{1}%
\section{#1}\label{info:#1}%
\ifcsname info#1Img\endcsname%
\begin{figure}
\includegraphics[width=\linewidth]{\csuse{info#1Img}}
\end{figure}%
\fi%
\csuse{info#1Cnt}%
\ifcsname info#1Ref\endcsname
\begin{longtable}{p{9cm}}
\csuse{info#1Ref}
\end{longtable}
\fi
}
\begin{document}
\newInfo[./images/myimage.eps]{Test}{Information content}
\printInfo{Test}
\end{document}
It says ./images/myimage.eps not found (regardless of the name, type, absolute/relative path, etc).
If you, however, replace line 21 by \includegraphics[width=\linewidth]{./images/myimage.eps}, then it displays the image correctly.
• Remove the file extension, i.e. use it as ./images/myimage – user31729 Oct 1 '17 at 15:04
• You should expand \csuse{info#1Img} before executing \includegraphics. But I can't understand several parts of the code which appear to do nothing useful at all. In particular, what's the purpose of \nullfont#3\normalfont? – egreg Oct 1 '17 at 15:13
In order to get the automatic conversion from EPS to PDF the file name in the argument to \includegraphics should be explicit, so you can change the code into
\ifcsname info#1Img\endcsname
\begin{figure}
\begingroup\edef\x{\endgroup
\noexpand\includegraphics[width=\linewidth]{\csuse{info#1Img}}%
}\x
\end{figure}
\fi
Some comments: your example code misses \makeatletter and \makeatother, because you want to define commands with @ in their name. By the way \edef\@creatingInfo{1} is exactly the same as \def\@creatingInfo{1}, just less efficient.
Moreover, several parts of the code seem to do nothing at all, probably because the shown code is just an excerpt. But
\nullfont#3\normalfont
is something I can't understand at all. Why typesetting the file name in \nullfont, so nothing appears?
Finally, never use the pdftex option for graphicx.
• Thank you for your comments. the \nullfont#3\normalfont is essential for my package. It forces latex to evaluate several commands (which are stored inside other commands) without printing the output. That's why there is the distinction between \@creatingInfo and \@printingInfo – Benedict Oct 14 '17 at 13:01 | 2020-02-24 19:05:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8561880588531494, "perplexity": 2283.9289679647018}, "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-2020-10/segments/1581875145966.48/warc/CC-MAIN-20200224163216-20200224193216-00488.warc.gz"} |
https://en.algorithmica.org/hpc/external-memory/list-ranking/ | List Ranking - Algorithmica
# List Ranking
In this section, we will apply external sorting and joining to solve a problem that seems useless on the surface but is actually a key primitive used in a large number of external memory and parallel algorithms.
Problem. Given a singly-linked list, compute the rank of each element, equal to its distance from the last element.
This problem can be trivially solved in the RAM model: you just traverse the entire list with a counter. But this pointer jumping wouldn’t work well in the external memory setting because the list nodes are stored arbitrarily, and in the worst case, reading each new node may require reading a new block.
### #Algorithm
Consider a slightly more general version of the problem. Now, each element has a weight $w_i$, and for each element, we need to compute the sum of the weights of all its preceding elements instead of just its rank. To solve the initial problem, we can just set all weights equal to 1.
The main idea of the algorithm is to remove some fraction of elements, recursively solve the problem, and then use these weight-ranks to reconstruct the answer for the initial problem — which is the tricky part.
Consider some three consecutive elements $x$, $y$ and $z$. Assume that we deleted $y$ and solved the problem for the remaining list, which included $x$ and $z$, and now we need to restore the answer for the original triplet. The weight of $x$ would be correct as it is, but we need to calculate the answer for $y$ and adjust it for $z$, namely:
• $w_y’ = w_y + w_x$
• $w_z’ = w_z + w_y + w_x$
Now, we can just delete, say, the first element, solve the problem recursively, and recalculate weights for the original array. But, unfortunately, it would work in quadratic time, because to make the update, we would need to know where its neighbors are, and since we can’t hold the entire array in memory, we would need to scan it each time.
Therefore, on each step, we want to remove as many elements as possible. But we also have a constraint: we can’t remove two consecutive elements because then merging results wouldn’t be that simple.
Ideally, we want to split our list into even and odd elements, but doing this is not simpler than the initial problem. One workaround is to choose the elements at random: toss a coin for each element, and then remove all “heads” after which a “tail” follows. This way no two consecutive elements will ever be selected, and on average we get rid of ¼ of the current list. The arithmetic complexity of this solution would still be linear, because
$$T(N) = T\left(\frac{3}{4} N\right) = O(N)$$
The only tricky part here is how to implement the merge step in external memory. To do it efficiently, we need to maintain our list in the following form:
• List of tuples $(i, j)$ indicating that element $j$ follows after element $i$
• List of tuples $(i, w_i)$ indicating that element $i$ currently has weight $w_i$
• A list of deleted elements
Now, to restore the answer after randomly deleting some elements and recursively solving the smaller problem, we need to iterate over all lists using three pointers looking for deleted elements. and for each such element, we will write $(j, w_i)$ to a separate table, which would signify that before the recursive step we need to add $w_i$ to $j$. We can then join this new table with initial weights, add these additional weights to them.
After coming back from the recursion, we need to update weights for the deleted elements, which we can do with the same technique, iterating over reversed connections instead of direct ones.
I/O complexity of this algorithm with therefore be the same as joining, namely $SORT(N) = O\left(\frac{N}{B} \log_{\frac{M}{B}} \frac{N}{M} \right)$.
### #Applications
List ranking is especially useful in graph algorithms.
For example, we can obtain the Euler tour of a tree in external memory by constructing a linked list from the tree that corresponds to its Euler tour and then applying the list ranking algorithm — the ranks of each node will be the same as its index $tin_v$ in the Euler tour. To construct this list, we need to:
• split each undirected edge into two directed ones;
• duplicate the parent node for each up-edge (because list nodes can only have one incoming edge, but we visit some vertices multiple times);
• route each such node either to the “next sibling,” if it has one, or otherwise to its own parent;
• and then finally break the resulting cycle at the root.
This general technique is called tree contraction, and it serves as the basis for a large number of tree algorithms.
The same approach can be applied to parallel algorithms, and we will cover that much more deeply in part II. | 2022-10-05 11:16:09 | {"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": 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": 0.6127281188964844, "perplexity": 367.858483811949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337625.5/warc/CC-MAIN-20221005105356-20221005135356-00764.warc.gz"} |
https://www.clutchprep.com/macroeconomics/average-propensity-to-consume-and-save | Clutch Prep is now a part of Pearson
Ch. 15 - Income and ConsumptionWorksheetSee all chapters
# Average Propensity to Consume and Save
See all sections
Sections
The Consumption Function
The Saving Function
Determinants of Consumption and Saving
Average Propensity to Consume and Save
Multiplier Effect of Investment Spending
Concept #1: Average Propensity to Consume and Save
Practice: If the Keynesian consumption function is C = 10 + 0.8 Yd then, when disposable income is $1000, what is the average propensity to consume? Practice: If the Keynesian consumption function is C = 10 + 0.8 Yd then, when disposable income is$1000, what is the marginal propensity to save? | 2022-06-27 06:32:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46880781650543213, "perplexity": 13297.849582297173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103328647.18/warc/CC-MAIN-20220627043200-20220627073200-00610.warc.gz"} |
https://lists.gnu.org/archive/html/pspp-users/2008-04/msg00004.html | pspp-users
[Top][All Lists]
## Re: pspp new bee
From: unknown-1 Subject: Re: pspp new bee Date: Mon, 14 Apr 2008 18:06:05 +0200
2008/4/14, Jason Stover
On Sat, Apr 12, 2008 at 04:36:15PM +0200, unknown-1 wrote:
> Question:
> Furthermore I tried to compile it for Windows. The newest version of pspp,
> 0.6, compiles nice using cygwin when the gui is disabled. And it seems to
> work nice to. (But I didn't a in dept test.) Did anybody succeed in
> compiling pspp for use in windows with the gui enabled? If so, how?
Since no one more knowledgeable has answered yet, I'll say what I think
is correct:
If you have cygwin, X11 and gtk+2.12.0 or later, you should be able to
install it with ./configure. If I remember cygwin correctly, you will
have to run the X under cygwin, then start psppire from a terminal
window.
In fact what I did, I hope I wrote it down correctly, is this little howto:
===========================
Howto compile PSPP for Windows.
This howto is used with
- Windows XP SP2
- PSPP 0.6.0 which uses directory 0.4.3
- Cygwin setup 2.573.2.2 as is on April 13, 2008
Steps:
- Quit your software firewalls, maybe this is not necessary for your firewall
- Start the setup program
- Keep the defaults except:
- I recommend to use a user specified "local package directory"
- Select "Install" for the Packages groups
- Libs
- Graphics
- finish the Cygwin installation
- Now you can start your firewall again
- copy /usr/include/ncurses/term.h to /usr/include/term.h (this looks as a bug somewhere)
- Download the PSPP software and place them in a directory inside the newly created Cygwin directory. For example in c:\cygwin\home\itsme\pspp*
- Start cygwin en go to the directory where the PSPP source is.
- type "./configure --without-gui"
- type "make"
- type "make install"
- now your pspp.exe is in /usr//local/bin
- you can start it when you are in the cygwin environment
- with cygcheck pspp.exe you can see with dll's are used in case you want to place them on another PC's
==============================
My problems:
1)
- copy /usr/include/ncurses/term.h to /usr/include/term.h (this looks as a bug somewhere)
No idea why cygwin places this file in the include/ncurses directory and the make of pspp is looking in the include directory. I guess a cygwin problem but if the pspp people can solve this....
2)
When I compile without "--without-gui" then the ./configure tells me:
=============
configure: error: The following required prerequisites are not installed.
You must install them before PSPP cab be built:
gtk+ 2.0 v2.12.0 or later (or use --without-gui)
==============
As far as I understand I have GTK+ 1.2.10-2 which is too old and libglade2 2.5.1-1 which should be new enough if libglade=libglade2.
So I guess I have to wait till the cygwin people update gtk+ and libglade. Or is there anyone who has an idea how to solve this.
BTW. If anyone can use them I can place the pspp.exe and it's dll's for download on a free upload server. | 2019-04-25 23:29:27 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8221344351768494, "perplexity": 10931.356804980825}, "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/1555578742415.81/warc/CC-MAIN-20190425213812-20190425235812-00324.warc.gz"} |
http://www.tutornext.com/math/type-of-numbers-natural-whole-integers-rational-irrational.html | Sales Toll Free No: 1-855-666-7440
# Type of Numbers
We need to use numbers in order to work in mathematics. A real number is a number used to represent quantities such as price, length, distance etc.
The set of real numbers is in turn made up of different types of numbers viz. Natural, Whole, Integers, Rational and Irrational.
## Natural Numbers
In the beginning, “number” meant something you could count, like how many sheep the farmer has. From this evolved the notion of the natural numbers, also called the counting numbers.
Examples of natural numbers are: 1, 2, 3, 4, 5… basically all the numbers used for counting.
This set of natural numbers is denoted by N.
## Whole Numbers
The natural numbers together with the number “zero” is called the set of Whole numbers.
Example: 0, 1, 2, 3, 4, 5…
The set of whole numbers is denoted by W.
So the set of whole numbers is basically the set of natural numbers combined with 0.
## Integers
The set of whole numbers combined with the negative numbers -1,-2,-3… forms the set of integers.
Example: …,-5,-4,-3,-2,-1,0,1,2,3,4,5…
This set is denoted by Z.
## Rational Numbers
A rational number can be defined as any number that can be expressed as $$\frac{m}{n}$$ where m and n are both integers and n is not equal to 0.
In other words, rational numbers include repeating decimals, terminating decimals and fractions.
Examples: $$\frac{2}{3},-3\frac{1}{2},0.6666$$
This set is denoted by R.
Hence, Rational numbers is the combination of integers, whole numbers, natural numbers and all the fractions in between.
## Irrational Numbers
Irrational numbers are defined as numbers that are not rational.
These are basically numbers that cannot be written as fractions or decimals that do not terminate or repeat (a rational will do at least one).
Examples: $$0.254787558785785775…… \pi, \sqrt{2}$$ etc.
This set is denoted by Q.
## Summary of Number System
We can summarize the set of real numbers by the following diagram. | 2017-11-20 01:54:45 | {"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": 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": 0.7819355130195618, "perplexity": 606.0547663798546}, "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/1510934805894.15/warc/CC-MAIN-20171120013853-20171120033853-00303.warc.gz"} |
https://labs.tib.eu/arxiv/?author=A.%20Yu.%20Barnyakov | • ### Measurement of $\Gamma_{ee}\times\mathcal{B}_{\mu\mu}$ for $\psi(2S)$ meson(1801.10362)
April 4, 2018 hep-ex
The product of the electronic width of the $\psi(2S)$ meson and the branching fraction of its decay to the muon pair was measured in the $e^{+}e^{-} \to \psi(2S) \to \mu^{+}\mu^{-}$ process using nine data sets corresponding to an integrated luminosity of about 6.5 pb$^{-1}$ collected with the KEDR detector at the VEPP-4M electron-positron collider: $\Gamma_{ee}\times\mathcal{B}_{\mu\mu} = 19.3 \pm 0.3 \pm 0.5 ~\text{eV}.$ Adding the previous KEDR results on hadronic and leptonic channels, the values of the $\psi(2S)$ electronic width were obtained under two assumptions: either with the assumption of lepton universality $\Gamma_{ee} = 2.279 \pm 0.015 \pm 0.042 ~\text{keV}$ or without it, summing up hadronic and three independent leptonic channels: $\Gamma_{ee} = 2.282 \pm 0.015 \pm 0.042 ~\text{keV}.$
• ### Measurement of the $e^+e^- \to \eta\pi^+\pi^-$ cross section with the SND detector at the VEPP-2000 collider(1711.08862)
Nov. 24, 2017 hep-ex
The $e^+e^- \to \eta\pi^+\pi^-$ cross section is measured at the SND detector in the $\eta$ decay mode $\eta\to 3\pi^0$. The analysis is based on the data sample with an integrated luminosity of 32.7 pb$^{-1}$ collected at the VEPP-2000 $e^+e^-$ collider in the center-of-mass energy range $\sqrt{s}=1.075-2.000$ GeV. The data obtained in the $\eta\to 3\pi^0$ decay mode are found to be in agreement with the previous SND measurements in the $\eta\to \gamma\gamma$ mode. Therefore the measurements in the two modes are combined.
• The exclusive charmonium production process in $\bar{p}p$ annihilation with an associated $\pi^0$ meson $\bar{p}p\to J/\psi\pi^0$ is studied in the framework of QCD collinear factorization. The feasibility of measuring this reaction through the $J/\psi\to e^+e^-$ decay channel with the PANDA (AntiProton ANnihilation at DArmstadt) experiment is investigated. Simulations on signal reconstruction efficiency as well as the background rejection from various sources including the $\bar{p}p\to\pi^+\pi^-\pi^0$ and $\bar{p}p\to J/\psi\pi^0\pi^0$ reactions are performed with PandaRoot, the simulation and analysis software framework of the PANDA experiment. It is shown that the measurement can be done at PANDA with significant constraining power under the assumption of an integrated luminosity attainable in four to five months of data taking at the maximum design luminosity.
• Simulation results for future measurements of electromagnetic proton form factors at \PANDA (FAIR) within the PandaRoot software framework are reported. The statistical precision with which the proton form factors can be determined is estimated. The signal channel $\bar p p \to e^+ e^-$ is studied on the basis of two different but consistent procedures. The suppression of the main background channel, $\textit{i.e.}$ $\bar p p \to \pi^+ \pi^-$, is studied. Furthermore, the background versus signal efficiency, statistical and systematical uncertainties on the extracted proton form factors are evaluated using two different procedures. The results are consistent with those of a previous simulation study using an older, simplified framework. However, a slightly better precision is achieved in the PandaRoot study in a large range of momentum transfer, assuming the nominal beam conditions and detector performance.
• ### Measurement of hadron cross sections with the SND detector(1609.01040)
Sept. 5, 2016 hep-ex
New results on exclusive hadron production in $e^+e^-$ annihilation obtained in experiments with the SND detector at the VEPP-2M and VEPP-2000 $e^+e^-$ colliders are presented.
• ### Measurement of the $\mathbf{e^+e^-\to K^+K^-}$ cross section in the energy range $\mathbf{\sqrt{s}=1.05-2.0}$ GeV(1608.08757)
Aug. 31, 2016 hep-ex
The $e^+e^-\to K^+K^-$ cross section is measured in the center-of-mass energy range $1.05-2.00$ GeV at the SND detector. The measurement is based on data with an integrated luminosity of 35 pb$^{-1}$ collected at the VEPP-2000 $e^+e^-$-collider. The obtained results are consistent with the previous most accurate data obtained in the BABAR experiment and have a comparable accuracy.
• ### Measurement of the $e^+e^- \to \omega\eta$ cross section below $\sqrt{s}=2$ GeV(1607.00371)
July 1, 2016 hep-ex
The cross section for the process $e^+e^- \to \omega\eta$ is measured in the center-of-mass energy range 1.34--2.00 GeV. The analysis is based on data collected with the SND detector at the VEPP-2000 $e^+e^-$ collider. The measured $e^+e^- \to \omega\eta$ cross section is the most accurate to date. A significant discrepancy is observed between our data and previous BABAR measurement.
• ### Crossing integer spin resonance at VEPP-4M with conservation of beam polarization(1508.05739)
July 2, 2019 physics.acc-ph
A method proposed to preserve the electron beam polarization at the VEPP-4M collider during acceleration with crossing the integer (imperfection) spin resonance at energy E=1763 MeV has been successfully applied. It is based on full decompensation of the $0.6\times3.3$ Tesla$\times$meter integral of the KEDR detector longitudinal magnetic field due to the anti-solenoids 'switched-off'.
• ### On a search for the $\eta \rightarrow e^+ e^-$ decay at the VEPP-2000 $e^+e^-$ collider(1507.02073)
July 15, 2015 hep-ex
A sensitivity of the VEPP-2000 $e^+e^-$ collider in a search for the rare decay $\eta \rightarrow e^+ e^-$ has been studied. The inverse reaction $e^+ e^- \rightarrow \eta$ is proposed for this search. We have analyzed a data sample with an integrated luminosity of 108 nb$^{-1}$ collected with the SND detector in the center-of-mass energy range 520-580 MeV and found no background events for the reaction $e^+ e^- \rightarrow \eta$ in the decay mode $\eta\to\pi^0\pi^0\pi^0$. In the absence of background, a sensitivity to ${\cal B}(\eta \rightarrow e^+ e^-)$ of $10^{-6}$ can be reached during two weeks of VEPP-2000 operation. Such a sensitivity is better than the current upper limit on ${\cal B}(\eta \rightarrow e^+ e^-)$ by a factor of 2.3.
• ### Search for the $\eta^{\prime}\to e^+e^-$ decay with the SND detector(1504.01245)
April 6, 2015 hep-ex
A search for the process $e^+e^- \to \eta^\prime$ has been performed with the SND detector at the VEPP-2000 $e^+e^-$ collider. The data were accumulated at the center-of-mass energy of $957.78\pm 0.06$ MeV with an integrated luminosity of about 2.9 pb$^{-1}$. For reconstruction of the $\eta^\prime$ meson five decay chains have been used: $\eta^{\prime}\to\eta\pi^+\pi^-$ followed by the $\eta$ decays to $\gamma\gamma$ and $3\pi^0$, and $\eta^{\prime} \to \eta\pi^0\pi^0$ followed by the $\eta$ decays to $\pi^+\pi^-\pi^0$, $\gamma\gamma$, and $3\pi^0$. As a result, the upper limit has been set on the $\eta^\prime$ electronic width: $\Gamma_{\eta^{\prime}\to e^+e^-} < 0.0020$ eV at the 90\% confidence level.
• ### Measurement of the $e^+e^- \to \eta\pi^+\pi^-$ cross section in the center-of-mass energy range 1.22--2.00 GeV with the SND detector at the VEPP-2000 collider(1412.1971)
March 24, 2015 hep-ex
In the experiment with the SND detector at the VEPP-2000 $e^+e^-$ collider the cross section for the process $e^+e^-\to\eta\pi^+\pi^-$ has been measured in the center-of-mass energy range from 1.22 to 2.00 GeV. Obtained results are in agreement with previous measurements and have better accuracy. The energy dependence of the $e^+e^-\to\eta\pi^+\pi^-$ cross section has been fitted with the vector-meson dominance model. From this fit the product of the branching fractions $B(\rho(1450)\to\eta\pi^+\pi^-)B(\rho(1450)\to e^+e^-)$ has been extracted and compared with the same products for $\rho(1450)\to\omega\pi^0$ and $\rho(1450)\to\pi^+\pi^-$ decays. The obtained cross section data have been also used to test the conservation of vector current hypothesis.
• ### Study of the process $e^+e^-\to n\bar{n}$ at the VEPP-2000 $e^+e^-$ collider with the SND detector(1410.3188)
Oct. 13, 2014 hep-ex
The process $e^+e^-\to n\bar{n}$ has been studied at the VEPP-2000 $e^+e^-$ collider with the SND detector in the energy range from threshold up to 2 GeV. As a result of the experiment, the $e^+e^-\to n\bar{n}$ cross section and effective neutron form factor have been measured.
• ### Measurement of the ratio of the leptonic widths $\Gamma_{ee}/\Gamma_{\mu\mu}$ for the $J/\psi$ meson(1311.5005)
Feb. 17, 2014 hep-ex
The ratio of the electron and muon widths of the $J/\psi$ meson has been measured using direct $J/\psi$ decays in the KEDR experiment at the VEPP-4M electron-positron collider. The result $\Gamma_{ee}(J/\psi)/\Gamma_{\mu\mu}(J/\psi)=1.0022\pm0.0044\pm0.0048\ (0.65\%)$ is in good agreement with the lepton universality. The experience collected during this analysis will be used for a $J/\psi$ lepton width determination with up to 1% accuracy.
• ### Study of the process $e^+e^-\to\eta\gamma$ in the center-of-mass energy range 1.07--2.00 GeV(1312.7078)
Dec. 26, 2013 hep-ph, hep-ex
The $e^+e^-\to\eta\gamma$ cross section has been measured in the center-of-mass energy range 1.07--2.00 GeV using the decay mode $\eta\to 3\pi^0$, $\pi^0\to \gamma\gamma$. The analysis is based on 36 pb$^{-1}$ of integrated luminosity collected with the SND detector at the VEPP-2000 $e^+e^-$ collider. The measured cross section of about 35 pb at 1.5 GeV is explained by decays of the $\rho(1450)$ and $\phi(1680)$ resonances.
• We report results of experiments performed with the KEDR detector at the VEPP-4M $e^+e^-$ collider. They include final results for the mass and other parameters of the $J/\psi$, $\psi(2S)$ and $\psi(3770)$ and $J/\psi\to\gamma\eta_c$ branching fraction determination.
• We report the final results of a study of the \psi(3770) meson using a data sample collected with the KEDR detector at the VEPP-4M electron-positron collider. The data analysis takes into account the interference between the resonant and nonresonant $D\bar{D}$ production, where the latter is related to the nonresonant part of the energy-dependent form factor $F_D$. The vector dominance approach and several empirical parameterizations have been tried for the nonresonant $F_D^{\NR}(s)$. Our results for the mass and total width of \psi(3770) are M = 3779.2 ^{+1.8}_{-1.7} ^{+0.5}_{-0.7} ^{+0.3}_{-0.3} MeV, \Gamma =24.9 ^{+4.6}_{-4.0} ^{+0.5}_{-0.6} ^{+0.2}_{-0.9} MeV, where the first, second and third uncertainties are statistical, systematic and model, respectively. For the electron partial width two possible solutions have been found: (1) \Gamma_{ee} = 154 ^{+79}_{-58} ^{+17}_{-9} ^{+13}_{-25} eV, (2) \Gamma_{ee} = 414 ^{+72}_{-80} ^{+24}_{-26} ^{+90}_{-10} eV. Our statistics are insufficient to prefer one solution to another. The solution (2) mitigates the problem of non-$D\bar{D}$ decays but is disfavored by potential models. It is shown that taking into account the resonance--continuum interference in the near-threshold region affects resonance parameters, thus the results presented can not be directly compared with the corresponding PDG values obtained ignoring this effect.
• ### Search for narrow resonances in e+ e- annihilation between 1.85 and 3.1 GeV with the KEDR Detector(1107.2824)
July 14, 2011 hep-ex
We report results of a search for narrow resonances in e+ e- annihilation at center-of-mass energies between 1.85 and 3.1 GeV performed with the KEDR detector at the VEPP-4M e+ e- collider. The upper limit on the leptonic width of a narrow resonance Gamma(R -> ee) Br(R -> hadr) < 120 eV has been obtained (at 90 % C.L.).
• We present a study of the inclusive photon spectrum from 6.3 million J/psi decays collected with the KEDR detector at the VEPP-4M e+e- collider. We measure the branching fraction of the radiative decay J/psi -> eta_c gamma, eta_c width and mass. Taking into account an asymmetric photon line shape we obtain: M(eta_c) = (2978.1 +- 1.4 +- 2.0) MeV/c^2, Gamma(eta_c) = (43.5 +- 5.4 +- 15.8) MeV, B(J/psi->eta_c gamma) = (2.59 +- 0.16 +- 0.31)%\$.
• ### Measurement of \Gamma_{ee}(J/\psi)*Br(J/\psi->e^+e^-) and \Gamma_{ee}(J/\psi)*Br(J/\psi->\mu^+\mu^-)(0912.1082)
Jan. 25, 2010 hep-ex
The products of the electron width of the J/\psi meson and the branching fraction of its decays to the lepton pairs were measured using data from the KEDR experiment at the VEPP-4M electron-positron collider. The results are \Gamma_{ee}(J/\psi)*Br(J/\psi->e^+e^-)=(0.3323\pm0.0064\pm0.0048) keV, \Gamma_{ee}(J/\psi)*Br(J/\psi->\mu^+\mu^-)=(0.3318\pm0.0052\pm0.0063) keV. Their combinations \Gamma_{ee}\times(\Gamma_{ee}+\Gamma_{\mu\mu})/\Gamma=(0.6641\pm0.0082\pm0.0100) keV, \Gamma_{ee}/\Gamma_{\mu\mu}=1.002\pm0.021\pm0.013 can be used to improve theaccuracy of the leptonic and full widths and test leptonic universality. Assuming e\mu universality and using the world average value of the lepton branching fraction, we also determine the leptonic \Gamma_{ll}=5.59\pm0.12 keV and total \Gamma=94.1\pm2.7 keV widths of the J/\psi meson. | 2020-11-30 22:04:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8266337513923645, "perplexity": 1432.4403518932963}, "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-2020-50/segments/1606141486017.50/warc/CC-MAIN-20201130192020-20201130222020-00524.warc.gz"} |
https://math.stackexchange.com/questions/357156/cycle-of-length | # Cycle of length
I'm learning permutations and came upon this question which made me freeze. So to say it in my own words, it asks that how many permutations in $S_n$ do not have a cycle of length one in their disjoint cycle notation. My guess at it would be just one but I don't know how to show it.
• Have you tried writing down all the permutations of $S_3$ as products of disjoint cycles? – Paul Gustafson Apr 10 '13 at 14:18
• Tough problem, if you are meeting it fresh. Look at the wikipedia article on Derangements. – André Nicolas Apr 10 '13 at 14:22
• @AndréNicolas I looked at Derangments but it confused me further. I can see the connection to a certain limit but Im not able to solve for this problem. – user65422 Apr 14 '13 at 23:48
• @user66345 Like this? $(a,b,c)=(a,c),(a,b)$ – user65422 Apr 15 '13 at 1:18 | 2019-08-17 10:49:39 | {"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": 0.7766680121421814, "perplexity": 312.7155749588558}, "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/1566027312128.3/warc/CC-MAIN-20190817102624-20190817124624-00140.warc.gz"} |
https://stats.stackexchange.com/questions/272397/prove-that-the-joint-density-of-independent-multivariate-normal-variables-is-a-m | # Prove that the joint density of independent multivariate normal variables is a matrix-normal
Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent.
Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ is the identity nxn.
The density of a matrix-normal variable with independent entries is $$p(W|M,\mathbb{I},\Sigma) =\frac{\exp\left(-\frac{1}{2}tr(\Sigma^{-1}(W-M)^T\mathbb{I}(W-M)\right))}{(2\pi)^{np/2}|\Sigma|^{n/2}}$$
Given that my variables $X_i$ are independent, the joint density will be the product of them, so:
$$f(W|M,\mathbb{I},\Sigma) =\prod^n_{i=1}\left\{\frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}}\exp\left[-\frac{1}{2}(X_i-\mu_i)^T\Sigma^{-1}(X_i-\mu_i)\right]\right\}$$ $$=\frac{1}{(2\pi)^{np/2}|\Sigma|^{n/2}}\exp\left[-\frac{1}{2}\sum_{i=1}^n(X_i-\mu_i)^T\Sigma^{-1}(X_i-\mu_i)\right]$$
Now, what I have to do is somehow turn the expression within the exp into a diagonal matrix in order to change the expression from a sum to the trace of a matrix. Any tips on how to do that? My professor suggested trying to use kronecker product in order to generate this matrix.
Easier ways to prove it or any books with the solution would also be really apreciated. Thanks in advance. | 2019-07-20 05:23:21 | {"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": 0.7303915619850159, "perplexity": 179.0036870988933}, "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/1563195526446.61/warc/CC-MAIN-20190720045157-20190720071157-00557.warc.gz"} |
https://askdev.io/questions/36912/what-am-i-missing-out-on-to-get-symlinks-to-collaborate-with | # What am I missing out on to get symlinks to collaborate with CIFS?
Circumstance
I have a brainless Ubuntu 10.10 RC box running a couple of solution applications on my residence network. I have a Windows 2008 Server organizing all my network shares and also disk drives. I am presently placing the network drives at boot - up making use of FSTAB with the adhering to alternatives set:
credentails=/etc/smbcredentials,. iocharset=utf8,uid=1000,gid=1000,file_mode=0777,dir_mode=0777,noserverino,sfu
Question
What alternative do I require to ready to get SYMLINKS to effectively register making use of CIFS? I need to confess the details in man mount.cifs does not appear to give a clear adequate definition of which alternatives I need to be making use of for correct assistance.
Trouble
When running RSYNC from the Ubuntu equipment to support picked folders to the Windows shares, it falls short attempting to recreate the SYMLINKS. I am worried that this will certainly create a trouble when later on attempting to recover these documents back need to I ever before require to.
0
2019-05-13 04:12:24
Source Share
Not certain, yet I are afraid that a cifs share, that in your instance is basically a folder on a ntfs dividing readily available via the network, can not take care of symbolic web links.
Various would certainly hold true if the cifs share were given by a samba web server on a linux equipment.
The remedy that enter your mind is:
• create a massive adequate documents on the share (with dd, as an example)
• create a ext4 filesystem on this documents
• mount the documents as a dividing photo, with - o loop
• usage this ext4 dividing as a location for your back-up
0
2019-05-17 16:50:26
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https://tanaikech.github.io/2018/07/02/benchmark-search-for-array-processing-using-google-apps-script/ | Gists
Kanshi Tanaike
# Introduction
Please be careful! This result can be only used for Google Apps Script.
There are a limit executing time for Google Apps Script (GAS). That is 6 minutes.1 So users always have to pay attention to reducing the process cost of the scripts. Especially, it is very important to know the process cost for the array processing, because the array processing is often used for spreadsheet and Google APIs. Recently, I have reported about the process cost of the loop for the array processing.2 Also I have reported “Improved Algorithms for Summation of Array Elements” as a method for reducing the process cost.3 From these reports, it has found that GAS shows much different process cost from other languages. So it is important to investigate the process cost for various scenes. In this report, the process cost of “searching strings in an array” for the array processing using GAS has been investigated.
Here, as a sample, it supposes the situation that it searches 3 strings from one dimensional array. When users use the process for this situation using GAS, the following 3 methods can be considered.
1. Search using linear search by for loop
2. Search using hash by creating an object
3. Search using indexOf() 4
From the report of loop cost, 2 it has already found that the loop by map and filter is the fastest at GAS. 2 But in this report, “for loop” which is the general loop was used.
As the result, it was found that the cost of search by indexOf() was the lowest of all methods. Namely, it was the fastest of all. 2nd and last one were the search by for loop and the search by the hash, respectively. In the case of the search by hash, it was fount that although the cost of search by the hash from the object is very low, the cost for creating the object to search the hash was the highest of all. By this, the search by the hash became the lowest rank. If the object for searching has already been created, the cost of search by the hash will be the lowest of all.
# Experimental procedure
In the experiment, the sample scripts for GAS were used. 1 dimensional array including 3 strings for searching was used as a source array. The 3 string values were put to the top, center and end of the array in order to measure the same condition. In this case, it uses all of elements in the array. 3 methods as mentioned above were used for searching the strings from the array. As the measurements of process cost, those 3 strings were searched from the source array by changing the number of elements of the array, and the process cost which is the search time was measured. In each search, the index of array which has the searched string was retrieved. The cost for creating array was not included. Only the cost for searching was measured. By the way, at GAS, the processing time is not stable as you know. So the average value for more than 250 times measurements was used for each data point which is shown by figures. At this time, the fluctuation of the average values was less than 1 %. I worry that each detailed-data point at my environment might be different from that at other user’s environment. But I think that the trend of this result can be used.
# Results
Fig 1. Number of elements vs. processing time (process cost) for searching 3 strings. The solid lines with the color of red, blue and green mean the result from the search using hash, the linear search and the search using indexOf, respectively.
Fig 2. Each method vs. processing efficiency time (process cost) for searching strings. Here, the chart of search using indexOf shown in Fig. 1 was magnified.
Figure 1 shows the number of elements vs. processing time (process cost) for searching 3 strings. At figure 2, the chart of search using indexOf shown in Fig. 1 was magnified. From Fig. 1, it was found that the search using indexOf is the fastest of all. The second is the linear search. The last is the search using hash. It is considered that the reason of this order is as follows.
1. From the sample script, both the linear search and the search using hash process all elements in the array using the for loop. On the other hand, the search using indexOf doesn’t process all elements using the for loop. The index of the element with the found string is directly retrieved using indexOf. When the string is found, the elements in the array are removed from the start index to the index of element with the found string. By this, the array length becomes small. This flow is repeated until to find all strings for searching. So it is considered that the search cost of indexOf which doesn’t process all elements using the for loop became the lowest of all. This may indicate that the scan for indexOf() is different from it for the general for loop.
2. In the case of the search by hash, it was found that although the cost of search by the hash from the object is very low, the cost for creating the object to search the hash was the highest of all. So it is considered that the search by hash became the lowest rank. If the object for searching has already been created, the cost of search by the hash will be the lowest of all.
3. The linear search processes all elements using the for loop as same as the search using hash. At the linear search, the elements are scanned and the searched string is retrieved using if. On the other hand, at the search using hash, the object is created to search while using if. After the object was created, the strings are searched. From the result of order of cost, it is considered that the cost for creating an object is higher than that of push(). So it is considered that the linear search became the second.
Fig 3. Number of elements which can be searched per unit time. Vertical axis is the logger scale. Horizontal axis is each method. These were calculated using the slopes of each method in Figs 1 and 2.
Figure 3 shows the number of elements which can be searched per unit time. The vertical axis and horizontal axis are the logger scale and each method, respectively. These values were calculated using the slopes of each method in Figs 1 and 2. From Fig. 3, it was found that the search using indexOf can reduce the process cost of more than 99 % from the linear search and the search using hash.
# Summary
In this report, the process cost of “searching strings in an array” for the array processing by 3 methods using GAS was investigated. As the result, it was found the following important features for GAS.
• Process cost of search by indexOf() was the lowest of all methods.
• 2nd and last one were the search by for loop and the search by the hash, respectively.
• About the search by hash, although the cost of search by the hash from the object is very low, the cost for creating the object to search the hash was the highest of all. By this, the search by hash became the lowest rank. If the object for searching has already been created, the cost of search by the hash will be the lowest of all.
• Search using indexOf can reduce the process cost of more than 99 % from the linear search and the search using hash.
• From these results, it is considered that the scan for indexOf() may be different from the general for loop.
As a note, I have to describe that this is the result for Google Apps Script. For other languages, this result might be difference. And also, the process cost of this report might be modified by future update of Google.
# References
## Appendix
### Scripts
#### Creating sample arrays
The function for creating an array is as follows. “searchText” is put to the center, top and end of the array. In this report, the script searches the index of array elements with “searchText” from the array including 3 “searchText”.
function createSampleArray(max, searchText) {
var r = Array.apply(null, new Array(max)).map(function(_, i) {return "val" + ('0000000000' + i).slice(-8)});
// Put the search value to center of the array.
Array.prototype.splice.apply(r, [Math.floor(r.length / 2), 0].concat([searchText]));
// Put the search value to top of the array.
r.unshift(searchText);
// Put the search value to end of the array.
r.push(searchText);
return r;
}
The array becomes like below.
[
"searchText",
"val00000000",
"val00000001",
"val00000002",
"val00000003",
,
,
,
"searchText",
,
,
,
"val00000097",
"val00000098",
"val00000099",
"val00000100",
"searchText"
]
#### Searching values
// Linear search
function linearSearch(array, searchText) {
var result = [];
for (var i = 0; i < array.length; i++) {
if (array[i] == searchText) {
result.push(i);
}
}
return result;
}
// Search using hash
function hashSearch(array, searchText) {
var hash = {};
for (var i = 0; i < array.length; i++) {
if (hash[array[i]]) {
hash[array[i]] = hash[array[i]].concat(i);
} else {
hash[array[i]] = [i];
}
}
return hash[searchText];
}
// Search using indexOf
function indexofSearch(array, searchText) {
var result = [];
var base = 0;
var len = array.length;
for (var i = 0; i < len; i++) {
var c = array.indexOf(searchText);
if (c > -1) {
base += c;
result.push(i + base);
array.splice(0, c + 1);
} else {
break;
}
}
return result;
} | 2021-11-29 16:56:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.18112392723560333, "perplexity": 1108.7959128143273}, "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-2021-49/segments/1637964358786.67/warc/CC-MAIN-20211129164711-20211129194711-00044.warc.gz"} |
https://puzzling.stackexchange.com/questions/48323/its-all-greek-to-me/48374 | It's all Greek to me
And this is barely a puzzle.
How can everything be so important?
Maybe there's an alternate way to find the answer to what this OP is craving?
HINT:
The sum of the parts is equal to the whole.
HINT 2:
Something does actually add up. But if at first you don't succeed, try again.
HINT 3:
There happens to be an entirely different and unrelated answer from what I had in mind that I will accept as well. By happenstance I stumbled upon it and it nearly fits perfectly.
• I added a hint above. If you want another let me know. Jan 22, 2017 at 11:23
• I sure do!!! 😄 Jan 23, 2017 at 14:57
• I just added another hint today. This one is a bit more important than the first. This puzzle does not have an obvious solution at first; however you will know when you come to the correct answer based on the puzzle itself. Jan 23, 2017 at 15:12
• This wouldn't happen to involve greek numerals now would it...? Apr 3, 2018 at 2:35
• This was my first puzzle written so it's a bit of a clumsy one, I'm afraid. The words used were very specific, but it may need another hint for people to get onto the right track. Apr 3, 2018 at 14:59
SETS
Considering only the first para is relevant-
Something doesn't add up ... Plus this is barely a puzzle . How can everything be so important ? There must be an alternate way to find the answer ...
Assuming there is a Morse Code hidden in the para. And, ?=Dash & .=DOT
we get ... . - ... -> SETS So OP is craving for SETS :)
• A good effort, but it's a bit more complicated than that. There are key words in the riddle that are used to derive the answer. Jan 22, 2017 at 11:17
I don't have hard evidence, but I think the OP is craving
Pie
Reasoning below, admittedly, this is essentially guesswork.
The terms "Add up" and "Plus" point to something mathematical.
Considering the title, "It's all Greek to me", we're looking for something Greek, mathematical and alimentary.
Therefore, Pi, as a Greek letter, a mathematical constant, and phonetically, a dish, fits pretty well.
• The words "plus" and "add up" are big hints; however, I may have made this riddle too ambiguous. It was one of my first. If you want another hint let me know. Feb 19, 2017 at 1:48
• I think this is the right track, considering (cv vf gur inyhr bs fbzr snzbhf nygreangvat vasvavgr fhzf!) Aug 25, 2021 at 14:45
All the clues seem to point towards it being:
Sigma (a greek letter)
For the reasons that
The symbol $\Sigma$ is used for summation in mathematics, also from hint number two "If you first don't succeed" seems to be talking about error, the lower case sigma $\sigma$ is usually used to denote errors.
However I'm not sure what this ties into as far as something the OP desires unless:
They want to be 18 (sigma is the 18th letter of the greek alphabet or make-up brushes.
• Good reasoning. As I stated in a comment above I may have made this a little too ambiguous, being my first riddle. There are important words that should be taken into consideration. Feb 19, 2017 at 1:49
Is OP craving
lamb?
Reasoning:
Take the number of words in each line and add them up: 5+6+6+6+9+6=38. Add 3 and 8 (suggested by hint 2): 3+8=11. The 11th letter in the Greek alphabet is lambda (lamb).
• So far your idea is the closest to what I had in mind. Not quite, but close. Feb 15, 2018 at 1:28
I think it's:
PITA
Reasoning:
And this is barely a puzzle.
If I count all the syllables in this sentence I get 15. Since it doesn't add up I assume one is missing to get 16 or P
How can everything be so important?
If I simply count all the syllables in this sentence I get 9 or I.
Each line must provide a clue.
Maybe there's an alternate way to find the answer...
Adding up the syllables in these two lines I find 20 or T.
Maybe there's an alternate way to find the answer...
This lead me to count all the syllables including 'To what OP is craving...' this is exactly 52. If I add the one added from the previous 'doesn't add up' I get 53. If I mod 26 then I get 1 or A.
Combined with the title It's all Greek to me. I have to assume PITA is what you crave.
I think it's:
Staggering Elk Lager
BTW it's a real beer, see:
http://www.pintley.com/beer/Staggering-Elk-Lager/5027/
Because:
Hints and text suggest the question didn't need much/any text. Treating the title as a cryptic clue...
It's all Greek to me
Anagram of "all Greek" is "Elk Lager"
So a staggering "Elk Lager" could be "all Greek"
• The title is certainly important, but this sadly is not the answer I'm looking for.. although I may indeed crave it later. Jan 22, 2017 at 15:44
Maybe its-
Η η eta, ήτα
Explanation
Counting all the letters from each line and adding them
COUNTS - 9 6 4 3 2
SUM -24
FURTHER SUM - 6
And this is barely a puzzle.
COUNTS - 3 4 2 6 1 6
SUM -22
FURTHER SUM - 4
How can everything be so important?
COUNTS - 3 3 10 2 2 9
SUM -29
FURTHER SUM - 11
AGAIN FURTHER SUM - 2
Each line must provide a clue.
COUNTS - 4 4 4 7 1 4
SUM -24
FURTHER SUM - 6
Maybe there's an alternate way to find the answer...
COUNTS - 5 6 2 9 3 2 4 3 6
SUM - 40
FURTHER SUM - 4
To what this OP is craving.
COUNTS - 2 4 4 2 2 7
SUM - 21
FURTHER SUM - 3
Adding all the FURTHER SUM 6 + 4 + 11 + 6 + 4 + 3 = 34
And 3 + 4 = 7
Alternatively
Adding all the FURTHER SUM and instead of 11 I added 2(11's digit sum) (for alternative approach) we get 25
2 + 5 = 7
By both way we get 7 and hence the 7th Greek alphabet.
• Very close! However, not every word needs to be part of the sum. E.g. "And" Apr 5, 2018 at 11:20
• Great idea on adding sums, definitely on the right track! The puzzle did not count the number of letters, but you're certainly the closest in concept for the solution. Apr 5, 2018 at 11:24
• Something entertaining about this answer is that the 7th letter of the Greek alphabet sounds like "eat a". You have the right idea, but not quite there. Dec 14, 2021 at 12:13 | 2022-09-25 08:07:00 | {"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": 0.6842629909515381, "perplexity": 1276.684458905579}, "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/1664030334515.14/warc/CC-MAIN-20220925070216-20220925100216-00692.warc.gz"} |
https://www.zbmath.org/?q=an%3A1386.05146 | # zbMATH — the first resource for mathematics
Semitotal domination in claw-free cubic graphs. (English) Zbl 1386.05146
W. Goddard et al. [Util. Math. 94, 67–81 (2014; Zbl 1300.05220)] introduced semitotal domination in graphs. Let $$V(G)$$ be the vertex set of a graph $$G$$ containing no isolated vertices. A set $$S \subseteq V(G)$$ is called a semitotal dominating set of $$G$$ if each vertex not in $$S$$ is adjacent to a vertex in $$S$$, and each vertex in $$S$$ is at distance at most two from another vertex of $$S$$. The semitotal domination number of $$G$$, denoted by $$\gamma_{t2}(G)$$, is the minimum cardinality over all semitotal dominating sets of $$G$$.
For $$i\in\{1,2\}$$, let $$D_i$$ be a complete graph of order $$4$$ minus one edge; let $$V(D_i)=\{a_i, b_i, c_i, d_i\}$$ and let $$a_ib_i$$ be the missing edge in $$D_i$$. We denote by $$N_2$$ the graph obtained from the disjoint union of $$D_1$$ and $$D_2$$ by adding two edges $$a_1b_2$$ and $$a_2b_1$$, and we denote by $$K_4$$ the complete graph on $$4$$ vertices.
M. A. Henning and A. J. Marcon [Ann. Comb. 20, No. 4, 799–813 (2016; Zbl 1354.05107)] showed that, if $$G$$ is a connected claw-free cubic graph of order $$n \geq 10$$, then $$\gamma_{t2}(G) \leq \frac{4n}{11}$$, and they posed the following conjecture.
(C1) If $$G \not\in \{K_4, N_2\}$$ is a connected claw-free cubic graph of order $$n$$, then $$\gamma_{t2}(G) \leq \frac{n}{3}$$.
In this paper, the authors prove the above conjecture (C1), and they propose the following conjecture.
(C2) If $$G \not\in \{K_4, N_2\}$$ is a connected claw-free graph of order $$n$$ with minimum degree at least $$3$$, then $$\gamma_{t2}(G) \leq \frac{n}{3}$$.
##### MSC:
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
semitotal domination; cubic graph; claw-free graph
Full Text:
##### References:
[1] Duckworth, W; Wormald, NC, Minimum independent dominating sets of random cubic graphs, Electron. J. Comb., 9, 147-161, (2002) · Zbl 1009.05106 [2] Goddard, W; Henning, MA; McPillan, CA, Semitotal domination in graphs, Util. Math., 94, 67-81, (2014) · Zbl 1300.05220 [3] Haas, R; Seyffarth, K, The k-dominating graph, Graph Comb., 30, 609-617, (2014) · Zbl 1294.05122 [4] Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998) · Zbl 0883.00011 [5] Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998) · Zbl 0890.05002 [6] Henning, MA, A survey of selected recent results on total domination in graphs, Discrete Math., 309, 32-63, (2009) · Zbl 1219.05121 [7] Henning, MA; Löwnstein, C, Locating-total domination in claw-free cubic graphs, Discrete Math., 312, 3107-3116, (2012) · Zbl 1252.05162 [8] Henning, MA; Marcon, AJ, Semitotal domination in claw-free cubic graphs, Ann. Comb., 20, 1-15, (2016) · Zbl 1354.05107 [9] Henning, M.A., Yeo, A.: Total Domination in Graphs. Springer, New York (2013) · Zbl 1408.05002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-02-28 04:44:36 | {"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": 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": 0.8419667482376099, "perplexity": 1216.8181283670153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360107.7/warc/CC-MAIN-20210228024418-20210228054418-00605.warc.gz"} |
https://chemistry.stackexchange.com/tags/chromatography/hot | # Tag Info
15
Building on my comment, marking the distance that the solvent traveled allows us to calculate the retention factor $R_f$ (or apparently retardation factor, according to IUPAC meddlers who need to rename things with perfectly good names that everyone else uses). The retardation factor is the ratio of the distance traveled by the spot to the distance traveled ...
9
Why do we use the Kováts index? The Kováts index is used to normalize GC data. This Wikipedia page(though not very long), sums up why you would want to do so. Retention times of the same compound on even two different versions of identical instrument with the same temperature and pressure program using the same column from the same manufacturer might not ...
9
I wish we would stop teaching chromatography in terms of "polar" and "nonpolar." The aspirin will interact fairly strongly with the silica due to hydrogen bonding/electrostatic interactions of the carboxylic acid and the ester with the silica. If you increased the "polar" component of the mobile phase, it would travel further due to the mobile phase ...
8
Question 1 Yes, of course! Something that is "relatively" non-polar would travel further with a non-polar mobile phase because it would have less attraction to the stationary phase. While a polar substance would not travel as far with the mobile phase because it would have a greater attraction to the stationary phase. Question 2 Now, that depends. Are you ...
8
Normal phase Normal phase HPLC systems are similar to the flash-column chromatography that you might be familiar with. A silica stationary phase is eluted with a non-polar solvent such as hexane, or a fairly non-polar solvent mixture such as 2-propanol in hexanes. In normal phase chromatography, only organic solvents are used. In the normal phase, polar ...
8
Yes, your question contains the answer. Think about capillarity. Without the little cuts the eluant would be "sucked" and flow from both the bottom and the left ( and right) edges of the thin plate, resulting in a tilted flow that tends to move the elute faster along the vertical edges while clumping the spots to the center; decrease the resolution at the ...
7
Is this even how GC/MS results work? As cbeleites said the method you described is a proper technique but not likely to be appropriate given the information you cited. In GC/MS you should have two sets of information. The first is the GC Total Ion Chromatograph (TIC) which will have time as the x-axis and response (abundance) as the y-axis. For each ...
7
Since aspirin has a carboxylic acid group on it, it would be considered polar. Silica gel, consisting of $\ce{SiO2}$, is also polar. Since polar molecules attract other polar molecules, the aspirin molecules will tend to bind to the silica and not move up the TLC plate in a nonpolar eluent, resulting in a low $R_f$ value. When the polarity of the eluent is ...
6
Here are some similarities: There is always a mobile (e. g. Gas - GC, Liquid - HPLC; GPC) and a stationary phase (liquid; gel - GC, solid - LCs, GPC). Compounds in the sample interact different with both phases and are therefore held back stronger or lesser. This results in the accumulation of compounds that interact similar at some point in the system $-$ ...
6
Dead time ($t_{\bf M}$, also called holdup time) is the time it takes for the mobile phase (eluent) to traverse one length of the column, for a given flow rate. Because moieties in the eluent do not interact chemically with the stationary phase, $t_{\bf M}$ is therefore a lower bound on retention times for the species that do interact (chemically) with the ...
6
The stationary phase in chromatography is the one that doesn’t move according to the eyes of a macroscopic (i.e. human researcher) observer. (That complicated way to put it was to prevent anybody raising any relativism arguments.) Obviously, the paper does not move through the water but the water does through the paper. You should discard the five-ish ...
6
The key to the answer is understanding how FID works. The hydrogen flame has a minimal flame ionisation, what is needed for the low signal baseline. Incoming organic molecules from the HPGC column create in the flame a lot of ions and increase the flame electric conductivity. Using alternatives causing higher ionisation would decrease FID sensitivity that ...
6
Not sure if I can fulfill Ed's "real answer". I like the word you used for tuning solvent polarity- a knob. Modern students may understand this better. There is no theoretical restriction in chromatography to use multiple solvents or a single solvent. For example in gas chromatography, you always use a pure gas. The reason for using a mixture of solvents in ...
5
I would expect it to depend on the pH of the mobile phase. The charge of your stationary phase and whether you want the azide to elute will determine what charge you want on the azide. Given that, you should set the pH of the mobile phase either acidic or basic enough to ensure that the azide is fully charged in the direction that you want. From Wikipedia: ...
5
Why, of course you can, but most of the time one of them would be "too strong" and the other "too weak", so neither would work well for you. That's why you have to mix them to the right proportion (which depends on your system, and is found by trial and error).
5
I know it is a relative old issue but I had the same problem with nucleosides during my PhD. I tried a wide range of reagents but all of them lead to the decomposition of compounds or no reaction occurred. The only one that worked was TBAF 1 M in THF. I could not extract them because they are too polar. FCC did not work at all. Solution (at least it worked ...
5
Slightly askew of what was asked, but still relevant. TLC is not dependent on conjugation. That reasoning suggests to me You are using UV light to visualize the TLC. UV light is used because it is clean, quick and non-destructive. There are destructive visualization techniques, such as vanillin stain. In this case, after elution one dips the tlc slide ...
5
The answer is the dimer formation of the benzoic acid. This is why higher concentrations lead to more extended dimer formation, thus higher $\mathrm{R_f}$ values as the carboxyl group "gets shielded." One thing of note is that $\mathrm{R_f}$ values are not directly dependent on the polarity. They also depend for example on the size of the molecule (Oswald ...
5
Mixture of solvents is used for 2 main reasons: Elution time For given HPLC column and set of analytes, the mobile phase must have the proper degree of general polarity, what is often called "solvent strength". For polar sorbents like silica or alumina, more polar solvents have generally bigger strength with shorter elution times. Note that the solvent ...
5
Who was responsible for this naming system and how can we change it? Michael Faraday was responsible for the terms anode and cathode more than hundred years ago. All the confusion regarding the nomenclature will vanish if you do not associate electrostatic signs with these two terms. One should identify the electrode labels with the redox processes rather ...
4
On a reverse phase C18 column (which itself is hydrophobic, it is an octadecyl hydrocarbon chain bound to the support material) the more hydrophobic the molecule, the more strongly it will bind to the stationary phase [like dissolves (or in this case binds to) like], and the longer it will take to elute with an organic solvent. Below are the structures of ...
4
One approach would be to use various solvents. You would require solvents in which your various stains were soluble in. Alcohols / water would be suitable to dissolve the soda. Whereas an organic solvent would be suitable for washing away the gasoline. Though gasoline is typically colourless. Another approach would be to expose the paper to copious UV ...
4
This sounds like A being an internal standard for the determination of B. Internal standards are used to get rid of (small) multiplicative variation due to certain influences, e.g. in GC, the injected volume in optical spectroscopy, the optical path length or the illuminated volume. Sometimes the calculation part of the idea is called normalization. The ...
4
Preconcentration means to increase the concentration of a sample prior to analysis or detection. For example you can do preconcentration for an non-volatile organic sample with evaporation its solvent. An operation (process) as the result of which microcomponents are transferred from the sample of larger mass into the sample of smaller mass, so that the ...
4
To answer your three questions in series: 1) The time is a measure of how strong the interaction of a compound is with the column used. The stronger the interaction the longer the compound will stay on the column and this is therefore a way to separate two compounds. For example one molecule with a weak interaction spending a minute on the column, the other ...
4
The OH group of phenol is indeed more polarized - and a stronger acid - than the OH group of an aliphatic alcohol - maybe that is what you are asking. There are two contributions to this: (1) inductive: the sp2 hybridized carbon participating in the sigma bond to the OH group is more electron-withdrawing than the sp3-hybridized carbon of an alcohol bonded ...
4
However, in my opinion, there should be a proportionality between the concentration and the peak intensity, not the peak area. There is a proportionality between both peak area vs. concentration and peak height vs. concentration. Peak height is proportional to the instantaneous amount of analyte that is transiting the detector. Peak area is proportional ...
4
Chromatography only works when the affinity for the paper vs. affinity for the water is in a dynamic equilibrium. Suppose there are three compounds in the ink, called $\ce{I}$, $\ce{N}$, and $\ce{K}$. Lets call the spots on the paper fiber that bind ink molecules as $\ce{P}$. Then let's write the dyanmic equilibria this way: \ce{I(aq) + P <=>>...
4
TLC in alkaloid research Both analytical and preparative TLC have been used to separate and identify alkaloids from plant material. Pretreatment of TLC plates and adsorbents If you are using silica plates, it is often advantageous to wave the plate above an open bottle with aqueous ammonia or pretreat the plates with an organic solvent containing some ...
4
When you mentioned column chromatography I had to assume you meant flash chromatography or biotage, which by implication is normal phase. When people mention HPLC they are normally defaulting to reverse phase. Now, if you can get reverse phase TLC plates (C18 are available) , you could make some advances. The issue though is then getting the silica and you ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-12-15 19:24:59 | {"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": 0.6045734882354736, "perplexity": 1596.7965122850949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541309137.92/warc/CC-MAIN-20191215173718-20191215201718-00341.warc.gz"} |
http://www.math.md/en/publications/basm/issues/y2020-n1/13165/ | RO EN
On self-adjoint and invertible linear relations generated by integral equations
Authors: V. M. Bruk
Abstract
We define a minimal operator $$L_{0}$$ generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator $$L_{0}$$ to be densely defined. In general, $$L^{*}_{0}$$ is a linear relation. We give a description of $$L^{*}_{0}$$ and establish that there exists a one-to-one correspondence between relations $$\widehat{L}$$ with the property $$L_{0}\!\subset\widehat{\!L}\!\subset \!L^{*}_{0}$$ and relations $$\theta$$ ente\-ring in boundary conditions. In this case we denote $$\widehat{L}=L_{\theta}$$. We establish conditions under which linear relations $$L_{\theta}$$ and $$\theta$$ together have the following properties: a linear relation $$(l.r)$$ is self-adjoint; $$l.r$$ is closed; $$l.r$$ is invertible, i.e., the inverse relation is an operator; $$l.r$$ has the finite-dimensional kernel; $$l.r$$ is well-defined; the range of $$l.r$$ is closed; the range of $$l.r$$ is a closed subspace of the finite codimension; the range of $$l.r$$ coincides with the space wholly; $$l.r$$ is continuously invertible. We describe the spectrum of $$L_{\theta}$$ and prove that families of linear rela\-tions $$L_{\theta(\lambda)}$$ and $$\theta(\lambda)$$ are holomorphic together.
Saratov State Technical University
77, Politehnicheskaja str., Saratov 410054
Russia
E-mail:
0.20 Mb | 2021-10-19 05:24:26 | {"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": 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": 0.8487401008605957, "perplexity": 254.11538792993858}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585242.44/warc/CC-MAIN-20211019043325-20211019073325-00090.warc.gz"} |
http://merenlab.org/data/spiroplasma-pangenome/ | Microbial 'omics
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# Pangenomics, Phylogenomics, and ANI of Spiroplasma genomes
Summary
The purpose of this document is to provide a reproducible workflow for the pangenomics, phylogenomics, and ANI of 31 Spiroplasma genomes. It also includes the detailed description of how we linked all these analyses together into the following figure using anvi’o:
doi:10.6084/m9.figshare.8201852 gives access to anvi’o files to visualize, update, and edit the Spiroplasma pangenome.
If you have any questions, please feel free to leave a comment down below, send an e-mail to us, or get in touch with other anvians through Slack:
## Introduction
This is a part of the collaboration we had with Carl Yeoman and his colleagues.
Our team here at the Meren Lab processed the raw metagenomic sequences Yeoman et al generated from Wheat Stem Sawfly, reconstructed and manually curated a novel population genome that resolved to the Spiroplasma genus, and put this genome into the context of other Spiroplasma genomes that are available in public databases.
The Spiroplasma genome had a GC% content of 24.56%, 754 open reading frames, components of a single ribosomal RNA operon, and 23 tRNA encoding genes. Full annotation and the genome are available through PATRIC and the NCBI Bioproject PRJNA540284.
To put this genome into the context of other Spiroplasma genomes we used genomes available through the NCBI and included recently published genomes from the Ixodetes clade of mollicutes by Sapountzis et al (named EntAcro1 and EntAcro10).
We tested this workflow on anvi’o v5.5 and it should work with anvi’o releases v5 or later. To see the installation instructions of anvi’o please visit here.
## Metagenome assembled genomes
If you wish to fully reproduce the workflow down below, you can download the genomes we have used for this study the following way:
# create a directory for MAGs
mkdir MAGs && cd MAGs
wget http://merenlab.org/data/spiroplasma-pangenome/files/Spiroplasma_MAG.fa.gz
wget http://merenlab.org/data/spiroplasma-pangenome/files/Entomoplasmatales_EntAcro1.fa.gz
wget http://merenlab.org/data/spiroplasma-pangenome/files/Entomoplasmatales_EntAcro10.fa.gz
# unpack all and go to the upper directory
gzip -d * && cd ..
## Pangenomics, Phylogenomics, and ANI
To determine the relationship of the Spiroplasma MAG to previously published Spiroplasma genomes we first downloaded all genomes from the NCBI that resolve to genus Spiroplasma (following the protocol described in the blog post “Accessing and including NCBI genomes in ‘omics analyses in anvi’o”:
ncbi-genome-download --assembly-level complete \
bacteria \
--genus Spiroplasma \
Then we processed the resulting GenBank files to prepare them for an anvi’o analysis:
anvi-script-process-genbank-metadata -m metadata.txt \
-o Spiroplasma \
--output-fasta-txt Spiroplasma-fasta.txt \
--exclude-gene-calls-from-fasta-txt
We then removed contigs that are shorter than 1,000 nucleotides from the two genomes EntAcro1 and EntAcro10,
anvi-script-reformat-fasta ./MAGs/Entomoplasmatales_EntAcro1.fa \
-l 1000 \
-o ./MAGs/Entomoplasmatales_EntAcro1-min1K.fa
anvi-script-reformat-fasta ./MAGs/Entomoplasmatales_EntAcro10.fa \
-l 1000 \
-o ./MAGs/Entomoplasmatales_EntAcro10-min1K.fa
And created a fasta.txt (the format of which is described here) to run the anvi’o pangenomic workflow:
echo -e "Spiroplasma_MAG.fa\tpwd/MAGs/Spiroplasma_MAG.fa" \
>> Spiroplasma-fasta.txt
echo -e "Entomoplasmatales_EntAcro10\tpwd/MAGs/Entomoplasmatales_EntAcro10-min1K.fa" \
>> Spiroplasma-fasta.txt
echo -e "Entomoplasmatales_EntAcro1\tpwd/MAGs/Entomoplasmatales_EntAcro1-min1K.fa" \
>> Spiroplasma-fasta.txt
After editing this file to make sure names look human-readable (a copy of the final input file is here), we generated a configuration that looked like this (a copy of it is here),
{
"fasta_txt": "Spiroplasma-fasta.txt",
"project_name": "Spiroplasma",
"external_genomes": "external-genomes.txt"
}
Note that the external-genomes.txt file will be generated automatically by the pangenomic workflow.
And finally ran the anvi’o pangenomic workflow (with 6 cores) to get the pangenome computed:
anvi-run-workflow -w pangenomics \
-c pan-config.json \
--jobs 6 \
--resources nodes=6
This took about 15 minutes on a laptop computer. This workflow generated a new directory in our working space called 03_PAN, which included the pan database and genomes storage files.
To add a heatmap that shows the average nucleotide identity estimates across genomes, we run the following command, which uses PyANI program in the background and adds the ANI information into the pan database automatically:
anvi-compute-ani -e external-genomes.txt \
-o ANI \
-p 03_PAN/Spiroplasma-PAN.db \
-T 6
If you are using anvio v6 or later, anvi-compute-ani has been replaced by anvi-compute-genome-similarity. The above command remains the same otherwise.
To infer evolutionary associations between 31 genomes in our pangenome, we used single-copy core genes (SCGs) across all genomes for a phylogenomic analysis. To recover a FASTA file for individually aligned and concatenated SCGs specific to the pangenome, we ran the following command:
anvi-get-sequences-for-gene-clusters -p 03_PAN/Spiroplasma-PAN.db \
-g 03_PAN/Spiroplasma-GENOMES.db \
--min-num-genomes-gene-cluster-occurs 31 \
--max-num-genes-from-each-genome 1 \
--concatenate-gene-clusters \
--output-file 03_PAN/Spiroplasma-SCGs.fa
This resulted in a FASTA file, which we first cleaned up by removing nucleotide positions that were gap characters in more than 50% of the seqeunces using trimAl,
trimal -in 03_PAN/Spiroplasma-SCGs.fa \
-out 03_PAN/Spiroplasma-SCGs-clean.fa \
-gt 0.50
And ran the phylogenomic analysis using IQ-TREE with the ‘WAG’ general matrix model to infer a maximum likelihood tree:
iqtree -s 03_PAN/Spiroplasma-SCGs-clean.fa \
-nt 8 \
-m WAG \
-bb 1000
In order to organize genomes in the pangenome during the visualization step, we generated a ‘layers order’ file (the format of which is explained here), and imported it into the pan database:
# generate the file
echo -e "item_name\tdata_type\tdata_value" \
> 03_PAN/Spiroplasma-phylogenomic-layer-order.txt
# add the newick tree as an order
echo -e "SCGs_Bayesian_Tree\tnewick\tcat 03_PAN/Spiroplasma-SCGs-clean.fa.contree" \
>> 03_PAN/Spiroplasma-phylogenomic-layer-order.txt
# import the layers order file
anvi-import-misc-data -p 03_PAN/Spiroplasma-PAN.db \
-t layer_orders 03_PAN/Spiroplasma-phylogenomic-layer-order.txt
At this point all the information was in the database, so we visualized it using the following command.
anvi-display-pan -p 03_PAN/Spiroplasma-PAN.db \
-g 03_PAN/Spiroplasma-GENOMES.db
Which opened an interactive interface with the pangenome.
## Polishing the pangenome
We find it critical to properly visualize complex data, and often put the extra effort to prepare and polish our visualizations for publication.
This figure shows the initial/raw display of the pangenome:
We polished this display by displaying (1) the phylogenomic tree and (2) the ANI estimates we computed and stored in the pan database prevoiusly through the following steps:
• Click Draw after the interface first opens.
• Click Settings > Layers > Layer Groups > ANI_percentage_identity to display the ANI heatmap.
• Select all ANI layers by selecting ANI_percentage_identity from Settings > Layers > Select all layers in a group.
• Increase the minimum values for each ANI layer all at once by entering 0.7 in Settings > Layers > Edit attributes for multiple layers > Min.
• Click Settings > Layers > Redraw layer data to see changes.
• Select ‘SCG_Phylogeny’ from Settings > Layers > Order by combo box to order genomes by the phylogenomic tree
• Increase the radius of the dendrogram in the center by entering 6000 in Settings > Main > Show Additional Settings > Dendrogram > Radius
• Increase the height of the phylogenomic tree on the right-top by entering 2000 in Settings > Layers > Tree/Dendrogram > Height
• Increase the size and selections layer by entering 400 in Settings > Main > Additional Settings > Selections > Height
• Click Settings > Main > Show Additional Settings > Custom margins to enable custom margins
• Use Settings > Main > Layers to increase genome / group distances using the margin column
• Command-right-click to the branch of singletons and click Collapse from the menu
• Reduce the opacity of layer bacgrounds by setting 0.15 to Settings > Main > Show Additional Settings > Layers > Background opacity
• Increase the maximum font size by entering 240 in Settings > Main > Show Additional Settings > Layer Labels > Max. Font Size
• Click Draw to see changes
Which gave us this display:
We then used the Save SVG button that is located in the Settings panel (bottom-right) to download the SVG file, and further polished labels in Inkscape to acquire this final figure:
doi:10.6084/m9.figshare.8201852 gives access to anvi’o files to visualize, update, and edit the Spiroplasma pangenome.
If you have any questions, please feel free to leave a comment down below, send an e-mail to us, or get in touch with other anvians through Slack: | 2020-04-03 03:42:54 | {"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": 0.2334602028131485, "perplexity": 13186.037101102647}, "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/1585370510287.30/warc/CC-MAIN-20200403030659-20200403060659-00548.warc.gz"} |
https://cs.stackexchange.com/questions/74610/propositional-logic-in-an-sma-algorithm | # Propositional logic in an SMA* algorithm
I'm trying to implement an SMA* graph search algorithm that I found in a paper here(Rong Zhou, Eric A. Hansen: Memory-Bounded A* Graph Search. FLAIRS-02 Proceedings 203-209) and I would like to clarify my understanding of the pseudo-code found in the appendices, specifically the set builder notation used:
What I understand this to mean is: make a set called B from all the elements of beta, given that beta and x are elements of OPEN, and only include the elements of beta where F(beta)<=F(x)
To test my understanding I have written this pseudo-code which I believe builds the necessary set.
x = {OPEN}
b = {OPEN}
B = {}
for i = 0; i<sizeof(b); i++
for j = 0; j<sizeof(x); j++
if F(b[i]) <= F(x[j])
B.append(b[i])
My question is:
Do my plain English sentence and pseudo-code demonstrate that I have interpreted the set builder notation correctly?
• What exactly is your question? "Please debug my code for me" is off-topic here and everywhere else on the Stack Exchange network. – David Richerby Apr 27 '17 at 17:04
• Welcome to CS.SE! Do you know what set comprehension notation $\{\beta \mid \cdots \}$ means? If not, it might help you to study that. Can you ask a more specific question than "I don't understand those two lines"? What parts of those lines do you understand, and what parts don't you? Can you infer anything from the text of the paper? It appears they're using nonstandard notation in at least one place: I have no idea what $\forall x \implies \cdots$ is supposed to mean. Can you edit the question to give a full reference for the paper, including title, authors, and where it was published? – D.W. Apr 27 '17 at 17:39
• The paper says that SMAG* came from an earlier paper, "Kaindl and Khorsand (1994)", and is based on SMA*, from "Russell (1992)". You should be reading those earlier papers! Presumably the paper that introduced SMAG* should have a clear description of how it works. Why don't you read those papers, then if there's something you're still unclear on, come back and edit your question to ask a more specific question and to address the feedback you've gotten so far? – D.W. Apr 27 '17 at 17:41
• Thanks for the feedback. I can see now that I included too much information in the hope of providing some context, but that it has confused the question. I had not heard of set comprehension notation before, but after reading the wikipedia article it seems to be the missing puzzle piece in my understanding. I'll now edit the question, and hopefully it'll be clearer what I am asking. – moremilopls Apr 28 '17 at 5:18
• I did look for those other referenced papers but I couldn't find them on google scholar or my University's database. I think I understand how the algorithm works and I can do it on paper, but my background is in engineering, so the logic and the code are confusing me – moremilopls Apr 28 '17 at 5:50 | 2019-08-20 01:46:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5986047387123108, "perplexity": 595.1800406762449}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315174.57/warc/CC-MAIN-20190820003509-20190820025509-00190.warc.gz"} |
http://encyclopedia.kids.net.au/page/mo/Morgoth | ## Encyclopedia > Morgoth
Article Content
# Morgoth
A fictional character from J. R. R. Tolkien's universe, Middle-earth, Morgoth Bauglir (Morgoth means 'Black Enemy', Bauglir is 'The Constrainer'), originally Melkor, was to begin with the most powerful of the Valar, and he contended with Eru himself in the Music of the Ainur. If he must be assigned a dominion (like Ulmo and water, Varda and stars, etc.), he would be the Vala of Knowledge (Tolkien was always skeptical of science).
Warning: Wikipedia contains spoilers
History Melkor ("he who arises in might") was jealous of Eru already before Arda was created, and wanted to be king of other wills himself. When Eru revealed the results of their song to the Ainur (Arda, as it was), Melkor was one of the first to descend into it, mainly from this desire.
Melkor fought with the other Valar for a long time for the control of Arda. It appears that while he was the single most powerful Vala, he was not able to stand up to the combined forces of the lawful Valar. However, these were busy ordering the new world, creating the mountains, the sky, the earth, the waters - so the fight was not even.
He was held at bay by the aid of Tulkas, who came late to the party, and the Valar ordered Arda to their pleasing. Melkor was only biding his time, however, so when the Valar finally rested, he and his followers (downfallen Ainur, like Sauron and the later Balrogs) attacked their dwellings and destroyed their Two Lamps (precursors to the Two Trees and the sun and the moon).
The Valar then retired to Valinor in the West, and Melkor held dominion over Middle-earth from his fortress of Utumno in the North. His reign ended, however, after Eru awoke the Elves in the East of Middle-earth, and the Valar resolved to rescue them from him. They made immediate and devastating war on him, and he was brought to Valinor in chains to serve a term in the Halls of Mandos for three Ages[?].
It was after this sentence was ended, and he used his newfound freedom to corrupt the Noldor (a people of the Elves who had relocated to Valinor) and steal the Silmarils, that Fëanor of the Noldor first named him Morgoth, "dark destroyer of the world". With the aid of Ungoliant he also managed to destroy the Two Trees and bring darkness to Valinor, before he fled.
Back in Middle-earth, he took up his reign in the North again, this time in Angband, which had not been destroyed as thoroughly by the Valar as Utumno had. This time however, there were Elves and after a time also Men and Dwarves who resisted him, so he was not the sole ruler of Middle-earth.
However, after building his strength, he soon dispatched his enemies, one by one, through violence or treachery, until only isolated pockets of resistance remained (such as the strongholds of the Dwarves and the Havens of Mithlond[?] (check this w sources). His mastery was again complete.
But it was not to last. This time the part Elf, part Maia, part Man Eärendil managed to plead with the Valar until they agreed to send an army to vanquish Morgoth. This time, the Valar themselves did not go, but many of the Maiar went, and most of the Calaquendi (Elves living in Valinor) ferried over into Middle-earth by the ships of the Teleri (another people of the Elves).
This time, he was utterly defeated, and his punishment was final. He was shut outside the gates of the world forever (or at least until the rumored Final Battle when he supposedly returns to fight a united army of Valar, Maiar, Elves and Men).
One legend of Middle-earth suggests that in the Last Battle, Morgoth will be slain by Turin Turambar, who will return from the dead to defeat him. This legend was included in one of Tolkien's many notes on The Silmarillion, and it was published in The Shaping of Middle-earth (Book 4 of the History of Middle-earth series).
Characteristics Appearance, powers, etc.
Politics Alliances, betrayals, lies.
Accomplishments Loss of the Silmarils, creation of the Orcs, fear of night and death in Men, Sauron, Balrogs, Dragons, etc.
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Quadratic formula ... is equivalent to $x^2+\frac{b}{a}x=-\frac{c}{a}.$ The equation is now in a form in which we can conveniently complete ... | 2021-01-25 12:05:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37990933656692505, "perplexity": 7146.704031590093}, "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/1610703565541.79/warc/CC-MAIN-20210125092143-20210125122143-00441.warc.gz"} |
https://swmath.org/?term=stabilized%20methods | • # SDPT3
• Referenced in 703 articles [sw04009]
• infeasible primal-dual predictor-corrector path-following method, with either ... Various techniques to improve the efficiency and stability of the algorithm are incorporated. For example ... semidefinite cones are calculated via the Lanczos method. Numerical experiments show that this general purpose...
• # SUPG
• Referenced in 31 articles [sw08373]
• finite element method for the computations. Without proper numerical stabilization, computation of coupled fluid mechanics ... computational challenges, we propose a stabilized finite element method based on the streamline-upwind/Petrov-Galerkin ... techniques. To demonstrate the effectiveness of the stabilized formulation, we present test computations with...
• # fpc
• Referenced in 27 articles [sw07105]
• package fpc: Flexible procedures for clustering. Various methods for clustering and cluster validation. Fixed point ... corrected Rand index. Cluster-wise cluster stability assessment. Methods for estimation of the number ... prediction strength, Fang and Wang’s bootstrap stability. Gaussian/multinomial mixture fitting for mixed continuous/categorical variables ... DBSCAN clustering. Interface functions for many clustering methods implemented in R, including estimating the number...
• # FODE
• Referenced in 296 articles [sw08377]
• combining the predictor-corrector approach with the method of lines, the algorithm is designed ... corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing...
• # SERK2
• Referenced in 12 articles [sw10426]
• SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems. Traditionally, explicit ... ODEs with very large dimension. Stabilized Runge-Kutta methods (also called Runge-Kutta-Chebyshev methods ... Runge-Kutta methods are explicit methods with extended stability domains, usually along the negative real ... stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs...
• # BiCGstab
• Referenced in 128 articles [sw04022]
• matrix are studied. The basis of these method forms various investigations of the BiCG part ... BiCG residue and for improving the numerical stability. Both parts are studied in detail...
• # ImplicitLNLMethods
• Referenced in 10 articles [sw23184]
• Implicit and implicit-explicit strong stability preserving Runge-Kutta methods with high linear order. Strong ... search for high order strong stability preserving time-stepping methods with high order and large ... also find implicit methods with large linear stability regions that pair with known explicit...
• # XTOR
• Referenced in 21 articles [sw03221]
• numerical method is discussed with particular emphasis on critical issues leading to numerical stability ... some neoclassical effects. The time advance method used in XTOR is unconditionally stable for linear ... nonlinear stability criteria. The robustness of the method is illustrated by some numerically difficult simulations ... like geometry about its nonlinear stability threshold...
• # MR and LTV Synthesis Tools
• Referenced in 36 articles [sw05190]
• discrete-time LTV systems using LMI synthesis methods System type conversion (i.e. multi-rate ... truncation LTV stability and stabilizability through eigenvalue and LMI solution methods Discrete system data structure...
• # Nek5000
• Referenced in 142 articles [sw08064]
• wavemaker. We concentrate on global linear stability analysis, which considers the linearised Navier--Stokes equations ... eigenvalue problems are solved using matrix-free methods adopting the time-stepping Arnoldi approach...
• # BlackHat
• Referenced in 73 articles [sw10450]
• make use of recently developed on-shell methods for evaluating coefficients of loop integrals, introducing ... means of improving efficiency and numerical stability. We illustrate the numerical stability of our approach...
• # SERK2v3
• Referenced in 8 articles [sw14473]
• very efficient for specific problems. Stabilized explicit Runge-Kutta methods (SERK ... class of explicit methods with extended stability domains along the negative real axis ... evaluate the function $s$ times, but the stability region is $O(s^2)$. Hence ... method of lines (MOL) discretizations of parabolic multi-dimensional PDEs. Additionally, the stability domain...
• # SSPTSmethods
• Referenced in 7 articles [sw38981]
• nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed ... hyperbolic PDEs, and the strong stability properties of these methods are of interest ... explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations ... call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that...
• # S-ROCK
• Referenced in 42 articles [sw11792]
• stiff stochastic differential equations (SDEs). These methods, called S-ROCK (for stochastic orthogonal Runge-Kutta ... strong order 1 and possess large stability domains in the mean-square sense. For mean ... much more efficient than the standard explicit methods proposed so far for stochastic problems...
• # TIGRESS
• Referenced in 9 articles [sw23826]
• TIGRESS: Trustful Inference of Gene REgulation using Stability Selection. BACKGROUND: Inferring the structure of gene ... popular feature selection method, least angle regression (LARS) combined with stability selection, for that purpose ... stability selection, which improves the performance of feature selection with LARS. The resulting method, which ... REgulation with Stability Selection), was ranked among the top GRN inference methods in the DREAM5... | 2022-07-05 22:29:16 | {"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": 0.17090286314487457, "perplexity": 3220.8259048394575}, "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/1656104628307.87/warc/CC-MAIN-20220705205356-20220705235356-00595.warc.gz"} |
https://homework.zookal.com/questions-and-answers/find-the-linear-approximation-of-fx--x14-at-x-945169186 | 1. Math
2. Calculus
3. find the linear approximation of fx x14 at x...
# Question: find the linear approximation of fx x14 at x...
###### Question details
Find the linear approximation of f(x) = x1/4 at x = 625 and use this to get an estimate for 4√610. | 2021-03-01 10:24:43 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9195887446403503, "perplexity": 3010.3685086539635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362481.49/warc/CC-MAIN-20210301090526-20210301120526-00176.warc.gz"} |
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Statistical_Thinking_for_the_21st_Century_(Poldrack)/29%3A_Comparing_Means_in_R/29.05%3A_Analysis_of_Variance_(Section_28.6.1) | # 29.5: Analysis of Variance (Section 28.6.1)
Often we want to compare several different means, to determine whether any of them are different from the others. In this case, let’s look at the data from NHANES to determine whether Marital Status is related to sleep quality. First we clean up the data:
NHANES_sleep_marriage <-
dplyr::select(SleepHrsNight, MaritalStatus, Age) %>%
drop_na()
In this case we are going to treat the full NHANES dataset as our sample, with the goal of generalizing to the entire US population (from which the NHANES dataset is mean to be a representative sample). First let’s look at the distribution of the different values of the MaritalStatus variable:
NHANES_sleep_marriage %>%
group_by(MaritalStatus) %>%
summarize(n=n()) %>%
kable()
MaritalStatus n
Divorced 437
LivePartner 370
Married 2434
NeverMarried 889
Separated 134
Widowed 329
There are reasonable numbers of most of these categories, but let’s remove the Separated category since it has relatively few members:
NHANES_sleep_marriage <-
NHANES_sleep_marriage %>%
dplyr::filter(MaritalStatus!="Separated")
Now let’s use lm() to perform an analysis of variance. Since we also suspect that Age is related to the amount of sleep, we will also include Age in the model.
lm_sleep_marriage <- lm(SleepHrsNight ~ MaritalStatus + Age,
data=NHANES_sleep_marriage)
summary(lm_sleep_marriage)
##
## Call:
## lm(formula = SleepHrsNight ~ MaritalStatus + Age, data = NHANES_sleep_marriage)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.016 -0.880 0.107 1.082 5.282
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.51758 0.09802 66.49 < 2e-16 ***
## MaritalStatusLivePartner 0.14373 0.09869 1.46 0.14536
## MaritalStatusMarried 0.23494 0.07094 3.31 0.00093 ***
## MaritalStatusNeverMarried 0.25172 0.08404 3.00 0.00276 **
## MaritalStatusWidowed 0.26304 0.10327 2.55 0.01090 *
## Age 0.00318 0.00141 2.25 0.02464 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.4 on 4453 degrees of freedom
## Multiple R-squared: 0.00458, Adjusted R-squared: 0.00347
## F-statistic: 4.1 on 5 and 4453 DF, p-value: 0.00102
This tells us that there is a highly significant effect of marital status (based on the F test), though it accounts for a very small amount of variance (less than 1%).
It’s also useful to look in more detail at which groups differ from which others, which we can do by examining the estimated marginal means for each group using the emmeans() function.
# compute the differences between each of the means
leastsquare <- emmeans(lm_sleep_marriage,
pairwise ~ MaritalStatus,
# display the results by grouping using letters
CLD(leastsquare\$emmeans,
alpha=.05,
Letters=letters)
## MaritalStatus emmean SE df lower.CL upper.CL .group
## Divorced 6.7 0.066 4453 6.5 6.8 a
## LivePartner 6.8 0.073 4453 6.7 7.0 ab
## Married 6.9 0.028 4453 6.8 7.0 b
## NeverMarried 6.9 0.050 4453 6.8 7.0 b
## Widowed 6.9 0.082 4453 6.8 7.1 ab
##
## Confidence level used: 0.95
## P value adjustment: tukey method for comparing a family of 5 estimates
## significance level used: alpha = 0.05
The letters in the group column tell us which individual conditions differ from which others; any pair of conditions that don’t share a group identifier (in this case, the letters a and b) are significantly different from one another. In this case, we see that Divorced people sleep less than Married or Widowed individuals; no other pairs differ significantly.
## 29.5.1 Repeated measures analysis of variance
The standard analysis of variance assumes that the observations are independent, which should be true for different people in the NHANES dataset, but may not be true if the data are based on repeated measures of the same individual. For example, the NHANES dataset involves three measurements of blood pressure for each individual. If we want to test whether there are any differences between those, then we would need to use a repeated measures analysis of variance. We can do this using lmer() as we did above. First, we need to create a “long” version of the dataset.
NHANES_bp_all <- NHANES_adult %>%
drop_na(BPSys1,BPSys2,BPSys3) %>%
dplyr::select(BPSys1,BPSys2,BPSys3, ID) %>%
gather(test, BPsys, -ID)
Then we fit a model that includes a separate intercept for each individual.
repeated_lmer <-lmer(BPsys ~ test + (1|ID), data=NHANES_bp_all)
summary(repeated_lmer)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: BPsys ~ test + (1 | ID)
## Data: NHANES_bp_all
##
## REML criterion at convergence: 89301
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -4.547 -0.513 -0.005 0.495 4.134
##
## Random effects:
## Groups Name Variance Std.Dev.
## ID (Intercept) 280.9 16.8
## Residual 16.8 4.1
## Number of obs: 12810, groups: ID, 4270
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 122.0037 0.2641 4605.7049 462.0 <2e-16 ***
## testBPSys2 -0.9283 0.0887 8538.0000 -10.5 <2e-16 ***
## testBPSys3 -1.6215 0.0887 8538.0000 -18.3 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) tsBPS2
## testBPSys2 -0.168
## testBPSys3 -0.168 0.500
This shows us that the second and third tests are significant different from the first test (which was automatically assigned as the baseline by lmer()). We might also want to know whether there is an overall effect of test. We can determine this by comparing the fit of our model to the fit of a model that does not include the test variable, which we will fit here. We then compare the models using the anova() function, which performs an F test to compare the two models.
repeated_lmer_baseline <-lmer(BPsys ~ (1|ID), data=NHANES_bp_all)
anova(repeated_lmer,repeated_lmer_baseline)
## Data: NHANES_bp_all
## Models:
## repeated_lmer_baseline: BPsys ~ (1 | ID)
## repeated_lmer: BPsys ~ test + (1 | ID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## repeated_lmer_baseline 3 89630 89652 -44812 89624
## repeated_lmer 5 89304 89341 -44647 89294 330 2 <2e-16
##
## repeated_lmer_baseline
## repeated_lmer ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
This shows that blood pressure differs significantly across the three tests. | 2021-10-27 07:37:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6024543046951294, "perplexity": 4170.637491516224}, "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-2021-43/segments/1634323588102.27/warc/CC-MAIN-20211027053727-20211027083727-00546.warc.gz"} |
https://east-centricarch.eu/en/how-do-you-solve-this-system-of-equations-using-substitution-3x-2y4-4x-3y7.2719703.html | micahsmith507
27
# How do you solve this system of equations using substitution 3x-2y=4 4x-3y=7
$\begin{cases} 3x-2y=4 \\ 4x-3y=7 \end{cases} \\ \begin{cases} -2y=4-3x \\ 4x-3y=7\end{cases} \\ \begin{cases} y= \frac{4-3x}{-2} \\ 4x-3y=7 \end{cases} \\ \begin{cases}y=\frac{4-3x}{-2}\\ 4x-3(\frac{4-3x}{-2})=7\end{cases} \\ \begin{cases}4x+6-4,5x=7\\y=\frac{4-3x}{-2} \end{cases} \\ \begin{cases}6- \frac{1}{2}x=7\\ y=\frac{4-3x}{-2} \end{cases} \\ \begin{cases} - \frac{1}{2}x=1\\y=\frac{4-3x}{-2} \end{cases} \\ \begin{cases}x=-2 \\y=\frac{4-3x}{-2} \end{cases}$ $\begin{cases}x=-2 \\ y= \frac{4-3(-2)}{-2} \end{cases} \\ \begin{cases}x=-2 \\ y=-5 \end{cases}$ | 2022-08-09 20:44:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.929729700088501, "perplexity": 1014.3910821750517}, "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/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00394.warc.gz"} |
http://rosalind.info/problems/dna/ | # Counting DNA Nucleotides solved by 10666
July 2, 2012, midnight by Rosalind Team
Topics: String Algorithms
## A Rapid Introduction to Molecular Biology
Figure 1. A 1900 drawing by Edmund Wilson of onion cells at different stages of mitosis. The sample has been dyed, causing chromatin in the cells (which soaks up the dye) to appear in greater contrast to the rest of the cell.
Figure 2. A sketch of DNA's primary structure.
Making up all living material, the cell is considered to be the building block of life. The nucleus, a component of most eukaryotic cells, was identified as the hub of cellular activity 150 years ago. Viewed under a light microscope, the nucleus appears only as a darker region of the cell, but as we increase magnification, we find that the nucleus is densely filled with a stew of macromolecules called chromatin. During mitosis (eukaryotic cell division), most of the chromatin condenses into long, thin strings called chromosomes. See Figure 1 for a figure of cells in different stages of mitosis.
One class of the macromolecules contained in chromatin are called nucleic acids. Early 20th century research into the chemical identity of nucleic acids culminated with the conclusion that nucleic acids are polymers, or repeating chains of smaller, similarly structured molecules known as monomers. Because of their tendency to be long and thin, nucleic acid polymers are commonly called strands.
The nucleic acid monomer is called a nucleotide and is used as a unit of strand length (abbreviated to nt). Each nucleotide is formed of three parts: a sugar molecule, a negatively charged ion called a phosphate, and a compound called a nucleobase ("base" for short). Polymerization is achieved as the sugar of one nucleotide bonds to the phosphate of the next nucleotide in the chain, which forms a sugar-phosphate backbone for the nucleic acid strand. A key point is that the nucleotides of a specific type of nucleic acid always contain the same sugar and phosphate molecules, and they differ only in their choice of base. Thus, one strand of a nucleic acid can be differentiated from another based solely on the order of its bases; this ordering of bases defines a nucleic acid's primary structure.
For example, Figure 2 shows a strand of deoxyribose nucleic acid (DNA), in which the sugar is called deoxyribose, and the only four choices for nucleobases are molecules called adenine (A), cytosine (C), guanine (G), and thymine (T).
For reasons we will soon see, DNA is found in all living organisms on Earth, including bacteria; it is even found in many viruses (which are often considered to be nonliving). Because of its importance, we reserve the term genome to refer to the sum total of the DNA contained in an organism's chromosomes.
## Problem
A string is simply an ordered collection of symbols selected from some alphabet and formed into a word; the length of a string is the number of symbols that it contains.
An example of a length 21 DNA string (whose alphabet contains the symbols 'A', 'C', 'G', and 'T') is "ATGCTTCAGAAAGGTCTTACG."
Given: A DNA string $s$ of length at most 1000 nt.
Return: Four integers (separated by spaces) counting the respective number of times that the symbols 'A', 'C', 'G', and 'T' occur in $s$.
## Sample Dataset
AGCTTTTCATTCTGACTGCAACGGGCAATATGTCTCTGTGTGGATTAAAAAAAGAGTGTCTGATAGCAGC
## Sample Output
20 12 17 21 | 2014-04-23 14:40:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "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": 0.29881608486175537, "perplexity": 1942.9136626674888}, "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/1398223202774.3/warc/CC-MAIN-20140423032002-00281-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/975953/the-limit-of-a-solution-of-the-logistic-equation-as-time-tends-to-infinity | # The limit of a solution of the logistic equation as time tends to infinity
$$\frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$
What value does $P$ approach as $t$ gets large, ie. as $t \to\infty$.
How do I solve this? Is the idea to this question to first rearrange the equation so that there is a constant on the Right Hand Side, then you can integrate both sides with respect to $dt$?
Thanks
• I think one should learn the easy way to find $\lim\limits_{t\to\infty} P$ without actually solving the differential equation. Thus my posted answer does not separate and integrate. One should also learn how to separate and integrate, but it would be wrong to think that's the simplest way to answer the question posed here. ${}\qquad{}$ – Michael Hardy Oct 16 '14 at 2:37
$$\frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$
Two posted answers so far have answered this by solving the differential equation.
If there is any reason for anyone to learn about differential equations, they should learn how to answer this question without solving this equation.
Notice that $P$ is intially between $2$ and $4$. That means $P(4-P)$ is positive. That means $P$ is increasing. As long as $P$ is between $2$ and $4$, then $P$ is increasing. But notice that if $P$ is more than $4$, then $P$ is decreasing. And when $P$ is exactly $4$, then $dP/dt$ is zero, so $P$ is constant.
So:
• If $P$ is less than $4$ (but more than $0$) then $P$ gets bigger;
• If $P$ is more than $4$ then $P$ gets smaller;
• If $P$ should reach exactly $4$ then $P$ remains constant.
That tells you what $P$ approaches as $t\to\infty$.
• If you read my post carefully, then you should see I started out with the "quick and dirty way", which is the same as yours. But yours is inaccurate. You cannot say "When $P$ is more than $4$" because $P$ never actually reaches $4$! And $P$ is always increasing. – Deepak Oct 16 '14 at 2:35
• @Deepak : Would you feel better if I had said if $P$ is ever more than $4$? It was a hypothetical to be used in a context in which one has not actually fully solved the equation. – Michael Hardy Oct 16 '14 at 2:39
• . . . . ok, I've changed it to "if". ${}\qquad{}$ – Michael Hardy Oct 16 '14 at 2:40
The rigorous way to solve this would be to solve the differential equation, get an expression for $P(t)$ and then take the limit $\lim_{t \to \infty}P(t)$.
But if you start out by assuming the limit exists, then at the limit, $\frac{dP}{dt}=0$ (i.e. loosely speaking, "P stops changing").
So solve $3P(4-P) = 0$ giving $P=4$ as the limiting value (ignore the other root $P=0$ because $P(0) = 2$ and $P$ is always increasing).
Note that what I did is a "quick and dirty" solution. If you're asked this in homework, a test or an examination, you should start by solving the differential equation. It's a separable first order differential equation.
You start by separating the variables:
$$\frac{dP}{3P(4-P)} = dt$$
Then integrate both sides. I find it easier to set the initial conditions as the lower bound of a definite integral.
$$\int_2^{P(t)}\frac{dP}{3P(4-P)} = \int_0^t 1dt$$
The integrand on the left hand side can be resolved quickly with partial fractions, and the rest is algebra.
• Definitely the hard way!! – Michael Hardy Oct 16 '14 at 2:28
• @MichaelHardy I started out by giving what I consider the "easy way" before going on to the rigorous "hard way". Is there an even easier way than my "easy way"? – Deepak Oct 16 '14 at 2:32
• You show that if $P$ reaches $4$ then $P$ remains constant, but you are less than explicit in saying what happens when $P\ne 4$. At any rate I disagree with the proposition that the ONLY rigorous way to do this is to solve the equation. Certainly you would solve the equation if you want to know how fast $P$ approaches $4$, but the argument that stops short of solving the equation can be stated rigorously. – Michael Hardy Oct 16 '14 at 2:34
• @MichaelHardy I pointed out a major semantic error in your own answer above. And the only rigorous way to prove that convergence does, in fact, happen as expected is to solve the equation and take the limit. – Deepak Oct 16 '14 at 2:37
• I've changed "when" to "if" in my answer, so I hope you'll be happier with that. – Michael Hardy Oct 16 '14 at 2:41
Rearrange first, as you say.
$$\frac{dP}{P(4-P)} = 3\, dt\\ \because \frac1{P(4-P)} = \frac1{4}\left( \frac1{P} + \frac1{4-P} \right)\\ \therefore 3\int_0^t dt = \int_{P(0)}^{P(t)} \frac{dP}{P(4-P)} = \frac1{4}\int_{2}^{P} \left( \frac1{P} + \frac1{4-P} \right) dP \\ 3t = \frac1{4}\left( \ln \left(\frac{P}{2}\right) - \ln \left(\frac{4-P}{2}\right)\right) = \frac1{4}\ln\frac{P}{4-P} \\ 12t = \ln \frac{P}{4-P} \\$$ Then rearrange this and we get $$P = \frac{4}{1+e^{-12t}}$$
Now as $t\to \infty, e^{-t} \to 0$, so what can you say about P?
• Definitely the hard way!! – Michael Hardy Oct 16 '14 at 2:26 | 2019-09-23 20:37:21 | {"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": 0.9237396717071533, "perplexity": 206.4877863691888}, "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/1568514578201.99/warc/CC-MAIN-20190923193125-20190923215125-00020.warc.gz"} |
http://math.stackexchange.com/questions/107224/what-is-the-reason-to-use-hypergeometric-functions | # What is the reason to use hypergeometric functions?
I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert z \bigg] = \frac{1}{1-z}$, but in case it doesn't, why bother rewriting it, e.g. $\sum\limits_{k \leq m}\binom{n}{k} = \sum\limits_{k \geq 0} \binom{n}{m-k} = \binom{n}{m} {}_2 F_1 \bigg[{{-m\; 1}\atop{n-m+1}} \vert 1 \bigg]$
since it doesn't yield a closed form or approximation of it?
-
Closed form is in the eye of the beholder. – André Nicolas Feb 8 '12 at 22:12
Let's take your second sum as an example; it turns out that knowing the $-m$ numerator parameter is a negative integer is crucial, since it is a known property of hypergeometric functions that they degenerate to polynomials whenever one or both of their numerator parameters are negative. – J. M. Feb 8 '12 at 22:37
## 1 Answer
Hmm, I don't know... because the Gaussian hypergeometric function satisfies a very convenient set of identities?
Also, what André said in the comments. Gauss and others spent a fair bit of time unraveling identities satisfied by this function, and it'd be a damn shame not to make use of our predecessors' effort.
-
Re: approximation, you honestly don't think that all hypergeometric functions are always evaluated through the series, do you? – J. M. Feb 8 '12 at 22:27
All I'm thinking is that hypergeometric transformation might help in finding asymptotic lower and upper bounds on the sum – user19821 Feb 9 '12 at 0:50
Right; since the asymptotic behavior of hypergeometric functions are well-studied (did you have a look at the links I gave?), among other things, being able to represent something in terms of a hypergeometric function is a great boon. Hypergeometric functions also satisfy well-studied difference and differential equations, and those viewpoints are also a gold mine for teasing out useful properties. – J. M. Feb 9 '12 at 0:58
Thanks. Can you recommend some articles where they are applied in this way? – user19821 Feb 9 '12 at 4:39
If you look through the DLMF, the bibliography there should get you started... – J. M. Feb 9 '12 at 4:44 | 2016-05-27 10:57:52 | {"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": 0.8251012563705444, "perplexity": 398.0235942539642}, "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-2016-22/segments/1464049276567.28/warc/CC-MAIN-20160524002116-00056-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3274494/find-the-limit-of-7n-9n1-n-when-n-goes-to-infty | # Find the limit of $(7^n + 9^n)^{(1/n)}$ when n goes to $\infty$
I tried this using L' Hopitals rule. But I always get a limit that is not defined.
$$\lim_{n \to\infty}(7^n +9^n)^{(1/n)}$$
Let $$y = (7^n +9^n)^{(1/n)}$$ then take the $$log$$ of both side, $$ln(y)= ln((7^n +9^n)^{(1/n)})$$ $$ln(y)= (1/n) * ln(7^n +9^n)$$ $$ln(y) = (ln(7^n +9^n))/n$$
then we find the limit of both side when n goes to $$\infty$$ $$\lim_{n \to\infty}(ln(y))= \lim_{n \to\infty}((ln(7^n +9^n))/n)$$
we can see the limit of numerator and the denominator is infinity. So, we apply L'Hopitals rule,
$$\lim_{n\to\infty}(ln(y)) = (1/(7^n +9^n))*(7^n*ln7+ 9^n*ln9) = (7^n*ln7+ 9^n*ln9)/(7^n +9^n)$$
Again this limit is not defined as we get $$\infty$$ by $$\infty$$ If I apply L'Hopitals again and the limit of the result will be the same.
Can someone help me to find the answer by L'Hopitals or an Alternative way.
This may be a bit more lengthy, but it is a good general method which is not so widespread.
Using the Cauchy-D'Alembert criterion(see https://math.stackexchange.com/a/3159844/629594), we have
$$\lim \limits _{n\to \infty}\sqrt[n]{7^n+9^n}=\lim \limits _{n\to \infty}\frac{7^{n+1}+9^{n+1}}{7^n+9^n}=\lim \limits _{n\to \infty}\frac{7\cdot\left(\frac{7}{9}\right)^n+9}{\left(\frac{7}{9}\right)^n+1}=9$$
$$\sqrt[n]{9^n+7^n}=9\sqrt[n]{1+\left(\frac{7}{9}\right)^n}\rightarrow9.$$
$$9^{n}<7^{n}+9^{n}<9^{n}+9^{n}=2\cdot 9^{n}$$
$$9<\bigg(7^{n}+9^{n}\bigg)^{\frac{1}{n}}<\bigg(2\cdot 9^{n}\bigg)^{\frac{1}{n}}$$
Using Squeeze Theorem
$$\lim_{n\rightarrow \infty}\bigg(7^{n}+9^{n}\bigg)^{\frac{1}{n}}=9.$$
In general, for $$a>0$$ and $$b>0$$, we have $$\lim_{n\to \infty}\sqrt[n]{a^n+b^n}=\max(a,b)$$ In this case, the given limit is equal to $$9$$ (just as they answered you above). | 2021-06-14 05:19:02 | {"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": 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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.986094057559967, "perplexity": 179.45935867588324}, "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/1623487611445.13/warc/CC-MAIN-20210614043833-20210614073833-00006.warc.gz"} |
https://dougraffle.com/111/week1/ch5/ch5.html | 5/20/2015
## Overview
In this chapter, we will discuss:
• Standardizing Values with Z-Scores
• Shifting and Scaling Distributions
• Normal Models
• Finding Normal Percentiles
• Assessing Normality
## The Data
For this chapter, we will compare the SAT and ACT scores of 10,000 students who took both exams.
## ID SAT ACT
## 1 338861 1417.255 28.87883
## 2 801180 1275.967 21.93852
## 3 252174 1509.345 20.92183
## 4 858407 1413.393 24.02147
## 5 819949 1606.584 24.25653
## 6 970656 1593.724 19.32590
## Comparing Distributions
Both variables are unimodal and symmetric, but looking at the values we see that they both have very different scales.
We may want to compare the distributions. Recall that we have two main ways of doing this for numeric variables:
• Side-by-Side Histograms
• Side-by-Side Boxplots
Because the variables have very different scales, comparing them may be difficult
## Numerical Summaries
Obviously, we can tell from the plots that the distributions are very different. Let's look at our quantitative measures to see just how different:
## Min. 1st Qu. Median 3rd Qu. Max.
## SAT 568.2000 1329.00 1497.0 1666.00 2439.0
## ACT 0.4642 17.56 20.9 24.32 41.4
Because the distributions are symmetric, we can also look at the sample means and standard deviations:
## Mean St.Dev
## SAT 1498.168 248.9613
## ACT 20.92007 5.008556
For simplicity's sake, let's call the means 1500 and 21 and the standard deviations 250 and 5, respectively.
## Standardizing with Z-Scores
What we've seen here is the problems that occur when we compare distributions with different scales. We typically deal with this by standardizing our variables with z-scores.
For a given distribution, we define as z-score as:
• $$z = \frac{x - \bar{x}}{s}$$
For our each of our students, we can find the standardized SAT and ACT scores as:
• $$z_{SAT} = \frac{SAT - \overline{SAT}}{s_{SAT}} = \frac{SAT - 1500}{250}$$
• $$z_{ACT} = \frac{ACT - \overline{ACT}}{s_{ACT}} = \frac{ACT - 21}{5}$$
## What did Standardizing Do?
To see exactly what happened, let's look at our numerical summaries.
Five Number Summaries:
## Min. 1st Qu. Median 3rd Qu. Max.
## Z.SAT -3.735 -0.6798 -0.003724 0.6724 3.778
## Z.ACT -4.084 -0.6704 -0.004263 0.6791 4.088
Means and Standard Deviations:
## Mean St.Dev
## Z.SAT 0 1
## Z.ACT 0 1
We can see that the five number summaries are almost identical, and both variables have a mean of 1 and standard deviation of 0.
## Why Standardizing?
For any symmetric distribution, standardizing by finding the z-scores:
• Forces the mean to 0 and the standard deviation to 1
This allows us:
• To compare two (or more) distributions on the same scale
• Directly compare observations from each distribution in terms of their z-scores
Z-Scores have other uses that we'll discuss later
## What is a Z-Score?
$z = \frac{x - \bar{x}}{s}$
Breaking it down:
• $$x - \bar{x}$$ is how far a point is from the mean
• Divinding by $$s$$ tells us how many standard deviations fit into the difference
Putting it together:
• For any observation, its z-score is its distance from the mean, measured in standard deviations
• If a z-score is positive, that observation is higher than the mean
• If a z-score is negative, that observation is below the mean
## Using Z-Scores to Compare Observations
Let's look at one particular student who scored 1725 on the SATs and 27 on the ACTs.
Which score should she send with her application?
• $$Z_{SAT} = \frac{SAT - \overline{SAT}}{s_{SAT}} = \frac{1725 - 1500}{200} = \frac{225}{250} = 0.9$$
• $$Z_{ACT} = \frac{ACT - \overline{ACT}}{s_{ACT}} = \frac{27 - 21}{5} = = \frac{6}{5} = 1.2$$
Compared to her peers:
• This student's SAT score was 0.9 standard deviations higher than average
• This student's ACT score was 1.2 standard deviations higher than average
• She did better on the ACTs than the SATs
## Using Z-Scores to Compare Observations
Let's look at another student who scored 1475 on the SATs and 20 on the ACTs.
Which score should he send with he application?
• $$Z_{SAT} = \frac{SAT - \overline{SAT}}{s_{SAT}} = \frac{1475 - 1500}{250} = -\frac{25}{250} = -0.1$$
• $$Z_{ACT} = \frac{ACT - \overline{ACT}}{s_{ACT}} = \frac{20 - 21}{5} = -\frac{1}{5} = -0.2$$
Compared to her peers:
• This student's SAT score was 0.1 standard deviations lower than average
• This student's ACT score was 0.2 standard deviations lower than average
• He did better on the SATs than the ACTs
## Why Z-Scores Work: Shifting and Scale
We call what z-scores do centering and scaling.
• We shift the mean so the scores are centered around zero
• We scale the variable so its standard deviation is one
In general:
• If we add (or subtract) a constant to every value, all measures of position (the mean and five number summary) are shifted by that constant
• If we multiply (or divide) by a constant, the standard deviation is scaled by that constant
• a constant is just a number that doesn't change
## Shifting Example
Consider the data set: $$1,\; 3,\; 5,\; 7,\; 9$$
• $$\bar{x} = \frac{\sum x}{n} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5$$
Now let's add 3 to every number: $$4,\; 6,\; 8,\; 10,\; 12$$
• $$\bar{x} = \frac{4 + 6 + 8 + 10 + 12}{5} = \frac{40}{5} = 8$$
Note that measures of spread will not change
• The distance between observations stays the same
What happened?
• By adding 3 to every number, we shifted the mean by 3
• Every number in the five number summary also goes up by 3
## Shifting in Practice
Say some Ivy League university will only accept students who score at least a 1750 on the SATs. We can look at our distribution of SAT scores in terms of how far above (or below) our students are by subtracting 1750 from every score.
What does this do to the mean (recall that $$\overline{SAT} = 1500$$)?
• The mean of the exam scores is now $$1500 - 1750 = -250$$ instead of $$1500$$, which means the average student does not meet their requirements
What will happen to the histogram?
## Histogram of SAT Scores
The red line represents the cut-off:
## Histogram of Shifted SAT Scores
The red line represents the cut-off:
## Shifting: Summaries
Let's see what happened to our summary stats:
## Min Q1 Median Q3 Max IQR Mean SD
## SAT 568.2 1328.9 1497.2 1665.6 2438.6 -336.7 1498.2 249
## Shifted -1181.8 -421.1 -252.8 -84.4 688.6 -336.7 -251.8 249
Notice:
• All measures of position (the five number summary and mean) were shifted down by 1750
• Both measures of spread (the SD and IQR) stayed the same
## Scaling
Consider the data set: $$1,\; 3,\; 5,\; 7,\; 9$$
• $$\bar{x} = 5$$
• $$s_{x} = 3.16$$
Now let's multiply all of them by 3: $$3,\; 9,\; 15,\; 21,\; 17$$
• $$\bar{x}_{3x} = 15$$
• $$s_{3x} = 3\times s_x = 9.487$$
When we scale by a constant:
• All measurements of center and scale are multiplied by that constant
## Scaling in Practice
Say we're interested in the weights of the cars from the Motor Trend data set we used in the previous chapter, but we're writing for a European magazine which expects the weights in kilograms.
• The standard deviation of weight was 978.5 lbs.
What's the standard deviation in kg?
• $$1 lb = 0.454kg$$
• $$s = 978.5 lbs = (978.5 \times 0.454) kg = 444.2 kg$$
How do the distributions compare if we change the units?
## Scaling: Summaries
Let's see what happened to the numerical summaries:
## Min Q1 Median Q3 Max IQR Mean SD
## Weight (lbs) 1513.0 2542.5 3325.0 3650.0 5424.0 -1107.5 3217.2 978.5
## Weight (kg) 686.9 1154.3 1509.5 1657.1 2462.5 -502.8 1460.6 444.2
Notice:
• All measures of position and scale we scaled down
• The oppostite would be true had we multiplied by a number larger than one
## The Normal Model
So far, we've talked generically about symmetric, unimodal distributions. The Normal Model or Normal Distribution is a special type of symmetric unimodal distribution.
• The shape of Normal Model is defined entirely by it's mean and standard deviation
• The Normal Distribution show up naturally almost any time we take measurements (height, weight, length, etc.)
• Many exams and tests (IQ, SATs, GREs, etc.) are designed so that their scores follow a Normal Distribution
• Most of the statistical tests we'll talk about in the course assume that the variable(s) follow a Normal Distribution
## The Shape of the Normal Model
The Normal Model is:
• Unimodal
• Bell-Shaped
Because the distribution is symmetric,
• The Mode = The Mean = The Median
## Describing a Normal Distribution
If a variable $$X$$ has a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, we write this as:
$X \sim N\left(\mu, \sigma\right)$
• "X is distributed normally with mean mu (mew) and standard deviation sigma"
• Typically, when we refer to a variable, we use upper-case letters
• When we refer to the variable take a particular value, we use lower-case letters
• $$X = x$$ means a particular value of $$X$$, e.g. $$X = 3$$
## Why the Greek Letters?
• The means and standard deviations we've calculated so far only describe a sample or a group of observed values.
• A number that describes a sample is called a statistic, and they are usually represented with Roman letters (e.g., $$\bar{x}$$, $$s$$)
• Samples are drawn from larger populations which we are usually trying to describe or study.
• When we write the distribution, we are describing the entire population
• Numbers that describe the population are called parameters
• The sample statistics are usually to estimate the parameters, but they are usually off by a bit because we don't have the entire population to use in the calculation
## Z-Scores for Populations
Earlier, we defined a z-score as:
• $$z = \frac{x - \bar{x}}{s}$$
If don't know anything about the population, this is the best we can do.
If we do know the population parameters, however, we can write:
• $$z = \frac{x - \mu}{\sigma}$$
What's the difference?
• The first one talks about where a value falls in the sample
• The second one tells us where the data falls in the population
• If we know $$\mu$$ and $$\sigma$$, we should use them
## Standard Normal Distributions
The problem with normal distributions is that there is a unique distribution for every possible combination of $$\mu$$ and $$\sigma$$. This means that there are an infinite number of normal distributions.
It turns out there's a way to convert any normal distribution to one that has $$\mu = 0$$ and $$\sigma = 1$$, which we call the Standard Normal Distribution. This lets us compare different normal distributions to each other more easily.
If $$X \sim N(\mu, \sigma)$$:
• For every value of $$X$$, find $$z = \frac{x - \mu}{\sigma}$$
• $$Z \sim N(0, 1)$$
## Statistics vs. Parameters: SAT
Recall that we rounded liberally when we said $$\overline{SAT} = 1500$$ and $$S_{SAT} = 250$$, the real values were:
## Mean St.Dev
## SAT 1498.168 248.9613
I did primarily to make the z-score calculations easier, but it turns out the SAT is created in such a way that:
$SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
The larger our sample is, the closer to the true value of the parameters our statistics will be.
In practice, we usually don't know what $$\mu$$ and $$\sigma$$ are supposed to look like.
## The 68-95-99.7 Rule
In addition to being bell-shaped, symmetric, and unimodal, the Normal Distribution has another nice feature, called the 68-95-99.7 Rule:
• About 68% of the data is within one standard deviation of the mean
• About 95% of the data is within two standard devations of the mean
• About 99.7% of the data is within three standard deviations of the mean
This holds for any variable that follows The Normal Model, no matter what $$\mu$$ and $$\sigma$$ are.
Important Notes:
• The percentages are approximate, but they can give us a good idea of what to expect
• Almost all individuals fall within three standard deviations of the mean
## 68-95-99.7 Rule in Practice (1)
As we've said, SAT scores are designed to follow a Normal Distribution: $SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
So what did the middle 68% of people score on the SATs?
• About 68% are within one standard deviation
• One standard deviation above the mean is $$\mu + \sigma = 1500 + 250 = 1750$$
• One standard deviation below the mean is $$\mu - \sigma = 1500 - 250 = 1350$$
• The middle 68% scored between 1350 and 1750 on the SATs
## 68-95-99.7 Rule in Practice (2)
$SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
What was the cut-off for the lowest 2.5% of people?
• 95% score within two standard deviation
• This means 5% are more than two standard deviations away from the mean on either side
• The Normal Distribution is symmetric about the mean, so only 2.5% scored more than two standard deviations below the mean
• Two standard deviations below the mean is: $$\mu - 2\sigma = 1500 - 2(250) = 1500 - 500 = 1000$$
• 2.5% of people scored less than 1000 on the SATs
## 68-95-99.7 Rule in Practice (3)
$SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
What percent of people scored higher than 2250?
• We need to know how many standard deviations above the mean 2250 is, which is the z-score for 2250
• $$z = \frac{x - \mu}{\sigma} = \frac{2250 - 1500}{250} = \frac{750}{250} = 3$$
• 2250 is three standard deviations above the mean
• 99.7% of people score within three standard deviations of the mean, so $$100\% - 99.7\% = 0.3\%$$ scored more than three away from the mean
• Because Normal Distributions are symmetric, this is split evenly above and below the mean
• Only $$0.3\% / 2 = 0.15\%$$ of people scored higher than 2250
## Finding Percentiles of Normal Distributions
A percentile is the value that cuts of some percentage of the distribution.
• For example, 25% of values are below $$Q1$$, so $$Q1$$ is the 25th Percentile
• As we saw, the 68-95-99.7 rule can give us percentiles (1000 was the 2.5th Percentile, 2250 was the 99.85th Percentile)
• The 68-95-99.7 only gives us approximate percentages or percentiles
• The 68-95-99.7 only lets us look at values that are exactly 1, 2, or 3 standard deviations away from the mean.
What's the alternative?
• We can use the shape of the Normal Model to find exact percentiles
• In this class, we use StatCrunch to do this
## Writing Percentiles
StatCrunch and other calculators look for percentiles in specific formats. In either case, we start by telling StatCrunch $$\mu$$ and $$\sigma$$.
How would I ask for the 95th Percentile?
• $$P(X \le x) = 0.95$$
• Note that we wrote the percentage as a proportion
• StatCrunch will fill in the value for $$x$$
If we wanted to know what percent of people scored higher than 2100:
• $$P(X \ge 2100) = p$$
• StatCrunch will solve for the proportion, which we need to turn into a percentage
## Percentages of Ranges
$SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
What percentage scored between 1480 and 1530?
• We write this as $$P(1480 \le X \le 1530) = p$$
• StatCrunch will find $$p$$ for us if we select Between at the top of the Normal Calculator
• $$P(1480 \le X \le 1530) = 0.0796 = 7.96\%$$
## Cut-offs for Percentages
$SAT \sim N\left(\mu = 1500, \sigma = 250\right)$
Say we wanted to find the IQR of the population of SAT scores. To do this, we need to find $$Q1$$ and $$Q3$$. Unfortunately, StatCrunch cannot compute this directly.
We need to break it down:
• $$P(Q1 \le X \le Q3) = 0.5$$
• $$P(X \le Q1) = 0.25$$
• $$P(X \ge Q3) = 0.25$$
• Let StatCrunch find $$Q1$$ and $$Q3$$, then $$IQR = Q3 - Q1$$
## Finding Population IQR from StatCruch
From StatCrunch, we found:
• $$P(X \le Q1) = 0.25 \to Q1 = 1331.4$$
• $$P(X \ge Q3) = 0.25 \to Q3 = 1668.6$$
• $$IQR = Q3 - Q1 = 1668.6 - 1331.4 = 337.2$$
We can use the same process for any middle percentage. Say we want the middle 80%:
• $$P(lower \le X \le upper) = 0.8$$
• $$P(X \le lower) = 0.1$$
• $$P(X \ge upper) = 0.1$$
• Use StatCrunch to find the lower and upper cut-offs
## Assessing Normality
The techniques we've been discussing only work if the data follow the Normal Model.
Before blindly assuming that a variable has a Normal Distribution, we should first check whether or not it does. We have two ways to do this:
• Make a histogram: is it unimodal, symmetric, and bell-shaped?
• Make a QQ Plot: do the points make a straight line?
Notes:
• For small sample sizes, histograms can be unreliable (as we've seen)
• It doesn't have to be perfect, we're looking for "close enough"
## QQ Plots
QQ is short for Quantile-Quantile
• A quantile is just a value that cuts the distribution at a certain place (quartiles are a specific type of quantile that cuts the data into 4$$^{th}$$s).
• QQ Plots, compare the quantiles we would see if the distribution were Normal to the quantiles that we actually have in our data.
• If our data is Normal, we'll be able to draw a (mostly) straight line through the points.
Making a QQ Plot:
• Each observation is plotted as a point
• The x and y coordinates are given by the theoretical (Normal) quantiles and the observed quantiles, respectively
• In StatCrunch: Graph $$\to$$ QQ Plot
## Assessing Normality
To show you how these work, I generated two samples that I know come from a $$N(\mu = 100, \sigma = 10)$$ distribution.
• Sample 1 has twenty observations $$(n = 20)$$
• Sample 2 has a sample size of two hundred $$(n = 200)$$ | 2020-02-19 01:58:15 | {"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": 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": 0.7385736107826233, "perplexity": 1423.0133879267464}, "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/1581875143963.79/warc/CC-MAIN-20200219000604-20200219030604-00351.warc.gz"} |
https://iclr-blog-track.github.io/2022/03/25/kgs/ | # Knowledge Graph Papers @ ICLR 2021
Hi! 👋 Today we are going to have a look at ICLR 2021 papers focusing on knowledge graphs (KGs), particularly in areas of graph representation learning and NLP. Among 860 accepted papers we highlight 10 particularly interesting and promising works that might influence the field in near future. This post is be structured as follows:
## Reasoning in KGs
Query embedding and neural query answering are quite hot topics today, and such systems are way more capable in complex reasoning than N+1’th KG embedding model.
Usually, in query embedding, you have to embed a lot of possible combinations of atoms which could easily be 50M points induced by 1-hop, 2-hop, AND, OR, etc queries. That is, starting from a relatively small graph (a subset of Freebase of 270K edges is a typical benchmark) you have to embed orders of magnitude more points. Is it really necessary? 🤨
Surprisingly, no! Arakelyan, Daza, Minervini, and Cochez show that it is pretty much enough to take any pre-trained KG embedding model (trained only on 1-hop queries in the form head, relation, ? ) and decode them in a smart way. The authors propose CQD (Continuous Query Decomposition) with two options: 1) model queries with t-norms (continuous optimization); 2) just use a beam search (combinatorial optimization) similar to that of your favourite NLG transformer. That is, you simply traverse the embedding space with beam search and you don’t need all those redundant millions of points. 👨🔬 In the experiments, the beam search strategy performs very well and leaves far behind the previous approaches that do model those millions explicitly. That’s a neat result and, in my opinion, will be a very strong baseline for all future works in this domain. Well deserved Outstanding ICLR’21 Paper award! 🙌
Continuing with rules and reasoning, Qu, Chen et al take another direction and propose RNNLogic which algorithm is depicted below 👇. RNNLogic employs relational paths that can be mined from the background KG and, hence, generated after some learning procedure. Given a query head, relation, ? , we first generate a set of rules (sequence of relations parameterized by LSTM, RNN-part of the name comes from here) from which we sample ‘most plausible’ rules and then send them into the predictor to produce scores over possible answers. That is, the generator tries to predict better and better rules pruning the search space for the predictor. The predictor can be parameterized by entity and relation embeddings akin to RotatE, which shows observable improvements in the experiments. During inference, RNNLogic not only predicts a target entity of a query, but also supports it with a bunch of relevant rules which positively impacts explainability, a common pitfall of embeddings-only algorithms.
RNNLogic. Source: Qu, Chen et al
## Temporal Logics and KGs
Inthe temporal setup, we add a time dimension to our KG. That is, now we have timestamped quadruples (head, relation, tail, time) as data points and hence our queries are (head, relation, ?, time). In other words, a model has to take into account when a particular relation happened.
At ICLR’21, Han, Chen, et al propose xERTE, an attention-based model capable of predicting future links 🧙♂️. The crux of xERTE is iterative subgraph expansion around head and this expansion keeps track of seen timestamps so that prior links do not have access to the posterior ones. In some sense, it can be seen as a temporal extension of GraIL (ICML’20). Each node embedding is obtained through a concatenation of entity embedding and time embedding (which happens to be a d-dimensional vector of cosines of varying frequencies). Then, in L steps, usually less than 4, xERTE computes attention over the neighbours, prunes the subgraph to retain only the most probable nodes, and yields a distribution of attention scores over candidates (🖼 👇). Thanks to the iterative nature, xERTE can visualize reasoning paths of ranked predictions which was well appreciated by 50+ participants of the user study!
xERTE. Source: Han, Chen, et al
I’d also put in this section a very interesting work by Hahn et al on learning to solve Linear Temporal Logic (LTL) formulas which are widely used in formal verification. LTL is based on propositional logic with temporal operators Next (some formula holds in the next position of a sequence), Until (some formula f holds until g holds), “every point in time”, and “future point in time”. The formulas might look like this, i.e., they are pretty long sequences of atoms and operators:
LTL examples. Source: Hahn et al
The authors pose the task of predicting a solution of LTL formulas by generating a satisfiable trace 👣. What do we do with sequences? Put them into the Transformer, of course.
Enhanced with tree-based positional encodings of such long formulas, the authors find that even a relatively small Transformer (8 layers, 8 heads, 1024 FC size) yields surprisingly good results, accurate both semantically and syntactically. Since verifying logical formulas is much simpler than finding them (usually log vs polynomial), the Transformer could generate plausible solutions which then can be verified by non-neural solvers. Furthermore, the authors observe that the Transformer can generalize to the semantics of LTL and perform well on larger/longer formulas compared to training formulas!
## NLP Perspective: Relations, PMI, Entity Linking
There is a good amount of NLP-related research involving KGs this year.
First, Allen, Balažević, and Hospedales study the nature of learnable relation embedding in KGs from the PMI (pointwise mutual information) point of view. Back in 2014, Levy and Goldberg showed (in their very influential paper) that learning word2vec implicitly factorizes a PMI matrix of words co-occurrences. Then it was shown that we could extract particular semantic concepts like relatedness, paraphrase, similarity, and analogy from that PMI matrix. Can we draw some parallels and observe such patterns in learnable KG relations?
Turns out yes! The authors identified 3 possible categories of relations: 1) those which signal about the relatedness of two nodes (eg, verb_group relation in Wordnet); 2) those which exhibit specialization (hyponym-hypernym); 3) most common context shift (eg, meronym). Furthermore, the matrices of relatedness-type relations tend to be more symmetric, and eigenvalues/norms of relation matrices/vectors indicate the strength of relatedness. The authors then demonstrate that multiplicative models like DistMult or TuckER better capture such relatedness relation types in KGs. 🏃♀ Chasing SOTA, current KG embedding literature lacks deep analysis of what is actually learned there, and it’s great to see such a long-needed qualitative study!
Source: Allen, Balažević, and Hospedales
Ding, Wang, et al also present a work focusing on relations, but this time in a context of relation extraction from raw texts and learning relation prototypes. That is, instead of learning to distinguish hundreds of unique relations (where some of them might be semantically similar), we’d rather learn a smaller set of centroids/prototypes which would group similar relations together on a manifold — the authors propose a unit sphere (see illustration). For pre-training, the authors use weak labels from Wikidata using their relations together with mapped entities from Wikipedia. The resulting approach performs particularly well in zero- and few-shot scenario with up to 10% of absolute improvement 💪.
Relation prototypes. Source: Ding, Wang, et al
Moving towards entities, De Cao et al propose another look onto the entity linking task. Usually, in retrievers and entity linkers such as DPR or BLINK, you have to keep in memory the whole index of named entities where lots of entities have certain tokens in common, eg., “Leonardo DiCaprio”, “Leonardo da Vinci”, “New York”, “New Jersey”, etc.
Of course, in large knowledge bases of millions of entities this leads to a large memory consumption and a necessity to have hard negative samples during training to be able to distinguish between “New York” and “New Jersey”. Instead, the authors propose GENRE (generative entity retrieval) to generate entity names autoregressively (token by token) given a context (check out an awesome illustration below 👇). As a backbone, the authors use BART to fine-tune on generating entity names. The inference process using a beam search is a bit more cumbersome: since we want to prune impossible combinations (eg, not sampling “Jersey” after “Leonardo”), the authors build a prefix tree (trie) which encodes 6M Wikipedia titles in a decent 600 MB index. GENRE is also parameter-efficient 🏋 : while DPR or BLINK require 30–70GB of memory and 6–15B (billion) parameters, GENRE only requires 2GB and 17M (million) parameters!
Generating entity names token by token. Source: GENRE github repo
By the way, a multilingual version, mGENRE, has been published and released either 😉
## Complex Question Answering: More Modalities
Research on open-domain QA often employs graph structures between documents as reasoning paths (whereas KG-based QA directly traverses a background KG). Open-domain QA immediately benefits from enormously large LMs and recent dense retrieval techniques, that’s why more efforts from big labs are put along this dimension.
First, Xiong, Li, et al extend the idea of Dense Passage Retriever to the multi-hop setup where complex questions are answered iteratively step by step. During training, you’d feed MDR (Multi-hop Dense Retriever) with a question and previously extracted passages together with positive and negative samples of possible passages, so it’s pretty close to the original DPR. At inference (check the illustration below), the authors apply beam search and MIPS to generate top-K passages, score them, and prepend best candidates to the query at the next iteration. Pretty much all existing multi-hop QA datasets can be solved in 2–3 steps, so it’s not a big burden on the system.
🧪 Experiments show that the graph structure is not necessary here. That is, you could omit mining and traversing links between paragraphs and resort to the dense index alone to get even better prediction quality! On average, MDR is 5–20 absolute points better and 10x faster than its contenders. Besides, does the chosen approach (beam search over pre-trained index) resemble conceptually CQD for KG query answering from the first section? 😉
MDR intuition. Source: Xiong, Li, et al
While MDR focuses on passages of pure text (being them extracted from Wikipedia or other text-only sources), a continuing trend is to cover more sources beyond flat texts. To this end, Chen et al study the problem of complex QA over both tabular and textual data and construct a new OTT-QA dataset (Open Table-and-Text Question Answering). The authors suggest an elegant solution of linearizing tables into table segments to be put into the transformer: split a table in rows and prepend to each row some common information about the whole table (eg, title, min/max values). Doing so, 400k original tables were transformed into 5M segments which is a difficult enough task for the table retriever. Conversely, the proposed model has to learn to retrieve both relevant segments and text passages.
In the experiments, the authors find that a traditional BERT-based iterative retriever-reader works quite poorly (10% F1 score) and instead propose to group connected passages and table segments together into fused blocks. Such an early fusion is achieved through entity linking of cells contents to textual mentions. Stacking all the goodies (fusion + long-range transformers + improved reader) the quality increases to 32% F1 💪. In the last sentence of the paper the authors ask: can we employ even more modalities into the QA..?
Source: Chen et al
… And Talmor, Yoran, Catav, Lahav et al answer this question right away in their work building MultiModalQA! A new dataset poses a multi-hop cross-modal reasoning objective over text 📚, tables 📊, and images 🖼. Cross-modal here means that at least one hop in a question implies querying another modality. In the example, a question consists of 3 hops, and each hop can be answered by a relevant source in its own modality. Overall, the dataset consists of ~30K QA-pairs spanning across 16 different compositional templates (eg, combining answers from one table and one image, a template will indicate which modalities have to be queried).
👩🔬 Empirically, the authors show that one-modality baselines yield only about 18 F1 points, while a fused model (called ImplicitDecomp) which derives modalities from classified templates returns ~56 F1 📈. Text and table QA modules use RoBERTa-Large while the visual QA module employs VILBERT-MT. It’s still far from a human score of 91 F1, so take a note — there is a new unsaturated benchmark 😉.
## Conclusion
That’s all for today! This conference year, we have seen a lot of examples of out-of-the-box thinking (eg, in KG reasoning, entity linking, drawing parallels to similar domains) which lead to actually cool results - and I would encourage you to try out the same! Maybe that unusual idea you’ve been dismissing recently is actually worth trying?
You will need to sign in to GitHub to add a comment! To edit or delete your comment, visit the discussions page and look for your comment in the right discussion. | 2023-02-08 00:48:53 | {"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": 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": 0.6178274750709534, "perplexity": 3569.9418693780485}, "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/1674764500664.85/warc/CC-MAIN-20230207233330-20230208023330-00442.warc.gz"} |
https://www.bartleby.com/questions-and-answers/cheyenne-corp.-was-organized-on-january-1-2017.-it-is-authorized-to-issue21000shares-of7percent-doll/6bbfd976-df8d-4df2-a39f-bb44d91c9edb | # Cheyenne Corp. was organized on January 1, 2017. It is authorized to issue 21,000 shares of 7%, $53 par value preferred stock and 454,000 shares of no-par common stock with a stated value of$2 per share. The following stock transactions were completed during the first year. Jan. 10 Issued 67,000 shares of common stock for cash at $4 per share. Mar. 1 Issued 1,140 shares of preferred stock for cash at$56 per share. May 1 Issued 112,000 shares of common stock for cash at $7 per share. Sept. 1 Issued 4,400 shares of common stock for cash at$8 per share. Nov. 1 Issued 2,400 shares of preferred stock for cash at $56 per share. Prepare the paid-in capital portion of the stockholders’ equity section at December 31, 2017. CHEYENNE CORP.Partial Balance Sheetchoose the accounting period December 31, 2017For the Month Ended December 31, 2017For the Year Ended December 31, 2017 select an opening section name Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital select an opening subsection name Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital select a name of the subordinate part one of this subsection Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital enter a balance sheet item$enter a dollar amount enter a balance sheet item enter a dollar amount select a summarizing line for the subordinate part one of this subsection Current Assets Current Liabilities Intangible Assets Long-term Investments Long-term Liabilities Property, Plant and Equipment Stockholders' Equity Total Assets Total Current Assets Total Current Liabilities Total Intangible Assets Total Liabilities Total Liabilities and Stockholders' Equity Total Long-term Investments Total Long-term Liabilities Total Property, Plant and Equipment Additional Paid-in Capital Paid-in Capital Capital Stock Total Capital Stock Total Paid-in Capital Total Paid-in Capital and Retained Earnings Total Stockholders’ Equity Total Additional Paid-in Capital $enter a subtotal of the two previous amounts select a name of the subordinate part two of this subsection Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital enter a balance sheet item enter a dollar amount enter a balance sheet item enter a dollar amount select a summarizing line for the subordinate part two of this subsection Current Assets Current Liabilities Intangible Assets Long-l Long-term Investments Total Long-term Liabilities Total Property, Plant and tal Stockholders’ Equity Total Additional Paid-in Capital enter a subtotal of the two previous amounts in Capital Capital Stock Total Capital Stock Total Paid-in Capital Total Paid-in Capital and Retained Earnings Total Stockholders’ Equity Total Additional Paid-in Capital$enter a total amount for this subsection
Question
Cheyenne Corp. was organized on January 1, 2017. It is authorized to issue 21,000 shares of 7%, $53 par value preferred stock and 454,000 shares of no-par common stock with a stated value of$2 per share. The following stock transactions were completed during the first year.
Jan. 10 Issued 67,000 shares of common stock for cash at $4 per share. Mar. 1 Issued 1,140 shares of preferred stock for cash at$56 per share. May 1 Issued 112,000 shares of common stock for cash at $7 per share. Sept. 1 Issued 4,400 shares of common stock for cash at$8 per share. Nov. 1 Issued 2,400 shares of preferred stock for cash at $56 per share. Prepare the paid-in capital portion of the stockholders’ equity section at December 31, 2017. CHEYENNE CORP. Partial Balance Sheet choose the accounting period December 31, 2017For the Month Ended December 31, 2017For the Year Ended December 31, 2017 select an opening section name Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital select an opening subsection name Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital select a name of the subordinate part one of this subsection Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital enter a balance sheet item$enter a dollar amount
enter a balance sheet item
enter a dollar amount
select a summarizing line for the subordinate part one of this subsection
Current Assets Current Liabilities Intangible Assets Long-term Investments Long-term Liabilities Property, Plant and Equipment Stockholders' Equity Total Assets Total Current Assets Total Current Liabilities Total Intangible Assets Total Liabilities Total Liabilities and Stockholders' Equity Total Long-term Investments Total Long-term Liabilities Total Property, Plant and Equipment Additional Paid-in Capital Paid-in Capital Capital Stock Total Capital Stock Total Paid-in Capital Total Paid-in Capital and Retained Earnings Total Stockholders’ Equity Total Additional Paid-in Capital
$enter a subtotal of the two previous amounts select a name of the subordinate part two of this subsection Current AssetsCurrent LiabilitiesIntangible AssetsLong-term InvestmentsLong-term LiabilitiesProperty, Plant and EquipmentStockholders' EquityTotal AssetsTotal Current AssetsTotal Current LiabilitiesTotal Intangible AssetsTotal LiabilitiesTotal Liabilities and Stockholders' EquityTotal Long-term InvestmentsTotal Long-term LiabilitiesTotal Property, Plant and EquipmentAdditional Paid-in CapitalPaid-in CapitalCapital StockTotal Capital StockTotal Paid-in CapitalTotal Paid-in Capital and Retained EarningsTotal Stockholders’ EquityTotal Additional Paid-in Capital enter a balance sheet item enter a dollar amount enter a balance sheet item enter a dollar amount select a summarizing line for the subordinate part two of this subsection Current Assets Current Liabilities Intangible Assets Long-l Long-term Investments Total Long-term Liabilities Total Property, Plant and tal Stockholders’ Equity Total Additional Paid-in Capital enter a subtotal of the two previous amounts in Capital Capital Stock Total Capital Stock Total Paid-in Capital Total Paid-in Capital and Retained Earnings Total Stockholders’ Equity Total Additional Paid-in Capital$enter a total amount for this subsection | 2021-01-15 15:24:12 | {"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": 0.4547552168369293, "perplexity": 10026.15193721471}, "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/1610703495901.0/warc/CC-MAIN-20210115134101-20210115164101-00169.warc.gz"} |
https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=19188&extra=longdesc_idm46461545336336&clicked=1 | This shows a rectangle. The lengths of the sides are marked as 5 m, 4 m, 5 m, and 4 m, working clockwise around the figure starting at the top edge.
2 Perimeters | 2021-02-26 15:19:18 | {"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": 0.9167624115943909, "perplexity": 618.8324054219437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357929.4/warc/CC-MAIN-20210226145416-20210226175416-00442.warc.gz"} |
https://math.stackexchange.com/questions/1809012/classical-logic-without-negation-and-falsehood | # Classical logic without negation and falsehood
It seems to me that Gerhard Gentzen's sequent calculus could just omit negation and falsehood, and still prove any classical tautology in a suitable form. (For a specific formula, falsehood gets replaced by the conjunction of all relevant propositional variables. For predicate logic, falsehood gets replaced by the conjunction of the universally quantified predicate symbols, including for example $\forall x\forall y\ x=y$. One could also introduce a constant $F$ and axioms like $F\to\forall x\forall y\ x=y$. More details about the consequences of removing falsehood can be found here.) This doesn't seem possible for his natural deduction calculus, where the law of excluded middle is used to arrive at classical logic.
Why hasn't he adapted the relevant rule from his sequent calculus to his natural deduction calculus: $\begin{array}{l} A\to(B\lor C) \\ \hline (A\to B)\lor C\end{array}$
At first sight, this rule doesn't look worse than $\begin{array}{l} \\ \hline A \lor \lnot A\end{array}$ or $\begin{array}{l} \lnot \lnot A \\ \hline A \end{array}$, and it would have mirrored his sequent calculus more closely. Of course, in the sequent calculus he didn't need to write $\lor$, and this made this deduction rule look even more attractive. But being able to omit both negation and falsehood seems attractive to me, independent of how attractive the rules themselves appear.
The only reason I could come up with is that he developed his natural deduction calculus first, became dissatisfied with it, then developed his sequent calculus, and didn't find it important to further improve his natural deduction calculus, because it had other irreparable flaws anyway.
• What reason would he have to want to omit negation and falsity? – Noah Schweber Jun 1 '16 at 23:59
• @NoahSchweber Robinson arithmetic omits induction, and is still recursively incompletable and essentially undecidable. This shows that induction is not responsible for those phenomena. You can omit negation and falsehood, and still get essentially the same classical logic. This shows that negation itself is not responsible for the interesting features of classical logic. Omitting also falsehood prevents the impression that you could just define negation by implication to falsehood. Still, falsehood is only a single element, and adjoining it back is easy and unproblematic. – Thomas Klimpel Jun 2 '16 at 3:41
• If you remove the rules for negation and $\bot$ from the logic, you will generally obtain a system known as minimal logic, which is strictly weaker than classical logic. en.wikipedia.org/wiki/Minimal_logic – Carl Mummert Jun 5 '16 at 11:38
• @CarlMummert Yes, minimal logic was my motivation for also removing falsehood. Minimal logic is actually minimally weaker than intuitionistic logic, but the difference is so small that I consider them still essentially the same (intuitionistic) logic. But just like for monoids and semigroups, the minimal gained generality is rarely worth the effort this causes in terms of more convolved definitions and theorem statements. (Noah Schweber's answer nicely demonstrates this, at least it wasn't as clear to me before.) – Thomas Klimpel Jun 5 '16 at 13:10
First, I would like to strongly disagree with the third sentence of your recent comment - just because (basically) the same proof system is complete for a restricted logical language when restricted appropriately, doesn't mean that that restricted logic is in any way similar to what you started with. This becomes especially clear once we consider the semantics - e.g. if $T$ is a first-order theory without negation, then it has a model (consisting of one element, with all relations total), so there are no inconsistent theories at all. It's also evident if we consider the algebraic structure of a logic.
• I admit that I haven't proved my statement: "You can omit negation and falsehood, and still get essentially the same classical logic." Especially my replacement (or translation) for falsehood was a bit weak. A better approach to see that they are essentially the same might be to translate negation as in $\lnot A$ by $A\to p$, where $p$ is a free propositional variable. So all the homomorphisms and properties with respect to homomorphisms will stay the same. (Note that Gentzen already has implication in his language, so negation becomes sort of redundant.) – Thomas Klimpel Jun 2 '16 at 6:12
• @ThomasKlimpel I think you're misunderstanding my comment about homomorphisms. Let me put it this way: for any fixed language, there's a single structure which satisfies all first-order sentences, in that language, without negation (one element, all relations are total)! That is, there are no inconsistent theories if we get rid of $\neg$. How is this the "essentially the same" as first-order logic? (Note that this also means that the consistency question - which Gentzen definitely cared about! - is trivial for first-order theories without negation, unless rephrased in a more convoluted way.) – Noah Schweber Jun 2 '16 at 17:57
• This comment shows an interesting fact about what is meant by consistency. If a classical first order theory is inconsistent, then the only possible Boolean valued models take values in the one element Boolean algebra (where $\bot=\top$). The same fact remains true if we remove negation from the language. If you exclude (=don't allow) the one element Boolean algebra in case that the language has negation, and suddently allow it if negation is removed from the language, then this has very little to with whether we still have essentially the same classical logic or not. – Thomas Klimpel Jun 2 '16 at 20:00
• It depends on whether you think that the sentence $(x=x) \to p$ is satisfied (or write $\top\to p$, if you prefer). If the logic has more than one value, then it has a value different from $\top$, and the sentence would be false because $p$ could assume that value. – Thomas Klimpel Jun 2 '16 at 20:32 | 2021-03-07 14:59:16 | {"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": 0.8032564520835876, "perplexity": 504.65825427062396}, "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-2021-10/segments/1614178377821.94/warc/CC-MAIN-20210307135518-20210307165518-00476.warc.gz"} |
https://wikimili.com/en/Lens_(optics) | # Lens (optics)
Last updated
A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (elements), usually arranged along a common axis. Lenses are made from materials such as glass or plastic, and are ground and polished or molded to a desired shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.
A light beam or beam of light is a directional projection of light energy radiating from a light source. Sunlight forms a light beam when filtered through media such as clouds, foliage, or windows. To artificially produce a light beam, a lamp and a parabolic reflector is used in many lighting devices such as spotlights, car headlights, PAR Cans and LED housings. Light from certain types of laser has the smallest possible beam divergence.
In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.
## History
The word lens comes from lēns , the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil plant also gives its name to a geometric figure. [1]
The lentil is an edible legume. It is a bushy annual plant known for its lens-shaped seeds. It is about 40 cm (16 in) tall, and the seeds grow in pods, usually with two seeds in each.
In 2-dimensional geometry, a lens is a convex set bounded by two circular arcs joined to each other at their endpoints. In order for this shape to be convex, both arcs must bow outwards (convex-convex). This shape can be formed as the intersection of two circular disks. It can also be formed as the union of two circular segments, joined along a common chord.
Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. [2] The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BC which may or may not have been used as a magnifying glass, or a burning glass. [3] [4] [3] [5] Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses". [6] [ verification needed ]
The Nimrud lens, also called Layard lens, is a 3000-year-old piece of rock crystal, which was unearthed in 1850 by Austen Henry Layard at the Assyrian palace of Nimrud, in modern-day Iraq. It may have been used as a magnifying glass, or as a burning-glass to start fires by concentrating sunlight, or it may have been a piece of decorative inlay.
Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. Hieroglyphs combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters. Cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts were derived from hieroglyphic writing, as was the Proto-Sinaitic script that later evolved into the Phoenician alphabet. Through the Phoenician alphabet's major child systems, the Greek and Aramaic scripts, the Egyptian hieroglyphic script is ancestral to the majority of scripts in modern use, most prominently the Latin and Cyrillic scripts and the Arabic script and Brahmic family of scripts.
The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BC) mentioning a burning-glass. [7] Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period. [8] Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness, though the reference is vague). [9] Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.
Aristophanes, son of Philippus, of the deme Kydathenaion, was a comic playwright of ancient Athens. Eleven of his forty plays survive virtually complete. These provide the most valuable examples of a genre of comic drama known as Old Comedy and are used to define it, along with fragments from dozens of lost plays by Aristophanes and his contemporaries.
The Clouds is a Greek comedy play written by the playwright Aristophanes. A lampooning of intellectual fashions in classical Athens, it was originally produced at the City Dionysia in 423 BC and was not as well received as the author had hoped, coming last of the three plays competing at the festival that year. It was revised between 420 and 417 BC and was thereafter circulated in manuscript form.
Pliny the Elder was a Roman author, a naturalist and natural philosopher, a naval and army commander of the early Roman Empire, and a friend of emperor Vespasian.
Ptolemy (2nd century) wrote a book on Optics , which however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received, by medieval scholars in the Islamic world, and commented upon by Ibn Sahl (10th century), who was in turn improved upon by Alhazen ( Book of Optics , 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century (Eugenius of Palermo 1154). Between the 11th and 13th century "reading stones" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock cystal Visby lenses may or may not have been intended for use as burning glasses. [10]
Claudius Ptolemy was a mathematician, astronomer, geographer and astrologer. He lived in the city of Alexandria in the Roman province of Egypt, under the rule of the Roman Empire, had a Latin name, which several historians have taken to imply he was also a Roman citizen, cited Greek philosophers, and used Babylonian observations and Babylonian lunar theory. The 14th-century astronomer Theodore Meliteniotes gave his birthplace as the prominent Greek city Ptolemais Hermiou in the Thebaid. This attestation is quite late, however, and there is no other evidence to confirm or contradict it. He died in Alexandria around AD 168.
Ptolemy's Optics is a work on geometrical optics, dealing with reflection, refraction, and colour.
Ibn Sahl was a Persian mathematician and physicist of the Islamic Golden Age, associated with the Buwayhid court of Baghdad. Nothing in his name allows us to glimpse his country of origin.
Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century. [11] This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, [12] and later in the spectacle-making centres in both the Netherlands and Germany. [13] Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses (probably without the knowledge of the rudimentary optical theory of the day). [14] [15] The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands. [16] [17]
The Netherlands is a country located mainly in Northwestern Europe. The European portion of the Netherlands consists of twelve separate provinces that border Germany to the east, Belgium to the south, and the North Sea to the northwest, with maritime borders in the North Sea with Belgium, Germany and the United Kingdom. Together with three island territories in the Caribbean Sea—Bonaire, Sint Eustatius and Saba—it forms a constituent country of the Kingdom of the Netherlands. The official language is Dutch, but a secondary official language in the province of Friesland is West Frisian.
The optical microscope, often referred to as the light microscope, is a type of microscope that commonly uses visible light and a system of lenses to magnify images of small objects. Optical microscopes are the oldest design of microscope and were possibly invented in their present compound form in the 17th century. Basic optical microscopes can be very simple, although many complex designs aim to improve resolution and sample contrast. Often used in the classroom and at home unlike the electron microscope which is used for closer viewing.
A refracting telescope is a type of optical telescope that uses a lens as its objective to form an image. The refracting telescope design was originally used in spy glasses and astronomical telescopes but is also used for long focus camera lenses. Although large refracting telescopes were very popular in the second half of the 19th century, for most research purposes the refracting telescope has been superseded by the reflecting telescope which allows larger apertures. A refractor's magnification is calculated by dividing the focal length of the objective lens by that of the eyepiece.
With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces. [18] Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in a 1758 patent.
## Construction of simple lenses
Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres. Each surface can be convex (bulging outwards from the lens), concave (depressed into the lens), or planar (flat). The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.
Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses that are used to correct astigmatism in someone's eye.
More complex are aspheric lenses. These are lenses where one or both surfaces have a shape that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less aberration than standard simple lenses, but they are more difficult and expensive to produce.
### Types of simple lenses
Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or just convex) if both surfaces are convex. If both surfaces have the same radius of curvature, the lens is equiconvex. A lens with two concave surfaces is biconcave (or just concave). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is convex-concave or meniscus. It is this type of lens that is most commonly used in corrective lenses.
If the lens is biconvex or plano-convex, a collimated beam of light passing through the lens converges to a spot (a focus) behind the lens. In this case, the lens is called a positive or converging lens. For a thin lens in air, the distance from the lens to the spot is the focal length of the lens, which is commonly represented by f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.
Biconvex lens
If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
Biconcave lens
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface and is thicker at the centre than at the periphery. An ideal thin lens with two surfaces of equal curvature would have zero optical power, meaning that it would neither converge nor diverge light. All real lenses have nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.
### Lensmaker's equation
The focal length of a lens in air can be calculated from the lensmaker's equation: [19]
${\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right],}$
where
${\displaystyle f}$ is the focal length of the lens,
${\displaystyle n}$ is the refractive index of the lens material,
${\displaystyle R_{1}}$ is the radius of curvature (with sign, see below) of the lens surface closer to the light source,
${\displaystyle R_{2}}$ is the radius of curvature of the lens surface farther from the light source, and
${\displaystyle d}$ is the thickness of the lens (the distance along the lens axis between the two surface vertices).
The focal length f is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, 1/f, is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (inverse metres).
Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions.
#### Sign convention for radii of curvature R1 and R2
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies, but in this article a positiveR indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while negativeR means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, R1 > 0 and R2 < 0 indicate convex surfaces (used to converge light in a positive lens), while R1 < 0 and R2 > 0 indicate concave surfaces. The reciprocal of the radius of curvature is called the curvature. A flat surface has zero curvature, and its radius of curvature is infinity.
#### Thin lens approximation
If d is small compared to R1 and R2, then the thin lens approximation can be made. For a lens in air, f is then given by
${\displaystyle {\frac {1}{f}}\approx \left(n-1\right)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right].}$ [20]
## Imaging properties
As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens. Conversely, a point source of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens is called the focal plane .
If the distances from the object to the lens and from the lens to the image are S1 and S2 respectively, for a lens of negligible thickness, in air, the distances are related by the thin lens formula: [21] [22] [23]
${\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}$ .
This can also be put into the "Newtonian" form:
${\displaystyle x_{1}x_{2}=f^{2},\!}$ [24]
where ${\displaystyle x_{1}=S_{1}-f}$ and ${\displaystyle x_{2}=S_{2}-f}$.
Therefore, if an object is placed at a distance S1 > f from a positive lens of focal length f, we will find an image distance S2 according to this formula. If a screen is placed at a distance S2 on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image .
This is the principle of the camera, and of the human eye. The focusing adjustment of a camera adjusts S2, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of S1 from the camera. Put another way, modifying S2 causes objects at a different S1 to come into perfect focus.
In some cases S2 is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they appear to form an image, this is called a virtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, S1 then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a real image on the retina.
A negative lens produces a demagnified virtual image.
A Barlow lens (B) reimages a virtual object (focus of red ray path) into a magnified real image (green rays at focus)
Using a positive lens of focal length f, a virtual image results when S1 < f, the lens thus being used as a magnifying glass (rather than if S1 >> f as for a camera). Using a negative lens (f < 0) with a real object (S1 > 0) can only produce a virtual image (S2 < 0), according to the above formula. It is also possible for the object distance S1 to be negative, in which case the lens sees a so-called virtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens) before the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.
Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.
A convex lens (fS1) forming a real, inverted image rather than the upright, virtual image as seen in a magnifying glass
For a thin lens, the distances S1 and S2 are measured from the object and image to the position of the lens, as described above. When the thickness of the lens is not much smaller than S1 and S2 or there are multiple lens elements (a compound lens), one must instead measure from the object and image to the principal planes of the lens. If distances S1 or S2 pass through a medium other than air or vacuum a more complicated analysis is required.
### Magnification
The linear magnification of an imaging system using a single lens is given by
${\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}$ ,
where M is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images M is positive, so the image is upright.
Linear magnification M is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the angular magnification—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote the plate scale, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as long-focus lenses or wide-angle lenses according to their focal lengths.
Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of 5 cm focal length, held 20 cm from the eye and 5 cm from the object, produces a virtual image at infinity of infinite linear size: M = ∞. But the angular magnification is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the moon using a camera with a 50 mm lens, one is not concerned with the linear magnification M−50 mm / 380000 km = −1.3×10−10. Rather, the plate scale of the camera is about 1°/mm, from which one can conclude that the 0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.
In the extreme case where an object is an infinite distance away, S1 = ∞, S2 = f and M = −f/∞= 0, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, since diffraction places a lower limit on the size of the point spread function. This is called the diffraction limit.
## Aberrations
Optical aberration
Defocus
Lenses do not form perfect images, and a lens always introduces some degree of distortion or aberration that makes the image an imperfect replica of the object. Careful design of the lens system for a particular application minimizes the aberration. Several types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.
### Spherical aberration
Spherical aberration occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be ground and polished, and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Lenses in which closer-to-ideal, non-spherical surfaces are used are called aspheric lenses. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses. Spherical aberration can be minimised by carefully choosing the surface curvatures for a particular application. For instance, a plano-convex lens, which is used to focus a collimated beam, produces a sharper focal spot when used with the convex side towards the beam source.
### Coma
Coma, or comatic aberration, derives its name from the comet-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays that pass through the centre of a lens of focal length f are focused at a point with distance f tan θ from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a comatic circle. The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called bestform lenses.
### Chromatic aberration
Chromatic aberration is caused by the dispersion of the lens material—the variation of its refractive index, n, with the wavelength of light. Since, from the formulae above, f is dependent upon n, it follows that light of different wavelengths is focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using an achromatic doublet (or achromat) in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope. An apochromat is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction. Apochromats are much more expensive than achromats.
Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal fluorite. This naturally occurring substance has the highest known Abbe number, indicating that the material has low dispersion.
### Other types of aberration
Other kinds of aberration include field curvature , barrel and pincushion distortion , and astigmatism .
### Aperture diffraction
Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the diffraction of light passing through the lens' finite aperture. A diffraction-limited lens is one in which aberrations have been reduced to the point where the image quality is primarily limited by diffraction under the design conditions.
## Compound lenses
Simple lenses are subject to the optical aberrations discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.
The simplest case is where lenses are placed in contact: if the lenses of focal lengths f1 and f2 are "thin", the combined focal length f of the lenses is given by
${\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}.}$
Since 1/f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.
If two thin lenses are separated in air by some distance d, the focal length for the combined system is given by
${\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}.}$
The distance from the front focal point of the combined lenses to the first lens is called the front focal length (FFL):
${\displaystyle {\mbox{FFL}}={\frac {f_{1}(f_{2}-d)}{(f_{1}+f_{2})-d}}.}$ [26]
Similarly, the distance from the second lens to the rear focal point of the combined system is the back focal length (BFL):
${\displaystyle {\mbox{BFL}}={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}.}$
As d tends to zero, the focal lengths tend to the value of f given for thin lenses in contact.
If the separation distance is equal to the sum of the focal lengths (d = f1+f2), the FFL and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam. This type of system is called an afocal system , since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by
${\displaystyle M=-{\frac {f_{2}}{f_{1}}},}$
which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (f1 > 0, f2 > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (f1 > 0 > f2) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.
## Other types
Cylindrical lenses have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a laser diode into a round beam. They are also used in motion picture anamorphic lenses.
A Fresnel lens has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic and are inexpensive.
Lenticular lenses are arrays of microlenses that are used in lenticular printing to make images that have an illusion of depth or that change when viewed from different angles.
A gradient index lens has flat optical surfaces, but has a radial or axial variation in index of refraction that causes light passing through the lens to be focused.
An axicon has a conical optical surface. It images a point source into a line along the optic axis, or transforms a laser beam into a ring. [27]
Diffractive optical elements can function as lenses.
Superlenses are made from negative index metamaterials and claim to produce images at spatial resolutions exceeding the diffraction limit. [28] The first superlenses were made in 2004 using such a metamaterial for microwaves. [28] Improved versions have been made by other researchers. [29] [30] As of 2014 the superlens has not yet been demonstrated at visible or near-infrared wavelengths. [31]
A prototype flat ultrathin lens, with no curvature has been developed. [32]
## Uses
A single convex lens mounted in a frame with a handle or stand is a magnifying glass.
Lenses are used as prosthetics for the correction of visual impairments such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the eyeballs; their curvature may not be axially symmetric to correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.
Other uses are in imaging systems such as monoculars, binoculars, telescopes, microscopes, cameras and projectors. Some of these instruments produce a virtual image when applied to the human eye; others produce a real image that can be captured on photographic film or an optical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up with curved mirrors to make a catadioptric system where the lens's spherical aberration corrects the opposite aberration in the mirror (such as Schmidt and meniscus correctors).
Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as burning-glasses for at least 2400 years. [7] A modern application is the use of relatively large lenses to concentrate solar energy on relatively small photovoltaic cells, harvesting more energy without the need to use larger and more expensive cells.
Radio astronomy and radar systems often use dielectric lenses, commonly called a lens antenna to refract electromagnetic radiation into a collector antenna.
Lenses can become scratched and abraded. Abrasion-resistant coatings are available to help control this. [33]
## Related Research Articles
In optics, aberration is a property of optical systems such as lenses that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with the nature of the distortion depending on the type of aberration. Aberration can be defined as a departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an object does not converge into a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.
The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal lengh indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.
An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths into focus on the same plane.
Spherical aberration is a type of aberration found in optical systems that use elements with spherical surfaces. Lenses and curved mirrors are most often made with surfaces that are spherical, because this shape is easier to form than non-spherical curved surfaces. Light rays that strike a spherical surface off-centre are refracted or reflected more or less than those that strike close to the centre. This deviation reduces the quality of images produced by optical systems.
Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. The closely related term spatial resolution refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments.
An optical telescope is a telescope that gathers and focuses light, mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct view, or to make a photograph, or to collect data through electronic image sensors.
A reflecting telescope is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century, by Isaac Newton, as an alternative to the refracting telescope which, at that time, was a design that suffered from severe chromatic aberration. Although reflecting telescopes produce other types of optical aberrations, it is a design that allows for very large diameter objectives. Almost all of the major telescopes used in astronomy research are reflectors. Reflecting telescopes come in many design variations and may employ extra optical elements to improve image quality or place the image in a mechanically advantageous position. Since reflecting telescopes use mirrors, the design is sometimes referred to as a "catoptric" telescope.
The Newtonian telescope, also called the Newtonian reflector or just the Newtonian, is a type of reflecting telescope invented by the English scientist Sir Isaac Newton (1642–1727), using a concave primary mirror and a flat diagonal secondary mirror. Newton's first reflecting telescope was completed in 1668 and is the earliest known functional reflecting telescope. The Newtonian telescope's simple design makes it very popular with amateur telescope makers.
Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in size, sometimes called minification or de-magnification.
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings it to focus creating an image. The eyepiece is placed near the focal point of the objective to magnify this image. The amount of magnification depends on the focal length of the eyepiece.
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, which is caused by diffraction from the optical system's aperture. Aberrations tend to get worse as the aperture diameter increases, while the Airy circle is smallest for large apertures.
A catadioptric optical system is one where refraction and reflection are combined in an optical system, usually via lenses (dioptrics) and curved mirrors (catoptrics). Catadioptric combinations are used in focusing systems such as searchlights, headlamps, early lighthouse focusing systems, optical telescopes, microscopes, and telephoto lenses. Other optical systems that use lenses and mirrors are also referred to as "catadioptric" such as surveillance catadioptric sensors.
The Cassegrain reflector is a combination of a primary concave mirror and a secondary convex mirror, often used in optical telescopes and radio antennas, the main characteristic being that the optical path folds back onto itself, relative to the optical system's primary mirror entrance aperture. This design puts the focal point at a convenient location behind the primary mirror and the convex secondary adds a telephoto effect creating a much longer focal length in a mechanically short system.
An aspheric lens or asphere is a lens whose surface profiles are not portions of a sphere or cylinder. In photography, a lens assembly that includes an aspheric element is often called an aspherical lens.
In optics, a thin lens is a lens with a thickness that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called thick lenses.
A curved mirror is a mirror with a curved reflecting surface. The surface may be either convex or concave. Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems, like spherical lenses, suffer from spherical aberration. Distorting mirrors are used for entertainment. They have convex and concave regions that produce deliberately distorted images.
Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which a flat object normal to the optical axis cannot be brought properly into focus on a flat image plane.
## References
1. The variant spelling lense is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.
• Brians, Paul (2003). Common Errors in English. Franklin, Beedle & Associates. p. 125. ISBN 978-1-887902-89-2 . Retrieved 28 June 2009. Reports "lense" as listed in some dictionaries, but not generally considered acceptable.
• Merriam-Webster's Medical Dictionary. Merriam-Webster. 1995. p. 368. ISBN 978-0-87779-914-6. Lists "lense" as an acceptable alternate spelling.
• "Lens or Lense – Which is Correct?". writingexplained.org. 30 April 2017. Analyses the almost negligible frequency of use and concludes that the misspelling is a result of a wrong singularisation of the plural (lenses).
2. Sines, George; Sakellarakis, Yannis A. (1987). "Lenses in antiquity". American Journal of Archaeology. 91 (2): 191–196. doi:10.2307/505216. JSTOR 505216.
3. Whitehouse, David (1 July 1999). "World's oldest telescope?". BBC News. Retrieved 10 May 2008.
4. "The Nimrud lens/The Layard lens". Collection database. The British Museum. Retrieved 25 November 2012.
5. D. Brewster (1852). "On an account of a rock-crystal lens and decomposed glass found in Niniveh". Die Fortschritte der Physik (in German). Deutsche Physikalische Gesellschaft. p. 355.
6. Kriss, Timothy C.; Kriss, Vesna Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery". Neurosurgery. 42 (4): 899–907. doi:10.1097/00006123-199804000-00116. PMID 9574655.
7. Aristophanes (22 January 2013) [First performed in 423 BC]. The Clouds. Translated by Hickie, William James. Project Gutenberg. EBook #2562.
8. Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 10.
9. Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 16
10. Tilton, Buck (2005). The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival. Menasha Ridge Press. p. 25. ISBN 978-0-89732-633-9.
11. Glick, Thomas F.; Steven John Livesey; Faith Wallis (2005). Medieval science, technology, and medicine: an encyclopedia. Routledge. p. 167. ISBN 978-0-415-96930-7 . Retrieved 24 April 2011.
12. Al Van Helden. The Galileo Project > Science > The Telescope. Galileo.rice.edu. Retrieved on 6 June 2012.
13. Henry C. King (28 September 2003). The History of the Telescope. Courier Dover Publications. p. 27. ISBN 978-0-486-43265-6 . Retrieved 6 June 2012.
14. Paul S. Agutter; Denys N. Wheatley (12 December 2008). Thinking about Life: The History and Philosophy of Biology and Other Sciences. Springer. p. 17. ISBN 978-1-4020-8865-0 . Retrieved 6 June 2012.
15. Vincent Ilardi (2007). Renaissance Vision from Spectacles to Telescopes. American Philosophical Society. p. 210. ISBN 978-0-87169-259-7 . Retrieved 6 June 2012.
16. Microscopes: Time Line, Nobel Foundation. Retrieved 3 April 2009
17. Fred Watson (1 October 2007). Stargazer: The Life and Times of the Telescope. Allen & Unwin. p. 55. ISBN 978-1-74175-383-7 . Retrieved 6 June 2012.
18. This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.
19. Greivenkamp 2004 , p. 14
Hecht 1987 , § 6.1
20. Hecht 1987, § 5.2.3.
21. Nave, Carl R. "Thin Lens Equation". Hyperphysics. Georgia State University. Retrieved 17 March 2015.
22. Colwell, Catharine H. "Resource Lesson: Thin Lens Equation". PhysicsLab.org. Retrieved 17 March 2015.
23. "The Mathematics of Lenses". The Physics Classroom. Retrieved 17 March 2015.
24. Hecht 2002, p. 120.
25. There are always 3 "easy rays". For the third ray in this case, see File:Lens3b third ray.svg.
26. Hecht 2002, p. 168.
27. Proteep Mallik (2005). "The Axicon" (PDF). Archived from the original (PDF) on 23 November 2009. Retrieved 22 November 2007.
28. Grbic, A.; Eleftheriades, G. V. (2004). "Overcoming the Diffraction Limit with a Planar Left-handed Transmission-line Lens". Physical Review Letters . 92 (11): 117403. Bibcode:2004PhRvL..92k7403G. doi:10.1103/PhysRevLett.92.117403. PMID 15089166.
29. Valentine, J.; et al. (2008). "Three-dimensional optical metamaterial with a negative refractive index". Nature . 455 (7211): 376–9. Bibcode:2008Natur.455..376V. doi:10.1038/nature07247. PMID 18690249.
30. Yao, Jie; Liu, Zhaowei; Liu, Yongmin; Wang, Yuan; Sun, Cheng; Bartal, Guy; Stacy, Angelica M.; Zhang, Xiang (15 August 2008). "Optical Negative Refraction in Bulk Metamaterials of Nanowires". Science. 321 (5891): 930. Bibcode:2008Sci...321..930Y. CiteSeerX . doi:10.1126/science.1157566. ISSN 0036-8075. PMID 18703734.
31. Nielsen, R.B.; Thoreson, M.D.; Chen, W.; Kristensen, A.; Hvam, J.M.; Shalaev, V. M.; Boltasseva, A. (2010). "Toward superlensing with metal–dielectric composites and multilayers" (PDF). Applied Physics B. 100 (1): 93. Bibcode:2010ApPhB.100...93N. doi:10.1007/s00340-010-4065-z. Archived from the original (PDF) on 9 March 2013.
32. Patel, Prachi. "Good-Bye to Curved Lens: New Lens Is Flat" . Retrieved 16 May 2015.
33. Schottner, G (May 2003). "Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives". Journal of Sol-Gel Science and Technology . pp. 71–79. Retrieved 28 December 2009. | 2021-05-10 07:21:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 17, "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": 0.6606041789054871, "perplexity": 1353.7803394685263}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989115.2/warc/CC-MAIN-20210510064318-20210510094318-00601.warc.gz"} |
https://socialsci.libretexts.org/Bookshelves/Communication/Book%3A_Media_Innovation_and_Entrepreneurship_(Ferrier_and_Mays)/07%3A_Startup_Funding/7.05%3A_Startup_Funding-_Crowdfunding | # 7.5: Startup Funding- Crowdfunding
## CJ Cornell
Crowdfunding, which barely existed a few years ago, spread to every aspect of society—from funding creative projects, local civic projects to causes and charities, and of course—funding startup companies. Today, crowdfunding is an essential part of every entrepreneurship conversation.
Modern crowdfunding has the following haiku-like definition:[1]
• An individual or organization,
• on behalf of a cause, a project, a product, or a company,
• solicits and collects money,
• usually in relatively small amounts,
• from a large number of people,
• using an online platform,
• where communications and transactions are managed over electronic networks.
Crowdfunding is one of the great movements of the twenty-first century, and one of the most powerful. Still in its infancy, crowdfunding moved beyond Kickstarter-like passion projects and became a major new and disruptive force in investment funding. And just in the last few years, the crowdfunding movement impelled the U.S. Congress, state legislatures in all 50 states and the Security Exchange Commission to overhaul investment laws that existed since 1934.
Broadly defined, crowdfunding is collecting small amounts of money from a large number of people. With the advent of social networks (abundance of people on the network), crowdfunding only recently became practical. But the concept has been around for centuries—except that with crowdfunding, the people willingly contribute money for a project, product, or cause that they believe in.
# Crowfunding—a Little History
The concept of crowdfunding is old. What’s new is applying technology—social networks—to the process. This allows interested supporters from around the globe to support any project, anywhere, as long as they share passion and see merit in the project’s goals.
In the early 1880s, France gave the United States a gift: the Statue of Liberty. It was shipped to New York City where remained packed in crates for over a year. Why? The New York state government would not allocate the $200,000 required to build and mount the statue onto a pedestal (today this would be over$2.5 million). Newspaper mogul Joseph Pulitzer ran a fundraising campaign through his “New York World” tabloid—offering to publish the name of everyone who donated, to the front page. Donors also would get little rewards: $1 got you a 6-inch replica statue and$5 got you a 12-inch statue. The campaign went viral. Within six months of the fundraising appeal, the pedestal was fully funded, with the majority of donations being under a dollar.
But this was not the first crowdfunding campaign. For centuries, book authors have appealed to the crowd for funding. Kickstarter (the largest crowdfunding site) proudly recounts one of the earliest examples:[2]
In 1713, Alexander Pope (the poet) set out to translate 15,693 lines of ancient Greek poetry into English. It took five long years to get the six volumes right, but the result was worth the wait: a translation of Homer’s Iliad that endures to this day. How did Pope go about getting this project off the ground? Turns out he kind of Kickstarted it.
In exchange for a shout-out in the acknowledgments an early edition of the book, and the delight of helping to bring a new creative work into the world, 750 subscribers pledged two gold guineas to support Pope’s effort before he put pen to paper. They were listed in an early edition of the book. | 2020-02-20 11:19:13 | {"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": 0.22460120916366577, "perplexity": 4731.483929784867}, "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-2020-10/segments/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00044.warc.gz"} |
https://physics.stackexchange.com/questions/400970/definition-of-creation-and-annihilation-operators | # Definition of creation and annihilation operators
An appropriately symmetrized (and normalized) $N$-particle physical ket can be expressed in the form: $$| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle \equiv \frac{1}{\sqrt{N! \prod_{i=1}^{\infty}(n_{\beta_{i}}!)}} \sum_{\mathcal{P} \in S_{N}} \zeta^{[1-\mathrm{sgn}(\mathcal{P})]/2} \, \mathcal{P} (|\beta_{1}\rangle \otimes |\beta_{2} \rangle \otimes \dots \otimes |\beta_{N} \rangle)\,,$$ with $\sum_{i=1}^{\infty} n_{i} = N$. In this equation:
• $n_{\beta_{i}}$ is the occupation number of the state $|\beta_{i}\rangle$ in the set of states $\{|\beta_{1}\rangle, |\beta_{2}\rangle, \dots, |\beta_{N}\rangle\}$
• for fermions $\zeta = -1$, while for bosons $\zeta = 1$
• $\mathcal{P}$ is a permutation operator belonging the the symmetric group on $N$ letters, $S_{N}$, and acting on the tensor product state $|\beta_{1}\rangle \otimes |\beta_{2} \rangle \otimes \dots \otimes |\beta_{N} \rangle$
• $\mathrm{sgn}(\mathcal{P})$ is the sign of the permutation $\mathcal{P}$, being +1 if $\mathcal{P}$ is even or -1 if it is odd.
In the book by Negele, J.W. and Orland, H., Quantum Many-Particle Systems, the creation operator $a_{\lambda}^{\dagger}$ is defined in a Fock space as: $$a_{\lambda}^{\dagger}| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle \equiv \sqrt{n_{\lambda} + 1}\,| \lambda, \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle\,,$$ where $n_{\lambda}$ is the occupation number of the state $|\lambda\rangle$ in $| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle$.
Following this definition, the annihilation operator $a_{\lambda}$ is simply the adjoint of $a_{\lambda}^{\dagger}$, and its action on a many-particle state is determined using the closure relation in the Fock space:
$$|0\rangle\langle0|+\sum_{M=1}^{\infty}\frac{1}{M!}\sum_{\alpha_{1}, \dots, \alpha_{M}} \prod_{i=1}^{\infty} (n_{\beta_{i}}!)\,|\alpha_{1}, \alpha_{2}, \dots, \alpha_{M}\rangle\langle\alpha_{1}, \alpha_{2}, \dots, \alpha_{M}| = \mathbb{1}\,.$$
Now, the problem is that when I try to derive an expression for $a_{\lambda}$, I am not able to obtain the prefactor $\frac{1}{\sqrt{n_{\lambda}}}$, as in equation 1.78b of the aforementioned book, which reads:
$$a_{\lambda} |\beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle = \frac{1}{\sqrt{n_{\lambda}}} \sum_{i = 1}^{N} \zeta^{i-1} \delta_{\lambda, \beta_{i}} |\beta_{1}, \dots, \overline{\beta_{i}}, \dots, \beta_{N}\rangle\,,$$ in which $\delta$ is the Kronecker delta and $\overline{\beta_{i}}$ indicates that the state $|\beta_{i}\rangle$ has been removed from the $N$-particle state $|\beta_{1}, \beta_{2}, \dots, \beta_{N}\rangle$.
Any help? Thanks.
The formula for the annihilator is written to be valid for bosons and fermions and assumes that $n_\lambda$ is not zero and (because $\frac{1}{\sqrt{n_\lambda}}$ will be $1/0$ then, but we can define that $0/\sqrt{0} = 0$). This generality causes the complicated structure (for bosons one would usually work with states labelled by occupation numbers, since all permutations of the state labels signify the same state).
In the Fermi case $n_\lambda$ is one or zero and only one term will contribute on the right hand side, the $\zeta^{i-1}$ produces the correct signs. (Note: even if the Fermi state is a zero written in a fancy way like $\left|1,0,1\right>$ or $\left|1,1,0\right>$ the formula will be correct, the two terms on the right hand side will cancel).
Now for bosons all permutations of the labels signify the same state, so we can think in terms of occupation numbers, and derive that $a_\lambda\left|n\right> = \sqrt{n}\left|n-1\right>$ by taking the adjoint of the formula for the creator (and that the occupation numbers of all other states do not matter).
Now there are $n_\lambda$ equal terms on the right hand side (since any permutation of the labels implies the same state) and $a_\lambda$ has to create a factor $\sqrt{n_\lambda}$ overall to be the adjoint to $a_\lambda^\dagger$, therefore you need the $n_\lambda^{-1/2}$ to arrive at the correct factor. | 2022-06-30 14:39:29 | {"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": 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": 0.9558578729629517, "perplexity": 111.58712158352553}, "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/1656103821173.44/warc/CC-MAIN-20220630122857-20220630152857-00021.warc.gz"} |
http://math.stackexchange.com/questions/166560/how-is-i-0-inftyt-defined | # How is $I_{[0,\infty)}(t)$ defined?
How is $I_{[0,\infty)}(t)$ defined? This must be a notation in probabilty theory.
-
It's likely an indicator function: it has value 1 on $[0,\infty)$ and 0 on $(-\infty,0)$. – David Mitra Jul 4 '12 at 12:55
Although indicator functions come up in probability, it doesn't seem that an indicator over an infinite interval would come up. – Michael Chernick Jul 4 '12 at 15:26
You can use the Heaviside Step Function $H(t)$:
$H(0) = 1$ is used when $H$ needs to be right-continuous. For instance cumulative distribution functions are usually taken to be right continuous, as are functions integrated against in Lebesgue–Stieltjes integration. In this case $H$ is the indicator function of a closed semi-infinite interval: $$H(t) = \mathbf{I}_{[0,\infty)}(t).\,$$
...or one could use Iverson brackets as well: $[t \geq 0]$. – J. M. Aug 9 '12 at 10:49 | 2016-07-24 05:16:59 | {"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": 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": 0.8997180461883545, "perplexity": 531.9855047692052}, "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-2016-30/segments/1469257823947.97/warc/CC-MAIN-20160723071023-00305-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:0795.35105 | # zbMATH — the first resource for mathematics
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On the (generalized) Korteweg-de Vries equation. (English) Zbl 0795.35105
The well-posedness of the initial value problem for the Korteweg-de Vries equation $u\sb t+ u\sb{xxx}+ uu\sb x =0$ and its generalized form $u\sb t+ u\sb{xxx}+ a(u) u\sb x=0$ in the classical Sobolev spaces and the regularity of their solutions in $L\sb s\sp p$ spaces are studied. A global smoothing effect of the solutions of these equations is also proved. See also a paper by {\it T. Kato} [Studies in applied mathematics, Adv. Math., Suppl. Stud., Vol. 8, 93-128 (1983; Zbl 0508.00010)].
##### MSC:
35Q53 KdV-like (Korteweg-de Vries) equations 35B65 Smoothness and regularity of solutions of PDE
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[1] J. Bergh and J. Löfström, Interpolation Spaces , Springer, 1970. [2] J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation , Roy. Soc. Lond. Ser A 278 (1975), no. 1287, 555-601. JSTOR: · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035 · http://links.jstor.org/sici?sici=0080-4614%2819750703%29278%3A1287%3C555%3ATIPFTK%3E2.0.CO%3B2-I&origin=euclid [3] J. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces , Duke Math. J. 43 (1976), no. 1, 87-99. · Zbl 0335.35032 · doi:10.1215/S0012-7094-76-04309-X [4] R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E , Beijing lectures in harmonic analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3-45. · Zbl 0623.47052 [5] J. Ginibre and Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equations , · Zbl 0702.35224 [6] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations , Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin, 1975, pp. 25-70. · Zbl 0315.35077 [7] T. Kato, On the Korteweg-de Vries equation , Manuscripta Math. 28 (1979), no. 1-3, 89-99. · Zbl 0415.35070 · doi:10.1007/BF01647967 · eudml:154631 [8] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation , Studies in Applied Math., Advances in Mathematics Supplementary Studies, vol. 8, Academic Press, New York, 1983, pp. 93-128. · Zbl 0549.34001 [9] T. Kato and G. Ponce, On nonstationary flows of viscous and ideal fluids in $L^ p_s(\mathbb R^2)$ , Duke Math. J. 55 (1987), no. 3, 487-499. · Zbl 0649.76011 · doi:10.1215/S0012-7094-87-05526-8 [10] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations , Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. · Zbl 0671.35066 · doi:10.1002/cpa.3160410704 [11] S. N. Kruzhkov and A. V. Framinskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation , Math. U.S.S.R. Sbornik 48 (1984), 93-138. · Zbl 0549.35104 · doi:10.1070/SM1984v048n02ABEH002682 [12] B. Marshall, Mixed norm estimates for the Klein-Gordon equation , Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 638-649. · Zbl 0516.35047 [13] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation , Math. Z. 185 (1984), no. 2, 261-270. · Zbl 0538.35063 · doi:10.1007/BF01181697 · eudml:173400 [14] J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation , Israel J. Math. 24 (1976), no. 1, 78-87. · Zbl 0334.35062 · doi:10.1007/BF02761431 [15] E. M. Stein, Oscillatory Integrals in Fourier Analysis , Beijing Lectures in Harmonic Analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton University Press, Princeton, NJ, 1986, pp. 307-355. · Zbl 0618.42006 [16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces , Princeton University Press, Princeton, N.J., 1971. · Zbl 0232.42007 [17] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations , Duke Math. J. 44 (1977), no. 3, 705-714. · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1 [18] R. Temam, Sur un problème non linéaire , J. Math. Pures Appl. (9) 48 (1969), 159-172. · Zbl 0187.03902 [19] P. Tomas, A restriction theorem for the Fourier transform , Bull. Amer. Math. Soc. 81 (1975), 477-478. · Zbl 0298.42011 · doi:10.1090/S0002-9904-1975-13790-6 | 2016-05-03 08:49:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7468899488449097, "perplexity": 2398.4174911996483}, "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/1461860121090.75/warc/CC-MAIN-20160428161521-00136-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://secretsheep.com/l1v4zv/cc181c-maximum-oxidation-state-is-shown-by-os-mn-co-cr | (-2 oxidation state). So if we do the maths, (letting the charge of the Manganese ion be X), X + 4(-2) = -1 X= +7 So the oxidation number of Mn in the MnO4 ion is +7. Chlorine can take one electron to form chloride anion. i) Mn Shows the highest oxidation state of +7 with oxygen but with fluorine, it shows the highest oxidation state of +4 because of the ability of oxygen to form multiple bonds with Mn metal. b) number of d-electrons. The oxidation number of all elements in the elemental state is zero. The oxidation states are also maintained in articles of the elements (of course), and systematically in the table {{Infobox element/symbol-to-oxidation-state}} (An overview is here). ii) Cr2+ is strongly reducing in nature. Highest (+7) oxidation state is shown by [MP PMT 1990, 2001; RPMT 1999; AIIMS 1999; JIPMER 2001; CBSE PMT 1994, 2002; MP PET 1989, 2003] Manganate ions, or MnO4, have a charge of -1. It has a d4 configuration. Higher oxidation states are shown by chromium, manganese and cobalt. Only Sc (+II) and Co(+V) are in doubt. In addition, several of the elements have zero-valent and other low-valent states in complexes. Mo(CO)6 or [Mn(CO)6]+, because the CO-to-metal Ï-donor electron transfer will be enhanced at the expense of the metal to CO back donation. (-1 oxidation state). The maximum oxidation states observed for the second- and third-row transition metals in groups 3â8 increase from +3 for Y and La to +8 for Ru and Os, corresponding to the formal loss of all ns and (n â 1)d valence electrons. c) determine which dominates, splitting energy or pairing energy (low spin or high spin) d) number of unpaired electrons. For which one of these metals the change in oxidation state from +2 to +3 is easiest? Until much more research has been performed, you should probably not attempt to predict maximum and minimum oxidation states of these elements. Sol: EAN = 25 (electrons from Mn atom) + 10 (electrons from fiveCO ligand) + 1 (electron from MnâMn bond) = 36 Thus, structure will be, complex formed with a cyclic polydentate ligand when See spectrochemical series in appendix for ligand abbreviation. Chlorine can give seven electrons to make chloric acid to show +7 oxidation number. K3 [Re(Ox)3], Ca3[Co(NO3)4CO3], [Os(bpy)2(CO)2]Cl3. ⢠In LnM(CO), the CO carbon becomes particularly δ+ in character if the L groups are good Ï acids or if the complex is cationic, e.g. Stability of oxidation states. The maximum oxidation states observed for the second- and third-row transition metals in groups 3â8 increase from +3 for Y and La to +8 for Ru and Os, corresponding to the formal loss of all ns and (n â 1)d valence electrons. These facts may be conveniently memorized, because the oxidation states form a regular âpyramidâ as shown in Table 18.2. Oxygen will usually have an oxidation number of -2. All of this complicates the analysis strongly. Example 6: The EAN of each Mn (Z = 25) in Mn 2 (CO) 10 is 36. e) wavelength of light absorbed. It does not show optical isomerism. Maintenance & ⦠As an example, $\ce{[Fe(CO)4]^2-}$ with an iron oxidation state of $\mathrm{-II}$ is known. The E °M 3+ / M 2+ values for Cr, Mn, Fe and Co are â 0.41, +1.57, 0.77 and +1.97 V respectively. Sulfur gives its all last six electrons to make sulfuric acid molecule (+6 oxidation state). a) Oxidation state. unpaired electron. Sulfur can take two electrons to form sulfide anion. 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https://www.homesteadingtoday.com/threads/pygmy-lamancha-cross.404927/ | # Pygmy/LaMancha Cross?
Discussion in 'Goats' started by Rechellef, Jul 17, 2011.
1. ### RechellefShow us your teats!!
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I know this sounds like a silly question, but some friends of our raise Pygmys and they are interested in breeding one of their does because they are interested in possibly the dairy end of the goat world. My buckling's momma is a heavy milker, so the genese are there. Would this cross result in a potential dairy goat? Would it be the same as a mini-mancha or does it have to be a ND crossed with a LaMancha to produce a mini-dairy breeed?
Last edited: Jul 17, 2011
2. ### Alice In TX/MOMore dharma, less drama.Supporter
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Mini Manchas are made with a LaMancha doe and a Nigerian Dwarf buck.
You should NOT breed a Pygmy doe with a full size buck. The resulting kids could be too big to pass through the pelvis of the doe at birth.
Pygmies are not dairy animals.
3. ### RechellefShow us your teats!!
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I didn't think so, but I am fairly new to this, so I always check on things like this to make sure.
4. ### grandmajoWell-Known Member
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If your friends are interested in a more dairy type of animal, tell them to look for pygmies that are milking, they are out there. A pygmy, if it comes from the right lines, can produce a respectable amount of milk for their size and it's extremely rich.
On a side note, there have been kinder breeders who have started out by breeding a Nubian buck to a pygmy doe. However, the doe is usually from the old style line, which is generally bigger than the pygmies that most people are breeding today.
5. ### goatkidWell-Known MemberSupporter
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I once had a doeling that was out of a Pygmy doe and a La Mancha buck. She was a sweet little thing, but infortunately, I nver got to see how she'd do as a milker. Our neighbor's dog killed her. Though her dam had no problems delivering her, I would not suggest breeding a Pygmy doe to a full size buck. Instead, sell her a La Mancha doeling to breed to one of her bucks.
6. ### LFRJWell-Known MemberSupporter
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Isn't that a pygmy BUCK over a Nubian DOE? The result would probably be similar, I'd imagine, depending on the parents, and I'm not sure, but I thought it was originally the other way around sire/dam wise.
7. ### chewieWell-Known MemberSupporter
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I am agreeing with LFRJ, i am perty sure it was the pygmy buck used. I would be concerned that the other way around would be hard on the doe as others have pointed out.
8. ### grandmajoWell-Known Member
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NW corner of Ohio
That is the way that the KGBA recommends that it be done, but if you look on their website they state that it has been done this way. They don't however recommend it. From their website:
"There are a few breeders who have begun successfully with a Pygmy doe and a Nubian buck. The Association feels, however, that it is a safer procedure to use the larger doe with the smaller buck." | 2017-06-25 01:59:50 | {"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": 0.8587861061096191, "perplexity": 5352.453919693969}, "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-2017-26/segments/1498128320386.71/warc/CC-MAIN-20170625013851-20170625033851-00538.warc.gz"} |
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Q. . Why first group elements are called alkali metals ?
Sol. Group I elements are highly reactive and with water (moisture in the atmosphere) form strong alkalies, so they are called alkali metals.
Q. . Write the chemical name and formula of washing soda.
Sol. Washing soda is sodium carbonate $\left(N a_{2} C O_{3}\right)$.
Q. . Why do alkali metals not occur in free state ?
Sol. They are highly reactive, therefore, they occur in combined state and do not occur in free state.
Q. . Write the important minerals of lithium.
Sol. The important minerals of lithium are:
(i) Lipidolite $(L i . N a . K)_{2} A l_{2}\left(S i O_{3}\right)_{3} \cdot F(O H)$
(ii) Spodumene $L i A I S i_{2} O_{3}$
(iii) Amblygonite $\operatorname{LiAl}\left(P O_{4}\right) F$
Q. . How sodium hydroxide is prepared at large scale ?
Sol. At large scale, sodium hydroxide is prepared by castner kellner cell.
Q. . Why $\mathrm{KHCO}_{3}$ is not prepared by Solvay process ?
Sol. Because solubility of $\mathrm{KHCO}_{3}$ is fairly large as compared to $\mathrm{NaHCO}_{3}$.
Q. . What is chemical composition of the Plaster of Paris ?
Sol. Plaster of paris is calcium sulphate hemihydrate :
$\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2} \mathrm{O}$ or $\left(\mathrm{CaSO}_{4}\right)_{2} \cdot \mathrm{H}_{2} \mathrm{O}$
Q. . Why do alkali metals have low density ?
Sol. Due to weak metallic bonds and large atomic size, their density is low.
Q. . Why is first ionization energy of alkali metals lower than those of alkaline earth metals ?
Sol. Alkali metals have bigger atomic size, therefore they have lower first I.E. than group 2 elements.
Q. . First group elements are strong reducing agents, why ?
Sol. Because they have a strong tendency to lose outer most electron.
Q. . Explain why ?
LiI is more soluble than KI in ethanol. [NCERT]
Sol. $K I$ In the chemical bond is ionic in character. On the other hand due to small size of lithium ion and its high polarising power the bond in is predominently covalent in character. Hence LiI is more soluble than in ethanol.
Q. . LiH is more stable than $\mathrm{NaH}$ Explain.
Sol. Both $L i^{+}$ and $H^{-}$ have small size and their combination has high lattice energy. Therefore LiH is stable as compared with $\mathrm{NaH}$
Q. . Why is $B e C l_{2}$ soluble in organic solvents ?
Sol. $B e C l_{2}$ is covalent, therefore soluble in organic salvents.
Q. . Name the metal which floats on water without any apparent reaction with water. [NCERT]
Sol. Lithium floats on water without any apparent reaction with it.
Q. . Name an element which is invariably bivalent acid and whose oxide is soluble in excess of NaOH and its dipositive ion has a noble gas core.
Sol. The element is beryllium, its oxide $B e O$ is soluble in excess of
$\mathrm{NaOH}$
$B e O+2 N a O H \longrightarrow N a_{2} B e O_{2}+H_{2} O$
Its dipositive ion has electronic configuration $\left(B e^{2+}=1 s^{2}\right)$
Q. . State reason for the high solubility of $B e C l_{2}$ in organic solvents.
Sol. Because $B e C l_{2}$ is covalent compound.
Q. . What is the cause of diagonal relationship ? [NCERT]
Sol. The charge over radius ratio, i.e., polarizing power is similar, that is the cause of diagonal relationship.
[esquestion]. Name the alkali metals which forms superoxide when heated in excess of air.
Q. . Which out of K, Mg,Ca & Al form amphoteric oxide ?
Sol. form amphoteric oxide, i.e., acids as well as basic in nature.
Q. . Explain the following :
Sodium wire is used to dry benzene but cannot be used to dry ethanol.
Sol. Sodium metal removes moisture from benzene by reacting with water. However, ethanol cannot be dried by using sodium because it reacts with sodium.
$2 \mathrm{Na}+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{ONa}+\mathrm{H}_{2}$
Q. . Why is $\mathrm{CaCl}_{2}$ added to $N a C l$ in extraction of $N a$ by Down cell ?
Sol. $\mathrm{CaCl}_{2}$ reduces melting point of $N a C l$ and increases electrical conductivity.
Q. . Carbon dioxide is passed through a suspension of limestone in water. Write balanced chemical equation for the above reaction.
Sol. $\mathrm{CaCO}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \longrightarrow \mathrm{Ca}\left(\mathrm{HCO}_{3}\right)_{2}$
Q. . What do we get when crystals of washing soda exposed to air ?
Sol. We get amorphous sodium carbonate becouse it loses water molecules.
Q. . $M g_{3} N_{2}$ when react with water gives off $\mathrm{NH}_{3}$ but $H C l$ is not obtained from $M g C l_{2}$on reaction with water at room temperature.
Sol. $M g_{3} N_{2}$ is a salt of a strong base, $M g(O H)_{2}$ and a weak acid
$\left(N H_{3}\right)$ and hence gets hydrolysed to give $N H_{3} .$ In contrast,
$M g C l_{2}$ is a salt of a strong base, and a strong acid, and hence does not undergo hydrolysis to give
Q. . Why caesium can be used in photoelectric cell while lithium cannot be ?
Sol. Caesium has the lowest while lithium has the highest ionization enthalpy among all the alkali metals. Hence, caesium can lose electron very easily while lithium cannot.
Q. . Which nitrates are used in pyrotechnics ?
Sol. Strontium and Barium nitrates are used in pyrotechnics for giving red and green flames.
Q. .What are s-block elements ? Write their electronic configuration.
Sol. The elements in which the last electron enters the -orbital of their outermost energy level are called -block elements.
It consists of Group 1 and Group 2 elements. Their electronic configuration is
Q. . Name the metals which are found in each of the following minerals :
(i) Chile Salt Petre (ii) Marble
(iii) Epsomite (iv) Bauxite.
Sol. (i) $\quad N a$(ii) $\mathrm{Ca}$ (iii) $M g \quad$ (iv) $\quad A l$
Q. . What is composition of Portland cement ? What is average composition of good quality cement ?
[NCERT]
Sol. $\mathrm{CaO}=50$ to $60 \% \quad \mathrm{SiO}_{2}=20$ to $25 \%, \quad \mathrm{Al}_{2} \mathrm{O}_{3}=5$ to $10 \%$
$M g O=2$ to $3 \%, \quad F e_{2} O_{3}=1$ to $2 \%, \quad S O_{2}=1 \quad$ to $2 \% \quad$ is The ratio of $S i O_{2}$ (silica) to alumina $\left(A l_{2} O_{3}\right)$ should be between
2.5 and 4.0 and the ratio of lime $(\mathrm{CaO})$ to total oxides of silicon
$\mathrm{SiO}_{2}, \mathrm{Al}_{2} \mathrm{O}_{3}$ and $\mathrm{Fe}_{2} \mathrm{O}_{3}$ should be as close to 2 as possible.
Q. . Write chemical reactions involved in Down process for obtaining Mg from sea water.
Sol.
Q. . What is the mixture of $\mathrm{CaCN}_{2}$ and carbon called ? How is it prepared ? Give its use.
Sol.
Q. . State the difficulties in extraction of alkaline earth metals from the natural deposits.
Sol. Like alkali metals, alkaline earth metals are also highly electropositive and strong reducing agents. Same difficulties, we face in the extraction of these metals. Therefore, these metals are extracted by electrolysis of their fused metal halides.
[esquestion]. Starting with sodium chloride how would you proceed to prepare (state the steps only) :
(i) sodium metal (ii) sodium hydroxide
(iii) sodium peroxide (iv) sodium carbonate
#tag# [NCERT]
Q. .Write three general characteristics of the elements of s-block of the periodic table which distinguish them from the elements of the other blocks.
[NCERT]
Sol. (i) They do not show variable oxidation states.
(ii) They are soft metals having low melting and boiling point.
(iii) They are highly electropositive and most reactive metals.
Q. . Why is $L i F$ almost insoluble in water whereas $L i C l$ is soluble not only in water but also in acetone ?
[NCERT]
Sol. $L i F$ is an ionic compound containing small ions and hence has very high lattice enthalpy. Enthalpy of hydration in this case is not sufficient to compensate for high lattice enthalpy. Hence, is insoluble in water. has partial covalent character due to polarization of chloride ion by ion. Thus, has partial covalent character and partial ionic character and hence is soluble in water as well as less polar solvents such as acetone.
Q. . When an alkali metal dissolves in liquid ammonia the solution acquires different colours. Explain the reasons for this type of colour change.
[NCERT]
Sol. When an alkali metal is dissolved in liquid ammonia it produces a blue coloured conducting solution due to formation of ammoniated cation and ammoniated electron as given below :
When the concentration is above the colour of solution is copper-bronze. This colour change is because the concentrated solution contains clusters of metal ions and hence possess metallic lustre.
Q. . Arrange the following in the decreasing order of the property mentioned :
Sol.
[esquestion]. Alkali metals have low ionisation enthalpies.Why is it so?
Q. . Which out of $L i, N a, K, B e, M g, C a$ has lowest ionisation enthalpy and why ?
Sol. $K^{\prime}$ has lowest ionisation energy due to larger atomic size among these elements. The force of attraction between valence electron and nucleus is less, therefore it can loose electron easily.
Q. . What is responsible for the blue colour of the solution of alkali metal in liquid ammonia ? Give chemical equation also.
Sol. The solvated electron, $e\left(N H_{3}\right)_{x}$ or ammoniated electron is responsible for blue colour of alkali metal solution in It absorbs light from visible regions and radiates complimentary colour:
$2 N a(s)+2 N H_{3}(l) \longrightarrow 2 N a N H_{2}(s)+H_{2}(g)+e\left(N H_{3}\right)_{x}$
Q. . The alkali metals follow the noble gases in their atomic structure. What properties of these metals can be predicted from the information ?
Sol. (i) They form unipositive ions.
(ii) Their second ionisation energy is very high.
(iii) They have weak metallic bonds due to larger size and only one valence electron.
Q. . Comment on each of the following observations :
(i) The mobilities of the alkali metal ions in aqueous solution are $L i^{+}<N a^{+}<R b^{+}<C s^{+}$
(ii) Lithium is the only alkali metal to form directly a nitride.
Sol. (i) $L i^{+}$ ions are smallest in size therefore most hydrated, that is why they have lowest mobility in aqueous solution. $C s^{+}$ ions are largest in size, least hydrated, therefore have highest mobility. Size of hydrated cation decreases, therefore, mobility of ions increases down the group.
(ii) $L i$ is smallest in size and best reducing agent, therefore, it forms nitride with $N_{2}$
$6 L i+N_{2} \longrightarrow 2 L i_{3} N$ (Lithium nitride)
(iii) It is due to less difference in their standard reduction potentials which is resultant of sublimation energy, ionisation energy and hydration energy.
(iv) is least ionic as compared to other fluorides of alkali metals. It has high lattice energy, therefore, it is least soluble.
[esquestion]. Why do alkali metals impart characteristic colours to the flame of a bunsen burner ? What is the colour imparted to the flame by each of the following metals ? Lithium, Sodium and Potassium.
Q. . Commercial aluminium always contains some magnesium. Name two such alloys of aluminium. What properties are imparted by the addition of magnesium in these alloys ?
[NCERT]
Sol. Duralium and Magnaliam are alloys of $\mathrm{Al}, \mathrm{Mg}$ is lighter in density than $A l$ therefore, it makes the alloys lighter. These alloys are used in automobile engines and aeroplanes.
Q. . Arrange the (i) hydroxides and (ii) Sulphates of alkaline earth metals in order of decreasing solubilities., giving a suitable reason for each.
Sol. $-B e(O H)_{2}<M g(O H)_{2}<C a(O H)_{2}<S r(O H)_{2}<B a(O H)_{2}$
Solubility of hydroxide goes on increasing down the group because hydration energy dominates over lattice energy.
$B e S O_{4}>M g S O_{4}>C a S O_{4}>S r S O_{4}>B a S O_{4}$
Solubility of sulphate goes on decreasing down the group because lattice energy dominates over hydration energy.
Q. . State the properties of beryllium different than other elements of the group.
Sol. Properties of beryllium different than other elements of the group
(i) Beryllium is harder than other elements.
(ii) Melting and boiling points are higher than that of other elements.
(iii) It does not react with water but other elements do.
(iv) It does not react with acids to form hydrogen.
(v) It forms covalent compounds but others form ionic compounds.
(vi) Beryllium oxide is amphoteric but other oxides are basic.
Q. . Mention the general trends in Group 1 and in Group 2 with increasing atomic number with respect to
(i) density (ii) melting point (iii) atomic size
(iv) ionization enthalpy.
[NCERT]
Sol.
Q. . How do the following properties change on moving from Group 1 to Group 2 in the periodic table ?
(i) Atomic size (ii) Ionization enthalpy
(iii) Density (iv) Melting points.
Sol. (i) Atomic size decrease from group 1 to group 2 due to increase in effective nuclear charge.
(ii) First ionisation enthalpy increases from group 1 to group 2 due to decrease in atomic size.
(iii) Density increases from group 1 to 2.
(iv) Melting points increase from group 1 to 2.
Q. . Compare and contrast the chemistry of Group 1 metals with that of Group 2 metals with respect to
(i) nature of oxides
(ii) solubility and thermal stability of carbonates
(iii) polarizing power of cations
(iv) reactivity and reducing power.
Sol. (i) Group 1 metals form oxides of strong basic nature (except $L i_{2} O$. Group 2 metal form oxide of less basic nature.
(ii) Carbonates : Carbonates of Group 1st are soluble and stable, solubility in case of Group 2nd decreases down the group.
(iii) Polarizing power of cations increases.
(iv) Reactivity and reducing power decreases.
Q. . Explain what happens when ?
(i) Sodium hydrogen carbonate is heated
(ii) Sodium amalgam reacts with water
(iii) Fused sodium metal reacts with ammonia.
Sol. (i) On heating sodium bicarbonate it forms sodium carbonate
$2 \mathrm{NaHCO}_{3} \stackrel{\Delta}{\longrightarrow} \mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2}$
(ii) When sodium amalgam, $N a / H g,$ is formed the vigourosity of reaction of sodium with water decreases.
$2 \mathrm{Na} / \mathrm{Hg}+2 \mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{NaOH}+\mathrm{H}_{2}$
(iii) Sodium reacts with ammonia to form amide.
$2 \mathrm{Na}+2 \mathrm{NH}_{3} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{NaNH}_{2}+\mathrm{H}_{2}$
Q. . State as to why ?
(i) An aqueous solution of sodium carbonate give alkaline tests.
$\mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{Na}^{+}+\mathrm{OH}^{-}+\mathrm{HCO}_{3}^{-}$
(ii) Sodium is prepared by electrolytic method and not by chemical method.
Sol. (i) An aqueous solution of sodium carbonate has a large concentration of hydroxyl ions making it alkaline in nature.
(ii) Sodium is a very strong reducing agent therefore it cannot be extracted by the reduction of its ore (chemical method). Thus the best way to prepare sodium is by carrying electrolysis of its molten salts containing impurities of calcium chloride.
Q. . Explain why :
(i) Lithium on being heated in air mainly forms the monoxide and not peroxide.
(ii) $K, R b$ and $C s$ on being heated in the presence of excess supply of air form superoxides in preference to oxides and peroxides.
(iii) An aqueous solution of sodium carbonate is alkaline in nature.
Sol. (i) $L i^{+}$ ions is smaller in size. It is stabilized more by smaller anion, oxide ion $\left(O^{2-}\right)$ as compared to larger anion, peroxide ion $\left(O_{2}^{2-}\right)$.
(ii) $\quad K^{+}, R b^{+}, C s^{+},$ are large cations. A large cation is more stabilized by large anions. since superoxide ion, $O_{2}^{-}$ is quite large, $K, R b$ and $C s$ form superoxides in preference to oxides and peroxides.
(iii) In aqueous solution sodium carbonate undergoes hydrolysis forming sodium hydroxide.
$\mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{NaHCO}_{3}+\mathrm{NaOH}$
[esquestion]. Write balanced equations or reactions between :
Q. . Name an alkali metal carbonate which is thermally unstable and why ? Give its decomposition reaction.
Sol. $L i_{2} C O_{3}$ is thermally unstable because it is covalent. It decomposes to form $L i_{2} O$ and $C O_{2}$
$L i_{2} C O_{3} \stackrel{\Delta}{\longrightarrow} L i_{2} O+C O_{2}$
[esquestion]. Why are ionic hydrides of only alkali metals and alkaline earth metals are known ? Give two examples.
Q. . Why does the following reaction :
proceed better with $K F$ than with $N a F ?$
Sol. It is because $K F$ is more ionic than $N a F$
Q. . The enthalpy of formation of hypothetical $\operatorname{CaCl}(s)$ theoretically found to be equal to $188 \mathrm{kJ} \mathrm{mol}^{-1}$ and the $\Delta H_{f}^{\circ}$ for $C a C l_{2}(s)$ is $-795 k J m o l^{-1} .$ Calculate the $\Delta H^{\circ}$ for the disproportionation reaction.
Sol.
Q. . Why is it that the s-block elements never occur in free state in nature ? What are their usual modes of occurrence and how are they generally prepared ?
[NCERT]
Sol. s-block elements are highly reactive, therefore they never occur in free state rather occur in combined state in the form of halides, carbonates, sulphates. They are generally prepared by electrolysis of their molten salts
.
Q. . Name the chief form of occurrence of magnesium in nature. How is magnesium extracted from one of it ores ?
[NCERT]
Sol. mg occurs in the form of $M g C l_{2}$ in sea water from which it can be extracted. Sea water containing $\mathrm{MgCl}_{2}$ is concentrated under the sun and is treated with $\mathrm{Ca}(\mathrm{OH})_{2} \cdot \mathrm{Mg}(\mathrm{OH})_{2}$ is thus precipitated, filtered and heated to give the oxide. The oxide is treated with $\mathrm{C}$ and $\mathrm{Cl}_{2}$ to get $\mathrm{MgCl}_{2}$.
$\mathrm{MgO}+\mathrm{C}+\mathrm{Cl}_{2} \stackrel{\text { heat }}{\longrightarrow} \mathrm{MgCl}_{2}+\mathrm{CO}$
$\mathrm{MgCl}_{2}$ is fused with $\mathrm{NaCl}$ and $\mathrm{CaCl}_{2}$ at $970-1023 \mathrm{K}$ and molten mixture is electrolysed. Magnesium is liberated at cathode and Chlorine is evolved at the anode.
At Cathode : $\mathrm{Mg}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Mg}$
At Anode : $2 \mathrm{Cl}^{-} \longrightarrow \mathrm{Cl}_{2}+2 \mathrm{e}^{-}$
A steam of coal gas is blown through the cell to prevent oxidation of Mg metal.
Q. . How is pure magnesium prepared from sea water ? What happens when mg is burned in air ? Write chemical equations of reactions involved.
Sol. Sea water contains $M g C l_{2}$ which is concentrated under the sun and is treated with calcium hydroxide, $\mathrm{Ca}(\mathrm{OH})_{2}$ , magnesium hydroxide is thus precipitated, filtered and heated to give oxide. The oxide is treated with carbon and $C l_{2}$ to get $M g C l_{2}$.
$M g O+C+C l_{2} \longrightarrow M g C l_{2}+C O$
is mixed with sodium chloride so as to reduce its melting point and increase its electrical conductivity. Molten mixture is electrolysed using steel cathode and carbon anode.
A steam of coal gas is blown through the cell to prevent oxidation. of metal.
At cathode Mg’t $+2 \mathrm{e}^{-} \longrightarrow \mathrm{Mg}$
At anode $\quad 2 \mathrm{Cl}^{-}-2 \mathrm{e}^{-} \longrightarrow \mathrm{Cl}_{2}$
mg obtained in liquid state is further distilled to give pure mg When burns in air, it forms magnesium oxide and magnesium nitride.
$3 M g+N_{2} \longrightarrow M g_{3} N_{2} ; 2 \mathrm{Mg}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{MgO}$
[esquestion]. (i) Draw a neat and labelled diagram of Castner-Kellner cell for the manufacture of caustic soda.
(ii) Give chemical equations of the reaction of caustic soda with (a) ammonium chloride, and (b) carbon dioxide.
Q. . What happens when :
(i) sodium metal is dropped in water.
(ii) sodium metal is heated in free supply of air.
(iii) sodium peroxide dissolves in water.
Sol. (i) $2 N a+2 H_{2} O \longrightarrow 2 N a O H+H_{2}$ ; sodium hydroxide is formed with evolution of $H_{2}(g)$ The hydrogen gas catches fire due to highly exothermic process.
(ii) $\quad 2 \mathrm{Na}+\mathrm{O}_{2} \longrightarrow \mathrm{Na}_{2} \mathrm{O}_{2}$ sodium peroxide is formed.
(iii) Sodium hydroxide and hydrogen peroxide are formed.
$\mathrm{Na}_{2} \mathrm{O}_{2}+2 \mathrm{H}_{2} \mathrm{O} \longrightarrow 2 \mathrm{NaOH}+\mathrm{H}_{2} \mathrm{O}_{2}$
Q. . The hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. Explain.
Sol. The lattice enthalpies of hydroxides and carbonates of magnesium and calcium are very high due to the presence of divalent cations. The enthalpy of hydration cannot compensate for the energy required to break the lattice in these compounds. Hence, they are sparingly soluble in water. On the otherhand, the lattice enthalpies of hydroxides and carbonates of sodium and potassium are low due to the presence of monovalent cations. The enthalpy of hydration in this case is sufficient to break the lattice in these compounds. Hence, hydroxides and carbonates of sodium and potassium are easily soluble in water.[esquestion]. What happens when :
(i) magnesium is burnt in air
(ii) quicklime is heated with silica
(iii) chlorine reacts with slaked lime
(iv) calcium nitrate is heated
#tag# [NCERT]
Q. . Like lithium in Group 1, beryllium shows anomalous behaviour in Group 2. Write three such properties of beryllium which make it anomalous in the group.
Sol. (i) Be forms amphoteric oxide whereas others form basic acids.
(ii) $B e C l_{2}$ is covalent, others form ionic halides.
(iii) does not react even with hot water where as others react easily with water.
has smallest atomic size, highest ionisation energy, high polarising power which makes it anomalous in this group.[esquestion]. Beryllium exhibits some similarities with aluminium. Point out three such properties. [NCERT]
Q. . Compare the solubility and thermal stability of the following compounds of the alkali metals with those of the alkaline earth metals :
(i) nitrates (ii) carbonates (iii) sulphates.
Sol. (i) Nitrates of alkali metals are more stable than that of alkaline earth metals. All nitrates are soluble in water.
(ii) Carbonates of alkali metals are more stable and more soluble in water than that of alkaline earth metals.
(iii) Sulphates of alkali metals are more stable and more soluble in water than that of alkaline earth metal.[esquestion]. Discuss the anomalous behaviour of Lithium. Give its diagonal relationship with Magnesium.
Q. . State, why :
(i) A solution of $\mathrm{Na}_{2} \mathrm{CO}_{3}$ is alkaline.
(ii) Alkali metals are prepared by electrolysis of their fused chlorides.
(iii) Sodium is found more useful than potassium.
Sol. (i) $\mathrm{Na}_{2} \mathrm{CO}_{3}$ is alkaline due to its hydrolysis, it forms $O H^{-}$ more than $H^{+}$ because $H_{2} C O_{3}$ is weak acid, therefore, is alkaline.
(ii) Alkali metals are strong reducing agents and highly reactive with water, therefore, they are prepared by electrolysis of their fused Chlorides.
(iii) Sodium is more useful than potassium because sodium is less reactive and found in more abundance than K
Q. . Contrast the action of heat on the following :
(i) $\quad \mathrm{Na}_{2} \mathrm{CO}_{3}$ and $\mathrm{CaCO}_{3}$
(ii) $\quad M g C l_{2} \cdot 6 H_{2} O$ and $C a C l_{2} \cdot 6 H_{2} O$
(iii) $\quad \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}$ and $\mathrm{NaNO}_{3}$
Sol. (i) Sodium carbonate does not decompose whereas calcium carbonate decomposes on heating.
$\quad \mathrm{CaCO}_{3} \stackrel{\Delta}{\longrightarrow} \mathrm{CaO}+\mathrm{CO}_{2}$
(ii) $\mathrm{MgCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} \stackrel{\Delta}{\longrightarrow} \mathrm{MgO}+2 \mathrm{HCl}+5 \mathrm{H}_{2} \mathrm{O}$
$\mathrm{CaCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{CaCl}_{2}+6 \mathrm{H}_{2} \mathrm{O}$
(iii) $2 \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{CaO}+4 \mathrm{NO}_{2}+\mathrm{O}_{2}$
$2 \mathrm{NaNO}_{3} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{NaNO}_{2}+\mathrm{O}_{2} |$
Q. . What happens when :
(i) Carbon dioxide gas is passed through an aqueous solution of sodium carbonate.
(ii) Potassium carbonate is heated with milk of lime.
(iii) Lithium nitrate is heated.
Give chemical equation for the reactions involved.
Sol.
Q. . What happen when
(i) Magnesium is burnt in air
(ii) Quick lime is heated with silica
(iii) Chlorine reacts with slaked lime
(iv) Calcium nitrate is heated
Sol.
Q. . ‘The chemistry of beryllium is not essentially ionic’. Justify the statement by making a reference to the nature of oxide, chloride and fluoride of beryllium.
[NCERT]
Sol. Be predominantly forms covalent compounds due to smaller size, higher ionisation energy and high polarising power.
$B e O, B e C l_{2}$ and $B e F_{2}$ are covalent and get hydrolysed by water.
BeO is least soluble in water due to covalent character.
$B e O+H_{2} O \longrightarrow B e(O H)_{2}$
$B e C l_{2}+2 H_{2} O \longrightarrow B e(O H)_{2}+2 H C l$
$B e F_{2}+2 H_{2} O \longrightarrow B e(O H)_{2}+2 H F$
They are less soluble in water but more soluble in organic solvents, which shows they are covalent in nature.
Q. . Compare and contrast the chemistry of group 1 metals with that of group 2 metals with respect to :
(i) nature of oxides
(ii) solubility and thermal stability of carbonates
(iii) polarizing power of cations
(iv) reactivity and reducing power.
Sol. (i) Oxides of group 1 elements are more basic than that of group 2.
(ii) Solubility and thermal stability of carbonates of group 1 is higher than that of group 2.
(iii) Polarizing power of cations of group 2 is higher than that of group 1.
(iv) Reactivity and reducing power of group 1 elements is higher than corresponding group 2 elements.
Q. . Describe the importance of the following in different areas :
(i) limestone (ii) cement
(iii) plaster of paris.
Sol. (i) Lime stone : (a) It is used in manufacture of glass and cement. (b) It is used as flux in extraction of iron.
(ii) Cement : (a) It is used as building material. (b) It is used in concrete and reinforced concrete, in plastering and in construction of bridges, dams and buildings.
(iii) Plaster of Paris : (a) It is used for manufacture of chalk. (b) It is used for plastering fractured bones. (c) It is used for making casts and moulds. (d) It is also used in dentistry, in ornaments work and for taking casts of statues.
Q. . Describe the general characteristics of group1 elements.
Sol. (i) Group-I consists of lithium, sodium, potassium, rubidium, caesium and francium.
(ii) Elements have electronic configuration $n s^{1}$
(iii) Elements have 1 electron in the outermost s-orbital and have a strong tendency to lose this electron, so :
(a) These are highly electropositive metals.
(b) Never found in free state due to their high reactivity.
(c) They form $M^{+}$ ion.
(iv) Atomic radii : Atomic radii of alkali metals are largest in their respective periods.
(v) Density : Their densities are quite low. Lithium is the lightest known metal.
(vi) Oxidation state : The alkali metal atoms show only +1 oxidation state.
(vii) Reducing agents : Due to very low value of ionisation energy, alkali metals are strong reducing agents and reducing character increase $N a$ to $C s$ but $L$ but is the strongest reducing agent.
(viii) When alkali metals are heated in air, Lithium forms normal oxide $\left(L i_{2} O\right)$ sodium forms peroxide $\left(N a_{2} O_{2}\right)$potassium, rubidium and caesium form superoxides with peroxides.
(ix) The alkali metals form hydrides of type reacting with hydrogen at about $673 K \quad \text { (Li forms hydride at } 1073 K)$ forms hydride at Ionic character of hydrides increases from to
Q. . Describe the manufacture process of sodium-carbonate.
Sol. Sodium carbonate $\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot \mathrm{H}_{2} \mathrm{O}$ or washing soda is manufactured by solvay-process.
Principle of process : Carbondioxide gas is passed through brine solution (about 28% $N a C l$ saturated with ammonia, sodium carbonate is formed.
Plant used for the manufacture of washing soda.
Process : It completes in the following steps :
(i) Saturation of brine with ammonia : Lime stone (Calcium carbonate is strongly heated to form carbon dioxide.
Ammonia and carbondioxide mixture is passed through a tower in which saturated brine is poured down. Ammoniated brine is filtered to remove impurities of calcium and magnesium carbonate.
(ii) The milky solution is removed and filtered with the help of a vacuum pump.
(iii) Ammonia recovery tower : The filtrate is step 2 is mixed with calcium hydroxide and heated with steam.
Ammonia obtained is recycled with carbondioxides.
Potassium carbonate cannot be prepared by Solvay process as the solubility of $\mathrm{KHCO}_{3}$ is fairly large as compared to $\mathrm{NaHCO}_{3}$
Q. . Give chemical equations for the various reactions taking place during the manufacture of washing soda by Solvay’s process. What are the raw materials used in this process ? What is the by-product in this process ?
Sol.
The raw materials used in this process are sodium chloride and lime stone Calcium chloride is the by-product in this process.
Q. . Describe three industrial uses of caustic soda. Describe one method of manufacture of sodium hydroxide. What happens when sodium hydroxides reacts with (i) aluminium metal (ii) $\mathrm{CO}_{2}$ (iii) $\mathrm{SiO}_{2} ?$
Sol. Industrial uses of caustic soda.
(i) It is used for manufacture of soap.
(ii) It is used in paper industry.
(iii) It is used in textile industries.
Sodium hydroxide can be prepared by electrolysis of saturted solution of brine $(N a C l)$
(iii) [esquestion]. (i) How is plaster of Paris prepared ? Describe its chief property due to which it is widely used.
(ii) How would you explain ?
(a) $\quad$ BeO is insoluble but $B e S O_{4}$ in soluble in water.
(b) $\quad B a O$ is soluble but $B a S O_{4}$ is insoluble in water.
(c) $\quad$ Lil is more soluble than $K I$.
(d) $\quad \mathrm{NaHCO}_{3}$ is known in solid state but $\mathrm{Ca}\left(\mathrm{HCO}_{3}\right)_{2}$ is not isolated in solid state. \
#tag# [NCERT]
Q. . (i) Name an element which is invariable bivalent and whose oxide is soluble in excess of $N a O H$ and its dipositive ion has a noble gas core.
(ii) Differentiate between (a) quick-lime (b) lime-water (c) slaked-lime.
Sol. (i) Beryllium : It is invariably divalent. Its oxide is soluble in and its dipositive ion has a noble gas core.
(ii) Quick lime : It is calcium oxide $(C a O)$ It is produced by heating $\mathrm{CaCO}_{3}$
Lime water : The clear aqueous solution of Calcium hydroxide in water is called lime water. It is formed when $\mathrm{Ca}(\mathrm{OH})_{2}$ is dissolved in excess of $H_{2} O$
Slaked Lime : Calcium hydroxide solid is known as Slaked Lime. It is produced when water is added to $\mathrm{CaO}$
Q. . What is the effect of heat on the following compounds ? (Write equations for the reactions).
(i) Calcium carbonate.
(ii) Magnesium chloride hexahydrate.
(iii) Gypsum.
(iv) Magnesium sulphate heptahydrate.
Sol. (i) Action of heat on calcium carbonate : When calcium carbonate is heated, a colourless gas carbon dioxide is given out and a white residue of calcium oxide is left behind.
(ii) Action of heat on magnesium chloride hexahydrate : It loses water and hydrogen chloride and yields a residue of magnesium oxide.
(iii) Action of heat on gypsum : It forms a hemi-hydrate, called plaster of paris.
(iv) Action of heat on magnesium sulphate heptahydrate : It loses water of crystallization and forms anhydrous salt.
Q. . (i) Describe two important uses of each of the following:
(a) caustic soda (b) sodium carbonate (c) quicklime.
Sol. (i) (a) Caustic soda is used in prepartion of pure fats and oils.
It is also used for preparation of rayon (artificial silk).
(b) Sodium carbonate is used for manufacture of glass.
It is used in softening of hard water.
(c) Quick lime is used for white washing.
It is used for manufacturing of glass and cement.
[Polymeric structure of $\left.B e C l_{2}\right]$ in solid state | 2020-05-28 04:41:21 | {"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": 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": 0.5864307284355164, "perplexity": 3892.35879781336}, "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/1590347396495.25/warc/CC-MAIN-20200528030851-20200528060851-00397.warc.gz"} |
http://mathoverflow.net/feeds/question/9731 | Polynomial representing all nonnegative integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:33:33Z http://mathoverflow.net/feeds/question/9731 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers Polynomial representing all nonnegative integers Bjorn Poonen 2009-12-25T06:40:19Z 2010-11-26T00:44:32Z <p>Lagrange proved that every nonnegative integer is a sum of 4 squares.</p> <p>Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.</p> <p>Is there a 2-variable polynomial $f(x,y) \in \mathbf{Q}[x,y]$ such that $f(\mathbf{Z} \times \mathbf{Z})=\mathbf{N}$?</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/9732#9732 Answer by Qiaochu Yuan for Polynomial representing all nonnegative integers Qiaochu Yuan 2009-12-25T07:19:03Z 2009-12-25T07:38:20Z <p>I don't think so (but I haven't checked this argument very thoroughly):</p> <p>First we claim that any such $f$ has degree two. Clearly the leading term of $f$ cannot be odd, so suppose by contradiction that $f$ has degree at least four. <strike>Pick a constant $R$ large enough so that in the region $D_1$ consisting of points satisfying $|x|, |y| \ge R$, there exists a constant $c$ such that $f(x, y) \ge c \text{ min}(|x|, |y|)^4$.</strike> (<strong>Edit:</strong> This isn't always possible, but I think it can be salvaged.) Then it's not hard to see that $\sum_{D_1} \frac{1}{f(x, y)}$ converges. But $D_2 = \mathbb{Z}^2 - D_1$ can be partitioned into $4R - 2$ not necessarily disjoint lines in which one of $x$ or $y$ is fixed. On any of these regions $f$ cannot be linear, so it either grows at least quadratically or is a constant; we can ignore the lines on which $f$ is constant. It follows that $\sum_L \frac{1}{f(x, y)}$ converges for any line $L$ on which $f$ is nonconstant, hence $\sum \frac{1}{f(x, y)}$ converges if we sum over every point of $\mathbb{Z}^2$ except the lines on which $f$ is constant. Since we have only thrown out finitely many of the values of $f$ in this sum, those values cannot contain every positive integer. </p> <p>But if $f$ is quadratic, it is a constant plus the sum of squares of two polynomials with rational coefficients and there are many integers not representable as the sum of squares of two rational numbers.</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/9796#9796 Answer by ougao for Polynomial representing all nonnegative integers ougao 2009-12-26T15:25:09Z 2009-12-26T15:25:09Z <p>I have thinked over this question for about one year occationally. I think it may be related to the Hilbert Polynomial, a reason is that the Gauss Number is a kind of this type.</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/11241#11241 Answer by Richard Stanley for Polynomial representing all nonnegative integers Richard Stanley 2010-01-09T19:03:40Z 2010-01-09T19:03:40Z <p>What can be said about the following stronger question? Let $f(x,y)\in \mathbf{Q}[x,y]$ such that $f(\mathbf{Z}\times \mathbf{Z})$ is a subset of $\mathbf{N}$. Let $g(n)$ be the number of elements of $f(\mathbf{Z}\times \mathbf{Z})\cap \lbrace 0,1,\dots,n\rbrace$. How fast can $g(n)$ grow? Is it always true that $g(n) =O(n/\sqrt{log(n)})$? If true this is best possible since if $f(x,y)=x^2+y^2$ then $g(n)\sim cn/\sqrt{log(n)}$. </p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/11245#11245 Answer by Zev Chonoles for Polynomial representing all nonnegative integers Zev Chonoles 2010-01-09T20:22:42Z 2010-01-09T20:22:42Z <p>If we can make a (single variable) polynomial function $g(x)$ from $\mathbb{Z}$ onto $\mathbb{N}$, we could compose it with the <a href="http://en.wikipedia.org/wiki/Cantor%5Fpairing%5Ffunction#Cantor%5Fpairing%5Ffunction" rel="nofollow">Cantor pairing function</a>, but such a $g(x)$ seems implausible for some reason...</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/11355#11355 Answer by Terry Tao for Polynomial representing all nonnegative integers Terry Tao 2010-01-10T18:15:31Z 2010-01-10T18:40:36Z <p>This is a cute problem! I toyed with it and didn't really get anywhere - I got the strong impression that it requires fields of mathematics that I am not expert in. </p> <p>Indeed, given that the problem seems related to that of counting integer solutions to the equation $f(x,y) = c$, one may need to use arithmetic geometry tools (e.g. Faltings' theorem). In particular if we could reduce to the case when the genus is just 0 or 1 then presumably one could kill off the problem. (One appealing feature of this approach is that arithmetic geometry quantities such as the genus are automatically invariant (I think) with respect to invertible polynomial changes of variable such as $(x,y) \mapsto (x,y+P(x))$ or $(x,y) \mapsto (x+Q(y),y)$ and so seem to be well adapted to the problem at hand, whereas arguments based on the raw degree of the polynomial might not be.)</p> <p>Of course, Faltings' theorem is ineffective, and so might not be directly usable, but perhaps some variant of it (particularly concerning the dependence on c) could be helpful. [Also, it is overkill - it controls rational solutions, and we only care here about integer ones.] This is far outside of my own area of expertise, though...</p> <p>The other thing that occurred to me is that for fixed c and large x, y, one can invert the equation $f(x,y) = c$ to obtain a Puiseux series expansion for y in terms of x or vice versa (this seems related to resolution of singularities at infinity, though again I am not an expert on that topic; certainly Newton polytopes seem to be involved). In some cases (if the exponents in this series expansion are favourable) one could then use Archimedean counting arguments to show that f cannot cover all the natural numbers (this is a generalisation of the easy counting argument that shows that a 1D polynomial of degree 2 or more cannot cover a positive density set of integers), but this does not seem to work in all cases, and one may also have to use some p-adic machinery to handle the other cases. One argument against this approach though is that it does not seem to behave well with respect to invertible polynomial changes of variable, unless one works a lot with geometrical invariants.</p> <p>Anyway, to summarise, it seems to me that one has to break out the arithmetic geometry and algebraic geometry tools. (Real algebraic geometry may also be needed, in order to fully exploit the positivity, though it is also possible that positivity is largely a red herring, needed to finish off the low genus case, but not necessary for high genus, except perhaps to ensure that certain key exponents are even.)</p> <p>EDIT: It occurred to me that the polynomial $f(x,y)-c$ might not be irreducible, so there may be multiple components to the associated algebraic curve, each with a different genus, but presumably this is something one can deal with. Also, the geometry of this curve may degenerate for special c, but is presumably stable for "generic" c (or maybe even all but finitely many c). </p> <p>It also occurs to me that one use of real algebraic geometry here is to try to express f as something like a sum of squares. If there are at least two nontrivial squares in such a representation, then f is only small when both of the square factors are small, which is a 0-dimensional set and so one may then be able to use counting arguments to conclude that one does not have enough space to cover all the natural numbers (provided that the factors are sufficiently "nonlinear"; if for instance $f(x,y)=x^2+y^2$ then the counting arguments barely fail to provide an obstruction, one has to use mod p arguments or something to finish it off...)</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/11359#11359 Answer by Ilya Nikokoshev for Polynomial representing all nonnegative integers Ilya Nikokoshev 2010-01-10T19:40:16Z 2010-01-10T21:32:46Z <p>The search turned up a 1981 paper by John S.Lew (in the <strong>Unsolved problems</strong> section)</p> <ul> <li><a href="http://www.jstor.org/stable/2320113" rel="nofollow">Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points</a></li> </ul> <p>which discusses related problems, and ends up stating this one. The author's problems are: </p> <ul> <li><strong>Problem A.</strong> Classify bijections $\mathbb N\times\mathbb N \to \mathbb N$.</li> <li><strong>Problem B.</strong> Classify bijections $\mathbb Z\times\mathbb Z \to \mathbb Z$.</li> <li><strong>Problem C.</strong> Classify surjections $\mathbb Z\times\mathbb Z \to \mathbb N$.</li> </ul> <p>His main conjecture is that the only solutions to <strong>A</strong> are Cantor's $x+ \frac12(x+y-1)(x+y-2)$, which apparently goes to the time of Polya. Lew states <strong>C</strong> independently from empirical observations.</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/19855#19855 Answer by Botong Wang for Polynomial representing all nonnegative integers Botong Wang 2010-03-30T17:41:18Z 2010-03-30T17:41:18Z <p>How about an alternative question: does there exist a polynomial $f\in\mathbb{Q}[x,y]$ with integer values at lattice points, and of degree at least two on each variable, such that for any prime $p$, the map $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ composing with $\mathbb{Z}\to\mathbb{F}_p$ is surjective? Or more specifically, does a degree two such polynomial exist? The last part shouldn't be too hard, but I don't know how to solve it.</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/47389#47389 Answer by George Lowther for Polynomial representing all nonnegative integers George Lowther 2010-11-26T00:44:32Z 2010-11-26T00:44:32Z <p>After thinking about this problem for a bit using rather a naive approach, looking at regions where f grows faster than quadratically (as mentioned in Qiaochu's attempt), it certainly appears that obtaining a negative answer to this problem is very difficult. To obtain a positive answer might be easier since we only need to exhibit a single polynomial with $f(\mathbb{Z}\times\mathbb{Z})=\mathbb{N}$ although, in all likelihood, there do not exist such examples. However, even using the naive approach, it quickly becomes clear what kind of behaviour could lead to a polynomial f having the required properties. Polynomials of degree less than four can be quickly dismissed. Then, assuming f has degree 2n, look at the leading order terms $f_{2n}$, which is a homogeneous polynomial of degree 2n. Away from the zeros of $f_{2n}$, it grows at rate $R^{2n}$ ($R=\sqrt{x^2+y^2}$) and dominates the lower order terms, so f cannot cover a strictly positive density of the integers. The difficulties occur when we look close to certain curves which are asymptotic to the zeros of $f_{2n}$ (being homogeneous, the zeros of $f_{2n}$ lie on a finite set of lines radiating out from the origin). On such curves, $f_{2n}$ will grow at a rate less than $O(R^{2n})$ and cancellation with lower order terms can occur. Looking at how much cancellation can occur seems to lead to difficult problems of Diophantine approximation.</p> <p>The kind of polynomial which are plausible candidates for a polynomial mapping onto the positive integers are as follows. $$f(x,y)=a\prod_{i=1}^d\left(x-\alpha_iy\right)^{2n} - q(x,y)\qquad\qquad{\rm(1)}$$ where $\alpha$ is a real algebraic integer with minimal polynomial of degree $d > 2$ over the rationals and conjugates $\alpha_1,\cdots,\alpha_d$, $a$ is a positive integer, and $q(x,y)$ is a polynomial of degree $2n(d-2)$. By <a href="http://en.wikipedia.org/wiki/Dirichlet%2527s_approximation_theorem" rel="nofollow">Dirichlet's theorem</a>, we know that there are infinitely many integer x,y such that $\vert x/y-\alpha_i\vert < y^{-2}$, so the leading order term of (1) is less than some fixed multiple of $R^{2n(d-2)}$ infinitely often. So there will be some cancellation with q. On the other hand, by the <a href="http://en.wikipedia.org/wiki/Thue%25E2%2580%2593Siegel%25E2%2580%2593Roth_theorem" rel="nofollow">Thue-Siegel-Roth theorem</a>, we know that $\vert x/y-\alpha\vert > y^{-2-\epsilon}$ for all large x,y, so the leading order terms of (1) grow at least as fast as $O(R^{2n(d-2)-\epsilon})$ which, at least, means that (1) cannot go negative very quickly. The question then, is there an algebraic number $\alpha$ such that $\vert x/y-\alpha\vert\ge cy^{-2}$ for some positive constant c and all integer x,y? In that case, the leading order term of (1) would be bounded below by a multiple $R^{2n(d-2)}$ and q could be chosen such that $f(\mathbb{Z}\times\mathbb{Z})\subseteq\mathbb{N}$. It is then possible, but still unlikely, that (1) gives a polynomial mapping onto the positive integers. Looking in my copy of Hindry & Silverman (Diophantine Geometry, An Introduction) it mentions that it is an open problem whether there exist such algebraic numbers (and there are no known examples or counterexamples with $d > 2$) but it is conjectured that there aren't any. So, polynomials such as (1) appear unlikely to do what we want, but proving this seems to be very difficult. Of course, that f actually covers $\mathbb{N}$ would be a much stronger statement than $\vert x/y-\alpha\vert \ge cy^{-2}$ so, maybe an expert in this area could actually rule out such examples, but it still looks like a very tricky problem.</p> <p>We can also try polynomials such as $$f(x,y)=a\prod_{i=1}^d\left((x-\alpha_iy)^r-p(\alpha_i)y^s\right)^{2n} - q(x,y)\qquad\qquad{\rm(2)}$$ where, now, p is a polynomial with integer coefficients,r,s are positive integers and q has degree less than $2n$. This is even more difficult than (1) to deal with and whether or not such polynomials can provide what the question is asking for depends on how small $(x/y-\alpha)-p(\alpha)^{1/r}y^{s/r-1}$ can be. This also looks like a very difficult problem in Diophantine approximation.</p> <p>So, although I expect that the answer to this is no, there are no such $f$, any method of proving this has to cope with possibilities such as (1) and (2). Just these two cases look extremely difficult to handle. Maybe it is possible though, and an expert on such areas would be able to say something more about them than I can.</p> | 2013-05-23 12:33:38 | {"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": 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": 0.8431944847106934, "perplexity": 256.9911638308592}, "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/1368703306113/warc/CC-MAIN-20130516112146-00014-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://hal.elte.hu/fij/h/?n=Main.Python-2016-05-10-a | + indep. WoS citations
Python and networks - 2016-05-10
# 1. Problem: Characteristic length in text and DNA
Download the plain text version of “The Origin of Species by Means of Natural Selection” by Charles Darwin. Replace any sequence of consecutive whitespaces with a single space character. Save the "Contents" section of the book in a text variable. Each group of $$n$$ characters appearing in this text is called an $$n$$-gram. For $$n=1,2,\ldots,20$$ compute $$f_\mathrm{txt}(n)$$, which is the number of occurrences of the most frequently used $$n$$-gram. Here "txt" means text. Similarly, compute $$f_\mathrm{dna}(n)$$ for the first 10,000 nucleotides in Chromosome 1 of baker's yeast (Saccharomyces cerevisiae).
Plot $$f_\mathrm{txt}(n)$$ and $$f_\mathrm{dna}(n)$$ together. How and why do they differ?
# 2. Solution
### 2.1 Python code (top-ngram.py)
# number of occurrences of the most frequently used n-gram
# as a function of n in two data sets
import random; import sys; import re
script, infileTXT, infileDNA, groupSizeMax, outfile = sys.argv
# === function defitions ===
def find_top_ngram_usage(data,n,n2top):
# the frequency (number of times) that each n-grams of length n is used
nGramFreq = {}
# take the starting position of each substring of length n
for i in (range(0, len(data)-int(n)+1)):
# extract the substring (group) that starts at this position
subStr = data[i:i+int(n)]
# if we have already seen this substring, then increment its counter
if subStr in nGramFreq:
nGramFreq[subStr] += 1
else:
# else: set its counter to 1
nGramFreq[subStr] = 1
# result: save the number of occurrences of the most frequently used n-gram
n2top[n] = max(nGramFreq.values())
# === main ===
# read text file into a single string
# save the "Contents" section of the book
txt = re.findall('CONTENTS\.\s+?([\d\D]+?INDEX\.)', txt).pop(0)
# replace any sequence of whitespaces with a single space character
txt = ' '.join(txt.split())
# txt_n2top{n}: most frequently used n-gram of size n
txt_n2top = {}
# read file containing the DNA sequence
# remove first line
dna = re.sub(r'^.+?\n', r'', dna)
# remove newlines and spaces
dna = re.sub(r'\s+', r'', dna)
# use only the first 10,000 nucleotides of the sequence
dna = dna[0:10000]
# dna_n2top{n}: most frequently used n-gram of size n
dna_n2top = {}
# open outfile unbuffered, print file header
o = open(outfile, "w", 0)
o.write("# Group size (n)\n")
o.write("#\tNumber of occurrences of the most frequently used n-gram of size n\n")
o.write("#\tText\n")
o.write("#\t\tDNA\n")
o.write("\n")
# list all possible group sizes, n
for n in range( 1, 1+int(groupSizeMax) ):
# find most frequently used n-gram of size n
find_top_ngram_usage(txt,n,txt_n2top)
find_top_ngram_usage(dna,n,dna_n2top)
# print output data
o.write("%d\t%d\t%d\n" % (n,txt_n2top[n],dna_n2top[n]))
### 2.2 Usage of python code
The two input files are 2009.txt.utf-8 and chr01.fsa
python top-ngram.py 2009.txt.utf-8 chr01.fsa 10 top-ngram.txt
### 2.3 Output (top-ngram.txt)
# Group size (n)
# Number of occurrences of the most frequently used n-gram of size n
# Text
# DNA
1 1071 3268
2 169 1206
3 131 473
4 128 189
5 78 80
6 32 45
7 32 28
8 29 17
9 15 12
10 14 10
11 9 9
12 9 8
13 9 7
14 8 6
15 8 5
16 8 4
17 8 3
18 8 2
19 5 2
20 5 1
### 2.4 Gnuplot command file (top-ngram.gnu)
# settings
se term posts lands color enh dash "Helvetica" 22
se xlab "n [size of n-gram]"
se ylab "Number of occurrences of the\nmost frequently used n-gram"
se grid
se o 'top-ngram.ps'
se key top right
se log y
# plot
p [-1:21][.7:5000] \
\
'top-ngram.txt' u 1:2 w p pt 1 ps 2 lt 3 lw 3 ti 'Text', \
\
'' u 1:3 w p pt 2 ps 2 lt 1 lw 3 ti 'DNA'
### 2.5 Using gnuplot and converting the image
gnuplot top-ngram.gnu
convert -geometry 320 -rotate 90 -sharpen 5 top-ngram.ps top-ngram.png | 2019-02-19 17:46:36 | {"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": 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": 0.4738713502883911, "perplexity": 9326.957486229432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247490806.45/warc/CC-MAIN-20190219162843-20190219184843-00116.warc.gz"} |
https://math.stackexchange.com/questions/58198/all-pairs-shortest-path-in-undirected-and-unweighted-graphs | # All pairs shortest path in undirected and unweighted graphs
I'm aware that the single source shortest path in a undirected and unweighted graph can be easily solved by BFS.
For the case of the all pairs shortest path problem, is there any better solution than running a BFS for each node?
• Standard books in algorithms cover various efficient algorithms for all-pairs shortest path. – MCH Aug 17 '11 at 21:24
• Can you cite any of those standard books in algorithms? Thanks! – aromero Aug 17 '11 at 23:26
• @aromero, this question is off-topic for cstheory, please refer to FAQ to understand the scope of cstheory. I am closing the question as off-topic and migrating it to Math.SE. – Kaveh Aug 18 '11 at 0:45
• FWIW, nowadays questions such as this are ontopic on Computer Science. – Raphael Sep 17 '14 at 15:50
## 3 Answers
I have no idea why this question was considered off-topic for CSTheory. Certainly this question is very interesting to those who work in graph algorithms.
To that group, asking if there is a better solution to APSP than running BFS from each node, is equivalent to asking if there is an algorithm that runs in asymptotically less than $O(mn+n^2)$ time where $m$ is the number of edges and $n$ is the number of nodes. It is a major open problem to improve significantly on this running time.
Timothy Chan found some small algorithmic improvements in SODA'06 that may have promise in practice (he implemented them). See the paper:
Timothy M. Chan: All-pairs shortest paths for unweighted undirected graphs in o(mn) time. SODA 2006: 514-523
In the undirected and unweighted case, one can solve the problem via reductions to matrix multiplication of $n \times n$ matrices (so theoretically, this means you can get $n^{2.376}$ time). If your graph is dense then this could be very useful. These algorithms are rather ingenious:
Zvi Galil, Oded Margalit: All Pairs Shortest Distances for Graphs with Small Integer Length Edges. Inf. Comput. 134(2): 103-139 (1997)
Zvi Galil, Oded Margalit: All Pairs Shortest Paths for Graphs with Small Integer Length Edges. J. Comput. Syst. Sci. 54(2): 243-254 (1997)
Raimund Seidel: On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs. J. Comput. Syst. Sci. 51(3): 400-403 (1995)
Hopefully, expositions of the last three (or the papers themselves) can be found freely on the Web.
• Great answer, thanks – aromero Aug 18 '11 at 19:55
I had to write a fast implementation of this to deal with large graphs, and I found the n BFS to be much better than the Floyd-Warshall algorithm. Their way of storing the results, though (a matrix of predecessors) remains a very good way to store the result ! way better than storing $\binom n 2$ paths (even when they are short) -- especially for compact use of memory.
Nathann
• You are right, the storing of results is also relevant to note. Thanks – aromero Aug 18 '11 at 19:58
It is possible to find all shortest paths (only distances) in $O(n^2 log\ n)$ time and $O(n^2)$ space.
Udaya Kumar Reddy K. R, and K. Viswanathan Iyer: All-pairs shortest-paths problem for unweighted graphs in $O(n^2 log\ n)$ time
World Academy of Science, Engineering and Technology Vol:3 (2009) | 2019-12-11 09:32:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3568594455718994, "perplexity": 693.4691842501328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540530452.95/warc/CC-MAIN-20191211074417-20191211102417-00184.warc.gz"} |
https://math.stackexchange.com/questions/3065530/the-expected-value-of-beta-function | The expected value of Beta Function
Estimate the probability of success
Suppose I send 10 tasks to my machine. 6 out of 10 tasks success, and 4 failed. These outcomes is summarized by $$X$$ as a binary variable, 1 is task success, and 0 if task fail. We know that $$X$$ is continuous random variable
The expected value of a continuous random variable is dependent on the probability density function used to model the probability that the variable will have a certain value. Therefor, I exploit Beta distribution to estimate the probability of success for next tasks. I will $${\alpha}$$ as input of the number past success tasks and $${\beta}$$ as the number of past fail tasks
Expected value
$$$$E(x) = \frac{\alpha+1}{\alpha+\beta+2}$$$$
In my example, $$\alpha = 6$$ and $$\beta = 4$$. Thus, the $$E(x)$$ = 0.58.
Does every think looks good? | 2019-06-16 21:10:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8516888618469238, "perplexity": 541.2323676835917}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998298.91/warc/CC-MAIN-20190616202813-20190616224813-00130.warc.gz"} |
http://openstudy.com/updates/55e81020e4b0227206128d21 | ## Michele_Laino one year ago Tutorial: Sign of a permutation and the Ricci tensor
1. Michele_Laino
Let's consider the symmetric group $$S_3$$, namely the group of bijective applications, also called \emph{permutations}, of the set $$\{ 1,2,3\}$$. A generic element of that group, can be written as below: $\sigma :\{ 1,2,3\} \to \{ 1,2,3\} ,\quad \sigma = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ i&j&k \end{array}} \right)$ where, of course, $$i,j,k$$ $$\in \{ 1,2,3\}$$. Each entry of the second row, is the image of the corresponding element of the first row: $\sigma \left( 1 \right) = i,\quad \sigma \left( 2 \right) = j,\quad \sigma \left( 3 \right) = k$ In general, we can write a permutation, omitting the first row, so we will write the images of a permutation $$\sigma$$, like below: $i\quad j\quad k$ being: $$i,j,k \in \{ 1,2,3\}$$. \\ To each permutation of ${S_3}$, can be assigned a number, namely its \emph{sign}, whose definition is: $\operatorname{sgn} \left( \sigma \right):\quad {S_3} \to \left\{ { + 1, - 1} \right\}$ we say that the sign of a permutation is 1, if we need of an \emph{even} number of swaps in order to go from that permutation to the identical permutation, and the sign of a permutation is ${ - 1}$, if we need of an \emph{odd} number of swaps, in order to go from that permutation to the identical permutation. A swap is the permutation which change only two elements, here is an example of swap: $\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&3&2 \end{array}} \right)$ \\ whereas, the identical permutation is denoted like below: $Id = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&2&3 \end{array}} \right)$ or, simply like this: $1\quad 2\quad 3$ As we well know, the elements of the set ${S_3}$ is equal to $3! = 6$, so we have 6 permutation in total, here they are: $\begin{array}{*{20}{c}} 1&2&3&{}&{}&2&1&3 \\ 2&3&1&{}&{}&3&2&1 \\ 3&1&2&{}&{}&1&3&2 \end{array}$ At the left we have written all permutations which whose sign is $$+ 1$$, whereas at the right, we have written all permutations whose sign is $$- 1$$. We can go from right to left and from left to right, by simply making one swap, as we can check immediately. In physics, and in particular in classical mechanics, the sign of a permutation, is denoted with the $$Ricci tensor$$: ${\varepsilon _{ijk}}$ so we can write this: $\begin{gathered} {\varepsilon _{123}} = {\varepsilon _{231}} = {\varepsilon _{312}} = + 1 \hfill \\ {\varepsilon _{213}} = {\varepsilon _{321}} = {\varepsilon _{132}} = - 1 \hfill \\ \end{gathered}$ which are the only non-zero components of such tensor, among the {27} components of $${\varepsilon _{ijk}}$$. $$Application$$ Let $$\mathbf{a},\;\mathbf{b}$$ two generic vectors, of the euclidean space $$E$$, then let's consider its vector product or cross product, namely the subsequent map: ${\mathbf{ \times }}\;:\;E \times E \to E\quad \left( {{\mathbf{a}},\;{\mathbf{b}}} \right) \to {\mathbf{a}} \times {\mathbf{b}}$ then, using the Ricci's tensor, we can write the $$i-th$$ component of the vector $$\mathbf{a} \times \mathbf{b}$$, like below: ${\left( {{\mathbf{a}} \times {\mathbf{b}}} \right)_i} = \sum\limits_{j,k = 1}^3 {{\varepsilon _{ijk}}\;{a_j}{b_k}}$ and since the indexes $$j,\; k$$ are repeated, we can omit the symbol of summation ($$Einstein \; convention)$$: ${\left( {{\mathbf{a}} \times {\mathbf{b}}} \right)_i} = {\varepsilon _{ijk}}\;{a_j}{b_k}$ The Ricci's tensor satisfies an important identity: ${\varepsilon _{ijk}}\;{\varepsilon _{ilm}} = {\delta _{jl}}{\delta _{km}} - {\delta _{jm}}{\delta _{kl}}$ where $$\delta _{ij}$$ is the $$Kronecker's symbol$$. Finally, from the previous discussion, we have this other property of the Ricci's tensor: $\begin{gathered} {\varepsilon _{ijk}} = - {\varepsilon _{jik}} \hfill \\ {\varepsilon _{ijk}} = - {\varepsilon _{ikj}} \hfill \\ \end{gathered}$
2. Michele_Laino
here is the corresponding PDF file:
3. zzr0ck3r
Can you prove that every permutation can be written in either all even transposes or all odd ? I was curious about that.
4. anonymous
whoa
5. Michele_Laino
thanks :) :) @77777jeannie77777
6. anonymous
lol youre welcome i will never understand that lol im not in that level of math and i hope i wont be anytime soon hahaha :P
7. Michele_Laino
thanks again!! :) yes! sure one day you will understand it and you will write another tutorial, better than mine @77777jeannie77777
8. Astrophysics
Awesome, thanks for sharing @Michele_Laino
9. Michele_Laino
thanks!! :) @Astrophysics
10. anonymous
Richi
11. Empty
I thought the Ricci tensor was a contraction of the Riemann-Christoffel tensor and this was the Levi-Civitta tensor. | 2016-10-21 18:45:44 | {"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833106994628906, "perplexity": 413.4778086014222}, "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/1476988718296.19/warc/CC-MAIN-20161020183838-00495-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610231 | Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610231
Title: Indices for supersymmetric quantum field theories in four dimensions
Author: Ehrhardt, Mathieu
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2012
Availability of Full Text:
Access from EThOS: Full text unavailable from EThOS. Please try the link below. Access from Institution:
Abstract:
In this thesis, we investigate four dimensional supersymmetric indices. The motivation for studying such objects lies in the physics of Seiberg's electric-magnetic duality in supersymmetric field theories. In the first chapter, we first define the index and underline its cohomological nature, before giving a first computation based on representation theory of free superconformal field theories. After listing all representations of the superconformal algebra based on shortening conditions, we compute the associated Verma module characters, from which we can extract the index in the appropriate limit. This approach only provides us with the free field theory limit for the index and does not account for the values of the $R$-charges away from free field theories. To circumvent this limitation, we then study a theory on $\mathbb{R}\times S^3$ which allows for a computation of the superconformal index for multiplets with non-canonical $R$-charges. We expand the fields in harmonics and canonically quantise the theory to analyse the set of quantum states, identifying the ones that contribute to the index. To go beyond free field theory on $\mathbb{R}\times S^3$, we then use the localisation principle to compute the index exactly in an interacting theory, regardless of the value of the coupling constant. We then show that the index is independent of a particular geometric deformation of the underlying manifold, by squashing the sphere. In the final chapter, we show how the matching of the index can be used in the large $N$ limit to identify the $R$-charges for all fields of the electric-magnetic theories of the canonical Seiberg duality. We then conclude by outlining potential further work.
Supervisor: Osborn, Hugh Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.610231 DOI:
Keywords: supersymmetric index ; Seiberg duality ; superconformal field theories ; harmonics ; electric-magnetic duality ; indices
Share: | 2018-07-18 22:40:53 | {"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": 0.4993950128555298, "perplexity": 809.4201782713403}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590329.62/warc/CC-MAIN-20180718213135-20180718233135-00556.warc.gz"} |
https://ncatlab.org/nlab/show/arithmetic+D-module | # Contents
## Idea
The theory of arithmetic D-modules was primarily developped by Berthelot to better understand the functoriality properties of rigid cohomology. It gives a theory of coefficients for the cohomology of quasi-projective algebraic varieties over finite fields that are stable by the six Grothendieck operations, after Kedlaya and Caro. This allows a purely p-adic proof of Deligne’s Weil II theorem, that generalized the Riemann hypothesis over finite fields to the category of coefficients for cohomology (i.e., motivic sheaves).
## References
Revised on February 16, 2014 08:11:54 by Urs Schreiber (89.204.155.248) | 2017-02-19 14:19:09 | {"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": 0.8008359670639038, "perplexity": 1029.9389615352463}, "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-2017-09/segments/1487501169776.21/warc/CC-MAIN-20170219104609-00332-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://www.julia-harz.de/publication/harz-theoretical-2016/ | # Theoretical uncertainty of the supersymmetric dark matter relic density from scheme and scale variations
### Abstract
For particle physics observables at colliders such as the LHC at CERN, it has been common practice for many decades to estimate the theoretical uncertainty by studying the variations of the predicted cross sections with a priori unpredictable scales. In astroparticle physics, this has so far not been possible, since most of the observables were calculated at Born level only, so that the renormalization scheme and scale dependence could not be studied in a meaningful way. In this paper, we present the first quantitative study of the theoretical uncertainty of the neutralino dark matter relic density from scheme and scale variations. We first explain in detail how the renormalization scale enters the tree-level calculations through coupling constants, masses and mixing angles. We then demonstrate a reduction of the renormalization scale dependence through one-loop SUSY-QCD corrections in many different dark matter annihilation channels and enhanced perturbative stability of a mixed on-shell/$barrm DR$ renormalization scheme over a pure $barrm DR$ scheme in the top-quark sector. In the stop-stop annihilation channel, the Sommerfeld enhancement and its scale dependence are shown to be of particular importance. Finally, the impact of our higher-order SUSY-QCD corrections and their scale uncertainties are studied in three typical scenarios of the phenomenological Minimal Supersymmetric Standard Model with eleven parameters (pMSSM-11). We find that the theoretical uncertainty is reduced in many cases and can become comparable to the size of the experimental one in some scenarios.
Type
Publication
Physical Review D | 2021-03-06 05:47:10 | {"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": 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": 0.6374918222427368, "perplexity": 683.8731100580949}, "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-2021-10/segments/1614178374391.90/warc/CC-MAIN-20210306035529-20210306065529-00027.warc.gz"} |
https://cs.stackexchange.com/questions/99804/quick-sort-with-first-element-as-pivot | Quick Sort with first element as pivot
I'm studying Quick-Sort and I am confused as to how it works when the first element is chosen as the pivot point.
I am trying to trace the first step in the Quick-Sort algorithm, to move the pivot S[1] (17) into its appropriate position.
Example: [17, -10, 7, 19, 21, 23, -13, 31, 59].
^# = pivot
^ pointer
My understanding:
17, -10, 7, 19, 21, 23, -13, 31, 59
^# ^
Comparison 1. No swap.
17, -10, 7, 19, 21, 23, -13, 31, 59
^# ^
Comparison 2. No swap.
17, -10, 7, 19, 21, 23, -13, 31, 59
^# ^
Comparison 3. Swap.
-13, -10, 7, 19, 21, 23, 17, 31, 59
^ ^#
Comparison 4. Swap.
-13, -10, 7, 19, 21, 17, 23, 31, 59
^ ^#
Comparison 5. Swap.
-13, -10, 7, 19, 17, 21, 23, 31, 59
^ ^#
Comparison 6. Swap.
-13, -10, 7, 17, 19, 21, 23, 31, 59
^ ^#
Comparison 7. No swap.
-13, -10, 7, 17, 19, 21, 23, 31, 59
^ ^#
Comparison 9. No swap.
-13, -10, 7, 17, 19, 21, 23, 31, 59
^ ^#
Comparison 10. No swap.
Is this how it works? Would it take 10 comparisons and 4 swaps to move pivot S[1] (17) into the correct position?
You only need to compare each elemnt with the pivot once.
You can do so by keeping the pivot in place and then swapping elements in the remainder of the array.
17, -10, 7, 19, 21, 23, -13, 31, 59
# ^ ^
start
17, -10, 7, 19, 21, 23, -13, 31, 59
# ^ ^
move left pointer to first element larger than pivot. 3 compares
17, -10, 7, 19, 21, 23, -13, 31, 59
# ^ ^
move right pointer to first element smaller than pivot. 3 compares
17, -10, 7, -13, 21, 23, 19, 31, 59
# ^ ^
swap elements at pointer
17, -10, 7, -13, 21, 23, 19, 31, 59
# ^ ^
move left pointer to first element larger than pivot. 1 compare
17, -10, 7, -13, 21, 23, 19, 31, 59
# ^^
move right pointer to first element smaller than pivot (but not past left pointer. 1 compare
-13, -10, 7, 17, 21, 23, 19, 31, 59
#
swap pivot with last element smaller than it. 1 swap
total = 8 compares and up to n/2 swaps per partition.
Quicksort doesn't swap the pivot into its correct position in that way, but it lies on the hypothesis that each recursive call sorts the sub-array and then merging sorted sub-arrays would provide a completely sorted array:
• let $$V$$ be the array to sort
• $$pivot \leftarrow pick()$$ picks a pivot, in your case it's always the first element in $$V$$
• let $$V_{\lt}$$ be a list $$s.t. \forall a \in V_{\lt} \ a \in V \ \wedge \ a \lt pivot$$
• let $$V_{=}$$ be a list $$s.t. \forall a \in V_{=} \ a \in V \ \wedge \ a = pivot$$
• let $$V_{\gt}$$ be a list $$s.t. \forall a \in V_{\gt} \ a \in V \ \wedge \ a \gt pivot$$
Quicksort then proceeds recursively calling itself on $$V_{\lt}$$ and $$V_{\gt}$$, thus assuming to get those two back with their values sorted.
The final step is then to return the concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$, in this order.
Providing a numerical example:
$$V = [17, -10, 7, 19, 21, 23, -13, 31, 59]$$
Steps:
1. Initial run
$$pivot = 17$$
$$V_{\lt} = [-10, 7, -13]$$
$$V_{=} = [17]$$
$$V_{\gt} = [19, 21, 23, 31, 59]$$
2. Recursion on $$V_{\lt} = [-10, 7, -13]$$
$$pivot = -10$$
$$V_{\lt} = [-13]$$
$$V_{=} = [-10]$$
$$V_{\gt} = [7]$$
skips recursion on $$V_{\lt}$$ and $$V_{\gt}$$ since they're of size 1, thus already sorted
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[-13, -10, 7]$$
3. Recursion on $$V_{\gt} = [19, 21, 23, 31, 59]$$
$$pivot = 19$$
$$V_{\lt} = []$$
$$V_{=} = [19]$$
$$V_{\gt} = [21, 23, 31, 59]$$
4. Recursion on $$V_{\gt} = [21, 23, 31, 59]$$
$$pivot = 21$$
$$V_{\lt} = []$$
$$V_{=} = [21]$$
$$V_{\gt} = [23, 31, 59]$$
5. Recursion on $$V_{\gt} = [23, 31, 59]$$
$$pivot = 23$$
$$V_{\lt} = []$$
$$V_{=} = [23]$$
$$V_{\gt} = [31, 59]$$
6. Recursion on $$V_{\gt} = [31, 59]$$
$$pivot = 31$$
$$V_{\lt} = []$$
$$V_{=} = [31]$$
$$V_{\gt} = [59]$$
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[31, 59]$$
7. Back from recursion in 5.
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[23, 31, 59]$$
8. Back from recursion in 4.
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[21, 23, 31, 59]$$
9. Back from recursion in 3.
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[19, 21, 23, 31, 59]$$
10. Back from recursion in 2.
returns $$V_{sort}$$ = concatenation of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$ = $$[-13, -10, 7, 21, 23, 31, 59]$$
As you can see, from the step 3 and onwards, the chosen pivot isn't an optimal one, since there only are elements at its right, preventing the algorithm to run in optimal time of $$\mathcal{O}(n log_2 n)$$
EDIT: following Kai's comment, I fixed the definitions of $$V_{\lt}$$, $$V_{=}$$ and $$V_{\gt}$$. I also acknowledge this is the simpler and less efficient Lomuto's partition.
• Thanks for attempting an explanation. In my view, your presentation could be improved by not conflating set notation (as e.g. for $V_<$) with list representation (which is what you use later when you talk about examples). Finally, there's a difference between Hoare's original quicksort and the one you describe (I think) in that yours is Lomuto's simpler but less efficient version. – Kai Nov 9 '18 at 10:54
place the pindex point to the last index in the array.Compare each and every element with pivot if the element is greater the pivot swap with pindex and decrement the pindex..The Following code helps you for the implementation of partition
def partioning(a,s,e):
pivot=a[s]
pi=e
for i in range(1,e):
if(a[i]>=pivot):
a[i],a[pi]=a[pi],a[i]
pi-=1
a[pi],a[s]=a[s],a[pi]
return pi | 2020-08-12 07:14:57 | {"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": 80, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49480825662612915, "perplexity": 1337.12135502476}, "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-2020-34/segments/1596439738878.11/warc/CC-MAIN-20200812053726-20200812083726-00249.warc.gz"} |
https://math.stackexchange.com/questions/894392/question-on-factoring | # Question on Factoring
I have very basic Question about factoring, we know that,
$$x^2+2xy+y^2 = (x+y)^2$$ $$x^2-2xy+y^2 = (x-y)^2$$
But what will
$$x^2-2xy-y^2 = ??$$ $$x^2+2xy-y^2 = ??$$
• Your second line should be $x^2-2xy+y^2$. – Jam Aug 11 '14 at 19:43
• I corrected the equation in the above question – Shoaibkhanz Aug 11 '14 at 19:47
• @Shoaibkhanz I like your question - it shows that you are thinking about what you know in a good way, and that will serve you well as you learn more. You should look carefully at the way that Thomas Andrews has used the other common identity $x^2-y^2=(x+y)(x-y)$ which works whenever you have the difference of two squares (or of two positive expressions even) - then you will really start to understand what is going on - even if it takes you a little time, it will be time well spent. – Mark Bennet Aug 11 '14 at 20:32
• Certainly Mark! , its interesting how he splits that into an identity, in addition to that John provides the Formula way of achieving it – Shoaibkhanz Aug 12 '14 at 17:07
You can use the quadratic formula:
$$x^2 + (-2y)x + (-y^2) = 0\to \\ x= \frac{2y \pm \sqrt{4y^2 - 4(1)(-y^2)}}{2} \\ =y \pm\sqrt{2}y = (1 \pm \sqrt{2})y.$$
So $x^2 - 2xy - y^2 = [x - (1+\sqrt{2})y][x - (1-\sqrt{2})y].$
For the other case,
$$x^2 + (2y)x + (-y^2) = 0\to \\ x= \frac{-2y \pm \sqrt{4y^2 - 4(1)(-y^2)}}{2} \\ =-y \pm\sqrt{2}y = (1 \pm \sqrt{2})y.$$
So $x^2 + 2xy - y^2 = [x - (1-\sqrt{2})y][x - (1+\sqrt{2})y].$
There is no simple factorization of $x^2+2xy-y^2$ nor $x^2-2xy-y^2$, although you can write:
\begin{align}x^2+2xy-y^2 &= (x+y)^2-2y^2 \\&= \left(x+y(1+\sqrt{2})\right)\left(x+y(1-\sqrt{2})\right) \end{align}
and similarly:
$$x^2-2xy-y^2 = \left(x-y(1+\sqrt{2})\right)\left(x-y(1-\sqrt{2})\right)$$
This is another way of solving this problem using Completing the Square method. $x^2+2xy-y^2=??$
First we have,
$x^2+2xy-y^2=0$
$x^2+2xy=y^2,$
By adding both sides by $y^2$ to make it perfect square then,
$x^2+2xy+y^2=y^2+y^2$
$(x+y)^2=2y^2$
$x+y=[2^(1/2)]y$
$x=(+ or -)[2^(1/2)]y-y$
$x=[2^(1/2)]y-y ; x=-[2^(1/2)]y-y$
$x=[2^(1/2)-1]y ; x=-[2^(1/2)-1]y$
therefore we have, ${x-[2^(1/2)-1]y}{x+[2^(1/2)-1]y}$
By solving $x^2-2xy-y^2=??$ Try to solve this by relying with this pattern. Thank you.
• I don't think the answer is complete. Also please check the formatting – Shailesh Mar 31 '16 at 11:01
You can factor a quadratic trinomial
$$ax^2+bxy+cy^2$$
by finding the roots of
$$\frac{ax^2+bxy+cy^2}{y^2}=a\left(\frac xy\right)^2+b\left(\frac xy\right)+c=0.$$
Then
$$ax^2+bx+c=y^2a\left(\frac xy-r_0\right)\left(\frac xy-r_1\right)=a\left(x-r_0y\right)\left(x-r_1y\right).$$ | 2021-06-19 03:10:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8670313954353333, "perplexity": 645.2890796849143}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487643380.40/warc/CC-MAIN-20210619020602-20210619050602-00095.warc.gz"} |
https://zbmath.org/?q=an:1129.53302 | ## On Chen invariant of CR-submanifolds in a complex hyperbolic space.(English)Zbl 1129.53302
Recently B. Y. Chen [Jap. J. Math., New Ser. 26, No. 1, 105–127 (2000; Zbl 1026.53009)] introduced an invariant for a Riemannian manifold and obtained a sharp inequality between his invariant and the squared mean curvature for a CR-submanifolds $$M$$ in real spaceforms.
In the present paper, by applying the Chen invariant the author has obtained some interesting new results for CR-submanifolds which satisfy $\delta(n_1, ...,n_k) = c(n_1, ..., n_k)H^2 - b(n_1, ..., n_k) -3n + \frac{3}{2} \Sigma^k_{i = 1} n_i$ where $$\delta (n_1, ..., n_k)$$ and $$H^2$$ are the Chen invariant and the square of the mean curvature of $$M$$.
### MSC:
53C40 Global submanifolds
### Keywords:
submanifold; complex hyperbolic space; invariants
Zbl 1026.53009
Full Text: | 2022-07-06 10:33:53 | {"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": 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": 0.7123879790306091, "perplexity": 1052.0111879444057}, "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-2022-27/segments/1656104669950.91/warc/CC-MAIN-20220706090857-20220706120857-00075.warc.gz"} |
https://www.physicsforums.com/threads/light-and-relativity.363050/ | # Light and Relativity
1. Dec 13, 2009
### Dragohunter
2 questions
Does light have a frame of reference?
Although there is no speed greater than c, can it be said that the change in distance between two identifiable objects changed at a rate greater than c?
For example one person I know said this:
Last edited: Dec 13, 2009
2. Dec 13, 2009
### bcrowell
Staff Emeritus
Are you asking whether it's possible to have a frame of reference that's moving along with a beam of light? If so, then the answer is no. For example, the $\gamma$ factor linking this frame to the frame of any material object would be infinite.
Your friend is correct. What's forbidden by relativity is much more specific. For example, suppose you have an observer who takes measurements in a certain reference frame. Relativity forbids a particle from whizzing right past the observer's location at a speed greater than c. It doesn't forbid distant objects from moving at greater than c relative to the observer. For example, you can say that galaxies beyond the edge of the observable universe are moving away from us at greater than c; however, it's impossible for us to observe those galaxies. | 2018-03-21 13:07:15 | {"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": 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": 0.3748804032802582, "perplexity": 434.6575174328328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647649.70/warc/CC-MAIN-20180321121805-20180321141805-00280.warc.gz"} |
https://www.khanacademy.org/math/algebra-basics/alg-basics-graphing-lines-and-slope/alg-basics-slope/v/introduction-to-slope | # Intro to slope
CCSS Math: HSF.LE.A.2
Tags
## Video transcript
- [Voiceover] As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this notion of steepness, how steep a line is, how quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics. So ideally, we'd be able to assign a number to each of these lines or to any lines that describes how steep it is, how quickly does it increase or decrease? So what's a reasonable way to do that? What's a reasonable way to assign a number to these lines that describe their steepness? Well one way to think about it, could say well, how much does a line increase in the vertical direction for a given increase in the horizontal direction? So let's write this down. So let's say if we an increase increase, in vertical, in vertical, for a given increase in horizontal for a given increase a given increase in horizontal. So, how can this give us a value? Well let's look at that magenta line again. Now let's just start at an arbituary point in that magenta line. But I'll start at a point where it's going to be easy for me to figure out what point we're at. So if we were to start right here, and if I were to increase in the horizontal direction by one. So I move one to the right. To get back on the line, how much do I have to increase in the vertical direction? Well I have to increase in the vertical direction by two. By two. So at least for this magenta line, it looks like our increase in vertical is two, whenever we have an increase in one in the horizontal direction. Let's see, does that still work if I were to start here, instead of increasing the horizontal direction by one, if I were increase in the horizontal direction... So let's increase by three. So now, I've gone plus three in the horizontal direction, then to get back on the line, how much do I have to increase in the vertical direction? I have to increase by one, two, three, four, five, six I have to increase by six. So plus six. So when I increase by three in the horizontal direction, I increase by six in the vertical. We were just saying, hey, let's just measure how much to we increase in vertical for a given increase in the horizontal? Well two over one is just two and that's the same thing as six over three. So no matter where I start on this line, no matter where I start on this line, if I take and if I increase in the horizontal direction by a given amount, I'm going to increase twice as much twice as much in the vertical direction. Twice as much in the vertical direction. So this notion of this increase in vertical divided by increase in horizontal, this is what mathematicians use to describe the steepness of lines. And this is called the slope. So this is called the slope of a line. And you're probably familiar with the notion of the word slope being used for a ski slope, and that's because a ski slope has a certain inclination. It could have a steep slope or a shallow slope. So slope is a measure for how steep something is. And the convention is, is we measure the increase in vertical for a given in increase in horizontal. So six two over one is equal to six over three is equal to two, this is equal to the slope of this magenta line. So let me write this down. So this slope right over here, the slope of that line, is going to be equal to two. And one way to interpret that, for whatever amount you increase in the horizontal direction, you're going to increase twice as much in the vertical direction. Now what about this blue line here? What would be the slope of the blue line? Well, let me rewrite another way that you'll typically see the definition of slope. And this is just the convention that mathematicians have defined for slope but it's a valuable one. What is are is our change in vertical for a given change in horizontal? And I'll introduce a new notation for you. So, change in vertical, and in this coordinate, the vertical is our Y coordinate. divided by our change in horizontal. And X is our horizontal coordinate in this coordinate plane right over here. So wait, you said change in but then you drew this triangle. Well this is the Greek letter delta. This is the Greek letter delta. And it's a math symbol used to represent change in. So that's delta, delta. And it literally means, change in Y, change in Y, divided by change in X, change in X. So if we want to find the slope of the blue line, we just have to say, well how much does Y change for a given change in X? So, the slope of the blue line. So let's see, let me do it this way. Let's just start at some point here. And let's say my X changes by two so my delta X is equal to positive two. What's my delta Y going to be? What's going to be my change in Y? Well, if I go by the right by two, to get back on the line, I'll have to increase my Y by two. So my change in Y is also going to be plus two. So the slope of this blue line, the slope of the blue line, which is change in Y over change in X. We just saw that when our change in X is positive two, our change in Y is also positive two. So our slope is two divided by two, which is equal to one. Which tells us however much we increase in X, we're going to increase the same amount in Y. We see that, we increase one in X, we increase one in Y. Increase one in X, increase one in Y. >From any point on the line, that's going to be true. You increase three in X, you're going to increase three in Y. It's actually true the other way. If you decrease one in X, you're going to decrease one in Y. If you decrease two in X, you're going to decrease two in Y. And that makes sense from the math of it as well Because if you're change in X is negative two, that's what we did right over here, our change is X is negative two, we went two back, then your change in Y is going to be negative two as well. Your change in Y is going to be negative two, and negative two divided by negative two, is positive one, which is your slope again. | 2018-11-21 08:07:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7829355001449585, "perplexity": 292.14534035200484}, "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-2018-47/segments/1542039747369.90/warc/CC-MAIN-20181121072501-20181121094501-00462.warc.gz"} |
https://en.wikipedia.org/wiki/Distinguishing_coloring | # Distinguishing coloring
Distinguishing coloring of a 4-hypercube graph
In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.
Distinguishing colorings and distinguishing numbers were introduced by Albertson & Collins (1996), who provided the following motivating example, based on a puzzle previously formulated by Frank Rubin: "Suppose you have a ring of keys to different doors; each key only opens one door, but they all look indistinguishable to you. How few colors do you need, in order to color the handles of the keys in such a way that you can uniquely identify each key?"[1] This example is solved by using a distinguishing coloring for a cycle graph. With such a coloring, each key will be uniquely identified by its color and the sequence of colors surrounding it.[2]
## Examples
Eight asymmetric graphs, each given a distinguishing coloring with only one color (red)
A graph has distinguishing number one if and only if it is asymmetric.[3] For instance, the Frucht graph has a distinguishing coloring with only one color.
In a complete graph, the only distinguishing colorings assign a different color to each vertex. For, if two vertices were assigned the same color, there would exist a symmetry that swapped those two vertices, leaving the rest in place. Therefore, the distinguishing number of the complete graph Kn is n. However, the graph obtained from Kn by attaching a degree-one vertex to each vertex of Kn has a significantly smaller distinguishing number, despite having the same symmetry group: it has a distinguishing coloring with ${\displaystyle \lceil {\sqrt {n}}\rceil }$ colors, obtained by using a different ordered pair of colors for each pair of a vertex Kn and its attached neighbor.[2]
A distinguishing coloring of a ring of six keys, using two colors (red and unpainted)
For a cycle graph of three, four, or five vertices, three colors are needed to construct a distinguishing coloring. For instance, every two-coloring of a five-cycle has a reflection symmetry. In each of these cycles, assigning a unique color to each of two adjacent vertices and using the third color for all remaining vertices results in a three-color distinguishing coloring. However, cycles of six or more vertices have distinguishing colorings with only two colors. That is, Frank Rubin's keyring puzzle requires three colors for rings of three, four or five keys, but only two colors for six or more keys or for two keys.[2] For instance, in the ring of six keys shown, each key can be distinguished by its color and by the length or lengths of the adjacent blocks of oppositely-colored keys: there is only one key for each combination of key color and adjacent block lengths.
Hypercube graphs exhibit a similar phenomenon to cycle graphs. The two- and three-dimensional hypercube graphs (the 4-cycle and the graph of a cube, respectively) have distinguishing number three. However, every hypercube graph of higher dimension has distinguishing number only two.[4]
The Petersen graph has distinguishing number 3. However other than this graph and the complete graphs, all Kneser graphs have distinguishing number 2.[5] Similarly, among the generalized Petersen graphs, only the Petersen graph itself and the graph of the cube have distinguishing number 3; the rest have distinguishing number 2.[6]
## Computational complexity
The distinguishing numbers of trees, planar graphs, and interval graphs can be computed in polynomial time.[7][8][9]
The exact complexity of computing distinguishing numbers is unclear, because it is closely related to the still-unknown complexity of graph isomorphism. However, it has been shown to belong to the complexity class AM.[10] Additionally, testing whether the distinguishing chromatic number is at most three is NP-hard,[9] and testing whether it is at most two is "at least as hard as graph automorphism, but no harder than graph isomorphism".[11]
A coloring of a given graph is distinguishing for that graph if and only if it is distinguishing for the complement graph. Therefore, every graph has the same distinguishing number as its complement.[2]
For every graph G, the distinguishing number of G is at most proportional to the logarithm of the number of automorphisms of G. If the automorphisms form a nontrivial abelian group, the distinguishing number is two, and if it forms a dihedral group then the distinguishing number is at most three.[2]
For every finite group, there exists a graph with that group as its group of automorphisms, with distinguishing number two.[2] This result extends Frucht's theorem that every finite group can be realized as the group of symmetries of a graph.
## Variations
A proper distinguishing coloring is a distinguishing coloring that is also a proper coloring: each two adjacent vertices have different colors. The minimum number of colors in a proper distinguish coloring of a graph is called the distinguishing chromatic number of the graph.[12]
## References
1. ^ Rubin, Frank (1979), "Problem 729: the blind man's keys", Journal of Recreational Mathematics, 11: 128. Solution in vol. 12, 1980. As cited by Albertson & Collins (1996). Instead of using colors, Rubin phrased the problem in terms of key handles that could be distinguished from each other by touch. More precisely, this problem assumes that each key is symmetric, so that the keys cannot be distinguished from each other by their orientations on the keyring.
2. Albertson, Michael O.; Collins, Karen L. (1996), "Symmetry breaking in graphs", Electronic Journal of Combinatorics, 3 (1): R18, MR 1394549.
3. ^ See, e.g., Imrich, Wilfried; Klavžar, Sandi (2006), "Distinguishing Cartesian powers of graphs", Journal of Graph Theory, 53 (3): 250–260, CiteSeerX 10.1.1.59.9242, doi:10.1002/jgt.20190, MR 2262268, If a graph has no nontrivial automorphisms its distinguishing number is 1. In other words, D(G) = 1 for asymmetric graphs.
4. ^ Bogstad, Bill; Cowen, Lenore J. (2004), "The distinguishing number of the hypercube", Discrete Mathematics, 283 (1–3): 29–35, doi:10.1016/j.disc.2003.11.018, MR 2061481.
5. ^ Albertson, Michael O.; Boutin, Debra L. (2007), "Using determining sets to distinguish Kneser graphs", Electronic Journal of Combinatorics, 14 (1): R20, MR 2285824.
6. ^ Lal, A. K.; Bhattacharjya, B. (2009), "Breaking the symmetries of the book graph and the generalized Petersen graph", SIAM Journal on Discrete Mathematics, 23 (3): 1200–1216, doi:10.1137/080728640, MR 2538646. Lal and Bhattacharjya (Theorem 4.3) credit this result to an unpublished masters thesis of K. S. Potanka (Virginia Polytechnic University, 1998).
7. ^ Cheng, Christine T. (2006), "On computing the distinguishing numbers of trees and forests", Electronic Journal of Combinatorics, 13 (1): R11, MR 2200539.
8. ^ Arvind, V.; Cheng, Christine T.; Devanur, Nikhil R. (2008), "On computing the distinguishing numbers of planar graphs and beyond: a counting approach", SIAM Journal on Discrete Mathematics, 22 (4): 1297–1324, arXiv:math/0703927, doi:10.1137/07068686X, MR 2443115.
9. ^ a b Cheng, Christine T. (2009), "On computing the distinguishing and distinguishing chromatic numbers of interval graphs and other results", Discrete Mathematics, 309 (16): 5169–5182, doi:10.1016/j.disc.2009.04.004, MR 2548918.
10. ^ Russell, Alexander; Sundaram, Ravi (1998), "A note on the asymptotics and computational complexity of graph distinguishability", Electronic Journal of Combinatorics, 5: R23, MR 1617449.
11. ^ Eschen, Elaine M.; Hoàng, Chính T.; Sritharan, R.; Stewart, Lorna (2011), "On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two", Discrete Mathematics, 311 (6): 431–434, arXiv:0907.0691, doi:10.1016/j.disc.2010.12.013, MR 2799894.
12. ^ Collins, Karen L.; Trenk, Ann N. (2006), "The distinguishing chromatic number", Electronic Journal of Combinatorics, 13 (1): R16, MR 2200544. | 2021-09-17 11:14:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.6741383671760559, "perplexity": 886.3275442951546}, "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/1631780055632.65/warc/CC-MAIN-20210917090202-20210917120202-00073.warc.gz"} |
https://forums.opera.com/topic/36190/high-cpu-usage-watching-youtube/106?lang=en-US | Do more on the web, with a fast and secure browser!
• battery saver
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# High cpu usage watching youtube
• @fragger911
Is unusable for those with the problem since Media is mostly everywhere, even without the ability to stop AutoPlay and not always you use or watch in FullScreen.
The Workaround about change between WorkSpaces, for daily use, it's annoying.
If it's not fixed even for W10, then...
• I may have found a fix.
For me, Twitch was using about 30% of my CPU per screen. I'm not going to go into detail or lay out my specs but I did find a way to fix my problem.
1. Make sure hardware acceleration is enabled or this won't work
(If you can't find hardware acc, go here opera://settings and type in "hardware acceleration")
2. Go to opera://flags/#ignore-gpu-blocklist
Enable that setting, relaunch and you should be good.
I think someone above already answered this but I was unable to find this setting using their instructions.
I really hope this helps cause I've been having to stress over this for about 2 weeks and it feels good to be back at minimal CPU usage.
• @gamedevchrs No, problem persist for me in v.75.
It still goes to about +40% higher cpu power consumption with favicon animation compared to mouse cursor hover on tab.
Spec: win8.1, fanless cpu i5 4570T, internal gpu.
• Hi @marjang, first of all go to "opera:flags#disable-accelerated-video-decode" and make sure it's set to "Disabled", and if not disable it, restart the browser and check if it helps.
Also, you can enable "opera:flags#enable-gpu-rasterization" (don't forget to restart the browser to make it works).
In addition to the above, check if running the Opera with the following switches (add them at the end of the shortcut which you use to run Opera): "Path\to\launcher.exe" --disable-gpu-driver-bug-workarounds --num-raster-threads=4 --use-gl=desktop
improves all over performance of the browser (including this playing videos).
Small note: in case "--use-gl=desktop" makes some artifacts (causes displaying issues in the browser UI), remove it and try to run the Opera with the other switches only.
• @l33t4opera Thanks for your help! I followed each step you wrote and tried each time how cpu power changed with youtube favicon animated.
Sadly, no change. cpu goes to about +40% when favicon is animated.
• @marjang Which "youtube favicon" do you mean? Can you show it in the screenshot?
Does playing YouTube videos consume now less CPU resources or not?
If not try the opposite, I mean go to "opera:flags#disable-accelerated-video-decode" and now set it to "Enabled" and restart the browser. Maybe there's something broken with the flag switch, and when you enable this it will work.
• @l33t4opera This is an old problem of the Opera and it is not solved by the flags of chromium. The problem is in the animation of the favicon.
• screenshot
The discussion is about the tab's equalizer
• Hi guys, @johnd78 yes I was already aware about the issue with "lagging" audio indicator since the beginning.
@andrew84 Thanks for the screenshot. Are you sure it's only about that? , it states in general "High CPU usage watching YouTube".
Also, are you completely sure that the lagging is caused by this indicator? As I remember it the indicator was causing a much more CPU use than 40%.
For example, when I'm listening to only music (without video) on some websites, the indicator is active, but it seems not to use much more CPU resources in this case.
• @marjang Can you re-check it in the latest Opera stable 76.0.4017.94, please?
• @l33t4opera I can't say for sure, but I'd say that I don't see much difference when I compare the same video (and same resolution) in Chrome/Edge vs Opera with muted tab.
• As I remember it the indicator was causing a much more CPU use than 40%
Yeah, Opera was fixed to lower the CPU usage. But, it can still be high enough (as much as 17% in some cases) to cause performance issues.
DNA-82936 is an unresolved bug for this to make things even better. However, users ultimately want an option to disable the thing completely.
• @l33t4opera I tried in v.76 and it is the same with your mentioned flags enabled or disabled. Cpu power goes to about +40% when the sound favicon on a tab is animated. Causing cpu temperature rise over acceptable values.
My "solution" is to watch videos in fullscreen or hover mouse cursor over tab.
Spec: win8.1, fanless cpu i5 4570T, internal gpu.
• @marjang I see. Start watching a video, and check two things:
1. Right click on it, select "Stats for nerds", and see which codec it displays "avc" (h.264), vp9 or any other,
2. Open "chrome:media-internals", and click on the item below "Recent Players" which contains "kPlay", press "Ctrl+F" and search for "videodecodername". What it shows on the right of it, "FFmpeg..." or something else?
• Issue still persist in version 76.0.4017.94.
• @l33t4opera
1. stats4nerds: it says vp09.
2. kVideoDecoderName = "VpxVideoDecoder"
• @marjang Ah, so that could be also the cause of some additional using of the CPU.
Can you install the h264ify extension, and check if the browser uses less CPU resources when you replaying videos?
(you can see that it forces replaying videos with H.264 codec, when it displays "avc1... /mp4a..." under "Stats for nerds")
You can install the extension under a fresh profile, if you don't want to clutter the current one as follows:
"Path\to\launcher.exe" --user-data-dir=\Path\to\writable_dir\fresh_profile" .
• @l33t4opera Interesting, it definitely is better with the extension. Animated icon raises cpu for about 25% now.
• @marjang Nice, so at least that helped to reduce the CPU use in your case ;-)
By the way: it should now also use less CPU resources on most of the other website where you watch videos, not only on YT.
What it's displaying now in "chrome:media-internals" on the right of "videodecodername" ?
• videodecodername
"MojoVideoDecoder" in my case | 2021-10-17 17:07:26 | {"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": 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": 0.22317557036876678, "perplexity": 4208.997169890895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585178.60/warc/CC-MAIN-20211017144318-20211017174318-00340.warc.gz"} |
https://www.physicsforums.com/threads/power-series-expansion.793983/ | # Homework Help: Power series expansion
1. Jan 24, 2015
### ngc2024
1. The problem statement, all variables and given/known data
"Use power series to evaluate the function at the given point"
$ln (x+ \sqrt{1+{x^2}}) - sin x$
at $x = 0.001$
2. Relevant equations
Relevant power series:
A: $ln (1+x) = \Big( \sum_{n=0}^\infty\frac{({(-1)^{n+1}}{x^n})}{n} \Big)$
B: ${(1+x)^p} = \sum_{n=0}^{\infty }\binom{p}{n}{x^n}$
C: $sin x = \Big ( \sum_{n=0}^\infty\frac{({(-1)^{n}{x}^{2n+1}})}{(2n+1)!} \Big)$
3. The attempt at a solution
Starting out with the logarithmic expression, there are two power series.
If $\sqrt{1+{x^2}} = y$
then $ln (x+y) = \Big( \sum_{n=0}^\infty\frac{({(-x)^{n+1}}{y^n})}{n} \Big)$
and $y = \sum_{n=0}^{\infty }\binom{(\frac{1}{2})}{n}{({(x^2)^n)}}$
I hope I am correct this far? Of course, I can now expand the series and multiply term by term, but this is quite tedious - especially since (using wolfram alpha), the first terms are in the power of 5 and 7. I also tried making a general expression, but then I end up with something like..
$ln (x+y) = \Big( \sum_{n=0}^\infty\frac{({(-x)^{n+1}})}{n} \Big) * (\sum_{n=0}^{\infty }\binom{(\frac{1}{2})}{n}{({(x^2)^n)}})$
.. and I don't know how to simplify this.
There must be some obvious mistake, or a simpler way to do this?
2. Jan 24, 2015
### Staff: Mentor
That does not look right and I don't see where this comes from. You cannot simply replace occurences of 1 by a variable. In particular, the two occurences of "1" in the expansion given above have a completely different meaning.
What is $\binom{(\frac{1}{2})}{n}$?
There is a power series for $\sqrt{1+x}$ that you can use.
With the correct power series, it is sufficient to consider the first few terms - I guess you just need the first non-vanishing order of x.
3. Jan 24, 2015
### ngc2024
"In particular, the two occurrences of "1" in the expansion given above have a completely different meaning."
Yes, I was unsure of this.
And maybe I was unclear - I used the binomial series to expand $\sqrt{1+{x^2}}$
$\sqrt{1+{x^2}} = \sum_{0}^{\infty }\binom{\frac{1}{2}}{n}{(x^2)^n} = 1 + \frac{x^2}{2} - \frac{x^4}{8} + \frac{x^6}{16} ...$ where $\binom{\frac{1}{2}}{n}$ is the binomial expression 0.5 choose n.
I think this is correct?
I don't fully understand, however - how do I make a power series of:
$ln (1 + x + \frac{x^2}{2} - \frac{x^4}{8} + \frac{x^6}{16}...)$ ... ?
4. Jan 24, 2015
### Staff: Mentor
Okay, I found the appropriate generalization of the standard binomial coefficients needed for that.
$\ln(1+x+\frac{x^2}{2}+...) = \ln(1+c)= ...$ with c coming from the sum. You won't need many terms to find one that does not cancel.
5. Jan 24, 2015
### ngc2024
Of course! Thanks!
6. Jan 24, 2015
### Ray Vickson
Write $1 + x + \frac{x^2}{2} - \frac{x^4}{8} + \frac{x^6}{16} + \cdots$ as $1+z$, and use the series for $\ln(1+z)$ in powers of $z$. | 2018-07-16 06:00:39 | {"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": 0.8398468494415283, "perplexity": 464.3056952870222}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589179.32/warc/CC-MAIN-20180716041348-20180716061348-00351.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=Fermat%27s_Last_Theorem&diff=next&oldid=5095 | # Difference between revisions of "Fermat's Last Theorem"
Fermat's Last Theorem is a long-unproved theorem stating that for non-zero integers $\displaystyle a,b,c,n$ with $n \geq 3$, there are no solutions to the equation: $\displaystyle a^n + b^n = c^n$
## History
Fermat's last theorem was proposed by Pierre Fermat in the margin of his book Arithmetica. The note in the margin (when translated) read: "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." Despite Fermat's claim that a simple proof existed, the theorem wasn't proven until Andrew Wiles did so in 1993. Interestingly enough, Wiles's proof was much more complicated than anything Fermat could have produced himself. | 2021-03-01 14:10:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "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": 0.7989507913589478, "perplexity": 392.924416910229}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362513.50/warc/CC-MAIN-20210301121225-20210301151225-00425.warc.gz"} |
https://socratic.org/questions/how-do-empirical-formulas-and-molecular-formulas-differ | # How do empirical formulas and molecular formulas differ?
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$.
SMARTERTEACHER
Jan 18, 2014
The empirical formula represents the ratio of atoms in a molecule in lowest terms, while the molecular formula is the actual atom number in the molecule.
For instance, carbohydrates have an empirical formula of $C {H}_{2} O$, while the carbohydrate glucose has a molecular formula of ${C}_{6} {H}_{12} {O}_{6}$ and the sugar ribose found in RNA has a molecular formula of ${C}_{5} {H}_{10} {O}_{5}$
Water has an empirical formula of ${H}_{2} O$ which is the same as the molecular formula, but hydrogen peroxide whose molecular formula is ${H}_{2} {O}_{2}$, it has an empirical formula of $H O$. | 2019-04-19 10:36:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 54, "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": 0.8460046052932739, "perplexity": 921.479236461569}, "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-2019-18/segments/1555578527566.44/warc/CC-MAIN-20190419101239-20190419123239-00344.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=59290 | ## Irreversible reactions and temperature
$\Delta U=q+w$
Jessica Li 4F
Posts: 115
Joined: Fri Aug 09, 2019 12:16 am
### Irreversible reactions and temperature
Can you have an irreversible reaction that involves no change in temperature or delta U, or could you have a reaction at constant temperature but a nonzero value for delta U?
For instance, 4.17 is at constant temperature but delta U does not equal to 0.
Angela Patel 2J
Posts: 110
Joined: Sat Aug 24, 2019 12:17 am
### Re: Irreversible reactions and temperature
It is possible to have a reaction where the temperature change is 0 but delta U is constant. Work and heat both affect temperature, so if their effects on temperature cancel out you could technically have a situation where there is no temp. change.
005324438
Posts: 51
Joined: Sat Aug 24, 2019 12:16 am
### Re: Irreversible reactions and temperature
An example of this would be solid vs liquid of the same material. Inputting energy into the system will make the solid turn to liquid, but they can be the same temperature.
705302428
Posts: 50
Joined: Sat Aug 24, 2019 12:16 am
### Re: Irreversible reactions and temperature
Yes it is possible. | 2020-08-09 09:30:07 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "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": 0.6032863259315491, "perplexity": 2705.4376411704407}, "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/1596439738523.63/warc/CC-MAIN-20200809073133-20200809103133-00092.warc.gz"} |
https://block.arch.ethz.ch/brg/content/publication/1104 | # Design of a funicular concrete bridge with knitted formwork
Popescu M., Bouten S., Ranaudo F., Mengeot P., Wyns K., Van Mele T. and Block P.
Proceedings of IABSE Congress 2021
Ghent
2021
The need for sustainable design, engineering and fabrication strategies for concrete construction is recognised as a key challenge in the building industry. Using principles of structural geometry and material effectiveness in design makes it possible to significantly reduce the amount of material used in a structure and its embodied emissions.
This paper presents the design, engineering and digital fabrication strategies for a circular pedestrian bridge to be built as part of “De Groene Boog” development of the A16 highway north of Rotterdam, The Netherlands. The bridge is designed as a lightweight funicular filigree concrete gridshell based on the principle of a three-hinge arch extrapolated to 3D geometry. In its realisation, it demonstrates a model of circular construction using novel material developments (such as recycled concrete) and an efficient flexible formwork system. A hybrid spline-supported 3D-knitted textile made of recycled and natural fibres that is easy and fast to assemble, will be the formwork to cast the complex structural geometry needing minimal scaffolding. An in-house developed computational pipeline, based on the open-source COMPAS framework, enables the efficient design, collaborative exchange and streamlined fabrication of the bridge.
The presented design and fabrication process are developed collaboratively by the Block Research Group at ETH Zurich and De Groene Boog, commissioned by the Dutch Ministry of Infrastructure and Water Management (Rijkswaterstaat).
BibTeX
@inproceedings{Popescu2021,
author = "Popescu, M. and Bouten, S. and Ranaudo, F. and Mengeot, P. and Wyns, K. and Van Mele, T. and Block, P.",
title = "Design of a funicular concrete bridge with knitted formwork",
booktitle = "Proceedings of IABSE Congress 2021",
year = "2021",
editor = "",
volume = "",
number = "",
pages = "",
publisher = "",
month = "September",
doi = "",
note = "abstract accepted",
}
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ETH Zurich
Institute of Technology in Architecture
Block Research Group
Stefano-Franscini-Platz 1, HIB E 45
8093 Zurich, Switzerland
[email protected]
block.arch.ethz.ch
+41 44 633 38 35 phone
+41 44 633 10 53 fax | 2021-02-27 13:25:36 | {"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": 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": 0.3102928102016449, "perplexity": 11993.003937750034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358956.39/warc/CC-MAIN-20210227114444-20210227144444-00537.warc.gz"} |
https://docs.opengosim.com/manual/input_deck/thermodynamic_props/eos_water/ | # EOS WATER¶
Defines the water properties such as density, viscosity and enthalpy, and the surface density. For example:
EOS WATER
SURFACE_DENSITY 1000.0 kg/m^3
END
The above is the minimum required input, where the surface density is specified, while the other properties are defaulted to use standard water tables, IFC67.
When the GAS_WATER mode is used, the water viscosity is modelled by default with the Batzle and Wang correlation to account for brine salinity.
The following modelling options are available for the water properties:
Different type of models can be selected for different properties, however mixed selections should be assessed carefully. Properties for which no models have been selected will default to the IFC67 water tables.
## SURFACE_DENSITY specification of water surface density¶
This keyword specifies the density of the aqueous phase at surface conditions. This may include any dissolved salts and the effect of the surface temperature. The surface density value is used to convert fluid flows in the reservoir into surface volume flows. Surface volumes are used to specify required well rates with targets like TARG_WSV, and surface volume rates and totals are reported in the Mass file and in the Eclipse files. The surface density must be entered or the software will return an error. The density units are optional, if not entered are defaulted to kg/m^3.
An example is:
SURFACE_DENSITY 1004.2 kg/m^3 | 2021-07-25 09:00:53 | {"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": 0.5231271386146545, "perplexity": 2592.636540193969}, "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-2021-31/segments/1627046151641.83/warc/CC-MAIN-20210725080735-20210725110735-00082.warc.gz"} |
https://courses.lumenlearning.com/mathforliberalartscorequisite/chapter/rewriting-expressions-using-the-commutative-and-associative-properties/ | ## Rewriting Expressions Using the Commutative and Associative Properties
### Learning Outcomes
• Identify the associative and commutative properties of addition and multiplication
• Use the associative and commutative properties of addition and multiplication to rewrite algebraic expressions
## Use the Commutative and Associative Properties
Think about adding two numbers, such as $5$ and $3$.
$\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill 3+5\hfill \\ \hfill 8\hfill & & & \hfill 8\hfill \end{array}$
The results are the same. $5+3=3+5$
Notice, the order in which we add does not matter. The same is true when multiplying $5$ and $3$.
$\begin{array}{cccc}\hfill 5\cdot 3\hfill & & & \hfill 3\cdot 5\hfill \\ \hfill 15\hfill & & & \hfill 15\hfill \end{array}$
Again, the results are the same! $5\cdot 3=3\cdot 5$. The order in which we multiply does not matter.
These examples illustrate the commutative properties of addition and multiplication.
### Commutative Properties
Commutative Property of Addition: if $a$ and $b$ are real numbers, then
$a+b=b+a$
Commutative Property of Multiplication: if $a$ and $b$ are real numbers, then
$a\cdot b=b\cdot a$
The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
### example
Use the commutative properties to rewrite the following expressions:
1. $-1+3=$
2. $4\cdot 9=$
Solution:
1. $-1+3=$ Use the commutative property of addition to change the order. $-1+3=3+\left(-1\right)$
2. $4\cdot 9=$ Use the commutative property of multiplication to change the order. $4\cdot 9=9\cdot 4$
### try it
What about subtraction? Does order matter when we subtract numbers? Does $7 - 3$ give the same result as $3 - 7?$
$\begin{array}{ccc}\hfill 7 - 3\hfill & & \hfill 3 - 7\hfill \\ \hfill 4\hfill & & \hfill -4\hfill \\ & \hfill 4\ne -4\hfill & \end{array}$
The results are not the same. $7 - 3\ne 3 - 7$
Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
$\begin{array}{ccc}\hfill 12\div 4\hfill & & \hfill 4\div 12\hfill \\ \hfill \frac{12}{4}\hfill & & \hfill \frac{4}{12}\hfill \\ \hfill 3\hfill & & \hfill \frac{1}{3}\hfill \\ & \hfill 3\ne \frac{1}{3}\hfill & \end{array}$
The results are not the same. So $12\div 4\ne 4\div 12$
Since changing the order of the division did not give the same result, division is not commutative.
Addition and multiplication are commutative. Subtraction and division are not commutative.
Suppose you were asked to simplify this expression.
$7+8+2$
How would you do it and what would your answer be?
Some people would think $7+8\text{ is }15$ and then $15+2\text{ is }17$. Others might start with $8+2\text{ makes }10$ and then $7+10\text{ makes }17$.
Both ways give the same result, as shown below. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)
When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.
The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:
$5\cdot \frac{1}{3}\cdot 3$
Changing the grouping of the numbers gives the same result.
When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.
If we multiply three numbers, changing the grouping does not affect the product.
You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties.
### Associative Properties
Associative Property of Addition: if $a,b$, and $c$ are real numbers, then
$\left(a+b\right)+c=a+\left(b+c\right)$
Associative Property of Multiplication: if $a,b$, and $c$ are real numbers, then
$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
### example
Use the associative properties to rewrite the following:
1. $\left(3+0.6\right)+0.4=$
2. $\left(-4\cdot \frac{2}{5}\right)\cdot 15=$
### try it
Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.
### example
Use the Associative Property of Multiplication to simplify: $6\left(3x\right)$.
### try it
The following video provides more examples of how to simplify expressions using the commutative and associative properties of multiplication and addition. | 2022-01-22 08:29:13 | {"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": 0.6782909035682678, "perplexity": 379.4070266183772}, "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-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00405.warc.gz"} |
https://zbmath.org/?q=an:0937.55003 | # zbMATH — the first resource for mathematics
Schur $$Q$$-functions and a Kontsevich-Witten genus. (English) Zbl 0937.55003
Mahowald, Mark (ed.) et al., Homotopy theory via algebraic geometry and group representations. Proceedings of a conference on homotopy theory, Evanston, IL, USA, March 23-27, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 220, 255-266 (1998).
The author gives a homotopy-theoretical interpretation of Virasoro operations in Witten’s theory of two-dimensional topological gravity as endomorphisms of an ordinary cohomology theory with coefficients in the localization $$\Delta[q_1^{-1}]$$ of I. Schur’s ring $$\Delta$$ of $$Q$$-functions. The central construction of topological gravity is a partition function which is defined by a family of maps from compactifications of Riemann moduli spaces (for each genus) to the complex cobordism spectrum tensored with the rational numbers. The main result of the paper is the construction of a morphism from the complex cobordism spectrum to the spectrum of cohomology theory with coefficients in the localization $$\Delta[q_1^{-1}]$$. This morphism (called the Kontsevich-Witten genus) when composed with the above family sends the fundamental class of the moduli space to a highest-weight vector for a naturally defined Virasoro action on $$\Delta$$ tensored with the rational numbers. The resulting theory has many of the features of a vertex operator algebra.
For the entire collection see [Zbl 0901.00044].
Reviewer: T.E.Panov (Moskva)
##### MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 55N35 Other homology theories in algebraic topology 55P42 Stable homotopy theory, spectra
Full Text: | 2021-10-22 09:38:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7670419812202454, "perplexity": 509.5646313553015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585504.90/warc/CC-MAIN-20211022084005-20211022114005-00365.warc.gz"} |
https://gamedev.stackexchange.com/questions/142667/why-is-the-movement-space-increasing | # Why is the movement space increasing?
This is a follow-up question to my previous question. I got my key holding to move, using boolean variables and making another method with a javax.swing.Timer for delays(Thank you to KingDolphin), but strangely when I move my Icon it seems to increase how many pixels it moves each time by 1. It moves one pixel the first keyPressed, then two for the next, then three, etc. This may not seem like much, but when you are moving by 30+ pixels, it gets out of control. I really have no Idea how to fix it. I have tried changing some things, and I have "done my homework"(Googled and searched StackExchange sites including this one and StackOverflow), but have found nothing even related to it. Here's my layoutGame and move methods(The problem is in there), and thank you for helping me out:
private void layoutGame() {
JLabel title = new JLabel("Dodge The Enemies!");
Font titleFont = new Font(Font.SERIF, Font.BOLD, 32);
title.setFont(titleFont);
title.setHorizontalAlignment(JLabel.CENTER);
title.setBackground(Color.BLACK);
title.setForeground(Color.WHITE);
title.setOpaque(true);
JLabel background = new JLabel();
background.setIcon(backDrop);
JLabel you = new JLabel();
you.setBounds(x, y, 50, 50);
JTextPane direction = new JTextPane();
direction.setText("Direction");
@Override
public void keyPressed(KeyEvent e) {
if(e.getKeyChar() == 'w') {
wPressed = true;
move(you);
} else if (e.getKeyChar() == 'a') {
aPressed = true;
move(you);
} else if (e.getKeyChar() == 's') {
sPressed = true;
move(you);
} else if (e.getKeyChar() == 'd') {
dPressed = true;
move(you);
}
direction.setText(null);
}
public void keyReleased(KeyEvent e) {
wPressed = false;
aPressed = false;
sPressed = false;
dPressed = false;
}
});
}
private void move(JLabel icon) {
int delay = 100; //milliseconds
ActionListener taskPerformer = new ActionListener() {
public void actionPerformed(ActionEvent evt) {
if (wPressed) {
y = y - 1;
} else if(aPressed) {
x = x - 1;
} else if(sPressed) {
y = y + 1;
} else if(dPressed) {
x = x + 1;
}
icon.setBounds(x, y, 50, 50);
icon.repaint();
}
};
}
• I would really recommend you to reduce your code to the actually relevant parts. There is a lot of fluff in there which definitely won't have anything to do with the problem. The more code you post, the less people will be willing to dig through all of it until they found your bug. – Philipp Jun 19 '17 at 8:37
• @Philipp Ok, I will shorten that. Thank you for your feedback – CodingNinja Jun 19 '17 at 17:34
The reason for this lies in the way you've implemented the move method: in there, you create a new javax.swing.Timer object that is responsible for increasing or decreasing the icon's x or y position.
However, note the following:
1. Timers execute their actionPerformed method not just once, but indefinitely until they're canceled.
2. You create a new Timer object every time a key is pressed.
3. Your timers are never canceled, i.e., they keep running forever.
This means before long, you've created a whole bunch of Timers that all run concurrently and that all increase/decrease your icons position. That's why you see an additive effect. You can see this quite easily if you add the following debug line to the actionPerformed method inside move:
public void actionPerformed(ActionEvent evt) {
System.out.println("timer " + hashCode() + " running");
Now, watch the terminal output while you play the game.
I don't think you actually need a timer at all, you can just move the code that's currently inside actionPerformed directly into move and the effect will disappear.
(However, an even better tip would be to read up on "game loops".)
• Thank you, this solved my problem. I still used the timer, but I declared it privately. All I had to do was start it in my move method, and stop it in my keyReleased method. So relieved! – CodingNinja Jun 19 '17 at 18:03 | 2021-06-17 23:49:49 | {"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": 0.19002650678157806, "perplexity": 2828.722270592488}, "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-2021-25/segments/1623487634576.73/warc/CC-MAIN-20210617222646-20210618012646-00003.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/in-corner-rectangular-field-dimensions-35m-22-m-well-14-m-inside-diameter-dug-8-m-deep-earth-dug-out-spread-evenly-over-remaining-part-concept-of-surface-area-volume-and-capacity_75209 | # In a Corner of a Rectangular Field with Dimensions 35m × 22 M, a Well with 14 M Inside Diameter is Dug 8 M Deep. the Earth Dug Out is Spread Evenly Over the Remaining Part - Mathematics
Sum
In a corner of a rectangular field with dimensions 35m × 22 m, a well with 14 m inside diameter is dug 8 m deep. The earth dug out is spread evenly over the remaining part of the field. Find the rise in the level of the field.
#### Solution
We have,
Length of the field, l = 35 m,
Width of the field, b = 22 m,
Depth of the well, H = 8 m and
Radius of the well, "R" = 14/7 = 7 "m"
Let the rise in the level of the field be h.
Now,
Volume of the earth on remaining part of the field = Volume of earth dug out
⇒ Area of the earth on remaining part of the field = Volume of earth
⇒ (Area of the field - Area of base of the well ) × h = πR2H
⇒ (lb - πR2) × h = πR2
rArr (35xx22-22/7xx7xx7)xx"h" =22/7xx7xx7xx8
⇒ (770 - 154) × h = 1232
⇒ 616 × h = 1232
⇒ "h" = 1232/616
∴ h = 2 m
So, the rise in the level of the field is 2 m.
Is there an error in this question or solution?
#### APPEARS IN
RS Aggarwal Secondary School Class 10 Maths
Chapter 19 Volume and Surface Area of Solids
Exercise 19B | Q 32 | Page 900 | 2021-04-15 13:59:22 | {"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": 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": 0.7663766145706177, "perplexity": 1474.4211426644931}, "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/1618038085599.55/warc/CC-MAIN-20210415125840-20210415155840-00144.warc.gz"} |
http://mathhelpforum.com/statistics/4800-independent-probability-print.html | # Independent probability
• Aug 7th 2006, 08:42 PM
kwtolley
Independent probability
E and F are independent with P(E)=0.35 and P(F)=0.45. P(E union F)=___.
ok this one I have some trouble with. P(E) - P(F) = 0.10 this is my answer is it right. this looks easy but hard for me. list of possible answers are.
1. 0.80
2. 0.10
3. 0.1575
d. 0.6425
• Aug 8th 2006, 06:29 AM
galactus
Cap'n, you're much more learned than me so correct me if I am wrong, but doesn't P(E union F) mean P(E or F), not P(E and F)?.
Since E and F are independent, P(E)P(F)=(.35)(.45)=.1575
P(E or F)=P(E)+P(F)-P(E or F)=.35+.45-.1575=.6425, assuming they're not mutually exclusive.
Then again, mutually exclusive events can not be independent events.
• Aug 8th 2006, 07:32 AM
kwtolley
ok I see
Ok I see both ways, but don't a P(E union F) mean it needs to be added. I only ask this because thats what my text book shows how to do it. So please let me know hich is right, and thanks so much again for all the help.
• Aug 8th 2006, 03:16 PM
Soroban
Hello, kwtolley!
I agree with Galactus . . .
Quote:
$E$ and $F$ are independent with: $P(E)=0.35$ and $P(F)=0.45.\qquad P(E \cup F) = \_\_\_$
$1)\;0.80\qquad 2)\;0.10\qquad 3)\;0.1575\qquad 4)\;0.6425$
Formula: . $\boxed{P(E \cup F)\;=\;P(E) + P(F) - P(E \cap F)}$
Since $E$ and $F$ are independent: . $P(E \cap F) \:=\:P(E)\cdot P(F) \:=\:(0.35)(0.45)\:=\:0.1575$
Therefore: . $P(E \cup F) \;=\;0.35 + 0.45 - 0.1575 \;= \;0.6425$ . . . answer (d)
• Aug 8th 2006, 08:45 PM
kwtolley
thanks for the help
ok I got it | 2016-10-22 00:15:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "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": 0.8898000121116638, "perplexity": 3090.4144182374066}, "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/1476988718311.12/warc/CC-MAIN-20161020183838-00345-ip-10-171-6-4.ec2.internal.warc.gz"} |
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Horror is enjoying a massive resurgence right now, with blockbusting franchises like Insidious and Paranormal Activity consistently pulling in big bucks worldwide (both have new installments due later this... | 2017-12-12 23:36:37 | {"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": 0.9182392358779907, "perplexity": 6746.988674396876}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948520042.35/warc/CC-MAIN-20171212231544-20171213011544-00399.warc.gz"} |
https://mathhelpboards.com/threads/question-about-linear-dependency.2899/ | # Question about Linear Dependency
#### Yankel
##### Active member
which one of the next statements is the correct one ?
Let v,u,w be linearly dependent vectors in a vector space over R.
u1 = 2u
v1 = -3u+4v
w1 = u+2v-aw (a scalar from R)
(1) the vectors u1, v1 and w1 are linearly dependent for every value of a
(2) the vectors u1, v1 and w1 are linearly independent for every value of a
(3) the vectors u1, v1 and w1 are linearly independent for every value of a apart from 0
(4) the vectors u1, v1 and w1 are linearly independent for every positive value of a
(5) there exists a value of a for which the vectors u1, v1 and w1 are linearly independent
Thanks a lot !!
#### Chris L T521
##### Well-known member
Staff member
which one of the next statements is the correct one ?
Let v,u,w be linearly dependent vectors in a vector space over R.
u1 = 2u
v1 = -3u+4v
w1 = u+2v-aw (a scalar from R)
(1) the vectors u1, v1 and w1 are linearly dependent for every value of a
(2) the vectors u1, v1 and w1 are linearly independent for every value of a
(3) the vectors u1, v1 and w1 are linearly independent for every value of a apart from 0
(4) the vectors u1, v1 and w1 are linearly independent for every positive value of a
(5) there exists a value of a for which the vectors u1, v1 and w1 are linearly independent
Thanks a lot !!
This is somewhat similar to the last one. Again, we know that $u,v,w$ are dependent, implying that there are constants $c_1,c_2,c_3$ not all zero such that $c_1u+c_2v+c_3w=0$. Now, we want to analyze when the following is true:
$d_1u_1+d_2v_1+d_3w_1=0$
where $d_1,d_2,d_3\in\mathbb{R}$ are arbitrary constants. The idea now is to express the above equation in terms of a linear combination of just $u,v,w$, then use the fact that $u,v,w$ are linearly dependent to come up with the appropriate conclusion.
I hope this helps!
#### Yankel
##### Active member
right, so if I am not mistaken I get:
(2d1-3d2+d3)u + (-4d2+2d3)v + (-ad3)w = 0
what does it tells me ? I know that at least one of the coefficients is not zero, because u,v and w are dependent...what can I say about u1,v1,w1 and what about a ?
#### Deveno
##### Well-known member
MHB Math Scholar
i think focusing on the coefficients overmuch is a mistake.
it is clear that:
$\{u_1,v_1,w_1\} \subset \text{Span}(\{u,v,w\})$.
since {u,v,w} is linearly dependent, this has dimension ≤ 2.
therefore $u_1,v_1,w_1$ cannot be linearly independent, else we have a subspace of greater dimension than a space which contains it.
("a" is a red herring). | 2021-01-27 00:14:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5640824437141418, "perplexity": 539.2294444441334}, "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-2021-04/segments/1610704804187.81/warc/CC-MAIN-20210126233034-20210127023034-00259.warc.gz"} |
https://www.risk.net/risk-quantum/6221071/new-occ-default-model-cuts-5-billion-off-clearing-fund | The Options Clearing Corporation (OCC) lopped 36% off its clearing fund requirement in the third quarter, following the introduction of a new methodology for sizing its default resources.
Clearing members’ mandatory contributions to the default fund stood at $9.5 billion at end-September, down from$14.8 billion at end-June, and are now at their lowest level since the third quarter of 2017. It made for a second consecutive quarterly reduction in the amount of required contributions, which | 2019-01-18 09:10:50 | {"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": 0.3689765930175781, "perplexity": 2640.576475681494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660020.5/warc/CC-MAIN-20190118090507-20190118112507-00226.warc.gz"} |
https://www.physicsforums.com/threads/two-blocks-collide-momentum-energy-conservation.902770/ | # Two blocks collide (Momentum-Energy conservation)
1. Feb 5, 2017
### Arman777
1. The problem statement, all variables and given/known data
Block 1 of mass 3.0 kg is sliding across a flooe with speed $v_1=2.00 \frac m s$ when it makes a head-on,one dimensional,elastic collision with initially stationary block 2 of mass 2.0 kg.The coefficient of kinetic friction between the blocks and the floor is $μ_k=0,30$
Find the speeds of (a) block 1 and 2 just after the collision.Also find (c) their final seperation after friction has stopped them and (d) the energy lost to thermal energy because of the friction.
2. Relevant equations
Energy-Momnetum conservation Equations
3. The attempt at a solution
$Δ\vec p =\vec p_f-\vec p_i=F_{external}Δt$
So If we assume Δt is too small we can say $Δ\vec p=0$ so from that
$m_1v_1=m_1v'_1+m_2v'_2$
$6kg\frac m s=3kg v'_1+2kgv'_2$
And we know theres friction so there must be energy lost.From the Energy conservation
$\frac 1 2m_2(v'_2)^2+\frac 1 2m_1(v'_1)^2-\frac 1 2m_1(v_1)^2=W_{friction}=F_fΔx$
I dont know how to calculate Δx here.Also theres friction between two blocks..Where I should add it in the energy conservation equation ? And How can I calculate it
Last edited: Feb 5, 2017
2. Feb 5, 2017
### haruspex
You have correctly assumed Δt is negligible during the collision. Would that not be true of Δx too?
3. Feb 5, 2017
### TomHart
The problem stated, "The coefficient of kinetic friction between the blocks and the floor is μ_k=0,30." I think what that meant was that there is friction between each block and the floor - not necessarily between the two blocks. The collision is elastic, which is really all you need to know about the interaction of the two blocks.
To me it makes sense to find the velocities immediately after impact. Once the velocities are known, it should be straightforward to solve the remainder of the problem (distance traveled, etc.).
4. Feb 5, 2017
### Arman777
Maybe it took some distance to get there...I mean can we exactly say $Δx=0$ ?
I see you are right about that thefriction betwwen blocks is kinda awkward idea for this quesiton.
Ok I ll solve as $Δx=0$ lets see
5. Feb 5, 2017
### haruspex
Δx=vΔt. If v is moderate and Δt infinitesimal then Δx is infinitesimal.
6. Feb 5, 2017
### Arman777
but $Δt$ in there is the impact time isnt it ?
and $Δx$ is the distance traveled by object ?
I was thinking like this If there were $5m$ between A and B and they were collide the $Δt$ could be zero or we can assume its zero. But $Δx=5m$ cause it took $5m$ to come there and it lost some kinetic energy ?
By the way,I found the velocities correctly I ll try to solve other parts
7. Feb 5, 2017
### haruspex
The energy equation you posted that involved Δx appeared to be relating energy just before collision to energy just after. Therefore I assumed your Δx represented the distance moved during the collision.
You do need to find the two velocities immediately after the collision.
8. Feb 5, 2017
### Arman777
Oh I see now.I never looked that way...
And I solved the question(all parts) Thanks for help | 2017-08-20 10:43:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7393282651901245, "perplexity": 1329.859244037918}, "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-2017-34/segments/1502886106367.1/warc/CC-MAIN-20170820092918-20170820112918-00574.warc.gz"} |
http://why-lambda.blogspot.fr/2014/05/observing-sql-queries-in-their-natural.html | ## Thursday, May 01, 2014
### Observing SQL queries in their natural habitat
Torsten Grust and Jan Rittinger. 2013. Observing SQL queries in their natural habitat. ACM Trans. Database Syst. 38, 1, Article 3 (April 2013), 33 pages.
This paper presents a system called Habitat, that allows highlighting a subexpression of an SQL query whose results are of interest (e.g. due to surprising/incorrect query results). The system then compiles it into another query whose results are rows containing the values of variables in the "closure" (that is, the values of variables bound outside the subexpression) and the associated values of the selected query.
As a simple example, suppose the query (with underlined part being the WHERE clause):
SELECT r.A, s.D FROM R,S WHERE B=C
$R = \begin{array}{c|c} A & B\\ \hline 1 & 2\\ 3 & 7 \end{array}\quad S = \begin{array}{c|c} A & B\\ \hline 3 & 4\\ 7 & 8 \end{array}$
Then the normal result would be something like:
$\begin{array}{cc} A & D \\ \hline 3 &8 \end{array}$
and the result of the debug query arising from the above highlighting would be:
$\begin{array}{ccccc} R.A & R.B & S.C & S.D & B = C\\ \hline 1 & 2 & 3 & 4 & false\\ 1 & 2 & 7 & 8 & false\\ 3 & 7 & 3 & 4 & false\\ 3 & 7 & 7 & 8 & true \end{array}$
Of course, this is a very simplistic example. Habitat supports much richer queries including aggregation and grouping operations, and deals with the case that some highlighed part of the query is never reached for some set of bindings.
Stray thoughts:
• The compilation process essentially generates one new column per subexpression, and postprocessing is needed to eliminate columns/queries that are not actually needed to answer the user's request. It could be interesting to generate just what is needed directly.
• it would be interesting to compare the results with the information provided by different provenance systems (which allow selecting a part of the result and asking for an explanation), or combine the information somehow to understand how "this" part of the query influenced "that" part of the result
• it might also be interesting to formulate the compilation in terms of a higher-level / comprehension-based query language (perhaps with some syntactic marking indicating the subexpression of interest).
Labels: , | 2017-05-27 15:47:49 | {"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": 0.4347553551197052, "perplexity": 1776.3145399085813}, "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-2017-22/segments/1495463608956.34/warc/CC-MAIN-20170527152350-20170527172350-00395.warc.gz"} |
https://www.clutchprep.com/physics/practice-problems/142436/assume-the-following-waves-are-propagating-in-aira-calculate-the-wavelength-1-fo | What is an Electromagnetic Wave? Video Lessons
Concept
# Problem: Assume the following waves are propagating in aira) Calculate the wavelength λ1 for gamma rays of frequency f1 = 6.30×1021 Hz.b) Calculate the wavelength λ2 for visible light of frequency f2 = 5.40×1014 Hz.
###### FREE Expert Solution
The wavelength for a wave propagating in air is given by:
$\overline{){\mathbf{\lambda }}{\mathbf{=}}\frac{\mathbf{c}}{\mathbf{f}}}$, where c is the speed of light and f is the frequency of the wave propagation.
79% (65 ratings)
###### Problem Details
Assume the following waves are propagating in air
a) Calculate the wavelength λ1 for gamma rays of frequency f1 = 6.30×1021 Hz.
b) Calculate the wavelength λ2 for visible light of frequency f2 = 5.40×1014 Hz. | 2021-04-16 07:40:08 | {"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": 0.8342364430427551, "perplexity": 1034.4028125980844}, "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-2021-17/segments/1618038088731.42/warc/CC-MAIN-20210416065116-20210416095116-00625.warc.gz"} |
https://www.researching.cn/articles/OJ5491be97ccbad667 | Search by keywords or author
• Chinese Optics Letters
• Vol. 20, Issue 9, 091602 (2022)
Yu Cao, Li Chong, Ke-Hui Wu, Lu-Qian You, Sen-Sen Li, and Lu-Jian Chen*
Author Affiliations
• Department of Electronic Engineering, School of Electronic Science and Engineering, Xiamen University, Xiamen 361005, China
• show less
Yu Cao, Li Chong, Ke-Hui Wu, Lu-Qian You, Sen-Sen Li, Lu-Jian Chen. Dynamic coloration of polymerized cholesteric liquid crystal networks by infiltrating organic compounds[J]. Chinese Optics Letters, 2022, 20(9): 091602 Copy Citation Text show less
Abstract
We demonstrate the dynamic coloration of polymerized cholesteric liquid crystal (PCLC) networks templated by the “wash-out/refill” method in the presence of organic compounds. The reflection colors were modulated by two key approaches, that is, the injection of mutually soluble organic fluids into a microfluidic channel and the diffusion of volatile organic compounds (VOCs). The reversible tuning of reflected colors with central wavelengths between $∼450 nm$ and $∼600 nm$ was achieved by alternative injection of nematic liquid crystal E7 (nav = 1.64) and benzyl alcohol (n = 1.54) using syringe pumps. The fascinating iridescence with reflection centers from $∼620 nm$ to $∼410 nm$ was presented from the volatilization and diffusion of alcohol as a model VOC. Additionally, the flow velocity of fluid and the diffusion time were adjusted to explore the underlying mechanism for the dynamic coloration of cholesteric networks. This work is expected to extend the study of PCLCs as a dynamically tunable optofluidic reflector, visually readable sensor, or compact anti-counterfeit label in response to organic compounds.
1. Introduction
During the past few decades, cholesteric liquid crystals (CLCs) with intrinsic helical configuration of molecular directors have great perspectives towards a wide range of advanced photonic applications such as brightness-enhancement devices of liquid crystal (LC) displays, diffractive optical elements, smart windows, mirrorless lasers, and sensors[15]. Thanks to the Bragg reflection of CLCs, which confers nontrivial optical functionalities, significant structural color can be generated to reflect circularly polarized (CP) light with identical handedness in the visible spectrum. In general, the colors of CLCs, exhibited by the selective reflection wavelength, depend on the helical pitch length ($p$) or/and the average refractive index (average RI, $nav$)[68] in response to the external stimuli, such as temperature, electric field, and light irradiation. Also, some organic compounds can be incorporated into CLCs to alter the optical properties directly. The molecular structure of CLCs can be further modified with recognition fragments and utilized to absorb specific analytes. The uptake of specific analytes will affect $p$ and $nav$ and reveal different reflected colors consequently[911]. This feature is beneficial for the detection of volatile samples such as alcohol, amine, and acetone[1214], providing a versatile sensing platform with several advantages like low cost, being power-free, and naked-eye detection.
Copy Citation Text
Yu Cao, Li Chong, Ke-Hui Wu, Lu-Qian You, Sen-Sen Li, Lu-Jian Chen. Dynamic coloration of polymerized cholesteric liquid crystal networks by infiltrating organic compounds[J]. Chinese Optics Letters, 2022, 20(9): 091602 | 2022-06-25 01:58:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "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": 0.3668203353881836, "perplexity": 8081.021800876956}, "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/1656103033925.2/warc/CC-MAIN-20220625004242-20220625034242-00161.warc.gz"} |
https://www.physicsforums.com/threads/trig-substitution-into-integrals.738601/ | # Trig substitution into integrals
1. Feb 15, 2014
I was testing for convergence of a series:
∑$\frac{1}{n^2 -1}$ from n=3 to infinity
I used the integral test, substituting n as 2sin(u)
so here's the question:
when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine.
Is it still possible to make the substitution? Or is there a restriction when this happens?
2. Feb 15, 2014
### Staff: Mentor
What does "fit inside the sine" mean?
Sure, you can make the substitution. The integral will be from 3 to, say b, and you take the limit as b → ∞.
Not that you asked, but it's probably simpler and quicker to break up 1/(n2 - 1) using partial fractions.
3. Feb 16, 2014
Inside the sine meaning, the argument of the 'arcsine' would only range from -1 to 1.
So I'm guessing you can't make the substitution because arcsin(infinity) = error?
4. Feb 16, 2014
### gopher_p
If you're looking for an appropriate trig substitution for the definite integral (and not just one that gets you a correct antidierivative), then $\sec u$ is the way to go. But like Mark44 said, partial fractions is really the "right" technique of integration for this particular integral. | 2016-02-14 23:09:39 | {"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": 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": 0.9362971186637878, "perplexity": 1172.2557329566232}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454702039825.90/warc/CC-MAIN-20160205195359-00300-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://homework.cpm.org/category/CON_FOUND/textbook/mc1/chapter/6/lesson/6.1.1/problem/6-4 | ### Home > MC1 > Chapter 6 > Lesson 6.1.1 > Problem6-4
6-4.
On grid paper:
• Draw a square that measures $5$ units on each side.
• Draw a design inside your $5×5$ square.
• Then draw a square that measures $15$ units on each side.
• Enlarge your picture as accurately as possible so that it fits inside of the $15 × 15$ square.
How much wider and how much longer is your new picture?
A simple design will be easiest to enlarge.
How much longer is a $15$ unit number line than a $5$ unit number line? | 2023-02-05 18:21:20 | {"extraction_info": {"found_math": true, "script_math_tex": 6, "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": 0.43649783730506897, "perplexity": 1970.1770628240195}, "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-2023-06/segments/1674764500273.30/warc/CC-MAIN-20230205161658-20230205191658-00183.warc.gz"} |
https://www.physicsforums.com/threads/formation-of-shock-waves-and-gas-dynamics-of-flow-in-a-nozzle.586879/ | # Formation of shock waves and gas dynamics of flow in a nozzle.
1. Mar 14, 2012
### Urmi Roy
Hi I have a few questions regarding gas dynamics....
1. when M=1, dA=0....what does this mean? Does it imply that if we have a converging (subsonic) nozzle with its exit mach no,say 0.8, and then direct the flow to a constant area tube, then we have to attain M=1?
2. When we know the length of a duct, and the entering mach no, can we predict from only this information whether there will be a shock or not? (i.e what is the minimum info we require to predict whether there will be a shock or not?).
3. Is it true that while a subsonic flow can be continuously accelerated to supersonic flow, the converse (i.e continuous deceleration of a supersonic flow to subsonic speed) is not true and has to take place via a shock wave?(but I thought this was possible in a diverging diffusor.....
Thanks a lot.
2. Mar 15, 2012
### Mech_Engineer
I can't answer your shock questions, but for the first one it seems to me when you're considering a nozzle and you are given the conditions M=1 and dA=0, this means in my mind it is a boundary condition where the speed at the nozzle's throat (where the rate of change of the nozzle's cross-sectional area approaches zero) is Mach 1. In other words, the nozzle is a subsonic nozzle accelerating the flow to Mach 1, a.k.a. choked flow.
Whether the speed MUST be M=1 is dependent on geometry, but a tube is not a nozzle so it seems to me it will not have any further acceleration of the flow.
3. Mar 15, 2012
### Urmi Roy
The inference in the first question comes from the equation in red on
http://exploration.grc.nasa.gov/education/rocket/nozzle.html
To me, it seems that for subsonic flow with increasing area causes flow velocity to decrease whereas for supersonic flow with increasing area, the flow velocity decreases whereas a kind of singular/detached phenomenon is when M=1, and from this, we simply gather that dA=0.....it doesn't seem to have any connection with the other cases.
dA could be zero even if we had a maximum area at a crossection, and the passage were diverging-converging.....
4. Mar 15, 2012
Is there a context for this? If this is in the context of a nozzle, then Mech_Engineer hit it on the head. If you have a subsonic converging nozzle that ends such that the exit Mach number is 0.8, it will never reach sonic conditions without converging further. So in the case you mention, $M=1$ where $dA=0$, the upstream stagnation pressure is such that at the exit of the converging portion of the nozzle, the Mach number is unity. This would be the throat of the nozzle in the case of a converging-diverging nozzle.
No. Imagine you have adiabatic flow through a constant-area duct and you have the conditions at the entrance (station 1) and the exit (station 2). For now, don't worry about if they are the same or not. Starting from conservation of mass, momentum, energy and the equation of state, we have between the two stations:
Mass:
$$\rho_1 u_1 = \rho_2 u_2$$
Momentum:
$$\rho_1 u_1^2 + p_1 = \rho_2 u_2^2 + p_2$$
Energy:
$$c_p T_1 + \frac{1}{2}u_1^2 = c_p T_2 + \frac{1}{2}u_2^2$$
State:
$$\frac{p_1}{\rho_1 T_1} = \frac{p_2}{\rho_2 T_2}$$
There is obviously the trivial solution where $u_1 = u_2$, etc., but we are more interested in shocks, which represent a discontinuity here. So let's assume then that $u_1 \neq u_2$. In compressible flow, we generally like to work in the variables $M$, $p$, and $T$, so we can change the previous four equations into three equations in terms of these variables and solve for $M_1^2/M_2^2$:
$$\frac{M_1^2}{M_2^2} = \frac{\left(1+\gamma M_1^2\right)^2\left(1+\frac{\gamma-1}{2}M_2^2\right)}{\left(1+\gamma M_2^2\right)^2\left(1+\frac{\gamma-1}{2}M_1^2\right)}$$
Which is a quadratic equation with two solutions. One is simply what I mentioned earlier, or $M_1 = M_2$. The other is a discontinuity, or the shock. In other words, if you know the area of a duct and the incoming Mach number, you still have two solutions: one where the exit mach number is the same and one where it is different. To know whether there is a shock, you need to know that you have a discontinuity there. Most often, you know because you have a stagnation pressure difference between the inlet and the outlet. The important thing, though, is that you just need something to indicate the need for a flow discontinuity.
In general, you can't continuously decelerate a supersonic flow. In theory this is possible, but practically it isn't the case. If you had an established supersonic flow, you could have your supersonic diffuser designed such that you move back to Mach 1 at the second throat and then either have an infinitesimally weak shock bring it subsonic speeds and slow down through the diverging section. Problem number one is that you can and often do have shocks in the converging section of the diffuser, which can alter your flow.
The bigger problem, however, is that in a real flow in a supersonic wind tunnel, you have to start the tunnel first. When you do this, the starting shock (itself a normal shock) makes its way into the test section, and downstream of this the flow is subsonic. That means that it actually speeds up in the converging section, so in order to pass the mass flow of the nozzle through the diffuser as well, the diffuser throat has to be sized so as to go at most sonic at the Mach number downstream of the starting shock. This means the diffuser throat is necessarily much larger than that needed to bring the steady flow down to sonic conditions. What you end up with is a supersonic diffuser that slows the flow down and reduces the strength of the normal shock at its exit, but can never practically slow it down without a shock.
5. Mar 15, 2012
### Urmi Roy
Yes, please see pg 84 in the pdf http://ebooksgo.org/engineering-technology/fluidmechanics.pdf .......it is within the chapter variable area ducts....
In any case, I think I understand Mech_Engineer's point.
So unless we look for things that are hard to find, like stagnation pressure loss etc, we can't know if there would be a shock or not?
Given a particular numerical problem in gas dynamics, even if its in real life, we couldn't just see the setup and tell if its design or off -design?
6. Mar 15, 2012
Those aren't even hard to measure. At any rate, you usually won't even need to other than to confirm your design is functioning properly. For example, in a supersonic wind tunnel, you know both stagnation pressures because you have to know what pressure you are putting in and what you are exhausting to (typically either atmosphere or near-vacuum) in order to design the tunnel. You can get the upstream total pressure from a simple Pitot probe or from a pressure transducer in the air supply line and you can get the stagnation pressure of your exhaust reservoir through another pressure transducer. Simple.
In a more general sense, you need to know the conditions. It isn't enough just to say "you have air moving at Mach 2 through a duct, is there a shock?" If there doesn't need to be a shock, there won't be, so you need some sort of indication that nature will require a shock. Just looking at a constant-area duct and knowing a Mach number gives you none of that insight.
7. Mar 16, 2012
### Urmi Roy
Right....so that includes stagnation pressure, temperature, perhaps the static values of these also.....we just need the values of these quantities at certain cross-sections (usually inlet and test sections)......
We have a problem in our book which says :
"Air at p0= 10 bar, T0= 400K is supplied to a 50mm diameter pipe (constant area). The friction factor for the pipe (Fanno flow) for the pipe surface is 0.002.......if the mach no changes from 3 at the entry to 1 at the exit, determine.....etc etc....."
So, as Mech_Engineer said, for a constant area tube, you'd expect no change in mach no. However, in this example, clearly the mach number changes......is this just because there is friction?
Also, since friction causes increase in entropy, wouldn't we expect the mach no. to increase along the pipe, instead of dropping from 3 to 1?
Similarly, is it possible to intuitively explain why heat addition in a supersonic flow results in decrease of velocity?
Last edited: Mar 16, 2012
8. Mar 16, 2012
### Urmi Roy
I think what I'm looking for is an intuitive explanation of these two lines:
" In other words, subsonic flow through a pipe with friction will accelerate, and supersonic flow will decelerate."
"Adding heat to a fluid flowing at subsonic velocities in a pipe will cause the flow to accelerate, and adding heat to supersonic flow in a pipe will cause the flow to decelerate" | 2018-08-20 17:00:40 | {"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": 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": 0.6651877164840698, "perplexity": 584.6850021802213}, "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-2018-34/segments/1534221216718.53/warc/CC-MAIN-20180820160510-20180820180510-00088.warc.gz"} |
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1831 | ## On Geometric Ergodicity of CHARME Models
• In this paper we consider a CHARME Model, a class of generalized mixture of nonlinear nonparametric AR-ARCH time series. We apply the theory of Markov models to derive asymptotic stability of this model. Indeed, the goal is to provide some sets of conditions under which our model is geometric ergodic and therefore satisfies some mixing conditions. This result can be considered as the basis toward an asymptotic theory for our model.
$Rev: 13581$ | 2016-05-29 19:29:49 | {"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": 0.5525369644165039, "perplexity": 392.4398344267837}, "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-2016-22/segments/1464049281876.4/warc/CC-MAIN-20160524002121-00238-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://theinfolist.com/php/SummaryGet.php?FindGo=Fiber_optic | TheInfoList
A bundle of optical fibers
Fiber crew installing a 432-count fiber cable underneath the streets of Midtown Manhattan, New York City
A TOSLINK fiber optic audio cable with red light being shone in one end transmits the light to the other end
A wall-mount cabinet containing optical fiber interconnects. The yellow cables are single mode fibers; the orange and aqua cables are multi-mode fibers: 50/125 µm OM2 and 50/125 µm OM3 fibers respectively.
An optical fiber is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair.[1] Optical fibers are used most often as a means to transmit light[a] between the two ends of the fiber and find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths (data transfer rates) than electrical cables. Fibers are used instead of metal wires because signals travel along them with less loss; in addition, fibers are immune to electromagnetic interference, a problem from which metal wires suffer.[2] Fibers are also used for illumination and imaging, and are often wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope.[3] Specially designed fibers are also used for a variety of other applications, some of them being fiber optic sensors and fiber lasers.[4]
Optical fibers typically include a core surrounded by a transparent cladding material with a lower index of refraction. Light is kept in the core by the phenomenon of total internal reflection which causes the fiber to act as a waveguide.[5] Fibers that support many propagation paths or transverse modes are called multi-mode fibers, while those that support a single mode are called single-mode fibers (
An optical fiber is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair.[1] Optical fibers are used most often as a means to transmit light[a] between the two ends of the fiber and find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths (data transfer rates) than electrical cables. Fibers are used instead of metal wires because signals travel along them with less loss; in addition, fibers are immune to electromagnetic interference, a problem from which metal wires suffer.[2] Fibers are also used for illumination and imaging, and are often wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope.[3] Specially designed fibers are also used for a variety of other applications, some of them being fiber optic sensors and fiber lasers.[4]
Optical fibers typically include a core surrounded by a transparent cladding material with a lower index of refraction. Light is kept in the core by the phenomenon of total internal reflection which causes the fiber to act as a waveguide.[5] Fibers that support many propagation paths or transverse modes are called multi-mode fibers, while those that support a single mode are called single-mode fibers (SMF). Multi-mode fibers generally have a wider core diameter[6] and are used for short-distance communication links and for applications where high power must be transmitted.[7] Single-mode fibers are used for most communication links longer than 1,000 meters (3,300 ft).[citation needed]
Being able to join optical fibers with low loss is important in fiber optic communication.[8] This is more complex than joining electrical wire or cable and involves careful cleaving of the fibers, precise alignment of the fiber cores, and the coupling of these aligned cores. For applications that demand a permanent connection a fusion splice is common. In this technique, an electric arc is used to melt the ends of the fibers together. Another common technique is a mechanical splice, where the ends of the fibers are held in contact by mechanical force. Temporary or semi-permanent connections are made by means of specialized optical fiber connectors.[9]
The field of applied science and engineering concerned with the design and application of optical fibers is known as fiber optics. The term was coined by Indian-American physicist Narinder Singh Kapany, who is widely acknowledged as the father of fiber optics.[10]
In the late 19th
In the late 19th and early 20th centuries, light was guided through bent glass rods to illuminate body cavities.[14] Practical applications such as close internal illumination during dentistry appeared early in the twentieth century. Image transmission through tubes was demonstrated independently by the radio experimenter Clarence Hansell and the television pioneer John Logie Baird in the 1920s. In the 1930s, Heinrich Lamm showed that one could transmit images through a bundle of unclad optical fibers and used it for internal medical examinations, but his work was largely forgotten.[11][15]
In 1953, Du
In 1953, Dutch scientist Bram van Heel [nl] first demonstrated image transmission through bundles of optical fibers with a transparent cladding.[15] That same year, Harold Hopkins and Narinder Singh Kapany at Imperial College in London succeeded in making image-transmitting bundles with over 10,000 fibers, and subsequently achieved image transmission through a 75 cm long bundle which combined several thousand fibers.[15][16][17] The first practical fiber optic semi-flexible gastroscope was patented by Basil Hirschowitz, C. Wilbur Peters, and Lawrence E. Curtiss, researchers at the University of Michigan, in 1956. In the process of developing the gastroscope, Curtiss produced the first glass-clad fibers; previous optical fibers had relied on air or impractical oils and waxes as the low-index cladding material.[15]
Kapany coined the term fiber optics, wrote a 1960 article in Scientific American that introduced the topic to a wide audience, and wrote the first book about the new field.[15][18]
The first working fiber-optic data transmission system was demonstrated by German physicist Manfred Börner at Telefunken Research Labs in Ulm in 1965, which was followed by the first patent application for this technology in 1966.[19][20] In 1968, NASA used fiber optics in the television cameras that were sent to the moon. At the time, the use in the cameras was classified confidential, and employees handling the cameras had to be supervised by someone with an appropriate security clearance.[21]
Charles K. Kao and George A. Hockham of the British company Standard Telephones and Cables (STC) were the first, in 1965, to promote the idea that the attenuation in optical fibers could be reduced below 20 decibels per kilometer (dB/km), making fibers a practical communication medium.[22] They proposed that the attenuation in fibers available at the time was caused by impurities that could be removed, rather than by fundamental physical effects such as scattering. They correctly and systematically theorized the light-loss properties for optical fiber and pointed out the right material to use for such fibers—silica glass with high purity. This discovery earned Kao the Nobel Prize in Physics in 2009.[23] The crucial attenuation limit of 20 dB/km was first achieved in 1970 by researchers Robert D. Maurer, Donald Keck, Peter C. Schultz, and Frank Zimar working for American glass maker Corning Glass Works.[24] They demonstrated a fiber with 17 dB/km attenuation by doping silica glass with titanium. A few years later they produced a fiber with only 4 dB/km attenuation using germanium dioxide as the core dopant. In 1981, General Electric produced fused quartz ingots that could be drawn into strands 25 miles (40 km) long.[25]
Initially, high-quality optical fibers could only be manufactured at 2 meters per second. Chemical engineer Thomas Mensah joined Corning in 1983 and increased the speed of manufacture to over 50 meters per second, making optical fiber cables cheaper than traditional copper ones.[26] These innovations ushered in the era of optical fiber telecommunication.
The Italian research center CSELT worked with Corning to develop practical optical fiber cables, resulting in the first metropolitan fiber optic cable being deployed in Turin in 1977.[27][28] CSELT also developed an early technique for splicing optical fibers, called Springroove.[29]
Attenuation in modern optical cables is far less than in electrical copper cables, leading to long-haul fiber connections with repeater distances of 70–150 kilometers (43–93 mi). The erbium-doped fiber amplifier, which reduced the cost of long-distance fiber systems by reducing or eliminating optical-electrical-optical repeaters, was developed by two teams led by David N. Payne of the University of Southampton[30][31] and Emmanuel Desurvire at Bell Labs[32] in 1986 and 1987.
The emerging field of photonic crystals led to the development in 1991 of photonic-crystal fiber,[33] which guides light by diffraction from a periodic structure, rather than by total internal reflection. The first photonic crystal fibers became commercially available in 2000.[34] Photonic crystal fibers can carry higher power than conventional fibers and their wavelength-dependent properties can be manipulated to improve performance.
Optical fiber is used as a medium for telecommunication and computer networking because it is flexible and can be bundled as cables. It is especially advantageous for long-distance communications, because infrared light propagates through the fiber with much lower attenuation compared to electricity in electrical cables. This allows long distances to be spanned with few repeaters.
10 or 40 Gbit/s is typical in deployed systems.[35][36]
Through the use of wavelength-division multiplexing (WDM), each fiber can carry many independent channels, each using a different wavelength of light. The net data rate (data rate without overhead bytes) per fiber is the per-channel data rate reduced by the FEC overhead, multiplied by the number of channels (usually up to 80 in commercial dense WDM systems as of 2008[35][36]
Through the use of wavelength-division multiplexing (WDM), each fiber can carry many independent channels, each using a different wavelength of light. The net data rate (data rate without overhead bytes) per fiber is the per-channel data rate reduced by the FEC overhead, multiplied by the number of channels (usually up to 80 in commercial dense WDM systems as of 2008).
For short-distance applications, such as a network in an office building (see fiber to the office), fiber-optic cabling can save space in cable ducts. This is because a single fiber can carry much more data than electrical cables such as standard category 5 cable, which typically runs at 100 Mbit/s or 1 Gbit/s speeds.
Fiber is also immune to electrical interference; there is no cross-talk between signals in different cables and no pickup of environmental noise. Non-armored fiber cables do not conduct electricity, which makes fiber useful for protecting communications equipment in high voltage environments, such as power generation facilities, or metal communication structures prone to lightning strikes, and also preventing problems with ground loops. They can also be used in environments where explosive fumes are present, without danger of ignition. Wiretapping (in this case, fiber tapping) is more difficult compared to electrical connections, and there are concentric dual-core fibers that are said to be tap-proof.[citation needed]
Fibers are often also used for short-distance connections between devices.
Fiber is also immune to electrical interference; there is no cross-talk between signals in different cables and no pickup of environmental noise. Non-armored fiber cables do not conduct electricity, which makes fiber useful for protecting communications equipment in high voltage environments, such as power generation facilities, or metal communication structures prone to lightning strikes, and also preventing problems with ground loops. They can also be used in environments where explosive fumes are present, without danger of ignition. Wiretapping (in this case, fiber tapping) is more difficult compared to electrical connections, and there are concentric dual-core fibers that are said to be tap-proof.[citation needed]
Fibers are often also used for short-distance connections between devices. For example, most high-definition televisions offer a digital audio optical connection. This allows the streaming of audio over light, using the S/PDIF protocol over an optical TOSLINK connection.
Information traveling inside the optical fiber is even immune to electromagnetic pulses generated by nuclear devices.[b][citation needed]
Copper cable systems use large amounts of copper and have been targeted for metal theft, since the 2000s commodities boom.
Fibers have many uses in remote sensing. In some applications, the sensor is itself an optical fiber. In other cases, fiber is used to connect a non-fiberoptic sensor to a measurement system. Depending on the application, fiber may be used because of its small size, or the fact that no electrical power is needed at the remote location, or because many sensors can be multiplexed along the length of a fiber by using different wavelengths of light for each sensor, or by sensing the time delay as light passes along the fiber through each sensor. Time delay can be determined using a device such as an optical time-domain reflectometer.
Optical fibers can be used as sensors to measure strain, temperature, pressure, and other quantities by modifying a fiber so that the property to measure modulates the intensity, strain, temperature, pressure, and other quantities by modifying a fiber so that the property to measure modulates the intensity, phase, polarization, wavelength, or transit time of light in the fiber. Sensors that vary the intensity of light are the simplest since only a simple source and detector are required. A particularly useful feature of such fiber optic sensors is that they can, if required, provide distributed sensing over distances of up to one meter. In contrast, highly localized measurements can be provided by integrating miniaturized sensing elements with the tip of the fiber.[43] These can be implemented by various micro- and nanofabrication technologies, such that they do not exceed the microscopic boundary of the fiber tip, allowing such applications as insertion into blood vessels via hypodermic needle.
Extrinsic fiber optic sensors use an optical fiber cable, normally a multi-mode one, to transmit modulated light from either a non-fiber optical sensor—or an electronic sensor connected to an optical transmitter. A major benefit of extrinsic sensors is their ability to reach otherwise inaccessible places. An example is the measurement of temperature inside aircraft jet engines by using a fiber to transmit radiation into a radiation pyrometer outside the engine. Extrinsic sensors can be used in the same way to measure the internal temperature of electrical transformers, where the extreme electromagnetic fields present make other measurement techniques impossible. Extrinsic sensors measure vibration, rotation, displacement, velocity, acceleration, torque, and torsion. A solid-state version of the gyroscope, using the interference of light, has been developed. The fiber optic gyroscope (FOG) has no moving parts and exploits the Sagnac effect to detect mechanical rotation.
Common uses for fiber optic sensors include advanced intrusion detection security systems. The light is transmitted along a fiber optic sensor cable placed on a fence, pipeline, or communication cabling, and the returned signal is monitored and analyzed for disturbances. This return signal is digitally processed to detect disturbances and trip an alarm if an intrusion has occurred.
Optical fibers are widely used as components of optical chemical sensors and optical biosensors.[44]
Optical fiber can be used to transmit power using a photovoltaic cell to convert the light into electricity.[45] While this method of power transmission is not as efficient as conventional ones, it is especially useful in situations where it is desirable not to have a metallic conductor as in the case of use near MRI machines, which produce strong magnetic fields.[46] Other examples are for powering electronics in high-powered antenna elements and measurement devices used in high-voltage transmission equipment.
### Other uses
light guides in medical and other applications where bright light needs to be shone on a target without a clear line-of-sight path. In some buildings, optical fibers route sunlight from the roof to other parts of the building (see nonimaging optics). Optical-fiber lamps are used for illumination in decorative applications, including signs, art, toys and artificial Christmas trees. Optical fiber is an intrinsic part of the light-transmitting concrete building product LiTraCon.
Optical fiber can also be used in structural health monitoring. This type of sensor is able to detect stresses that may have a lasting impact on structures. It is based on the principle of measuring analog attenuation.
Use of optical fiber in a decorative lamp or nightlight
Optical fiber is also used in imaging optics. A coherent bundle of fibers is used, sometimes along with lenses, for a long, thin imaging device called an endoscope, which is used to view objects through a small hole. Medical en
Optical fiber can also be used in structural health monitoring. This type of sensor is able to detect stresses that may have a lasting impact on structures. It is based on the principle of measuring analog attenuation.
Optical fiber is also used in imaging optics. A coherent bundle of fibers is used, sometimes along with lenses, for a long, thin imaging device called an endoscope, which is used to view objects through a small hole. Medical endoscopes are used for minimally invasive exploratory or surgical procedures. Industrial endoscopes (see fiberscope or borescope) are used for inspecting anything hard to reach, such as jet engine interiors. Many microscopes use fiber-optic light sources to provide intense illumination of samples being studied.
In spectroscopy, optical fiber bundles transmit light from a spectrometer to a substance that cannot be placed inside the spectrometer itself, in order to analyze its composition. A spectrometer analyzes substances by bouncing light off and through them. By using fibers, a spectrometer can be used to study objects remotely.[47][48][49]
An optical fiber doped with certain rare-earth elements such as erbium can be used as the spectroscopy, optical fiber bundles transmit light from a spectrometer to a substance that cannot be placed inside the spectrometer itself, in order to analyze its composition. A spectrometer analyzes substances by bouncing light off and through them. By using fibers, a spectrometer can be used to study objects remotely.[47][48][49]
An optical fiber doped with certain rare-earth elements such as erbium can be used as the gain medium of a laser or optical amplifier. Rare-earth-doped optical fibers can be used to provide signal amplification by splicing a short section of doped fiber into a regular (undoped) optical fiber line. The doped fiber is optically pumped with a second laser wavelength that is coupled into the line in addition to the signal wave. Both wavelengths of light are transmitted through the doped fiber, which transfers energy from the second pump wavelength to the signal wave. The process that causes the amplification is stimulated emission.
Optical fiber is also widely exploited as a nonlinear medium. The glass medium supports a host of nonlinear optical interactions, and the long interaction lengths possible in fiber facilitate a variety of phenomena, which are harnessed for applications and fundamental investigation.[50] Conversely, fiber nonlinearity can have deleterious effects on optical signals, and measures are often required to minimize such unwanted effects.
Optical fibers doped with a wavelength shifter collect scintillation light in physics experiments.
Fiber-optic sights for handguns, rifles, and shotguns use pieces of optical fiber to improve visibility of markings on the sight.
An optical fiber is a cylindrical dielectric waveguide (nonconducting waveguide) that transmits light along its axis, by the process of total internal reflection. The fiber consists of a core surrounded by a cladding layer, both of which are made of dielectric materials.[51] To confine the optical signal in the core, the refractive index of the core must be greater than that of the cladding. The boundary between the core and cladding may either be abrupt, in step-index fiber, or gradual, in graded-index fiber. Light can be fed into optical fibers using lasers or LEDs.
### Index of refraction
The index of refraction (or refractive index) is a way of measuring the speed of light in a material. Light travels fastest in a vacuum, such as in outer space. The speed of light in a vacuum is about 300,000 kilometers (186,000 miles) per second. The refractive index of a medium is calculated by dividing the speed of light in a vacuum by the speed of light in that medium. The refractive index of a vacuum is therefore 1, by definition. A typical single-mode fiber used for telecommunications has a cladding made of pure silica, with an index of 1.444 at 1500 nm, and a core of doped silica with an index around 1.4475.[51] The larger the index of refraction, the slower light travels in that medium. From this information, a simple rule of thumb is that a signal using optical fiber for communication will travel at around 200,000 kilometers per second. To put it another way, the signal will take 5 milliseconds to travel 1,000 kilometers in fiber. Thus a phone call carried by fiber between Sydney and New York, a 16,000-kilometer distance, means that there is a minimum delay of 80 milliseconds (about ${\displaystyle {\tfrac {1}{12}}}$ of a second) between when one caller speaks and the other hears. (The fiber, in this case, will probably travel a longer route, and there will be additional delays due to communication equipment switching and the process of encoding and decoding the voice onto the fiber).
Most modern optical fiber is weakly guiding, meaning that the difference in refractive index between the core and the cladding is very small (typically less than 1%).[52]
### Total internal reflection
When light traveling in an optically dense medium hits a boundary at a steep angle (larger than the critical angle for the boundary), the light is completely reflected. This is called total internal reflection. This effect is used in optical fibers to confine light in the cor
The index of refraction (or refractive index) is a way of measuring the speed of light in a material. Light travels fastest in a vacuum, such as in outer space. The speed of light in a vacuum is about 300,000 kilometers (186,000 miles) per second. The refractive index of a medium is calculated by dividing the speed of light in a vacuum by the speed of light in that medium. The refractive index of a vacuum is therefore 1, by definition. A typical single-mode fiber used for telecommunications has a cladding made of pure silica, with an index of 1.444 at 1500 nm, and a core of doped silica with an index around 1.4475.[51] The larger the index of refraction, the slower light travels in that medium. From this information, a simple rule of thumb is that a signal using optical fiber for communication will travel at around 200,000 kilometers per second. To put it another way, the signal will take 5 milliseconds to travel 1,000 kilometers in fiber. Thus a phone call carried by fiber between Sydney and New York, a 16,000-kilometer distance, means that there is a minimum delay of 80 milliseconds (about ${\displaystyle {\tfrac {1}{12}}}$ of a second) between when one caller speaks and the other hears. (The fiber, in this case, will probably travel a longer route, and there will be additional delays due to communication equipment switching and the process of encoding and decoding the voice onto the fiber).
Most modern optical fiber is weakly guiding, meaning that the difference in refractive index between the core and the cladding is very small (typically less than 1%).[52]
### Total internal reflection
parabolic relationship between the index and the distance from the axis.
### Single-mode fiber
wavelength of the propagating light cannot be modeled using geometric optics. Instead, it must be analyzed as an electromagnetic waveguide structure, by solution of Maxwell's equations as reduced to the electromagnetic wave equation. The electromagnetic analysis may also be required to understand behaviors such as speckle that occur when coherent light propagates in multi-mode fiber. As an optical waveguide, the fiber supports one or more confined transverse modes by which light can propagate along the fiber. Fiber supporting only one mode is called single-mode or mono-mode fiber. The behavior of larger-core multi-mode fiber can also be modeled using the wave equation, which shows that such fiber supports more than one mode of propagation (hence the name). The results of such modeling of multi-mode fiber approximately agree with the predictions of geometric optics, if the fiber core is large enough to support more than a few modes.
The waveguide analysis shows that the light energy in the fiber is not completely confined in the core. Instead, especially in single-mode fibers, a significant fraction of the energy in the bound mode travels in the cladding as an evanescent wave.
The most common type of single-mode fiber has a core diameter of 8–10 micrometers and is designed for use in the near infrared. The mode structure depends on the wavelength of the light used, so that this fiber actually supports a small number of additional modes at visible wavelengths. Multi-mode fiber, by comparison, is manufactured with core diameters as small as 50 micrometers and as large as hundreds of micrometers. The normalized frequency V for this fiber should be less than the first zero of the Bessel function J0 (approximately 2.405).
### Special-purpose fiber
Some special-purpose optical fiber is constructed with a non-cylindrical core and/or cladding layer, us
The waveguide analysis shows that the light energy in the fiber is not completely confined in the core. Instead, especially in single-mode fibers, a significant fraction of the energy in the bound mode travels in the cladding as an evanescent wave.
The most common type of single-mode fiber has a core diameter of 8–10 micrometers and is designed for use in the near infrared. The mode structure depends on the wavelength of the light used, so that this fiber actually supports a small number of additional modes at visible wavelengths. Multi-mode fiber, by comparison, is manufactured with core diameters as small as 50 micrometers and as large as hundreds of micrometers. The normalized frequency V for this fiber should be less than the first zero of the Bessel function J0 (approximately 2.405).
Some special-purpose optical fiber is constructed with a non-cylindrical core and/or cladding layer, usually with an elliptical or rectangular cross-section. These include polarization-maintaining fiber and fiber designed to suppress whispering gallery mode propagation. Polarization-maintaining fiber is a unique type of fiber that is commonly used in fiber optic sensors due to its ability to maintain the polarization of the light inserted into it.
Photonic-crystal fiber is made with a regular pattern of index variation (often in the form of cylindrical holes that run along the length of the fiber). Such fiber uses diffraction effects instead of or in addition
Photonic-crystal fiber is made with a regular pattern of index variation (often in the form of cylindrical holes that run along the length of the fiber). Such fiber uses diffraction effects instead of or in addition to total internal reflection, to confine light to the fiber's core. The properties of the fiber can be tailored to a wide variety of applications.
Attenuation in fiber optics, also known as transmission loss, is the reduction in intensity of the light beam (or signal) as it travels through the transmission medium. Attenuation coefficients in fiber optics usually use units of dB/km through the medium due to the relatively high quality of transparency of modern optical transmission media. The medium is usually a fiber of silica glass that confines the incident light beam to the inside. For applications requiring spectral wavelengths especially in the mid-infrared ~2–7 μm, a better alternative is represented by fluoride glasses such as ZBLAN and InF3. Attenuation is an important factor limiting the transmission of a digital signal across large distances. Thus, much research has gone into both limiting the attenuation and maximizing the amplification of the optical signal. In fact, the four order of magnitude reduction in the attenuation of silica optical fibers over four decades (from ~1000 dB/km in 1965 to ~0.17 dB/km in 2005), as highlighted in the adjacent image (black triangle points; gray arrows), was the result of constant improvement of manufacturing processes, raw material purity, preform and fiber designs, which allowed for these fibers to approach the theoretical lower limit of attenuation. [53] Empirical research has shown that attenuation in optical fiber is caused primarily by both scattering and absorption. Single-mode optical fibers can be made with extremely low loss. Corning's SMF-28 fiber, a standard single-mode fiber for telecommunications wavelengths, has a loss of 0.17 dB/km at 1550 nm.[54] For example, an 8 km length of SMF-28 transmits nearly 75% of light at 1,550 nm. It has been noted that if ocean water was as clear as fiber, one could see all the way to the bottom even of the Marianas Trench in the Pacific Ocean, a depth of 36,000 feet.[55]
### Light scattering
Specular reflection
Diffuse reflection
The propagation of light through the core of an optical fiber is based on total internal reflection of the lightwave. Rough and irregular surfaces, even at the molecular level, can cause light rays to be reflected in random directions. This is called diffuse reflection or scattering, and it is typically characterized by wide variety of reflection angles.
Light scattering depends on the wavelength
The propagation of light through the core of an optical fiber is based on total internal reflection of the lightwave. Rough and irregular surfaces, even at the molecular level, can cause light rays to be reflected in random directions. This is called diffuse reflection or scattering, and it is typically characterized by wide variety of reflection angles.
Light scattering depends on the wavelength of the light being scattered. Thus, limits to spatial scales of visibility arise, depending on the frequency of the incident light-wave and the physical dimension (or spatial scale) of the scattering center, which is typically in the form of some specific micro-structural feature. Since visible light has a wavelength of the order of one micrometer (one millionth of a meter) scattering centers will have dimensions on a similar spatial scale.
Thus, attenuation results from the incoherent scattering of light at internal surfaces and interfaces. In (poly)crystalline materials such as metals and ceramics, in addition to pores, most of the internal surfaces or interfaces are in the form of grain boundaries that separate tiny regions of crystalline order. It has recently been shown that when the size of the scattering center (or grain boundary) is reduced below the size of the wavelength of the light being scattered, the scattering no longer occurs to any significant extent. This phenomenon has given rise to the production of transparent ceramic materials.
Similarly, the scattering of light
Light scattering depends on the wavelength of the light being scattered. Thus, limits to spatial scales of visibility arise, depending on the frequency of the incident light-wave and the physical dimension (or spatial scale) of the scattering center, which is typically in the form of some specific micro-structural feature. Since visible light has a wavelength of the order of one micrometer (one millionth of a meter) scattering centers will have dimensions on a similar spatial scale.
Thus, attenuation results from the incoherent scattering of light at internal surfaces and interfaces. In (poly)crystalline materials such as metals and ceramics, in addition to pores, most of the internal surfaces or interfaces are in the form of grain boundaries that separate tiny regions of crystalline order. It has recently been shown that when the size of the scattering center (or grain boundary) is reduced below the size of the wavelength of the light being scattered, the scattering no longer occurs to any significant extent. This phenomenon has given rise to the production of transparent ceramic materials.
Similarly, the scattering of light in optical quality glass fiber is caused by molecular level irregularities (compositional fluctuations) in the glass structure. Indeed, one emerging school of thought is that a glass is simply the limiting case of a polycrystalline solid. Within this framework, "domains" exhibiting various degrees of short-range order become the building blocks of both metals and alloys, as well as glasses and ceramics. Distributed both between and within these domains are micro-structural defects that provide the most ideal locations for light scattering. This same phenomenon is seen as one of the limiting factors in the transparency of IR missile domes.[56]
At high optical powers, scattering can also be caused by nonlinear optical processes in the fiber.[57][58]
In addition to light scattering, attenuation or signal loss can also occur due to selective absorption of specific wavelengths, in a manner similar to that responsible for the appearance of color. Primary material considerations include both electrons and molecules as follows:
• At the electronic level, it depends on whether the electron orbitals are spaced (or "quantized") such that they can absorb a quantum of light (or photon) of a specific wavelength or frequency in the ultraviolet (UV) or visible ranges. This is what gives rise to color.
• At the atomic or molecular level, it depends on the fr
The design of any optically transparent device requires the selection of materials based upon knowledge of its properties and limitations. The Lattice absorption characteristics observed at the lower frequency regions (mid IR to far-infrared wavelength range) define the long-wavelength transparency limit of the material. They are the result of the interactive coupling between the motions of thermally induced vibrations of the constituent atoms and molecules of the solid lattice and the incident light wave radiation. Hence, all materials are bounded by limiting regions of absorption caused by atomic and molecular vibrations (bond-stretching)in the far-infrared (>10 µm).
Thus, multi-phonon absorption occurs when two or more phonons simultaneously interact to produce electric dipole moments with which the incident radiation may couple. These dipoles can absorb energy from the incident radiation, reaching a maximum coupling with the radiation when the frequency is equal to the fundamental vibrational mode of the molecular dipole (e.g. Si–O bond) in the far-infrared, or one of its harmonics.
The selective absorption of infrared (IR) light by a particular material occurs because the selected frequency of the light wave matches the frequency (or an integer multiple of the frequency) at which the particles of that material vibr
Thus, multi-phonon absorption occurs when two or more phonons simultaneously interact to produce electric dipole moments with which the incident radiation may couple. These dipoles can absorb energy from the incident radiation, reaching a maximum coupling with the radiation when the frequency is equal to the fundamental vibrational mode of the molecular dipole (e.g. Si–O bond) in the far-infrared, or one of its harmonics.
The selective absorption of infrared (IR) light by a particular material occurs because the selected frequency of the light wave matches the frequency (or an integer multiple of the frequency) at which the particles of that material vibrate. Since different atoms and molecules have different natural frequencies of vibration, they will selectively absorb different frequencies (or portions of the spectrum) of infrared (IR) light.
Reflection and transmission of light waves occur because the frequencies of the light waves do not match the natural resonant frequencies of vibration of the objects. When IR light of these frequencies strikes an object, the energy is either reflected or transmitted.
Attenuation over a cable run is significantly increased by the inclusion of connectors and splices. When computing the acceptable attenuation (loss budget) between a transmitter and a receiver one includes:
• dB loss due to the type and length of fiber optic cable,
• dB loss introduced by connectors, and
• dB loss introduced by splices.
Connectors typically introduce 0.3 dB per connector on well-polished connectors. Splices typically introduce less than 0.3 dB per splice.
The total loss can be calculated by:
Loss = dB loss per connector × number of connectors + dB loss per splice × number of splices + dB loss per kil
Connectors typically introduce 0.3 dB per connector on well-polished connectors. Splices typically introduce less than 0.3 dB per splice.
The total loss can be calculated by:
Loss = dB loss per connector × number of connectors + dB loss per splice × number of splices + dB loss per k
The total loss can be calculated by:
where the dB loss per kilometer is a function of the type of fiber and can be found in the manufacturer's specifications. For example, typical 1550 nm single mode fiber has a loss of 0.4 dB per kilometer.
The calculated loss budget is used when testing to confirm that the measured loss is within the normal operating parameters. | 2021-02-27 21:39:53 | {"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": 0.4656969904899597, "perplexity": 1139.5267750283697}, "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-2021-10/segments/1614178359497.20/warc/CC-MAIN-20210227204637-20210227234637-00444.warc.gz"} |
http://www.cje.net.cn/CN/abstract/abstract19983.shtml | • 研究报告 •
### 基于突变级数法的广东省资源环境承载力动态
1. (1广州大学地理科学学院， 广州 510006； 2广东省地理国情监测与综合分析工程技术研究中心， 广州 510006)
• 出版日期:2019-06-10 发布日期:2019-06-10
### Dynamics of resource and environment carrying capacity of Guangdong Province based on catastrophe progression method.
SUN Duan1,2, CHEN Ying-biao1,2*, CAO Zhen1,2, HU Ying-long1,2
1. (1College of Geographical Sciences, Guangzhou University, Guangzhou 510006,China; 2Guangdong Province Engineering Technology Research Center for Geographical Condition Monitoring and Comprehensive Analysis, Guangzhou 510006, China).
• Online:2019-06-10 Published:2019-06-10
Abstract: The carrying capacity of resources and environment is important basis for sustainable development. Effective evaluation of carrying capacity of resources and environment is of great significance to the construction of ecological civilization society. Based on catastrophe progression method in catastrophe theory, we used 14 evaluation indices from four aspects of economy, society, resources and environment to evaluate the carrying capacity of urban comprehensive resources and environment of 21 prefecturelevel cities in Guangdong Province from 2000 to 2015. The results showed that in the 15 years, the index of comprehensive resources and environmental carrying capacity of Guangdong Province slightly decreased on the whole. Guangdong gradually changed from a high bearingcapacity province in most areas to a relatively high comprehensive bearing capacity index in the Pearl River Delta region and relatively weak comprehensive bearing capacity in Dongguan and Shenzhen. Each subsystem had different influence on the comprehensive carrying capacity of the city. The composite system of resources and environment is the supporting foundation of carrying capacity, while the composite system of economy and society is the powerful guarantee of carrying capacity. Among all cities, Jiangmen, Shantou, Huizhou and Foshan had the highest comprehensive carrying capacity index of resources and environment, indicating that the more balanced the economic level and resources and environment, the higher the comprehensive carrying capacity of cities. The bearing capacity index of each subsystem changed with different forms. The economic subsystem highlighted the status of urban economic development. The higher the economic level, the higher the carrying capacity, the smaller the overall growth pattern. Due to the aggravation of floating population, the carrying capacity index of the social subsystem declined slightly. Based on the premise of relative stability of resources, the carrying capacity index of resource subsystem was relatively stable. The carrying capacity index of environmental subsystem showed a downward trend due to environmental pollution. Spatially, the differences of bearing capacity index of the subsystems in Guangdong were significant. The index of carrying capacity of economic and social composite system was basically matched with the economic development level of each city. Meanwhile, the index of carrying capacity of the composite system of resources and environment was roughly consistent with the topography of Guangdong, with the index being high in eastern and western Guangdong, and gradual decline in the Pearl River Delta in Dongguan and Shenzhen. | 2022-01-21 19:47:26 | {"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": 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": 0.5293475389480591, "perplexity": 3485.0012378625747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303709.2/warc/CC-MAIN-20220121192415-20220121222415-00166.warc.gz"} |
http://newartisans.com/2007/11/ready-lisp-for-os-x-leopard/ | Lost in Technopolis
by John Wiegley
# Ready Lisp for OS X Leopard
Posted by John Wiegley on November 9, 2007 with labels: SBCL, SLIME
After upgrading my system to Leopard this weekend, I decided to refresh Ready Lisp as well. It now contains both 32-bit and 64-bit builds of SBCL (which has been bumped to 1.0.11), so if you have a Core 2 Duo machine, you’ll be running Lisp at full 64-bit! Alas, Emacs itself cannot support 64-bit as a Carbon app, because there are no 64-bit Carbon libraries. SLIME has also been updated, to CVS latest as of today. Aquamacs is still the same version at 1.2a.
I did spend several hours trying to build a fully Universal package that would run on PowerPC as well (I have a PowerBook G4 in addition to this MacBook Pro), but it seems Leopard has broken the PowerPC port of SBCL. Some of the core OS structures have changed, such as os_context_t.
Ready Lisp is now being versioned according to the SBCL version it contains, which makes today’s release ReadyLisp-1.0.11-10.5-x86.dmg. The older version, which still works on 10.4, can be downloaded here.
NOTE: The recent loading bug for Leopard users has been fixed. Please re-download. Also, it still does not work on OS X 10.4 (Tiger) at the moment. I will have to create a separate build of SBCL for that version this weekend. | 2015-10-10 19:40:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.24036988615989685, "perplexity": 4340.896726409309}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737962531.78/warc/CC-MAIN-20151001221922-00040-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://openstudy.com/updates/4dded6b1ee2c8b0bc3ee45e8 | ## anonymous 5 years ago The formula V=pie r^2h represents the volume of a cylinder where V represents the volume, r represents the radius of the base of the cylinder, and h represents the height of the cylinder. Solve this formula for h. Show all work.
$V=\pi r^2 h$ $h=\frac{V}{\pi r^2}$ | 2017-01-20 20:46:40 | {"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": 0.918541669845581, "perplexity": 542.2586693422169}, "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-2017-04/segments/1484560280872.69/warc/CC-MAIN-20170116095120-00470-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-the-inequality-3x-4-2x-1 | # How do you solve the inequality | 3x - 4 | > | 2x + 1 |?
$\textcolor{red}{\left(- \infty , \frac{3}{5}\right] \cup \left[5 , + \infty\right)}$
#### Explanation:
Given $\left\mid 3 x - 4 \right\mid > \left\mid 2 x + 1 \right\mid$
$+ \left(3 x - 4\right) > 2 x + 1$
$+ \left(3 x - 4 + 4 - 2 x\right) > 2 x + 1 + 4 - 2 x$
$\textcolor{red}{x > 5}$
$- \left(3 x - 4\right) > 2 x + 1$
$- 3 x + 4 > 2 x + 1$
$- 3 x - 2 x + 4 - 4 > 2 x - 2 x + 1 - 4$
$- 5 x$>$- 3$
Dividing by $- 5$ will change $>$ to $<$
$\frac{- 5 x}{- 5}$<$\frac{- 3}{- 5}$
$\textcolor{red}{x}$$\textcolor{red}{<}$$\textcolor{red}{\frac{3}{5}}$
God bless....I hope the explanation is useful. | 2019-11-17 17:01:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "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": 0.7785952091217041, "perplexity": 1633.1292448660824}, "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/1573496669225.56/warc/CC-MAIN-20191117165616-20191117193616-00455.warc.gz"} |
http://mathhelpforum.com/algebra/210658-algebra-2-can-someone-show-me-how-you-would-go-about-solving-g-x.html | # Math Help - Algebra 2:Can someone show me how you would go about solving for G(x)?
1. ## Algebra 2:Can someone show me how you would go about solving for G(x)?
The equation goes: 2G(x)-1/(G(x)+1=2√x2+1 -1/(√x2+1 +1)
I need to know how would you go about solving for the G(x) because i'm so stuck!
Note: The "x2+1" is under the square root. It jus didn't look right so I had to make a note.
2. ## Re: Algebra 2:Can someone show me how you would go about solving for G(x)?
is it (2G(x)-1)/(G(x)) or 2G(x)-(1/G(x))??
3. ## Re: Algebra 2:Can someone show me how you would go about solving for G(x)?
Because if it's (2G(x)-1)/(G(x)) then just split up the numerator and you'll get 2-1/G(x)= 2√x 2+1 -1/(√x2+1 +1) and that's the equivalent of 2-[2√x2+1 -1/(√x2+1 +1) ]=1/G(x)
So that'll give you G(x)=1/(2-[2√x2+1 -1/(√x2+1 +1)])
4. ## Re: Algebra 2:Can someone show me how you would go about solving for G(x)?
Hello, EJdive43!
How is this any different from any other "solve for G" problem?
$\text{Solve for G: }\:\frac{2G-1}{G+1} \:=\:\frac{2\sqrt{x^2+1} - 1}{\sqrt{x^2+1} + 1}$
We have: . $(2G-1)(\sqrt{x^2+1} + 1) \:=\:(G+1)(2\sqrt{x^2+1} - 1)$
. . $2G\sqrt{x^2+1} + 2G - \sqrt{x^2+1} - 1 \:=\:2G\sqrt{x^2+1} - G + 2\sqrt{x^2+1} - 1$
x . . . . . . . . . . . . . . . . . . . . $3G \:=\:3\sqrt{x^2+1}$
. . . . . . . . . . . . . . . . . . . . . . $G \:=\:\sqrt{x^2+1}$ | 2014-04-20 21:43:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "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": 0.9557715058326721, "perplexity": 544.2921495135248}, "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-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00116-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1504347/is-there-a-general-formula-to-compute-the-number-of-integer-solutions-of-an-equa | # Is there a general formula to compute the number of integer solutions of an equation?
recently, I asked a question concerning the number of solutions of a diophantine equation that used the rounding function. This question, however, dealt with a linear function, and I was wondering if the method or the answer could be generalized to include larger families of functions. I was trying to use the same technique given to me in the answer of that question to solve: $$\lceil x(\ln (x \ln x))\rceil+ \lceil y(\ln (y \ln y))\rceil = N$$ However, I do not think the same method can be applied, given that this equation is non linear. I have tried but I got stuck in the spot with the minimums and maximums. Is there a function $f(N)$ that counts how many integer solutions this equation has?
Furthermore, is there a function $f(g(x),m,N)$ that counts the number of integer solutions of the following equation? $$\sum_{i=1}^m g(x_i)=N$$
• You know the study of integer or rational solutions of diophantine equations is the major motivating force behind much of modern number theory and arithmetic geometry right? After all, Fermat's last theorem was only settled in the last 20 years and studies solutions to $x^n + y^n = z^n$ (for some fixed $n$). Pretty much as soon as you start considering nonlinear equations you go from linear algebra (easy) to algebraic geometry (hard). I'm sure estimates exist for certain types of equations, but there is pretty much nothing that can be said of nonlinear equations in general – oxeimon Oct 30 '15 at 0:50
• @oxeimon alright, so forget the last part of the question, what about the first? Is there an answer to the question regarding the first equation? – Guacho Perez Oct 30 '15 at 1:00
• @WillJagy why does anyone want to know anything in mathematics? :) – Guacho Perez Oct 30 '15 at 2:04
## 1 Answer
Of course there is such a function: you just defined it. The question is whether there is an algorithm for computing this function.
In general there is no way to tell whether a polynomial Diophantine equation (in several variables) has any solutions: see Hilbert's 10th Problem.
In your case, because the left side is greater than $x + y$ for $x, y \ge 3$, any solution must have $x + y \le N$, so there are only finitely many possibilities to try.
• And are there any methods to be used so I can find an explicit formula or even an algorithm? Maybe even properties of the function, i.e. bounds, asymptotes, etc.? Are there some books I can refer to? – Guacho Perez Oct 30 '15 at 2:05 | 2021-03-02 10:57:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7340121865272522, "perplexity": 126.30563841285931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363809.24/warc/CC-MAIN-20210302095427-20210302125427-00256.warc.gz"} |
https://www.physicsoverflow.org/611/higgs-field-is-its-discovery-truly-around-the-corner?show=2582 | # Higgs Field - Is its discovery truly "around the corner"?
+ 11 like - 0 dislike
32 views
Rather surprised I haven't seen many questions or discussion regarding the rumored confirmation of the Higgs field. As I understand it, the energies where they saw things were actually quite a bit higher than they had predicted (guessed?).
• What does it mean that the energies where it was detected are higher than anticipated?
• Does it impact the way we understand the standard model to work?
EDIT (Moshe, December 13): Now that the announcement is out, these questions and the more general one of potential implications of this for various ideas of BSM physics can be answered here.
This post has been migrated from (A51.SE)
retagged Apr 19, 2014
This won't be a discovery, but perhaps a strong hint. We will know much more in 72 hours, so maybe it's best to ask then. For now, though: it would be perfectly compatible with the Standard Model and nothing else. It would also be compatible with supersymmetry, although rather heavier than one might have expected for "typical" low-energy supersymmetry.
This post has been migrated from (A51.SE)
In my mind there are already plenty of places for both rumour mongering and complaining about said rumour mongering. Where we can do something new is a discussion of the implications of the announcement (on Tuesday), including that of the Higgs mass range preferred by the LHC. So, personally I’d love to see a question (even this question) focused on the physics of the Higgs and forgetting about the sociology and the “inside baseball”. Maybe a reformulation of the question, or one good answer, could set the right tone.
This post has been migrated from (A51.SE)
Dear @Matt, nice mini-answer, +1. One could also say that if it is "perfectly compatible with the SM", the mass is lighter than one would expect for a "typical" non-SUSY Standard Model (by 15 orders of magnitude, in fact, to add a funny twist to it). Incidentally, there will be evidence and "candidates" but the press release won't contain the word "hint". Do you want to make a bet? ;-)
This post has been migrated from (A51.SE)
@Moshe I got rid of the speculative social part of the question. Feel free to go in and edit more to bring it in line with the expectations of this site (remember, I am just fascinated by physics, but I am only an engineer and management type). Did I just, even tangentially, compare myself to a PHB?
This post has been migrated from (A51.SE)
@Larian: My comment is a mixture of what I see as the site’s expectations (still work in progress) and my own personal distaste for the relentless “there might be news” news stories. Thanks for making those changes, I think there is an interesting question there and this brings it out more clearly. I’d love for a few answers concentrating on the implications of a Higgs mass being in the indicated range, along the line of what Matt and Lubos started above.
This post has been migrated from (A51.SE)
@Moshe once the news conference happens, feel free to edit my question. You have a diamond after your name for a reason on this site. :) I just want to learn more, even if a lot of it requires me to study other documents to appreciate the answer.
This post has been migrated from (A51.SE)
@Larian: I think you have a good question there, no need for me to rephrase it. Most of the people able to give a good answer probably know already what is to be announced on Tuesday.
This post has been migrated from (A51.SE)
@LarianLeQuella: I think perhaps you will not get answers before tomorrow. I imagine those in the know are embargoed from talking about it until tomorrow.
This post has been migrated from (A51.SE)
"I'll soon be turning around corner now" (Queen, *Show must go on*).
This post has been migrated from (A51.SE)
+ 3 like - 0 dislike
I do not think it is fair to say "the Higgs looks like it's going to be at higher energies then anticipated". In fact, my money was on 160 GeV, based on models coming from noncommutative geometry. But the basic constraints on the Higgs mass from the standard model were really not very good (the following comes from the review article of Djouadi, 0503172v2). Short story: Unitarity starts failing around 900 GeV, perturbation theory fails around 700 GeV. A lower bound can be gotten by requiring that the Higgs quartic coupling remain positive, which gives $M_H> 70$ GeV. This depends on the cutoff-scale; the 70 GeV comes from assuming a 1 TeV cutoff scale. If the SM is valid up to GUT scales, this rises to $M_H>130$ GeV.
So, although the current values for Higgs are actually just below that 130 GeV, I think it's not fair to make any statement except that "it seems fine for the standard model" - it's too early to say "the Higgs mass implies new physics". All of these estimates are based on measured parameters such as the Top mass, which has it's own uncertainty associated with it. There is also the fine-tuning problem, but the above bounds generally give the same or slightly better estimates then that.
If someone wants to mention SUSY implications of a $\sim 125$ GeV Higgs, be my guest - there certainly are some. But SUSY can't possibly be real anyway, so I'm ok not knowing them ;-)
This post has been migrated from (A51.SE)
answered Jan 16, 2012 by (160 points)
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register. | 2018-06-19 08:26:54 | {"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": 0.40219762921333313, "perplexity": 869.1117296409337}, "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/1529267861981.50/warc/CC-MAIN-20180619080121-20180619100121-00223.warc.gz"} |
https://d2mvzyuse3lwjc.cloudfront.net/doc/LabTalk/ref/OPack-obj | # 3.7.5.44 OPack
LabTalk Object Type:
External Object
The OPack object is used to pack and unpack user-specified files.
## Properties:
Property Access Description
numeric
Controls whether the user is prompted to replace existing files. 0: (Default) Ask the user before replacing existing files. 1: Automatically replace existing files.
opack.file$Read/write string List of files need to be packed. Used by all methods. If the file name begins without '\', go to the Origin directory to find the corresponding file. opack.fileName$ Read/write
string
Archive file name. Use OPK as the extension name. Used by the opack.getFileNames( ), opack.pack( ), and opack.unpack() methods. If file name begins without '\', the archive file will be put in the corresponding path under the Origin directory.
Only used in the opack.pack( ) method. 1: (Default) Save the full path name when pack into archive file. 0: Save the file name only.
Only used in the opack.unpack( ) method. 1: (Default) Use the directory structure saved in OPK file when unpacking. 0: Use the file name only.
opack.unpack.targetFolder$Read/write, string This is the unpack directory. This property is only used in the opack.unpack( ) method. If it is not empty, use this as the target directory. opack.unpack.uninstall$ Read/write, string
LabTalk script to be executed when the package is uninstalled. This property is used by the Unpack method. During the installation of a package the LabTalk script in this property is saved by Origin. When Origin is asked to uninstall the package it will execute this script before any files are uninstalled/removed. This is where you can perform any custom tasks necessary to fully uninstall your package from Origin. OPack will only uninstall/remove files and will not undo any changes you made during installation. Remember that during installation OPack will execute script code in the setup.ogs file's OnBeforeSetup and OnAfterSetup sections. To use this property you should set it in your OnBeforeSetup script and clear it in the OnAfterSetup script.
## Methods:
Method Description
opack.getFileNames(TextFileName)
TextFileName is the name of text file containing the destination archive file name and the names of the files need to be packed. This method reads the first line to FileName$, then reads the following lines into the File$ property.
opack.findFile(FileName [, StrVar])
FileName is the file name to be found. StrVar is the string variable (optional). This method searches from the beginning of the File$property. If the required file name is found, it assigns it to the LabTalk string variable. Otherwise, it leaves the variable unchanged. When the search is performed, only the last part of the file name will be compared. For example: opack.findFile(.dat, S) will find the first file name with "dat" as the extension from the current position of the File$ property and then assign it to a LabTalk string variable S. This method's return value is the element index if found, or 0 if not found.
opack.findNext([StrVar])
StrVar is the LabTalk string variable (optional). This method searches from the current index of the File$property for the file name passed by the opack.findFile( ) method. If found, it is assigned to the LabTalk string variable. Otherwise, the variable is left unchanged. This method returns the element index if found, or 0 if not found. opack.pack([UserAccessCode]) UserAccessCode can be: 0 = All, 1 = Non-registered, and 2 = Registered. If UserAccessCode is not included, then 0 (All) is used. This method packs all the files listed in the File$ property to the file FileName$. If opack.pack.SaveFolderInfo = 1, save the full path name for every source file. If opack.pack.SaveFolderInfo = 0, save the file name only. This method returns 0 for success, 2 for a File I/O error, or 6 for a File creation error. opack.uninstall(ModuleName) Uninstall OPK file moduleName. If found, removes the button group from the ini file. Returns 0 = success, 1 = failure. opack.unpack( ) Unpacks the archive file FileName$. If opack.unpack.UseFolderInfo = 1, use the directory structure saved in the OPK file. If no folder information was saved, use the Origin directory instead. If opack.unpack.UseFolderInfo = 0, use the file name only. Return values are: 0 Success, 1 User access error, 2 File I/O error, 3 File header error, and 4 Expanding file error.
opack.reset( )
Sets opack.pack.SaveFolderInfo = 1, and opack.unpack.UseFolderInfo = 1. Sets all the other arguments to empty. | 2022-06-27 12:09:09 | {"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": 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": 0.28230300545692444, "perplexity": 6638.986857850263}, "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/1656103331729.20/warc/CC-MAIN-20220627103810-20220627133810-00149.warc.gz"} |
https://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-1.3 | Number systems
This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.
Free course
# 1.3 Further exercises
## Exercise 4
Solve the following linear equations.
• (a) 5x + 8 = −2
• (b) x + 2 = −5
• (c) 2x −1 = 5
• (d) 5x + 4 = 3
State in each case whether the solution belongs to , , and/or .
### Solution
• (a) x = −2, which belongs to (and hence to and ).
• (b) , which belongs to .
• (c) x = 3, which belongs to (and hence to , and ).
• (d) , which belongs to (and hence to ).
Remark Each linear equation with coefficients in has a solution in (because the equation ax + b = 0 where a, b and a ≠ 0 has solution x = − b/a, which involves only additive inverses and division of rational numbers). Similarly each linear equation with coefficients in has a solution in . The same is not true of , as part (d) illustrates.
## Exercise 5
Show that there is no rational number x such that x2 = 3.
### Solution
Suppose there is a rational number x such that x2 = 3. Then we can write x = p/q, where p and q are positive integers whose greatest common factor is 1.
Then the equation x2 = 3 becomes
Now p is either divisible by 3 or has remainder 1 or 2 on division by 3;
that is, p = 3k or 3k + 1 or 3k + 2 for some integer k.
But if p = 3k + 1, then p2 = 9k2 + 6k + 1 is not divisible by 3, and if p = 3k + 2, then
is not divisible by 3. So, since 3q2 is divisible by 3, we conclude that p = 3k.
Hence (3k)2 = 3q2, so q2 = 3k2.
But then the same argument applies to q to show that q must also be divisible by 3.
Hence 3 is a common factor of p and q.
This is a contradiction, so we conclude that the assumption must have been false. Hence there is no rational number x such that x2 = 3.
M208_6 | 2019-06-27 07:36:29 | {"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": 0.8609722256660461, "perplexity": 536.3455920288221}, "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-2019-26/segments/1560628000894.72/warc/CC-MAIN-20190627055431-20190627081431-00502.warc.gz"} |
https://www.parabola.unsw.edu.au/2020-2029/volume-57-2021/issue-3/article/two-simple-theorems-and-their-applications | Two simple theorems and their applications
What is the greatest product of $n$ numbers with some fixed sum?
What is the least sum of $n$ numbers with some fixed product?
These questions are answered, and applications are given. | 2022-01-29 13:01:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5601059794425964, "perplexity": 621.7346066531156}, "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-2022-05/segments/1642320306181.43/warc/CC-MAIN-20220129122405-20220129152405-00587.warc.gz"} |
https://www.physicsforums.com/threads/show-that-potential-energy-is-conserved.911943/ | # Show that potential energy is conserved
Tags:
1. Apr 19, 2017
### gelfand
1. The problem statement, all variables and given/known data
potential energy function of :
$$U(x) = 4x^2 + 3$$
And have to
i) Work out the equation of motion
ii) Prove explicitly that the total energy is conserved
2. Relevant equations
$$F = \frac{dU}{dt}$$
3. The attempt at a solution
I would say that I have the force of
$$F = 8x$$
By differentiating the given potential energy function. I need to work out the
equation of motion, what I have an object with mass $m$.
So this means that I have
$$F = 8x = ma$$
Then I have that
$$a = \frac{8x}{m}$$
Is this an equation of motion? I mean, it's acceleration, or should I find for
$v(t)$ and $x(t)$ as well as this?
In which case I would have
$$v(t) = \int a(t) dt$$
Which in this case is found as (having the mass in the equation seems unusual?)
$$v(t) = v_0 + \frac{1}{2m}8x^2 = v_0 + \frac{4}{m} x^2$$
So then from this I have that
$$x(t) = x_0 + v_0t + \frac{4}{3m}x^3$$
And this would be all of the equations of motion for this 1D case?
Then I need to prove that energy is conserved here, and I've no idea how to go
I've not been given any frictional forces, so it seems like it's just a given
that I'm going to have
$$W + PE_0 + KE_0 = PE_f + KE_f + \text{Energy(Lost)}$$
Here I can remove work $W$ and the energy lost for
$$PE_0 + KE_0 = PE_f + KE_f$$
And I need to do something with these?
Potential energy - I have the potential energy function given as part of the
problem which is
$$U(x) = 4x^2 + 3$$
Then I can sub this into the energy expression as
$$4x_0^2 + 3 + KE_0 = 4x_f^2 + 3 + KE_f$$
Getting rid of the constants seems pretty harmless
$$4x_0^2 + KE_0 = 4x_f^2 + KE_f$$
Now I'm really not sure what I should do from here, sub in kinetic formulas of
$K = \frac{1}{2}mv^2$?
$$4x_0^2 + \frac{1}{2}mv_0^2 = 4x_f^2 + \frac{1}{2}mv_f^2$$
I'm not sure if I can arrange this to be 'nicer' in any way either, I'm purely
thinking in algebra at the moment though not physics :S
$$8(x_0^2 - x_f^2) = m(v_f^2 - v_0^2)$$
I'm not sure if differentiation should do anything nice here, but I really have
no idea what I'm doing with this.
Thanks
2. Apr 19, 2017
### haruspex
Dividing energy by time gives power, not force.
3. Apr 20, 2017
### gelfand
OK $F = - \frac{dU}{dx}$ sorry , i'm still unsure about the question
4. Apr 20, 2017
### haruspex
You got a=8x/m ok, but you cannot integrate that wrt t directly. The expression you got for v(t) was the integral wrt x (which just gets you back to U).
There is a useful trick for solving equations like $\ddot x=f(x)$. Multiply both sides by $\dot x$, then integrate dt. | 2018-03-20 17:59:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8572453856468201, "perplexity": 643.6267685268238}, "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/1521257647519.62/warc/CC-MAIN-20180320170119-20180320190119-00166.warc.gz"} |
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