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# Unequal fringes fringe benefits in the United Kingdom by Francis Green
Written in English
## Subjects:
• Employee fringe benefits -- Great Britain.
Edition Notes
Bibliography, p123-128. - Includes index.
## Book details
Classifications The Physical Object Statement Francis Green, George Hadjimatheou, Robin Smail. Series Occasionalpapers on social administration -- no.75, Occasional papers on social administration -- no.75. Contributions Hadjimatheou, G., Smail, Robin G. M., National Council for Voluntary Organisations. LC Classifications HD4928.N6 Pagination 130p. : Number of Pages 130 Open Library OL17347618M ISBN 10 0719911419
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Jan 23, · An interesting thing happens if you pass light through a large number of evenly spaced parallel slits, called a diffraction wrcch2016.com interference pattern is created that is very similar to the one formed by a double slit (see).A diffraction grating can be manufactured by scratching glass with a sharp tool in a number of precisely positioned parallel lines, with the Unequal fringes book regions acting Author: OpenStaxCollege.
Labor on the Fringes of Empire: Voice, Exit and the Law (Palgrave Series in Indian Ocean World Studies) [Alessandro Stanziani] on wrcch2016.com *FREE* shipping on qualifying offers.
After the abolition of slavery in the Indian Ocean and Africa, the world of labor remained unequal, exploitativeAuthor: Alessandro Stanziani.
This book explains how economic inequality continues to grow in the United Unequal fringes book. For the past three decades, America has steadily become a nation of haves and have-nots.
Our incomes are increasingly drastically unequal: the top 1% of Americans collect almost 20% of the nation’s income-more than double their share in The Michelson interferometer is a common configuration for optical interferometry and was invented by Albert Abraham wrcch2016.com a beam splitter, a light source is split into two wrcch2016.com of those light beams is reflected back toward the beamsplitter which then combines their amplitudes using the superposition wrcch2016.com resulting interference pattern that is not directed back toward.
This book presents a sweeping, the world of labor remained unequal, exploitative, and violent, straddling a fine line between freedom and unfreedom. This book explains why. Labor on the Fringes of Empire is a pioneering global history of nineteenth-century labor.
Show all. About the authors. Jan 23, · Young’s double slit experiment gave definitive proof of the wave character of light. An interference pattern is obtained by the superposition of light from two slits.
There is constructive interference when, where is the distance between the slits, is the angle relative to the incident direction, and is the order of the wrcch2016.com: OpenStaxCollege. Think like an economist. For Nobel Prize-winner Paul Krugman, economics is not a set of answers—it’s a way of understanding the world.
In his economics MasterClass, Paul teaches you the principles that shape political and social issues, including access to health care, the tax debate, globalization, and political polarization.
With a diffraction grating (lots of slits), the fringes are highly focused, with small widths and unequal spacing. Conclusion from repeating the arguments above: small width, sure, unequal spacing, nope (maybe if you plot against $\lambda$, but that would be inconsistent with.
Jul 19, · Global justice requires that we look away from Geneva and New York to the outer fringes of global power.
Human Rights in an Unequal World Samuel Moyn The Belknap Press of Harvard University Press, \$ (cloth) the book is a wide-ranging offensive against both criticism of human rights and the most widely read critics of human rights.
Jan 13, · Deeply Divided: Racial Politics and Social Movements in Postwar America [Doug McAdam, Karina Kloos] on wrcch2016.com *FREE* shipping on qualifying offers.
By many measures--commonsensical or statistical--the United States has not been more divided politically or economically in the last hundred years than it is now.
How have we gone from the striking bipartisan cooperation and Cited by: Interferometry is a family of techniques in which waves, usually electromagnetic waves, are superimposed, causing the phenomenon of interference, which is used to extract information.
Interferometry is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy (and its applications to. Preface. Since the publication of the second edition of this book, many important advances have taken place in the field of optical testing.
On one hand, the requirements for faster and more precise tests are stronger than ever; on the other hand, the new technological tools permit us to do these tasks much better than before.
Strong optical asymmetry of an interference wedge with unequal-reflectivity mirrors and its use in unidirectional ring laser designs order fringes is investigated by finite-aperture.
View Robin Smail’s profile on LinkedIn, the world's largest professional community. Robin has 9 jobs listed on their profile. See the complete profile on LinkedIn and discover Robin’s connections and jobs at similar companies.
Co-author of book: "Unequal Fringes", December Research Assistant University of Aberdeen. – 2 Title: Training & Consultancy, Regional. This phenomenon is known as localization of the fringes and is related to the lack of spatial coherence of the illumination.
With an extended source and a plane-parallel plate, these interference fringes are fringes of equal inclination, localized at infinity. Get this from a library. Labor on the fringes of empire: voice, exit and the law. [Alessandro Stanziani] -- After the abolition of slavery in the Indian Ocean and Africa, the world of labor remained unequal, exploitative, and violent, straddling a fine line between freedom and unfreedom.
This book explains. Jul 15, · Entrepreneur Media, Inc. values your privacy. In order to understand how people use our site generally, and to create more valuable experiences for you, we may collect data about your use of Author: Melissa Loble.
Above all, he considers how the meaning and practices of freedom in the colonial world differed profoundly from those in the mainland. Arguing for a multi-centered view of imperial dynamics, Labor on the Fringes of Empire is a pioneering global history of nineteenth-century labor.
Books in Sociology: General Sociology published or distributed by the University of Chicago Press. The first five chapters of this book were written by Jane Austen around and the rest by "another" before its publication in Although I am sure that those who have studied the book in depth would notice the transition, I did not/5.
When two light waves from different coherent sources meet together, then the distribution of energy due to one wave is disturbed by the other.
This modification in the distribution of light energy due to super- position of two light waves is calle. Gender inequality refers to unequal treatment or perceptions of individuals based on their gender. It arises from differences in socially constructed gender roles.
[1] Gender systems are often dichotomous and hierarchical; gender binary systems may reflect the inequalities that manifest in numerous dimensions of daily life.
Jul 16, · In this new contribution to “Sociolinguistic Frontiers,” John Baugh provides an analytical overview of how the field has addressed issues of power and wrcch2016.com addresses how both social hierarchies and the legal system affect the standing of different languages and their wrcch2016.com: Rodrigo Ugarte.
BOOKS Encountering Poverty: Thinking and Acting in an Unequal World. Co-editors: Genevieve Negrón-Gonzales, Kweku Opoku-Agyemang, Clare Talwalker. University of California Press. Poverty, Interrupted Series. Encountering Poverty challenges mainstream frameworks of global poverty by going beyond the claims that poverty is a problem that can be solved through economic resources or.
Start studying soc ch. Learn vocabulary, terms, and more with flashcards, games, and other study tools. the unequal distribution of resources between countries.
gross national income (GNI) plus income earned by citizens and corporations headquartered in that country. peripheral nations. nations on the fringes of the global economy.
Project Gutenberg's The Last of the Mohicans, by James Fenimore Cooper This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever.
and ornamented with fringes and tassels. The colors are intended to imitate the hues of the wood, with a view to concealment.
The book was open at a hymn not ill. book has made me loose respect for sociology. I will stick with my STEM major where facts are actually fact. Racial and Ethnic Groups (14th Edition) Racial and Ethnic Relations, Census Update (9th Edition) There Goes the Neighborhood: Racial, Ethnic, and Class Tensions in Four Chicago Neighborhoods.
Reflection and transmission of the unequal mirrors interference wedge distance from the wedge and yields as a boundary case the fringes at plane-wave illumination.
influential book in. Mar 14, · NAMED BY THE NEW YORK TIMES AS ONE OF "6 BOOKS TO HELP UNDERSTAND TRUMP'S WIN". From the bestselling author of What’s the Matter With Kansas, a scathing look at the failures of liberal politics, a book that helps explain the shocking outcome of the presidential election.
It is a widespread belief among liberals that if only Democrats can continue to dominate /5. Jul 25, · According to Ian Bremmer in his book, Now the discussions gravitate to the radical fringes (on both sides). due to different talents, are unequal, but it can also help close the capital.
A Chicana from El Paso, Texas, Pat Mora has written three books of poetry, a children’s book, and a collection of essays. She earned both a B.A. and an M.A. in English from the University of Texas at El Paso while she raised three children who all currently attend universities. She lives and writes in.
Augustin-Jean Fresnel (also called Augustin Jean or simply Augustin), born in Broglie, Normandy, on 10 Maywas the second of four sons of the architect Jacques Fresnel (–) and his wife Augustine, née Mérimée (–).
Infollowing the Revolution, Broglie became part of the département of wrcch2016.com: 10 MayBroglie, Kingdom of France (now.
The wealth gap is just one of a multitude of indicators that reveal a country still deeply unequal along racial lines. This data begs for explanations of why and how. Our public discourse offers only frustration in its answers. Liberals often harken all the way back to slavery and elide the century and a Author: Justin Gomer.
It's one of her first books and it's true, the prose and development of characters is not as mature. The book is more of a homage/satire of Gothic lit, mixed with the comedy of manners style that she would be famous for later. But I LOVE this book. Seriously, this book is so wonderful.
The voice on this book/5(K). Jun 05, · The Uses of Pessimism and the Danger of False Hope by Roger Scruton The villains that stalk Roger Scruton's new book are those convinced that. Jun 04, · Labor on the Fringes of Empire is a tantalizing volume to engage with.
It draws on a wealth of important material and discusses major issues of current international research interest regarding abolitionism and unfree and free labor. It is also a difficult book, whose language is often unidiomatic and challenging for the wrcch2016.com: Preben Kaarsholm.
Aug 08, · Every goal is an event, no matter how unequal the contest. Frankly, a Major League baseball game that ends is a bit of a bore, but a Premier League. We should have mental eyelashes as well as physical ones, which can be used like shields, and let no evil thing in; or, like a stockade camp in the woods, which repels the first assault of the enemy.
This is the use of the fringes to our eyes, and so it Rev. Simpson— Days of Heaven Upon Earth. Nov 11, · Hi, Is there an equation that gives you the number of interference fringes if you know the light wave length, size and separation of the slits, and length Number of interference fringes based on n open slits | Physics Forums.
The Dance of Ideology and Unequal Riches. By Nolan McCarty, Keith T. Poole, and Howard Rosenthal they should be donating to like-minded legislators on the fringes of the ideological spectrum. Although much evidence suggests that extreme interest groups do donate heavily to the political process, the authors demonstrate that donations from.Dec 15, · The Great Recession is a speck in the rear-view mirror for America’s financial markets.
They’ve advanced far beyond pre-crisis levels. In fact, Goldman Sachs says you can go back a century.Sep 12, · How did the United States become the land of the unequal—and how do we find our way back? groups that speak only for the selfish fringes of the whole community.” and has written a book.
23899 views Friday, November 20, 2020
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# Obtaining field equations from an action
## Homework Statement
Provided an action:
$$S[A_\nu] = \int\left(\frac{1}{4}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta A_\mu J^\mu\right)\sqrt{-\eta}~d^4x$$
How would one go about finding the field equations for the same? I do understand that using the Euler-Lagrange method is how one should start out.
Does it tell us what kind of field we're looking at, at a glance?
## The Attempt at a Solution
I know that the terms in the parentheses are just ##F_{\mu\nu}F^{\mu\nu}##, but am unsure of how to proceed.
Does anyone know any good resources online or books that derive field equations and extract physics from a given action in steps, giving a good detailed process from which one can observe and learn how the Euler Lagrange method is being applied, in terms of the 4-vector notation. My struggle is essentially in decoding the physics, largely due to problems with understanding the notation.
Last edited:
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# Add or Modify SORT and DIST Keys
Abstract
SummaryOverview of possible Redshift cluster performance enhancements via configuring SORT and DIST keys.
You can improve the performance of your Redshift queries through the proper use of SORT and DIST keys.
The SORT key determines the order in which rows are loaded when initially loaded into a table. Choosing the right key for sorting can lead to faster query times as large sections of data can be skipped while processing the query. Alooma does not create any SORT keys automatically.
For more information about SORT keys, including how to choose the best key to use, see https://docs.aws.amazon.com/redshift/latest/dg/c_best-practices-sort-key.html
The DIST key defines the distribution of rows to compute nodes as they're loaded into a table. Optimizing the distribution can improve query time by minimizing the impact of the query optimizer's redistribution step.
Alooma creates the DIST key equal to the primary key that we receive from the source. This is the same key consolidations are based on.
For more information about choosing the DIST key, see https://docs.aws.amazon.com/redshift/latest/dg/c_best-practices-best-dist-key.html
Redshift does not allow modifications of SORT or DIST keys in existing tables. If you want to add or modify SORT or DIST keys in an existing Redshift table, you’ll need to create a new table. The basic process is:
1. Rename the existing table.
2. Create a new table with the SORT/DIST keys (re)defined as part of the schema.
3. Copy the data from the original table to the new table.
4. Rename the new table to the original table name.
5. Load the new table.
6. Drop the old table.
For example:
Copy
ALTER TABLE target_table RENAME TO old_target_table;
CREATE TABLE updated_target_table(/*[replicate the original table schema.]*/);
/* The CREATE TABLE is where you copy the schema of the original
table into the updated table and also where you would add/edit
your SORT and DIST keys */
INSERT INTO updated_target_table (SELECT * FROM old_target_table);
ALTER TABLE updated_target_table RENAME TO target_table;
DROP TABLE old_target_table;
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# Fourier transform on finite groups
Jump to: navigation, search
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
## Definitions
The Fourier transform of a function $f : G \rightarrow \mathbb{C}\,$ at a representation $\varrho : G \rightarrow GL(d_\varrho, \mathbb{C})\,$ of $G\,$ is
$\widehat{f}(\varrho) = \sum_{a \in G} f(a) \varrho(a).$
So for each representation $\varrho\,$ of $G\,$, $\widehat{f}(\varrho)\,$ is a $d_\varrho \times d_\varrho\,$ matrix, where $d_\varrho\,$ is the degree of $\varrho\,$.
Let $\varrho_i\,$ be a complete set of inequivalent irreducible representations of $G$. Then, $|G| = \sum_i d_{\varrho_i}^2$. Then the inverse Fourier transform at an element $a\,$ of $G\,$ is given by
$f(a) = \frac{1}{|G|} \sum_i d_{\varrho_i} \text{Tr}\left(\varrho_i(a^{-1})\widehat{f}(\varrho_i)\right),$
where $d_{\varrho_i}\,$ is the degree of the representation $\varrho_i.\,$
## Properties
### Transform of a convolution
The convolution of two functions $f, g : G \rightarrow \mathbb{C}\,$ is defined as
$(f \ast g)(a) = \sum_{b \in G} f(ab^{-1}) g(b).$
The Fourier transform of a convolution at any representation $\varrho\,$ of $G\,$ is given by
$\widehat{f \ast g}(\varrho) = \widehat{f}(\varrho)\widehat{g}(\varrho).$
### Plancherel formula
For functions $f, g : G \rightarrow \mathbb{C}\,$, the Plancherel formula states
$\sum_{a \in G} f(a^{-1}) g(a) = \frac{1}{|G|} \sum_i d_{\varrho_i} \text{Tr}\left(\widehat{f}(\varrho_i)\widehat{g}(\varrho_i)\right),$
where $\varrho_i\,$ are the irreducible representations of $G.\,$
## Fourier transform on finite abelian groups
Since the irreducible representations of finite abelian groups are all of degree 1 and hence equal to the irreducible characters of the group, Fourier analysis on finite abelian groups is significantly simplified. For instance, the Fourier transform yields a scalar- and not matrix-valued function.
Furthermore, the irreducible characters of a group may be put in one-to-one correspondence with the elements of the group.
Therefore, we may define the Fourier transform for finite abelian groups as
$\widehat{f}(s) = \sum_{a \in G} f(a) \bar{\chi_s}(a).$
Note that the right-hand side is simply $\langle f, \chi_s\rangle$ for the inner product on the vector space of functions from $G\,$ to $\mathbb{C}\,$ defined by
$\langle f, g \rangle = \sum_{a \in G} f(a) \bar{g}(a).$
The inverse Fourier transform is then given by
$f(a) = \frac{1}{|G|} \sum_{s \in G} \widehat{f}(s) \chi_s(a).$
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply $\delta_{a,0},\,$ where 0 is the group identity and $\delta_{i,j}\,$ is the Kronecker delta.
## Applications
This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized quickly using the fast Fourier transform, and this yields a fast method for solving systems of linear equations with circulant matrices. Similarly, the Fourier transform on arbitrary groups can be used to give fast algorithms for matrices with other symmetries (Åhlander & Munthe-Kaas 2005). These algorithms can be used for the construction of numerical methods for solving partial differential equations that preserve the symmetries of the equations (Munthe-Kaas 2006).
## References
• Åhlander, Krister; Munthe-Kaas, Hans Z. (2005), "Applications of the generalized Fourier transform in numerical linear algebra", BIT 45 (4): 819–850, doi:10.1007/s10543-005-0030-3, MR 2191479.
• Diaconis, P. (1988). Group Representations in Probability and Statistics. Lecture Notes — Monograph Series, Vol. 11. Hayward, California: Institute of Mathematical Statistics.
• Diaconis, P. (1991). "Finite Fourier Methods: Access to Tools." In Probabilistic Combinatorics and its Applications, Proceedings of Symposia in Applied Mathematics, Vol. 44. Bollobás, B., and Chung, F. R. K. (ed.).
• Munthe-Kaas, Hans Z. (2006), "On group Fourier analysis and symmetry preserving discretizations of PDEs", Journal of Physics A 39 (19): 5563–5584, doi:10.1088/0305-4470/39/19/S14, MR 2220776.
• Terras, A. (1999). Fourier Analysis on Finite Groups and Applications. Cambridge: Cambridge University Press.
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# PrintWriter font colour
Hi there!
I am trying to write via my groovy PrintWriter a debug text written in the regular black text, and an error text written in red. My googling was fruitless, if someone can help, it would be great!!
hi,
why not use inbuild keyword
KeywordUtil log = new KeywordUtil()
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Indico search will be reestablished in the next version upgrade of the software: https://getindico.io/roadmap/
# 36th Annual International Symposium on Lattice Field Theory
22-28 July 2018
Kellogg Hotel and Conference Center
EST timezone
## SMOM - $\overline{\mbox{MS}}$ Matching for $B_K$ at Two-loop Order
Jul 26, 2018, 8:50 AM
20m
104 (Kellogg Hotel and Conference Center)
### 104
#### Kellogg Hotel and Conference Center
219 S Harrison Rd, East Lansing, MI 48824
Standard Model Parameters and Renormalization
### Speaker
Ms Sandra Kvedaraite (University of Sussex)
### Description
The Kaon bag parameter, $B_K$, is a key non-perturbative ingredient in the search for new physics through CP-violation. It parameterizes the QCD hadronic matrix element of the effective weak $\Delta S=2$ four quark operator which can only be computed non-pertubatively on the lattice. The perturbative matching of $B_K$ between the lattice renormalization schemes and $\overline{\mbox{MS}}$ scheme has been done before at one-loop order. In this talk I am going to present a calculation of the conversion factors for $B_K$ between the four non-exceptional RI-SMOM schemes and the $\overline{\mbox{MS}}$ scheme at two-loop order in pertubation theory. The calculation is performed using the loop integral solving techniques such as integration by parts and sector decomposition.
### Primary authors
Ms Sandra Kvedaraite (University of Sussex) Dr Sebastian Jaeger (University of Sussex)
Slides
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How do you graph y\leq - 2x + 12?
Jun 15, 2017
See the explanation.
Explanation:
Graph:
$y \le - 2 x + 12$
Determine several points on the line by choosing values for $x$ and solving for $y$.
$\text{Points}$
$x = 4 ,$$y = 4$
$x = 3 ,$$y = 6$
$x = 0 ,$$y = 12$
Plot the points and draw a solid straight line through them. The inequality is $<$, so the area below the line should be shaded.
graph{y<=-2x+12 [-16.29, 15.73, -2.63, 13.39]}
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# 4 Strategies for Multi-Step Time Series Forecasting
Last Updated on August 21, 2019
Time series forecasting is typically discussed where only a one-step prediction is required.
What about when you need to predict multiple time steps into the future?
Predicting multiple time steps into the future is called multi-step time series forecasting. There are four main strategies that you can use for multi-step forecasting.
In this post, you will discover the four main strategies for multi-step time series forecasting.
After reading this post, you will know:
• The difference between one-step and multiple-step time series forecasts.
• The traditional direct and recursive strategies for multi-step forecasting.
• The newer direct-recursive hybrid and multiple output strategies for multi-step forecasting.
Kick-start your project with my new book Time Series Forecasting With Python, including step-by-step tutorials and the Python source code files for all examples.
Let’s get started.
• Update May/2018: Fixed typo in direct strategy example.
Strategies for Multi-Step Time Series Forecasting
Photo by debs-eye, some rights reserved.
## Multi-Step Forecasting
Generally, time series forecasting describes predicting the observation at the next time step.
This is called a one-step forecast, as only one time step is to be predicted.
There are some time series problems where multiple time steps must be predicted. Contrasted to the one-step forecast, these are called multiple-step or multi-step time series forecasting problems.
For example, given the observed temperature over the last 7 days:
A single-step forecast would require a forecast at time step 8 only.
A multi-step may require a forecast for the next two days, as follows:
There are at least four commonly used strategies for making multi-step forecasts.
They are:
1. Direct Multi-step Forecast Strategy.
2. Recursive Multi-step Forecast Strategy.
3. Direct-Recursive Hybrid Multi-step Forecast Strategies.
4. Multiple Output Forecast Strategy.
Let’s take a closer look at each method in turn.
### Stop learning Time Series Forecasting the slow way!
Take my free 7-day email course and discover how to get started (with sample code).
Click to sign-up and also get a free PDF Ebook version of the course.
## 1. Direct Multi-step Forecast Strategy
The direct method involves developing a separate model for each forecast time step.
In the case of predicting the temperature for the next two days, we would develop a model for predicting the temperature on day 1 and a separate model for predicting the temperature on day 2.
For example:
Having one model for each time step is an added computational and maintenance burden, especially as the number of time steps to be forecasted increases beyond the trivial.
Because separate models are used, it means that there is no opportunity to model the dependencies between the predictions, such as the prediction on day 2 being dependent on the prediction in day 1, as is often the case in time series.
## 2. Recursive Multi-step Forecast
The recursive strategy involves using a one-step model multiple times where the prediction for the prior time step is used as an input for making a prediction on the following time step.
In the case of predicting the temperature for the next two days, we would develop a one-step forecasting model. This model would then be used to predict day 1, then this prediction would be used as an observation input in order to predict day 2.
For example:
Because predictions are used in place of observations, the recursive strategy allows prediction errors to accumulate such that performance can quickly degrade as the prediction time horizon increases.
## 3. Direct-Recursive Hybrid Strategies
The direct and recursive strategies can be combined to offer the benefits of both methods.
For example, a separate model can be constructed for each time step to be predicted, but each model may use the predictions made by models at prior time steps as input values.
We can see how this might work for predicting the temperature for the next two days, where two models are used, but the output from the first model is used as an input for the second model.
For example:
Combining the recursive and direct strategies can help to overcome the limitations of each.
## 4. Multiple Output Strategy
The multiple output strategy involves developing one model that is capable of predicting the entire forecast sequence in a one-shot manner.
In the case of predicting the temperature for the next two days, we would develop one model and use it to predict the next two days as one operation.
For example:
Multiple output models are more complex as they can learn the dependence structure between inputs and outputs as well as between outputs.
Being more complex may mean that they are slower to train and require more data to avoid overfitting the problem.
See the resources below for further reading on multi-step forecasts.
## Summary
In this post, you discovered strategies that you can use to make multiple-step time series forecasts.
Specifically, you learned:
• How to train multiple parallel models in the direct strategy or reuse a one-step model in the recursive strategy.
• How to combine the best parts of the direct and recursive strategies in the hybrid strategy.
• How to predict the entire forecast sequence in a one-shot manner using the multiple output strategy.
## Want to Develop Time Series Forecasts with Python?
#### Develop Your Own Forecasts in Minutes
...with just a few lines of python code
Discover how in my new Ebook:
Introduction to Time Series Forecasting With Python
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### 186 Responses to 4 Strategies for Multi-Step Time Series Forecasting
1. anthony March 8, 2017 at 10:05 pm #
Thanks Jason for a wonderful post. Why does your model skips the value at “t”?
• Jason Brownlee March 9, 2017 at 9:54 am #
Just a choice of terminology, think of t+1 as t.
I could have made it clearer, thanks for the note.
• Pavlo Fesenko September 11, 2019 at 12:40 am #
Hi Jason,
Using t+1 instead of t is super confusing. 🙁 I don’t know about other users but it made it very difficult for me to grasp the idea with this terminology. If possible, please reconsider changing it to the traditional way in the future. Thanks a lot!
• Katie November 17, 2019 at 1:15 pm #
Hi Jason,
which strategy would you recommend for recursive models like ARIMA ? I originally thought recursive but now I’m wondering if the hybrid would make more sense. I have the same question for moving averages and exponential smoothing models. I was using the strictly recursive approach and repeating the entire training process for several models on several folds. This was really computationally expensive, though, and I don’t know if was really necessary. Not sure if this matters, but models from a few different families (arima, ets, ..) were pre-tuned/configured on the smallest possible subset of the training set (I used hyperopt) and then walk forward validation was applied for each of the candidate models. As mentioned, Multistep forecasts were estimated using a strictly recursive approach with the RMSE being calculated over all time steps in the horizon (t+1,t+2,..) for each iteration of walk forward validation. To reiterate, models were pre-tuned so the same exact models were applied to predict each value in the horizon for a given iteration, but it was recursive since each new time step in the horizon in a given iteration was predicted using a training set with previous predcited time steps appended.
• Katie November 17, 2019 at 1:22 pm #
Oh, I forgot to provide some important details for context:
1. I’m working with small samples
2. The frequency is monthly
3. The data is volatile
4. The context is inventory optimization (specifically, we’re predicting quantity of products issued by warehouses)
5. Forecasting is done at the SKU level and separate forecasts need to be made for each product and for each warehouse at my company
6. Some SKUs are sparse and most are extremely volatile
7. (slightly) Negative quantities do occur (indicating returns or adjustments) but are rare
7. Solution is being developed in R and Python in Azure ML
• Shiv March 4, 2020 at 6:59 am #
Hi Katie, Wondering if you were able to build and test any models. Please share your findings if you are able to.
• Jason Brownlee November 18, 2019 at 6:42 am #
I would recommend testing a suite of methods on your dataset and use the approach that results in the lowest error.
• Roise September 5, 2018 at 3:42 pm #
good question.thanks
2. Dylan March 14, 2017 at 2:31 am #
Hi Jason, it is always helpful to read your post. I have some confusion related to Time Series Forecasting.
There is traffic data (1440 pieces in total, and 288 pieces each day) I collected to predict traffic flow. The data is collected every 5 min in five consecutive working days. I am going to use the traffic data of the first four day to train the prediction model, while the traffic data of the fifth day is used to test the model.
Here is my question, if I want to predict the traffic flow of the fifth day, do I only need to treat my prediction as one-step forecast or do I have to predict 288-step?
• Jason Brownlee March 14, 2017 at 8:25 am #
Hi Dylan,
If you want to predict an entire day in advance (288 observations), this sounds like a multi-step forecast.
You could use a recursive one-step strategy or something like a neural net to predict the entire sequence one a one-shot manner.
Predicting so many steps in advance is hard work (a hard problem) and results may be poor. You will do better if you can use data as it comes in to continually refine your forecast.
Does that help?
• Dylan March 15, 2017 at 1:54 am #
Yes, your response is very helpful. Thank you very much. Now I realize my prediction is a multi-step forecast.
Could you recommend me some more detailed materials related to the multi-step forecast, like the recursive one-step strategy or the neural net?
Best regards
• Jason Brownlee March 15, 2017 at 8:10 am #
I am working on this type of material at the moment, it should be on the blog in coming weeks/months.
You can use an ARIMA recursively by taking forecasts as inputs to make the next prediction.
You can use a neural network to make a multi-step forecast by setting the output sequence length as the number of neurons in the output layer.
I hope that helps as a start.
• Patricia July 11, 2017 at 8:18 am #
Hi Jason,
Thank you for great posts! they’re awesome!
I have the same problem as Dylan and decided to use statsmodel’s SARIMAX. It takes some time to do the prediction for the entire next day (288 steps), and have been wondering if I’m doing this wrong or should I use a different approach.
Currently, I’m looking into LSTM RNN as a possible approach, but not sure.
The thing is, with my data, I have to predict the entire 288 steps in one shot and detect an anomaly if there’s any, then predict the type of anomaly that occured….
My question is, am I going in the right direction by looking into LSTM RNN?
Thanks Jason 🙂
• Jason Brownlee July 11, 2017 at 10:38 am #
I am not high on LSTMs for autoregression models:
http://machinelearningmastery.com/suitability-long-short-term-memory-networks-time-series-forecasting/
• Andreas P March 11, 2019 at 4:37 am #
Hi Jason! Thank you for your great posts!
I have a similar problem as Dylan. I have a dataset containing system metrics collected every 5 min. I have a total of 12 features (CPU utilization, network activity, disk operations etc.) which makes this a multivariate problem.
What would be the best approach if I would like to predict all 12 features for let’s say the next hour with a LSTM? Since every time step represent 5 min would it mean I have to predict t+12 (since 5 * 12 makes one hour)? Would it be beneficial to perhaps down sample from the per-5-min-observations to hours instead, making it possible to forecast the next hour with just one step instead of multiple?
Looking forward to hear your input on this!
Best regards,
Andreas
3. Kunpeng Zhang March 17, 2017 at 12:45 am #
Yes, I will. Discussing with you is always helpful. Look forward to reading your new post on Time Series Forecast.
4. Abhishek March 18, 2017 at 4:18 pm #
Hi Jason, just another brilliant post. Can you show up a working example for first or second method like you have always shown in other tutorials. It would be immense help to a novice like me. Thanks…
• Jason Brownlee March 19, 2017 at 6:09 am #
I do hope to have many examples on the blog in the coming weeks.
5. mary March 31, 2017 at 4:57 am #
Thank you Jason for your wonderful articles ! you are a life saver!
But I suppose you did a mistake in the example for number2 and 3. both has the same value as
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(prediction(t-1), obs(t-2), …, obs(t-n))
I believe that one of them should be
prediction(t+2) = model2(prediction(t+1), obs(t-1), …, obs(t-n-1))
• Jason Brownlee March 31, 2017 at 5:59 am #
Hmmm, I guess you’re right. I was thinking from the frame of the second prediction rather than the frame of both predictions.
Fixed.
• Jim January 17, 2021 at 11:12 pm #
Thanks Mary. So in effect what should both step be?
many thanks
6. Fatima Abu Salem April 7, 2017 at 3:29 pm #
Hello Jason,
What kind of Multiple output models would you recommend if we are opting for the fourth strategy?
• Jason Brownlee April 9, 2017 at 2:52 pm #
Neural networks, such as MLPs.
• Vipul September 21, 2017 at 7:38 pm #
• Jason Brownlee September 22, 2017 at 5:38 am #
Sure, try them, but contrast results to an MLP.
• Taha February 9, 2020 at 9:39 am #
Sir, Is MLP performs better than RNN for multi-output or multivariate ?
• Jason Brownlee February 9, 2020 at 1:04 pm #
It really depends on the specific dataset. I recommend testing a range of algorithms and discover what works best on your data.
7. Masum May 13, 2017 at 8:20 pm #
Sir,
Would you please come up with a blog where we would love to see all these strategies have been applied to an example (dataset) and their result comparisons.
• Jason Brownlee May 14, 2017 at 7:26 am #
Perhaps in the future, thanks for the suggestion.
• masum May 22, 2017 at 10:12 pm #
sir,
would you be kind enough to post soon?
I am stuck with my theoretical knowledge need to apply on my data to see the result and their comparative analysis.
• Akhil March 28, 2020 at 7:40 am #
Thanx for great explanations.
I have one little question.
Is the accuracy same for all these 4 strategies.?
If No, then which one gives more accuracy.
• Jason Brownlee March 29, 2020 at 5:47 am #
No, you must test a suite of models and strategies and discover what works best for your specific dataset.
8. Hans June 12, 2017 at 12:40 am #
What is a decent one-step prediction of unseen data? How would it looks like?
Let’s say I have 100 rows in a data set and do the following in R:
I write ‘=’ instead of the arrows because of the forum parser:
1. I split the 100 rows of raw data in 99 training rows and 1 testing row:
inTrain=createDataPartition(y=dataset$n12,p=1,list = FALSE) training=dataset[inTrain-1,]; testing=dataset[-inTrain+1,] 2. I train the model: modFit=train(n12~., data=training, method = ‘xxx’) 3. I get the final model of Caret finMod<-modFit$finalModel
4. I predict one step with the final model of the training and the one row of testing.
newx=testing[,1:11]
unseenPredict=predict(finMod, newx)
Now, do I have a decent prediction of one step unseen data in point 4 ???
And why there are libraries like forecast for R, if everything can have been coded to a one-step forecast by default?
https://github.com/robjhyndman/forecast/
• Jason Brownlee June 12, 2017 at 7:09 am #
Sorry, I don’t have examples of time series forecasting in R, I cannot offer good advice.
9. Hans June 12, 2017 at 6:18 pm #
I know there is also the option to use the time series object(s) in R.
But could you answer my question in general?
10. Hans June 14, 2017 at 1:57 am #
I don’t understand the difference between regression forecast and time series forecast.
Or what are the benefits from each over the other.
• Jason Brownlee June 14, 2017 at 8:47 am #
A time series forecast can predict a real value (regression) or a class value (classification).
11. Leonildo June 16, 2017 at 10:35 am #
Hello Jason,
How to prepare dataset for train models using with Direct Multi-step Forecast Strategy ?
For the serie: 1,2,3,4,5,6,7,8,9
Model 1 will forecast t+1 using window of size 3 , then the dataset would be:
1,2,3->4
2,3,4->5
3,4,5->6
4,5,6->7
5,6,7->8
6,7,8->9
Model 2 will forecast t+2 using window of size 3 , then the dataset would be:
1,2,3->5
2,3,4->6
3,4,5->7
4,5,6->8
5,6,7->9
Model 3 will forecast t+3 using window of size 3 , then the dataset would be:
1,2,3->6
2,3,4->7
3,4,5->8
4,5,6->9
and so on. Is it right ? Thanks
• Jason Brownlee June 17, 2017 at 7:18 am #
Great question.
For the direct approach, the input will be the available lag vars and the output will be a vector of the prediction.
I can see that you want to use different models for each step in the prediction.
You could structure it as follows:
Try many approaches and see what works best on your problem.
I hope that helps.
• Leonildo June 17, 2017 at 10:17 am #
thank you so much! This answers my question.
• Jason Brownlee June 18, 2017 at 6:29 am #
I’m happy to help (if I can).
• Davood Raoofsheibani July 29, 2018 at 11:05 pm #
Dear Jason,
In Direct Multi-step Forecast Strategy, for model 2
why haven’t you used obs(t-1) as well?
i.e.
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(obs(t-2), obs(t-2), obs(t-3), …, obs(t-n))
=>
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(obs(t-1), obs(t-2), …, obs(t-n))
The latter seems more compatible with the example you provided in this comment.
thanks!
• Jason Brownlee July 30, 2018 at 5:49 am #
t-1 for model2 would be the predicted value of model1. It could use them too if you wish.
• Shellder September 6, 2019 at 2:24 pm #
Hi, Jason,
Can you explain more about why t-1 for model2 would be the predicted value of model1?
I think there is no information leakage since the model 1 is “older” than the model 2. The model 2 can use all the data used by the model 1.
• Jason Brownlee September 7, 2019 at 5:16 am #
Sorry, I don’t follow, what’s the context exactly?
• Pavlo Fesenko September 11, 2019 at 12:37 am #
Hi Shellder,
I think that’s indeed a mistake in Jason’s formulation of the direct multi-step approach. The component obs(t-1) should also be included in the training set for the model2. 😉 It made me very confused the first time I read it but then after checking other sources I realized that there is no reason not to include obs(t-1) in the model2.
Jason, could you please have a look at it and correct it for the future blog visitors? =) Thanks!
12. Jisun August 30, 2017 at 7:00 am #
I have a question.. I am building a forecasting model with my timeseries dataset, which is a daily number of some cases, I have 3 years past data (so will be records of 3*365 days.) I’d like to forecast 2 months future data (60 days.)
I already built a multi-step LSTM model for this, however, it doesn’t seem to work well… For example, 3 years past data clearly has a pattern like Nov/Dec high peak seasonality and increasing trend, but 60 steps of LSTM gave me poor forecasting like decreasing trend with no seasonality… and even the base is decreasing too.. which is so not understandable.
My question is:
1. Do you think my parameter tuning could be wrong? I mean, LSTM multi step forecasting cannot be this much poor..?
2. Is there any recommendation for one model approach for my problem..? I used ARIMA, but I wanted to use algorithmic model rather than a statistical model, so that’s why I’m trying to build LSTM… Do you think I need to go back to ARIMA..?
(After building one model, I will use ensemble method to improve current model though. For now, I need a decent model giving me the understandable result.)
Thank you so much, your any opinion on this will be really appreciated.
• Jason Brownlee August 30, 2017 at 4:17 pm #
Thanks Jisun,
I generally would recommend using an MLP, LSTMs do not seem to perform well on straight autoregression problems.
13. dan November 23, 2017 at 4:09 pm #
Hi Jason,
There is a question above asking you “What kind of Multiple output models would you recommend if we are opting for the fourth strategy”?
Then I try to use mlp to get a one-shot sequence, but I keep getting error…
Below is my code and scenario,
x_train.shape:
(4, 5, 29)
y_train.shape:
(4, 28)
I wish to use prior 5 timesteps and 29 features to get the 28 timesteps ahead forecast sequence.
only 4 training data for illustrative purpose.
model = Sequential()
model.add(Dense(units = 100, input_shape = (5, 29)))
Error when checking target: expected dense_168 to have 3 dimensions, but got array with shape (4, 28)
\
How can I rectify it? Thank you very much
• Jason Brownlee November 24, 2017 at 9:33 am #
With an MLP, the prior observations will be features.
If you have 29 features and 5 time steps, this will in fact be 5 x 29 (145) input features to the MLP.
14. Amalka January 4, 2018 at 5:25 pm #
Hi Jason
I am trying to fit a LSTM model which is a multivariate (input and output) and multi step.
So I need to predict multiple steps and multiple features in one model.
Temp : [1,2,3,4],Rain[1,2,3,4] = predict(Temp : [5,6], Rain[5,6])
What is your recommended architecture to do this in one model ?.
I have daily selling values for 5 years with 167000(per item per store) features to predict 15 days for 167000 features
15. Charles Lang February 13, 2018 at 1:33 pm #
Hi Jason,
Thank you very much for sharing a great article again. I have read many your posts these days, and learned a lot from them.
My project is one-step forecast on time series data. Do you think which model is the best to it?
Charles
• Jason Brownlee February 14, 2018 at 8:13 am #
I would recommend testing a suite of models to see what works best for your data.
16. pezhman February 20, 2018 at 10:15 am #
Hi Jason,
Is there anyway to reduce the propagated error during Multi step ahead prediction with recurrent neural network?
Thanks
17. Roshan March 26, 2018 at 6:37 pm #
Hi Jason,
Thanks for the great post. Your post uses Direct Strategy. I would like to apply Direct-Recursive Hybrid Strategy using RNN-LSTM on a time series data that has trend and seasonality. What I need is a multistep forecast where the prediction for the prior time step is used as an input for making a prediction on the following time step. How to go about this ? Because recursion for multi-step would be highly computationally expensive. What changes do I need in the existing code for multi-step using hybrid method?
Thanks
18. Kaushal Shetty March 30, 2018 at 8:20 pm #
Hi Jason,
I am going with the 4th strategy you mentioned that is one model predicting forecasts in one shot.
I have two models in my mind for this.
1) Multi-Output Network : Output layer of this architecture has ‘forecast’ number of dense layer. In this model there are ‘forecast’ number of weight matrices each trained on predicting individual forecast? Each output dense layer is optimized(adam) for MSE.
2)Single-Output network: This architecture has one dense layer as output with ‘forecast’ number of neurons. So in this case there is only one weight matrix trained for all forecasts and one cost function across all forecasts as opposed to first approach.
Are both the architecture valid? Which architecture works best?
Also one more question Jason. What is the best way to add regularization on time series model.Dropouts or kernel regularizer?Both would do?
Thanks,
Kaushal
19. MLT May 17, 2018 at 10:03 pm #
Hi Jason, it is a great article and very helpful summary. I have one trivial question.
1. Direct Multi-step Forecast Strategy
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model1(obs(t-2), obs(t-3), …, obs(t-n))
==>
do you mean model2, not model1 for t+2?
prediction(t+2) = model2(obs(t-2), obs(t-3), …, obs(t-n))
why it starts t+1 not t
prediction(t) = model1(obs(t-1), obs(t-2), …, obs(t-n))
• Jason Brownlee May 18, 2018 at 6:24 am #
Yes, I meant model2, thanks – fixed.
• MLT May 19, 2018 at 12:31 am #
sorry to disturb you again.
Why it is t+1, not t? Thanks.
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
=>
prediction(t) = model1(obs(t-1), obs(t-2), …, obs(t-n))
• Jason Brownlee May 19, 2018 at 7:43 am #
It could be and probably should be t. Just a chosen notation of t-1, t+1.
20. Klaus May 21, 2018 at 2:22 pm #
Hi jason,
Thanks a lot for the great post. I am going with the 2th strategy you mentioned that is recursive multi-step forecast but having difficulty in implementing the recursive forecasting part.
How to put the prior time step to be used as an input for making a prediction on the following time step, for example with SVR or MLP. It would be immense help to a novice like me. Thanks…
• Jason Brownlee May 21, 2018 at 2:31 pm #
Thanks Klaus.
You will have to store it in a variable and then create an array that includes the value and use it as input for the next prediction.
21. Thada June 4, 2018 at 1:57 am #
Hi Jason,
Thank you for your great post.
If I use Recursive Multi-step Forecast with ARMA model, the effect of MA predictions will reduce over the steps of prediction or not?
Due to the nature of multi-step forecast, the error terms of the previous unobserved samples will become zero when they are used as an inputs for the further prediction. Howevers, MA will estimate based on the previous errors if I do not missunderstand. Thus, if the previous error terms become 0, will only AR terms affect the prediction results?
Sorry if I misunderstand about the ARMA model. I’m quite new for this topic.
• Jason Brownlee June 4, 2018 at 6:30 am #
Recursive prediction will result in compounding error. Why would MA go to zero?
• Thada June 5, 2018 at 12:14 am #
Sorry, I didn’t mean the real error but the unobserved residuals. For example in the following website, no observed values for residuals ε106, ε107 are considered as zero in the equation ŷ107 = f(ε106, ε107, y106).
http://www.real-statistics.com/time-series-analysis/arma-processes/forecasting-arma/
“This time, there are no observed values for ε106, ε107, or y106. As before, we estimate ε106 and ε107 by zero, but we estimate y106 by the forecasted value ŷ106.”
22. Sam June 13, 2018 at 10:54 am #
Hello Jason,
I am trying to build an LSTM model. My training set has 580 timesteps and 12000 features.
I wanted to use 10 timesteps to predict next 5 timesteps. In this case my train_x.shape will be (87,10,12000). However I am confused about train_y.shape. Should it be (87,10)?
23. MLT June 27, 2018 at 5:50 am #
Hi Jason,
In my understanding, Direct-Recursive Hybrid Strategies can be implemented in below three steps. Could you help me to check if it is correct please? Thanks in advance.
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(prediction(t+1), obs(t-1), …, obs(t-n))
1. Use train data to train model1
2. Predict t+1 for all train data
3. Use predicted t+1 plus train data to train model2
24. Davood Raoofsheibani July 29, 2018 at 11:12 pm #
Dear Jason,
In Recursive Multi-step Forecast:
I guess to predict the value at (t+2), the observed value of (t-n) is not needed but the one at (t-n+1).
therefore:
prediction(t+1) = model(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model(prediction(t+1), obs(t-1), …, obs(t-n))
should be :
prediction(t+1) = model(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model(prediction(t+1), obs(t-1), …, obs(t-n-1))
Am I right?
25. Ling July 31, 2018 at 6:09 am #
Dear Jason ,
I have some experience in python and machine learning and now I am trying to learn predicting a value ( Regression ) based on Timestamp ( Every 5 mins ). So please suggest me the appropriate model for this. Also, let me know if your book on forecasting helps me in this.
26. Kumar Avinash August 21, 2018 at 4:27 am #
HI JASON,
Very informative and nice one. I have one problem related to LSTM forecasting model. I am making a model on call volume forecasting. I want to forecast 3 months ahead of today, let’s say I have built a model and scoring today (in Aug’18), so the forecasted month should of Dec’18 3 months ahead of scoring month. And how should I be proceeding while building model (training dataset) and validation (test dataset) and scoring unseen data(as discussed above). Do I have to build the model on training dataset in similar way, like forecasting 3 months in advance? If yes, how to proceed.
• Jason Brownlee August 21, 2018 at 6:21 am #
Yes, frame the historical data in the way that you intend to use the model.
E.g. if you need a model to predict n months ahead, frame all historical data this way and fit a model, then score the fit model.
27. Ahmed R. Elshami September 29, 2018 at 5:45 am #
Hello Dr. Jason,
Thank you for your amazing post.
Suppose I have my training data D_pred = [1,2,3,4,5,6]
and my corresponding target data, D_trgt = [7,8,9,10,11,12,13]
I am using lag = 4 and I want to predict 5 step-ahead
My D_pred would be like this
1 2 3 4
2 3 4 5
3 4 5 6
I used my D_pred to get my prediction result D_out in one shot like this
x7 x8 x9 x10 x11
x8 x9 x10 x11 x12
x9 x10 x11 x12 x13
and my D_trgt would be like this
7 8 9 10 11
8 9 10 11 12
9 10 11 12 13
now, how to calculate the SMAPE error between D_out and D_trgt for horizon 5?
– get SMAPE error between [x7, x8, x9, x10, x11, x12, x13] and [7, 8, 9, 10, 11, 12, 13]
or
– get SMAPE error between [x11, x12, x13] and [11, 12, 13]
which way is the right way?
Thank you so much
28. Mike October 20, 2018 at 12:48 am #
Hi, master. What is the lag timesteps used for？ I still can’t understand，my doctor.
• Jason Brownlee October 20, 2018 at 5:55 am #
Lag time steps are observations at prior times. They are used as inputs to the model.
• Mike October 20, 2018 at 6:43 pm #
Can you explain more exactly?
• Jason Brownlee October 21, 2018 at 6:11 am #
Sure, which part are you having difficulty with exactly?
29. mk December 26, 2018 at 12:46 pm #
Is there any criteria for choosing the four main strategies?
For example,How many steps?How many datas?
• Jason Brownlee December 27, 2018 at 5:36 am #
Project requirements may impose constraints, alternately, choose the approach that results in the best skill.
30. Kristen January 30, 2019 at 4:31 pm #
Hi Jason,
Thanks for your post. I’m implementing the Multiple Output Strategy using a Neural Network approach. I have about 5 years worth of historical data at weekly granularity, and I want to predict 6, 12, 18 and 24 months into the future.
I’ve read your “How To Backtest” post, but am still not sure about how to do the train/test split in this case, ie. predicting 2 years into the future with only 5 years historical data. Even if I only wanted to forecast one year ahead (at say 3, 6, 9 and 12 months), how would you backtest?
Thanks.
• Jason Brownlee January 31, 2019 at 5:27 am #
You would use walk-forward validation as described in the back test tutorial.
31. Larry February 25, 2019 at 8:33 pm #
I do not understand the direct approach. I have only found vague examples to explain. Let us consider a problem in which I want to do a 1 step prediction.
history = [1,2,3,4,5,6,7,8,9,10]
Yt = 10
problem: predict Yt+1
My current understanding of how to formulate the training data is to use a sliding window of size 1.
X_train = [1,2,3,4,5,6,7,8,9]
Y_train = [2,3,4,5,6,7,8,9,10]
Then I train the first model using X_train and Y_train
In a single step prediction scenario, to predict what comes after 10 I would call
Predict(model1, 10), and the output should be 11 + some error, depending on the model
Now, for the 2-tep reccursive method I would call
Predict(model1,Predict(model1,10)) to get 12 + some error
The direct method for the 2 step prediction will be
a = Predict(model1, 10) to get 11 + some error
b = Predict(model2, K) to get 12 + some error
predictions = [ ]
predicions.append(a)
predictions.append(b)
Finally, a 2-step prediction is accomplished:
predictions == [11 + error, 12 + error]
My questions:
1)What is K? Which value do I need to send to the predict method for the first model
2) What are the values of the X_train an Y_train used to train the second model.
Thanks.
• Jason Brownlee February 26, 2019 at 6:20 am #
Looks good.
K would be an input for the required output, e.g. the last available observation, or the prior time step prediction, or a mix. You must test and see what is appropriate for your specific dataset. Same for input, you can test/choose.
32. xuxing February 27, 2019 at 2:51 pm #
Hi Jason,
I have two products, both of which have their historical data, and how do I model so that both products can predict future sequences from their historical data.Construct two models or just one model?
33. bara April 15, 2019 at 4:00 pm #
dear sir, i follow all your tutorials and combined them. i have more than thousand sequences. i choose vanilla lstm according this
https://machinelearningmastery.com/how-to-develop-lstm-models-for-time-series-forecasting/
i also add train and test, 80% 20%. i got RMSE and MAPE and the Graph.
i want to forecast next value, and i follow this.
https://machinelearningmastery.com/multi-step-time-series-forecasting/
according https://machinelearningmastery.com/how-to-develop-lstm-models-for-time-series-forecasting/
you forecasting 1 next value with 3 last value.
so want to forecast 2 and 3 next value using Recursive Multi-step Forecast, and my code like this
#one
x_input = array([Y_test[-1], Y_test[-2], Y_test[-3]])
x_input = x_input.reshape((1,1,3))
yhat = model.predict(x_input, verbose=0)
#two
x_input1 = array([yhat,Y_test[-1], Y_test[-2]])
x_input1 = x_input1.reshape((1,1,3))
yhat1 = model.predict(x_input1, verbose=0)
#three
x_input2 = array([yhat1,yhat, Y_test[-1]])
x_input2 = x_input2.reshape((1,1,3))
yhat2 = model.predict(x_input2, verbose=0)
my question is dowhat i am doing is right? do i follow the principle LSTM & Recursive Multi-step?
that all, thank you sir
• Jason Brownlee April 16, 2019 at 6:44 am #
Nice work.
I’m eager to help but I don’t have the capacity to review/debug your code. Sorry.
34. Yessica Chen April 28, 2019 at 12:16 pm #
Thank you Jason for sharing it. I want to know which methods are more helpful in timeseries problem. Did you do some contrast experiment with them?
35. Joachim May 2, 2019 at 3:17 am #
Hi Jason! Writing a thesis on this right now and your examples are very much appreciated.
I have a question about the Direct-Recursive Hybrid, as we have been able to test out all the other methods to a certain degree.
How would you go about writing the programming logic for this? Especially when using time-series cross validation.
What exactly do i fit model2 on at time t?
• Jason Brownlee May 2, 2019 at 8:08 am #
Perhaps review the example in the tutorial and try to map your data onto it, also perhaps check the paper for an elaboration on the approach.
You will have to write custom code to prepare data and fit models.
36. Yeqi Liu May 17, 2019 at 5:14 pm #
I have a question: is it possible for the same model to have different dimensions of input variables in Recursive Multi-step Forecast (i.e., from obs(t-n) to (t-1) with R_n dimension, and from t-n to prediction(t+1) with R_n+1 dimension)?
Also, is it better whether the input dimension should be the same in Direct-Recursive Hybrid Strategies?
• Jason Brownlee May 18, 2019 at 7:32 am #
Yes, you could have a multi-input model, like a neural net.
Perhaps experiment with your dataset/model and see what works well?
37. Leo June 21, 2019 at 1:34 pm #
Hi Jason,
I have a question regarding error propagation in different multi-step forecast models that I post on StackExchange before reading this post. (so the terminology I used is a bit non-standard) I would like to understand the theory behind error propagation. Could you shed some lights, please?
https://datascience.stackexchange.com/questions/54130/error-propagation-in-time-series-forecast-with-many-to-many-multi-steps-rnn-lstm
Many thanks!
• Jason Brownlee June 21, 2019 at 2:03 pm #
Perhaps you can summarize the gist of your question?
38. Lopa July 3, 2019 at 8:53 pm #
Hi Jason,
I have been able to implement the recursive strategy. However for Direct-Recursive Hybrid Strategy if I understood correctly we train & predict the entire training data & append those predictions to the initial train data & retrain the model.
Having said that if my initial train data has 100 observations & I predict all 100 then I append these predictions to my initial train data making 200 observations ? Is my understanding correct or am I missing out something?
• Jason Brownlee July 4, 2019 at 7:45 am #
Not quite, the predictions become inputs for subsequent predictions.
• Lopa July 5, 2019 at 9:59 pm #
Sorry to ask again Jason but can you please explain because I tried finding it in your books but couldn’t find & also tried understanding it based on the example & the example of the household electricity but could not really grasp it entirely.
• Jason Brownlee July 6, 2019 at 8:36 am #
Sure, what are you having trouble understanding exactly?
39. Lopa July 10, 2019 at 12:05 am #
Suppose I have 100 observations in total after training & validating my model I train my model on the entire data (all 100 observations) & predict one step ahead i.e; 101st observation.
Do I use this predicted value to replace the 1st observation in my original data to have 100 observations & predict the next step & repeat the process ?
Thanks again.
• Jason Brownlee July 10, 2019 at 8:14 am #
It really depends on how your model is defined.
You have defined your model to map some number of inputs to an output, you must provide data in that format to make a prediction.
40. Chao August 6, 2019 at 8:57 am #
Hi Jason, thanks for making this great tutorial.
I am not sure I’m fully understood the difference between recursive multi-step and direct-recursive hybrid. These two look exactly same in your example code.
prediction(t+1) = model1(obs(t-1), obs(t-2), ..., obs(t-n)) prediction(t+2) = model2(prediction(t+1), obs(t-1), ..., obs(t-n))
If I understood correctly, the main difference is hybrid is each model may use or may not use the models at prior time, and the recursive multi-step is each model will use the prior model?
Thanks!
• Chao August 6, 2019 at 9:00 am #
If the above description is correct, how do you decide when should use or not use? I assumed that requires to be training?
• Jason Brownlee August 6, 2019 at 2:01 pm #
Test a few different approaches and see what works best for your choice of model and the dataset.
• Jason Brownlee August 6, 2019 at 2:01 pm #
Recursive uses predictions as inputs.
Direct recursive hybrid uses the same idea, but separate models for each time step to be forecasted.
Does that help?
41. Matteo P. August 6, 2019 at 7:47 pm #
Hi Jason, your articles are great and they helped me a lot!
I’m working on predictive maintenance and given a long time series of data, each of them with 15 features, I should predict the next X time steps.
Basically I thought of using 400 time steps as input and predicting 20 steps as output. As a result I’m using your 4.Multiple Output Forecast Strategy.
My Net is this:
n_steps_in = 400
n_steps_out = 20
n_epochs = 20
batch_size = 128
model = Sequential()
activation=’relu’,
input_shape=(n_steps_in, n_features),
return_sequences=True))
model.compile(loss=’mse’,
metrics=[‘accuracy’])
model.fit(X,
y,
epochs=n_epochs,
validation_split=0.25,
batch_size=batch_size)
But sometimes i’ve got Nan in Loss while training, but i don’t know why. Do you have any explanation?
Thanks
• Jason Brownlee August 7, 2019 at 7:51 am #
Nice work.
Try scaling the data prior to fitting?
Try relu?
• Amit Krishna Baral August 21, 2019 at 12:40 pm #
Did you think about applying the concept of Survival model here?
42. Sneha Mitta September 19, 2019 at 6:46 am #
Hi Jason,
I wanted to know if you have any posts that have implemented each of the strategies you discussed above?
I also wanted to know if one strategy is better than the other by any chance? I’d like to get a deeper insight on what kind of strategy would work for a particular kind of data.
43. Sam October 18, 2019 at 1:06 pm #
Hi Jason, First of all, thanks for all your nice posts. People has asked this question before, but I was wondering if there might be an update! I was wondering do you have an example code for “Direct-Recursive Hybrid Strategies” similar to what you have for “Time Series Prediction With Deep Learning in Keras”. Or if your E-Book has it?
Thanks again, Sam
44. SOUALHi November 25, 2019 at 6:57 am #
Hi Mr. Jason,
I’m working on forecasting time series, i use LSTM as model to forecast.This is the main steps i used to structure my data in oder to predict one step:
1) The model takes 1 day of data as “training X”
2) The model takes the VALUE of 1 day + 18hours after as “trainingY”
3) I build a slliding window as well as the sequences are shifted by one value, fore example:
XTrain{1} = data(1:24) –> YTrain{1} = data(42)
XTrain{2} = data(2:25) –> YTrain{1} = data(43)
XTrain{3} = data(3:26) –> YTrain{1} = data(44)
XTrain{4} = data(4:27) –> YTrain{1} = data(45)
.
.
.
4) The test data are also constructed as the same way of training data, fore example:
XTtest{1} = data_test(1:24) –> YTest{1} = data_test(42)
XTtest{2} = data_test(2:25) –> YTest{2} = data_test(43)
.
.
.
First, to sumurize, my objective id to predict each time 18h after. Is this structed cited above is correct?
If yes, I have the problem when i try to predict the XTest{1} the obtained predicted value is the corresponding value of data_test(25) instead of d ata_test(42)? For this purpuse, why the predicted value is shifted? Where is the problem and how to remedy to this problem?
• Jason Brownlee November 25, 2019 at 2:04 pm #
There are many ways to frame your problem, your approach is one possible way.
Perhaps the model is not skillful?
Perhaps try alternate architectures or training configurations?
Perhaps try alternate framings of the problem?
Perhaps try alternate models?
• SOUALHi² November 26, 2019 at 1:01 am #
Thank you very your reativity. Hence, i have tried several architechtures of the model and LSTM gives the best reults, which suitable for for learning many sequences with different lengths.
For my part, i use this architechture to learn the model. However, could you please inform me or show me the orther architechtures to prepare the training data that can help me to solve this problem of shiftting results? Because, i use 3 test sets, when try t predict, the results are shifted by the number of difference steps used for target prediction. i.e.
model_1: data(1:24) –>data(30) the difference is 6 points. Therefore the predicted curve is shifted by 6 steps earlier
model_2: data(1:24) –>data(36) the difference is 12 points. Therefore the predicted curve is shifted by 12 steps earlier
I don’t think that this is logical response?
Best regards
45. Yass T November 27, 2019 at 9:45 pm #
Hello, could there be a typo in the explanation of Direct Multi-step Forecast Strategy ?
I was expecting the second row in the code to be :
prediction(t+2) = model2(obs(t-2), obs(t-4), obs(t-6), obs(t-8), obs(t-n)) – so basically no (t-3), and two time steps increments ?
Otherwise, I still do not understand that correctly 🙂
Thank you
• Jason Brownlee November 28, 2019 at 6:38 am #
No, I believe it is correct.
There are 2 models and both models only use available historic data.
model1 predicts t+1 and model2 predicts t+2.
Does that help?
46. Morteza March 4, 2020 at 3:58 am #
Hi Jason,
I have a question that what is the difference between these two cases when we want to have multi-step time series forecast:
1. There is one neuron in output layer (T+1). After extracting weights, we iteratively use weights and (T+1) forecast to get (T+2) forecast and so on.
2. We have multiple neurons for each horizon
My question is mostly related to LSTM, however, a general reply is also appreciated. I need a mathematical explanation if there is any.
Thanks a lot,
Morteza
• Jason Brownlee March 4, 2020 at 6:00 am #
Yes, in the first case you are reusing the same model recursively in the second you are using a single direct model.
Yes, you can describe each approach using math or pseudocode.
Perhaps I don’t understand the problem you’re having?
47. Ben March 11, 2020 at 5:18 am #
Hi Jason, can CNN LSTM do multi-step?
48. Xin April 13, 2020 at 12:02 am #
hello, very nice doc but there are several tiny typos:
in almost all the code, you wrote:
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(obs(t-2), obs(t-3), …, obs(t-n))
the prediction(t+2) is wrong, which should be
prediction(t+2) = model2(obs(t-2), obs(t-3), …, obs(t-n+1))
49. Samrat May 30, 2020 at 11:17 pm #
Hello.
Thanks for the awesome article.
Do you have any post on multivariate multi-step time series forecasting ??
Thanks.
50. Dr S Balasubramanian June 30, 2020 at 9:53 pm #
good very nice concept description
51. Jose Q July 11, 2020 at 8:18 am #
Hi Jason,
Great post! You are always clear in your concepts
Let me tell you this concern.
I have seen notebooks trying to analyze the spreadth of COVIT-19 using regression analysis. They treat X as the consecutive number of days since the first case [1,2,3,..,n-1,n], and y as the number of daily cases since the first case [1,3,7,…, 3821,4213].
Then they predict future values for X=[n+1, n+2, …n+20] trying to forecast daily cases for the next 20 days.
I think this is not correct, because these models do not consider the effect of time, and also, they are doing extrapolation in a regression analysis model.
I guess I saw one of your posts saying that supervised learning can intrapolate, but time series can extrapolate.
Is this correct so far?
Now, if you arrange time series data as a supervised learning problem with input sequence = [t-lags, …, t-1] and forecast sequence = [t+1, …,t+steps], now you can use supervised learning algorithms or MLP to make predictions, right?
We can now evaluate performance using backtesting.
Is this comparable to ARIMA/SES forecasting?
Thank you
52. tuttoaposto July 20, 2020 at 7:46 am #
If I understand this Autoreg model_fit.predict example correctly, it is an example of multi-step forecast strategy: https://machinelearningmastery.com/autoregression-models-time-series-forecasting-python/
But I got nan’s following this method. I guess that happened because the model has missing lag values as the input. Referring to this example code, it doesn’t explicitly update the historic lags for subsequent forecasts. I wonder if .predict() already handles that and something in my code caused the nan’s, or the example code is missing the recursively updating steps?
• tuttoaposto July 20, 2020 at 9:32 am #
I figured why: use series.values instead of series solved the problem.
• Jason Brownlee July 20, 2020 at 1:51 pm #
Sorry to hear that you’re having trouble.
You can make a multistep prediction directly by first fitting the model on all available data and calling predict() and specifying the interval to forecast, or calling forecast() and specifying the number of steps to predict.
53. saranraj k July 27, 2020 at 12:01 am #
H, Jason thank for the great article,
For the direct multi-step forecast, we build separate model for each forecast time step,
Let’s say for two days, model m1 for day 1 and model m2 for day2, and you have given the
following example
Example
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(obs(t-2), obs(t-3), …, obs(t-n))
In the above example, for predicting day t+1 we used data from precious day t-1 to t-n.
for predicting second t+2 we used t-2 to t-n.
If so,
It’s seems like for forecasting tomorrow’s data use the historical data from last day to
last day -n.
for forecasting day after tomorrow’s use the data from two days before to day -n, how it make
sense?
• Jason Brownlee July 27, 2020 at 5:47 am #
Yes, the other model can use historical data and output of another model to make a prediction.
Perhaps I don’t understand your question?
54. Zahra October 9, 2020 at 12:59 pm #
Hello and thanks for the tutorial,
I wonder for the second and third approaches for forecasting you mentioned above, which you said:
prediction(t+1) = model(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model(prediction(t+1), obs(t-1), …, obs(t-n))
if the prediction(t+1) uses some kind of a supervised time series approach where the model actually have seen a part of data at (t+1), is that going to create information leakage to the second model?
I
• Jason Brownlee October 9, 2020 at 1:49 pm #
Not leakage as it does not violate the design. It is by design.
55. Priya October 10, 2020 at 3:53 pm #
Hi,
First of all thanks for the tutorial.
For the direct multi-step forecast, you have given the
following example
Example
prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))
prediction(t+2) = model2(obs(t-2), obs(t-3), …, obs(t-n))
My questions are
1. For prediction(t+2) why you are not taking obs(t-1) as input like for prediction(t+1).
2. I am using direct multi-step forecast for my project and it is expected with machine learning that by increasing forecast horizon(time step) error should increase. Am I right? If yes then in my project, error is continuously increasing upto 7 time step but after that error is fluctuating. Can you suggest me how can I improve this error?
• Jason Brownlee October 11, 2020 at 6:44 am #
You’re welcome.
We do not use the observation at t-1 because it is not available in that framing of the problem.
Yes, the further into the future you want to predict, the more error.
You can reduce the error with different/better models or by simplifying the problem.
56. Sandy October 18, 2020 at 4:42 am #
Hi Jayson,
I recently bought the math bundle and considering the forecasting books.
I was wondering If you cover a full example of a Direct-Recursive Hybrid model? And in which e-book?
Best,
A fan
57. Adrien November 1, 2020 at 7:59 am #
Hi Jason,
great article, there is very little content on the internet on prediction strategies for using the model once trained.
I have defined a fairly simple recursive prediction method.
My model uses a 160 hour rollback window to predict the next hour on 10 features outputs.
Model inputs: 10 features
Model outputs: 10 features
Timesteps: 160h
Output timesteps: 1h
My recursive prediction method therefore consists of taking the last 160 hour window of my data to predict the next hour and re-integrating the prediction into the last window with an offset of 1 to then predict the second hour. Etc …
I understand that a recursive prediction method increases the error over time but I still get very bad results while on the evaluation of the model under test the results are very good (green series and red)
Here is an image to illustrate my point:
https://ibb.co/R0hs2NQ
The recursive prediction in blue seems smooth and totally wrong compared to the reality in orange.
Do you have any idea where to dig? I went through all your articles on this subject.
Thanks to you Jason.
• Jason Brownlee November 1, 2020 at 8:25 am #
Well done!
Perhaps you can vary the amount of input data?
Perhaps you can try alternate models? alternate configs? alternate data preparations?
Perhaps you can try direct prediction methods?
Perhaps you can try algorithms that support input sequences, like LSTMs?
Perhaps you can benchmark with a naive forecasting method (persistence) to see if any methods have skill?
Perhaps you can use a linear methods like SARIMA or ETS?
• Adrien November 2, 2020 at 3:40 am #
I’m actually already using a simple LSTM model with 3 layers .
I get very good results in MAE, MAPE and R2.
This is confirmed when I plot the test series with the predicted series.
The problem is with future predictions now, with my recursive method I got bad results, I tried to increase and decrease the rollback window but it doesn’t change much. The same for the model parameters (number of layers or neurons), the evaluation is good but the prediction on future data remains bad.
However, I managed to improve the prediction by adding an additional Dense layer with Relu activation in addition to the output Dense layer:
model = Sequential ()
model.add (LSTM (units = 50, return_sequences = True, input_shape = (X_train.shape [1], X_train.shape [2])))
model.add (LSTM (units = 50, return_sequences = True))
model.add (Dense (units = 50, activation = ‘relu’)) # New Dense layer !!!
model.add (Dense (units = X_train.shape [2], activation = ‘linear’))
model.compile (optimizer = ‘adam’, loss = ‘mean_squared_error’
On the other hand, I can’t quite understand why this is better with a second Dense layer in Relu activation.
Do you have an idea ?
• Jason Brownlee November 2, 2020 at 6:42 am #
Nice work!
We can’t answer “why” questions in deep learning (e.g. why does this config work and another one not work), the best we can do is run experiments and discover what works well for a given dataset.
• Adrien November 3, 2020 at 2:22 am #
Hi Jason,
I tried with the direct forecast and the results are much better!
I was using the wrong window to make my prediction that’s why.
• Jason Brownlee November 3, 2020 at 6:55 am #
Well done!
58. Bhaskar Tripathi January 11, 2021 at 5:30 am #
Hi Jason,
Is this apporach applicable for only linear models or it can be used with non-linear models like SVR , RandomForestRegressor etc as well?
59. shafi April 20, 2021 at 10:34 am #
Hi Jason,
I have a basic question. I have data from 1969Q1 to 2020Q4, train (1969q1- 2018q4) and test (2019q1-2020q4). Is the forecast on test data is one-step/single step or multip step. It is static no rolling window is used.
• shafi April 20, 2021 at 9:02 pm #
I think it is a multi-step where we apply the one-step strategy as you mentioned in one of your comments. Actually, somebody said to me, it is one step ahead but it is not one step ahead rather multiple steps using a one-step strategy.
• Jason Brownlee April 21, 2021 at 5:50 am #
You can choose how to frame your problem and what you want to use to evaluate the model.
60. N.G May 11, 2021 at 8:32 pm #
Hi Jason,
thanks for the wonderful posts you have published.
I’m new at machine learning, just completed some courses. I have a question: Is there any function that can automatically calculate different strategies for a given model?
• Jason Brownlee May 12, 2021 at 6:11 am #
You could implement such a library.
61. Aka May 21, 2021 at 3:05 am #
Hi Jason! Thanks a lot for all this admirable endeavour of yours – we deeply appreciate it!
Regarding the recursive Multi-step Forecast approach, would it be applicable in the case of datasets with multiple ‘predictor’ variables as well?
Say for example, one is interested in forecasting the temperature for the next two days but based both on current temperature and humidity:
temp(t+1) = model(temp(t-1), hum(t-1), temp(t-2), hum(t-2))
temp(t+2) = model(temp(t+1), hum(t+1), temp(t-1), hum(t-1))
Which would then be the humidity value to be used in the 2nd prediction step i.e. hum(t+1)? A humidity predicted value would not be available as the model of interest only predicts temperature.
I seem to be missing something…
• Jason Brownlee May 21, 2021 at 6:03 am #
Sure if you want.
Perhaps try it and see what works well or best for your data and model.
• Jeff July 8, 2021 at 7:40 am #
Hi Jason, thanks for putting this together.
As a follow-up to this, what would you advise when you have a lot of features and want to try out a recursive style model? At some point it would become impractical to build predictive models for each feature to use in the t+1 and beyond steps.
Thank you!!
• Jason Brownlee July 9, 2021 at 5:01 am #
I try to not be prescriptive – intuitions are often less effective.
Perhaps you can prototype a few approaches and discover what seems like a good fit for your specific dataset.
62. Konstantinos July 10, 2021 at 1:57 am #
Hi jason
i have a confusion about Recursive Multi-step Forecast.
Considering that we have 10 days and we use the first 6 for training our model to predict the temperature for day 7 and 8 with Recursive Multi-step Forecast.
To predict day 9 and 10 a new model is trained with the real temperatures of days 1-8?
Thank you!!
• Jason Brownlee July 10, 2021 at 6:12 am #
You can define the input and output any way that you want based on the data that is avaialble.
63. Miranda August 25, 2021 at 4:44 am #
Hi Jason,
Thanks for the tutorial! I have a problem of predicting the temperature flow in a specific process. I have multiple measurements from sensors and controllers. The measurements are recorded every 5 minutes. The system has to be stopped every now and then for an specific maintenance. The goal is to predict when the system needs to be shut down for this specific maintenance by predicting the value of temperature flow two weeks a head. Is this considered to be a multi step or one-step time series prediction?
Another question: I want to use the given sensor measurements for predicting the Temperature flow using a machine learning approach. Is there any specific approach that you would suggest for such a problem?
Thank you!
• Adrian Tam August 25, 2021 at 6:18 am #
It can be one-step to predict whether the system will be down for maintenance. It can also be multi-step to predict whether the system will be down for maintenance in the next N steps.
Hope this helps explain that.
64. Haitao September 10, 2021 at 11:09 am #
Hi Jason,
I am wondering if there is a model that takes a varying number of time steps as one sample that can be labelled ‘1’ or ‘0’. Suppose I have thousands of such samples to train the model and let it predict on new data (also of varying number of time steps). For example,
[ [1, 1.5, 1.45, 1.60, 1.60, 0.1, 1000],
[2, 1.55, 1.5, 1.82, 1.63, 0.06, 1200],
[3, 1.6, 1.61, 1.86, 1.72,0.06, 1150],
]
its label is ‘1’
[ [1, 1.50, 1.45, 1.60, 1.60, 0.10, 1000],
[2, 1.60, 1.50, 1.82, 1.63, 0.06, 1200],
[3, 1.63, 1.61, 1.56, 1.47, -0.06, 1150],
[4, 1.47, 1.50, 1.65, 1.55, 0.055, 1320]
]
its label is ‘0’
etc
after training, we want the model to predict for
[ [1, 1.50, 1.45, 1.60, 1.60, 0.10, 1000],
[2, 1.60, 1.50, 1.82, 1.63, 0.06, 1200],
[3, 1.63, 1.61, 1.56, 1.47, -0.06, 1150],
[4, 1.47, 1.50, 1.65, 1.55, 0.055, 1320],
[5, 1.67, 1.56, 1.98, 1.77, 0.061, 1350]
]
Is this possbile? Thank you.
• Adrian Tam September 11, 2021 at 6:29 am #
Depends on the model, you may do it. If you use a LSTM, you can train it to read from varying number of steps. Otherwise, people usually use padding to fill up the time steps to a fixed width.
|
## Upper Bound on Hot Dark-Matter Density from So(10) Yukawa Unification
We study low-energy consequences of supersymmetric SO(10) models with Yukawa unification h(t) = h(N) and h(b) = h(tau). We find that it is difficult to reproduce the observed m(b)/m(tau) ratio when the third-generation right-handed neutrino is at an intermediate scale, especially for small tan beta. We obtain a conservative lower bound on the mass of the right-handed neutrino M(N) > 6 x 10(13) GeV for tan beta < 10. This bound translates into an upper bound on the tau-neutrino mass, and therefore on its contribution to the hot dark matter density of the present universe, OMEGA(nu)h2 < 0.004. Our analysis is based on the full two-loop renormalization group equations with one-loop threshold effects. However, we also point out that physics above the GUT-scale could modify the Yukawa unification condition h(b) = h(tau) for tan beta less-than-or-similar-to 10. This might affect the prediction of m(b)/m(tau) and the constraint on M(N).
Published in:
Physics Letters B, 335, 3-4, 345-354
Year:
1994
ISSN:
0370-2693
Keywords:
Laboratories:
|
# Glossaries: parenthesis around page numbers
I have a glossary. Let us say:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[acronym]{glossaries}
\newacronym{gcd}{GCD}{Greatest Common Divisor}
\newacronym{lcm}{LCM}{Least Common Multiple}
\makeglossaries
\begin{document}
Given a set of numbers, there are elementary methods to compute
its \acrlong{gcd}, which is abbreviated \acrshort{gcd}. This process
is similar to that used for the \acrfull{lcm}.
\printglossary[type=\acronymtype]
\end{document}
I like the page numbers after the entry. But how can I enclose them in a parenthesis, or possibly prepend them with some text such as "Page(s): 1"?
\renewcommand{\glossaryentrynumbers}[1]{(#1)}
|
# Open Explorer and Highlight Specific File with PowerShell
There are programs allows us to "View File in Explorer" on a file it opened or is listing (e.g. iTunes). Upon selecting that function, the file explorer is opened to the path containing that file and the corresponding file is highlighted. This is the feature I would like to emulate with PowerShell.
It is simple to open to a directory with PowerShell (i.e. explorer $path). However, attempting to include a file just opens that file with the associated program (e.g. doing explorer$path/$file where $file is a .txt file will just open $file in notepad or whatever program is defined to handle .txt files). Can PowerShell open the File Explorer to a specific path AND highlight a specific file? ## 1 Answer It is possible to select a specific file by sending the select command to explorer.exe. This is actually native to cmd. So, the $path is restricted to the traditional DOS format (e.g. using \ only instead of both \ and /).
# Both Works on PowerShell 5
explorer /select,$path Invoke-Expression "explorer '/select,$path'"
This info is from this answer by Alex (thanks to AEonAX for pointing this out).
• Also works with double quotes explorer.exe "/select,C:\path\to\file.txt". I've found that the important thing is to wrap the whole /select and file name in quotes. – TechSpud Feb 16 '17 at 7:39
|
## FANDOM
68 Pages
The Shelter is one of the two primary locations from which users can obtain new eggs for their party, the other being the Lab. Run by Alex and Thomas, it is divided into two sections: the Egg Shelter, which holds eggs released by users, and the Safari Zone, containing released, hatched Pokémon. All users begin with six Shelter Grabs a day, permitting them the ability to obtain a total of six eggs or Pokémon from the Shelter. Aside from this, the two sections operate independently, and are subject to limitations on the kinds of Pokémon that can be adopted. A Shelter Pass allows people who bought one get an additional four Pokémon from the Shelter. You can get more Pokémon depending how many Shelter Passes you buy.
## Egg ShelterEdit
Eggs "released" by players of GPX+ are deposited to the Egg Shelter. Both excess eggs created by their Breeding Pairs (including those created by Professor Cypress) and unwanted, previously adopted eggs will appear in this section, typically between 15 and 20 minutes after they are released. These eggs cannot be adopted by anyone who has ever possessed the egg.
The Egg Shelter displays up to 60 different eggs at one time, excluding the restrictions described below. The eggs displayed at any one time can be changed by clicking the "Show more" button or pressing the Space bar. Any eggs that are not adopted within 36 hours of arrival into the Egg Shelter are deleted every half hour; excess eggs of a particular species may be deleted sooner.
## Safari ZoneEdit
The Safari Zone contains all released, hatched Pokémon, again appearing between 15 and 20 minutes after they were abandoned. Pokémon depicted in the Safari Zone will bounce around the page randomly, at speeds correlated to their base Speed stat; thus, an Accelgor will shoot about rapidly, while Shuckle will crawl at a snail's pace. Like the Egg Shelter, Safari Zone Pokémon cannot be adopted by anyone who appears in their timeline.
Because of the larger size of Pokémon sprites and their ability to move around, only fifteen different species will be shown at any one time (though members of the same evolutionary family can appear simultaneously); like in the Egg Shelter, users can change which Pokémon are displayed at a time. Unadopted Pokémon will be deleted after 72 hours.
## Shelter EnhancementsEdit
Although initially limited to six shelter grabs, users can expand this limit by purchasing Shelter Passes, which provide four additional shelter grabs, up to a maximum of 20. Additional shelter grabs are awarded according to the following rough formula:
$G = S * floor({I \over 2943} + {8P \over 2943}) + 4S + 6$
where G is the number of shelter grabs, I is the number of total interactions, P is the number of proper interactions, and S is the number of owned Shelter Passes. In other words, users gain extra grabs for every 2943 total interactions or every 327 proper interactions.
Both the initial number of grabs and those gained by interactions are enhanced by any multipliers active on the site. During a Site-wide Shiny Hunt, so long as the user has at least one shelter grab, adopting eggs of the target Pokémon will not count against their total shelter grabs.
The Sweet Honey item is also used during Shiny Hunts, Site-wide or otherwise. After hatching 15 of a Shiny Hunt Pokémon, a Sweet Honey can be used to bring up only that species' eggs in the Egg Shelter; if multiple Pokémon qualify, then the eggs displayed will be a combination of all qualifying Pokémon eggs.
## RestrictionsEdit
To prevent hoarding of special eggs, users are limited in the number of Very Rare and Novelty Pokémon that can be adopted from the Shelter, as outlined below:
• Users are limited to two adopted Very Rare or Novelty Pokémon from the Shelter or the Lab. Slime Slugma and self-summoned Pokémon do not count toward this limit.
• Users who possess two or more of any Very Rare or Novelty Pokémon, including evolutionary relatives, will be unable to find them in the Shelter. The exceptions are Deoxys, Bidofo, Dracowymsy, and MissingNo.; users can continue to find them and their eggs until they have one for each of their different forms. Slime Slugma is also exempt from this rule.
## The Dream WorldEdit
On November 27th, 2010, GPX+ introduced the Dream World, a special, limited-time section that exclusively featured Fifth Generation Pokémon eggs in commemoration of Pokémon Black and Pokémon White. 44 of the 156 Unova Pokémon were initially made available, with new Pokémon added in batches up until January 7th, 2011. Users were permitted to take six eggs from the Dream World every day. Eggs not in a user's Pokédex would appear as "Mystery" eggs in the Dream World, depicted with the generic egg icon until it was adopted. Because the games had not been released in North America at the time, users could opt to hide the appearance of Unova Pokémon and eggs, replacing their images with the generic egg sprite on their end[citation needed]. Once the games were released in North America, the Dream World was removed, along with the option to hide the egg and Pokémon images; any eggs left in the Dream World were added to the Lab.
The Punishing A Poacher exploration was initially created to support the Dream World; the Black and White Token Account Upgrades allowed users to take four additional eggs from the area each day. Unova-based Achievements were also rolled out to commemorate the Dream World, starting with Unova Master and Pokédex Master Plus, which were added to the Base Set.
## AchievementsEdit
Two Achievements are associated with the Shelter:
• Open-Hearted: Unlocked by adopting a total of 100 eggs and/or Pokémon from the Shelter
• Safari Warden: Unlocked by adopting a Scyther, Tauros, Kangaskhan, and Chansey (or any of their evolutions) from the Safari Zone.
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# Do You Know Your Sorting?
Let $k$ be a fixed constant. You are given a set of $n$ positive integers less than $k$, and you are tasked to sort it.
Which of the following is the asymptotic running time of the fastest possible algorithm?
×
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Ricci flow with surgery on manifolds with positive isotropic curvature
Abstract
We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short).
In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$.
In the second part, we develop a theory of ancient solutions which have bounded curvature; are $\kappa$-noncollapsed; are weakly PIC2; and are uniformly PIC. This is an extension of Perelman’s work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1.
In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman’s surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ which do not contain non-trivial incompressible $(n-1)$-dimensional space forms.
Authors
Simon Brendle
Department of Mathematics, Columbia University, New York, NY 10027
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Type
Database
Creator
Date
Thumbnail
# Search results
1,072 records were found.
## Overview of the SiLC R&D Activities
The R&D Collaboration SiLC (Silicon tracking for Linear Colliders) is based on generic R&D aiming to develop the next generation of large Silicon tracking systems for the Linear collider experiments; it serves all three ILC detector concepts. There is a strong involvement in ILD, a natural collaboration with SiD and recent 4th concept interest to use Silicon tracking technology as well. Here is a very brief summary of the latest results on sensors, Front End Electronics, Mechanics and Integration issues, test bench and test beam results and where to go from there.
## Large Silicon Tracking Systems for ILC: Role, Design and Main issues
The roles, the designs, the main issues and the current status of the R&D on large Silicon Tracking Systems for the ILC are discussed in this paper. The R&D work presented here is performed within the SiLC (Silicon tracking for the Linear Collider) R&D Collaboration.
## A new 130nm F.E readout chip for microstrip detectors
In the context of the Silicon tracking for a Linear Collider (SiLC) R&D collaboration, a highly compact mixed-signal chip has been designed in 130nm CMOS technology intended to read Silicon strip detectors for the experiments at the future International Linear Collider. The chip includes eighty eight channels of a full analog signal processing chain and analog to digital conversion with the corresponding digital controls and readout channels. The chip is 5x10mm2 where the analog implementation represents 4/5 of the total Silicon area.
## Silicon Data Acquisition and Front-End Electronics
A highly integrated Front-End readout and Data Acquisition scheme for Silicon trackers is presented. In this context, a 16-channel readout chip for Silicon strips detector has been designed in 180nm CMOS technology, having in view a highly multiplexed and sparsified readout global strategy. First results are presented.
## Extraction of the x-dependence of the non-perturbative QCD b-quark fragmentation distribution component
Using recent measurements of the b-quark fragmentation distribution obtained in $e^+e^- \to b \bar{b}$ events registered at the Z pole, the non-perturbative QCD component of the distribution has been extracted independently of any hadronic physics modelling. This distribution depends only on the way the perturbative QCD component has been defined. When the perturbative QCD component is taken from a parton shower Monte-Carlo, the non-perturbative QCD component is rather similar with those obtained from the Lund or Bowler models. When the perturbative QCD component is the result of an analytic NLL computation, the non-perturbative QCD component has to be extended in a non-physical region and thus cannot be described by any hadronic modelling. In the two examples used to characterize these two situations, which are studied at present, it ...
## Extraction of the x-dependence of the non-perturbative QCD b-quark fragmentation distribution component
Comment: 13 pages
## A 16-channel Silicon Strips Readout Chip in 180nm CMOS technology
A highly integrated readout scheme for Silicon trackers making use of Deep Sub-Micron CMOS electronics (DSM) and analog sampling techniques is presented. In the context of the International Linear Collider (ILC) tracking detectors developments, a 16-channel readout chip for Silicon strips detector has been designed in 180nm CMOS technology, each channel comprising a low noise amplifier, a pulse shaper, a sample and hold and a comparator operated at low power. Test results are presented.
## Front-End and Readout Electronics for Silicon Trackers at the ILC
A highly integrated readout scheme for Silicon trackers making use of Deep Sub-Micron CMOS electronics is presented. In this context,a 16-channel readout chip for Silicon strips detector has been designed in 180nm CMOS technology, each channel comprising a low noise amplifier, a pulse shaper, a sample and hold and a comparator. First results are presented.
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# What is a surface area of a cube with the volume of $8cm^3$?
Right, a cube with a volume of $8cm^3 =(2 \times 2 \times 2)$, what is the surface area? I think it's $8cm$, but I'm not sure and how do you work it out? My tutor sys it's $24cm$ but how is it? I've searched on the internet and it says you should do $l \times h \times w$. Confused?
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You mean a cube with volume $8cm^3$? – Lost1 Apr 11 '13 at 12:28
yeah volume lol, sorry – user61406 Apr 11 '13 at 12:29
Hi, Honky! Since you've asked a few questions now, and received some nice answers, I thought I'd encourage you to accept and upvote helpful answers. You can accept only one answer per question, just click on the $\large \checkmark$ to the left of the answer you'd like to accept. (You get two reputation points for each answer you accept.) AND you can Upvote as many answers as you'd like. – amWhy Apr 13 '13 at 18:00
that helps, thanks – user61406 Apr 14 '13 at 10:28
I think what you mean is that you have a cube of $\textbf{volume}$ $\ 8\text{cm}^3$=2cm x 2cm x 2cm.
A cube has 6 faces of equal area.
Therefore, the total surface area of the cube is equal to 6 multiplied by the area of one of the faces.
The area of one of the faces is 4$\text{cm}^2$ = 2cm x 2cm.
Therefore the total area of the cube is 24$\text{cm}^2$ = 6 x 4$\text{cm}^2$.
One thing you must remember is your units of measurement, writing 8cm2 (2x2x2) really makes no sense, since you are equating an area with a single number without units.
The result of your internet search has returned the method for working out the volume of the cube (or in fact a cuboid).
Volume = Length x Width x Height $\ \$ (cm$^3$ = cm x cm x cm)
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Hint. A cube is composed of 6 same squares, each of area $2\times2\mathrm{cm}^2$.
Also I asume you mean it has volume of $8\mathrm{cm}^3$.
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yes! the volume is 8cm3 – user61406 Apr 11 '13 at 12:29
You're confused. If a cube is $2\text{cm}\times2\text{cm}\times2\text{cm},$ then it has a volume of $8\text{cm}^3$, not an area of $8\text{cm}^2$. Note that each of its faces is $2\text{cm}\times2\text{cm}$, so has an area of $4\text{cm}^2$. How many faces does a cube have?
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6, yeah it's 24, thanks – user61406 Apr 11 '13 at 12:30
The actual formula for the surface area of a box in general is $$2(lw+wh+lh),$$ if you're curious. – Cameron Buie Apr 11 '13 at 12:42
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# Fortran weakness: byte wise I/O
In the Crm² lab, in Nancy (France), we have prototype of an hybrid pixel detector from the company Imxpad. The data from the detector are raw ASCII files that I need to convert into files suitable for integration software. We are working at the moment with Crysalis and the Eval suite.
The esperanto format used in Crysalis is a header with the frame written in binary as signed 32bits integer which translates into Fortran:
open(newunit=filehandled, form="unformatted", access="stream", file='filename')
write(filehandled) esperanto
close(filehandled)
This is the most efficient way to write an array in Fortran. The overhead on the write statement is minimum.
The cbf format used in Eval is a bit more complicated as Eval asks for byte offset compression on the data (signed integer). The result is that data have to be written byte by byte.
open(newunit=filehandled, form="unformatted", access="stream", file='filename')
basepixel=0
byteswritten=0
! origin is at the bottom
do j = ubound(cbf,2), 1, -1
do k = 1, ubound(cbf,1)
deltai=cbf(k,j)-basepixel
if(deltai>=-127 .and. deltai<=127) then
write(scratchfile) int(deltai,1)
byteswritten=byteswritten+1
basepixel=cbf(k,j)
else if(deltai>=-32767 .and. deltai<=32767) then
write(scratchfile) char(128) ! 0x80
write(scratchfile) int(deltai,2)
byteswritten=byteswritten+3
basepixel=cbf(k,j)
else if(deltai>=-2147483647 .and. deltai<=2147483647) then
! 0x8000 split in 2 character and manually switch for little_endian
write(scratchfile) char(128)//char(0)//char(128) ! 0x80 and 0x8000 little_endian
write(scratchfile) int(deltai,4)
byteswritten=byteswritten+7
basepixel=cbf(k,j)
else
! 0x80, 0x8000 and 0x80000000 little_endian
write(scratchfile) char(128)//char(0)//char(128)//char(0)//char(0)//char(0)//char(128)
write(scratchfile) deltai
byteswritten=byteswritten+15
basepixel=cbf(k,j)
end if
end do
end do
Comparing the two versions, the conversion to esperanto takes 78ms and the cbf version 130ms. When the number of threads are increased using OpenMP (several images at the same time), the cbf version does not scale at all (I guess with 4 different processes instead, it would scale) while the esperanto is ok. On 4 threads, the conversion to esperanto takes 23ms per image and the cbf 110ms. There is lot of information on the Internet about the slowness of Fortran on stream I/O. The testing was done on an Intel quad core Q9505 and the hard disk was a solid state Samsung SSD 840 PRO Series.
It is not possible to overcome the overhead on the write statement and I did not want to write a C function instead so I used a buffer as a character string to hold hold data before writing them to a file. There is no byte or binary type in Fortran. Which means I need to convert signed integer of different size into characters. This is done via bit wise shift and ‘and’ operator
The previous loop has then been written into:
basepixel=0
byteswritten=0
totalbyteswritten=0
! origin is at the bottom
do j = ubound(cbf,2), 1, -1
do k = 1, ubound(cbf,1)
if(byteswritten>buffer_length-15) then
write(scratchfile) longbuffer(1:byteswritten)
totalbyteswritten=totalbyteswritten+byteswritten
!print *, totalbyteswritten, byteswritten
byteswritten=0
end if
deltai=cbf(k,j)-basepixel
if(deltai>=-127 .and. deltai<=127) then
longbuffer(byteswritten+1:byteswritten+1) = char(iand(deltai, z'FF'))
byteswritten=byteswritten+1
basepixel=cbf(k,j)
else if(deltai>=-32767 .and. deltai<=32767) then
longbuffer(byteswritten+1:byteswritten+1) = char(128) ! 0x80
longbuffer(byteswritten+2:byteswritten+2) = char(iand(deltai, z'FF'))
longbuffer(byteswritten+3:byteswritten+3) = char(iand(ishft(deltai, -8), z'FF'))
byteswritten=byteswritten+3
basepixel=cbf(k,j)
else if(deltai>=-2147483647 .and. deltai<=2147483647) then
! 0x8000 split in 2 character and manually switch for little_endian
longbuffer(byteswritten+1:byteswritten+3) = char(128)//char(0)//char(128) ! 0x80 and 0x8000 little_endian
longbuffer(byteswritten+4:byteswritten+4) = char(iand(deltai, z'FF'))
longbuffer(byteswritten+5:byteswritten+5) = char(iand(ishft(deltai, -8), z'FF'))
longbuffer(byteswritten+6:byteswritten+6) = char(iand(ishft(deltai, -16), z'FF'))
longbuffer(byteswritten+7:byteswritten+7) = char(iand(ishft(deltai, -24), z'FF'))
byteswritten=byteswritten+7
basepixel=cbf(k,j)
else
! 0x80, 0x8000 and 0x80000000 little_endian
longbuffer(byteswritten+1:byteswritten+7) = char(128)//char(0)//&
& char(128)//char(0)//char(0)//char(0)//char(128)
longbuffer(byteswritten+8:byteswritten+8) = char(iand(deltai, z'FF'))
longbuffer(byteswritten+9:byteswritten+9) = char(iand(ishft(deltai, -8), z'FF'))
longbuffer(byteswritten+10:byteswritten+10) = char(iand(ishft(deltai, -16), z'FF'))
longbuffer(byteswritten+11:byteswritten+11) = char(iand(ishft(deltai, -24), z'FF'))
longbuffer(byteswritten+12:byteswritten+12) = char(iand(ishft(deltai, -32), z'FF'))
longbuffer(byteswritten+13:byteswritten+13) = char(iand(ishft(deltai, -40), z'FF'))
longbuffer(byteswritten+14:byteswritten+14) = char(iand(ishft(deltai, -48), z'FF'))
longbuffer(byteswritten+15:byteswritten+15) = char(iand(ishft(deltai, -56), z'FF'))
byteswritten=byteswritten+15
basepixel=cbf(k,j)
end if
end do
end do
if(byteswritten>0) then
write(scratchfile) longbuffer(1:byteswritten)
totalbyteswritten=totalbyteswritten+byteswritten
byteswritten=0
end if
The buffered version (longbuffer is 16384bytes) is much faster and as fast as the esperanto conversion. It now takes 72ms on one thread and 22ms on 4 threads which is 5 times faster than before. The remaining bottleneck is the reading of the ASCII data but nothing can be done apart using binary storage from the detector and not ASCII.
There is a C library for manipulating cbf files: CBFlib (Same on Sourceforge) I did not used it because I do not like to link to external libraries not available in official repositories of common linux distribution. Especially when just a few lines of codes are necessary to have my own function.
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## Locally Supported Kernels for Spherical Spline Interpolation
• By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.
Author: Michael Schreiner urn:nbn:de:hbz:386-kluedo-5411 Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (140) Preprint English 1995 1995 Technische Universität Kaiserslautern 2000/06/07 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011
$Rev: 13581$
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# dask.delayed - parallelize any code
## Contents
You can run this notebook in a live session or view it on Github.
# dask.delayed - parallelize any code¶
What if you don’t have an array or dataframe? Instead of having blocks where the function is applied to each block, you can decorate functions with @delayed and have the functions themselves be lazy.
This is a simple way to use dask to parallelize existing codebases or build complex systems.
Related Documentation
As we’ll see in the distributed scheduler notebook, Dask has several ways of executing code in parallel. We’ll use the distributed scheduler by creating a dask.distributed.Client. For now, this will provide us with some nice diagnostics. We’ll talk about schedulers in depth later.
[1]:
from dask.distributed import Client
client = Client(n_workers=4)
## A Typical Workflow¶
Typically if a workflow contains a for-loop it can benefit from delayed. The following example outlines a read-transform-write:
import dask
def process_file(filename):
data = do_a_transformation(data)
destination = f"results/{filename}"
write_out_data(data, destination)
return destination
results = []
for filename in filenames:
results.append(process_file(filename))
## Basics¶
First let’s make some toy functions, inc and add, that sleep for a while to simulate work. We’ll then time running these functions normally.
In the next section we’ll parallelize this code.
[2]:
from time import sleep
def inc(x):
sleep(1)
return x + 1
sleep(1)
return x + y
We time the execution of this normal code using the %%time magic, which is a special function of the Jupyter Notebook.
[3]:
%%time
# This takes three seconds to run because we call each
# function sequentially, one after the other
x = inc(1)
y = inc(2)
CPU times: user 168 ms, sys: 44.6 ms, total: 213 ms
Wall time: 3 s
### Parallelize with the dask.delayed decorator¶
Those two increment calls could be called in parallel, because they are totally independent of one-another.
We’ll make the inc and add functions lazy using the dask.delayed decorator. When we call the delayed version by passing the arguments, exactly as before, the original function isn’t actually called yet - which is why the cell execution finishes very quickly. Instead, a delayed object is made, which keeps track of the function to call and the arguments to pass to it.
[4]:
import dask
def inc(x):
sleep(1)
return x + 1
sleep(1)
return x + y
[5]:
%%time
# This runs immediately, all it does is build a graph
x = inc(1)
y = inc(2)
CPU times: user 474 µs, sys: 0 ns, total: 474 µs
Wall time: 1.05 ms
This ran immediately, since nothing has really happened yet.
To get the result, call compute. Notice that this runs faster than the original code.
[6]:
%%time
# This actually runs our computation using a local thread pool
z.compute()
CPU times: user 253 ms, sys: 63.6 ms, total: 317 ms
Wall time: 2.2 s
[6]:
5
## What just happened?¶
The z object is a lazy Delayed object. This object holds everything we need to compute the final result, including references to all of the functions that are required and their inputs and relationship to one-another. We can evaluate the result with .compute() as above or we can visualize the task graph for this value with .visualize().
[7]:
z
[7]:
Delayed('add-5a3755ad-272e-4be5-ada0-2df88367edec')
[8]:
# Look at the task graph for z
z.visualize()
[8]:
Notice that this includes the names of the functions from before, and the logical flow of the outputs of the inc functions to the inputs of add.
### Some questions to consider:¶
• Why did we go from 3s to 2s? Why weren’t we able to parallelize down to 1s?
• What would have happened if the inc and add functions didn’t include the sleep(1)? Would Dask still be able to speed up this code?
• What if we have multiple outputs or also want to get access to x or y?
## Exercise: Parallelize a for loop¶
for loops are one of the most common things that we want to parallelize. Use dask.delayed on inc and sum to parallelize the computation below:
[9]:
data = [1, 2, 3, 4, 5, 6, 7, 8]
[10]:
%%time
# Sequential code
def inc(x):
sleep(1)
return x + 1
results = []
for x in data:
y = inc(x)
results.append(y)
total = sum(results)
CPU times: user 476 ms, sys: 87.3 ms, total: 564 ms
Wall time: 8.01 s
[11]:
total
[11]:
44
[12]:
%%time
CPU times: user 3 µs, sys: 0 ns, total: 3 µs
Wall time: 7.39 µs
[13]:
@dask.delayed
def inc(x):
sleep(1)
return x + 1
results = []
for x in data:
y = inc(x)
results.append(y)
total = sum(results)
print("Before computing:", total) # Let's see what type of thing total is
result = total.compute()
print("After computing :", result) # After it's computed
Before computing: Delayed('add-338882cd649de04d93e2228cb9a6a6cf')
After computing : 44
How do the graph visualizations compare with the given solution, compared to a version with the sum function used directly rather than wrapped with delayed? Can you explain the latter version? You might find the result of the following expression illuminating
inc(1) + inc(2)
## Exercise: Parallelize a for-loop code with control flow¶
Often we want to delay only some functions, running a few of them immediately. This is especially helpful when those functions are fast and help us to determine what other slower functions we should call. This decision, to delay or not to delay, is usually where we need to be thoughtful when using dask.delayed.
In the example below we iterate through a list of inputs. If that input is even then we want to call inc. If the input is odd then we want to call double. This is_even decision to call inc or double has to be made immediately (not lazily) in order for our graph-building Python code to proceed.
[14]:
def double(x):
sleep(1)
return 2 * x
def is_even(x):
return not x % 2
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[15]:
%%time
# Sequential code
results = []
for x in data:
if is_even(x):
y = double(x)
else:
y = inc(x)
results.append(y)
total = sum(results)
print(total)
Delayed('add-f49c2865d305a580ecda9807c1eed4ce')
CPU times: user 293 ms, sys: 53 ms, total: 346 ms
Wall time: 5.01 s
[16]:
%%time
# TODO: parallelize the sequential code above using dask.delayed
# You will need to delay some functions, but not all
CPU times: user 14 µs, sys: 0 ns, total: 14 µs
Wall time: 17.6 µs
[17]:
@dask.delayed
def double(x):
sleep(1)
return 2 * x
results = []
for x in data:
if is_even(x): # even
y = double(x)
else: # odd
y = inc(x)
results.append(y)
total = sum(results)
[18]:
%time total.compute()
CPU times: user 227 ms, sys: 24 ms, total: 251 ms
Wall time: 3.06 s
[18]:
90
[19]:
total.visualize()
[19]:
### Some questions to consider:¶
• What are other examples of control flow where we can’t use delayed?
• What would have happened if we had delayed the evaluation of is_even(x) in the example above?
• What are your thoughts on delaying sum? This function is both computational but also fast to run.
## Exercise: Parallelize a Pandas Groupby Reduction¶
In this exercise we read several CSV files and perform a groupby operation in parallel. We are given sequential code to do this and parallelize it with dask.delayed.
The computation we will parallelize is to compute the mean departure delay per airport from some historical flight data. We will do this by using dask.delayed together with pandas. In a future section we will do this same exercise with dask.dataframe.
## Create data¶
Run this code to prep some data.
This downloads and extracts some historical flight data for flights out of NYC between 1990 and 2000. The data is originally from here.
[20]:
%run prep.py -d flights
### Inspect data¶
[21]:
import os
sorted(os.listdir(os.path.join("data", "nycflights")))
[21]:
['1990.csv',
'1991.csv',
'1992.csv',
'1993.csv',
'1994.csv',
'1995.csv',
'1996.csv',
'1997.csv',
'1998.csv',
'1999.csv']
### Read one file with pandas.read_csv and compute mean departure delay¶
[22]:
import pandas as pd
[22]:
Year Month DayofMonth DayOfWeek DepTime CRSDepTime ArrTime CRSArrTime UniqueCarrier FlightNum ... AirTime ArrDelay DepDelay Origin Dest Distance TaxiIn TaxiOut Cancelled Diverted
0 1990 1 1 1 1621.0 1540 1747.0 1701 US 33 ... NaN 46.0 41.0 EWR PIT 319.0 NaN NaN 0 0
1 1990 1 2 2 1547.0 1540 1700.0 1701 US 33 ... NaN -1.0 7.0 EWR PIT 319.0 NaN NaN 0 0
2 1990 1 3 3 1546.0 1540 1710.0 1701 US 33 ... NaN 9.0 6.0 EWR PIT 319.0 NaN NaN 0 0
3 1990 1 4 4 1542.0 1540 1710.0 1701 US 33 ... NaN 9.0 2.0 EWR PIT 319.0 NaN NaN 0 0
4 1990 1 5 5 1549.0 1540 1706.0 1701 US 33 ... NaN 5.0 9.0 EWR PIT 319.0 NaN NaN 0 0
5 rows × 23 columns
[23]:
# What is the schema?
df.dtypes
[23]:
Year int64
Month int64
DayofMonth int64
DayOfWeek int64
DepTime float64
CRSDepTime int64
ArrTime float64
CRSArrTime int64
UniqueCarrier object
FlightNum int64
TailNum float64
ActualElapsedTime float64
CRSElapsedTime int64
AirTime float64
ArrDelay float64
DepDelay float64
Origin object
Dest object
Distance float64
TaxiIn float64
TaxiOut float64
Cancelled int64
Diverted int64
dtype: object
[24]:
# What originating airports are in the data?
df.Origin.unique()
[24]:
array(['EWR', 'LGA', 'JFK'], dtype=object)
[25]:
# Mean departure delay per-airport for one year
df.groupby("Origin").DepDelay.mean()
[25]:
Origin
EWR 10.854962
JFK 17.027397
LGA 10.895592
Name: DepDelay, dtype: float64
### Sequential code: Mean Departure Delay Per Airport¶
The above cell computes the mean departure delay per-airport for one year. Here we expand that to all years using a sequential for loop.
[26]:
from glob import glob
filenames = sorted(glob(os.path.join("data", "nycflights", "*.csv")))
[27]:
%%time
sums = []
counts = []
for fn in filenames:
# Groupby origin airport
by_origin = df.groupby("Origin")
# Sum of all departure delays by origin
total = by_origin.DepDelay.sum()
# Number of flights by origin
count = by_origin.DepDelay.count()
# Save the intermediates
sums.append(total)
counts.append(count)
# Combine intermediates to get total mean-delay-per-origin
total_delays = sum(sums)
n_flights = sum(counts)
mean = total_delays / n_flights
CPU times: user 71.3 ms, sys: 146 µs, total: 71.4 ms
Wall time: 78.3 ms
[28]:
mean
[28]:
Origin
EWR 12.500968
JFK NaN
LGA 10.169227
Name: DepDelay, dtype: float64
### Parallelize the code above¶
Use dask.delayed to parallelize the code above. Some extra things you will need to know.
1. Methods and attribute access on delayed objects work automatically, so if you have a delayed object you can perform normal arithmetic, slicing, and method calls on it and it will produce the correct delayed calls.
2. Calling the .compute() method works well when you have a single output. When you have multiple outputs you might want to use the dask.compute function. This way Dask can share the intermediate values.
So your goal is to parallelize the code above (which has been copied below) using dask.delayed. You may also want to visualize a bit of the computation to see if you’re doing it correctly.
[29]:
%%time
CPU times: user 3 µs, sys: 0 ns, total: 3 µs
Wall time: 8.82 µs
If you load the solution, add %%time to the top of the cell to measure the running time.
[30]:
%%time
# This is just one possible solution, there are
# several ways to do this using dask.delayed
sums = []
counts = []
for fn in filenames:
# Groupby origin airport
by_origin = df.groupby("Origin")
# Sum of all departure delays by origin
total = by_origin.DepDelay.sum()
# Number of flights by origin
count = by_origin.DepDelay.count()
# Save the intermediates
sums.append(total)
counts.append(count)
# Combine intermediates to get total mean-delay-per-origin
total_delays = sum(sums)
n_flights = sum(counts)
mean, *_ = dask.compute(total_delays / n_flights)
CPU times: user 177 ms, sys: 43.5 ms, total: 220 ms
Wall time: 1.19 s
[31]:
(sum(sums)).visualize()
[31]:
[32]:
# ensure the results still match
mean
[32]:
Origin
EWR 12.500968
JFK NaN
LGA 10.169227
Name: DepDelay, dtype: float64
### Some questions to consider:¶
• How much speedup did you get? Is this how much speedup you’d expect?
• Experiment with where to call compute. What happens when you call it on sums and counts? What happens if you wait and call it on mean?
• Experiment with delaying the call to sum. What does the graph look like if sum is delayed? What does the graph look like if it isn’t?
• Can you think of any reason why you’d want to do the reduction one way over the other?
Visit the Delayed documentation. In particular, this delayed screencast will reinforce the concepts you learned here and the delayed best practices document collects advice on using dask.delayed well.
[33]:
client.close()
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Select Page
# Goundaga and Lengewal Latrine Project – Senegal
Location
Goundaga and Lengewal, Kolda Region, Senegal
Community Description
The villages of Goundaga and Lengewal (population 350 and 300 respectively) are located in the Kolda Region of Senegal. The villages are comprised of mostly farmers whose fields look onto the seasonal river nearby.
The children of the villages study either at the school in Goundaga or in the larger town of Kunkane about five kilometers away.
As you walk through the villages, you can’t help but notice the large expanse of trees and agroforestry techniques adopted by the villagers, evidence of previous Peace Corps volunteers in the area. It is clear the people of these villages are hard workers and know how to best use volunteers to their advantage.
Sanitation within Goundaga and Lewngewal is lacking significantly. The few latrines in these villages have filled up, requiring people to go to the fields or forest to relieve themselves.
A common problem seen is allowing young children to relieve themselves in the middle of the compound because the alternative of taking them far away is too much work. People often neglect washing their hands because it is not readily available after they relieve themselves.
Project Description
This project will bring 15 latrines to individual households in Goundaga and Lewngewal.
Water Charity funds will pay for materials, including cement and rebar, and the labor of skilled masons. The masons will be employed within the village to build the latrine heads. In the process they will gain knowledge and experience, and pass the technique on to villagers, allowing for continued proliferation of latrines in the area.
The beneficiaries will do the unskilled labor, including the digging of the holes. They will also gather their own sand and gravel to be mixed with the cement, as well as transport the cement and rebar at their own cost.
In addition to the latrine construction, the Peace Corps volunteers, along with their health worker counterpart, will conduct a series of sanitation trainings, including washing with soap, water sanitation, and the construction of Bidon Handwashing Stations.
Project Impact
650 people will benefit from the project.
Peace Corps Volunteer Directing Project
Stephanie McAlexander
Monitoring and Maintenance
At the project’s completion, the Peace Corps volunteer will ensure each latrine has been built accordingly and that people are implementing proper sanitation practices.
Over the long term, the community health worker will report to the Peace Corps volunteer about latrine quality and sanitation within the community.
$482.65 Donations Collected to Date$482.65
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Difference between revisions of "The Seqed and Modern Trigonometry Exploration"
The Seqed and the Cotangent
Several of the problems in the Rhind papyrus compute the so called seqed of a pyramid. The Egyptians computed the ratio (half the base)/height.
In the image of the triangle here the seqed is actually the run over the rise of the smaller right triangle that makes up the left half of the image. Note that in this case the seqed measures the inverse of the slope (which would be the rise over run). Recall that for a right triangle we can define several trigonometric functions. the best known of those are:
• sin(x) = opposite / hypothenuse
• cos(x) = adjacent / hypothenuse
• tan(x) = opposite / adjacent
• cotangent(x) = adjacent / opposite
So for the small right triangle that takes up half of the larger triangle - which can be thought of as a cross section of a pyramid - we can look at the cotangent and notice that it is exactly the seqed computed by the ancient Egyptians.
cotangent(x) = adjacent / opposite = half the base / height = seqed
So what exactly is the seqed (or if you prefer the cotangent) measuring? It gives a ratio of the base versus the height and hence gives a measuer of the steepness of the triangle. And because the triangle really represents the pyramid, it measures how steep the pyramid is.
Note that a very steep pyramid would have a relatively great height in comparison to the size of the base. So smaller seqeds correspond to higher pyramids, which in turn correspond to steeper pyramids
Example:
Rhind Papyrus 56: Example of reckoning a pyramid 360 cubits wide and 250 cubits high. Cause thou that I know the seked of it. You are to take half of 360; It becomes 180. You are to reckon with 250 to find 180. Result: 1/2 + 1/5 + 1/50. (slight adjustment of the text to make it easier to read)
What the Egyptian scribe is doing here is showing how to compute the seqed.
The first part is straight forward:
You are to take half of 360; It becomes 180. We need to use half the base length, so we divide the given measurement by 2.
You are to reckon with 250 to find 180. Result: 1/2 + 1/5 + 1/50. This part means divide 250 into 180 and the answer is 1/2 + 1/5 + 1/50, which we in modern days would simplify to 18/25, or if we use a calculator would be represented as .72.
Question 1:
Compute problem 58 from the Rhind papyrus: In a pyramid whose height is 93 1/3 [cubits] make known the seqed of it when its base side is 140 [cubits].
Question 2:
Solve problem 57 from the Rhind papyrus: If the seqed is 3/2 and the base is 280 [cubits], what is the height of the pyramid?
Question 3:
Solve problem 59(a) from the Rhind papyrus: In a pyramid whose base side is 12 [cubits] and whose altitude is 8 [cubits]; what is its seqed?
Real Pyramids, right triangles and their Seqeds
One theory for how the Egyptians planned the construction of their pyramids and maintain a constant slope is that they used right triangles. The idea is that they may have constructed wooden right triangles that could have been used as a guide during the construction of the monument. Egyptians may have known about perfect triangles from experimenting with their version of a measuring tape: knotted cords. It may have been noted for instance that if one has pieces of ropes of length 3, 4, and 5 that they form a right triangle. There is no real evidence that the Egyptians knew this back in the day of the pyramid builders, but it seems reasonable that they may have noticed that there are some perfect triangles out there.
Question 4:
Pythagorean triples refer to the side measures of right triangles, where the sides have integer length. One of the most well known Pythagorean triples is (3,4,5) which corresponds to a 3-4-5 right triangle. The side of length 5 is the hypothenuse. When it comes to identifying the base and the height we have a choice.
a) If we let 3 be the base and 4 be the height, compute the seqed (base/height).
b) If we let 4 be the base and 3 be the height, compute the seqed.
(This information will be used in problem 5)
Next suppose we have an isosceles triangle with both sides length 1. The hypothenuse would be $\sqrt{2}$, so this right triangle does not correspond to a Pythagorean triple, but it's interesting in it's own right.
c) Compute the seqed of the isosceles triangle.
Next, consider the triple ( 8, 15, 17)
d) If we let 8 be the base and 15 be the height, compute the seqed.
e) If we let 15 be the base and 8 be the height, compute the seqed.
Question 5:
For the following pyramids do the following:
(i) Compute the Seqed if possible.
(ii)Could the pyramid have been constructed using a technique involving right triangles?
Pyramid Height base Seqed of the pyramid Slope Possible right triangle
Great Pyramid (Pharaoh Khufu) 146.6 230.3 51.8 degrees
Pharaoh Khafre 143.5 215.25 .75 53.1 degrees 3-4-5 Triangle
Pharaoh Menkaure 66.45 103.4 .79 51.3 degrees Maybe 3-4-5 Triangle?
Pharaoh Userkaf 49 73.3 53 degrees
Pharaoh Sahure 48 78.5 50.5 degrees
Queen Khentkawes 72 25 52 degrees
Pharaoh Djedkare 52.5 78.75 52 degrees
Djedkare's Queen 21 41 ca 45 degrees
Pharaoh Teti 52.5 78.5 53.1 degrees
Teti's Queen (Iput) 21 21 63 degrees
Handin: A sheet with answers to all questions.
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# Quadratic utility: monotonicity and risk aversion
I am taking macro class this fall. One of the problems asks whether decreasing absolute risk-aversion and ever-increasing consumption are two unattractive implications of the quadractic utility function. Could anyone please help with this?
• I think quadratic utility is associated with increasing absolute risk-aversion. The assumption being that there is reduced risk-taking from wealthier folk, because the marginal utility on conducting risk is decreasing. – EB3112 Nov 26 '20 at 8:23
Quadratic utility is given by $$u(w) = w - b w^2$$ which has derivative $$u'(w) = 1- 2b w$$ such that for high levels of $$w, u'(w)<0$$. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less. The second derivative is $$u'(w) = -2b$$ such that absolute risk aversion is $$\frac{- u''(w)}{u'(w)} = \frac{ 2b}{1- 2b w},$$ which is increasing in wealth. This contradicts evidence that wealthier people take more financial risks instead of less.
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# Consider the following line integral, where C is the line segment from (2, 4) to (1, 5)....
## Question:
Consider the following line integral, where {eq}C {/eq} is the line segment from {eq}(2,\ 4) {/eq} to {eq}(1,\ 5) {/eq}.
{eq}\displaystyle \int_C (9 x - 6 y)\ dx + (10 y - 6 x)\ dy {/eq}.
(a) Show that the line integral is path independent by finding a scalar potential function.
(b) Evaluate the integral along the oriented curve by any appropriate method.
## Line Integral:
We will find the potential function by integrating the partial derivatives and then adding the integration constant and to solve the problem we will use the line integral.
Let us find the potential function:
{eq}f_{x}=9x-6y\\ f_{y}=10y-6x\\ f=\int (9x-6y)dx=\frac{9x^{2}}{2}-6xy+c_{x}\\ f=\int (10y-6x)dy=5y^{2}-6xy+c_{y}\\ f=\frac{9x^{2}}{2}-6xy+5y^{2}+c {/eq}
b) Now let us solve the line integral:
{eq}\frac{x-2}{-1}=\frac{y-4}{1}=t\\ x=-t+2\\ y=t+4\\ dx=-dt\\ dy=dt {/eq}
The integral will be:
{eq}=\int_{0}^{1}\left (9(-t+2)-6(t+4) \right )(-dt)+\left ( 10(t+4)-6(-2t) \right )dt\\ =\int_{0}^{1}(31t+34)dt {/eq}
Integrating we get:
{eq}=\frac{31t^{2}}{2}+34t {/eq}
Now let us plug-in the bounds:
{eq}=\frac{99}{2} {/eq}
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E01 Chapter Contents
E01 Chapter Introduction
NAG Library Manual
# NAG Library Routine DocumentE01THF
Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
## 1 Purpose
E01THF evaluates the three-dimensional interpolating function generated by E01TGF and its first partial derivatives.
## 2 Specification
SUBROUTINE E01THF ( M, X, Y, Z, F, IQ, LIQ, RQ, LRQ, N, U, V, W, Q, QX, QY, QZ, IFAIL)
INTEGER M, IQ(LIQ), LIQ, LRQ, N, IFAIL REAL (KIND=nag_wp) X(M), Y(M), Z(M), F(M), RQ(LRQ), U(N), V(N), W(N), Q(N), QX(N), QY(N), QZ(N)
## 3 Description
E01THF takes as input the interpolant $Q\left(x,y,z\right)$ of a set of scattered data points $\left({x}_{r},{y}_{r},{z}_{r},{f}_{r}\right)$, for $\mathit{r}=1,2,\dots ,m$, as computed by E01TGF, and evaluates the interpolant and its first partial derivatives at the set of points $\left({u}_{i},{v}_{i},{w}_{i}\right)$, for $\mathit{i}=1,2,\dots ,n$.
E01THF must only be called after a call to E01TGF.
This routine is derived from the routine QS3GRD described by Renka (1988).
## 4 References
Renka R J (1988) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software 14 151–152
## 5 Parameters
1: M – INTEGERInput
2: X(M) – REAL (KIND=nag_wp) arrayInput
3: Y(M) – REAL (KIND=nag_wp) arrayInput
4: Z(M) – REAL (KIND=nag_wp) arrayInput
5: F(M) – REAL (KIND=nag_wp) arrayInput
On entry: M, X, Y, Z and F must be the same values as were supplied in the preceding call to E01TGF.
6: IQ(LIQ) – INTEGER arrayInput
On entry: must be unchanged from the value returned from a previous call to E01TGF.
7: LIQ – INTEGERInput
On entry: the dimension of the array IQ as declared in the (sub)program from which E01THF is called.
Constraint: ${\mathbf{LIQ}}\ge 2×{\mathbf{M}}+1$.
8: RQ(LRQ) – REAL (KIND=nag_wp) arrayInput
On entry: must be unchanged from the value returned from a previous call to E01TGF.
9: LRQ – INTEGERInput
On entry: the dimension of the array RQ as declared in the (sub)program from which E01THF is called.
Constraint: ${\mathbf{LRQ}}\ge 10×{\mathbf{M}}+7$.
10: N – INTEGERInput
On entry: $n$, the number of evaluation points.
Constraint: ${\mathbf{N}}\ge 1$.
11: U(N) – REAL (KIND=nag_wp) arrayInput
12: V(N) – REAL (KIND=nag_wp) arrayInput
13: W(N) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{U}}\left(\mathit{i}\right)$, ${\mathbf{V}}\left(\mathit{i}\right)$, ${\mathbf{W}}\left(\mathit{i}\right)$ must be set to the evaluation point $\left({u}_{\mathit{i}},{v}_{\mathit{i}},{w}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.
14: Q(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{Q}}\left(\mathit{i}\right)$ contains the value of the interpolant at $\left({u}_{\mathit{i}},{v}_{\mathit{i}},{w}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$. If any of these evaluation points lie outside the region of definition of the interpolant the corresponding entries in Q are set to the largest machine representable number (see X02ALF), and E01THF returns with ${\mathbf{IFAIL}}={\mathbf{3}}$.
15: QX(N) – REAL (KIND=nag_wp) arrayOutput
16: QY(N) – REAL (KIND=nag_wp) arrayOutput
17: QZ(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{QX}}\left(\mathit{i}\right)$, ${\mathbf{QY}}\left(\mathit{i}\right)$, ${\mathbf{QZ}}\left(\mathit{i}\right)$ contains the value of the partial derivatives of the interpolant $Q\left(x,y,z\right)$ at $\left({u}_{\mathit{i}},{v}_{\mathit{i}},{w}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$. If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in QX, QY and QZ are set to the largest machine representable number (see X02ALF), and E01THF returns with ${\mathbf{IFAIL}}={\mathbf{3}}$.
18: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6 Error Indicators and Warnings
If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{M}}<10$, or ${\mathbf{LIQ}}<2×{\mathbf{M}}+1$, or ${\mathbf{LRQ}}<10×{\mathbf{M}}+7$, or ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=2$
Values supplied in IQ or RQ appear to be invalid. Check that these arrays have not been corrupted between the calls to E01TGF and E01THF.
${\mathbf{IFAIL}}=3$
At least one evaluation point lies outside the region of definition of the interpolant. At all such points the corresponding values in Q, QX, QY and QZ have been set to the largest machine representable number (see X02ALF).
## 7 Accuracy
Computational errors should be negligible in most practical situations.
The time taken for a call to E01THF will depend in general on the distribution of the data points. If X, Y and Z are approximately uniformly distributed, then the time taken should be only $\mathit{O}\left({\mathbf{N}}\right)$. At worst $\mathit{O}\left({\mathbf{M}}{\mathbf{N}}\right)$ time will be required.
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Nikita Markarian’s mathblog
Koszul duality for n-algebras
Posted in Mathematics by nikitamarkarian on February 19, 2011
Below there is a construction that from any augmented dg algebra over the operad of chains on framed little $n$-discs produces a coalgebra over the same operad.
This construction generalizes the Koszul duality, for $n=1$ this is the usual Koszul duality for algebras.
Another case of this duality is from topology. Take a topological space $X$. Then $\Omega^n X$ is a module over framed little $n$-discs. The union of symmetric powers of $X$ is a comodule over the (trivial) operad $Com$ and thus over framed discs (I am grateful to Victor Turchin for this observation). (Co)chains of this (co)modules over operads are dual to each other in our sence. This explains why one may calculate homology of mapping spaces from a $n$-manifold to $X$ either by means of the generalization of the higher Hochschild homology or the spherical homology.
It seems that it is connected with work in progress of M. Ching and P. Salvatore “Geometric self-duality for the little discs operads” (I found it on the homepage of Michael Ching).
UPD: This Koszul duality is mentioned in the draft of J. Lurie’s “Higher algebra”(see e.g. example 7.3.6.7). Unfortunately, details are postponed for future work. See also “Moduli Problems for Ring Spectra”.
Suppose $n=2$, the generalization is straightforward. Let $A$ be a dg $f2$(=chains on framed little 2-discs)-algebra and $e$ be an augmentation that is a one-dimensional module over it. “Module” means that $\ker e$ is closed under all operations. Let $HH(A)$ be the Hochschild complex of $A$ considered as an algebra, see this post.Then $A$ is a left and right module over $HH(A)$ (ibid.) and the composition gives a morphism $HH(A) \to e$ that is also an augmentation. Define $A^!=Tor^\bullet_{HH(A)}(A,e)$.
Let us show that this is a $f2$-coalgebra. The reasoning is similar to the one about the action of the operad of framed little discs on a multiloop space. Let $D$ be the closed disc. One may present $Tor^\bullet_{HH(A)}(A,e)$ as the complex $\oplus_k (C_*(fD^2(k))\otimes A^{\otimes k})$ (we use notations from here) modulo relations given by the operad action and modulo the augmentation ideal for copies of $A$ lying on the border:
Instead of placing the augmentation on the border on can put it on a closed subset of the disc that is the supplement to the union of non-intersecting discs:
The corresponding homology is the tensor product of $A^!$‘s by the number of white discs. The augmentation map gives the map from $A^!$ to the tensor product:
This defines the map $C_{*}(fD(n))\otimes A^!\to \underbrace{A^!\otimes\cdots\otimes A^!}_n$, where $C_*(fD(n))$ is the complex of chains on the space of framed discs. Dually this defines a $f2$-structure on $A^{!\vee}$. In fact it gives only a $2$-algerbra, but it is natural to propose that it be lifted to $f2$. In presence of an augmentation the difference between $2$ and $f2$ is not so big, I hope to discuss it somewhere.
Natural constructions of dual $f2$-algebras are dual to each others: S-homology for a compact surface $S$ are dual (call it the Poincare-Koszul duality), Hochschild homology are Koszul dual and so on. As an example consider a Lie bialgebra. Its Chevalley complex is (nearly) a BV algebra, the formality theorem gives a $f2$-algebra. The spherical homology of this $f2$-algebra equals to the tensor product of the Chevalley complex and the complex dual to the Chevalley complex of the dual bialgebra.The dual $f2$-algebra is the one corresponding to the dual bialgebra. Thus spherical homologies of dual algebras are dual, that fits in with the statement above.
I guess that this duality plays an important role in something like string theory. For example, one may propose, that BV algebras coming from A and B models for a given Calabi-Yau manifold are dual to each other.
An interesting question is to describe an algebra Koszul dual to the one coming from the string topology.
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### Qi Rao
• PhD, University of Technology Sydney
[email protected]
Qi Rao is currently a second-year Ph.D. student at AAII, University of Technology, Sydney, under the supervision of Prof. Yi Yang.
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# MATLAB: How to apply Otsu method to a stacked images
Image Processing Toolboxotsu stackedthresholding
Dear all,
I have 21 RGB microscopic images located in one folder. I want to apply the Otsu thresholding to each image in the folder (each image has different Otsu thresholding value).
How to make a "for loop" applied to the 21 images?
I tried this code but it finds the threshold value of the first image and applies it to all 21 images.
Any idea?
Meshoo
*****************************************************************
files = dir('*.tif');
for k = 1:(numel(files));
thresh = []; image = imread(files(k).name); thresh = graythresh(image); newImage = im2bw(image, thresh); imwrite(newImage, ['NewImage\', files(k).name,]);
end
thresh = graythresh(image)
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PDF e-Pub
## Section: Overall Objectives
### Overall Objectives
Keywords: Reformulation techniques in Mixed Integer Programming (MIP), Polyhedral approaches (cut generation), Robust Optimization, Approximation Algorithms, Extended formulations, Lagrangian Relaxation (Column Generation) based algorithms, Dantzig and Benders Decomposition, Primal Heuristics, Graph Theory, Constraint Programming.
Quantitative modeling is routinely used in both industry and administration to design and operate transportation, distribution, or production systems. Optimization concerns every stage of the decision-making process: long term investment budgeting and activity planning, tactical management of scarce resources, or the control of day-to-day operations. In many optimization problems that arise in decision support applications the most important decisions (control variables) are discrete in nature: such as on/off decision to buy, to invest, to hire, to send a vehicle, to allocate resources, to decide on precedence in operation planning, or to install a connection in network design. Such combinatorial optimization problems can be modeled as linear or nonlinear programs with integer decision variables and extra variables to deal with continuous adjustments. The most widely used modeling tool consists in defining the feasible decision set using linear inequalities with a mix of integer and continuous variables, so-called Mixed Integer Programs (MIP), which already allow a fair description of reality and are also well-suited for global optimization. The solution of such models is essentially based on enumeration techniques and is notoriously difficult given the huge size of the solution space.
Commercial solvers have made significant progress but remain quickly overwhelmed beyond a certain problem size. A key to further progress is the development of better problem formulations that provide strong continuous approximations and hence help to prune the enumerative solution scheme. Effective solution schemes are a complex blend of techniques: cutting planes to better approximate the convex hull of feasible (integer) solutions, extended reformulations (combinatorial relations can be formulated better with extra variables), constraint programming to actively reduce the solution domain through logical implications along variable fixing based on reduced cost, Lagrangian decomposition methods to produce powerful relaxations, and Bender's decomposition to project the formulation, reducing the problem to the important decision variables, and to implement multi-level programming that models a hierarchy of decision levels or recourse decision in the case of data adjustment, primal heuristics and meta-heuristics (greedy, local improvement, or randomized partial search procedures) to produce good candidates at all stage of the solution process, and branch-and-bound or dynamic programming enumeration schemes to find a global optimum, with specific strong strategies for the selection on the sequence of fixings. The real challenge is to integrate the most efficient methods in one global system so as to prune what is essentially an enumeration based solution technique. The progress are measured in terms of the large scale of input data that can now be solved, the integration of many decision levels into planning models, and not least, the account taken for random (or dynamically adjusted) data by way of modeling expectation (stochastic approaches) or worst-case behavior (robust approaches).
Building on complementary expertise, our team's overall goals are threefold:
$\left(i\right)$ Methodologies:
To design tight formulations for specific combinatorial optimization problems and generic models, relying on delayed cut and column generation, decomposition, extended formulations and projection tools for linear and nonlinear mixed integer programming models. To develop generic methods based on such strong formulations by handling their large scale dynamically. To generalize algorithmic features that have proven efficient in enhancing performance of exact optimization approaches. To develop approximation schemes with proven optimality gap and low computational complexity. More broadly, to contribute to theoretical and methodological developments of exact and approximate approaches in combinatorial optimization, while extending the scope of applications and their scale.
$\left(ii\right)$ Problem solving:
To demonstrate the strength of cooperation between complementary exact mathematical optimization techniques, dynamic programming, robust and stochastic optimization, constraint programming, combinatorial algorithms and graph theory, by developing “efficient” algorithms for specific mathematical models. To tackle large-scale real-life applications, providing provably good approximate solutions by combining exact, approximate, and heuristic methods.
$\left(iii\right)$ Software platform & Transfer:
To provide prototypes of modelers and solvers based on generic software tools that build on our research developments, writing code that serves as the proof-of-concept of the genericity and efficiency of our approaches, while transferring our research findings to internal and external users.
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[ "article:topic", "showtoc:no", "balanced", "balanced chemical equation", "not balanced", "coefficients", "subscripts", "odd", "license:ccbysa" ], https://chem.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FIntroductory_Chemistry%2FBook%253A_Introductory_Chemistry_Online_(Young)%2F05%253A_Chemical_Reactions%2F5.3%253A_Balancing_Chemical_Equations, information contact us at [email protected], status page at https://status.libretexts.org. Write your knowns and unknowns, then your develop a plan.) The balanced reaction between sodium metal and oxygen is: {eq}4Na(s) + O_2(g) \to 2Na_2O(s) {/eq} From the balanced reaction, 4 moles of sodium metal reacts to form 2 moles of sodium oxide. However, we can assume that different numbers of reactant molecules or product molecules may be involved. Balancing the hydrogens by inserting “2” in front of H2O2 in the reactants gives us an equation with four hydrogens on both sides on four oxygens on both sides; the equation is now balanced. i) Hydrogen + Chloride —-> Hydrogen chloride. Home. An abbreviated way of using symbols to represent a chemical change. An aqueous solution of barium chloride reacts with an aqueous solution of sodium sulfate to form solid barium sulfate and a solution of sodium chloride. water react to form sodium d. Methane (CHO reacts with excess oxygen to answer. Inspection of this equation, however, shows that, while there is one sodium atom on each side of the arrow, there are two chlorine atoms in the reactants and only one in the products. This equation is not balanced. 1)Carbon monoxide gas reacts oxygen gas to produce carbon dioxide gas. Sodamide is called as Sodium amide and sodium azanide. The simplest methods, where you examine and modify coefficients in some systematic order, is generally called “balancing by inspection”. These methods are generally useful for most simple chemical equations, although mathematical algorithms are often necessary for highly complex reactions. (hint: be sure to calculate theoretical yield first)4Na + O2 - - > Legal. B 2O 3(s) + 6Mg(s) → 3MgO(s) + Mg 3B 2(s) b.
(ii) Magnesium reacts with nitrogen upon heating to form magnesium nitride. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. … Test-Taking Tip Anticipate the guess what you think the answer should be. 2Fe 2 O 3 + 3C → 4Fe + 3CO 2. iii. Image Transcriptionclose. Sodium reacts with oxygen to form sodium oxide and has the following balanced chemical equation: 4 Na + O 2--> 2 Na2O. we might choose to count the carbon atoms first, finding that both sides are balanced with one carbon atom. Booster Classes. (a) 4 P + 5O2 (g) → P4O10 (b) Synthesis 6. What are the parts of a chemical equation? Of all the options above only option listed above, only … We will see examples of this soon. isopropanol + oxygen carbon dioxide and water C 3H 8O + O 2 CO 2 + H Figure 5.1 "The Formation of Water" shows a rather dramatic example of this very reaction. The statement. The diatomic oxygen has a coefficient of 1, which typically is not written but assumed in balanced chemical equations. Answer Save. The substances on the left side of the arrow are called reactantsA substance on the left side of the arrow in a chemical equation., and the substances on the right side of the arrow are called productsA substance on the right side of the arrow in a chemical equation.. write a balanced equation for the reaction of the magnesium and the oxygen, 3(s), reacts when strongly heated with magnesium powder to give a mixture of magnesium oxide and magnesium boride, Mg 3B 2(s). So option D is correct. (Write the balanced equation.) We can also use symbols to represent other words in the reaction. Example: Aluminum metal reacts with oxygen to form solid aluminum oxide. It implies that two water molecules are formed. Na + NH 3 → NaNH 2 + H 2. Now the equation is balanced, and no “trial and error” was required. Sodium metal reacts rapidly with water to form a colourless solution of sodium hydroxide (NaOH) and evolve hydrogen gas (H2). A balanced equation for the decomposition of sodium chlorate to produce sodium chloride and elemental oxygen is 2 NaClO3 -> 2 NaCl + 3 O2. The chemical equation is balanced. Problem: Sodium reacts with oxygen to form sodium oxide and has the following balanced chemical equation: 4 Na + O2 → 2 Na 2O.How many mole(s) of oxygen gas (O2) are needed to react with 2.0 moles of Na?a. is one way to represent that process, which is called a chemical reactionA representation of a chemical change.. Be sure to count all atoms on each side of the chemical equation . CH2 + 2 0, - CO2 + 2 H,O Determine whether each statement describing the reaction is true or false. In a popular classroom demonstration, solid sodium is added to liquid water and reacts to produce hydrogen gas and aqueous sodium hydroxide. (b) Sodium metal reacts with water to form sodium hydroxide and hydrogen gas. By counting, we find two sodium atoms and two oxygen atoms in the reactants and four sodium atoms and two oxygen atoms in the products. Chemical reactions are represented by chemical equations that list reactants and products. (a) FeCl3 +3NaOH →Fe(OH)3 +3NaCI (b) Double Replacement 7. For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. Because elemental chlorine is a diatomic molecule, we need four chlorine molecules to get a total of eight chlorine atoms. This equation is not balanced. The reactants have one carbon atom, four hydrogen atoms, and four oxygen atoms. Use what you know to hydroxide. he carefully weighs out of oxalic acid , a diprotic acid that can be purchased inexpensively in high purity, and dissolves it in of distilled water. There are many different strategies that people use in order to balance chemical equations. Zn reacts with sodium hydroxide to form sodium zincate and the liberation of hydrogen gas occurs. You must use the proper chemical formula of the substance. By counting, we find two sodium atoms and two oxygen atoms in the reactants and four sodium atoms and two oxygen atoms in the products. Add produced Na 2 O 2 to distilled water to produce NaOH and H 2 O 2 aqueous solution. Hydrogen reacts with nitrogen to give ammonia, according to the equation shown below; Zinc metal reacts with aqueous HCl to give hydrogen gas and zinc chloride, according to the equation shown below; Iron(III) oxide reacts with chlorine gas to give iron(III) chloride and oxygen gas, according to the equation shown below; Sodium metal reacts with ammonia to give sodium amide and hydrogen gas, according to the equation shown below; Ethane reacts with oxygen gas to give carbon dioxide and water vapor, according to the equation shown below. How many mole(s) of oxygen gas (O 2) are needed to react with 8.0 moles of Na? Sodium hydroxide reacts with iron (III) nitrate to create a precipitate of iron (III) hydroxide in a solution of sodium nitrate. Switch to. If 5.98g of sodium react with water to form .26g of hydrogen and 10.40g of sodium hydroxide, what mass of water was consumed in the reaction? How many moles of oxygen gas are . The balanced equation for this reaction is: If 12 moles of oxygen react, The reaction consumes moles of glucose (C6H12O6). Then check another element, going back and forth from one side of the equation to another, until each element has the same number of atoms on both sides of the arrow. When sodium reacts with oxygen each sodium atom loses an electron, which means sodium is oxidized, and each oxygen gains two electrons, meaning it's reduced. The law of conservation of matter is satisfied because we now have the same number of atoms of each element in the reactants and in the products. OFeeo 3. _____Na (s) + _____NH 3 (l) → _____H 2 (g) + _____NaNH 2 (s) Ethane reacts with oxygen gas to give carbon dioxide and water vapor, according to the equation shown below; balance this equation. Another example where the “odd-even” approach works well is the decomposition of hydrogen peroxide to yield water and oxygen gas, as shown below. Write an equation expressing the fact that solid potassium and fluorine gas react to make solid potassium fluoride. To make this chemical equation conform to the law of conservation of matter, we must revise the amounts of the reactants and the products as necessary to get the same number of atoms of a given element on each side. What’s the mass of sodium oxide produced if 1.5 g of sodium reacts with excess gaseous oxygen? To balance this reaction means we need to equalize the number of these above atoms and polyatomic ion. Word Equations and Balanced Chemical Equations Translate each of the following word equations into a balanced chemical reactions by writing the correct chemical formulas (including physical states) for the reactants and products. Because every substance has a characteristic chemical formula, we cannot change the chemical formulas of the individual substances. Write a chemical equation to express the fact that hydrogen gas and solid iodine react to make gaseous hydrogen iodide. Write a balanced equation for the following chemical reactions. Since cesium (Z=55) is in the same family as sodium, it reacts in a similar manner and therefore should form similar compounds. Favorite Answer. In this equation, there are two sodiums in the reactants, two sodiums in the products, two chlorines in the reactants and two chlorines in the products; the equation is now balanced. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. A number that gives the number of molecules of a substance in a balanced chemical equation. This point is so important that we should repeat it. What’s the mass of sodium oxide produced if 1.5 g of sodium reacts with excess gaseous oxygen? To create a balanced equation when aluminium reacts with copper (II) sulphate, provide the correct coefficient for each compound. Write a balanced chemical equation that summarizes this reaction. The equation is Al203 + 2NaOH +3H2O -----> 2NaAl(OH)4 Aluminum oxide is amphoteric and is acing as an acid in this case. You know, sodium is a very reactive material and reacts with oxygen to produce sodium oxide. How many mole(s) of oxygen gas (O 2 ) … Zinc hydroxide reacts with phosphoric acid (H 3 PO 4 ) to produce zinc phosphate and water. Balanced equation for the reaction between aluminium oxide and sodium hydroxide solution is : A. Science ... What is the equation for sodium and chlorine react to form sodium chloride? (Hint: consider coefficient ratios. Get the detailed answer: Sodium reacts with oxygen to form sodium oxide and has the following balanced chemical equation. The solid sodium reacts with liquid water to produce molecular hydrogen gas and the ionic compound sodium hydroxide (a solid in pure form, but readily dissolved in water). 2Na + Cl_2 -> 2NaCl The skeletal equation: Na + Cl_2 -> NaCl Balanced: 2Na + Cl_2 -> 2NaCl . write a balanced equation for the reaction of the magnesium and the oxygen, In order to prevent further reactions with Oxygen, it is therefore necessary to clean the ribbon by to remove the layer of MgO. In the left side ,we have 6 oxygen and on the right,we have six oxygen.So the number of oxygen is already balanced.So we are getting the following equation that is ... Sulfuric acid reacts with sodium hydroxide and produce sodium sulphate(Na₂SO₄) and water(H₂O).Here sodium ... 【 】Balanced Chemical Equation for Sodium and Chlor We fix this by putting a 4 in front of the HCl: Now each side has four hydrogen atoms. The products have one carbon atom, four hydrogen atoms, and four oxygen atoms. Write the word equation, formula equation, and balance. A. For instance, perhaps two water molecules are produced, not just one: The 2 preceding the formula for water is called a coefficientA number that gives the number of molecules of a substance in a balanced chemical equation.. Octane (C8H18) reacts with oxygen to make carbon dioxide and water. Choose. The product side has a total of eight chlorine atoms (four from the CCl4 and four from the four molecules of HCl), so we need eight chlorine atoms as reactants. None of the given option is correct. The balanced equation for this reaction is When potassium hydroxide reacts with phosphoric acid, potassium phosphate and water are 3KOH(aq)H3PO4(aq) -» K3PO4 (aq) +3H20(1) If 12 moles of potassium hydroxide react moles of phosphoric acid The reaction consumes moles of potassium phosphate and The reaction produces moles of water Source: Photo courtesy of the US Navy, http://commons.wikimedia.org/wiki/File:Hindenburg_burning.jpg. Aqueous sulfurous acid H2SO3 and aqueous sodium chloride are formed by the reaction of aqueous sodium sulfite Na2SO3 and aqueous hydrochloric acid HCl . Sodium reacts with ammonia gas and produce sodamide and hydrogen gas as products. The above reaction is an example of (a)Oxidation of hydrogen (b)Reduction of oxygen (c)Reduction of hydrogen (d)Redox reaction When chlorine reacts with sodium iodide, sodium chloride and iodine are produced. When sodium metal reacts with oxygen gas to form sodium oxide, a balanced chemical equation will be… On the MathsGee STEM Question and Answer Bank, learners can … Make sure the equation satisfies the law of conservation of matter. You stated that the reaction of Na with O₂ is: 4 Na + O₂ → 2 Na₂O. Write an equation expressing the fact that hydrogen gas and fluorine gas react to make gaseous hydrogen fluoride. Chemical equations need to be balanced to satisfy the law of conservation of matter. Your dashboard and recommendations. Smartie10. Explain why chemical equations need to be balanced. write a balanced equation for the reaction of the magnesium and the oxygen, 3(s), reacts when strongly heated with magnesium powder to give a mixture of magnesium oxide and magnesium boride, Mg 3B 2(s). Aqueous sodium sulfide reacts with aqueous copper(II) nitrate to form aqueous sodium nitrate and solid copper(II) sulfide. Na(s) + O2(g) = NaO2(s) Looking at the first equation that we wrote for the sodium-chlorine reaction, we note that there are an odd number of chlorines in the products and an even number of chlorines in the reactants. This can be easily fixed, however, by putting a coefficient of 2 in front of the diatomic hydrogen reactant: Now we have four hydrogen atoms and two oxygen atoms on each side of the equation. The chemical equation of sodamide preparation is below. 4Na(s)+O2(g)-->2Na2O(s) Write a balanced chemical equation that summarizes this reaction. How does one balance a chemical equation, starting with the correct formulas of the reactants and products? The products have one silver atom, one chlorine atom, one potassium atom, one nitrogen atom, and three oxygen atoms. 2. 2. Write a balanced chemical equation that summarizes this reaction. The balanced equation is , Z n + 2 N a O H → N a 2 Z n O 2 + H 2 . The equation … Sulfuric acid reacts with sodium hydroxide and produce sodium sulphate(Na₂SO₄) and water(H₂O).Here sodium sulphate(Na₂SO₄) is a salt beacause its positive part has come from base and negative part has Na (s) + Cl 2(g) → NaCl ... generate a chemical equation. H₂SO₄ +2NaOH ----->Na₂SO₄ +H₂O. 4.0 moles Chemistry Chemical Reactions Chemical Equations. or below does not react with oxygen. Hydrogen gas and iron(lll) Oxide powder react to form liquid water and solid iron powder. When iron rusts, solid iron reacts with gaseous oxygen to form solid iron (III) oxide. Review and Assessment 381 3.0 molesc. In a popular classroom demonstration, solid sodium is added to liquid water and reacts to produce hydrogen gas and aqueous sodium hydroxide. When hydrogen gas reacts is combined with oxygen gas and the mixture ignited with a spark, water is formed in a violent reaction. 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# Oppenheim's Inequality for triangles, American Mathematical Monthly problems
I did a proof for the inequality below, and I would like know if anyone also has a trigonometric proof for this inequality.If you have a trigonometric demonstration, please post your solution. This problem appeared in the American Mathematical Monthly magazine in 1965, the inequality was proposed in that form by Sir Alexander Oppenheim:
Let $x,y,z$ positive real numbers and $\Delta ABC$ a triangle. $\displaystyle [ABC]$ denotes the triangle area and $\displaystyle a,b,c$ the sides of the triangle. The inequality below is true: $$a^2x+b^2y+c^2z\geq 4[ABC]\sqrt{xy+xz+yz}$$
Various inequalities can be deduced through this inequality, for example, Weitzenböck's inequality, Neuberg-Pedoe inequality, Hadwiger-Finsler inequality, and so on. I'll post my solution right below. $$Proof$$
Let $\alpha,\beta,\gamma$ denote the opposite angles to the sides $a, b, c$, respectively. $R$ is the circumradius of $\Delta ABC$. Observe that: $$a^2x+b^2y+c^2z\geq 4[ABC]\sqrt{xy+xz+yz}$$ $$a^2x+b^2y+c^2z\geq \frac{abc}{R}\sqrt{xy+xz+yz}$$ $$\frac{aRx}{bc}+\frac{bRy}{ac}+\frac{cRz}{ab}\geq \sqrt{xy+xz+yz}$$ $$\frac{1}{2}\left(\frac{4aR^2x}{2Rbc}+\frac{4bR^2y}{2Rac}+\frac{4cR^2z}{2Rab}\right)\geq \sqrt{xy+xz+yz}$$ $$x\frac{\sin\alpha }{\sin\beta \sin \gamma}+y\frac{\sin\beta }{\sin\alpha \sin \gamma}+z\frac{\sin\gamma }{\sin\alpha \sin \beta}\geq 2\sqrt{xy+xz+yz}$$
$$x\frac{\sin(\pi-\alpha) }{\sin\beta \sin \gamma}+y\frac{\sin(\pi-\beta )}{\sin\alpha \sin \gamma}+z\frac{\sin(\pi-\gamma )}{\sin\alpha \sin \beta}\geq 2\sqrt{xy+xz+yz}$$
$$x\frac{\sin(\alpha+\beta+\gamma-\alpha) }{\sin\beta \sin \gamma}+y\frac{\sin(\alpha+\beta+\gamma-\beta )}{\sin\alpha \sin \gamma}+z\frac{\sin(\alpha+\beta+\gamma-\gamma )}{\sin\alpha \sin \beta}\geq 2\sqrt{xy+xz+yz}$$
$$x\frac{\sin(\beta+\gamma) }{\sin\beta \sin \gamma}+y\frac{\sin(\alpha+\gamma )}{\sin\alpha \sin \gamma}+z\frac{\sin(\alpha+\beta )}{\sin\alpha \sin \beta}\geq 2\sqrt{xy+xz+yz}$$
$$x\frac{(\sin\beta \cos\gamma+\sin\gamma \cos \beta) }{\sin\beta \sin \gamma}+y\frac{(\sin\alpha \cos\gamma+\sin\gamma \cos \alpha) }{\sin\alpha \sin \gamma}+z\frac{(\sin\alpha \cos\beta+\sin\beta \cos \alpha) }{\sin\alpha \sin \beta}\geq 2\sqrt{xy+xz+yz}$$
$$(\cot\beta+\cot\gamma)x+(\cot\alpha+\cot\gamma)y+(\cot\alpha+\cot\beta)z\geq 2\sqrt{xy+xz+yz} \tag{1}$$
Since inequality is homogeneous in the variables $x,y,z$, do it $\displaystyle xy+xz+yz=1$ and take the substitution $\displaystyle x=\cot\alpha',y=\cot\beta',z=\cot\gamma'$, we have que $\displaystyle \alpha',\beta',\gamma'$ are angles of a triangle, and our inequality will be equivalent to the inequality below:
$$(\cot\beta+\cot\gamma)\cot\alpha'+(\cot\alpha+\cot\gamma)\cot\beta'+(\cot\alpha+\cot\beta)\cot\gamma'\geq 2 \tag{2}$$ Suppose without loss of generality that (the reverse case is analogous) :
$$\cot\alpha \geq \cot \alpha' \tag{3}$$ $$\cot\beta \geq \cot \beta' \tag{4}$$ $$\cot\gamma'\geq \cot \gamma \tag{5}$$ Because these variables are angles of a triangle, we can not have $\cot \alpha \geq \cot \alpha' , \cot\beta \geq \cot \beta', \cot \gamma\geq \cot\gamma'$.In fact, this can not occur, since it supposes without loss of generality that $\displaystyle \alpha'\geq\alpha$ and $\displaystyle \beta'\geq\beta$(as the cotangent is decreasing, this implies that $\displaystyle \cot\alpha \geq \cot \alpha'$ and $\displaystyle \cot\beta \geq \cot \beta'$), summing up these first two inequalities we have:
$\\ \\ \displaystyle \alpha'+\beta'\geq \alpha+\beta \Rightarrow \cot(\alpha+\beta)\geq \cot(\alpha'+\beta')\Rightarrow -\cot(\pi-\alpha+\beta)\geq- \cot(\pi-\alpha'+\beta') \Rightarrow -\cot(\alpha+\beta+\gamma-(\alpha+\beta))\geq- \cot(\alpha'+\beta'+\gamma'-(\alpha'+\beta')) \Rightarrow -\cot(\gamma)\geq- \cot(\gamma') \Rightarrow \cot(\gamma')\geq \cot(\gamma)\\ \\$
Now set the $\displaystyle f_1(\alpha,\beta,\gamma,\alpha',\beta',\gamma'):\mathbb{R}^6\rightarrow \mathbb{R}$ and $\displaystyle f_2(\alpha,\beta,\gamma,\alpha',\beta',\gamma'):\mathbb{R}^6\rightarrow \mathbb{R}$ such that:
\begin{equation*} f_1(\alpha,\beta,\gamma,\alpha',\beta',\gamma')= \end{equation*} $$(\cot\beta+\cot\gamma)(\cot\alpha'-\cot\alpha)+(\cot\alpha+\cot\gamma)(\cot\beta'-\cot\beta)+(\cot\alpha+\cot\beta)(\cot\gamma'-\cot\gamma) \tag{6}$$
\begin{equation*} f_2(\alpha,\beta,\gamma,\alpha',\beta',\gamma')= \end{equation*} $$(\cot\beta'+\cot\gamma')(\cot\alpha-\cot\alpha')+(\cot\alpha'+\cot\gamma')(\cot\beta-\cot\beta')+(\cot\alpha'+\cot\beta')(\cot\gamma-\cot\gamma') \tag{7}$$
Note now that by inequalities (3), (4) and (5) it follows that:
$$0 \geq \cot\alpha'-\cot\alpha \tag{8}$$
$$0 \geq \cot \beta' -\cot\beta \tag{9}$$
$$\cot\gamma'-\cot\gamma \geq 0 \tag{10}$$ We know that $\displaystyle \alpha',\beta',\gamma'$ are angles of a triangle, so there exists $\displaystyle a',b',c'$ such that $\displaystyle a'^2=b'^2+c'^2-2b'c'\cos\alpha',b'^2=a'^2+c'^2-2a'c'\cos\beta',c'^2=a'^2+b'^2-2a'b'\cos\gamma'$.Let $\displaystyle R'$ the circumradius of the triangle of sides $\displaystyle a',b',c'$.
Let
$\displaystyle k_{\alpha',\beta',\gamma'}:=\frac{a'}{b'c'}+\frac{b'}{a'c'}+\frac{c'}{a'b'}$, therefore:
$$\frac{a'}{b'c'}+\frac{b'}{a'c'}+\frac{c'}{a'b'}=k_{\alpha',\beta',\gamma'} \tag{11}$$
Where $\displaystyle k_{\alpha',\beta',\gamma'}$ is a real variable of any kind.And since our original inequality is homogeneous in the variables a, b, c, suppose without loss of generality that the equality below occurs:
$$\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}=k_{\alpha',\beta',\gamma'} \tag{12}$$
For each real value of fixed $\displaystyle k_{\alpha',\beta',\gamma'}$.Since x, y, z do not depend of the circumradius R ', suppose that $\displaystyle R'\geq R$.Take the inequality (3) and consider the development (applying the law of cosines and law of sines): $\\ \displaystyle \cot\alpha \geq \cot\alpha' \Rightarrow \frac{(b^2+c^2-a^2)R}{abc} \geq \frac{(b'^2+c'^2-a'^2)R'}{a'b'c'} \Rightarrow \left(\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}-2\frac{a}{bc}\right)R\geq \left(\frac{a'}{b'c'}+\frac{b'}{a'c'}+\frac{c'}{a'b'}-2\frac{a'}{b'c'}\right)R' \Rightarrow Rk_{\alpha',\beta',\gamma'}-2\frac{aR}{bc}\geq Rk_{\alpha',\beta',\gamma'}-2\frac{a'R'}{b'c'} \Rightarrow \frac{a'R'}{b'c'}\geq \frac{aR}{bc} \Rightarrow$
$$\frac{a'R'}{b'c'}\geq \frac{aR}{bc} \tag{13}$$
Applying the same rationale for inequality (4), we conclude: $$\frac{b'R'}{a'c'}\geq \frac{bR}{ac} \tag{14}$$
Suppose by contradiction that it occurs:
$$\cot\alpha+\cot\gamma> \cot\alpha'+\cot\gamma' \tag{15}$$
See that:
$\\ \displaystyle \cot\alpha+\cot\gamma> \cot\alpha'+\cot\gamma' \Rightarrow \frac{(b^2+c^2-a^2)R}{abc}+\frac{(a^2+b^2-c^2)R}{abc}> \frac{(b'^2+c'^2-a'^2)R'}{a'b'c'}+\frac{(a'^2+b'^2-c'^2)R'}{a'b'c'} \Rightarrow \frac{bR}{ac}>\frac{b'R'}{a'c'} \\$
This contradicts the inequality (14). On the other hand, suppose by contradiction that it occurs:
$$\cot\beta+\cot\gamma> \cot\beta'+\cot\gamma' \tag{16}$$
See that:
$\\ \displaystyle \cot\beta+\cot\gamma> \cot\beta'+\cot\gamma' \Rightarrow \frac{(a^2+c^2-b^2)R}{abc}+\frac{(a^2+b^2-c^2)R}{abc}> \frac{(a'^2+c'^2-b'^2)R'}{a'b'c'}+\frac{(a'^2+b'^2-c'^2)R'}{a'b'c'} \Rightarrow \frac{aR}{bc}>\frac{a'R'}{b'c'} \\$
$$\cot\alpha+\cot\gamma \leq \cot\alpha'+\cot\gamma' \tag{17}$$
$$\cot\beta+\cot\gamma \leq \cot\beta'+\cot\gamma' \tag{18}$$
Multiplying (17) by $\displaystyle \cot\beta'-\cot\beta$ and multiplying (18) by $\displaystyle \cot\alpha'-\cot\alpha$, note that these inequalities will reverse, since we are multiplying by non-positive quantities, we will have, respectively:
$$(\cot\alpha+\cot\gamma) (\cot\beta'-\cot\beta)\geq (\cot\alpha'+\cot\gamma')(\cot\beta'-\cot\beta) \tag{19}$$
$$(\cot\beta+\cot\gamma) (\cot\alpha'-\cot\alpha)\geq (\cot\beta'+\cot\gamma')(\cot\alpha'-\cot\alpha) \tag{20}$$
On the other hand of inequalities (3) and (4) we know that: $$\cot\alpha+\cot\beta \geq \cot\alpha'+\cot\beta' \tag{21}$$ Multiplying the above inequality by $\displaystyle \cot\gamma'-\cot\gamma$, that by the inequality (10) we know to be greater than or equal to zero, we will have:
$$(\cot\alpha+\cot\beta) (\cot\gamma'-\cot\gamma)\geq (\cot\alpha'+\cot\beta')(\cot\gamma'-\cot\gamma) \tag{22}$$ Adding (19), (20) and (22), we will have:
\begin{equation*} f_1(\alpha,\beta,\gamma,\alpha',\beta',\gamma')\geq \end{equation*} $$(\cot\alpha'+\cot\gamma')(\cot\beta'-\cot\beta)+(\cot\beta'+\cot\gamma')(\cot\alpha'-\cot\alpha)+(\cot\alpha'+\cot\beta')(\cot\gamma'-\cot\gamma) \tag{23}$$
Adding the LHS of (23) with the LHS of (7) and the RHS of (23) with the RHS of (7), the terms will cancel and we will have:
$$f_1(\alpha,\beta,\gamma,\alpha',\beta',\gamma')+f_2(\alpha,\beta,\gamma,\alpha',\beta',\gamma')\geq 0$$ And this implies, finally, that:
$$(\cot\beta+\cot\gamma)\cot\alpha'+(\cot\alpha+\cot\gamma)\cot\beta'+(\cot\alpha+\cot\beta)\cot\gamma'\geq 2$$ That is precisely the inequality (2), which is equivalent to the desired inequality.Thus, the inequality yields.
• What do you mean by "trigonometric proof" if you don't count this as trigonometric? It is hard to tell what the point of including your (long) proof is. – Thomas Andrews Dec 26 '16 at 3:05
• A proof that does not use algebraic inequalities... – Israel Meireles Chrisostomo Dec 26 '16 at 3:09
• What a work since your recent post ! – Jean Marie Dec 26 '16 at 3:44
• @Israel Meireles Chrisostomo There is a smooth and an easy algebraic proof. – Michael Rozenberg Dec 26 '16 at 7:17
• @Michael Rozenberg,The problem of algebraic proof's is the restriction of on sign variable, if you take a trigonometric proof you can see that at least one variable can be negative, since xy+xz+yz are positive. – Israel Meireles Chrisostomo Dec 26 '16 at 7:26
Here is my algebraic proof.
We need to prove that:
$$(a^2x+b^2y+c^2z)^2\geq\sum\limits_{cyc}(2a^2b^2-a^4)(xy+xz+yz)$$ or $$c^4z^2-\left(\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2a^2c^2\right)x+\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2b^2c^2\right)y\right)z+$$ $$+a^4x^2+b^4y^2-\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2a^2b^2\right)xy\geq0,$$ for which it's enough to prove that $$\left(\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2a^2c^2\right)x+\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2b^2c^2\right)y\right)^2-$$ $$-4c^4\left(a^4x^2+b^4y^2-\left(\sum\limits_{cyc}(2a^2b^2-a^4)-2a^2b^2\right)xy\right)\leq0$$ or $$\sum\limits_{cyc}(2a^2b^2-a^4)\left((a^2+c^2-b^2)x-(b^2+c^2-a^2)y\right)^2\geq0.$$ Done!
A "sloppy" but quite straightforward proof. If someone comes up with a simple argument to solve the two points at the end I would be happy!
Writing the expression as $$\frac{a^2x+b^2y+c^2z}{\sqrt{xy+yz+zx}}\geq 4[ABC]$$ the RHS is independent of $x,y,z$, therefore it is necessary and sufficient to prove the inequality in the worst possible case, i.e. when the LHS is minimized in $x,y,z$.
In particular, by homogeneity we can fix $a,b,c$ and consider the problem $$\min\{a^2x+b^2y+c^2z:xy+yz+zx=1,\,x,y,z\geq0\}.$$
If two of $x,y,z$ are zero the inequality is trivially proven. If just one of them is zero, say $z$, then the problem becomes $$\min \{a^2x+b^2y:xy=1,\,x,y\geq 0\}=2ab$$ by AM-GM, and clearly $2ab\geq 2ab\sin\gamma=4[ABC]$.
The last case is $x,y,z>0$. By the Lagrange method we obtain a critical point in the interior where $$(a^2,b^2,c^2)=\lambda(y+z,z+x,x+y)$$ that is \begin{align} &x=b^2+c^2-a^2\\ &y=c^2+a^2-b^2\\ &z=a^2+b^2-c^2 \end{align} up to a multiplicative constant. Substituting above we obtain $$a^2x+b^2y+c^2z=2(a^2b^2+b^2c^2+c^2a^2)-(a^4+b^4+c^4)=16[ABC]^2$$ by Heron's formula, and $$\sqrt{xy+yz+zx}=\sqrt{2(a^2b^2+b^2c^2+c^2a^2)-(a^4+b^4+c^4)}=4[ABC]$$ again by Heron. In particular, in the critical point the equality holds.
The sloppyness comes from the fact that:
1) we don't know the critical point is actually a minimum (or do we?)
2) the infimum could be at infinity, think of $(x,y,z)=(\epsilon,\epsilon,\frac{1-\epsilon^2}{2\epsilon})$ with $\epsilon$ small
Given two triangles $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and positive numbers $x,y,z$.
Let $a_i,b_i,c_i$; $A_i, B_i, C_i$ and $\Delta_i$ be the sides, angles and area of triangle $\triangle A_iB_iC_i$.
We are going to show${}^{\color{blue}{[1]}}$ a two-triangle version of inequality in question. $$\bbox[padding: 1em;border:1px solid blue]{x a_1a_2 + y b_1 b_2 + z c_1 c_2 \ge 4\sqrt{(xy+yz+zx) \Delta_1 \Delta_2}}\tag{*1}$$
Let $u = xa_1a_2$, $v = yb_1b_2$, $w = zc_1c_2$ and $A_{\pm} = A_1 \pm A_2$, $B_{\pm} = B_1 \pm B_2$, $C_{\pm} = C_1 \pm C_2$.
Notice
$$2\Delta_1 = b_1c_1\sin A_1 = c_1a_1\sin B_1 = a_1b_1\sin C_1\\ 2\Delta_2 = b_2c_2\sin A_2 = c_2a_2\sin B_2 = a_2b_2\sin C_2$$ We have \begin{align} {\rm LHS}^2 - {\rm RHS}^2 &= (u + v + w)^2 - 4(uv\sin C_1\sin C_2 + vw \sin A_1\sin A_2 + wu \sin B_1\sin B_2)\\ &= u^2 + v^2 + w^2 + 2(uvW + vwU + wuV ) \end{align} where $U = 1 - 2\sin A_1\sin A_2$, $V = 1 - 2\sin B_1\sin B_2$ and $W = 1 - 2\sin C_1 \sin C_2$. Notice $$U = 1 - 2\sin A_1\sin A_2 = 1 + \cos A_+ - \cos A_- = \cos A_+ + \frac12 \sin^2 \frac{A_-}{2} \ge \cos A_+$$ and similar inequalities $V \ge \cos B_+$, $W \ge \cos C_+$, we obtain
$${\rm LHS}^2 - {\rm RHS}^2 \ge u^2+v^2+w^2 + 2uv\cos A_+ + 2vw \cos B_+ + 2uv\cos C_+\tag{*2}$$
Consider following $3$ vectors in $\mathbb{R}^2$,
$$\vec{u} = (u,0),\quad \vec{v} = (v\cos C_+,v\sin C_+),\quad \vec{w} = (w\cos B_+,-w\sin B_+)$$
It is easy to see $$\vec{u}\cdot\vec{v} = uv \cos C_+\quad\text{ and }\quad\vec{w}\cdot\vec{u} = wu \cos B_+$$ Using the fact $A_+ + B_+ + C_+ = 2\pi$, we find \begin{align}\vec{v}\cdot\vec{w} &= vw (\cos B_+\cos C_+ - \sin B_+\sin C_+)\\ &= vw\cos(B_+ + C_+) = vw\cos(2\pi - A_+) = vw\cos A_+\end{align}
Substitute these back into $(*2)$, we obtain
$${\rm LHS}^2-{\rm RHS}^2 = |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 + 2\vec{u}\cdot\vec{v} + 2\vec{v}\cdot\vec{w} + 2\vec{w}\cdot\vec{u} = |\vec{u} + \vec{v} + \vec{w}|^2 \ge 0$$ From this, inequality $(*1)$ follows.
When $a_1 = a_2 = a, b_1 = b_2 = b, c_1 = c_2 = c$, we have $\Delta_1 = \Delta_2 = [ABC]$.
Inequality $(*1)$ reduces to the desired inequality: $$\bbox[padding: 1em;border:1px solid blue]{ xa^2 + yb^2 + zc^2 \ge 4 \sqrt{xy+yz+zx} [ABC]}$$
Notes
• $\color{blue}{[1]}$ - proof adapted from a chinese book 不等式探秘 (Questing for the Secrets of inequalities) by 李世杰, 李盛 (ISBN 978-7-5603-6228-1).
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## Step-by-step explanation
Problem to solve:
$\lim_{x\to0}\left(\frac{5x}{\ln\left(1-3x\right)}\right)$
Learn how to solve limits problems step by step online.
$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5x\right)}{\frac{d}{dx}\left(\ln\left(1-3x\right)\right)}\right)$
Learn how to solve limits problems step by step online. Evaluate the limit of (5x)/(ln(1-3*x) as x approaches 0. If we try to evaluate the limit directly, it results in indeterminate form. Then we need to apply L'Hôpital's rule. The derivative of the linear function times a constant, is equal to the constant. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. The derivative of a sum of two functions is the sum of the derivatives of each function.
$-\frac{5}{3}$$\,\,\left(\approx -1.6666666666666665\right)$
### Problem Analysis
$\lim_{x\to0}\left(\frac{5x}{\ln\left(1-3x\right)}\right)$
Limits
~ 0.09 seconds
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Home > Burst Error > Burst Error Length
# Burst Error Length
## Contents
What we will get is a λ × n {\displaystyle \lambda \times n} matrix. The methods used to correct random errors are inefficient to correct burst errors. Delay line is basically an electronic circuit used to delay the signal by certain time duration. Flooding – What is flooding? get redirected here
In addition to basic error correction provided by RS codes, protection against burst errors due to scratches on the disc is provided by a cross interleaver.[3] Current compact disc digital audio Coaxial Cable - Write Short Note on Coaxial Cable Twisted-Pair : What is Twisted-Pair Cable? Generated Wed, 05 Oct 2016 01:38:21 GMT by s_hv995 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection Definition.
## Burst Error Correction Codes
Input for the encoder consists of input frames each of 24 8-bit symbols (12 16-bit samples from the A/D converter, 6 each from left and right data (sound) sources). Finally, it also divides: x k − p − 1 = ( x − 1 ) ( 1 + x + … + x p − k − 1 ) {\displaystyle length of burst error is 8, although some bits are unchanged in between. Type of Errors There are two main types of errors in transmissions: 1.
In other words, since burst errors tend to occur in clusters, there is a strong possibility of several binary errors contributing to a single symbol error. Finally one byte of control and display information is added.[5] Each of the 33 bytes is then converted to 17 bits through EFM (eight to fourteen modulation) and addition of 3 At the transmitter, the random interleaver will reposition the bits of the codewords. Burst Error Correcting Codes Ppt What is Ethernet? 100VG-Any LAN What is transmission Baseband?
Transmission Media Transmission Modes - What are the different Transmission Modes? Burst Error Detection And Correction Suppose that we want to design an ( n , k ) {\displaystyle (n,k)} code that can detect all burst errors of length ⩽ ℓ . {\displaystyle \leqslant \ell .} A First we observe that a code can detect all bursts of length ⩽ ℓ {\displaystyle \leqslant \ell } if and only if no two codewords differ by a burst of length Thus, these factors give rise to two drawbacks, one is the latency and other is the storage (fairly large amount of memory).
## Burst Error Correction Using Hamming Code
Each symbol of the alphabet can be represented by m {\displaystyle m} bits. Location of burst - Least significant digit of burst is called as location of that burst. 2. Burst Error Correction Codes Every second of sound recorded results in 44,100×32 = 1,411,200 bits (176,400 bytes) of data.[5] The 1.41 Mbit/s sampled data stream passes through the error correction system eventually getting converted to Burst Error Correction Example Lemma 1.
Thus, the main function performed by the interleaver at transmitter is to alter the input symbol sequence. Get More Info By using this site, you agree to the Terms of Use and Privacy Policy. We now consider a fundamental theorem about cyclic codes that will aid in designing efficient burst-error correcting codes, by categorizing bursts into different cosets. Microwave Transmission – What is a Microwave Transmission? Burst And Random Error Correcting Codes
Convolutional interleaver OR Cross interleaver Cross interleaver is a kind of multiplexer-demultiplexer system. In this case, when the input multiplexer switch completes around half switching, we can read first row at the receiver. Generally, N {\displaystyle N} is length of the codeword. useful reference Facebook Twitter LinkedIn Google+ Link Public clipboards featuring this slide × No public clipboards found for this slide × Save the most important slides with Clipping Clipping is a handy
Say the code has M {\displaystyle M} codewords, then there are M n 2 ℓ − 1 {\displaystyle Mn2^{\ell -1}} codewords that differ from a codeword by a burst of length Burst Error Correcting Convolutional Codes The noise occurring for a longer duration affects multiple bits. This motivates burst error correcting codes.
## In burst error, it is not necessary that only consecutive bits are changed.
Let n be the number of delay lines and d be the number of symbols introduced by each delay line. Proof : Consider two different burst errors e1 and e2 of length l or less which lie in same coset of codeword C. Now suppose e1 is a received vector. Single Bit Error And Burst Error Describe the Congestion Control Algorithm commonly used Types of Routers Data Communication - What is Data Communication?
Single bit error can happen in parallel transmission where all the data bits are transmitted using separate wires. Definitions A burst : Consider a binary representation of length l such that l > 1. Efficiency of block interleaver ( γ {\displaystyle \gamma } ): It is found by taking ratio of burst length where decoder may fail to the interleaver memory. http://onewebglobal.com/burst-error/burst-of-error.php Also a burst error detection simulator is provided.
If 1 ⩽ ℓ ⩽ 1 2 ( n + 1 ) {\displaystyle 1\leqslant \ell \leqslant {\tfrac {1}{2}}(n+1)} is a binary linear ( n , k ) , ℓ {\displaystyle (n,k),\ell What is Transfer rate? ISO Architecture What is Ethernet Frame? Burst error is most likely to occur in a serial transmission.
Assume deg ( d ( x ) ) ≠ 0 , {\displaystyle \deg(d(x))\neq 0,} then p ( x ) = c d ( x ) {\displaystyle p(x)=cd(x)} for some constant O. The number of errors can be obtained by adding e1+e2+....+et Example: First provide the length of the burst ("t") you would like to detect Enter a message word (binary) The resulting Simulation: (The below steps depict the Random Block Interleaver code algorithm): 1.
Following graph represents the result of the simulation.
Observations : We can see from the above that for all the test messages, the bit error rate with interleaver model is Type of Unbound transmission media Optical Fibers: What is a Optical Fibers? Conversely, if h > λ ℓ , {\displaystyle h>\lambda \ell ,} then at least one row will contain more than h λ {\displaystyle {\tfrac {h}{\lambda }}} consecutive errors, and the ( Therefore, k = n − r {\displaystyle k=n-r} for cyclic codes.
We can further revise our division of j − i {\displaystyle j-i} by g ( 2 ℓ − 1 ) {\displaystyle g(2\ell -1)} to reflect b = 0 , {\displaystyle b=0,} Why not share! The basic idea behind the use of interleaved codes is to jumble symbols at the receiver. of CSE Southeast University 4.
Corollary : Let C be an [n, k]-linear l-burst-error-correcting code. Every cyclic code with generator polynomial of degree r {\displaystyle r} can detect all bursts of length ⩽ r . {\displaystyle \leqslant r.} Proof. If one bit has an error, it is likely that the adjacent bits could also be corrupted. Encode message without interleaver 6.
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The Weighted Mean & Median , Screen 6
In the exercise code :
rooms <- houses\$TotRms AbvGrd
rooms <- as.numeric(stringr::str_replace(rooms, ‘10 or more’, ‘10’))
rooms_sorted <- sort(rooms)
Find the median
middle_indices <- c(length(rooms_sorted) / 2,
** (length(rooms_sorted) / 2) + 1**
** ) # 2930 is even so we need two indices.**
middle_values <- rooms_sorted[middle_indices]
median <- mean(middle_values)
If length(vector) is odd, middle indices will be a float . Float indices is not acceptable.
Also, I didn’t understand : middle_values <- rooms_sorted[middle_indices]
Hey there!
Totally understand the confusion here — happy to help clear things up a bit on my end
The purpose of this exercise is to show how the median can be calculated manually if there are an even number if elements in the vector — there’s absolutely more efficient ways of calculating the median which we go over in later missions.
middle_values <- rooms_sorted[middle_indices] indexes the original vector by the two indices that you calculated in the 2 lines prior.
Hope that helps!
1 Like
If length(vector) is odd, we only need the middle index which can be computed as follows avoiding floating index
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## Simplest Calculation of Half-band Filter Coefficients
November 20, 20179 comments
Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency fs and odd symmetry about fs/4 [1]*. And it so happens that almost half of the coefficients are zero. The passband and stopband bandwiths are equal, making these filters useful for decimation-by-2 and interpolation-by-2. Since the zero coefficients make them computationally efficient, these filters are ubiquitous in DSP systems.
Here we will compute half-band...
## There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle
October 23, 20179 comments
Reference [1] has some 3D plots of frequency response magnitude above the unit circle in the Z-plane. I liked them enough that I wrote a Matlab function to plot the response of any digital filter this way. I’m not sure how useful these plots are, but they’re fun to look at. The Matlab code is listed in the Appendix.
This post is available in PDF format for easy...
## Modeling a Continuous-Time System with Matlab
June 6, 20172 comments
Many of us are familiar with modeling a continuous-time system in the frequency domain using its transfer function H(s) or H(jω). However, finding the time response can be challenging, and traditionally involves finding the inverse Laplace transform of H(s). An alternative way to get both time and frequency responses is to transform H(s) to a discrete-time system H(z) using the impulse-invariant transform [1,2]. This method provides an exact match to the continuous-time...
## Canonic Signed Digit (CSD) Representation of Integers
February 18, 2017
In my last post I presented Matlab code to synthesize multiplierless FIR filters using Canonic Signed Digit (CSD) coefficients. I included a function dec2csd1.m (repeated here in Appendix A) to convert decimal integers to binary CSD values. Here I want to use that function to illustrate a few properties of CSD numbers.
In a binary signed-digit number system, we allow each binary digit to have one of the three values {0, 1, -1}. Thus, for example, the binary value 1 1...
## Matlab Code to Synthesize Multiplierless FIR Filters
October 31, 20163 comments
This article presents Matlab code to synthesize multiplierless Finite Impulse Response (FIR) lowpass filters.
A filter coefficient can be represented as a sum of powers of 2. For example, if a coefficient = decimal 5 multiplies input x, the output is $y= 2^2*x + 2^0*x$. The factor of $2^2$ is then implemented with a shift of 2 bits. This method is not efficient for coefficients having a lot of 1’s, e.g. decimal 31 = 11111. To reduce the number of non-zero...
## The Power Spectrum
October 8, 2016
Often, when calculating the spectrum of a sampled signal, we are interested in relative powers, and we don’t care about the absolute accuracy of the y axis. However, when the sampled signal represents an analog signal, we sometimes need an accurate picture of the analog signal’s power in the frequency domain. This post shows how to calculate an accurate power spectrum.
Parseval’s theorem [1,2] is a property of the Discrete Fourier Transform (DFT) that...
## Digital PLL's -- Part 2
June 15, 20162 comments
In Part 1, we found the time response of a 2nd order PLL with a proportional + integral (lead-lag) loop filter. Now let’s look at this PLL in the Z-domain [1, 2]. We will find that the response is characterized by a loop natural frequency ωn and damping coefficient ζ.
Having a Z-domain model of the DPLL will allow us to do three things:
Compute the values of loop filter proportional gain KL and integrator gain KI that give the desired loop natural...
## Digital PLL's -- Part 1
June 7, 201622 comments
1. Introduction
Figure 1.1 is a block diagram of a digital PLL (DPLL). The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal. The loop includes a phase detector to compute phase error and a loop filter to set loop dynamic performance. The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.
One application of the DPLL is to recover the timing in a digital...
## Peak to Average Power Ratio and CCDF
May 17, 20162 comments
Peak to Average Power Ratio (PAPR) is often used to characterize digitally modulated signals. One example application is setting the level of the signal in a digital modulator. Knowing PAPR allows setting the average power to a level that is just low enough to minimize clipping.
However, for a random signal, PAPR is a statistical quantity. We have to ask, what is the probability of a given peak power? Then we can decide where to set the average...
## Filter a Rectangular Pulse with no Ringing
May 12, 201610 comments
To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients: they must all be positive. However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical. What we are describing is basically a window function.
Consider a rectangular pulse 32 samples long with fs = 1 kHz. Here is the Matlab code to generate the pulse:
N= 64; fs= 1000; % Hz sample...
## Filter a Rectangular Pulse with no Ringing
May 12, 201610 comments
To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients: they must all be positive. However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical. What we are describing is basically a window function.
Consider a rectangular pulse 32 samples long with fs = 1 kHz. Here is the Matlab code to generate the pulse:
N= 64; fs= 1000; % Hz sample...
## There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle
October 23, 20179 comments
Reference [1] has some 3D plots of frequency response magnitude above the unit circle in the Z-plane. I liked them enough that I wrote a Matlab function to plot the response of any digital filter this way. I’m not sure how useful these plots are, but they’re fun to look at. The Matlab code is listed in the Appendix.
This post is available in PDF format for easy...
## Model Signal Impairments at Complex Baseband
December 11, 20192 comments
In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise. To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code. Although our example system is QAM, the models can be used for any complex-baseband signal.
I used a very simple complex-baseband model of a QAM system in my last
## A Direct Digital Synthesizer with Arbitrary Modulus
June 3, 20195 comments
Suppose you have a system with a 10 MHz sample clock, and you want to generate a sampled sinewave at any frequency below 5 MHz on 500 kHz spacing; i.e., 0.5, 1.0, 1.5, … MHz. In other words, f = k*fs/20, where k is an integer and fs is sample frequency. This article shows how to do this using a simple Direct Digital Synthesizer (DDS) with a look-up table that is at most 20 entries long. We’ll also demonstrate a Quadrature-output DDS. A note on...
## Design Square-Root Nyquist Filters
July 13, 2020
In his book on multirate signal processing, harris presents a nifty technique for designing square-root Nyquist FIR filters with good stopband attenuation [1]. In this post, I describe the method and provide a Matlab function for designing the filters. You can find a Matlab function by harris for designing the filters at [2].
Background
Single-carrier modulation, such as QAM, uses filters to limit the bandwidth of the signal. Figure 1 shows a simplified QAM system block...
## Compute Images/Aliases of CIC Interpolators/Decimators
November 1, 20202 comments
Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators. For these filters, all coefficients are equal to 1, and there are no multipliers. They are typically used when a large change in sample rate is needed. This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and the aliases of CIC decimators.
1. CIC Interpolators
Figure 1 shows three interpolate-by-M...
## Third-Order Distortion of a Digitally-Modulated Signal
June 9, 2020
Analog designers are always harping about amplifier third-order distortion. Why? In this article, we’ll look at why third-order distortion is important, and simulate a QAM signal with third-order distortion.
In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude. With this assumption, the output y of a non-ideal amplifier can be written as a power series of the input signal x:
y=...
## Find Aliased ADC or DAC Harmonics (with animation)
January 11, 2021
When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics. If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias. In this case, it is not at once obvious if a given spur is a harmonic, and if so, its order. In this article, we’ll present Matlab code to simulate the data converter nonlinearities and find the harmonic alias frequencies. Note that Analog Devices has an online tool for...
## Add a Power Marker to a Power Spectral Density (PSD) Plot
February 7, 2021
Perhaps we should call most Power Spectral Density (PSD) calculations relative PSD, because usually we don’t have to worry about absolute power levels. However, for cases (e.g., measurements or simulations) where we are concerned with absolute power, it would be nice to be able to display it on a PSD plot. Unfortunately, you can’t read the power directly from the plot. For example, the plotted spectral peak of a narrowband signal, such as a sinewave, is lower than the...
## Setting Carrier to Noise Ratio in Simulations
April 11, 2021
When simulating digital receivers, we often want to check performance with added Gaussian noise. In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N). I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.
Definition of C/N
The Carrier to noise ratio is defined as the ratio of average signal power to noise power for a modulated...
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# How to normalize by the covariance matrix? [duplicate]
I am trying to understand an image processing research paper [1] that calls for normalizing a distance between an object's center point and the center of a cluster of points by the covariance matrix of the cluster.
Given
$m^f_c$ - center of cluster $c$ at frame $f$ (2 dimensional $x,y$ point)
$m^f_t$ - center of cluster $t$ at frame $f$ (2 dimensional $x,y$ point)
$\Psi_c$ - covariance of cluster c
For each object $t$, the distance to each cluster $c$ is calculated $D_{ct}=\frac{1}{f}\sum_f(m^f_c-m^f_t)(\Psi^f_c)^{-1}(m^f_c-m^f_t)^T$
The paper says "It is important to normalize the cluster-to-[object] distance by the covariance matrix of the cluster to handle the scale variation of objects"
What is meant by "normalize by the covariance matrix"? I understand normalization by a scalar, but by a covariance matrix is confusing me.
[1] Anjulan, A., & Canagarajah, N. (2009). A Unified Framework for Object Retrieval and Mining. IEEE Transactions on Circuits and Systems for Video Technology, 19(1), 63–76. doi:10.1109/TCSVT.2008.2005801
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## Algebra 2 (1st Edition)
The student should have gotten a zero on the right side of the equation before factoring. Done correctly, the solution should be: $$x^2+7x-8=0 \\ (x+8)(x-1) = 0 \\ x=1,-8$$
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Here’s another set of papers! These ones are all about utilizing the tremendous amount of monolingual corpora out there. One of my interests here is in applying them to aid our comprehension across languages.
1. Common Representation Learning is the task of embedding different descriptions of data into a common subspace. An example is two sentences in different languages that mean the same thing. It would be good to reliably be able to associate one with the other without explicitly telling a machine that, for example, “cats” and “dogs” are alike.
This paper does this a deep autoencoder model with a well-considered objective function. It then applies this model to tasks like cross-language document classification where a document is represented by the tf-idf sum of its bag of words representation. Here’s the objective function it’s trying to minimize:
Above, L is the reconstruction loss. They use cross-entropy error if the input takes binary values and squared error loss otherwise. z_i is the paired inputs from the different descriptions (sentences), (x_i, y_i), and h(z) = f(Wx + Vy + b), where f is a non-linear activation function. W and V are the learned projection matrices for the different descriptions, and b is a learned bias vector. Then z' = g([W'h(z), V'h(z)] + b') is the reconstruction that we compare to z while learning reconstruction matrices W', V', and b'. Function g is any activation function. Note that [a, b] is the concatenation of vectors a and b.
The sum is made of four parts. The first part, L(z_i, g(h(z_i))), teaches the autoencoder to reconstruct its input (both languages). The second and third terms train it to reconstruct the input from just one description. So, if we are training on French and English, it must learn to output the French language with just the English sentence. These three together imply that the model learns to embed the sentences in the same vector space. The fourth term maximizes the correlation between the vector embeddings of similar sentences across languages. This constrains the vectors in different languages to be close to each other. Pretty slick addition.
An important note is that the inputs are sentence-level binary bags of words. This means that word order is not taken into effect and changing this to include word order would be a reasonable next thing to try.
Overall, I really like this idea. It’s simple but makes sense and the results suggest that it’s working well for document classification. They are able to assemble deep models and can scale up the language representation with lots of monolingual data without compromising the model.
2. A couple of months before the above paper, this one achieved even better results on document classification with a different method and a slightly different objective.
The goals are similar. Can a single method:
• Contrain the word-level representations to be compositional?
• Leverage both monolingual and bilingual data.
• Scale to large vocabulary sizes without greatly impacting time.
While very favorable towards the [Chandar et al] paper, they describe a deficiency:
[Chandar et al]’s method represents each sentence as a bag of words vector with the size of the whole vocabulary. This leads to computational scaling issues and necessitates a vocabulary cut-off which may hamper performance for compounding languages such as German.
Accordingly, their neural network representing sentences as an addition composition over word vectors can be trained on seven million sentences in under six hours on a single core CPU. They did show a composition function using bigrams to somewhat consider word order, but left it as future work:
To increase the expressiveness of our method we plan to investigate more complex composition functions, possibly based on convolution to preserve word order information.
I mentioned a different objective as well. Here it is:
The first term is the bilingual objective. It’s calculated by taking aligned sentences, computing their vector representation with the composition function, and then minimizing the squared euclidean distance between them:
The monolingual objective is very different. It’s based on a notion that phrases are more similar to their sub-phrases than to randomly sampled phrases. For example, if the phrase is My friend went to the park after getting ready, then the sub-phrase went to the park is more relevant than took a shower. Given a large enough monolingual corpus, another plausible sub-phrase like went to the movies would also appear and be embedded close to the larger phrase. This would consequently embed it close to went to the park. Here is the loss function that achieves this:
Above, m is a margin, the outer a is a phrase sampled from a sentence, the inner a is a sub-phrase of the outer, and the noise b is a phrase extracted from a sentence uniformly sampled from the corpus. All extracted phrases have length larger than three.
3. This work was inspired by Mikolov et al’s paper4 in 2013. It abstracts the skip-gram model for word embeddings to the sentence-level. The result are generic sentence representations that perform robustly across a wide variety of tasks, including semantic relatedness, paraphrase detection, image-sentence ranking, and classification.
How does it induce the vectors? Take a large corpus of contiguous sentences (e.g. books). For each consecutive sentence tuple (x, y, z), use an RNN with GRUs to encode the midde sentence, y. Now use an RNN (with GRU) to decode this encoding and try to predict x, the sentence before. Do the same thing with another decoder for predicting z, the sentence after. These two decoders have separate weight matrices but share the weight matrix for the vocabulary.
The objective function is the sum of the log-probabilities of the previous and next sentences conditioned on the encoder representation:
Another nice takeaway from this paper was that they were able to expand the vocabulary for the language model from 20k words to 930k words using pre-trained word2vec representations. This was done by first developing the RNN as above and then finding a linear mapping matrix W where v' ~ Wv for v in the word2vec vocabulary and v' in the RNN vocabulary. Now any word represented in word2vec can also be mapped to the RNN.
1. Chandar et al, University of Montreal
2. Soyer et al, National Institute of Informatics, Tokyo, Japan
3. Kiros et al, University of Toronto
4. Google, Efficient Estimation of Word Representations in Vector Space
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# Whats Special about this Crop Circle
This crop circle has hidden mathematical meaning, Can you find it out?
Note by Vikram Pandya
3 years, 3 months ago
## Comments
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Starting from the innermost circle, each expansion of the radius occurs after a number of tenths of a full circle that encodes the decimal representation of π. · 3 years, 3 months ago
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but many people says that they are hoax. · 3 years, 3 months ago
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I found this article: Crop Circles · 3 years, 3 months ago
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Can you summarize? Staff · 3 years, 3 months ago
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Its about value of PI. Every bumped cut in crop circle represents a digit of PI. It gives PI upto 10 digits · 3 years, 3 months ago
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Yes that 's right. Interesting isn't it? · 3 years, 3 months ago
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# f(x+C)/f(x) = const identifies potentiation?
I couldn't find this on the Internet.
We know this property of potential function $f(x) = k \cdot a^x$:
$\displaystyle \frac{f(x+C)}{f(x)} = P$ for any real $x$, constant $C$, and constant $P$.
Can we state the reverse?
If $\displaystyle \frac{f(x+C)}{f(x)} = P$ for any real $x$, constant $C$, and constant $P$, is $f(x) = k \cdot a^x$? If not, what are the other solutions?
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your "const" is not constant, it is $C$-dependent. – Carlo Beenakker Nov 10 '13 at 15:06
If $C$ and $P$ fixed constants, then there are infinitely many functions satisfying your condition. E.g., for $P=1$ and $C$ arbitrary, take $f(x)=\sin(2\pi x/C) + d$, with any $d>1$ (the $d$ is there to make sure $f$ has no zeros). In fact, take any function on the interval $[0,C]$ with $f(C)=Pf(0)$ (and no zeros), then your condition allows to extend it to all of $R$. – UwF Nov 10 '13 at 20:04
## migrated from mathoverflow.netNov 10 '13 at 15:18
This question came from our site for professional mathematicians.
If you do not add any regularity condition (such as monotonicity, continuity etc.), then of course, there plenty of counterexamples. Take $f(x)=e^{g(x)}$ where $g(x)$ is an additive function that is not continuous. Then, $$\frac{f(x+c)}{f(x)}=e^{g(x+c)-g(x)}=e^{g(c)}.$$ Once you add some regularity one can get an effective description.
Added later: Since the OP specified that we are interested in continuous positive functions, here is the proof that all such functions are of the form $f(x)=ke^{bx}.$ Indeed, since $f>0$ we can let $f(x)=e^{g(x)}.$ Our functional equation can be rewritten as $$e^{g(x+c)-g(c)}=e^{h(c)},$$ where $e^{h(c)}$ is the constant that depends on $c.$ In other words, $$g(x+c)=g(x)+h(c).$$
The rest depends on whether your $c$ is fixed or you let it vary. If $c$ is fixed there are plenty of counter examples. Namely, take $h(c)=0$ and let $g(x)$ to be any periodic function with period $c.$
Now, if your equality holds for all $c,$ then we have $$g(x+y)=g(x)+h(y)$$ for all $x,y\in\mathbb{R}.$ Taking $x=0$ we get $g(y)=g(0)+h(y)$ and thus $$g(x+y)=g(x)+g(y)-g(0).$$
Introduce, $g_1(x)=g(x)- g(0),$ and note that the last equation implies $$g_1(x+y)=g_1(x)+g_1(y).$$
This is Cauchy functional equation and once you have continuity, it implies that $g_1(x)=kx.$
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thank you very much. Well, $f(x)$ should be both continuous and monotonically increasing. Besides, it denotes frequency, so it will always be non-negative. – t5d Nov 10 '13 at 17:03
then, just reverse the the previous example. Namely, you get that $g$ is continuous and thus linear. – Alvin Nov 10 '13 at 17:12
thanks again! I understand, that I asked that incorrectly, but: is the equation $f(x+C)/f(c) = const$ an indicator that $f(x)$ is exponential function? And if it is, then why? – t5d Nov 10 '13 at 17:14
In ohter words, I know that exponential functions match this condition, and I need to prove, e.g. that it is the only family of functions that match, or if some other family of functions match, I need to prove that only exponential and this family of functions match, etc. – t5d Nov 10 '13 at 17:26
Thank you! That is just what I really needed. I have found that this solution is the only one for rational numbers. Can we state the same for real and complex numbers? – t5d Nov 12 '13 at 1:31
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## Electronic Journal of Probability
### Clustering Behavior of a Continuous-Sites Stepping-Stone Model with Brownian Migration
Xiaowen Zhou
#### Abstract
Clustering behavior is studied for a continuous-sites stepping-stone model with Brownian migration. It is shown that, if the model starts with the same mixture of different types of individuals over each site, then it will evolve in a way such that the site space is divided into disjoint intervals where only one type of individuals appear in each interval. Those intervals (clusters) are growing as time $t$ goes to infinity. The average size of the clusters at a fixed time $t$ is of the order of square root of $t$. Clusters at different times or sites are asymptotically independent as the difference of either the times or the sites goes to infinity.
#### Article information
Source
Electron. J. Probab., Volume 8 (2003), paper no. 11, 15 p.
Dates
First available in Project Euclid: 23 May 2016
https://projecteuclid.org/euclid.ejp/1464037584
Digital Object Identifier
doi:10.1214/EJP.v8-141
Mathematical Reviews number (MathSciNet)
MR1986843
Zentralblatt MATH identifier
1064.60074
#### Citation
Zhou, Xiaowen. Clustering Behavior of a Continuous-Sites Stepping-Stone Model with Brownian Migration. Electron. J. Probab. 8 (2003), paper no. 11, 15 p. doi:10.1214/EJP.v8-141. https://projecteuclid.org/euclid.ejp/1464037584
#### References
• Cox, J.T., Durrett, R. and Perkins, E.: Rescaled voter models converge to super-Brownian motion. Ann. Probab., 28, 185–234 (2000).
• Cox, J.T. and Griffeath, D.: Occupation time limit theorems for the voter model. Ann. Probab., 11, 876–893 (1983).
• Dawson, D.A., Greven, A. and Vaillancourt, J.: Equilibria and quasi–equilibria for infinite collections of interacting Fleming–Viot processes. Trans. Amer. Math. Soc., 347, 2277–2360 (1995).
• Donnelly, P., Evans, S.N., Fleischmann, K., Kurtz T.G. and X. Zhou, X.: Continuum-sites stepping-stone models, coalescing exchangeable partitions, and random trees. Ann. Probab., 28, 1063–1110 (2000).
• Evans, S.N.: Coalescing Markov labeled partitions and continuous sites genetics model with infinitely many types. Ann. Inst. H. Poincaré Probab., 33, 339–358 (1997).
• Evans, S.N. and Fleischmann, K.: Cluster formation in a stepping stone model with continuous, hierarchically structured sites. Ann. Probab., 24, 1926–1952 (1996).
• Fleischmann, K. and Greven, A.: Time-space analysis of the cluster-formation in interacting diffusions. Electron. J. Probab., 1 (1994).
• Handa, K.: A measure-valued diffusion process describing the stepping stone model with infinitely many alleles. Stochastic Process. Appl., 36, 269–296 (1990).
• Kimura, M.: "Stepping-stone" models of population. Technical report 3, Institute of Genetics, Japan, 1953.
• Klenke, A.: Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab., 24, 660–697 (1996).
• Liggett, T.M.: Interacting Particle Systems. New York: Springer-Verlag, 1985.
• Müller, C. and Tribe, R.: Stochastic p.d.e.'s arising from the long range contact and long range voter processes. Probab. Th. Rel. Fields, 102, 519–546 (1995).
• Sawyer, S.: Results for the stepping stone models for migration in population genetics. Ann. Probab., 4, 699–728 (1976).
• Shiga, T.: Stepping stone models in population genetics and population dynamics. In S. Albeverio et al., ed, Stochastic Processes in Physics and Engineering, Mathematics and Its Applications, 345–355. D. Reidel Publishing Company, 1988.
• Walters, P.: An introduction to ergodic theory. New York: Springer-Verlag, 1982.
• Zhou X.: Duality between coalescing and annihilating Brownian motions. Preprint.
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# What are the orbits of tensor product under the action of a product of groups?
Suppose that group $$G_i$$ acts on finite dimensional complex vector space $$V_i$$ with finite number of orbits for $$i=1,2$$. Let $$G=G_1\times G_2, V=V_1\otimes_{\mathbb C} V_2$$ and let $$G$$ act on $$V$$ by $$(g_1,g_2).v_1\otimes v_2=(g_1v_1)\otimes (g_2v_2)$$. [Edit: Here a $$G$$ action on $$V$$ means a linear representation.] Is there a general way to describe the orbits of the group action of $$G$$ on $$V$$ in terms of the orbits of the $$G_i$$ action on $$V_i$$? (Edit: if it is easier, one can assume the representation $$V_i$$ of $$G_i$$ is irreducible.)
If the above question has no general answer, here is a more concrete one. Suppose that $$G_1=GL_2(\mathbb C)$$ and $$V_1=\mathbb C^2$$. Suppose that $$G_2$$ acts on a complex vector space $$V_2$$ with 4 orbits $$0, C_1,C_2,C_3$$, then is there a way to describe the orbits of the natural $$G_1\times G_2$$ action on $$V_1\otimes_{\mathbb C}V_2$$ in terms of $$C_1,C_2,C_3$$?
The following are some efforts I tried for the concrete example. The space $$V_1\otimes V_2$$ can be interpreted as $$Hom_{\mathbb C}(V_1^*,V_2)\cong M_{2\times n}(\mathbb C)$$, where $$n=dim V_2$$. The action of $$G_1\times G_2$$ on $$Hom(V_1^*,V_2)$$ is given by $$(g_1,g_2).f=g_2fg_1^{-1}.$$ We can consider the ranks of $$f$$. If $$rk(f)=0,$$ there is only one orbit. If $$rk(f)=1$$, there are 3 orbits depending on the image of $$f$$. But I don't know what happened if $$rk(f)=2$$. I am very confused about the last case. I am wondering if there is a general method to solve this kind problem.
• Do you know that the action of $G_1\times G_2$ on $V_1\otimes V_2$ is well defined? Dec 7, 2019 at 7:07
• @OliverJones The tensor product representation of $G_1 \times G_2$ from representations of $G_1$ and $G_2$ is a standard construction in representation theory. Why are querying it? But I am not sure whether you can enumerate the orbits of its action just from a knowledge of the numbers of the orbits of $G_1$ and $G_2$. Dec 8, 2019 at 11:56
• @QingZhang The number of orbits is irrelevant. What complicates the description of an orbit is the action of $G_1\times G_2$ on the indecomposable vectors in $V_1\otimes V_2$. Dec 10, 2019 at 0:05
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# Positivity of a rational function
A rational function is called positive if all its Taylor coefficients are positive. Friedrichs-Lewy conjecture states the positivity of the rational function \begin{eqnarray*}\frac{1}{ (1-x)(1- y)+(1- y)(1-z)+(1-z)(1-x)} = \sum\limits_{ k,m,n\ge0} a_{k,m, n }x^k y^mz^n. \end{eqnarray*} The conjecture was first proved by G. Szego.
Let $P_n=\prod\limits_{i=1}^n(1-x_i)$, is it ture that the following rational function is positive \begin{eqnarray*}\frac{1}{ \sum\limits_{i=1}^n\frac{P_n}{1-x_i}}=\sum\limits_{i_1,i_2\cdots, i_n\ge 0}a_{i_1,i_2\cdots, i_n }x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}? \end{eqnarray*}
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Posts Tagged ‘Triangle Inequality’
A Small Substitution. Goes a Long Way!
Now, many of us (including me) are really troubled by some geometric inequalities which seem more difficult than many algebraic inequalities. But actually, as we shall see later, many of them are even weaker than AM-GM inequality. Now, many questions arise when I say this, but I hope those will be clarified as you read further.
Also, in a triangle ,
Now, here comes our main substitution
In a triangle, Let
So, we have .
Substituting these in our earlier identities, we get
These little ideas really do go a very long way in proving many geometric inequalities.
We begin with the Hadwiger-Finsler Inequality.
In a triangle ,
By the extended law of sines, we have
Also,
Now, putting these values in our inequality, we get
HERE COMES THE SUBSTITUTION!Performing our substitution, we get
Now,
which is trivial and follows from
Now we solve another geometric inequality by the same method, but we have to do some work before employing it. This inequality was proposed to me by Aravind Srinivas L.
Given triangle with cevians not necessarily concurrent, lie on the sides respectively. Suppose are extended to meet the circumcircle of triangle at respectively. Prove that:
Note that , where denotes area.
Therefore, we only have to prove,
Now, it is well known that area of a triangle maximizes (given that it is on a fixed base and circle and the third point lies on a particular arc i.e. major arc or minor arc) when the third point is the midpoint of the arc.
Therefore, it suffices to prove the given statement when are the midpoints of the arcs(minor) respectively. This happens if and only if are the angle bisectors of .
Therefore,
Also, .
Notice also that have the same circumradius. Let it be R.
Therefore, since area of any
and similarly for others.
Thus our inequality becomes ready to be killed with our substitution. It becomes,
Thus, we can apply our substitution now. (if you don’t know how to go about the half angle sines then use the formula )
Employing our substitution, we get
Now, taking , this turns into
For positive reals with ,
which is an easy application of Cauchy-Schwarz Inequality.
Indeed,
due to Cauchy-Schwarz and Nesbitt’s inequality.
(Alternately you may try doing it by Jensen’s too!)
So…we are done
Now we do another problem. More harder. But easy with this method.
(Source: Vietnam TST 1993 for the 34nd IMO, problem 4)
In a triangle , if are the orthocentre, circumcentre, incentre respectively, then prove that
Now, rearranging and dividing throughout by we get,
(where )
Since it is well known that , now we can apply our substitution !
Now, and so on to get
Since we also know how to substitute for , so our inequality becomes,
Note that we’ve used the identity: . Now we also use
Which holds by Newton’s Inequality. .
We are done!
The last question I’m gonna do is not an inequality but an identity. This shows that we can prove trigonometrical identities also with it.
is acute and denote its altitudes. denote the radii of its circumcircle, incircle and incircle of its orthic triangle (whose vertices are the feet of its altitudes). Prove the relation:
where are the feet of altitudes.
Now, using , we get
,
Now, using the well known formulas, , , , and , our equation reduces to
Now, let
Therefore , and
Plugging these into the equation, bringing it to a common denominator, it reduces to
After some (not SO tedious) expansion(Actually, I had written it, but it wasn’t coming out fully when I previewed it )…..as many terms cancel out…we get,
Which is true.
So, we are done!
So….we can see how this opens up various new possibilities….
One can simply pick up a strong geometric inequality in which this inequality can be employed….and then make this substitution to turn it into an algebraic inequality….
Ok, So, here are some exercises to make you further in touch with this useful method.
Prove that in triangle
In triangle , let be the points where the incircle touches the sides. Let be the inradius, circumradius, semiperimeter, respectively, of the triangle. Prove that
and determine when equality occurs.
(MOP 1998) Let be an acute triangle with circumcentre , orthocentre and circumradius . Show that for any point on the segment ,
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Definition:Set Equality
Definition
Let $S$ and $T$ be sets.
Definition 1
$S$ and $T$ are equal if and only if they have the same elements:
$S = T \iff \paren {\forall x: x \in S \iff x \in T}$
Definition 2
$S$ and $T$ are equal if and only if both:
$S$ is a subset of $T$
and
$T$ is a subset of $S$
Axiom of Extension
The concept of set equality is axiomatised as the Axiom of Extension in the axiom schemata of all formulations of axiomatic set theory:
Let $A$ and $B$ be sets.
The axiom of extension states that $A$ and $B$ are equal if and only if they contain the same elements.
That is, if and only if:
every element of $A$ is also an element of $B$
and:
every element of $B$ is also an element of $A$.
This can be formulated as follows:
$\forall x: \paren {x \in A \iff x \in B} \iff A = B$
Equality as applied to Classes
In the context of class theory, the same definition applies:
Let $A$ and $B$ be classes.
Definition 1
$A$ and $B$ are equal, denoted $A = B$, if and only if:
$\forall x: \paren {x \in A \iff x \in B}$
where $\in$ denotes class membership.
Definition 2
$A$ and $B$ are equal, denoted $A = B$, if and only if:
$A \subseteq B$ and $B \subseteq A$
where $\subseteq$ denotes the subclass relation.
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Kattis
The Shibuya scramble crossing in Tokyo is infamous for being heavily used, resulting in people bumping into each other. The crossing can be modeled as a convex polygon, where the $n$ people about to cross initially stand at a point that is on the perimeter of the polygon and in its lower half. When the traffic lights change, each person starts to walk towards a unique point on the perimeter in the upper half of the polygon. The path each person takes may look like spaghetti (it may even cross itself), but it will never leave the polygon and no two paths will cross more than once.
Oskar who is a badass geek observes the crossing from the Starbucks nearby. He has numbered the people in the crossing consecutively $1$ through $n$ in counter-clockwise order (starting with the person at the very left). Sadly he doesn’t know the intended paths of the people at the crossing, but he has gathered some intelligence telling him exactly which persons’ paths will cross one another (and this information is consistent with the physical reality).
Being a nerd he obviously knows about Murphy’s Law saying “Anything that can go wrong, will go wrong!”. So all people who could possibly bump into each other, i.e., all people whose paths cross, will actually bump into each other! He now asks himself, “After all the $n$ people have crossed, what is the size of the largest group of people where everyone has bumped into each other?”. Now that is a geeky and tough question, can you help him?
Input
The first line contains an integer $1 \leq n \leq 800$, the number of people at the crossing, and an integer $0 \leq m \leq 10\, 000$, the number of paths that will cross, i.e., intersect one another. The next $m$ lines each contain two integers $a$ and $b$, $1 \leq a < b \leq n$, meaning that the path taken by person $a$ will cross the path taken by person $b$. (No pair will occur twice in the input.)
Output
Output a single integer giving the size of the largest group of people where everyone has bumped into one another.
Sample Input 1 Sample Output 1
3 1
1 2
2
Sample Input 2 Sample Output 2
5 7
1 3
1 5
1 4
2 4
3 4
2 5
2 3
3
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# Python
## Computing the CRC8-ATM CRC in Python
The 8-bit CRC8-ATM polynomial is used in many embedded applications, including Trinamic UART-controlled stepper motor drivers like the TMC2209:
$$\text{CRC} = x^8 + x^2 + x^1 + x^0$$
The following code provides an example on how to compute this type of CRC in Python:
def compute_crc8_atm(datagram, initial_value=0):
crc = initial_value
# Iterate bytes in data
for byte in datagram:
# Iterate bits in byte
for _ in range(0, 8):
if (crc >> 7) ^ (byte & 0x01):
crc = ((crc << 1) ^ 0x07) & 0xFF
else:
crc = (crc << 1) & 0xFF
# Shift to next bit
byte = byte >> 1
return crc
This code has been field-verified for the TMC2209.
Posted by Uli Köhler in Algorithms, Embedded, MicroPython, Python
## MicroPython ESP32 minimal UART example
This example shows how to use UART on the ESP32 using MicroPython. In this example, we use UART1 which is mapped to pins GPIO9 (RX) and GPIO10 (TX).
from machine import UART
uart = UART(1, 115200) # 1st argument: UART number: Hardware UART #1
# Write
uart.write("test")
Don’t know how to upload the file to MicroPython so it is automatically run on boot?
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to fix ESP32 MicroPython ‘ValueError: pin can only be input’
### Problem:
You are trying to initialize an ESP32 pin in MicroPython using
import machine
machine.Pin(34, machine.Pin.OUT)
but you see an error message like
>>> machine.Pin(34, machine.Pin.OUT)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: pin can only be input
### Solution:
On the ESP32, pins with numbers >= 34 are input-only pins!
You need to use other pins < 34 if you need output capability!
For reference, see the relevant MicroPython source code section:
// configure mode
if (args[ARG_mode].u_obj != mp_const_none) {
mp_int_t pin_io_mode = mp_obj_get_int(args[ARG_mode].u_obj);
if (self->id >= 34 && (pin_io_mode & GPIO_MODE_DEF_OUTPUT)) {
mp_raise_ValueError("pin can only be input");
} else {
gpio_set_direction(self->id, pin_io_mode);
}
}
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to upload files to MicroPython over USB/serial
In this post we will investigate how to connect to a wireless network on boot
First, install ampy – a tool to modify the MicroPython filesystem over a serial connection.
sudo pip3 install adafruit-ampy
Now prepare your script – we’ll use main.py in this example.
Upload the file to the board:
ampy -p /dev/ttyUSB* put main.py
This only takes about 2-4 seconds. In case ampy is still running after 10 seconds, you might need to
• Stop ampy (Ctrl+C), reset the board using the RESET button and retry the command
• Stop ampy (Ctrl+C). Detach USB and ensure the board is powered off (and not powered externally). Re-Attach USB and retry the command.
• In case that doesn’t help, try re-flashing your board with the most recent version of MicroPython. See How to flash MicroPython to your ESP32 board in 30 seconds. This will also clear the internal filesystem and hence remove any file that might cause failure to boot properly.
Posted by Uli Köhler in Embedded, MicroPython, Python
This MicroPython code blinks GPIO2 which is connected to the LED on most ESP32 boards.
import machine
import time
led = machine.Pin(2, machine.Pin.OUT)
while True:
led.value(1)
time.sleep(1)
led.value(0)
time.sleep(1)
Don’t know how to upload the file to MicroPython so it is automatically run on boot?
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to run WebREPL without webrepl_setup in MicroPython
### Problem:
You want to enable WebREPL on your MicroPython board using
import webrepl
webrepl.start()
but it is only showing this error message:
WebREPL is not configured, run 'import webrepl_setup'
However, you want to configure WebREPL programmatically instead of manually running it on every single board.
### Solution:
Use
import webrepl
webrepl.start(password="Rua8ohje")
This will circumvent webrepl_setup completely and is compatible with an automated setup process.
Note: At the time of writing this you can only use passwords with 8 characters max! (see How to fix MicroPython WebREPL ValueError in File „webrepl.py“, line 72, in start )
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to fix MicroPython WebREPL ValueError in File “webrepl.py”, line 72, in start
### Problem:
You want to configure your MicroPython WebREPL programmatically using webrepl.start(password="...") but you see a stacktrace like
>>> webrepl.start(password="Rua8ohjedo")
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "webrepl.py", line 72, in start
ValueError:
### Solution:
Use a shorter password with 8 characters max:
webrepl.start(password="Rua8ohje")
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to autoconnect to Wifi using MicroPython on your ESP32 board
In this post we will investigate how to connect to a wireless network on boot
First, install ampy – a tool to modify the MicroPython filesystem over a serial connection.
sudo pip3 install adafruit-ampy
Now download main.py and save it in your current working directory and insert your wifi credentials:
import network
station = network.WLAN(network.STA_IF)
station.active(True)
station.connect("YourWifiName", "EnterYourWifiPasswordHere")
Upload the file to the board:
ampy -p /dev/ttyUSB* put main.py
In case ampy shows no output within 5 seconds, try resetting the board, waiting for 5-10 seconds and retrying the upload using ampy
Note: You can list the files on the board’s filesystem using
ampy -p /dev/ttyUSB0 ls
You can verify the content of main.py using
ampy -p /dev/ttyUSB0 get main.py
Posted by Uli Köhler in Embedded, MicroPython, Python
## How to read IDF diabetes statistics in Python using Pandas
The International Diabetes Foundation provides a Data portal with various statistics related to diabetes.
In this post we’ll show how to read the Diabetes estimates (20-79 y) / People with diabetes, in 1,000s data export in CSV format using pandas.
First download IDF (people-with-diabetes--in-1-000s).csv from the data page.
Now we can parse the CSV file:
import pandas as pd
# Parse year columns to obtain floats and multiply by thousands factor. Pandas fails to parse values like "12,345.67"
for column in df.columns:
try:
int(column)
df[column] = df[column].apply(lambda s: None if s == "-" else float(s.replace(",", "")) * 1000)
except:
pass
As you can see in the postprocessing step, the number of diabetes patients are given in 1000s in the CSV, so we multiply them by 1000 to obtain the actual numbers.
If you want to modify the data columns (i.e. the columns referring to year), you can use this simple template:
for column in df.columns:
try:
int(column) # Will raise ValueError() if column is not a year number
# Whatever you do here will only be applied to year columns
df[column] = df[column] * 0.75 # Example on how to modify a column
# But note that if your code raises an Exception, it will be ignored!
except:
pass
Let’s plot some data:
regions = df[df["Type"] == "Region"] # Only regions, not individual countries
from matplotlib import pyplot as plt
plt.style.use("ggplot")
plt.gcf().set_size_inches(20,4)
plt.ylabel("Diabetes patients [millions]")
plt.xlabel("Region")
plt.title("Diabetes patients in 2019 by region")
plt.bar(regions["Country/Territory"], regions["2019"] / 1e6)
Note that if you use a more recent dataset than the version I’m using the 2019 column might not exist in your CSV file. Choose an appropriate column in that case.
Posted by Uli Köhler in Bioinformatics, Python
## Parsing World Population Prospects (WPP) XLSX data in Python
The United Nations provides the Word Population Prospects (WPP) dataset on geographic and age distribution of mankind as downloadable XLSX files.
Reading these files in Python is rather easy. First we have to find out how many rows to skip. For the 2019 WPP dataset this value is 16 since row 17 contains all the column headers. The number of rows to skip might be different depending on the dataset. We’re using WPP2019_POP_F07_1_POPULATION_BY_AGE_BOTH_SEXES.xlsx in this example.
We can use Pandas read_excel() function to import the dataset in Python:
import pandas as pd
df = pd.read_excel("WPP2019_INT_F03_1_POPULATION_BY_AGE_ANNUAL_BOTH_SEXES.xlsx", skiprows=16, na_values=["..."])
This will take a few seconds until the large dataset has been processed. Now we can check if skiprows=16 is the correct value. It is correct if pandas did recognize the column names correctly:
>>> df.columns
Index(['Index', 'Variant', 'Region, subregion, country or area *', 'Notes',
'Country code', 'Type', 'Parent code', 'Reference date (as of 1 July)',
'0-4', '5-9', '10-14', '15-19', '20-24', '25-29', '30-34', '35-39',
'40-44', '45-49', '50-54', '55-59', '60-64', '65-69', '70-74', '75-79',
'80-84', '85-89', '90-94', '95-99', '100+'],
dtype='object')
Now let’s filter for a country:
russia = df[df["Region, subregion, country or area *"] == 'Russian Federation']
This will show us the population data for multiple years in 5-year intervals from 1950 to 2020. Now let’s filter for the most recent year:
russia.loc[russia["Reference date (as of 1 July)"].idxmax()]
This will show us a single dataset:
Index 3255
Variant Estimates
Region, subregion, country or area * Russian Federation
Notes NaN
Country code 643
Type Country/Area
Parent code 923
Reference date (as of 1 July) 2020
0-4 9271.69
5-9 9350.92
10-14 8174.26
15-19 7081.77
20-24 6614.7
25-29 8993.09
30-34 12543.8
35-39 11924.7
40-44 10604.6
45-49 9770.68
50-54 8479.65
55-59 10418
60-64 10073.6
65-69 8427.75
70-74 5390.38
75-79 3159.34
80-84 3485.78
85-89 1389.64
90-94 668.338
95-99 102.243
100+ 9.407
Name: 3254, dtype: object
How can we plot that data? First, we need to select all the columns that contain age data. We’ll do this by manually inserting the name of the first such column (0-4) into the following code and assuming that there are no columns after the last age column:
>>> df.columns[df.columns.get_loc("0-4"):]
Index(['0-4', '5-9', '10-14', '15-19', '20-24', '25-29', '30-34', '35-39',
'40-44', '45-49', '50-54', '55-59', '60-64', '65-69', '70-74', '75-79',
'80-84', '85-89', '90-94', '95-99', '100+'],
dtype='object')
Now let’s select those columns from the russia dataset:
most_recent_russia = russia.loc[russia["Reference date (as of 1 July)"].idxmax()]
age_columns = df.columns[df.columns.get_loc("0-4"):]
russian_age_data = most_recent_russia[age_columns]
Let’s have a look at the dataset:
>>> russian_age_data
0-4 9271.69
5-9 9350.92
10-14 8174.26
15-19 7081.77
20-24 6614.7
25-29 8993.09
30-34 12543.8
35-39 11924.7
40-44 10604.6
45-49 9770.68
50-54 8479.65
55-59 10418
60-64 10073.6
65-69 8427.75
70-74 5390.38
75-79 3159.34
80-84 3485.78
85-89 1389.64
90-94 668.338
95-99 102.243
100+ 9.407
That looks useable, note however that the values are in thousands, i.e. we have to multiply the values by 1000 to obtain the actual estimates of the population. Let’s plot it:
from matplotlib import pyplot as plt
plt.style.use("ggplot")
plt.title("Age composition of the Russian population (2020)")
plt.ylabel("People in age group [Millions]")
plt.xlabel("Age group")
plt.gcf().set_size_inches(15,5)
# Data is given in thousands => divide by 1000 to obtain millions
plt.plot(russian_age_data.index, russian_age_data.as_matrix() / 1000., lw=3)
The finished plot will look like this:
Here’s our finished script:
#!/usr/bin/env python3
import pandas as pd
# Filter only russia
russia = df[df["Region, subregion, country or area *"] == 'Russian Federation']
# Filter only most recent estimate (1 row)
most_recent_russia = russia.loc[russia["Reference date (as of 1 July)"].idxmax()]
# Retain only value columns
age_columns = df.columns[df.columns.get_loc("0-4"):]
russian_age_data = most_recent_russia[age_columns]
# Plot!
from matplotlib import pyplot as plt
plt.style.use("ggplot")
plt.title("Age composition of the Russian population (2020)")
plt.ylabel("People in age group [Millions]")
plt.xlabel("Age group")
plt.gcf().set_size_inches(15,5)
# Data is given in thousands => divide by 1000 to obtain millions
plt.plot(russian_age_data.index, russian_age_data.as_matrix() / 1000., lw=3)
# Export as SVG
plt.savefig("russian-demographics.svg")
Posted by Uli Köhler in Bioinformatics, Data science, Python
## How to save Matplotlib plot to string as SVG
You can use StringIO to save a Matplotlib plot to a string without saving it to an intermediary file:
from matplotlib import pyplot as plt
plt.plot([0, 1], [2, 3]) # Just a minimal showcase
# Save plot to StringIO
from io import StringIO
i = StringIO()
plt.savefig(i, format="svg")
# How to access the string
print(i.getvalue())
Note that unless you save to a file you need to set the format=... parameter when calling plt.savefig(). If saving to a file, Matplotlib will try to derive the format from the filename extension (like .svg)
Posted by Uli Köhler in Python
## How to map MeSH ID to MeSH term using Python
Our MeSH-JSON project provides a pre-compiled MeSH-ID-to-term map as JSON:
wget "http://techoverflow.net/downloads/mesh.json.gz"
How to use in Python:
#!/usr/bin/env python3
import json
import gzip
with gzip.open("mesh.json.gz", "rb") as infile:
# Usage example
print(mesh_id_to_term["D059630"]) # Prints 'Mesenchymal Stem Cells'
Posted by Uli Köhler in Bioinformatics, Python
## How to parse all PubMed baseline files in parallel using Python
In our previous post How to parse PubMed baseline data using Python we investigate how to use the pubmed_parser library to parse PubMed medline data using Python.
In this follow-up we’ll provide an example of how to use glob to select all PubMed baseline files in a directory and use concurrent.futures with tqdm to provide a convenient yet easy-to-use process parallelism using ProcessPoolExecutor and a progress bar UI for the command line.
First, install the requirements using
pip install git+git://github.com/titipata/pubmed_parser.git six numpy tqdm
Now download this script, ensure some files like pubmed20n0002.xml.gz or pubmed20n0004.xml.gz are in the same directory and run it:
#!/usr/bin/env python3
import pubmed_parser as pp
import glob
import os
from collections import Counter
import concurrent.futures
from tqdm import tqdm
# Source: https://techoverflow.net/2017/05/18/how-to-use-concurrent-futures-map-with-a-tqdm-progress-bar/
def tqdm_parallel_map(executor, fn, *iterables, **kwargs):
"""
Equivalent to executor.map(fn, *iterables),
but displays a tqdm-based progress bar.
Does not support timeout or chunksize as executor.submit is used internally
**kwargs is passed to tqdm.
"""
futures_list = []
for iterable in iterables:
futures_list += [executor.submit(fn, i) for i in iterable]
for f in tqdm(concurrent.futures.as_completed(futures_list), total=len(futures_list), **kwargs):
yield f.result()
def parse_and_process_file(filename):
"""
This function contains our parsing code. Usually, you would only modify this function.
"""
# Don't parse authors and references for this example, since we don't need it
dat = pp.parse_medline_xml(filename, author_list=False, reference_list=False)
# For this example, we'll build a set of count of all MeSH IDs in this file
ctr = Counter()
for entry in dat:
terms = [term.partition(":")[0].strip() for term in entry["mesh_terms"].split(";")]
for term in terms:
ctr[term] += 1
return filename, ctr
if __name__ == "__main__":
# Find all pubmed files in the current directory
all_filenames = glob.glob("pubmed*.xml.gz")
# but a ProcessPoolExecutor is a good default
executor = concurrent.futures.ProcessPoolExecutor(os.cpu_count())
# Iterate results as they come in (the order is not the same as in the input!)
for filename, ctr in tqdm_parallel_map(executor, parse_and_process_file, all_filenames):
# NOTE: If you print() here, this might interfere with the progress bar,
# but we accept that here since it's just an example
print(filename, ctr)
Now you can start modifying the example, most notably the parse_and_process_file() function to do whatever processing you intend to do.
Posted by Uli Köhler in Bioinformatics, Python
## How to parse PubMed baseline data using Python
Want to parse more than one PubMed file? Also see our follow-up at How to parse all PubMed baseline files in parallel using Python
PubMed provides a data dump of metadata from all PubMed articles on the NCBI Servers.
In this example, we’ll parse one of the compressed metadata XML files using the pubmed_parser Python library.
First, download one of the .xml.gz files. For this example, we’ll use pubmed20n0001.xml.gz.
Now we can install the required libraries:
pip install git+git://github.com/titipata/pubmed_parser.git six numpy
Now you can download & run our script. For this example, we will extract a list of MeSH terms for every and print PubMed ID and a list of MeSH IDs.
#!/usr/bin/env python3
import pubmed_parser as pp
# Don't parse authors and references for this example, since we don't need it
dat = pp.parse_medline_xml("pubmed20n0001.xml.gz", author_list=False, reference_list=False)
# Iterate PubMed entries from that file
for entry in dat:
# entry["mesh_terms"] is like "D000818:Animal; ..."
# In this example, we are only interested in the MeSH ID, like D000818.
# Print the PubMed ID, followed by a list of MeSH terms.
print(entry["pmid"], [
term.partition(":")[0].strip() for term in entry["mesh_terms"].split(";")
])
Running this script takes 13.3 seconds on my Notebook which is equivalent to about 1.4 MBytes of GZipped input data per second. When running the script, you will see lines like
30957 ['D000319', 'D001794', 'D003864', 'D006801', 'D006973', 'D007676', 'D010869']
which means that the PubMed Article with ID 30957 has the MeSH terms ['D000319', 'D001794', 'D003864', 'D006801', 'D006973', 'D007676', 'D010869'].
See the pubmed_parser documentation, or just try it out interactively for more information on what fields are available in the entries you can iterate.
Posted by Uli Köhler in Bioinformatics, Python
## How to install PyPy3 + virtual environment in 30 seconds
#### TL;DR:
Run this
wget -qO- https://techoverflow.net/scripts/pypy3-installer.sh | bash
then run vpypy every time you want to activate (you might need to restart). The script currently assumes you are running Linux x86_64 and have installed virtualenv (sudo apt install virtualenv or similar if you don’t have it installed)
#### Full description:
PyPy is an alternate Python implementation that can be used to speed up many workloads. However, installing it is a somewhat cumbersome process, especially if you don’t have too much experience with virtual environments and related concepts.
We provide a script that automatically downloads PyPy3, installs it to ~/.pypy3 and creates a virtual environment in ~/.pypy3-virtualenv. After that, it creates a shell alias vpypy that aliases to source ~/.pypy3-virtualenv/bin/activate and hence provides an easily memoizable way of activating the environment without requiring the user to memoize the directory.
Also, since both pypy3 itself and the virtual environment are installed in the user’s home directory, running this script does not require admin permissions.
After running the script using
wget -qO- https://techoverflow.net/scripts/pypy3-installer.sh | bash
you can activate the virtual environment using the vpypy alias that is automatically added to ~/.bashrc and ~/.zshrc. Restart your shell for the alias definition to load, then run vpypy:
uli@uli-laptop ~ % vpypy
(.pypy3-virtualenv) uli@uli-laptop ~ %
You can see that the prompt has changed. Now you can use pip (which will install packages locally to the PyPy3 virtualenv), python (which maps to pypy3) and other related executables. In order to run a script using PyPy, just run python myscript.py
#### Full source code:
#!/bin/bash
# TechOverflow's 30-second Pypy3 virtual environment generator
# This script is released under CC0 1.0 Universal
DIRECTORY=~/.pypy3
VENV_DIRECTORY=~/.pypy3-virtualenv
VERSION=pypy3.6-v7.3.0-linux64
if [ -d "$DIRECTORY" ]; then echo "Skipping PyPy download, already exists" else echo "Downloading PyPy to$DIRECTORY"
wget https://techoverflow.net/downloads/${VERSION}.tar.bz2 -O /tmp/${VERSION}.tar.bz2
bash -c "cd /tmp && tar xjvf ${VERSION}.tar.bz2" mv /tmp/${VERSION} $DIRECTORY rm /tmp/${VERSION}.tar.bz2
fi
# Create virtualenv
if [ -d "$VENV_DIRECTORY" ]; then echo "Skipping to create pypy3 virtualenv, already exists" else echo "Creating PyPy virtual environment in$VENV_DIRECTORY"
virtualenv -p ${DIRECTORY}/bin/pypy3${VENV_DIRECTORY}
fi
# Create "vpypy" shortcut
set -x
vpypy
result="$?" set +x if [ "$result" -ne 127 ]; then
echo "Skipping to create vpypy shortcut, already exists in current shell"
else
echo "Creating bash/zsh shortcut 'vpypy'"
if [ -f ~/.bashrc ]; then
echo -e "\n# TechOverflow PyPy installer\nalias vpypy='source ${VENV_DIRECTORY}/bin/activate'\n" >> ~/.bashrc fi if [ -f ~/.zshrc ]; then echo -e "\n# TechOverflow PyPy installer\nalias vpypy='source${VENV_DIRECTORY}/bin/activate'\n" >> ~/.zshrc
fi
# Activate shortcut in current shell (but do not automatically activate virtual environment)
alias vpypy='source ${VENV_DIRECTORY}/bin/activate' fi echo -e "\n\nPyPy installation finished. Restart your shell, then run 'vpypy' to activate the virtual environment" Posted by Uli Köhler in Linux, Python ## Is pypng 16-bit PNG encoding faster using pypy on the Raspberry Pi? In our previous post How to save Raspberry Pi raw 10-bit image as 16-bit PNG using pypng we investigated how to use the pypng library to save 10-bit raw Raspberry Pi Camera images to 16-bit PNG files. However, saving a single image took ~26 seconds using CPython 3.7.3. Since pypy can provide speedups to many Python workloads, we tried using pypy3 7.0.0 (see How to install pypy3 on the Raspberry Pi) to speed up the PNG encoding. ### Results pypng PNG export seems to be one of the workloads that are much slower using pypy3. • CPython 3.7.3: Encoding took 24.22 seconds • pypy3 7.0.0: Encoding took 266.60 seconds Encoding is more that 10x slower when using pypy3! Hence I don’t recommend using pypy3 to speed up pypng encoding workloads, at least not on the Raspberry Pi! ### Full example This example is derived from our full example previously posted on How to save Raspberry Pi raw 10-bit image as 16-bit PNG using pypng: #!/usr/bin/env python3 import time import picamera import picamera.array import numpy as np import png # Capture image print("Capturing image...") with picamera.PiCamera() as camera: with picamera.array.PiBayerArray(camera) as stream: camera.capture(stream, 'jpeg', bayer=True) # Demosaic data and write to rawimg # (stream.array contains the non-demosaiced data) rawimg = stream.demosaic() # Write to PNG print("Writing 16-bit PNG...") t0 = time.time() with open('16bit.png', 'wb') as outfile: writer = png.Writer(width=rawimg.shape[1], height=rawimg.shape[0], bitdepth=16, greyscale=False) # rawimg is a (w, h, 3) RGB uint16 array # but PyPNG needs a (w, h*3) array png_data = np.reshape(rawimg, (-1, rawimg.shape[1]*3)) # Scale 10 bit data to 16 bit values (else it will appear black) # NOTE: Depending on your photo and the settings, # it might still appear quite dark! png_data *= int(2**6) writer.write(outfile, png_data) t1 = time.time() print(f"Encoding took {(t1 - t0):.2f} seconds") Posted by Uli Köhler in Python, Raspberry Pi ## How to install pypy3 on the Raspberry Pi This post shows you an easy way of getting pypy3 running on the Raspberry Pi. I used Raspbian Buster on a Raspberry Pi 3 for this example. On Raspbian buster this will install pypy3 7.x! First install pypy3 and virtualenv: sudo apt update && sudo apt -y install pypy3 pypy3-dev virtualenv Now we can create a virtualenv to install pypy packages into: virtualenv -p /usr/bin/pypy3 ~/pypy3-virtualenv Now we can activate the virtualenv. You need to do this every time you want to use pypy, for each shell / SSH connection separately: source ~/pypy3-virtualenv/bin/activate If your shell prompt is now prefixed by (pypy3-virtualenv) you have successfully activated the virtualenv: (pypy3-virtualenv) uli@raspberrypi:~$
Now python points to pypy3 and pip will install packages locally to ~/pypy3-virtualenv.
Now you can use e.g.
python3 myscript.py
to run your script (both python and python3 will point to pypy3 if you activated the virtual environment!).
Note: Installing pypy3-dev is not strictly neccessary to get pypy3 running, but you need it in order to compile native librarie like numpy.
Posted by Uli Köhler in Python, Raspberry Pi
## How to capture RaspberryPi camera 10-bit raw image in Python
You can use the picamera Python library to capture a raw sensor image of a camera attached to the Raspberry Pi via CSI:
#!/usr/bin/env python3
import picamera
import picamera.array
import numpy as np
# Capture image
print("Capturing image...")
with picamera.PiCamera() as camera:
with picamera.array.PiBayerArray(camera) as stream:
camera.capture(stream, 'jpeg', bayer=True)
# Demosaic data and write to rawimg
# (stream.array contains the non-demosaiced data)
rawimg = stream.demosaic()
rawimg is a numpy uint16 array of dimensions (w, h, 3), e.g. (1944, 2592, 3) and contains integer values from 0 to 1023.
You can, for example, save it in a NumPy file using
np.save("rawimg.npy", rawimg) # Reload with np.load("rawimg.npy")
or save it in a compressed format using
np.savez_compressed("rawimg.npz", rawimg) # Reload with np.load("rawimg.npz")
Posted by Uli Köhler in Python, Raspberry Pi
## How to fix ModuleNotFoundError: No module named ‘picamera’
### Problem:
You want to run a Python script using the Raspberry Pi camera but you see an error message like
Traceback (most recent call last):
File "mycamera.py", line 2, in <module>
import picamera
ModuleNotFoundError: No module named 'picamera'
### Solution:
You need to install the picamera Python module using pip:
sudo pip3 install picamera
or, if you are still using Python 2.x:
sudo pip install picamera
In case you see
sudo: pip3: command not found
install pip3 using
sudo apt install -y python3-pip
Posted by Uli Köhler in Python, Raspberry Pi
## Simulating survival data for Kaplan-Meier plots in Python
Libraries like lifelines provide a plethora of example datasets that one can work with. However, for many tasks you need to simulate specific behaviour in survival curves.
In this post, we demonstrate a simple algorithm to generate survival data in a format comparable to the one used in the lifelines example datasets like load_leukemia().
The generation algorithm is based on the following assumptions:
• There is a strict survival plateau with a given survival probability starting at a given point in time
• The progression from 100% survival, t=0 to the survival plateau is approximately linear (i.e. if you would generate an infinite number of datapoints, the survival curve would be linear)
• No censoring events shall be generated except for censoring all surviving participants at the end point of the timeline.
Code:
import numpy as np
import random
from lifelines import KaplanMeierFitter
def simulate_survival_data_linear(N, survival_plateau, t_plateau, t_end):
"""
Generate random simulated survival data using a linear model
Keyword parameters
------------------
N : integer
Number of entries to generate
survival_plateau : float
The survival probability of the survival plateau
t_plateau : float
The time point where the survival plateau starts
t_end : float
The time point where all surviving participants will be censored.
Returns
-------
A dict with "Time" and "Event" numpy arrays: 0 is censored, 1 is event
"""
data = {"Time": np.zeros(N), "Event": np.zeros(N)}
for i in range(N):
r = random.random()
if r <= survival_plateau:
# Event is censoring at the end of the time period
data["Time"][i] = t_end
data["Event"][i] = 0
else: # Event occurs
# Normalize where we are between 100% and the survival plateau
p = (r - survival_plateau) / (1 - survival_plateau)
# Linear model: Time of event linearly depends on uniformly & randomly chosen position
# in range (0...tplateau)
t = p * t_plateau
data["Time"][i] = t
data["Event"][i] = 1
return data
# Example usage
data1 = simulate_survival_data_linear(250, 0.2, 18, 24)
data2 = simulate_survival_data_linear(250, 0.4, 17.2, 24)
Given data1 and data2 (see the usage example at the end of the code) you can plot them using
# Plot bad subgroup
kmf1 = KaplanMeierFitter()
ax = kmf1.plot()
# Plot good subgroup
kmf2 = KaplanMeierFitter()
kmf2.fit(data2["Time"], event_observed=data2["Event"], label="Good subgroup")
ax = kmf2.plot(ax=ax)
# Set Y axis to fixed scale
ax.set_ylim([0.0, 1.0])
Thi
#### Do not want a survival plateau?
Just set t_end = t_survival:
# Example usage
data1 = simulate_survival_data_linear(250, 0.2, 24, 24)
data2 = simulate_survival_data_linear(250, 0.4, 24, 24)
# Code to plot: See above
#### What happens if you have a low number of participants?
Let’s use 25 instead of 250 as above:
# Example usage
data1 = simulate_survival_data_linear(25, 0.2, 24, 24)
data2 = simulate_survival_data_linear(25, 0.4, 24, 24)
# Plot code: See above
Although we generated the data with the same data, the difference is much less clear in this example, especially towards the end of the timeline (note however that the data is generated randomly, so you might see a different result). You can see a large portion of the confidence intervals overlappings near t=24. In other words, based on this data it is not clear that the two groups of patients are significantly different (in other words, $P \geq 0.05$)
Posted by Uli Köhler in Python, Statistics
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# Doubt in maths
Q6, option B doubt
Q3
For Q3 you can use Newton sums, or you can write it as product of two matrices.
For Q6, M^{2}+M N^{2}=N^{4}+N^{2} M=N^{2}\left(N^{2}+M\right).
We know that
0=N^{4}-M^{2}=\left(N^{2}+M\right)\left(N^{2}-M\right) (given in question).
So multiply both sides of first equation by N^{2}-M and it will come out to be zero matrix. So one possible value of the matrix “U” is N^{2}-M
4 Likes
bhai ye kya hota hai? Theory bhej sakte ho?
2 Likes
Ques 25 without putting values…
2 Likes
2 Likes
The number of permutations of n different objects, taken all at a time, when m specified
objects always come together is m! × (n – m + 1)!.
Koi iska proof batado by example…
@Dexter27 @Rade_Woosh @Anorak @VictoryGod and anyone
4 Likes
proof toh yaad nahi try taking n=3 differently coloured objects where m=2 specified colours e.g red and blue together and subtitue in formula and also try same ques normal method you’ll get same answer
4 Likes
dekho…ham n me se m ko hamesha saath rakhna chahte hai…so what we do is take the m objects together always and consider it as a single unit instead…so we get (n-m+1) from here…uske baad the internal permutation amongst that single unit is taken account by m! term…
@Achyut_Singh
5 Likes
Ok samjh gya Thanks @Dexter27 @Anorak
4 Likes
mujhe number of solutions ko accurately count krne me bahaut dikkat aati hai…
pls help
@Anorak @Achyut_Singh @LaveshGupta @VictoryGod @Deadpool and everyone…
1 Like
@Dexter27
Try some methods like:-
1)Graphical methods
2)Equating Solutions
3)Comparing Domains …etc.
1 Like
bro like in 2nd question its evident that rhs will always assume negative value…so i get range for x…but what to after that??
1)take e^modsinx=t and solve using determinant D and sinx range condition
2.)only possible by graph plotting
3.)since they didnt give range of x i assumd 0,2pi and solved using general solution of sinpix^2/3=1
4.) we can solve by graph of tan{x}=1 but that is harder to visualise so so change {x}=x-[x] and expand and simplify then draw graph
3 Likes
ln(sinx) ka graph kaise banega???
i took aproximation
we know that modsinx is oscilaatory and ranges between (0,1]
and since we have to find points of intersection my graph wasnt accurate
also we can further simplify to write x-1=(1-lnsinx)^1/2 and take condition from here as well
2 Likes
Chamka?
but this we will not be able to know without graph?!
i am still struggling bro…
1 Like
ig this question will be easier if we write in form of e…because it will be e^(1-(x+1)^2) and the power is negative…so will be like e^(-x) form graph…and will cut |sinx| at 2 points…
@Anorak
yeah that is easier lol
2 Likes
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# Can hyperbolic space be bounded?
There are many visualisations of hyperbolic geometry using Poincaré disks.
1. What are their purpose?
2. Can hyperbolic space be bounded?
3. Can we endow the disk with the structure described by the FLRW metric?
4. Does it have constant curvature?
5. Could our universe be bounded, but yet still infinite like this?
• I don't think it's 'does the Poincaré disk satisfy the FLRW metric?' Rather, I would say, can we endow the disk with the structure described by the FLRW metric? – JamalS Nov 9 '14 at 21:07
• Oh, and yes, the Poincaré disks have constant curvature; in general it's $d(d-1)$. – JamalS Nov 9 '14 at 21:23
What are their purpose?
The "purposes" of Hyperbolic geometries are many and varied in mathematics, but one stands out far beyond all others, at least historically as the purpose. Hyperbolic geometries were constructed to prove that the Euclid parallel postulate (see "Parallel Postulate" Wiki page) was logically independent of Euclid's other axioms of geometry. Before János Bolyai and Nikolai Lobachevsky discovered concrete examples of geometries that fulfilled all the other Euclid postulates, but not the parallel postulate in the 1820s, there were many notable supposed (but later shown to be flawed) "proofs" of the parallel postulate from Euclid's others (these are discussed on the Wiki page). But the concrete demonstration of a geometry fulfilling the other axioms but wherein the parallel postulate did not hold decisively showed that it could not be derived from the others alone: otherwise it would be in logical contradiction with the exhibited concrete models (see "Model Theory" Wiki page) that Bolyai and Lobachevsky discovered.
Another, probably the main, modern purpose (outside the study of hyperbolic geometry for its own sake) is as a local approximation to the geometry of a general manifold in a neighbourhood where the curvature can be taken as approximately constant and negative. It is "one step up" from the Euclidean / Minkowskian (signatured flat) local approximation to a manifold given by the tangent space. If you like, hyperbolic geometry (and henceforth I mean constant curvature hyperbolic geometry by these words) is like taking Taylor approximation to a suface to second order (in a region of negative curvature) where the tangent space is the first order Taylor approximation. In two dimensions, for example, hyperbolic geometry is a good approximation to the geometry on a surface in the neighbourhood of a saddle point.
Can hyperbolic space be bounded?
Truly constant curvature hyperbolic space cannot be compact in the topology that makes it hyperbolic: take the Poincaré disk model and witness that for any distance $$d_0$$, no matter how big, there are always points $$u,\,v$$ for which global minimum distance between them is greater. Take $$u$$ to be the centre of the Poincaré disk and $$v$$ to be the point given by $$y=z=0$$ and $$x = \sqrt{\frac{\cosh(d_0+\epsilon)-1}{\cosh(d_0+\epsilon)+1}}$$, where $$\epsilon>0$$ for example. However, manifolds which are locally hyperbolic can certainly be compact: intuitively this is obvious if you blow a balloon up and poke two fingers into its surface to give it two concave dimples. The saddle region in between the dimples is locally hyperbolic, but the global manifold is diffeomorphic to the compact 2-sphere.
However, you seem to be thinking something slightly different from my paragraph above, i.e. that the Poincaré disk is homeomorphic to a bounded but open i.e noncompact subspace of Euclidean space: let's hold this thought until I answer your last question.
Can we endow the disk with the structure described by the FLRW metric?
Yes you can. The Poincaré disk models constant negative curvature hyperbolic space, so one can think of the FLRW metric as a kind of dilation of the Poincaré disk by a function of the FLRW scale factor $$a(t)$$. See my calculations at the end of my answer to see this more clearly.
Could our universe be bounded, but yet still infinite like this?
As in Doetoe's answer, a nonlinear transformation maps an FLRW constant negative curvature onto a "finite" set - finite in the ambient Euclidean space. But the Minkowskian distance is what a being belonging to and living in this universe would measure. It is the only "physical" distance function in this universe, and such a universe always contains points arbitrarily distant from one another. So if you think of such a structure as bounded, then the answer is "yes", but this is a wholly artificial construction and has nothing to do with physics. You can always find a nonlinear transformation to map infinite regions to open, bounded ones. It is like mapping all time - the unbounded real line $$\mathbb{R}$$ to a finite interval by the transformation $$\tau:\mathbb{R}\to(-1,\,1);\,\tau(x) = \tanh(x)$$.
It may be helpful to you to understand that the Poincaré disk is the bijective ("information preserving" or "invertible") and isometric ("length and angle preserving") Stereographic Projection (see the "Relation to the hyperboloid model" section on the Poincaré disk Wiki page)) of the hyperboloid, an unbounded geometric object.
# Relationship to FLRW Metric
To see how the Poincaré disk fits in FLRW Metricwith the Reduced-circumference polar coordinates for the FLRW metric, we begin with:
$$\mathrm{d}\mathbf{\Sigma}^2 = \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\mathbf{\Omega}^2, \quad \text{where } \mathrm{d}\mathbf{\Omega}^2 = \mathrm{d}\theta^2 + \sin^2 \theta \, \mathrm{d}\phi^2\tag{1}$$
as on the Wiki page. Exactly as for the Schwarzschild metric, here $$r = const$$ parameterises the hypersphere centred on the origin such that the length of a geodesic around the hypersphere is $$2\,\pi\,r$$. $$r$$ does not correspond to the length of a geodesic joining a point on the hypersphere and the origin, aside from when the curvature $$k$$ is nought.
In terms of the ambient Euclidean co-ordinates $$(x,\,y,\,z)$$ for points on the Poincaré disk, we have:
$$\mathrm{d}\mathbf{\Sigma}_P^2 = 4\,\frac{\mathrm{d}\,x^2+\mathrm{d}\,y^2+\mathrm{d}\,z^2}{(1-R^2)^2}\tag{2}$$
where $$R=x^2+y^2+z^2$$; take careful heed of the difference between little $$r$$ and big $$R$$. $$R$$ is the polar radial co-ordinate in the ambient Euclidean space and $$\mathrm{d}\mathbf{\Sigma}_P^2$$ is the line element on the Poincaré disk. Therefore, the length of a great circle on the Poincaré disk is:
$$C(R)=\int_0^{2\,\pi}\, 2\,\frac{R}{1-R^2}\,\mathrm{d}\,\theta = 4\,\pi\,\frac{R}{1-R^2} = 2\,\pi\,r\tag{3}$$
the last step following from the definition of the reduced circumference radius, and so:
$$r = \frac{2\,R}{1-R^2}\tag{4}$$
and so:
$$\mathrm{d}r^2 = \frac{4\,(1+R^2)^2}{(1-R^2)^4}\,\mathrm{d}R^2\tag{5}$$
On substituting (4) and (5) into (1), but now (i) letting $$\phi$$ stand for the azimuthal co-ordinate on the plane wherein the tangent vector along which we measure the line element lies (i.e. without loss of generality we think of our tangent vector lying in the appropriate equatorial plane) and (ii) setting the constant curvature to be $$k=-1$$ we find:
$$\begin{array}{lcl}\mathrm{d}\mathbf{\Sigma}^2 &=& \frac{\frac{4\,(1+R^2)^2}{(1-R^2)^4}}{1-k\,\left(\frac{2\,R}{1-R^2}\right)^2}\,\mathrm{d}R^2 + \frac{4\,R^2}{(1-R^2)^2}\,\mathrm{d}\Omega^2\\&=&\frac{4}{(1-R^2)^2}\left(\mathrm{d}R^2+R^2\,\mathrm{d}\,\phi^2\right)\\&=&4\frac{\mathrm{d}\,x^2+\mathrm{d}\,y^2+\mathrm{d}\,z^2}{(1-R^2)^2}\\&=&\mathrm{d}\mathbf{\Sigma}_P^2\end{array}\tag{6}$$
i.e. is equal to the line element measured on the Poincaré disk.
1. The purpose of any model of the hyperbolic plane is that some aspect in it will be easy to work with computationally or intuitively, e.g. writing out certain isometries, identifying geodesics, computing volumes, etc.
2. Hyperbolic space is unbounded, a hyperbolic manifold can be bounded.
3. I don't know
4. Yes
5. If you are not using the terms bounded and infinite to be mutually exclusive by definition you are using non-standard terminology. Maybe you are under the impression that the hyperbolic plane in the Poincaré disk model is bounded, but that is not the case. It would be if we were using the Euclidean metric, but then it wouldn't be the hyperbolic plane anymore.
• Well, I am not using the terminology. Obviously, poincaré disk is defined on |x|<1, so it is bounded. By the standard definition, it is finite. However, if you converge to the boundary, you have to take infinite steps, so by my intuitive definition it is infinite. I seem to find self-defining and self-proving the definition that something is bounded from the bounded set. Off course, then there is a problem that we can always create an "off-world" "outer space", from which point our space can be anything we like. Or can we? – mimok Nov 10 '14 at 10:21
• (Btw, usually infinite means that it has an infinite number of points, so in that sense they're obviously not mutually exclusive.) The Poincaré disk is a model of the hyperbolic plane on a set that happens to be bounded in the Euclidean metric, but it is endowed with a different metric. With this metric, it is unbounded, and it is only this metric that has a meaning for the Poincaré disk. In Rob's example the interval $(-1,1)$ is made isometric to $\Bbb R$ through the $\tanh$ function, but that doesn't make $\Bbb R$ bounded. – doetoe Nov 10 '14 at 13:02
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# variable separable method
## Differential Equations in Variable Separable Form
Here you will learn how to find the solution of the differential equations in variable separable form with examples. Let’s begin – Differential Equations in Variable Separable Form If the differential equation can be put in the form f(x) dx = g(y) dy, we say that the variables are seperable and such equations can be …
## Differential Equations Reducible to Variable Separable Form
Here you will learn how to find the solution of the differential equations reducible to variable separable form with examples. Let’s begin – Differential Equations Reducible to Variable Separable Form Differential Equations of the form $$dy\over dx$$ = f(ax + by + c) can be reduce to variable separable form by the substitution ax + …
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Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?
1. When constructing the Riemann surfaces, why does the space of germs construction correspond to the gluing construction?
2. Why is the connected component of the space of germs over $\mathbb{C}^*$ corresponding to the complex logarithm analytically isomorphic to $\mathbb{C}$?
I have read the page with this related question About the Riemann surface associated to an analytic germ which is basically asking a similar thing, so read it please if you find some of the terms I'm using unclear.
Whenever I've asked this questions to anybody, they have answered me with questions and "think about it" which is very annoying.
My view on this: Suppose we have a space of germs $G = \{[f]_z: z \in G\}$ over a domain $D$ ($\mathbb{C}^*$ in case of the complex logarithm function). Then we define a projection map $$p: G \rightarrow D$$ $$p:[f]_z \rightarrow z$$ $$p:[f]_A \rightarrow A$$ where $A$ is a domain in $D$. I'm able to show that $p$ is continuous and a homeomorphism. It seems clear that} $p$ is also a bijection.
Furthermore, we can also find a unique equivalence class of atlases on $G$ so that the projection $p$ is analytic.
$p$ is then analytic and a bijection (and a homeomorphism). Can we take $p$ as our analytic isomorphism then? Do we need to show $p^{-1}$ analytic (?) or are we done already? Or do we need something completely else altogether?
Can also define an evaluation map $\varepsilon: G \rightarrow \mathbb{C}$ with $$\varepsilon: [f]_z \rightarrow f(z)$$ which can be shown to be analytic as a map between Riemann Surfaces. Does this then answer the second question?
I'm suspicious of my answers above since they don't contain a word about "cut & paste" constructions (gluing) mentioned in the original problem. Anyone care to share their thoughts about this?
• Did you try reading Springer's book "Introduction to Riemann surfaces" mentioned in the 2nd answer in the link? Springer spends the whole chapter 3 addressing just this issue. How can you expect to get a more detailed answer here at MSE? – Moishe Kohan Mar 30 '14 at 5:24
• In the same vein as studiousus response, these exact questions are answered in Freitag's book Complex Analysis 2, chapter 1, section 2. – Alex Youcis Mar 30 '14 at 7:02
• @studiosus, I'm going to get that book in the library tomorrow. Is it wrong of me to ask a question on MSE, huh? – user138878 Mar 30 '14 at 12:23
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# Pattern condition involving multiple arguments to a function
I can define conditions on individual arguments to a function:
f[a_List/;condition(a),b_List/;condition(b)]:=(...)
I am wondering if there is a way of declaring something like this:
f[a_List,b_List/;condition(a,b)]:=(...)
so for instance, I might want to be sure that a and b are of the same size or that {a,b} follows a certain pattern for example.
I know I can leave the condition cases inside the body of the function but I am wondering if there is any way of avoiding that.
• Note that patt /; test is a shorthand for Condition[patt, test]. You are writing f[a_, Condition[b_, test[a,b]]], but there is no a in the pattern in the first argument of Condition in this case. Note also that function definitions in Mathematica are nothing but patterns, so it is harmless to put the entire left hand side in the first argument of Condition, then you will have access to all symbols in the second argument of Condition. This is what eldo does. – C. E. Dec 23 '15 at 5:08
f[a_List,b_List/;condition(a,b)]:=(...) is not possible
fun[a_List, b_List] /; Length@a == Length@b := 1
fun[a_List, b_List] := 2
fun[Range@3, Range@3]
1
fun[Range@3, Range@4]
2
Also possible:
fun[a_List, b_List] := 1 /; Length@a == Length@b
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# Histone H3 binding to the PHD1 domain of histone demethylase KDM5A enables active site remodeling
## Abstract
Histone demethylase KDM5A removes methyl marks from lysine 4 of histone H3 and is often overexpressed in cancer. The in vitro demethylase activity of KDM5A is allosterically enhanced by binding of its product, unmodified H3 peptides, to its PHD1 reader domain. However, the molecular basis of this allosteric enhancement is unclear. Here we show that saturation of the PHD1 domain by the H3 N-terminal tail peptides stabilizes binding of the substrate to the catalytic domain and improves the catalytic efficiency of demethylation. When present in saturating concentrations, differently modified H3 N-terminal tail peptides have a similar effect on demethylation. However, they vary greatly in their affinity towards the PHD1 domain, suggesting that H3 modifications can tune KDM5A activity. Furthermore, hydrogen/deuterium exchange coupled with mass spectrometry (HDX-MS) experiments reveal conformational changes in the allosterically enhanced state. Our findings may enable future development of anti-cancer therapies targeting regions involved in allosteric regulation.
## Introduction
Post-translational modifications of histone proteins are important regulators of chromatin structure and function and are controlled by proteins that write, read and erase these marks1,2. A common and functionally diverse histone modification is lysine methylation, which regulates many cellular processes, including heterochromatin formation, regulation of transcription and DNA repair3,4. Lysine methylation is a reversible modification, and its removal is catalyzed by lysine histone demethylases (KDMs). The KDMs are grouped into several subfamilies depending on their domain composition, substrate specificity and reaction mechanism. The KDM1 family (LSD1 and LSD2) uses a flavin-dependent mechanism, and acts on mono- or di-methylated lysines5,6. A broader range of demethylation is possible by the jumonji C (JmjC) domain-containing family of KDMs (KDM2-9) that utilize a Fe(II)- and α-ketoglutarate (α-KG)-dependent mechanism as they are able to demethylate mono-, di- and tri- methylated lysines7. They predominantly act on histone proteins, but in some instances also catalyze demethylation of non-histone substrates8,9. Understanding the role of the chromatin environment in regulating activities of these enzymes is critical to elucidation of context-dependent spatial and temporal regulation of chromatin methylation. Several reports in recent years have pointed out the critical role of chromatin reader domains in regulation of demethylase activities, substrate specificities, and localization10,11,12,13.
The human KDM5 subfamily of JmjC demethylases consists of four family members, KDM5A-D, which demethylate H3K4me1/2/3 marks. The proteins in the KDM5 family share common structural features, such as an iron containing active site comprised of the JmjN and JmjC domains14,15,16, a DNA binding ARID domain, a zinc-finger domain, and either two (for KDM5C and D) or three (for KDM5A and B) plant homeodomain (PHD) chromatin reader domains7,17,18,19 (Fig. 1a). There has been a considerable amount of interest in the KDM5 family due to their roles in many disorders as all four members have been shown to be involved in various cancers20,21,22,23,24. Specifically, KDM5A is overexpressed in breast cancer25 and its fragment is known to form a fusion with NUP98 in acute leukemia20. Additionally, there is evidence for overexpression of KDM5A in cancer drug resistance in lung cancer models26 as well as osteoporosis27. KDM5B is overexpressed in hepatocellular carcinoma where it promotes metastasis28. Additionally, this enzyme is involved in drug resistance in melanoma treatments29 and regulation of genes involved in stem cell differentiation22,30. KDM5C is highly expressed in neuronal tissues and mutations in this enzyme have been associated with X-linked intellectual disability disorders17,31. KDM5D has been suggested to have a role in spermatogenesis32. It is for these reasons that the KMD5 family are of clinical interest, prompting investigations into development of small molecule inhibitors of these enzymes15,16,33,34,35,36,37.
KDM5A contains three PHD domains (Fig. 1a), which are commonly recognized as chromatin readers and are often found in chromatin modifying enzymes. PHD domains in demethylases have traditionally been associated with the recruitment of demethylases to chromatin38. For example, PHD domain protein BHC80 is a component of LSD1 co-repressor complex that stabilizes the recruitment of LSD1 to chromatin39. In addition, PHD domains can also regulate substrate specificity of demethylases, as in the case of PHF8 and KIAA171810. In vitro studies on PHD domains in KDM5A have shown that PHD1 preferentially binds unmodified H3 N-terminal peptides40 and that PHD3 binds H3K4me3 marks20, while no function for PHD2 has yet been determined. Binding of the H3K4me3 mark by PHD3 enables the recruitment of the demethylase to its substrate, a role consistent with the prototypical function of reader domains. In contrast to this canonical role, our earlier in vitro studies on the PHD1 domain have uncovered a function of the PHD1 domain in the allosteric regulation of the demethylase activity of KDM5A40. Specifically, we found that the engagement of the PHD1 domain by unmodified H3 peptide enhances the activity of the catalytic domain. Since H3 peptide unmodified at Lys4 is the product of demethylation, these findings suggested a positive feedback regulatory mechanism that may allow for demethylation to be propagated along nucleosomes. Such feedback regulation could be important for removal of broad H3K4me3 domains in early embryo development, a process mediated by KDM5A41 and KDM5B42,43.
The molecular basis of allosteric stimulation by the product peptide binding to the PHD1 domain in KDM5A remains poorly understood. In addition, it remains unclear how the allostery is propagated between the PHD1 and JmjC domains. Current crystal structures of KDM5 enzymes are missing the structure of PHD1 and its surrounding regions due to either the PHD1 domain being removed from the construct or being disordered in these structures14,15,16,35. With limited structural information, here we aim to determine the mechanistic rationale for the catalytic enhancement that results from binding of the peptide ligand to the PHD1 domain. Using purified KDM5A, we demonstrate that this allosteric enhancement is caused by improved substrate binding rather than enhanced catalytic turnover. Differently modified H3 tail peptides bound to the PHD1 domain promote similar effects, indicating that affinity of the modified H3 ligand, which would affect occupancy of the PHD1 domain, drives ligand-induced allosteric stimulation. Hydrogen/Deuterium exchange MS (HDX-MS) was used to probe conformational changes and highlighted regions of the catalytic domain that become less solvent protected in the allosterically enhanced state.
## Results
### Asp292 is important for PHD1 in recognizing H3 peptides
The allosteric regulation of KDM5A by its PHD1 domain highlights a more complex role of this domain rather than simple recognition of chromatin marks. Therefore, it is important to understand the molecular basis of how the PHD1 domain interacts with its ligand. PHD1KDM5A binds the N-terminal tail of histone H3 with a preference for unmodified H3, and binding affinity decreases with an increase in H3K4 methylation40. Structural investigations into PHD1KDM5B and several other PHD1 domains, which like PHD1KDM5A preferentially recognize unmodified H3K4 residues, demonstrate electrostatic interactions between the unmodified H3K4 residue and a conserved carboxylate residue39,44,45,46 (Fig. 1b). However, our initial NMR studies of PHD1KDM5A have not detected any interactions between the equivalent conserved carboxylate residue (Asp292) and H3K4 in the histone peptide40, necessitating further investigations into binding of H3K4 by PHD1 of KDM5A.
We suspected that the discrepancy regarding the role of the carboxylate residue in PHD1KDM5A and PHD1KDM5B could be due to the Asp292 residue in PHD1KDM5A being close to the N-terminus of the construct used in our previous studies (V291-K347 used in binding assays, D292-E344 used in NMR experiments). A new construct was generated (S287-K347) (Supplementary Figure 1) and used to measure binding affinity for the H3 peptide via a fluorescence polarization assay (Fig. 1c). A D292A variant was also generated to determine the contribution of the conserved Asp residue to peptide binding. Importantly, this mutation does not impair the overall fold or stability of the PHD1 domain (Supplementary Figure 2).
Compared to the PHD1V291-K347 construct, the PHD1S287-K347 construct shows an approximate twofold increase in affinity for the unmodified and mono-methylated, while di- and tri-methylated H3K4 peptides have similar binding affinity to both PHD1 constructs (Table 1). Our observations suggest that the additional N-terminal residues (S287-F290) are contributing to a higher affinity for unmodified and mono-methylated H3K4 peptides. This is further supported by the observation that only the longer PHD1S287-K347 shows a 2.3-fold preference for WT H3 peptide over the H3K4A peptide, whereas the binding affinities of WT H3 and H3K4A peptides to the shorter PHD1V291-K347 are highly similar (Table 1). Our findings with the extended construct are similar to the 2.5-fold discrimination of WT H3 over H3K4A peptide that has been previously observed for PHD1KDM5B44.
We further probed the PHD1-H3 interaction using a PHD1S287-K347 D292A variant. Compared to the WT PHD1S287-K347, the mutant PHD1 domain shows an approximately eightfold decrease in affinity for the unmodified H3 peptide (Fig. 1c, Table 1) and a 10–16-fold decrease in affinity for methylated H3K4 peptides (Table 1) highlighting importance of Asp292 in contributing to PHD1-H3 tail interaction. These large fold reductions in affinity suggest that contribution of D292 extends beyond its likely electrostatic interaction with Lys4 in H3 and into structural organization of the N-terminus of the PHD1 for histone binding. This conclusion is consistent with structural analysis of the interaction of the highly homologous PHD1KDM5B with H3, where the equivalent carboxylate residue facilitates recognition of H3T644. Together, our findings suggest that the proper orientation of the Asp292 residue in the longer PHD1 construct significantly contributes to binding of the H3 peptide to the PHD1 domain.
### PHD1 occupancy improves the catalytic efficiency of KDM5A
We have previously shown that binding of a H3 peptide to the PHD1 domain enhances the rate of demethylation by KDM5A in vitro40. To dissect contributions of changes in substrate binding and enzymatic turnover to the overall allosteric enhancement, we investigated how the Michaelis–Menten kinetic parameters were affected by ligand binding to the PHD1 domain. A challenge for such an investigation lies in the nature of the ligand for each of the two sites, the PHD1 domain and the catalytic domain. While the PHD1 domain preferentially interacts with unmodified H3, it also binds H3K4me3, the substrate of the catalytic domain, albeit with a decreased affinity (Table 1). This cross-binding between two sites first prompted us to develop a discrete set of peptide ligands for the two sites in KDM5A (Fig. 2a). After studying the available structural information, N-terminal acetylation of the H3 peptide would likely prevent its binding to the PHD1 domain, as is the case in PHDUHRF146. Indeed this was the case, as an N-terminally acetylated 21mer H3K4me2 peptide (Ala1-Ala21) does not bind to the PHD1 domain at the concentrations used in our assays (Supplementary Figure 3a). Importantly, although activity is reduced relative to corresponding peptide with a free N-terminus (Supplementary Figure 3c), N-terminal acetylation is tolerated by the catalytic domain (Fig. 2b, c). To develop a peptide selective for the PHD1 domain, we relied on the observation that C-terminal truncation of the K4 methylated H3 tail peptide leads to a dramatic increase in Km for peptide substrates (Supplementary Figure 3c). This increase in Km was particularly high for 10mer peptides, both in the di- and trimethylated state (Km > 750 µM; Supplementary Figure 3b, c). Combined with our observation that H3 10mer peptides bind efficiently to the PHD1 domain (Fig. 1, Table 1), we selected the H3 10mer peptide (Ala1-Ser10) as ideal effector peptide for our experiments. Based on this information, a discrete occupancy activity assay was developed that utilizes a N-terminal H3 tail 10mer peptide as PHD1 effector ligand (H3 Ala1-Ser10) and a N-terminally acetylated 21mer H3K4me3 peptide as the catalytic domain substrate (Ac-H3K4me3 Ala1-Ala21) (Fig. 2).
We measured the Michaelis–Menten kinetic parameters for substrate demethylation in the presence and absence of the PHD1 domain ligand. The Km and kcat values were determined by measuring the initial rate of demethylation by KDM5A using the formaldehyde release assay at varying concentrations of acetylated substrate. With the Ac-H3K4me3 21mer peptide alone, we observe a Km value of 95.5 ± 7.6 µM and a kcat of 2.82 ± 0.06 min−1 (Fig. 2b, Table 2). However, in the presence of the H3 10mer effector peptide at a concentration that allows for the saturation of the PHD1 domain with the effector (38 µM, 20 × Kd) the Km is reduced to 16.8 ± 1.1 µM while kcat remains similar (Fig. 2b, Table 2). This approximate fivefold reduction in Km is also observed when the Ac-H3K4me2 21mer peptide is used as a substrate (Fig. 2c, Table 2). Importantly, the presence of the effector peptide does not lead to a significant change in Km of α-KG co-substrate (Supplementary Figure 4). The reduction in Km for the peptide substrate alone suggests that effector peptide binding to the PHD1 domain allosterically promotes binding of peptide substrates in KDM5A.
### Substrate binding is improved upon PHD1 occupation
A decrease in Km upon effector peptide binding to the PHD1 domain suggests that there is improved substrate binding to the catalytic domain under these assay conditions. To further investigate this possibility the direct binding of an N-terminally acetylated substrate peptide was determined using fluorescence polarization assay. Binding assays were conducted in the presence and absence of the effector peptide and FP signal monitored over time (Supplementary Figure 5).
We observe that at 1 min the substrate binds with similar affinity to KMD5A, both in the absence (Kd = 56.9 ± 7.9 µM) and presence (Kd = 62.5 ± 5.4 µM) of H3 10mer peptide (Fig. 3). However, with no effector peptide, present the fluorescence polarization signal decreases over time. In contrast, when the H3 10mer peptide is present the signal is stable for at least 1 h. This striking difference between the two conditions suggests that the substrate peptide likely dissociates from the catalytic domain over time in the absence of the effector peptide, while the effector peptide occupied PHD1 domain stabilizes the interaction of substrate and the catalytic domain. Our data is consistent with a model where the demethylase is in a slow equilibrium between an open, substrate binding-competent state, and a closed state that does not bind the substrate. Binding of the effector peptide to the PHD1 domain could shift the equilibrium between the two states toward the substrate binding-competent state (Supplementary Figure 6). Our observations that the substrate binding is stabilized in the presence of the PHD1 ligand could explain the decrease in Km that we observe when an effector peptide is present.
### Impact of H3 modifications on allostery
The impact of post-translational modifications of H3 on binding of reader domains has been investigated thoroughly11,12,40,47,48,49,50,51. However, it is largely unknown how these modifications may regulate processes beyond effector-reader interactions, such as allostery.
To begin to investigate how modifications impact allostery in KMD5A, we performed competition-based FP assays to determine affinities of a panel of modified peptides (Fig. 4a). In analogy to our earlier findings the PHD1V291-K347 construct40, we observe that both methylation and mutation of H3R2 reduces binding affinities, emphasizing the importance of unmodified H3R2 in the PHD1 and H3 interaction. Phosphorylation of either the H3T3 or H3T6 residue greatly reduces the affinity, with H3T3 phosphorylation having a more pronounced effect. Methylation of H3R8 or H3K9 as well as acetylation of H3K9 has a modest or no effect on the affinity for the PHD1 domain. This agrees with the previous observation that the PHD1 domain predominantly interacts with the first five residues of the H3 peptide40,44.
We investigated what impact modifications of the effector peptide have on demethylase activity using our discrete occupation activity assay. Specifically we determined the Michaelis–Menten parameters for demethylation of the Ac-H3K4me3 21mer substrate in the presence of saturating concentrations of various PHD1 ligand peptides (Fig. 4a). Similar to unmodified H3 (Fig. 3), we observed a significant decrease in the Km value of the substrate peptide for all modified effector peptides, with no significant to modest changes in the kcat value (Fig. 4a, b). The reduction in Km ranged from 3 to 6.5-fold, with the highest reduction in Km observed with the K9me3 and K9ac effector peptides, in addition to the unmodified H3 10mer peptide. This could be due to the lack of any H3 modifications that could interfere with optimal ligand engagement with the PHD1 domain in these effector peptides.
Together, our investigations of the impact of H3 modifications on PHD1 binding and KDM5A catalysis indicate a similar reduction in the Km of demethylation when PHD1 is saturated with effector peptide (Fig. 4b). However, modifications have a profound effect on the interaction of the PHD1 with H3 ligands, which would determine the concentrations of the PHD1 ligand required to saturate. Our findings indicate that the chromatin modifications context can impact the catalytic activity of KDM5A through modulation of the degree to which PHD1 is bound by a ligand.
### PHD1 domain deletion affects substrate binding to KDM5A
The observation that effector peptide binding to the PHD1 domain reduces the Km for demethylation raises questions about how the liganded PHD1 domain enables enhanced interaction of the demethylase with its substrate. In order to determine the effect of the PHD1 domain deletion on the kinetics of demethylation, a ΔPHD1 KDM5A construct was generated. While deletion constructs of KDM5 enzymes have been generated previously, these constructs lack both the ARID and PHD1 domains14,16,35. Our ΔPHD1 construct retains both the necessary JmjN and JmjC domains, as well as the ARID and ZF domains. The demethylase activity of the resulting ΔPHD1 construct toward the Ac-H3K4me3 21mer substrate was evaluated, and compared to the WT KDM5A1-797 construct.
Despite similar kcat values of the two constructs, an approximate eightfold increase in Km is observed for ΔPHD1 KDM5A-catalyzed demethylation of acetylated substrate (Fig. 5, Table 3). These findings suggest that the binding of the substrate to the catalytic domain is impaired in the absence of PHD1. While we cannot exclude the possibility that the introduction of a GS linker may obstruct substrate binding, our observation that kcat is unaffected indicates that the linker does not inhibit catalysis. Our findings suggest that the PHD1 domain could restructure the catalytic domain to enhance substrate binding, or alternatively directly contribute to substrate binding.
### Assessing active site changes in the enhanced state
Currently available structures of KDM5 enzymes lack the PHD1 domain, which is either disordered15 or absent in the crystallized construct16,35. Without structural information, it is difficult to determine how effector peptide PHD1 binding-induced allosteric regulation is communicated between the PHD1 and catalytic domains.
In order to investigate the conformational mobility of KDM5A upon effector peptide binding to the PHD1 domain we performed HDX-MS analysis. HDX-MS measures the rate at which amide hydrogen atoms exchange with deuterium in the solvent and can report on regions of a protein that undergo changes in hydrogen bonding networks upon ligand binding. We applied HDX-MS to KDM5A1-797 under conditions reflecting those of the discrete occupancy assay (Figs. 2 and 4). The conditions assayed were: KDM5A1-797 bound to the PHD1-specific H3 10mer peptide; KDM5A1-797 bound to the catalytic domain-specific Ac-H3K4me3 21mer substrate; and KDM5A1-797 bound to both the PHD1-specific H3 10mer peptide and catalytic domain-specific Ac-H3K4me3 21mer substrate.
For all conditions tested, >90% coverage of KDM5A1-797 was obtained thereby enabling us to confidently assign changes in HDX kinetics to various protein domains. While the majority of peptides assigned to KDM5A1-797 showed statistically insignificant changes (gray regions) in the levels of deuterium uptake, various regions of KDM5A1-797 undergo differential deuterium uptake upon peptide binding (Figs. 6a, b, S7, S8). When only the PHD1-specific H3 10mer peptide is present, residues Gly300-Ala342 of the PHD1 domain showed between 2 and 8% reduced deuterium uptake (referred to as protection from deuterium exchange) (Fig. 6a, green regions; Supplementary Figures 7a and 8a). The observation that the majority of the PHD1 domain (Fig. 6c), rather than just the previously mapped peptide binding residues40, undergoes a change suggests a broad stabilization of the PHD1 domain upon H3 peptide binding. Conversely, when only the catalytic domain-specific Ac-H3K4me3 21mer substrate peptide was tested no perturbations are observed in the protein (Supplementary Figure 7b), despite using saturating concentrations of the substrate (25-fold over protein concentration, ~5 × Km). While it may be expected that residues in the catalytic domain would undergo protection from exchange with solvent upon substrate binding, this lack of perturbation may be due to weak binding of the acetylated substrate peptide, consistent with its high Km measured (Fig. 2b, Table 2). Alternatively, substrate interactions with the catalytic domain could be predominantly hydrophobic or hydrogen bonds from the peptide to KMD5A1-797 could be replacing hydrogen bonds in apo KDM5A1-797. Previous HDX studies have also shown no perturbation in hydrogen/deuterium exchange upon ligands binding52. However, when both the PHD1-specific H3 10mer peptide and the catalytic domain-specific Ac-H3K4me3 21mer substrate peptide are present, two regions of KDM5A1-797 undergo significant changes (Fig. 6b, Supplementary Figures 7c, 8c). In addition to the increased protection observed in the PHD1 domain (residues Gly300-Ala342), as in the presence of the H3 10mer peptide alone, a protein segment within the catalytic domain (residues Ser528-Leu545) experiences a statistically significant increase in deuterium uptake (3–4%) suggesting decreased protection (Fig. 6b, yellow regions). These observed perturbations map onto a helical loop region between the fourth and the fifth β-strand of the DSBH motif in the JmjC domain at the interface with the ARID domain (Fig. 6d). These findings reveal a conformation alteration in KDM5A1-797, only present in the PHD1 effector bound higher activity state of KDM5A1-797. Furthermore, the observation that helical loop region becomes more exposed only in the presence of both the effector and substrate peptide indicates a high degree of cooperativity between occupancy of these two sites. Such cooperativity helps explain the coupled energetic effects on catalysis seen in the context of substrate and effector binding (Fig. 2).
While currently available structural data does not allow for clear identification of the role of the helical loop region, we hypothesize that this region may be involved in binding of the methylated peptide substrate. This hypothesis comes from an overlay of structures of KDM5A and the structurally related Arabidopsis thaliana demethylase JMJ14 (53.2% sequence identity, 73.4% similarity between catalytic domains of KDM5A and JMJ14 with an RMSD of 0.613) in complex with a H3K4me3 peptide substrate53. Many residues of JMJ14, shown to interact with the H3K4me3 peptide substrate, are conserved in KDM5A53 and comparison between the structures suggests a potential substrate binding groove in KDM5A (Supplementary Figure 9). The difference in length between the 21mer peptide substrate used in our experiments and the 7mer peptide substrate co-crystallized with JMJ14 makes it challenging to compare the residues that participate in binding of the C-terminal end of the peptide substrate. However, several residues conserved in between JMJ14 and KDM5A—Gln535, Leu536 and Val537—bind the H3K4me3 peptide in JMJ14 and are part of the helical loop region that we observe to undergo increased deuterium uptake when both PHD1 effector peptide and the substrate peptide are present. This region has reduced solvent protection only in the presence of both the substrate and effector peptides, and could be responsible for the stabilized substrate binding we have observed in direct binding experiments (Fig. 3).
## Discussion
Here we show that the allosteric stimulation of KDM5A activity, caused by binding of H3 N-terminal tail effector peptides to the PHD1 domain, is due to a reduced Km value of peptide substrate demethylation. This was determined by using a novel in vitro activity assay that allows us to discretely occupy the two H3 binding sites in KDM5A. Furthermore, upon assessing the direct binding of peptide substrate to the demethylase we attribute this change to stabilization of the KDM5A-substrate complex only when the PHD1 domain is occupied. The enhancement in demethylation catalysis is largely independent of the type of modification present on the effector peptide. However, it is strongly dependent on the occupancy of the PHD1 domain which directly correlates to the affinity of the effector peptide to the PHD1 domain. Therefore it is the affinity for the PHD1 domain that will dictate activity enhancement. Using HDX-MS, we identified a region in the catalytic domain that becomes more exposed only when both the PHD1 domain ligand and the substrate peptide are present. This structural change could be responsible for the stabilized substrate binding and the decrease in Km that we observe in demethylation assays when effector peptide is present.
Active regulation of the catalytic activity of KDM5A by H3 tail as an allosteric ligand has important implications. The observation that the allosteric stimulation is preserved with modified peptides, as long as the PHD1 domain is saturated, suggests that both the nature of the modification and the local concentration of the effector peptide can influence catalysis. Despite the low affinity of PHD1 for H3K4me3 marks40, the local concentration of H3K4me3 marks at the sites of recruitment20,55,56, can enable allosteric enhancement. Once generated, products H3K4me2, H3K4me1, and non-methylated H3 peptides would further enhance the catalytic efficiency of KDM5A due to their higher affinities for the PHD1 domain. Our previous evaluation of nucleosome demethylation suggests that the magnitude of stimulation in the context of chromatin substrate can be far greater than what is observed with the peptides in this study40. This positive feedback mechanism, where the reaction is stimulated by the product of the reaction, could contribute to rapid spreading of demethylation of H3K4me3-rich regions, which has been shown to occur in development, during maternal to zygotic transition41,42,43 and in regulation of HOX gene expression18,57. However, rapid removal of these marks and the consequent repression of genes must be tightly regulated as dysregulation of boundary regions between active and silenced genes can disrupt cellular functions and lead to disease58. Our findings implicate PHD1 domain as a possible regulator of demethylation spread.
There are additional examples of allosteric stimulation of enzymes involved in epigenetic modifications59. For example, propagation of H3K27 methylation by the PRC2 complex is mediated by product binding to an EED domain within this complex60. Similar positive feedback regulation is observed in the Suv39h class of histone methyl transferases where H3K9 methylation is enhanced by the binding of the product of the reaction to a chromodomain within the methyltransferase61,62,63. Additional examples of allosteric regulation in epigenetic enzymes include acetyl transferases p300/CBP, Gcn559, DNA methyltransferase DNMT3A64, and DNA demethylase Tet265. Our studies have provided mechanistic rationale for allosteric regulatory role of the PHD1 domain in KDM5A, which expands the function of the PHD class of reader domain in regulation of chromatin methylation. PHD domain mediated allostery has also been shown to occur in the RAG-1/RAG-2 complex where H3K4me3 binding by the RAG-2 PHD finger changes the conformation of RAG-1 leading to enhanced substrate binding and cleavage of DNA at recombination signal sequences66,67.
Our investigation of catalysis in PHD1 deletion construct indicate that the PHD1 domain plays an important role either directly or indirectly, in organizing the catalytic domain for optimal substrate binding. We show that KDM5A retains its catalytic activity after deletion of the PHD1 domain, but that its substrate binding is impaired as reflected in a higher Km value. This finding is also mirrored in studies performed on a ΔAP KDM5B construct which shows a large increase in Km, although the precise domain responsible could not be pinpointed16. Together, these data further highlight the importance of PHD1 in KDM5 enzymes, and parallel in vivo observations that this domain is necessary for activity of KDM5B44,68 and its drosophila homolog Lid56.
Our HDX-MS experiments have uncovered residue level dynamics information that cannot be deduced from currently available structures, thereby providing insights into the structural changes in KDM5A that are induced by effector peptide binding to the PHD1 domain15,16,35. The binding of the effector peptide to the PHD1 domain does not only induce changes in the peptide binding region that we previously mapped40 but rather affects protection of backbone amide protons throughout the PHD1 domain. Additionally, only in the presence of both the substrate and the effector peptide a helical loop region between the fourth and the fifth β-strand of the DSBH motif in the catalytic domain becomes less protected from solvent. Overlaying the structures of KDM5A and plant homolog JMJ14, bound to a H3K4me3 peptide substrate, suggests that this helical loop region is involved in substrate binding (Supplementary Figure 9). These data could therefore suggest that the improved substrate binding is caused by a potential effector peptide binding-induced repositioning of the PHD1 domain that is coupled to the conformation of the helical loop region which orients itself to optimally bind the substrate.
This possibility is yet to be validated, given a lack of a co-crystal structure between any member of KDM5 family and its H3K4me3 substrate. However, our direct binding experiments lend support to this hypothesis and suggest that occupation of the PHD1 favors a conformation that is able to better bind substrate peptides. While our studies have determined regions of conformational change within KDM5A, additional studies are needed in order to further define the interactions of the PHD1 domain and the catalytic domain, as well as their interaction with chromatin. Our data cannot exclude the possibility that portions of the helical loop region, revealed by HDX-MS, may participate in domain communication within the demethylase.
Our findings contribute to the growing knowledge about allosteric regulation of epigenetic enzymes and highlight the complexity of mechanisms that enable context-dependent regulation of chromatin marks. We hypothesize that the regulation of KDM5A activity by its PHD1 domain could be a mechanism of controlling demethylation spread. Recognition of H3 modifications by PHD1 could also be synergistic with other epigenetic regulatory enzymes. For example, H3R8 methylation supports optimal allosteric stimulation of KDM5A, but has a negative effect on activity of H3K4 methyltransferase MLL1. This could suggest a role where the chromatin context allows demethylation to occur while simultaneously lowering the opposing methylating activity of MLL169, keeping Lys4 in its unmethylated state. Further studies will be needed to address how allostery of KDM5A affects epigenetic regulation in a cellular context. Additionally, understanding the interface and conformational changes that occur in the higher affinity state of KDM5A could lead to future prospects of development of small molecule inhibitors of KDM5A that target its allosteric regulation.
## Methods
### Expression of recombinant WT and ΔPHD1 KDM5A
WT KDM5A1-797 was expressed in Sf21 cells following Invitrogen Bac-to-Bac baculovirus expression system protocol. KDM5A1-797 was cloned into a pFASTBAC HTA vector. Purified bacmid was transfected in Sf21 cells. Approximately 0.8 × 10−5 cells per well of a six-well dish were allowed to attach in 2 ml of SF-900 II SFM media containing 50 U ml−1 penicillin and 50 μg ml−1 streptomycin. While cells attached, 8 μl of Cellfectin II reagent (Invitrogen) in 100 μl of unsupplemented Grace’s medium was mixed with ~2–5 μg of bacmid in 100 μl of unsupplemented Grace’s medium and incubated for 15–30 min at 25 °C. Once cells were attached, media was removed and cells were washed with 2 ml of Grace’s unsupplemented media. The bacmid DNA:Cellfectin mixture was then diluted to 1 ml with Grace’s unsupplemented media and added to the well. Cells with bacmid:Cellfectin II mixture were incubated for 5 h at 27 °C. After 5 h of incubation, bacmid:Cellfectin mixture was removed and replaced with 2 ml of SF-900 II SFM 50 U ml−1 penicillin and 50 μg ml−1 streptomycin. Transfected cells were incubated 3–5 days or until signs of viral infection were observed. After transfection, the supernatant was spun down to remove the dead cells. The supernatant was then sterile filtered to obtain the P1 viral stock. To make P2, 20 ml of viral stock at ~2 × 106 cells per ml of sf21 was infected with 2 ml of P1 virus and incubated for 48–60 h. After 56 h, the cells were spun down and the supernatant was collected and sterile filtered to obtain P2 viral stock. Similarly, P3 viral stock from P2 viral stock was obtained. Generally, 1 l of Sf21 at 2 × 106 cells per ml was infected with ~40 ml of P3 virus for ~48–56 h. Cells were then collected and resuspended in the lysis buffer (25 mM HEPES pH 7.9, 350 mM NaCl, 5 mM KCl, 1.5 mM MgCl2, 10 mM imidazole, aprotinin 2 μg ml−1, leupeptin 3 μg ml−1, pepstatin 3 μg ml−1, and 1 mM PMSF). Cells were homogenized by emulsiflex. After lysis, the supernatant was recovered by centrifuging at 35k r.p.m. for 45 min, and incubated with cobalt resin equilibrated in lysis buffer for 1 h at 4 °C. After incubation the resin was washed with wash buffer (25 mM HEPES, pH 7.9, 350 mM NaCl, 0.5 mM MgCl2, 10% glycerol, 10 mM imidazole, aprotinin 2 μg ml−1, leupeptin 3 μg ml−1, pepstatin 3 μg ml−1, and 1 mM PMSF). His-KDM5A was eluted with elution buffer (25 mM HEPES pH 7.9, 100 mM NaCl, 0.5 mM MgCl2, 10% glycerol, 100 mM imidazole, aprotinin 2 μg ml−1, leupeptin 3 μg ml−1, pepstatin 3 μg ml−1, and 1 mM PMSF). The His-tag was removed by overnight incubation with TEV protease at 4 °C in the dialysis buffer (25 mM HEPES, pH 7.9, 100 mM NaCl, and 2 mM DTT). After cleavage, protein was further purified by size-exclusion chromatography using a S200 column. Purified KDM5A was eluted, aliquoted and stored at −80 °C in 40 mM HEPES, pH 7.9, 50 mM KCl.
The KDM5A1-797 ΔPHD1 construct was generated after removing the residues Ser287-Lys347 and replacing them with a (GS)4 linker before cloning into a pFASTBAC HTA vector. The protocol for expression and purification of the KDM5A1-797 ΔPHD1 construct was the same as WT.
### Expression of GST-PHD1 (Ser287-Lys347)
All PHD1 constructs used (Val291-Lys247, S287-K347, and S287-K347 D292A) (Supplementary Figure 1) were cloned into a pET41a vector and expressed in BL21(DE3) E. coli cells. Expression and purification of all mutants followed the same protocol. Specifically, cells were induced with 0.4 mM IPTG and grown at 18 °C overnight before the pellet was collected. The cells were resuspended in lysis buffer (140 mM NaCl, 2.7 mM KCl, 10 mM Na2HPO4, 1.8 mM KH2PO4, 0.5 mM TCEP, 50 µM ZnCl2, 1 mM phenylmethylsulfonyl fluoride (PMSF), pH 7.3), lysed by sonication and centrifuged. The supernatant was purified using Glutathione Sepharose 4B resin, washed with high salt buffer (50 mM HEPES, 700 mM KCl, 10% glycerol, 0.5 mM TCEP, 50 mM ZnCl2, 1 mM PMSF, pH 8) and low salt buffer (50 mM HEPES, 150 mM KCl, 10% glycerol, 0.5 mM TCEP, 50 mM ZnCl2, 1 mM PMSF, pH 8) and recovered by elution using low salt buffer with 30 mM glutathione. The sample was then concentrated and further purified by size-exclusion chromatography using a Hiload 26/60 Superdex 75 gel filtration column. Samples were eluted into 40 mM HEPES, 50 mM KCl, 0.5 mM TCEP, and 50 mM ZnCl2 and flash-frozen.
### Fluorescence polarization binding assays
The binding of GST-PHD1 to H3 10mer peptides was measured by either direct or competition-based fluorescence polarization (FP). For direct FP binding assay, 10 nM of C-terminal fluorescently labeled H3 peptide were incubated with varying concentrations of GST-PHD1. Data from direct FP measurements were fitted to equation 1:
$${\mathrm{FP}}_{{\mathrm{obs}}} = \frac{{{\mathrm{FP}}_{{\mathrm{max}}}\left[ {{\mathrm{PHD1}}} \right] + {\mathrm{FP}}_{{\mathrm{min}}}K_{\mathrm{d}}}}{K_{\mathrm{d}} + [{\mathrm{PHD}}1]}$$
For competition-based FP assays, 2 μM GST-PHD1 was incubated with 10 nM of C-terminal fluorescently labeled H3 peptide and different concentrations of unlabeled peptides were used as competitors. Data from competition-based FP assays were fitted to equation 2:
$${\mathrm{FP}}_{{\mathrm{obs}}} = \frac{{K_{i}\left( {{\mathrm{FP}}_{{\mathrm{max}}}\left[ {{\mathrm{PHD1}}} \right] + {\mathrm{FP}}_{{\mathrm{min}}}K_{\mathrm{d}}} \right) + {\mathrm{FP}}_{{\mathrm{min}}}K_{\mathrm{d}}[{\mathrm{I}}]}}{{K_{i}\left( {K_{\mathrm{d}} + [{\mathrm{PHD1}}]} \right) + K_{\mathrm{d}}[{\mathrm{I}}]}}$$
where FPobs is the observed FP, FPmax is the maximum FP value, FPmin is the minimum FP value, [PHD1] is concentration of PHD1, Kd is the dissociation constant, Ki is the inhibition constant referring to the competing peptide and [I] is the competing peptide concentration.
The binding of peptide substrates to KDM5A1-797 was measured using the direct FP assay. A Ac-H3K4me3-fluorescein 31mer peptide was used, rather than the 21mer used in kinetic assays. This was in order to reduce interference of the C-terminal fluorescein in binding to KDM5A, since KDM5A is sensitive to changes in C-terminal tail of the H3 peptide (Supplementary Figure 3c). Four hundred micromolar MnCl2 and N-OG were included to inhibit demethylase activity and 350 µM H3 10mer effector peptide was present when stated. A schematic showing the setup of measuring substrate binding is shown in Supplementary Figure 5.
### KDM5A demethylation assay
A formaldehyde dehydrogenase (FDH)-coupled assay was used to monitor lysine 4 demethylation of histone H3 by following the production of formaldehyde during the KDM5A demethylation reaction. Reactions were performed in 50 mM HEPES, 50 mM KCl, 1 mM α-ketoglutarate, 50 μM Fe(NH4)2(SO4)2, 2 mM ascorbic acid, 2 mM NAD+ and 0.05 U FDH (Sigma) at pH 7.5 at room temperature. Different concentrations of H3K4me3/2/1 peptides were incubated with 1 μM KDM5A. For assays measuring the stimulation by the effector peptide on the Michaelis–Menten kinetics, the effector peptide (differently modified H3 10mers Ala1-Ser10) was included in the reaction mix at 20× Kd concentrations. There was an exception for the H3K4me2 10mer which was used at 10× Kd due to its ability to act as a substrate at 20× Kd (Supplementary Figure 3). Reactions were initiated by the addition of substrate and followed in 20 s intervals on a SpectraMax M5e (Molecular Devices) using 350 nm excitation and 460 nm emission wavelengths. An NADH standard curve was used to convert fluorescence to concentration of product formed. The initial 3 min were used to calculate initial velocities, which were plotted against substrate concentration. Michaelis–Menten parameters were determined using Graphpad Prism.
### Hydrogen/deuterium exchange-mass spectrometry (HDX-MS)
Solution-phase amide HDX experiments were carried out with a fully automated system (CTC HTS PAL, LEAP Technologies, Carrboro, NC; housed inside a 4 °C cabinet).
### Peptide identification for HDX-MS
Peptides were identified using tandem MS (MS2 or MS/MS) experiments performed with a LTQ linear ion trap mass spectrometer (LTQ Orbitrap XL with ETD, Thermo Fisher, San Jose, CA) over a 70 min gradient. Product ion spectra were acquired in a data-dependent mode and the five most abundant ions were selected for the product ion analysis per scan event. The MS/MS *.raw data files were converted to *.mgf files and then submitted to MASCOT ver2.3 (Matrix Science, London, UK) for peptide identification. The maximum number of missed cleavage was set at 4 with the mass tolerance for precursor ions +/−0.6 Da and for fragment ions ± 8 ppm. Oxidation of methionine was selected for variable modification. Pepsin was used for digestion and no specific enzyme was selected in the MASCOT during the search. Peptides included in the peptide set used for HDX detection had a MASCOT score of 20 or greater. The MS/MS MASCOT search was also performed against a decoy (reverse) sequence and false positives were ruled out. The MS/MS spectra of all the peptide ions from the MASCOT search were further manually inspected and only the unique charged ions with the highest MASCOT score were used in estimating the sequence coverage and included in HDX peptide set.
### HDX-MS analysis
For differential HDX experiments, 5 μl of 20 μM KDM5A1-797 was diluted to 25 μl with D2O-containing HDX buffer (40 mM HEPES, 50 mM KCl, pH 7.9) and incubated at 4 °C for 10, 30, 60, 900, and 3600 s. HDX conditions aimed to mimic the demethylation assay conditions as described in Fig. 2. Apo conditions were compared to the conditions with substrate and/or effector peptides. Apo refers to KDM5A with 50 µM FeSO4 and 100 µM N-OG. The complex with substrate and/or effector peptides refers to KDM5A with 50 µM FeSO4, 100 µM N-OG, and 500 µM Ac-H3K4me3 21mer substrate (25× [KDM5A]) with or without 100 µM H3 10mer effector peptide. Following on-exchange, unwanted forward or backward exchange is minimized, and the protein is denatured by dilution to 50 μl with 0.1% TFA in 5 M urea and 50 mM TCEP (held at 4 °C, pH 2.5). Samples are then passed across an immobilized pepsin column (prepared in house) at 50 μl min−1 (0.1% TFA, 15 °C), and the resulting peptides are trapped onto a C8 trap cartridge (Thermo Fisher, Hypersil Gold). Peptides were then gradient eluted (5% CH3CN to 50% CH3CN, 0.3% formic acid over 6 min, 4 °C) across a 1 mm × 50 mm C18 analytical column (Hypersil Gold, Thermo Fisher) and electrosprayed directly into a high resolution orbitrap mass spectrometer (LTQ Orbitrap XL with ETD, Thermo Fisher). Percent deuterium exchange values for peptide isotopic envelopes at each time point were calculated and processed using HDX Workbench. Each HDX experiment was carried out in triplicate with a single preparation of each protein-ligand complex. The intensity weighted mean m/z centroid value of each peptide envelope was calculated and subsequently converted into a percentage of deuterium incorporation. Statistical significance for the differential HDX data is determined by an unpaired t-test for each time point, a procedure that is integrated into the HDX Workbench software. Corrections for back-exchange were made on the basis of an estimated 70% deuterium recovery and accounting for 80% final deuterium concentration in the sample (1:5 dilution in D2O HDX buffer).
Hundred micrograms of KDM5A1-797 was crosslinked with 1 mM BS3-H12 for 1 h at 4 °C. KMD5A in the presence of the H3 effector peptide (40 µM) was crosslinked with 1 mM BS3-D12 (Creative Molecules) under the same conditions. After quenching with 50 mM Tris-base, the light and heavy crosslinked reaction mixtures were combined and precipitated with ice cold acetone. Crosslinked proteins were pelleted by spinning at 21,000 × g, the supernatant was removed, and the pellet was washed once with cold acetone. The precipitate was brought up in 8 M Urea, 10 mM TCEP, heated at 56 °C for 20 min, and then alkylated with 15 mM iodoacetamide (45 min at room temperature). The sample was diluted to 2 M Urea and digested overnight with 0.8 µg trypsin for 4 h at 37 °C. A second aliquot of trypsin was then added and digestion was allowed to proceed overnight. The digestion mixture was acidified to 1% TFA, desalted using a 100 µl OMIX C18 tip (Agilent), and evaporated to dryness. Crosslinked products were brought up in 10 µl of SEC buffer (70:30 H2O:ACN with 0.1% TFA) and enriched by size-exclusion chromatography (Superdex Peptide, GE Healthcare Life Sciences). Hundred-microliter fractions eluting between 0.9 and 1.4 ml were dried, resuspended in 0.1% formic acid for MS analysis. The fractions starting at 0.9 ml and 1.3 ml were combined prior to evaporation to make four total SEC fractions.
LC-MS analysis was performed with a Q-Exactive Plus mass spectrometer (Thermo Scientific) coupled with a nanoelectrospray ion source (Easy-Spray, Thermo) and NanoAcquity UPLC system (Waters). Enriched fractions were separated on a 15 cm × 75 μm ID PepMap C18 column (Thermo) using a 70 min gradient from 2 to 23% solvent B (A: 0.1% formic acid in water, B: 0.1% formic acid in acetonitrile) followed by a 10 min gradient from 23 to 40%. Precursor MS scans were measured in the Orbitrap scanning from 350 to 1500 m/z (mass resolution: 70,000). The ten most intense triply charged or higher precursors were isolated in the quadrupole (isolation window: 4 m/z), dissociated by HCD (normalized collision energy: 23), and the product ion spectra were measured in the Orbitrap (mass resolution: 17,500). A dynamic exclusion window of 15 s was applied and the automatic gain control targets were set to 3e6 (precursor scan) and 5e4 (product scan).
Peaklists were generated using Proteome Discoverer 2.2 (Thermo) and searched for crosslinked peptides with Protein Prospector 5.20.23. In total, 85 of the most intense peaks from each product ion spectrum were searched against a database containing the KDM5A construct and the H3 effector peptide concatenated with 10 randomized decoy versions of each of these sequences. Search parameters were as follows: mass tolerance of 10 ppm (precursor) and 25 ppm (product); fixed modifications of carbamidomethylation on cysteine; variable modifications of peptide N-terminal glutamine conversion to pyroglutamate, oxidation of methionine, and “dead-end” modification of lysine and the protein N-terminus by semi-hydrolyzed heavy and light BS3; crosslinking reagents were BS3-H12 and BS3-D12; trypsin specificity was used with two missed cleavages and three variable modifications per peptide were allowed. Unique, crosslinked residue pairs were reported at a 0.5% FDR threshold, corresponding to a Score Difference cutoff of 10.
For quantitative analysis, precursor ion filtering in Skyline 4.1 was used to extract light:heavy crosslinked precursor ion signals. Skyline does not directly support crosslinking analysis, so the elemental composition of each crosslinked peptide species, precursor charge state, and retention time were imported as a small molecule transition list (generated in the R programming environment). Transitions were generated for every light or heavy crosslinked peptide species discovered in the Protein Prospector search and paired with its corresponding heavy or light counterpart. Theoretical elemental compositions were generated if the counterpart was not identified in the search results. Precursor ion transitions matching the first three isotopes were extracted across all four LC-MS fractions. Each extracted ion chromatogram was manually inspected and the start and end points were adjusted to ensure that the correct peaks were detected and that there were no interfering signals. The peak areas were re-imported into R and summarized at the level of crosslinked residues for each light and heavy crosslink. A weighted mean was used as the summarization function with weights being given by the peak areas (e.g., signals with larger peak area were weighted more heavily). Finally, log2 ratios of the heavy (crosslinked in the presence of effector peptide) to light (absence of effector peptide) peak areas were determined.
### Reporting Summary
Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
All relevant data supporting the key findings of this study are available within the article and its Supplementary Information files or from the corresponding author upon reasonable request. A reporting summary for this Article is available as a Supplementary Information file.
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## Acknowledgements
We would like to thank Christian Freund, John Gross, Mark Kelly, Geeta Narlikar, Bruce Pascal and Fatima Ugur for their comments on the paper. This work is supported by National Institutes of Health (R01 GM114044 and R01 GM114044-03S1 to DGF and 1S10D016229 to AB) and the Dr. Miriam and Sheldon G. Adelson Medical Research Foundation (AB).
## Author information
Authors
### Contributions
J.E.L. and D.G.F. wrote the manuscript with input from V.D., M.T., A.B., and P.R.G. J.E.L., C.M.C., M.T., and V.D. designed and performed the experiments. I.O.T., D.G., and K.R. assisted in experimental design and initial experiments. All authors contributed to the interpretation and discussion of the results.
### Corresponding author
Correspondence to Danica Galonić Fujimori.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Longbotham, J.E., Chio, C.M., Dharmarajan, V. et al. Histone H3 binding to the PHD1 domain of histone demethylase KDM5A enables active site remodeling. Nat Commun 10, 94 (2019). https://doi.org/10.1038/s41467-018-07829-z
• Accepted:
• Published:
• ### Exploring the Ligand Preferences of the PHD1 Domain of Histone Demethylase KDM5A Reveals Tolerance for Modifications of the Q5 Residue of Histone 3
• Sarah E. Anderson
• , James E. Longbotham
• , Patrick T. O’Kane
• , Fatima S. Ugur
• , Danica Galonić Fujimori
• & Milan Mrksich
ACS Chemical Biology (2021)
• ### Histone tails as signaling antennas of chromatin
• Yunhui Peng
• , Shuxiang Li
• , David Landsman
• & Anna R Panchenko
Current Opinion in Structural Biology (2021)
• ### The emerging role of KDM5A in human cancer
• Guan-Jun Yang
• , Ming-Hui Zhu
• , Xin-Jiang Lu
• , Yan-Jun Liu
• , Jian-Fei Lu
• , Chung-Hang Leung
• , Dik-Lung Ma
• & Jiong Chen
Journal of Hematology & Oncology (2021)
• ### Recognition of Histone H3 Methylation States by the PHD1 Domain of Histone Demethylase KDM5A
• James E. Longbotham
• , Mark J. S. Kelly
• & Danica Galonić Fujimori
ACS Chemical Biology (2021)
• ### Enzyme Dynamics: Looking Beyond a Single Structure
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## Empty one year ago Is there some function f(x) that satisfies this:
1. Empty
$a^{f(a)}=e$ $a^{f(b)}=1$ for $$a \ne b$$
2. amistre64
f(b) = 0 f(a) = 1/ln(a) we have 2 points to play with, can we make a line? m = 1/[(a-b) ln(a)] f(x) = m(x-b) +f(b)
3. amistre64
can a=1? 1^k = e doesnt seem plausible to me, so a=1 is not an issue.
4. Empty
Ahh well the main thing that they have to satisfy is that a and b are not the same whole number and that they can be any whole number not just 1. Kind of the crux of why this is bothering me haha.
5. anonymous
I'm not sure there's anything that works besides the trivial function, $f(x)=\begin{cases}\log_x e&\text{for }x=a,\quad a\in\mathbb{R}_+\backslash\{1\}\\0&\text{for }x\neq a\end{cases}$(where $$\mathbb{R}_+$$ is the set of all positive reals) As amistre mentioned, $$x\neq1$$ because that would suggest $$\log_1e=\frac{\ln e}{\ln1}$$ exists, which it doesn't.
6. amistre64
$\Huge a^{\frac{x-b}{(a-b)ln(a)}}$ $\Huge a^{\frac{b-b}{(a-b)ln(a)}}=a^{f(b)\color{red}{=0}}=1$ $\Huge a^{\frac{a-b}{(a-b)ln(a)}}=a^{f(a)\color{red}{=\frac{1}{ln(a)}}}=e$
7. amistre64
log_a(e) change of bases to ln(e)/ln(a) hence the 1/ln(a)
8. Empty
Hmmm well the point of this is so that I can take any number, and do stuff like this: $\log 200 = 3 \log 2 + 2 \log 5$ Multiplying by f(2) or f(5) gives: $f(2) \log 200 = 3$$f(5) \log 200 = 2$ Really the point is that it is essentially doing a dot product with a projection on a vector space to get that component $\bar e_i \cdot \bar v = v_i$ Except the components are the exponents on the prime factorization.
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View Voting Results: Smartest and Funniest
Does road rage piss him off?
Does not approve
It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
"Did you get the tag of the car that ran you over?"
Na na na na na na na
I know that conventional fark knowledge holds cyclists pretty low on the food chain, but road ragers are at a circle of hell far below that. Saw a woman get cut off last week and she drove into the oncoming traffic lane for about 2 miles to get alongside the person who cut her off and literally try to run them off the road (sideswiping), I think I was not the only driver getting their phone out to call the cops on her crazy ass. Finally the other driver managed to pull off onto a side street and the psycho got stuck in an intersection, but she was out for literal blood it seems.
I have my problems with the Spandex Assholes that ride around Boulder, but I'd stand with them against this kind of bullshiat. I was on my commuter bike in a 6ft wide bike lane a few weeks ago and some woman on a cell phone drifted into my lane to the point where I could slap her back windshield. I wish I had had my bike lock in hand to give her a good thwack. Not the same thing, I know, but I'm still pissed.
My name is Richie Cunningham. And this is my wife, Oprah.
How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
That's what he gets for going for a swim on the highway
when will motorists learn roads were built for bicyclists, joggers, skateboarders, groups of teens walking in numbers abreast and idiots paying attention to their texting?
The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
Yogimus: How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
People I know that have it don't seem to be able to turn it off.
Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used.
So, I decided to run in the bicycle lane.
Cyclists are constantly riding with cars. Cyclists are smaller and slower, and they constantly complain about how a**hole drivers don't give them space or treat them like they belong. Seeing that I was slower than them, and without a bicycle, I expected that they would be understanding. That they would give me plenty of space, and wait until it was safe to pass me. Sure, I can't run as fast as a cyclist, but a cyclist can't cycle as fast as car. If they expect cars to slow down and go their speed until they can safely pass - certainly, they wouldn't begrudge doing the same to me. Right?
Apparently, I was wrong. Yay for double standards!
Go find a closed course somewhere to practice and stop bothering the rest of us with your Lance Armstrong fantasies.
TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
(Not So ) CSB....
I was bike commuting to work on a paved bike trail that has several street level crossings. Stopped at one, and traffic on the right stopped. I held up my hand to acknowledge the driver, and waited for the left traffic to stop. Since it was a 2-lane street, and both sides were stopped, I clipped in and rolled forward. No sooner did I clear the car on the right, than a guy from three cars back came flying around the cars, passing on the right, and flipped me up on the hood like Mannix. I went up the hood, smacked my ribs on the A pillar of the car, and slid back down to the pavement still clipped into the pedals.
The guy at first said, "I didn't see him." Well, yeah - when you pass a line of stopped cars on the right at a highly marked crosswalk, you're not likely to see someone actually IN the crosswalk.
He initially pled 'not guilty' to the citation, probably hoping that the officer wouldn't show. They did, as did I. At first, he said he was trying to park the car (at 25MPH). He then said he was looking of a museum (in Vienna, Va....where there are no museums within 10 miles). THEN, he said that I ran out in front of him, using the 'spandex' defense (that all cyclists are dangerous)....When it was my turn, I just handed the judge a printout of my GPS/HR monitor data.....which showed my exact location, slowing my speed, heart rate and pedal cadence approaching the intersection...then being stopped for 45 seconds before moving forward at 5 MPH......then suddenly moving sideways at 20MPH.
Guilty as charged. Took three years to settle out for my bike, doctor visits and X-rays/MRI's and PT (thanks to bruised ribs, a hip injury and a cracked vertebrae). Not fun at all.
bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
Chances are, those drivers are likely oblivious to other cars as well. So, you are not alone when it comes to idiots who should not have been given their licenses.
Too bad we don't have the other side of the story. Could have stopped for a dog in the road and they were just following too close/tailgating.
Charge the driver with assault with a deadly weapon. Maybe if drivers start doing some jail time for running cyclist off the road they'll calm the fark down.
astro716: TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Yes, I believe the guy in the car most likely had road rage but from the cyclist's own story, he was passed before they started going downhill which means he was going substantially below the speed limit. (unless these guys can pedal 45mph up a hill outside Boulder)
/glad the guy wasn't hurt too bad
//and I don't have any reason not to believe the cyclist's story but he is going to have to prove he wasn't following too close (although the motorist is going to be cited for leaving the scene of an accident)
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
Fark_Guy_Rob: Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used.
So, I decided to run in the bicycle lane.
Cyclists are constantly riding with cars. Cyclists are smaller and slower, and they constantly complain about how a**hole drivers don't give them space or treat them like they belong. Seeing that I was slower than them, and without a bicycle, I expected that they would be understanding. That they would give me plenty of space, and wait until it was safe to pass me. Sure, I can't run as fast as a cyclist, but a cyclist can't cycle as fast as car. If they expect cars to slow down and go their speed until they can safely pass - certainly, they wouldn't begrudge doing the same to me. Right?
Apparently, I was wrong. Yay for double standards!
In most areas cyclists have a legal right to the road and are allowed to 'take the lane; as necessary.
Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
kokomo61: There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
(Not So ) CSB....
I was bike commuting to work on a paved bike trail that has several street level crossings. Stopped at one, and traffic on the right stopped. I held up my hand to acknowledge the driver, and waited for the left traffic to stop. Since it was a 2-lane street, and both sides were stopped, I clipped in and rolled forward. No sooner did I clear the car on the right, than a guy from three cars back came flying around the cars, passing on the right, and flipped me up on the hood like Mannix. I went up the hood, smacked my ribs on the A pillar of the car, and slid back down to the pavement still clipped into the pedals.
The guy at first said, "I didn't see him." Well, yeah - when you pass a line of stopped cars on the right at a highly marked crosswalk, you're not likely to see someone actually IN the crosswalk.
He initially pled 'not guilty' to the citation, probably hoping that the officer wouldn't show. They did, as did I. At first, he said he was trying to park the car (at 25MPH). He then said he was looking of a museum (in Vienna, Va....where there are no museums within 10 miles). THEN, he said that I ran out in front of him, using the 'spandex' defense (that all cyclists are dangerous)....When it was my turn, I just handed the judge a printout of my GPS/HR monitor data.....which showed my exact location, slowing my speed, heart rate and pedal cadence approaching the intersection...then being stopped for 45 seconds before moving forward at 5 MPH......then suddenly moving sideways at 20MPH.
Guilty as charged. Took three years to settle out for my bike, doctor visits and X-rays/MRI's and PT (thanks to bruised ribs, a ...
This is the primary reason I rarely ride the bicycle on highways or city streets. Just lucky the multi-use trails systems around here are extensive.
/going to be curious to what the cyclist's GPS data shows also
The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I watch a group of cyclists go down my road every few days and the rules don't apply to them. Stop signs? Perish the thought!
I hate that. In Amsterdam if you pull that shiat you get a ticket. The Dutch may be very protective of bikers, but you better follow the rules of the road.
BunkyBrewman: astro716: TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Yes, I believe the guy in the car most likely had road rage but from the cyclist's own story, he was passed before they started going downhill which means he was going substantially below the speed limit. (unless these guys can pedal 45mph up a hill outside Boulder)
/glad the guy wasn't hurt too bad
//and I don't have any reason not to believe the cyclist's story but he is going to have to prove he wasn't following too close (although the motorist is going to be cited for leaving the scene of an accident)
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Swoop and squat in vegas? That'll cost you double at least
Carth: Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
So you are saying double standards are stated in the law? Because it does sound like a really not well thought out idea to make the runners run in traffic, but the cyclists can have their own private real estate.
Fark_Guy_Rob: Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used.
So, I decided to run in the bicycle lane.
Well, there's your problem. Roads are assumed shared. Marked bike lanes, not so much.
BunkyBrewman:
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
If it wasn't 8am on a sunday I'd do the math: E=\frac{1}{2}m\Delta v^{2},
Depending on the tires the triathlete was using I'm betting the coefficient of friction was so low there is no way he could stop if he was going downhill at speeds around 30-40mph.
This is why Go Pro cameras are awesome on bikes. It is great having video proof when this shiat happens.
If you cannot keep up with the pace of traffic you do not belong on the road.
Just like a slow car, they cause massive problems in traffic also.
It's my guess that anyone who drives an automobile experiences road rage to different degrees, due to other drivers who are aggressive, heedless, and downright dangerous. It depends on how well that adrenalin burst is controlled, which translates into how your own vehicle is controlled when you encounter these assholes.
As some Farker put it in an unrelated thread, you can't drink poison and expect the other person to die.
nucal: Does not approve
Where is the Fonz when you need him?
lack of warmth: Carth: Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
So you are saying double standards are stated in the law? Because it does sound like a really not well thought out idea to make the runners run in traffic, but the cyclists can have their own private real estate.
You aren't allowed to run in the road either. You have to run against traffic on the shoulder. I wish they'd made it illegal to run with headphones in too since too many runners are oblivious to what is going on around them.
I'd love to hear the other side of the story.
I like to imagine it went like this...
I was running late for work, going 5 mph over the speed limit. As I go around a blind corner these three idiots dressed in spandex were peddling 3 wide across the entire lane. I swerved, honked and had to swerve back for oncoming traffic. As one of them flipped me the bird, he lost control and wiped out. It was hilarious.
kokomo61: There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
(Not So ) CSB....
I was bike commuting to work on a paved bike trail that has several street level crossings. Stopped at one, and traffic on the right stopped. I held up my hand to acknowledge the driver, and waited for the left traffic to stop. Since it was a 2-lane street, and both sides were stopped, I clipped in and rolled forward. No sooner did I clear the car on the right, than a guy from three cars back came flying around the cars, passing on the right, and flipped me up on the hood like Mannix. I went up the hood, smacked my ribs on the A pillar of the car, and slid back down to the pavement still clipped into the pedals.
The guy at first said, "I didn't see him." Well, yeah - when you pass a line of stopped cars on the right at a highly marked crosswalk, you're not likely to see someone actually IN the crosswalk.
He initially pled 'not guilty' to the citation, probably hoping that the officer wouldn't show. They did, as did I. At first, he said he was trying to park the car (at 25MPH). He then said he was looking of a museum (in Vienna, Va....where there are no museums within 10 miles). THEN, he said that I ran out in front of him, using the 'spandex' defense (that all cyclists are dangerous)....When it was my turn, I just handed the judge a printout of my GPS/HR monitor data.....which showed my exact location, slowing my speed, heart rate and pedal cadence approaching the intersection...then being stopped for 45 seconds before moving forward at 5 MPH......then suddenly moving sideways at 20MPH.
Guilty as charged. Took three years to settle out for my bike, doctor visits and X-rays/MRI's and PT (thanks to bruised ribs, a ...
Hopefully, having to pay you for the bike & medical bills might cure that driver of acute douchebaggery.
/ probably not
// despite the judgement of the Court, I bet he blames you for everything. That's just how douchebags roll...
a motorist suffering from road rage
Carth: lack of warmth: Carth: Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
So you are saying double standards are stated in the law? Because it does sound like a really not well thought out idea to make the runners run in traffic, but the cyclists can have their own private real estate.
You aren't allowed to run in the road either. You have to run against traffic on the shoulder. I wish they'd made it illegal to run with headphones in too since too many runners are oblivious to what is going on around them.
Seriously, I enjoy running, and nothing irks me more than seeing other runners or walkers walking with traffic, headphones on, oblivious to everything. Especially the large groups of middle aged women gabbing while walking. The neighborhood I grew up in had no sidewalks, so I was always very careful and alert while running there. My new neighborhood has plenty of sidewalks, which makes me happy.
I would never assume the bike lane is for runners. It's a narrow lane, where the hell are they supposed to go when you're taking it up? If its too crowded on the sidewalk, either get out on your run earlier (or later), or find a new route.
It would be poetic justice if the driver probably had to slam on their brakes to avoid hitting a prairie dog....for some reason the asshats in Boulder seem to believe that prairie dogs are on the same level as FSM and deserve to be worshiped. Really, they're nothing more than cuter form of ground rats.
Also, how much do you want to bet these 3 asshats were riding 3 wide instead of single file? It's always the ones "in training" that seem to have the believe that road rules no longer apply to them.
BunkyBrewman: astro716: TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Yes, I believe the guy in the car most likely had road rage but from the cyclist's own story, he was passed before they started going downhill which means he was going substantially below the speed limit. (unless these guys can pedal 45mph up a hill outside Boulder)
/glad the guy wasn't hurt too bad
//and I don't have any reason not to believe the cyclist's story but he is going to have to prove he wasn't following too close (although the motorist is going to be cited for leaving the scene of an accident)
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Thing is noone hit the motorist. He wasnt actually in an accident.
I personally do not think cyclists should be on the roads with cars. No insurance for the reckless ones and no visible identification for the ones that cruise through 4 way stops. No lights when all other new vehicles have daytime running lights as a requirement.
I am a commercial driver that has a vehicle that is literally the legal maximum width. Cyclists hopping curbs at lights to get around me essentially barricade moderate traffic at their speed and endanger pedestrians when they get in the middle of them but they do it regularly.
They cut people off with no signals, come past at high speed in heavy traffic when I have a reduced rear visibility vehicle and crowd me.
Screw all cyclists. Make bike paths and go there. 9 years of accident free driving all my most dangerous situations are with cyclists.
The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
Oddly, I've had more problems running than biking. Most of the time it involves crossing a street at a T intersection. Someone coming off a side street attempting to make a right turn onto the main road will approach the intersection while staring out their left window for oncoming traffic. If there isn't any, they immediately start their right turn without ever checking the crosswalk for pedestrians. Happens at least once every week. On the rare occasion they cut so close to me could give their car a kick, but usually just manage to yell a "HEY!" and spend the next mile or two being all annoyed.
Fark_Guy_Rob: Apparently, I was wrong.
Do you think you should be able to go for a run down the middle of the traffic lane as well? Cycle lanes are usually there because it's been determined safety issues exist with cars and bikes sharing the road, or there is a large population of cycles in the area requiring their own lane. If I was in an area with a cycle lane and there were runners using it, I'd probably be annoyed too-- especially if there was a sidewalk.
Cyclists... the scourge of anyone trying to get anywhere on time using the roads. This guy's luck probably caught up with him after blocking the road for many, many other people and enjoying it.
A guy in Los Angeles got prison time for a similar move.
Something to think about when you're working out your ITG "brake check a tailgater and win the lottery" plan.
I'm no fan of road rage assholes, but to be fair, Mr Ironman didn't hit the car. He misjudged if he could get around the car, had to stop too fast, and flipped over the handlebars. He was operating his bike too fast then. If you cannot come to a complete stop safely, then you are going too fast.
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
LemSkroob: Go find a closed course somewhere to practice and stop bothering the rest of us with your Lance Armstrong fantasies.
I live with my mom
Nobody fantasizes about being Lance Armstrong anymore.
The two lane road leaving my office has a recently added bike lane. For about 4 blocks. Then it abruptly ends at a crowded stoplight, and is followed by painted signs on the road of a bike and arrows, indicating the bike lane continues further ahead (it doesn't). People are always cutting into the bike lane in order to be closer to the light, where the street opens into four lanes. You couldn't pay me to ride my bike in that death-trap of a lane.
Publikwerks: I
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
If the biker intentionally stopped with the goal of creating an unsafe situation? Yes. That is what the cyclist seems to be accusing the driver of and if it actually happened would make the car at fault.
According to the article from the drivers point of view he passed the bikes, later had to hit the brakes for some reason, and one of the bikes behind him flipped (if he even saw that). He was not even involved with any kind of accident (no contact with his car) and may not have even seen the bikes behind him.
Carth: Charge the driver with assault with a deadly weapon. Maybe if drivers start doing some jail time for running cyclist off the road they'll calm the fark down.
No, charge them with attempted murder. Upon conviction, hang them. Road rage will vanish overnight either from people no longer being raging assholes or we kill all the raging assholes. Either way, no more raging assholes.
Very strange people at fark. They scream about global warming and carbon footprints yet they side with the evil polluters! LOL
Yogimus: How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
I wish I knew. I know I certainly don't suffer from road rage, though the people around me seem to.
Zeno-25: Cyclists... the scourge of anyone trying to get anywhere on time using the roads. This guy's luck probably caught up with him after blocking the road for many, many other people and enjoying it.
Every time I see someone like you cresting a hill or coming around a blind corner straddling the center line in their truck I think of these bike threads. Oh sure in these threads you are all ragging on cyclists because you don't look good in spandex and get winded carrying the big bag of tater tots out of the freezer but once you get behind the wheel of your truck its time to drive down the middle of the road like an encephalitic cataract sufferer on Ambien.
Happy Baseless Assumption Day to you too!!
Fark_Guy_Rob: Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used.
So, I decided to run in the bicycle lane.
Cyclists are constantly riding with cars. Cyclists are smaller and slower, and they constantly complain about how a**hole drivers don't give them space or treat them like they belong. Seeing that I was slower than them, and without a bicycle, I expected that they would be understanding. That they would give me plenty of space, and wait until it was safe to pass me. Sure, I can't run as fast as a cyclist, but a cyclist can't cycle as fast as car. If they expect cars to slow down and go their speed until they can safely pass - certainly, they wouldn't begrudge doing the same to me. Right?
Apparently, I was wrong. Yay for double standards!
You are not a vehicle. Stay out of the street. Bikes should stay off the sidewalk for the same reason. If it's too crowded to run on a sidewalk, find a different running route.
Re: TFA. It's quite possible the car driver was just a dick, but I'd be interested to know if the cyclists were riding abreast. Because that is a total dick move; share the road goes both ways.
Dead for Tax Reasons:
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Swoop and squat in vegas? That'll cost you double at least
They all got out of the car holding their necks and screaming they were hurt. The cop arrived and found no damage on either car. (literally not a scratch) The cop did not even fill out an actual accident report since there was no damage. They refused any type of medical treatment. I never did ask how much my insurance company settled for, but quite sure it would've been enough to make my blood boil. After they took the bumper off of their car, they claimed there damage. Not sure how much there could have been since the vehicle I was in had nothing (it was fully inspected along with the frame behind the bumper) I guess their Mercedes was no match for my awesome Chrysler sebring convertible rental car.
stirfrybry: Very strange people at fark. They scream about global warming and carbon footprints yet they side with the evil polluters! LOL
Everything said in a Fark thread will eventually become a reductio. There is no facrtoid or philosophy that this place does not render into a ridiculous mental goo. But much like a Salvador Dali clock painting, it persists, so what else is there to do?
Carth: Publikwerks: I
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
If the biker intentionally stopped with the goal of creating an unsafe situation? Yes. That is what the cyclist seems to be accusing the driver of and if it actually happened would make the car at fault.
Bikes are inherently unsafe when on the road with cars.
BTW - Here's the problem with bikers:
http://www.bostonglobe.com/metro/2013/06/14/waltham-man-killed-maine -w hile-riding-his-bike-for-charity/sD9DDRiWJyln4b5LS6tOyL/story.html
Here's a part of the article:
Police said David LeClair was cycling along Route 2 in Hanover just before 9 a.m. with a group of other employees from Athenahealth Inc. of Watertown when the rear of a tractor-trailer hit LeClair, killing him instantly.
The truck did not stop, police said in a statement, although it was unclear if the driver knew his vehicle had struck LeClair.
LeClair's companions called police with a description of the truck, and police stopped a driver operating a truck matching the description a short time later in nearby Rumford. Police said they were interviewing the driver Friday and examining the vehicle.
Here's the follow up:
http://bangordailynews.com/2013/06/17/news/state/bicyclist-drinking- wa ter-at-time-of-fatal-collision-with-canadian-truck-in-trek-across-main e-police-say/
A bicyclist who died Friday while participating in the Trek Across Maine lost control of his bike while drinking from a water bottle as a tractor-trailer passed him, according to police.
So part of me wonders whether or not they were operating as they should, because the fact of the matter is that he was not in control of his bicycle. If you cannot come to a complete stop safely, you are not in control
stirfrybry: Very strange people at fark. They scream about global warming and carbon footprints yet they side with the evil polluters! LOL
Every single bicyclist I know doesn't also own a car, and they get all their spandex clothing from naturally occurring fairy rings where the forest druids call upon the earth mother to grant them the gift of brightly colored spandex outfits. Every single one.
nucal: Does not approve
I would punch my momma right in da mouf....
Carth: BunkyBrewman:
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
The bike at the same speed takes longer, definitely. Bike brakes suck. Cars weigh a lot more, but they have giant disc brakes gripped hydraulically by ceramic plates, while bikes have itty bits of rubber hand-squeezed onto a tiny strip of often-damp tire rim.
/Have a permanent dent in my left thigh from where a car with old-fashioned door handles passed me and immediately turned right into a parking lot, in the rain, assuming I could stop.
Publikwerks: I'm no fan of road rage assholes, but to be fair, Mr Ironman didn't hit the car. He misjudged if he could get around the car, had to stop too fast, and flipped over the handlebars. He was operating his bike too fast then. If you cannot come to a complete stop safely, then you are going too fast.
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
You know, that's actually a pretty good point. He did say his first choice was to try to get around the car. I'm not defending the motorist, but it's a valid argument.
BunkyBrewman: Dead for Tax Reasons:
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Swoop and squat in vegas? That'll cost you double at least
They all got out of the car holding their necks and screaming they were hurt. The cop arrived and found no damage on either car. (literally not a scratch) The cop did not even fill out an actual accident report since there was no damage. They refused any type of medical treatment. I never did ask how much my insurance company settled for, but quite sure it would've been enough to make my blood boil. After they took the bumper off of their car, they claimed there damage. Not sure how much there could have been since the vehicle I was in had nothing (it was fully inspected along with the frame behind the bumper) I guess their Mercedes was no match for my awesome Chrysler sebring convertible rental car.
That story suffered from a severe lack of hookers
I commute by car every morning through Boulder. It's peak time for bike commuters and cyclists taking advantage of cooler temperatures, and there are a lot of them on my route.
Even the rare riders not obeying the law and riding two or three abreast are such a momentary inconvenience I'm having a hard time understanding the contempt in this thread, let alone the people who would act aggressively towards them while driving in a car.
STAY. AWAY. FROM. CARS.
If you can't, KEEP YOUR EYES ON THEM.
13 years now I've been biking thru Philly. No road rage incidents. No injuries. Why? Ride against traffic, stay on the sidewalk as much as you can, don't be a dick when pedestrians are around. Simple. Works. Saved by ass NUMEROUS times.
"I was having one of my best seasons. I'm 39 soon to be 40. I don't have that much longer to keep competing."
But doesn't he know that at 40 he's in better shape than at 20?
Where are the Middle Aged Delusion Syndrome farkers?
Ker_Thwap: idiots dressed in spandex were peddling 3 wide across the entire lane.
It is dangerous to sell bikes on the highway, yes. Perhaps one day someone will sell you a dictionary.
WinoRhino: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
Oddly, I've had more problems running than biking. Most of the time it involves crossing a street at a T intersection. Someone coming off a side street attempting to make a right turn onto the main road will approach the intersection while staring out their left window for oncoming traffic. If there isn't any, they immediately start their right turn without ever checking the crosswalk for pedestrians. Happens at least once every week. On the rare occasion they cut so close to me could give their car a kick, but usually just manage to yell a "HEY!" and spend the next mile or two being all annoyed.
Fark_Guy_Rob: Apparently, I was wrong.
Do you think you should be able to go for a run down the middle of the traffic lane as well? Cycle lanes are usually there because it's been determined safety issues exist with cars and bikes sharing the road, or there is a large population of cycles in the area requiring their own lane. If I was in an area with a cycle lane and there were runners using it, I'd probably be annoyed too-- especially if there was a sidewalk.
I'm too much a klutz for road or mountain biking, so I run. In Colorado at least, it is legal for a pedestrian to walk or run in the roadway. Not recommended, but legal. When I run roads (very rarely) I run in the bike lane facing traffic - and share. It's easy to get around a bike.
Have almost been hit 3 times this calendar year by people turning right without looking. And I've been run off the road in my car by a broad from South Dakota who didn't understand that you need to look before changing lanes. Pickups and minivans seem to be the vehicle of choice for idiots.
Now: As for the crowd that thinks carbon fiber is a substitute for penis length - yes, there are too many to count. I stopped at a midblock crosswalk to let a family on bikes cross. Asshole in spandex passed me on the right, nearly clipped a 7 year old girl, and ran the red. I followed him at 20-25 mph for 3 miles, never honking, never saying anything, never buzzing him, obeying all traffic laws, just calmly following him. By the time he found a path I couldn't drive on the guy was screaming and shaking; he had no idea what I was up to, which was nothing except pure psychological warfare.
mark12A: STAY. AWAY. FROM. CARS.
If you can't, KEEP YOUR EYES ON THEM.
13 years now I've been biking thru Philly. No road rage incidents. No injuries. Why? Ride against traffic, stay on the sidewalk as much as you can, don't be a dick when pedestrians are around. Simple. Works. Saved by ass NUMEROUS times.
That's fine if you're going a short distance, slowly. Bikes don't belong on the sidewalk. I commuted in DC, and on many streets I was faster than the cars, because I could pass a whole line of them at a light. The sidewalks aren't an option there; crowded and in poor repair.
You stay hard to the right, and if you're slowing up cars because of a narrow lane with cars parked along it, you stand on the pedals and sprint. Never had an incident.
WinoRhino: Publikwerks: I'm no fan of road rage assholes, but to be fair, Mr Ironman didn't hit the car. He misjudged if he could get around the car, had to stop too fast, and flipped over the handlebars. He was operating his bike too fast then. If you cannot come to a complete stop safely, then you are going too fast.
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
You know, that's actually a pretty good point. He did say his first choice was to try to get around the car. I'm not defending the motorist, but it's a valid argument.
That's what I'm saying. I give Bikes a wide berth, and I try to chill with the little things. But the problem is that some bicyclists seem to have no issue with operating right up to and beyond the line of safe operation, and when something goes wrong, the car operator has to make up the difference or bad things happen.
blacksho89: Now: As for the crowd that thinks carbon fiber is a substitute for penis length - yes, there are too many to count. I stopped at a midblock crosswalk to let a family on bikes cross. Asshole in spandex passed me on the right, nearly clipped a 7 year old girl, and ran the red. I followed him at 20-25 mph for 3 miles, never honking, never saying anything, never buzzing him, obeying all traffic laws, just calmly following him. By the time he found a path I couldn't drive on the guy was screaming and shaking; he had no idea what I was up to, which was nothing except pure psychological warfare.
Quantum Apostrophe: "I was having one of my best seasons. I'm 39 soon to be 40. I don't have that much longer to keep competing."
But doesn't he know that at 40 he's in better shape than at 20?
Where are the Middle Aged Delusion Syndrome farkers?
That's a real thing. When I was 43, I got down to 155 pounds, and inline-skated 38 miles in 3:20. At 45, I ran a 10k in under an hour. When I was 20, I couldn't do endurance sports. I could run three miles in under 25 minutes for my Navy ROTC test, but I felt like I was dying the whole time.
/Got fat; can't do crap in my 50s.
"Youre goin' 30 feet up in the FARKing air, that's where you're going!" -G. Carlin
Yogimus: How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
It's like having an itchy trigger finger I think.
I've been in the driver's position before. Idiot cyclers going half the speed limit but riding in a way that makes it really hard to pass. And it sounds like the driver finally got a chance to pass and the cyclists took offense to the way he passed and started riding him. Oops.
//If you live in SF and you ever see Critical Mass, you will lose all ability to ever feel bad for a cyclist who gets nailed by a car
Quantum Apostrophe: "I was having one of my best seasons. I'm 39 soon to be 40. I don't have that much longer to keep competing."
But doesn't he know that at 40 he's in better shape than at 20?
Where are the Middle Aged Delusion Syndrome farkers?
Ker_Thwap: idiots dressed in spandex were peddling 3 wide across the entire lane.
It is dangerous to sell bikes on the highway, yes. Perhaps one day someone will sell you a dictionary.
Go look at Ironman race winners, and look at their ages. 40 year olds are faster than 20 year olds. And when you get into ultrarunning (100 miles +), women are minutes away from beating men. Ann Trason has won some ultras outright.
You might want to check your assumptions; they don't agree with the facts. And make sure you polish the brush guard on your Ram this weekend; you'll need it at the grocery store.
kokomo61: There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
I live in NoVa too (Live in Herndon, work in Reston). Very close to the W&OD trail in fact. I've got mad love for the bikers around here. Butttt....Nothing infuriates me more than encountering a group of 8 to 12, or even 20+, heavily logoed, spandex wearing, Lance Armstrong wannabees riding 4-5 abreast down a 2 lane street that also happens to have a dedicated bike lane, or at least a section that is large enough for a bike, hell...even two bike riding side by side.
I know I am supposed to "share the road," but that does not give the bikers the right to be road hogging dicks too. If the speed limit is 40, and you cannot sustain that kind of speed...GET THE FARK OUT OF THE FARKING WAY AND LET CARS PASS YOU, YOU SELF CENTERED DICKS!!! There is no reason that they need to ride 4-5 abreast. I know you may take issue with what I am saying here...but the bikers, like it or not, have some responsibility in the matter here too. Just because you have the right to do something, depending on the circumstance, or location, may not make it correct.
Reminds me of The Dude saying: "You're not wrong, you're just an asshole"
You pickin' up what I'm throwin' down?
CSB:
A few years ago, I (among others) called the cops on a biker. He was on Fairfax County Parkway...not riding on the shoulder/bike lane, no...in the actual car lane, at like 15MPH. He had created a line of traffic that went back at least a 1/2 mile or more. Once I got to the front and had the opportunity to discover the reason for the slowing, and pass him, I called 911. The dude was going to get himself hurt due to the rage that was building behind him. I was told that I was like the 20th person to call in the last 15 minutes, and that they already had a cop at the next intersection waiting to "deal with the moron" as the dispatcher put it.
I honestly cannot fathom why anyone sees it as being a good idea riding a bike on a, not perfectly straight, often curvy and somewhat hilly, 4 lane road that has 50MPH posted speed limits, where drivers regularly drive 60+MPH, and cross into each others lanes and the shoulder too.
I just "buzzed" a bicyclist riding half of the way into the road, when there was a good 6 feet to his right for biking (and bike lane markings). Sorry, I'm not going to cross the double yellow lines into oncoming traffic, "share the road" works both ways.
lazymojo: Yogimus: How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
It's like having an itchy trigger finger I think.
I have road rage pretty bad sometimes. I'm at my worst when I'm running late and in general need to get my shiat together. I assume it's a control issue. But I'm not such a shiat that I would take it out on some dude on a bike.
Hey, if you want to inconvenience hundreds of motorists for your own gay little hobby, then you take your chances. Cry me a river, a**holes, and pray I still believe there's a chance I could get caught.
I'm getting a kick because I'm also training for an Ironman at the end of this summer...
I sympathize quite a bit with the biker. As long as he doesn't ride like an asshole. Most cyclists don't, but the few who do ruin it for the rest of us. On the other hand, 9 out of 10 drivers don't drive like assholes either.
Articles like this are why this will probably be my only Ironman. I'll stick to running and shorter triathlons. I've come to realize that it's only a matter of time before some asshole who hates cyclists runs me off the road/hits me and seriously injures or kills me.
/It's always some dumbass redneck in a giant truck getting as close as he can to you and then flooring it to scare you.
//last week I had to jump off the road to avoid a crop dusting plane taking off on it. That was a new experience...
Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
The guy may have slammed in his brakes to be a dick, but it wouldn't be an issue if the bike wasn't following so close.
Quantum Apostrophe: blacksho89: Now: As for the crowd that thinks carbon fiber is a substitute for penis length - yes, there are too many to count. I stopped at a midblock crosswalk to let a family on bikes cross. Asshole in spandex passed me on the right, nearly clipped a 7 year old girl, and ran the red. I followed him at 20-25 mph for 3 miles, never honking, never saying anything, never buzzing him, obeying all traffic laws, just calmly following him. By the time he found a path I couldn't drive on the guy was screaming and shaking; he had no idea what I was up to, which was nothing except pure psychological warfare.
In that ferocious and self-annihilating way in which so many men of his generation served their families, my father served my mother, my sister Hannah, but particularly me.
Also, there's a damn good reason cyclist wear those spandex shorts. They have a pad on the crotch of the shorts and are a million times more comfortable on a long ride. It's not because they're "Lance Armstrong wannabes". Would you call a high school football player wearing his pads and uniform a "Tom Brady" wannabe? No, it's a farking uniform.
FTFA:
"To be perfectly honest, there are enough cyclists and courteous drivers that we have it pretty good here," he said. "But occasionally you have a bad apple... It's just tough when something like this happens. It seems completely senseless."
And by "bad apple", he means someone who doesn't think it's okay to just let the rest of a large group of bicyclists through on red light, in clear violation of the law.
farking dicks.
Endive Wombat: kokomo61: There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
I live in NoVa too (Live in Herndon, work in Reston). Very close to the W&OD trail in fact. I've got mad love for the bikers around here. Butttt....Nothing infuriates me more than encountering a group of 8 to 12, or even 20+, heavily logoed, spandex wearing, Lance Armstrong wannabees riding 4-5 abreast down a 2 lane street that also happens to have a dedicated bike lane, or at least a section that is large enough for a bike, hell...even two bike riding side by side.
Cycle racing definitely seems to appeal to self-entitled dicks. They're even worse on multi-use trails. You see idiots drafting on each other at 30+ mph, dodging around little kids on tricycles and old couples out for a walk. I was skating on a trail in NoVa when some jerk leading a bike pace line told me not to use so much of my lane. They're skates, asshole; they require a much wider track than a bicycle. I sent him off with a few choice comments about his cranial anatomy and parentage.
Spandex, though; spandex rules. If you're on a bike or skates for more than an hour, you REALLY appreciate how much more comfortable spandex is than running shorts.
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
No. Car turning yields both to traffic from the left, and traffic abreast in the bike lane. If the cyclist is coming up from behind the car, the car should still should yield because he's crossing the bike lane in order to turn. In practice, the cyclist should be ready to yield, because car drivers are generally oblivious to bikes.
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
Traffic laws suck. Not sure of the legality, but in the case you mention, common sense would dictate that the cyclist stage to the left of and in front of the right-turning car, much as if he were a vehicle. Otherwise, as a cyclist is technically a pedestrian, he has the right of way if he is at the crosswalk..
If cycles behaved as vehicles ALL OF THE TIME or as pedestrians ALL OF THE TIME there would be less angry interaction.
SirGeorgeBurkelwitzIII: Also, there's a damn good reason cyclist wear those spandex shorts. They have a pad on the crotch of the shorts and are a million times more comfortable on a long ride. It's not because they're "Lance Armstrong wannabes". Would you call a high school football player wearing his pads and uniform a "Tom Brady" wannabe? No, it's a farking uniform.
Eh, not really. The padding helps initially when you first start riding. Once you build up some muscle in the area, you don't really need the pads. I've been riding about a year, when I do bother to wear the padded liner, I wear some baggy shorts over it. No one wants to see my 50 year old butt with that much detail.
I also pull over when cars approach me, screw my rights, I'm not going to trust half blind elderly drivers, texting idiots, distracted and drunk drivers with my life. I just get right off the road when any car at all approaches.
blacksho89: Go look at Ironman race winners, and look at their ages. 40 year olds are faster than 20 year olds. And when you get into ultrarunning (100 miles +), women are minutes away from beating men. Ann Trason has won some ultras outright.
You might want to check your assumptions; they don't agree with the facts. And make sure you polish the brush guard on your Ram this weekend; you'll need it at the grocery store.
You might want to learn how to read.
Maybe some 40 year olds are faster than some *OTHER* 20 year olds, but barring disease, there is NO WAY that the same person is better at 40 than 20.
The guy in the article said it himself.
Why people want to desperately deny that youth is better than oldth, I don't know.
I wonder why the Olympics aren't packed with paunchy balding 40 year olds.
mbillips: You see idiots drafting on each other at 30+ mph, dodging around little kids on tricycles and old couples out for a walk.
Out for a *walk* on a designated bike path?
Here in Montreal there are places where there are pictures of bikes spray-painted on the ground.
At what point can we say we've done enough to tell people it's a bike path?
phaseolus: Traffic law question: Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Starting from a red light or in moving traffic? In moving traffic it should be a "yield" type situation where you aren't supposed to disrupt the movement of bicycle traffic to make your turn. Same as changing lanes in front of a car.
I recently heard about ongoing experiments with "bike boxes" to try to improve safety at red lights. If you tell me where I misplaced my notes I'll transcribe them for you. The idea is you put bikes ahead of cars at the red light. Now Ms. Right Turner On Her Cell Phone is going to have to look at the pack of bicycles which are in her path. Maybe she will notice them. They didn't work as well as hoped.
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
The car should move over into the bike lane and treat it like a right turn lane.
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
Incorrect, in theory. When you're driving a car you should not expect a bike to yield any more than another motorist. Doing this to another motorist is called "cutting them off".
In practice, the bike has to farking yield and may end up going head over handlebars in doing so, like the gentleman in TFA.
ggecko: [www.allfunnies.com image 400x400]
You'd need a deep neutrino scan to see where the bike seat is...
ggecko: [www.allfunnies.com image 400x400]
BunkyBrewman: He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Sounds like a swoop and squat to me
FTA:
Cunningham said a car passed them over a double-yellow line, honked its horn and "buzzed" them.
"It just went flying by as we were starting to go down the hill," said Cunningham, who was at the head of the group. "Then halfway down the hill, all of a sudden he just slammed on the brakes."
Intentional? Only one person knows...
runescorpio: BunkyBrewman: astro716: TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Yes, I believe the guy in the car most likely had road rage but from the cyclist's own story, he was passed before they started going downhill which means he was going substantially below the speed limit. (unless these guys can pedal 45mph up a hill outside Boulder)
/glad the guy wasn't hurt too bad
//and I don't have any reason not to believe the cyclist's story but he is going to have to prove he wasn't following too close (although the motorist is going to be cited for leaving the scene of an accident)
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Thing is noone hit the motorist. He wasnt actually in an accident.
I personally do not think cyclists should be on the roads with cars. No insurance for the reckless ones and no visible identification for the ones that cruise through 4 way stops. No lights when all other new vehicles have daytime running lights as a requirement.
I am a commercial driver that has a vehicle that is literally the legal maximum width. Cyclists hopping curbs at lights to get around me essentially barricade moderate traffic at their speed and endanger pedestrians when they get in the middle of them but they do it regularly.
They cut people off with no signals, come past at high speed in heavy traffic when I have a reduced rear visibility vehicle and crowd me.
Screw all cyclists. Make bike paths and go there. 9 years of accident free driving all my most dangerous situations are with cyclists.
People like you should take the bus. That you have a CDL proves that there should be some sort of psych evaluation for all license holders.
But can we all agree that electric scooters don't belong on the bike path?
There's a reason I avoid bike paths in downtown Montreal.
Meh. I ride 17 miles each way in plain old walking shorts. And a tshirt. I look like some south philly goomba riding his bike to his contracting job 'cause he lost his license after his third DUI....
If I rode thru south philly in spandex I'd get my ass kicked. Word.
Last week was a typical week. Rode my usual way, unscathed. I cut thru the construction site at the bottom of Richmond where they're shifting Richmond over and building a new onramp onto I-95. There a long section of new roadbed there, paralleling Richmond, but not connected or open to traffic, separated by dirt and orange traffic barrels, so I was cruising on it, completely separate from traffic, AND STILL some asshole in a van was yelling something at me as he drove by.
Coming back in the afternoon, there were people on the sidewalk in front of the shops at Penn's Landing, so I bailed out into the street (Columbus Blvd., into a bike lane, GOING WITH TRAFFIC!) to go around them. Of course, the usual happened when I venture into traffic. This time it was some numbnut on a crotch rocket decided to blow by me going 60 mph while doing a wheelie. Common event on Columbus, one of the reasons I stick to the sidewalk. If he had screwed up, I would have been dead meat.
One thing that continues to amaze me is all the Hipsters on old school Scwhinn type bikes that ride in traffic listening to Ipods. THAT is suicidal. You need to hear the enemy approaching....
Quantum Apostrophe Why people want to desperately deny that youth is better than oldth, I don't know
If I had to guess, at least in this context, is how often competitive cyclist are still competing and winning in some cases into their 30s, even late 30s and 40s. I think the average age of major event winners are late 20s to early 30s. Young 20s generally do not have the experience or stamina to win yet.
You can argue older is better but peak age is not 40s. Of course traditional sports like football and basketball is a totally different situation, unless you're Teemu Selanne.
Quantum Apostrophe: blacksho89: Go look at Ironman race winners, and look at their ages. 40 year olds are faster than 20 year olds. And when you get into ultrarunning (100 miles +), women are minutes away from beating men. Ann Trason has won some ultras outright.
You might want to check your assumptions; they don't agree with the facts. And make sure you polish the brush guard on your Ram this weekend; you'll need it at the grocery store.
You might want to learn how to read.
Maybe some 40 year olds are faster than some *OTHER* 20 year olds, but barring disease, there is NO WAY that the same person is better at 40 than 20.
The guy in the article said it himself.
Why people want to desperately deny that youth is better than oldth, I don't know.
I wonder why the Olympics aren't packed with paunchy balding 40 year olds.
mbillips: You see idiots drafting on each other at 30+ mph, dodging around little kids on tricycles and old couples out for a walk.
Out for a *walk* on a designated bike path?
Here in Montreal there are places where there are pictures of bikes spray-painted on the ground.
At what point can we say we've done enough to tell people it's a bike path?
Oh. Montreal. Never mind with facts, then, continue with your ignorant snobbery.
mark12A: 13 years now I've been biking thru Philly. No road rage incidents. No injuries. Why? Ride against traffic, stay on the sidewalk as much as you can, don't be a dick when pedestrians are around. Simple. Works. Saved by ass NUMEROUS times.
As a runner, cyclist and driver, I can't stand people like you. It's good to be wary of cars and defer to them whenever possible, but dont break the law and be a dick to everyone else.
Really Twowheeling Tim? You're going to dump on the guy with the CDL who's driven responsibly for 9 years to earn a living delivering products that you have in your house, all the while taking care to keep inconsiderate cyclists alive?
I was upset when I read the headline as I thought it was someone jogging. When I read the article and realized it was a bicyclists, a smile slowly crept across my face!
I constantly listen and look back for approaching cars, especially near intersections. If I am going straight and car is going to turn right I will slow down and let the car pass, even stop before the intersection. My goal is to do my route and make it home, I don't care how long it takes. I usually slow and move over to let cars pass and will give a friendly wave as drivers slow a bit or move over some as they pass.
Drivers have to use indicators though, the sooner the better, otherwise it looks like you are going to go straight.
I lived in boulder for seven years most of which was up a mountain pass that cyclist really liked to ride. Most were fine, but like drivers, some were idiots. I nearly hit at least 20 cyclists, and the occasional skate boarder, when they flew around blind corners in the on coming traffic lane. The most annoying cyclists are the ones who ride in large groups and then get mad when you pass them. Or the ones who are not in good enough shape to ride the mountain they are exhausted, not paying attention, and are weaving back and forth with every pedal push.
The laws in boulder are wired when it comes to cyclist and cars. Bikes are supposed to ride as far to the right as they can, cars are to give three feet when passing, this makes it technically illegal to pass a bike on double yellow one lane roads. Bikes are also considered vehicles and have to abide by all traffic laws, and dismount on sidewalks. This means impeding the flow of traffic is a ticketable offence, but I've never seen a ticket given to a cyclist for this.
If both cyclists and cars abide by the general rules of the road and are courteous to each other things would be fine. Instead cyclist kick and whip chains at cars, and drivers run cyclist off the road. It's a fight the cyclist won't win. I would rather be alive then right.. The saying I lived by in boulder as a cyclist and pedestrian. I had friend get hit two times while riding a bike in boulder, both times he thought he was in the right but it ended up being his fault. He is a big guy and broke his collarbone and totaled a metro in one of the wreaks.
Zeno-25: Cyclists... the scourge of anyone trying to get anywhere on time using the roads. This guy's luck probably caught up with him after blocking the road for many, many other people and enjoying it.
You mean the scourge of those who only give themselves a marginal amount of time to get where they want to be rather than ample time?
I always give myself an hour to ride my bike to work though it only takes thirty five minutes or so. I'm never late, even when I get a flat tire.
Maybe you should get your slovenly ass out of the car more often.
I read the whole article and what a gem! No contact between the cyclist and the car, meaning the vehicle isn't at fault and no tickets will be issued.
Learn how to use your damn brakes correctly! How hilarious, flipping yourself over the bars because you braked too hard! I love it
"Yeah I think he had some road rage," Cunningham said. "I don't know if he got angry cause it was a large group or what."
I'm guessing the other side of the story starts there.
Quantum Apostrophe: Maybe some 40 year olds are faster than some *OTHER* 20 year olds, but barring disease, there is NO WAY that the same person is better at 40 than 20.
Define better.
The 20 yr old is going to be bale to recover, and gain muscle faster.
If they regularly exercised from their 20's on there is a good chance they will be stronger and or faster (depending on their exercise).
I do a lot of local 10k's and would routinely win or place when I was in my 20's. I got faster in my 30's but no longer placed as everyone who did them got faster. Generally the 40's are the fastest group.
WRXminion: I lived in boulder for seven years most of which was up a mountain pass that cyclist really liked to ride. Most were fine, but like drivers, some were idiots. I nearly hit at least 20 cyclists, and the occasional skate boarder, when they flew around blind corners in the on coming traffic lane. The most annoying cyclists are the ones who ride in large groups and then get mad when you pass them. Or the ones who are not in good enough shape to ride the mountain they are exhausted, not paying attention, and are weaving back and forth with every pedal push.
The laws in boulder are wired when it comes to cyclist and cars. Bikes are supposed to ride as far to the right as they can, cars are to give three feet when passing, this makes it technically illegal to pass a bike on double yellow one lane roads. Bikes are also considered vehicles and have to abide by all traffic laws, and dismount on sidewalks. This means impeding the flow of traffic is a ticketable offence, but I've never seen a ticket given to a cyclist for this.
If both cyclists and cars abide by the general rules of the road and are courteous to each other things would be fine. Instead cyclist kick and whip chains at cars, and drivers run cyclist off the road. It's a fight the cyclist won't win. I would rather be alive then right.. The saying I lived by in boulder as a cyclist and pedestrian. I had friend get hit two times while riding a bike in boulder, both times he thought he was in the right but it ended up being his fault. He is a big guy and broke his collarbone and totaled a metro in one of the wreaks.
CRS 42-4-1005(d) allows passing a bicyclist moving the same direction and in the same lane when such movement can be made in safety and without interfering with, impeding, or endangering other traffic lawfully using the highway.
shaddix: I read the whole article and what a gem! No contact between the cyclist and the car, meaning the vehicle isn't at fault and no tickets will be issued.
Learn how to use your damn brakes correctly! How hilarious, flipping yourself over the bars because you braked too hard! I love it
So if one motorist runs the other off the road, but there is no contact between the cars, it's all cool then?
Sounds like the cyclist is mostly at fault in the article. If the motorist passing them was acting at all antagonistic, it's time to slow down and avoid that car like the plague. Just because someone may have the right of the road doesnt mean they have the right to be oblivious to changing conditions of the road. As others have stated in thread, in the battle car vs bicycle, the bicycle will always lose.
mark12A:One thing that continues to amaze me is all the Hipsters on old school Scwhinn type bikes that ride in traffic listening to Ipods. THAT is suicidal. You need to hear the enemy approaching....
I figure every asshole out there running or riding with ear buds in is completely insane. They're afraid to be alone with their own thoughts so they drown them out.
Zeno-25: Cyclists... the scourge of anyone trying to get anywhere on time using the roads. This guy's luck probably caught up with him after blocking the road for many, many other people and enjoying it.
Leave more time. If you can't get to where you need to be without a guarantee of being able to go the speed limit or higher all the time, then fark you. It's nice to be able to justify grievous bodily harm by inventing a vendetta in the cyclist's head.
wichitaleaf: If you cannot keep up with the pace of traffic you do not belong on the road.
Just like a slow car, they cause massive problems in traffic also.
No. They don't. It's only a problem to those drivers that can't stand being delayed a few seconds. The impatience of the average driver is immense. Add bike hate to the equation and they just can't stand the idea of having to slow down for anyone or anything.
Anyone who mentions what a cyclist is wearing is a complete asshat.
blacksho89: Oh. Montreal. Never mind with facts, then, continue with your ignorant snobbery.
That's an interesting argument. Care to expound?
I'd like to know what's ignorant, and what's snobbery.
Obviously the educational system here is lacking.
liam76: Define better.
The 20 yr old is going to be bale to recover, and gain muscle faster.
I'd say that's "better". The 20s will also be lighter, learn faster, have less AGE molecules in their body.
Less wrinkles, more hair.
Better hearing, eyesight.
Is it really that hard to believe that being younger is better?
As a runner, cyclist and driver, I can't stand people like you. It's good to be wary of cars and defer to them whenever possible, but dont break the law and be a dick to everyone else.
13 years. Oh, and by the way, 13 YEARS. 13 years I've been doing this, and if I was a dick I would have gotten my ass kicked loooong time ago by the good, but rough hewn, citizens of Philly. If what I was doing was dangerous, I would have been tagged years ago.
Is simples: Stay away from pedestrians when they're on sidewalks. And riding against traffic takes up NOT ONE SINGLE SQUARE INCH more space than riding with traffic. As for the oncoming motorists having less reaction time, that's OK, because now I HAVE REACTION TIME, unlike when I'm not facing traffic. So now I'm not completely vulnerable to an endless stream of total strangers coming up behind me.
squegeebooo: phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
The car should move over into the bike lane and treat it like a right turn lane.
I encounter this every day on the way to work, enough so that I actually took the time to look up the laws. This is correct.
SoupJohnB: As some Farker put it in an unrelated thread, you can't drink poison and expect the other person to die.
Unfortunately, that seems to be exactly what the Millenial generation expects, and even demands. Otherwise It's Not Fair and We Must Have Justice For (insert name of latest an hero).
Why hasn't he been cited for reckless driving yet?
mark12A: As a runner, cyclist and driver, I can't stand people like you. It's good to be wary of cars and defer to them whenever possible, but dont break the law and be a dick to everyone else.
13 years. Oh, and by the way, 13 YEARS. 13 years I've been doing this, and if I was a dick I would have gotten my ass kicked loooong time ago by the good, but rough hewn, citizens of Philly. If what I was doing was dangerous, I would have been tagged years ago.
Is simples: Stay away from pedestrians when they're on sidewalks. And riding against traffic takes up NOT ONE SINGLE SQUARE INCH more space than riding with traffic. As for the oncoming motorists having less reaction time, that's OK, because now I HAVE REACTION TIME, unlike when I'm not facing traffic. So now I'm not completely vulnerable to an endless stream of total strangers coming up behind me.
Since the bike is a vehicle, you are violating the law by driving against the flow of traffic and you put cyclists who do follow the law in an awkward and dangerous position when you pass them. If Philly is really that violent I'm sure more than one rider has wanted to knock you off your bike, but that would probably risk them getting injured too.
As for riding in the sidewalk, it is illegal in your city, but I'm sure people dont care if dont actually share it with pedestrians. It's still stupid. If a person stays deep in the bike lane and is a cautious rider, it's safer to follow traffic laws because it makes your behavior predictable.
Yogimus: How does one "Suffer" from road rage? It isn't cancer. It is a voluntary state of being.
This. Suffer implies no fault. Should be douchnozzle with road rage
Quantum Apostrophe: I'd say that's "better". The 20s will also be lighter, learn faster, have less AGE molecules in their body.
farking age molecules!
Quantum Apostrophe: Is it really that hard to believe that being younger is better
If that is your definition of better, no.
But in a thread about a guy doing an iron-man tri, discussing biking, etc speed would be a sensible measuring stick.
liam76: farking age molecules!
Tell me about it. The human body basically cooks itself sous vide for decades.
liam76: But in a thread about a guy doing an iron-man tri, discussing biking, etc speed would be a sensible measuring stick.
Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him?
Maybe you should email him.
Do dirtbag biker was:
Following to close.
Traveling too fast for conditions.
Ticket that SOB.
kokomo61: There are asses on both sides of the rider / car equation, but in every case, the cyclist will lose. Yeah, we have a right to ride on the roads. No, cars don't have the right to put cyclists in danger. Still, the odds are stacked in favor of the driver. I try to minimize my time on the road to ones with wide shoulders just for safety.
(Not So ) CSB....
I was bike commuting to work on a paved bike trail that has several street level crossings. Stopped at one, and traffic on the right stopped. I held up my hand to acknowledge the driver, and waited for the left traffic to stop. Since it was a 2-lane street, and both sides were stopped, I clipped in and rolled forward. No sooner did I clear the car on the right, than a guy from three cars back came flying around the cars, passing on the right, and flipped me up on the hood like Mannix. I went up the hood, smacked my ribs on the A pillar of the car, and slid back down to the pavement still clipped into the pedals.
The guy at first said, "I didn't see him." Well, yeah - when you pass a line of stopped cars on the right at a highly marked crosswalk, you're not likely to see someone actually IN the crosswalk.
He initially pled 'not guilty' to the citation, probably hoping that the officer wouldn't show. They did, as did I. At first, he said he was trying to park the car (at 25MPH). He then said he was looking of a museum (in Vienna, Va....where there are no museums within 10 miles). THEN, he said that I ran out in front of him, using the 'spandex' defense (that all cyclists are dangerous)....When it was my turn, I just handed the judge a printout of my GPS/HR monitor data.....which showed my exact location, slowing my speed, heart rate and pedal cadence approaching the intersection...then being stopped for 45 seconds before moving forward at 5 MPH......then suddenly moving sideways at 20MPH.
Guilty as charged. Took three years to settle out for my bike, doctor visits and X-rays/MRI's and PT (thanks to bruised ribs, a hip injury and a cracked vertebrae). Not fun at all.
What do you use for GPS that gives such accurate results?
Quantum Apostrophe: Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him?
Because he is a whiny douche?
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
Problem is the speeds. If the car and bike are both stopped, then I would yeild to the bike as a car driver.
But if the bike is overtaking in the bike lane, I think the bike should yield. it may not be the law, but it's the smart thing - Bikes are hard enough to see, and if you are overtaking in his blind spot, you're asking for trouble assuming he's going to spot you.
If the car is overtaking, he's being a jerk. Slow up for your turn anyway, and stay behind the bike.
Shang-High: I just "buzzed" a bicyclist riding half of the way into the road, when there was a good 6 feet to his right for biking (and bike lane markings). Sorry, I'm not going to cross the double yellow lines into oncoming traffic, "share the road" works both ways.
That makes you both assholes. Congratulations.
liam76: Quantum Apostrophe: Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him?
Because he is a whiny douche?
Oh I see. Just discard contrary opinions. Just remember to let fresh air into your bubble regularly.
If 76 refers to your birth year, you have some rude awakening ahead, in just a few years... It's downhill from here, and very rapidly at that.
Maybe if we can figure out how to clear out AGE molecules, aging wouldn't be such a problem.
ggecko: [www.allfunnies.com image 400x400]
It should be illegal to make certain types of clothing in certain sizes.
I've had motorists do the exact same thing to me (pull into the bike lane and then slam on the brakes). Fortunately I've always managed to stop in time. Since I'm in Fark's favorite state of Florida I generally expect this sort of behavior.
/always stays as far to the right as possible and stops at traffic lights
/rides alone or in a small group that rides single-file
/loves cars and bikes
Shang-High: I just "buzzed" a bicyclist riding half of the way into the road, when there was a good 6 feet to his right for biking (and bike lane markings). Sorry, I'm not going to cross the double yellow lines into oncoming traffic, "share the road" works both ways.
If a cyclist pushes all the way into the middle of the road, there's probably a good reason for it. There might be gravel or debris on the side, the road might be torn up over there, or there might be parked cars that create a 'door zone' you have to avoid. Cyclists typically don't ride in the middle of the road if they don't need to.
In my experience, 2/3 of bike lanes are unusable as places to ride due to being too narrow, too close to parked cars, too full of gravel and trash, too slanted, too uneven, or too torn up. They make you harder to see as well. I kind of wish they wouldn't make them; just make the road a bit wider with a good shoulder and have cyclists ride in the lane. It's actually safer to ride in the lane because drivers are more likely to see you there.
Quantum Apostrophe: liam76: farking age molecules!
Tell me about it. The human body basically cooks itself sous vide for decades.
liam76: But in a thread about a guy doing an iron-man tri, discussing biking, etc speed would be a sensible measuring stick.
Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him?
Maybe you should email him.
He competes in the Pro class. After he hits forty it is just about over for him: even at 39 he is at a huge disadvantage. He'll still be able to compete, and likely win, the master's age group but the purses aren't as big so he'll need a day job probably working as a coach or open his own Tri shop in the area.
johnny queso: Shang-High: I just "buzzed" a bicyclist riding half of the way into the road, when there was a good 6 feet to his right for biking (and bike lane markings). Sorry, I'm not going to cross the double yellow lines into oncoming traffic, "share the road" works both ways.
That makes you both assholes. Congratulations.
In Boise the driver could be charged with a misdomeaner
foo monkey: What do you use for GPS that gives such accurate results?
Garmin Edge 305 - since upgraded to an Edge 500. Sample data image below:
Like other GPS's, it's only accurate to within a few feet....but it DOES show speed, altitude, cadence, heart rate (and does averages, max, etc.) It showed my entire route up to that point, including all intersections, and how much time I stopped, etc. owed my location at that intersection, stopped for nearly a full minute, as well as the approach, where I coasted, stopped pedaling, slowed and stopped, with my HR falling the whole time. You can get a watch-based GPS for a little over $100. A bike mounted one for 2X that or so. The only other thing I wish I had on a bike commute would be a GoPro Hero. look what i can do OnlyM3: Do dirtbag biker was: Following to close. Traveling too fast for conditions. Ticket that SOB. So, when you're driving, and some road raging fellow motorist cuts you off and brake checks you, it's still your fault if you rear end him? Even though he was intentionally trying to cause an accident? Publikwerks: phaseolus: Traffic law question: Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight. Bike yields to car, correct? Problem is the speeds. If the car and bike are both stopped, then I would yeild to the bike as a car driver. But if the bike is overtaking in the bike lane, I think the bike should yield. it may not be the law, but it's the smart thing - Bikes are hard enough to see, and if you are overtaking in his blind spot, you're asking for trouble assuming he's going to spot you. If the car is overtaking, he's being a jerk. Slow up for your turn anyway, and stay behind the bike. If you're catching up to traffic stopped at a light when you're on a bicycle, and you see a car with a right turn signal on, the only sensible thing to do is to yield (stop behind them). And if they don't have a signal on, but are edging to the right, just assume they are a slob who doesn't use their signals, and still yield. And if you're cruising along at twenty plus MPH on your bicycle, catching up to a long line of cars stopped at a light, slow down. Sometimes there will be a car that does not have their signal on, has not edged over to the right, and has had an opportunity to do a legal right turn on a red for sometime (no cross traffic) who will instantly make a right turn when the light changes. I had that happen to me once - I think the guy in the truck saw me coming in his side view mirror, and decided he would try to make me crash. But I was able to turn inside of him. I give cyclists wide berth when able, and slow to pass when I can't. There's a few jerks out there pedaling, but by and large they're a decent group. Besides, I appreciate that they're not burning fossil fuel and getting a little exercise as I drive by. /fat. zinny: I give cyclists wide berth when able, and slow to pass when I can't. There's a few jerks out there pedaling, but by and large they're a decent group. Besides, I appreciate that they're not burning fossil fuel and getting a little exercise as I drive by. /fat. Um, are you new to the internet? You post exudes absolutely no toughness, and doesn't nothing to infuriate anyone. You're supposed to brag about how you just harassed a cyclist. Wait someone just did that--how you just killed a cyclist. And you don't care, man, cause he looked like an illegal anyways. That's FARKing. /Also, very reasonable post //I drive the same way around cyclists ///Had to stop cycling here because of the real life psychopaths on the road On July 2008, Thompson was driving a sedan in Brentwood when he came across two bicyclists. Thompson pulled out in front of the two bicyclists, and slammed the brakes, causing the two bicyclists to hit the back of the car. One of the bicyclists Ron Peterson, crashed through the back of the Infiniti. Peterson's nose was broken, and he suffered several other injuries. The other bicyclist fell to the pavement, and was also injured. Thompson went on trial in a case that was watched around the country. He was found guilty, and last week sentenced to 5 years in prison. The sentence came after much support from bicyclists around the country. The court received more than 270 messages calling for a tough sentence on the doctor. The two bicyclists had also pleaded on bicycle forums calling on bicyclists to write in to show their support for bicyclists in Los Angeles, who are now involved in increasing confrontations with motorists. Letters of support have poured in from as far as the UK and China. The judge said that he did not consider the messages of support that called for a stronger sentence, while delivering his decision. But he expressed his belief that Thompson has showed no remorse in his behavior after the accident. During the trial, Thompson had said that he and other residents in Brentwood were angry with bicyclists, because they frequently broke traffic rules. Why he choose to take that anger and misplaced frustration on two innocent bicyclists who were following traffic rules and riding safely, is not clear. Los Angeles Doctor in Road Rage Bicycle Accident Sentenced to 5 years in Prison zinny: I give cyclists wide berth when able, and slow to pass when I can't. There's a few jerks out there pedaling, but by and large they're a decent group. Besides, I appreciate that they're not burning fossil fuel and getting a little exercise as I drive by. /fat. This is an outrage, you can't possibly come into a bike thread with that attitude. There are standards to uphold. Quantum Apostrophe: liam76: Quantum Apostrophe: Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him? Because he is a whiny douche? Oh I see. Just discard contrary opinions. Just remember to let fresh air into your bubble regularly. If 76 refers to your birth year, you have some rude awakening ahead, in just a few years... It's downhill from here, and very rapidly at that. Maybe if we can figure out how to clear out AGE molecules, aging wouldn't be such a problem. As someone you'd consider old I'm cracking up reading your posts about being old sucking. It does. I hope you live a long life ;) Something you might not be aware of is that age has advantages in the more difficult endurance sports due to increased pain management. You become mentally tougher and are able to make your body do things you couldn't when it was younger and stronger. In cycling it is well known to fear the old men. They are retired so can train all day and they have mastered pain control. I've seen guys in their 70s post times equal to the fastest 20-30s guys but only at time trial distances. Quantum Apostrophe: mbillips: You see idiots drafting on each other at 30+ mph, dodging around little kids on tricycles and old couples out for a walk. Out for a *walk* on a designated bike path? What part of "multi-use path" do you not understand? It's not a bike path, it's a walking/cycling/running/skating trail, and labeled as such. The Googles Do Nothing: /When I ride my bike I stop at all the stop signs and lights and stay to the far right. Lane position is something that comes up a lot when I'm teaching a Basic Rider's Course (for motorcycles). One concern I have with a far-right lane position is this: You get the moron that wants to pass you and either: 1. Doesn't get over far enough and clips you, or 2. Sees and oncoming car and figures he can squeeze by you without getting over and clips you. I usually ride in the left or middle of the lane: that way if someone wants to pass: it discourages the second type, and gives you space to get over for the first type. phaseolus: Traffic law question: Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight. Bike yields to car, correct? I believe the law of physics trumps traffic law in this situation Smeggy Smurf: mark12A:One thing that continues to amaze me is all the Hipsters on old school Scwhinn type bikes that ride in traffic listening to Ipods. THAT is suicidal. You need to hear the enemy approaching.... I figure every asshole out there running or riding with ear buds in is completely insane. They're afraid to be alone with their own thoughts so they drown them out. I had a cop stop me in Osaka once for listening to headphones on a bicycle. Different culture there. Bikes rule the road and sidewalk too, due to sheer numbers. Quantum Apostrophe: liam76: Quantum Apostrophe: Then why is the guy in TFA bemoaning that at 40 it's pretty much over for him? Because he is a whiny douche? Oh I see. Just discard contrary opinions. Try taking your own advice. Or maybe show up to some non pro endurance races. Quantum Apostrophe: If 76 refers to your birth year, you have some rude awakening ahead, in just a few years... It's downhill from here, and very rapidly at that It is in the ballpark. It has been downhill for a number of things since 30. But I am still putting up faster times in races (if I actively train) than I used to. Quantum Apostrophe: But can we all agree that electric scooters don't belong on the bike path? There's a reason I avoid bike paths in downtown Montreal. No we can't agree....Let the scooters ride 4 or 5 abreast at 5mph and see how many of the bikers get a taste of road rage So if one motorist runs the other off the road, but there is no contact between the cars, it's all cool then? Without witnesses, yes. Nothing you can do. If there are witnesses or the driver confesses, depends on the situation. My ex-girlfriend ran a car off the road by making an unsafe lane change. She stopped to help and got a ticket. If she kept going she might have gotten away clean. In my area a bicyclist got crushed under a rear wheel of a truck. Driver kept going. Grand jury refused to indict. It was likely the driver never knew he hit anything. The bicycle was to the right of the truck. Fark_Guy_Rob: Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used. So, I decided to run in the bicycle lane. Cyclists are constantly riding with cars. Cyclists are smaller and slower, and they constantly complain about how a**hole drivers don't give them space or treat them like they belong. Seeing that I was slower than them, and without a bicycle, I expected that they would be understanding. That they would give me plenty of space, and wait until it was safe to pass me. Sure, I can't run as fast as a cyclist, but a cyclist can't cycle as fast as car. If they expect cars to slow down and go their speed until they can safely pass - certainly, they wouldn't begrudge doing the same to me. Right? Apparently, I was wrong. Yay for double standards! From a traffic law standpoint, yes, you were wrong. Fallout Zone: squegeebooo: phaseolus: Traffic law question: Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight. Bike yields to car, correct? The car should move over into the bike lane and treat it like a right turn lane. I encounter this every day on the way to work, enough so that I actually took the time to look up the laws. This is correct. Yes it is, but from the description it sounds like the road markings at this particular intersection are wrong. I cant speak for everywhere, but around here at least there arent any bike lanes on roads that dont also have parking lanes. Approaching intersections, the bike lane becomes dotted and cars are supposed to move over the bike lane into the parking lane which has become a turn lane at the end of the block to make a right hand turn. Repo Man: Publikwerks: phaseolus: Traffic law question: Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight. Bike yields to car, correct? Problem is the speeds. If the car and bike are both stopped, then I would yeild to the bike as a car driver. But if the bike is overtaking in the bike lane, I think the bike should yield. it may not be the law, but it's the smart thing - Bikes are hard enough to see, and if you are overtaking in his blind spot, you're asking for trouble assuming he's going to spot you. If the car is overtaking, he's being a jerk. Slow up for your turn anyway, and stay behind the bike. If you're catching up to traffic stopped at a light when you're on a bicycle, and you see a car with a right turn signal on, the only sensible thing to do is to yield (stop behind them). And if they don't have a signal on, but are edging to the right, just assume they are a slob who doesn't use their signals, and still yield. And if you're cruising along at twenty plus MPH on your bicycle, catching up to a long line of cars stopped at a light, slow down. Sometimes there will be a car that does not have their signal on, has not edged over to the right, and has had an opportunity to do a legal right turn on a red for sometime (no cross traffic) who will instantly make a right turn when the light changes. I had that happen to me once - I think the guy in the truck saw me coming in his side view mirror, and decided he would try to make me crash. But I was able to turn inside of him. The car should blend into the bike lane for a right hand turn. If the car isn't in the bike lane, it's on them for turning across a lane of traffic and causing an accident. Just like a cyclist making a left turn needs to merge over into the left turn lane for left turns, instead of just turning from a bike lane. That said, if you think someone is turning right, and they're not in the bike lane, you might want to consider yielding. Being technically correct, while the best correct, isn't worth risking a hospital visit. If he wants to be accorded the same rights and responsibilities as the driver of a motor vehicle, he should be issued a citation for following too close! Seriously, I feel for the guy, but the highway is no place to train for a bike race. He might be an upstanding citizen, but many (other) cyclists don't wear safety equipment, follow traffic rules or maintain their bicycles (or even sobriety!). Get off your high horse, and off the road. For what it's worth I don't harass cyclists, and don't pass them illegally no matter how slow they are, but they don't pay taxes to maintain the highways and they don't belong there. Bicycles being moving vehicles, subject to the same laws as motor vehicles in most states in the U.S., and having been hit 9 times in 3 years, 4 in my current neighborhood, once ending up on the hood of a guys car for 2 blocks (and when he figured it out, stopped fast, knocking me off, then told me that he didn't need to give me insurance info because I appeared fine, and the police took 3 hours to get there)- I find this funny. Not just because I've been hit twice by people on bicycles, including once last month as some hipster woman ignored a stop sign and ran over my right foot (witnesses screamed and screamed at her to stop), but because the guy seems like a huge cry baby, douchebag. If you're on a moving vehicle you should operate your vehicle defensively, not aggressively, which is how this fellow was riding his bike. The assumption that someone *has* to look out for you doesn't even apply to pedestrians (even if it IS the law, reality stands). I walk, so I risk getting hit by morons. Someone in my neighborhood was killed three blocks from me: a pedestrian, hit and run, but they caught the guy a day later (go go HPD slackers). He took the same risk willing me. We walk defensively. We'll jump out of the way. We're subject to road rage from drivers- they pull over, threaten us, chase us, run over us. This guy was being a dick, maybe he'll be more aware next time. Carth: After he hits forty it is just about over for him: I know. I'm just saying that there is no way that anyone at 40 is in better shape than at 20. It's just so much delusional BS. liam76: Try taking your own advice. Or maybe show up to some non pro endurance races. Why? I'm 41. I'd be dead in the first kilometer. liam76: It has been downhill for a number of things since 30. But I am still putting up faster times in races (if I actively train) than I used to. There we go. That wasn't so hard was it? So, are you going to give some money to SENS? I think aging is a program, a disease, and if we have enough knowledge it'll be controlled like diabetes. kokomo61: foo monkey: What do you use for GPS that gives such accurate results? Garmin Edge 305 - since upgraded to an Edge 500. Sample data image below: Like other GPS's, it's only accurate to within a few feet....but it DOES show speed, altitude, cadence, heart rate (and does averages, max, etc.) It showed my entire route up to that point, including all intersections, and how much time I stopped, etc. owed my location at that intersection, stopped for nearly a full minute, as well as the approach, where I coasted, stopped pedaling, slowed and stopped, with my HR falling the whole time. You can get a watch-based GPS for a little over$100. A bike mounted one for 2X that or so. The only other thing I wish I had on a bike commute would be a GoPro Hero.
Bit fancier than Strava on my iPhone.
Quantum Apostrophe: There we go. That wasn't so hard was it?
No. And I never had a problem with that sentiment, nor did anything I said counter it.
But when you say dumb things like this,
Quantum Apostrophe: I'm just saying that there is no way that anyone at 40 is in better shape than at 20
you clearly don't know what you are talking about.
Quantum Apostrophe: Carth: After he hits forty it is just about over for him:
I know. I'm just saying that there is no way that anyone at 40 is in better shape than at 20. It's just so much delusional BS.
What if you are morbidly obese at age 20 and then take up a sport at a later age? I was in terrible shape in my teens and 20s but I probably qualify as an athlete now. My boyfriend runs a lot faster in his 30's than he did in high school running sports because he had terrible coaching back then. Just because someone is younger doesn't make them in better shape or a better athlete.
The main limiting factor is that people who train to their peak performance throughout their development can lose a bit of speed as they age or they can just tear up their body with the sport and gradually lose performance due to injuries. To generalize beyond that is just not correct.
blacksho89: WRXminion: I lived in boulder for seven years most of which was up a mountain pass that cyclist really liked to ride. Most were fine, but like drivers, some were idiots. I nearly hit at least 20 cyclists, and the occasional skate boarder, when they flew around blind corners in the on coming traffic lane. The most annoying cyclists are the ones who ride in large groups and then get mad when you pass them. Or the ones who are not in good enough shape to ride the mountain they are exhausted, not paying attention, and are weaving back and forth with every pedal push.
The laws in boulder are wired when it comes to cyclist and cars. Bikes are supposed to ride as far to the right as they can, cars are to give three feet when passing, this makes it technically illegal to pass a bike on double yellow one lane roads. Bikes are also considered vehicles and have to abide by all traffic laws, and dismount on sidewalks. This means impeding the flow of traffic is a ticketable offence, but I've never seen a ticket given to a cyclist for this.
If both cyclists and cars abide by the general rules of the road and are courteous to each other things would be fine. Instead cyclist kick and whip chains at cars, and drivers run cyclist off the road. It's a fight the cyclist won't win. I would rather be alive then right.. The saying I lived by in boulder as a cyclist and pedestrian. I had friend get hit two times while riding a bike in boulder, both times he thought he was in the right but it ended up being his fault. He is a big guy and broke his collarbone and totaled a metro in one of the wreaks.
CRS 42-4-1005(d) allows passing a bicyclist moving the same direction and in the same lane when such movement can be made in safety and without interfering with, impeding, or endangering other traffic lawfully using the highway.
You familiar with bill 148 which passed in 2009?? Must have three feet to pass.
Also the whole "without interfering with, impeding, or endangering other traffic lawfully using the highway" means you can't break a law: cross a double yellow, speed, or drive recklessly to pass a ciclist.
whfolsom4: If he wants to be accorded the same rights and responsibilities as the driver of a motor vehicle, he should be issued a citation for following too close!
Seriously, I feel for the guy, but the highway is no place to train for a bike race. He might be an upstanding citizen, but many (other) cyclists don't wear safety equipment, follow traffic rules or maintain their bicycles (or even sobriety!). Get off your high horse, and off the road.
For what it's worth I don't harass cyclists, and don't pass them illegally no matter how slow they are, but they don't pay taxes to maintain the highways and they don't belong there.
This is such a moronic talking point. The overwhelming majority of bicyclists also drive a motor vehicles, and pay registration fees and gasoline taxes. You might be an upstanding citizen, but many motorists fail to obey traffic laws, and even drive drunk. When your fellow motorists break traffic laws and drive drunk they can quite easily kill you! But you are worried about sharing the road with a 170 pound rider on a twenty four pound bicycle?
TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
IIRC TFA said that the driver passed and and then changed into their lane immediately before the sudden, hard deceleration.
Look, I realize everyone wants to be an Ironman, but seriously, if you're getting in the way of motorists while you're training for it, it's clearly your fault. You should never be studying advanced physics in traffic, it's just irresponsible.
Ned Overend, Tinker Juarez go 1-2 on Mount Washington as Marti Shea defends 2010 crown
A pair of 50-somethings - Ned Overend and Tinker Juarez - went one-two on Saturday at the 39th Mount Washington Auto Road Bicycle Hillclimb in Pinkham Notch, New Hampshire.
Overend, who was celebrating his 56th birthday, pulled away less than two miles into the 7.6-mile ascent and rode to the victory in 55 minutes and three seconds.
Juarez, 50, took the runner-up spot just over a minute later with Timothy Ahearn third at 3:19.
"It's good to win," said Overend as he cooled off at the summit. "I wanted to win, and I wouldn't travel this far for it if I didn't think I had a chance. Comparing past times with the others, I figured I should be the favorite, but I'm not getting any younger!"
It was the second consecutive runner-up finish for Juarez, who finished second to Nico Toutenhoofd in 2010.
These guys are outliers, but they are a demonstration about what happens when you speak in sweeping generalities.
BunkyBrewman: astro716: TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
3/10. I got halfway through typing a response.
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
Yes, I believe the guy in the car most likely had road rage but from the cyclist's own story, he was passed before they started going downhill which means he was going substantially below the speed limit. (unless these guys can pedal 45mph up a hill outside Boulder)
/glad the guy wasn't hurt too bad
//and I don't have any reason not to believe the cyclist's story but he is going to have to prove he wasn't following too close (although the motorist is going to be cited for leaving the scene of an accident)
///let me tell you about the time someone did a "swoop and squat" on me in Vegas... guess who was at fault? (not them)
Why is it not possible that the vehicle was speeding when he passed? Not saying that the cyclists were going the full 45, but getting passed is hardly proof of going "too slow".
Carth: Fark_Guy_Rob: Mostly unrelated story - but early last year I was training for a 10k run. I was out for a run, down a road I hadn't been on before - it was really crowded though (busy city here) and I couldn't run on the sidewalk. But there was a bicycle lane that was being *very lightly* used.
So, I decided to run in the bicycle lane.
Cyclists are constantly riding with cars. Cyclists are smaller and slower, and they constantly complain about how a**hole drivers don't give them space or treat them like they belong. Seeing that I was slower than them, and without a bicycle, I expected that they would be understanding. That they would give me plenty of space, and wait until it was safe to pass me. Sure, I can't run as fast as a cyclist, but a cyclist can't cycle as fast as car. If they expect cars to slow down and go their speed until they can safely pass - certainly, they wouldn't begrudge doing the same to me. Right?
Apparently, I was wrong. Yay for double standards!
In most areas cyclists have a legal right to the road and are allowed to 'take the lane; as necessary.
Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
The one "slowing down to their speed until they could safely pass" in that scenario should have been you with respect to the other pedestrians on the sidewalk.
WinoRhino: Publikwerks: I'm no fan of road rage assholes, but to be fair, Mr Ironman didn't hit the car. He misjudged if he could get around the car, had to stop too fast, and flipped over the handlebars. He was operating his bike too fast then. If you cannot come to a complete stop safely, then you are going too fast.
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
You know, that's actually a pretty good point. He did say his first choice was to try to get around the car. I'm not defending the motorist, but it's a valid argument.
It's just the same thing that I see everytiem a car encounters something that is way too slow way too soon. They swerve to the shoulder to prevent impact is they might not be able to stop in time. It's an emergency maneuver. Being forced to consider doing so is in no way proof that the cyclist did anything wrong. Sounds more like a "swoop and squat" on the drivers behalf-it's the same tactic people use to commit insurance fraud. It's simple, just blow by your victim, enter their lane as close to them as you can, and then hit the breaks hard. Who here thinks they are immune to this?
Yeea, it's THIS thread again.
I've had a whiskey bottle chucked at me while cycling, among other things, and that was on a road with 8' paved shoulders on both sides, specifically to give cyclists several feet from high-speed traffic. Something of that mass traveling 30MPH could be fatal, but I'm sure the Good Ol' Boy who threw it never considered that. I've had motorcyclists yell at me as they pass; you'd think they would be more understanding.
Using the roads is like some vehicular food chain: the drivers hate the motorcyclists, both hate the bicyclists, all three hate the joggers, and everyone hates the gaggle of women wammering it up on the sidewalk, oblivious to all but their gossip.
Repo Man: whfolsom4: If he wants to be accorded the same rights and responsibilities as the driver of a motor vehicle, he should be issued a citation for following too close!
Seriously, I feel for the guy, but the highway is no place to train for a bike race. He might be an upstanding citizen, but many (other) cyclists don't wear safety equipment, follow traffic rules or maintain their bicycles (or even sobriety!). Get off your high horse, and off the road.
For what it's worth I don't harass cyclists, and don't pass them illegally no matter how slow they are, but they don't pay taxes to maintain the highways and they don't belong there.
This is such a moronic talking point. The overwhelming majority of bicyclists also drive a motor vehicles, and pay registration fees and gasoline taxes. You might be an upstanding citizen, but many motorists fail to obey traffic laws, and even drive drunk. When your fellow motorists break traffic laws and drive drunk they can quite easily kill you! But you are worried about sharing the road with a 170 pound rider on a twenty four pound bicycle?
It's actually not a talking point. I drive past bike lanes and bike-related signage every day. I know which one of us paid for it, and it wasn't the guy on the bike. And since a family member almost hit a cyclist riding on a 2-lane highway after dark, without lights or reflectors, yeah, it worries me.
Chaghatai: WinoRhino: Publikwerks: I'm no fan of road rage assholes, but to be fair, Mr Ironman didn't hit the car. He misjudged if he could get around the car, had to stop too fast, and flipped over the handlebars. He was operating his bike too fast then. If you cannot come to a complete stop safely, then you are going too fast.
If the roles were reversed, and the car ran him over because it couldn't come to a complete stop fast enough, would that then be the bikers fault?
You know, that's actually a pretty good point. He did say his first choice was to try to get around the car. I'm not defending the motorist, but it's a valid argument.
It's just the same thing that I see everytiem a car encounters something that is way too slow way too soon. They swerve to the shoulder to prevent impact is they might not be able to stop in time. It's an emergency maneuver. Being forced to consider doing so is in no way proof that the cyclist did anything wrong. Sounds more like a "swoop and squat" on the drivers behalf-it's the same tactic people use to commit insurance fraud. It's simple, just blow by your victim, enter their lane as close to them as you can, and then hit the breaks hard. Who here thinks they are immune to this?
Well, he'd just finished that bottle off, what else was he supposed to do with it? Can't leave it in the cab, that's for sure!
kokomo61: bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
[www.thecycler.net image 400x400]
Bikes aren't pedestrians and cars aren't required to yield at pedestrian crosswalks unless you get off your bike and walk.
whfolsom4: Repo Man: whfolsom4: If he wants to be accorded the same rights and responsibilities as the driver of a motor vehicle, he should be issued a citation for following too close!
Seriously, I feel for the guy, but the highway is no place to train for a bike race. He might be an upstanding citizen, but many (other) cyclists don't wear safety equipment, follow traffic rules or maintain their bicycles (or even sobriety!). Get off your high horse, and off the road.
For what it's worth I don't harass cyclists, and don't pass them illegally no matter how slow they are, but they don't pay taxes to maintain the highways and they don't belong there.
This is such a moronic talking point. The overwhelming majority of bicyclists also drive a motor vehicles, and pay registration fees and gasoline taxes. You might be an upstanding citizen, but many motorists fail to obey traffic laws, and even drive drunk. When your fellow motorists break traffic laws and drive drunk they can quite easily kill you! But you are worried about sharing the road with a 170 pound rider on a twenty four pound bicycle?
It's actually not a talking point. I drive past bike lanes and bike-related signage every day. I know which one of us paid for it, and it wasn't the guy on the bike. And since a family member almost hit a cyclist riding on a 2-lane highway after dark, without lights or reflectors, yeah, it worries me.
There is no way that you could know this, unless you know the person personally, and know their income. For all you know, that person has a collection of cars at home. As for your family member's close call, while it would be upsetting to hit a cyclist while driving a car, it would likely be lethal for the cyclist. Under the described circumstances, there is almost no way the motorist would get in trouble for the accident, unless they were drunk or driving recklessly.
mbillips: Carth: BunkyBrewman:
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
The bike at the same speed takes longer, definitely. Bike brakes suck. Cars weigh a lot more, but they have giant disc brakes gripped hydraulically by ceramic plates, while bikes have itty bits of rubber hand-squeezed onto a tiny strip of often-damp tire rim.
/Have a permanent dent in my left thigh from where a car with old-fashioned door handles passed me and immediately turned right into a parking lot, in the rain, assuming I could stop.
Uhm, I got hydraulic disc brakes on my bike. Less to do with the brakes and more to do with the light frame (and your spandexed ass) not providing enough traction with the road. You skid.
change1211: kokomo61: bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
[www.thecycler.net image 400x400]
Bikes aren't pedestrians and cars aren't required to yield at pedestrian crosswalks unless you get off your bike and walk.
That is true, and one more reason that bicycles should not be on the sidewalk, or in the crosswalk. Those are for pedestrians.
ReverendJynxed: mbillips: Carth: BunkyBrewman:
He might be trolling, but he does have a point. The bicyclist rear-ended the car. He has to prove why he was following to close. The question here is simple; What takes longer to stop when going downhill... a bicycle or a two ton vehicle?
The bike at the same speed takes longer, definitely. Bike brakes suck. Cars weigh a lot more, but they have giant disc brakes gripped hydraulically by ceramic plates, while bikes have itty bits of rubber hand-squeezed onto a tiny strip of often-damp tire rim.
/Have a permanent dent in my left thigh from where a car with old-fashioned door handles passed me and immediately turned right into a parking lot, in the rain, assuming I could stop.
Uhm, I got hydraulic disc brakes on my bike. Less to do with the brakes and more to do with the light frame (and your spandexed ass) not providing enough traction with the road. You skid.
I think the real culprit (the main limiting factor in a bicycle's deceleration) is the tire's contact patch. At least when speaking of modern, quality bicycles with good brakes (caliper or disc) and aluminum rims. Cheap ass bicycles (known in the trades as CABs) have very poor quality caliper brakes, that when combined with the friction characteristics of their cheap ass chromed steel rims, make for very long stopping distances when dry, and even worse when wet.
Fark_Guy_Rob:
So, I decided to run in the bicycle lane.
...
Apparently, I was wrong. Yay for double standards!
Runner & cyclist here. So wait, you were running in the bike lane, in the DIRECTION of traffic? That's the problem. You should always run into oncoming traffic, so you can see the maniacal/drunk/raging/distracted person that is about to kill you. Bike lanes are perfect for this. And when you see a cyclist coming, you jump out of the lane and onto the border/sidewalk for the 3-4 strides it takes him to pass.
Seriously. I've run thousands of miles in bike lanes and never gotten anything but "good morning" from a cyclist. Have had lots of cars, and a few city buses, try to kill me, though.
The Googles Do Nothing:
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I agree with this part, and I also make it a habit to stop, unclip, and tap the ground with one foot at every stop sign or light. I want to meet my grandkids and see them graduate high school, and I know that coasting through a stop sign is asking for a killin'.
However, I also habitually count the number of cars that coast through stop signs when there's no oncoming traffic, and it's pretty close to 100%. If someone sees me do that on my bike, they think the appropriate punishment is summary execution by any bystander with a motor vehicle. Yet we all see motorists do it every day, and nobody seems to even notice.
elysive: What if you are morbidly obese at age 20 and then take up a sport at a later age?
You'd still be in worse shape than if you had been in shape at 20. Good on you for getting in better shape, but overall, being old sucks.
verbivore: The Googles Do Nothing:
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I agree with this part, and I also make it a habit to stop, unclip, and tap the ground with one foot at every stop sign or light. I want to meet my grandkids and see them graduate high school, and I know that coasting through a stop sign is asking for a killin'.
However, I also habitually count the number of cars that coast through stop signs when there's no oncoming traffic, and it's pretty close to 100%. If someone sees me do that on my bike, they think the appropriate punishment is summary execution by any bystander with a motor vehicle. Yet we all see motorists do it every day, and nobody seems to even notice.
This has become so commonplace that this happens to me from time to time. I'm approaching a four way stop in my car, and another car is also. I arrive at the intersection before the other vehicle, and come to a full stop. Apparently, a full stop confuses the other motorist, who was expecting me to do a rolling stop. So they think I'm yielding the right of way to them!
born_yesterday: The two lane road leaving my office has a recently added bike lane. For about 4 blocks. Then it abruptly ends at a crowded stoplight, and is followed by painted signs on the road of a bike and arrows, indicating the bike lane continues further ahead (it doesn't).
That sounds like you're describing the road signs that indicate "bikes may use the full lane" like in the below article/photo:
http://burke.patch.com/groups/alan-youngs-blog/p/bp--bicycling-burke -a nd-beyond-new-bicycles-may-use-f36fa442da2
Meesterjojo: Bicycles being moving vehicles, subject to the same laws as motor vehicles in most states in the U.S., and having been hit 9 times in 3 years, 4 in my current neighborhood, once ending up on the hood of a guys car for 2 blocks (and when he figured it out, stopped fast, knocking me off, then told me that he didn't need to give me insurance info because I appeared fine, and the police took 3 hours to get there)- I find this funny. Not just because I've been hit twice by people on bicycles, including once last month as some hipster woman ignored a stop sign and ran over my right foot (witnesses screamed and screamed at her to stop), but because the guy seems like a huge cry baby, douchebag.
If you're on a moving vehicle you should operate your vehicle defensively, not aggressively, which is how this fellow was riding his bike. The assumption that someone *has* to look out for you doesn't even apply to pedestrians (even if it IS the law, reality stands)...
That's what my father always taught me.
The second piece of wisdom he taught me was that the best defense is a good offense.
verbivore: born_yesterday: The two lane road leaving my office has a recently added bike lane. For about 4 blocks. Then it abruptly ends at a crowded stoplight, and is followed by painted signs on the road of a bike and arrows, indicating the bike lane continues further ahead (it doesn't).
That sounds like you're describing the road signs that indicate "bikes may use the full lane" like in the below article/photo:
http://burke.patch.com/groups/alan-youngs-blog/p/bp--bicycling-burke -a nd-beyond-new-bicycles-may-use-f36fa442da2
That's exactly what it is. I had no idea that that is what it meant, and I bet most of the people around here don't know either. I sure as hell wouldn't trust that sign as protection versus the idiots around here.
kokomo61: foo monkey: What do you use for GPS that gives such accurate results?
Garmin Edge 305 - since upgraded to an Edge 500. Sample data image below:
[bombaydigital.com image 850x511]
Like other GPS's, it's only accurate to within a few feet....but it DOES show speed, altitude, cadence, heart rate (and does averages, max, etc.) It showed my entire route up to that point, including all intersections, and how much time I stopped, etc. owed my location at that intersection, stopped for nearly a full minute, as well as the approach, where I coasted, stopped pedaling, slowed and stopped, with my HR falling the whole time. You can get a watch-based GPS for a little over \$100. A bike mounted one for 2X that or so. The only other thing I wish I had on a bike commute would be a GoPro Hero.
I've used a Garmin ForeRunner 305 running watch to show the same type of data to the Austin metro bus company, to show where and when I had to jump off the street for the maniacal bus driver who decided to run me off the road, and where another (on the same route) tried to mow down a cyclist for coasting through a stop. I asked them to correlate it with their logs to verify that bus [number] was at [address] at [time] to verify that I wasn't making it up. I got the standard "your input is important to us, we strive for safety and excellence" form email back, but at the very least they know people have the data to check their homicidal nutjob drivers.
I've never had any issues with bicycle riders. I do always try to keep more than proper distance between myself and motorcycle riders. It really doesn't matter much if they're driving like bats out of hell, or all around assholes. Sometimes a person has to watch out for the more simple minded on the road. It's just the right thing to do. Although my will was tested once when some older guy on a piece of trash motorcycle decided to drive right between myself and another car on a two lane interstate. My passenger ould have reached out and punched them off their trashy motorcycle. Here there are a lot of "Look twice and save a life" signs for motorcycles. Waste of taxpayer money, really, because they shouldn't feel entitled just because they make the mistake of driving an inferior mode of transportation where wrecking almost always means certain death. Not everyone watches out for the simpletons like I do.
whfolsom4: [ ] they don't pay taxes to maintain the highways and they don't belong there.
[snip]
It's actually not a talking point. I drive past bike lanes and bike-related signage every day. I know which one of us paid for it, and it wasn't the guy on the bike. And since a family member almost hit a cyclist riding on a 2-lane highway after dark, without lights or reflectors, yeah, it worries me.
ORLY? I own two houses, two cars, and have a household income solidly in the top 5%. You think I don't pay TAXES??? Because I occasionally ride my bike to work, or ride through the hill country on a beautiful spring day? Because there's some sort of form that cyclists fill out that gives them a giant tax rebate? You really think this?
Repo Man: change1211: kokomo61: bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
[www.thecycler.net image 400x400]
Bikes aren't pedestrians and cars aren't required to yield at pedestrian crosswalks unless you get off your bike and walk.
That is true, and one more reason that bicycles should not be on the sidewalk, or in the crosswalk. Those are for pedestrians.
....unless you're talking about a MUP (Multi-Use Path) that's built for bikes, peds and other non-motorized use. The street level crossings are called crosswalks no matter who uses them. In Virginia, (where I live and was hit in a crosswalk), cars are required to yield to 'pedestrians' in a crosswalk. If a bike is using a crosswalk, they're treated as pedestrians, even when they don't dismount.
Prudence and Darwin dictate that you never just go into a crosswalk without looking both ways....which I did...and made sure that traffic was STOPPED in both directions before I moved ahead. I was stopped for 45 seconds waiting for traffic to stop and/or clear before entering the crosswalk.
I was riding on a bike path, not a sidewalk. The yahoo who hit me was either texting or just impatient, but instead of stopping, he decided to go into the parking spaces to get around the 2-3 stopped cars in front of him....and if he wasn't driving a Honda Civic (thank God for sloping hoods), I probably wouldn't be around to type this.
kokomo61: Repo Man: change1211: kokomo61: bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
[www.thecycler.net image 400x400]
Bikes aren't pedestrians and cars aren't required to yield at pedestrian crosswalks unless you get off your bike and walk.
That is true, and one more reason that bicycles should not be on the sidewalk, or in the crosswalk. Those are for pedestrians.
....unless you're talking about a MUP (Multi-Use Path) that's built for bikes, peds and other non-motorized use. The street level crossings are called crosswalks no matter who uses them. In Virginia, (where I live and was hit in a crosswalk), cars are required to yield to 'pedestrians' in a crosswalk. If a bike is using a crosswalk, they're treated as pedestrians, even when they don't dismount.
Prudence and Darwin dictate that you never just go into a crosswalk without looking both ways....which I did...and made sure that traffic was STOPPED in both directions ...
And yet another reason to avoid MUPs. Leave them for small children, pedestrians, runners, skaters.
Ker_Thwap: I'd love to hear the other side of the story.
I like to imagine it went like this...
I was running late for work, going 5 mph over the speed limit. As I go around a blind corner these three idiots dressed in spandex were peddling 3 wide across the entire lane. I swerved, honked and had to swerve back for oncoming traffic. As one of them flipped me the bird, he lost control and wiped out. It was hilarious.
CSB, I would've laughed my ass off too. YDI, stupid bikers!
If the guys FTA were doing that, I would immediately stop feeling bad for the guy. There is ZERO excuse to be walking, biking, or doing anything else on the road not in single file.
BarkingUnicorn: ggecko: [www.allfunnies.com image 400x400]
It should be illegal to make certain types of clothing in certain sizes.
All companies should be like Speedo and TYR are with racing swimsuits. The largest they come in (last I checked on Amazon) was a size 40, which is still pretty small.
Repo Man: And yet another reason to avoid MUPs. Leave them for small children, pedestrians, runners, skaters.
...and cars, too, evidently.
http://articles.washingtonpost.com/2013-06-02/local/39698135_1_trail -u sers-police-car-vienna
I'm 39. I'm much stronger than I was at 20 in terms of my weight lifting ability. I can't run short distances or sprints as fast as I could when I was on the high school track team, but I can run long races (10K+) whereas back then I'd be winded after a mile. You can be better at things physically at 40 than at 20. Not everything, but many things.
The Googles Do Nothing: I'm 39. I'm much stronger than I was at 20 in terms of my weight lifting ability. I can't run short distances or sprints as fast as I could when I was on the high school track team, but I can run long races (10K+) whereas back then I'd be winded after a mile. You can be better at things physically at 40 than at 20. Not everything, but many things.
An individual might be in better shape at 40 than 20 but there is a reason you don't see people in their 40s setting world records. After your mid 30s your genetic potential starts to drop off. If you never reached your potential in your late 20s and early 30s your times might still go down but you'll never be as good as you could have been when you were younger.
Quantum Apostrophe: elysive: What if you are morbidly obese at age 20 and then take up a sport at a later age?
You'd still be in worse shape than if you had been in shape at 20. Good on you for getting in better shape, but overall, being old sucks.
I wasn't morbidly obese, but I was totally out of shape and what I said still stands. There's no rule that young people are always in good shape. Being in shape is something that one has to work at and it has no age limits. Older people tolerate pain due to endurance exercise better and often develop skills that make them perform better than their younger counterparts.
There are some performance variables that definitely decline with age (reaction time etc), but that's not what you said. You just should have been more specific in your verbiage.
yay, my bike story: Riding down a main drag in a college town, going the speed limit(downhill), a car in the opposite lane thinks he can make it in front of me into a side street. The one way side street that another car is coming out of, at the same time, so the illegally left turning car has to stop. I hit the brakes a little as I see the turn start, then throw them on all the way as he stops in the street. My front wheel hits the front tire and I lose contact with my bike and get a bird's eye view of a happy family as I pass over the hood of the car. I remember the little kid start to wave as I go sailing past.
The impact with the car didn't hurt. The impact with the pavement after my little flying lesson was another story. I'm on my back, regaining consciousness as I see the car pull back into now-stopped traffic and pull away. Classic hit and run. Permanently distended collarbone on the right side, which makes rehab fun and continued exercising ... interesting because what my right arm used to be able to do, I no longer can do. I used to be able to throw a football on target the length of a football field. Playing catch with the nieces and nephews is permanently out. I will never be able to play catch with any children of my own. The morphine drip was fun, though.
Denver bicyclists, for the most part are insane to ride downtown. I have a friend who rides to work, and he's been run off the road more times than I care to recall. On the flipside, I've seen more red lights run by cyclists than I care to count. If you choose not to heed the rules of the road, you deserve what happens, bike or car. They are the rules of the road, and you are in/on the road. T-boned because you didn't see the red light? Get off the phone and look where you are going. Run over and road rash covered because you didn't think a red light applied to you? Wake up to reality. Lost the settlement because you were in an argument with the wife while driving and didn't see the bicyclist? Pay up, jerk off.
/They never found the guy.
//also marks the last time I was on a bike
///nine years and counting
The Googles Do Nothing: /When I ride my bike I stop at all the stop signs and lights and stay to the far right.
Don't go so far to the right that you risk running off the road, into drain grates, or get forced off the road by people passing too close. There's a such thing as riding too far right, and I get worried about folks who do that when I'm driving. Particularly if they're riding too far over on a narrow roadway, thus encouraging motorists to cross the centerline into oncoming traffic.
TheGregiss: Sounds like the dude on the bike was following too close. He should be issued a ticket in addition to his injuries.
Sounds like he got cut off by the driver and the driver stopped hard intentionally.
If youre on a bike on the road, do the 45/50 mph speed limit or get the fark out of the way, and don't get mad and tailgate on a hill when you get passed. Youll be made to look like the douche ftfa.
Sounds like you shouldn't be driving. The posted speed is a maximum, not a minimum. If you want minimum speed limits, only drive on roads with posted minimum speed limits. If you don't want to deal with pedestrians, cyclists, farm implements and equestrians, stick to roads that ban those modes. Otherwise, that's life, driver, DMV told you there'd be slow moving traffic when you got your license. Deal with it or don't drive. I don't need you making my job needlessly more dangerous with your anger issues and lack of time management skills.
tccomp: Too bad we don't have the other side of the story. Could have stopped for a dog in the road and they were just following too close/tailgating.
Drivers don't flee after a collision unless they know they were in the wrong and think they'll get away with it.
kokomo61: bearcats1983: The Googles Do Nothing: It is for this reason that I don't ever want to do a triathlon. Too much time on the road training with the bike and running into drivers like this. Drivers usually don't road rage at runners, but when they see a cyclist it seems to trigger something in them.
/When I ride my bike I stop at all the stop signs and lights and stay to the far right.
//Cyclists that don't give every one else a bad name and cause this kind of problem
I always enjoy the idiot drivers who are oblivious to anything that isn't a car. I can't count the number of times I've come inches from getting run down. Even when the runner/cyclist isn't being a dick, some drivers still just can't seem to understand the concept of sharing the road.
On my commuter bike, I have a route that crosses several intersections...that even with marked crosswalks, drivers will plow right through them to make a RTOR without stopping (or even slowing down). If I'm crossing with the green, and a car is going into the crosswalk...sometimes they'll get a dose of this - 115 dB of USCG-rated air horn goodness.
[www.thecycler.net image 400x400]
Those things aren't nearly loud enough for some cities, like Portland, Oregon. Hell, the MAX horns, audible from two miles away when used on "high" mode, are barely loud enough in that city.
BunkyBrewman: This is the primary reason I rarely ride the bicycle on highways or city streets. Just lucky the multi-use trails systems around here are extensive.
Depends on the region. Portland's MUPs suck because they're bad at maintaining them, the centerlines, lane lines and shoulder lines are faded or missing, and there's no obvious hints for pedestrians on where to go (so they often walk edge to edge, blocking both directions). Tulsa is lightyears ahead of Portland on the cycleways, except navigational signage (which, if it's existant at all, is just the old school "BIKE ROUTE" signs that expect you to be a psychic and know the route's destination and midpoints): Lane markings, roadway surface, and signage is well maintained, pedestrians have an idea which side to walk on and are good about sticking to that side so other traffic can pass easily (if not given their own sidewalk), and it's all around an easier system to ride on. Wish we had the first/last mile situation figured out, getting from anywhere and to anywhere from the cycleway network requires riding on some section line roads, which isn't for the faint hearted. Reminds me of Portland 20 years ago before it had bike lanes to speak of, except with an actually decent cycleway network.
inglixthemad: I watch a group of cyclists go down my road every few days and the rules don't apply to them. Stop signs? Perish the thought!
What state? Idaho and most of the midwest allow bicycles (and some cases like Oklahoma and Missouri, motorcycles) to treat stop signs the same as yield signs.
lack of warmth: Carth: Did you check your local bylaws to see if walkers and runners were allowed to use the bike lane? I know in our city it is pretty clear that cars and pedestrians aren't allowed to travel in bike lanes but are allow in 'multi use paths'
So you are saying double standards are stated in the law? Because it does sound like a really not well thought out idea to make the runners run in traffic, but the cyclists can have their own private real estate.
Pretty common, though. Oregon does this, though on the otherhand, Oregon does not have jaywalking laws, so crosswalks and sidewalks are merely suggestions, and not one that's ever taken by any jogger or runner I've seen in Oregon.
runescorpio: I am a commercial driver that has a vehicle that is literally the legal maximum width.
Goes to show there's a huge difference between merely being a commercial driver, and being a professional driver. I fall into both categories by driving a bit more defensively and taking more time to deal with disorderly operators, both motorized or not, and I don't let it stress me out. This shiat happens on the road. If you can't handle it, you're part of the problem.
Zeno-25: Cyclists... the scourge of anyone with no time management skills trying to get anywhere on time using the roads.
Fixed that for you. Traffic happens. They told you this at the DMV. Always good to leave extra time. If it wasn't a cyclist, it'd be something else, like some impatient jerk going too fast in the rain and causing a wreck on the expressway. Or someone trying to beat a light, failing at it, and t-boning cross traffic. It's not rocket surgery. Any five year old with a watch can figure it out.
born_yesterday: Then it abruptly ends at a crowded stoplight, and is followed by painted signs on the road of a bike and arrows, indicating the bike lane continues further ahead (it doesn't).
Want to know how I know you haven't been keeping current on the latest road laws? That symbol indicates that it's a designated bike lane open to motorists, not that there's a bike lane ahead. If it's positioned correctly, it also indicates where in the lane cyclists should be riding. If there's no or inadequate shoulder, that'd be the center of the lane or close to it.
You might want to crack open that driver's manual for a refresher course every year or two. You're expected to do that anyway even if they don't test you on it.
Publikwerks: Bikes are inherently unsafe when on the road with cars.
More like motor vehicles are inherently unsafe when operated by impatient, incompetent or incapacitated people. It's a logical fallacy to try to make excuses for any of those three things. If it wasn't a bike, it'd be a pedestrian, a rock, an animal, a disabled motor vehicle...these things happen, and the most important safety feature is the one that is theoretically between the seat and the steering wheel.
balloot: //If you live in SF and you ever see Critical Mass, you will lose all ability to ever feel bad for a cyclist who gets nailed by a car
If you're in SF and you're driving in the first place, I don't really have much sympathy for you in the first place, since you're intentionally making your own life miserable by doing so. Just admit you're a masochist and you get off on pain. You'd fit right in down on Castro Street.
Shang-High: I just "buzzed" a bicyclist riding half of the way into the road, when there was a good 6 feet to his right for biking (and bike lane markings). Sorry, I'm not going to cross the double yellow lines into oncoming traffic, "share the road" works both ways.
If the bike lane is covered in debris, it's not exactly a viable option. Being a douchebag won't change the condition roadway. Maybe you should call the city and ask for a sweeper to service the road. Unless you're in Oregon, where such technology doesn't exist yet, in which case, sucks to be you.
phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
Nope, car yields to through traffic, then shoots the asshole traffic planner who doesn't know what a right turn pocket is. Turning traffic always yields to any intermediate lanes, left or right.
blacksho89: Otherwise, as a cyclist is technically a pedestrian, he has the right of way if he is at the crosswalk..
Cyclist is technically a vehicle when operating above walking speeds or when on the roadway, as a pedestrian at walking speeds on a sidewalk. Yes, this means you're guilty of "driving on the sidewalk" if you're riding on the sidewalk faster than walking speed.
ZAZ: Starting from a red light or in moving traffic? In moving traffic it should be a "yield" type situation where you aren't supposed to disrupt the movement of bicycle traffic to make your turn. Same as changing lanes in front of a car.
Half right, this applies at a red light situation, too, though. Bad idea to proceed off a red light across another lane without at least making eye contact with other drivers, even if they're driving a bicycle.
I recently heard about ongoing experiments with "bike boxes" to try to improve safety at red lights. If you tell me where I misplaced my notes I'll transcribe them for you. The idea is you put bikes ahead of cars at the red light. Now Ms. Right Turner On Her Cell Phone is going to have to look at the pack of bicycles which are in her path. Maybe she will notice them. They didn't work as well as hoped.
That's not to say they're a total failure, they've made some significant improvements in terms of incidents and reducing traffic conflict where they've been placed. Still funny when I see someone get popped for making a right turn on red across one, ignoring the driver's manual and typically at least three signs saying "NO RIGHT TURN ON RED EXCEPT BICYCLES" signs. Or stopping in the bike box, ignoring a big green area with a bike logo, the stop bar with WAIT HERE painted in advance of the line, and a "STOP HERE ON RED EXCEPT BICYCLES" sign.
squegeebooo: phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
The car should move over into the bike lane and treat it like a right turn lane.
Only if it says that on the sign, ie " RIGHT LANE - BICYCLES AND RIGHT TURNS ONLY". Otherwise, you stay out of the lane and turn across it when clear.
Ker_Thwap: Really Twowheeling Tim? You're going to dump on the guy with the CDL who's driven responsibly for 9 years to earn a living delivering products that you have in your house, all the while taking care to keep inconsiderate cyclists alive?
He's not exactly making my job easier. I'm a professional driver, and it's hotheads like him that gives the rest of us a bad name.
jurzdevil: If I am going straight and car is going to turn right I will slow down and let the car pass, even stop before the intersection.
Don't do this. It's confusing and unpredictable. Signal a lane change and fall in behind the turning vehicle, then move back over after the intersection if you're that concerned.
WRXminion: This means impeding the flow of traffic is a ticketable offence, but I've never seen a ticket given to a cyclist for this.
Probably because the laws regarding impeding state that you're not impeding if you're going as fast as you can, and you're using slow vehicle turnouts to let people by. If they're doing the first, and there's no place to legally and safely do the second, then they're not legally impeding traffic.
Fallout Zone: squegeebooo: phaseolus: Traffic law question:
Intersection, car in right lane wishing to turn right. Bicycle to his right in bike lane, going straight.
Bike yields to car, correct?
The car should move over into the bike lane and treat it like a right turn lane.
I encounter this every day on the way to work, enough so that I actually took the time to look up the laws. This is correct.
Not sure you got that right. It's not the case in Oregon, Washington or Oklahoma, and such a move would be contraindicated by the US manual on uniform traffic control (the sign would have to say that's an allowed movement unless your traffic engineers just stopped giving a shiat about doing their job right).
Publikwerks: But if the bike is overtaking in the bike lane, I think the bike should yield. it may not be the law, but it's the smart thing - Bikes are hard enough to see, and if you are overtaking in his blind spot, you're asking for trouble assuming he's going to spot you.
Use your mirrors, signal, don't assume the cyclist is going to yield when you're the one entering his lane, not the other way around. Smart money is letting the cyclist in the bike lane by regardless of whether you just passed them or they're about to pass you. Shoulder check that shiat, man!
Mitrovarr: In my experience, 2/3 of bike lanes are unusable as places to ride due to being too narrow, too close to parked cars, too full of gravel and trash, too slanted, too uneven, or too torn up. They make you harder to see as well. I kind of wish they wouldn't make them; just make the road a bit wider with a good shoulder and have cyclists ride in the lane. It's actually safer to ride in the lane because drivers are more likely to see you there.
The bike lane is a lane, and the difference in visibility between riding in the same lane and the next lane over is negligible compared to the severe disadvantage riding on the sidewalk puts you. That said, streets with bike lanes should at least have enough shoulder to put the curb and gutter pan, some cities really have idiots planning these things. I don't give Oregon much credit, but they get props for this: Oregon holds the legal minimum width for a bike lane adjacent to motorized traffic in the same direction or any kind of traffic in the opposite direction at six feet, as measured from the edge of pavement, shoulder line or the outside edge of the gutter pan, whichever is closest to the center of the roadway, but recommends more (though sets a maximum at 9 feet, since 10 feet would be enough for two bike lanes, a 6 foot inner lane and a 4 foot outer lane).
mbillips: What part of "multi-use path" do you not understand? It's not a bike path, it's a walking/cycling/running/skating trail, and labeled as such.
There are some cycleways that are not MUPs, even in the US.
mllawso: The Googles Do Nothing: /When I ride my bike I stop at all the stop signs and lights and stay to the far right.
Lane position is something that comes up a lot when I'm teaching a Basic Rider's Course (for motorcycles). One concern I have with a far-right lane position is this: You get the moron that wants to pass you and either:
1. Doesn't get over far enough and clips you, or
2. Sees and oncoming car and figures he can squeeze by you without getting over and clips you.
I usually ride in the left or middle of the lane: that way if someone wants to pass: it discourages the second type, and gives you space to get over for the first type.
That's also why pretty much every state has language stating cyclists are to either ride in the center of the right lane or as far right as practical/practicible.
Eponymous: Quantum Apostrophe: But can we all agree that electric scooters don't belong on the bike path?
There's a reason I avoid bike paths in downtown Montreal.
No we can't agree....Let the scooters ride 4 or 5 abreast at 5mph and see how many of the bikers get a taste of road rage
Motor vehicle. Not allowed where signs says "No Motorized Vehicles."
whfolsom4: For what it's worth I don't harass cyclists, and don't pass them illegally no matter how slow they are, but they don't pay taxes to maintain the highways and they don't belong there.
Yes, actually, they do pay taxes to maintain the highways. Highways are maintained through income tax, or in Oklahoma, the sales tax. Gas taxes don't even pay the interest on bonds made to maintain existing motorist infrastructure, and registration only covers the cost of maintaining the registration database, if it even fully covers that. Motorists pay usage taxes on top of the taxes everyone pays into the highway fund whether they drive or not because motorists are actively destroying the highway just by using them with a heavy vehicle. Even as a professional driver, I wish that the fuel taxes at least covered the additional damage motorists cause to the highways compared to non motorized modes or mass transit; not exactly fair to expect the public to subsidize my privately owned machine like that.
whfolsom4: He might be an upstanding citizen, but many (other) cyclists don't wear safety equipment, follow traffic rules or maintain their bicycles (or even sobriety!). Get off your high horse, and off the road.
I've lived and travelled to a lot of places. I can say with confidence that the difference in the rate of compliance between motorists and cyclists anywhere is exactly zero. Which should scare you if you see a lot of cyclists doing this, because there's going to be the same ratio of motorists doing the same shiat.
born_yesterday: That's exactly what it is. I had no idea that that is what it meant, and I bet most of the people around here don't know either. I sure as hell wouldn't trust that sign as protection versus the idiots around here
It's shiat like this that makes me wish we had mandatory driver testing every other year in this country. The sad thing is driving takes more responsibility than food safety and we expect the guy running the fries at McDo's to re-test to keep a food handler's card every other year...
Vector R: The largest they come in (last I checked on Amazon) was a size 40, which is still pretty small.
Please tell me that's not waist size...I weigh 200 pounds and 40 is still about a foot more than I need...
Meesterjojo: Bicycles being moving vehicles, subject to the same laws as motor vehicles in most states in the U.S., and having been hit 9 times in 3 years, 4 in my current neighborhood, once ending up on the hood of a guys car for 2 blocks (and when he figured it out, stopped fast, knocking me off, then told me that he didn't need to give me insurance info because I appeared fine, and the police took 3 hours to get there)- I find this funny. Not just because I've been hit twice by people on bicycles, including once last month as some hipster woman ignored a stop sign and ran over my right foot (witnesses screamed and screamed at her to stop), but because the guy seems like a huge cry baby, douchebag.
Meesterjojo, if you have been hit that many times, you sir are a farking moron who needs to kill himself, or jump out in traffic and try again.
Im glad dude threw you off his hood after two blocks, jackass.
Baloo Uriza: born_yesterday: Then it abruptly ends at a crowded stoplight, and is followed by painted signs on the road of a bike and arrows, indicating the bike lane continues further ahead (it doesn't).
Want to know how I know you haven't been keeping current on the latest road laws? That symbol indicates that it's a designated bike lane open to motorists, not that there's a bike lane ahead. If it's positioned correctly, it also indicates where in the lane cyclists should be riding. If there's no or inadequate shoulder, that'd be the center of the lane or close to it.
You might want to crack open that driver's manual for a refresher course every year or two. You're expected to do that anyway even if they don't test you on it.
Thanks. Someone already explained that to me, only they didn't feel the need to be an asshole about it.
Baloo Uriza: born_yesterday: That's exactly what it is. I had no idea that that is what it meant, and I bet most of the people around here don't know either. I sure as hell wouldn't trust that sign as protection versus the idiots around here
It's shiat like this that makes me wish we had mandatory driver testing every other year in this country. The sad thing is driving takes more responsibility than food safety and we expect the guy running the fries at McDo's to re-test to keep a food handler's card every other year...
Over ten posts in a row. Nevermind; you're exactly what FARK is looking for.
mark12A: As a runner, cyclist and driver, I can't stand people like you. It's good to be wary of cars and defer to them whenever possible, but dont break the law and be a dick to everyone else.
13 years. Oh, and by the way, 13 YEARS. 13 years I've been doing this, and if I was a dick I would have gotten my ass kicked loooong time ago by the good, but rough hewn, citizens of Philly. If what I was doing was dangerous, I would have been tagged years ago.
Is simples: Stay away from pedestrians when they're on sidewalks. And riding against traffic takes up NOT ONE SINGLE SQUARE INCH more space than riding with traffic. As for the oncoming motorists having less reaction time, that's OK, because now I HAVE REACTION TIME, unlike when I'm not facing traffic. So now I'm not completely vulnerable to an endless stream of total strangers coming up behind me.
You sound like the sort of person that used to argue that you're safer not wearing a seatbelt because it's better to be thrown clear of the car.
I recently read that a cyclist riding against traffic is ~3x more likely to be in an accident. So on top of being illegal and a real dick move, it's also considerably more dangerous.
/Took up cycling recently.
//Steadily increasing my miles/week.
///Understand things from the cyclists' perspective much better than I did before.
Repo Man: And yet another reason to avoid MUPs. Leave them for small children, pedestrians, runners, skaters.
Amen. As I said in a previous post, I recently started cycling and thought the extensive trail networks around here would be an awesome way to spend some time outdoors and improve my fitness. But a few weeks into riding, I figured out that those paths are for biking, but cyclists belong on the road. The cycling club I ride with avoids the paths.
born_yesterday: Baloo Uriza: born_yesterday: That's exactly what it is. I had no idea that that is what it meant, and I bet most of the people around here don't know either. I sure as hell wouldn't trust that sign as protection versus the idiots around here
It's shiat like this that makes me wish we had mandatory driver testing every other year in this country. The sad thing is driving takes more responsibility than food safety and we expect the guy running the fries at McDo's to re-test to keep a food handler's card every other year...
Over ten posts in a row. Nevermind; you're exactly what FARK is looking for.
Not my fault nobody else was posting at that time.
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## Object inheritance
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
ivan
Party member
Posts: 1454
Joined: Fri Mar 07, 2008 1:39 pm
Contact:
### Re: Object inheritance
I'll put it to you this way, if your examples managed to confuse both myself and pgimeno - it doesn't speak good for the code.
Basically you're doing a simple __index fallback to another table.
It won't work for more than one level of inheritance, so you can't make a subclass of "subclass".
4vZEROv
Prole
Posts: 36
Joined: Wed Jan 02, 2019 8:44 pm
### Re: Object inheritance
What I use :
class.lua
Code: Select all
local Class = {}
function Class:new() end
function Class:extend() local obj = {} obj.__call, obj.__index, obj.super = self.__call, obj, self return setmetatable(obj, self) end
function Class:__index(v) return Class[v] end
function Class:__call(...) local obj = setmetatable({}, self) obj:new(...) return obj end
return Class
vehicle.lua
Code: Select all
Vehicle = Class:extend()
function Vehicle:new(nb_of_wheels, name)
self.nb_of_wheels = nb_of_wheels
self.name = name
end
function Vehicle:update(dt) print(self.name, self.nb_of_wheels) end
function Vehicle:draw() love.graphics.rectangle("line", 10, 10, 10, 10) end
car.lua
Code: Select all
Car= Vehicle:extend()
function Car:new(name) self.super.new(self, 4, name) end
function Car:update(dt) self.super.update(self, dt) end
function Car:draw() self.super.draw() end
main.lua
Code: Select all
function love.load()
require("class")
require("vehicle")
require("car")
my_car = Car("Ferrari")
end
function love.update(dt) my_car:update(dt) end
function love.draw() my_car:draw() end
pgimeno
Party member
Posts: 1690
Joined: Sun Oct 18, 2015 2:58 pm
### Re: Object inheritance
To clarify, I was confused by Schwender.exe's code, not by TheHUG's code, especially regarding initialization of members. It made me think that Player was an instance, not a class.
TheHUG
Citizen
Posts: 51
Joined: Sun Apr 01, 2018 4:21 pm
### Re: Object inheritance
ivan wrote:
Mon Jul 08, 2019 1:44 pm
I'll put it to you this way, if your examples managed to confuse both myself and pgimeno - it doesn't speak good for the code.
Basically you're doing a simple __index fallback to another table.
It won't work for more than one level of inheritance, so you can't make a subclass of "subclass".
Of course you can, anything created with subobject:new will have subobject as it's metatable,and subobject __index attribute will be itself. You can then use that object as a subclass, defining new methods for it, which anything made with subsubobject:new will inherit.
It was the first result back when I googled 'classes in lua' :https://www.lua.org/pil/16.1.html . it's seems to be a fairly standard way to do it, and I still prefer it since it's simple and explicit.
ivan
Party member
Posts: 1454
Joined: Fri Mar 07, 2008 1:39 pm
Contact:
### Re: Object inheritance
Oops sorry pgimeno was talking about somebody else. And no Lua doesn't have classes or instances.
I don't follow your explanation so you have to show me a code example with more than one layer of inheritance.
zorg
Party member
Posts: 2620
Joined: Thu Dec 13, 2012 2:55 pm
Location: Absurdistan, Hungary
Contact:
### Re: Object inheritance
Does this qualify for multiple inheritance? I typed it up and tested it in 5 mins:
Code: Select all
-- "class" A
local A = {} -- "methods/members" for "class instances"
A.bar = function() print "A!" end -- example of above
local mtA = {__index = A} -- one metatable to make instances access these methods
local newA = function() return setmetatable({}, mtA) end -- set instance to access class methods/members if it itself doesn't contain such a key.
-- "class" B
local B = {}
B.baz = function() print "B!" end
local mtB = {__index = B}
setmetatable(B, mtA) -- set class B's method/member table to have class A's method/member table as a fallback.
local newB = function() return setmetatable({}, mtB) end
local foo
foo = newA()
foo:bar()
foo = newB()
foo:baz()
foo:bar() -- method "bar" in class "A" called through fallback index metamethod defined on class "B"'s own methodlist
-- This can be extended to more than one "round", e.g. an instance of class C falling back to class B falling back to class A's
-- class methods or members.
Me and my stuff True Neutral Aspirant. Why, yes, i do indeed enjoy sarcastically correcting others when they make the most blatant of spelling mistakes. No bullying or trolling the innocent tho.
ivan
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### Re: Object inheritance
Looks fine and yes this would work thanks to the following line:
Code: Select all
setmetatable(B, mtA) -- set class B's method/member table to have class A's method/member table as a fallback.
So you can have as many layers of inheritance as you want as long as the metatables fallback sequentially to the parent object.
I would call this "layers of inheritance" since "multiple inheritance" usually refers to something different.
zorg
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### Re: Object inheritance
Yeah, the diamond of death thing; i guess even that could be possible, depending on whether you can set __index to be a function and check multiple "parent" tables defined by the class itself.. but then you'd probably also want a way to programatically set which other "classes" one should search for potentially missing members/methods, the order of the definitions mattering and such. (or you could just hard-code it)
Me and my stuff True Neutral Aspirant. Why, yes, i do indeed enjoy sarcastically correcting others when they make the most blatant of spelling mistakes. No bullying or trolling the innocent tho.
ivan
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Posts: 1454
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### Re: Object inheritance
Never had any use for multiple inheritance myself. But simple inheritance like in your code example can be very useful.
TheHUG
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Posts: 51
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### Re: Object inheritance
An example of more levels of inheritance would be:
Code: Select all
Object = {}
function Object:new(x, y)
local o = {x=x, y=y}
setmetatable(o, self)
self.__index = self
return o
end
Collider = Object:new() -- Collider inherits Object's :new method
function Collider:collide(...)
...
end
a_collider = Collider:new() -- an instance of Collider
BouncingCollider = Collider:new(new_default_an_arg) -- creates BouncingCollider and sets its metatable to Collider, inheriting :new and :collide
--override :collide
function BouncingCollider:collide(...)
--bounce and then
getmetatable(self).collide(self, ...)
end
a_bouncing_collider = BouncingCollider:new() -- works
internally when BouncingCollider:new() is called it finds that BouncingCollider.new == nil, so it looks in getmetatable(BouncingCollider).__index (i.e. Collider) for .new. Since that is also nil, it will check getmetatable(Collider).__index (i.e. Object) and find the .new function there.
There are several variations you can do, including some where the boundary between instance and class is clearer and/or enforceable, though I don't really see the need. That's one advantage of being explicit - it preserves customizability so you can code with your preference.
One variation I used for a while was to leave out self.__index = self in :new and specifically set the metatable and __index for each subclass, such that :new would only work when invoked on the class, not on an instance, but I ended up liking this better, they're all just tables after all.
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Image text transcribed for accessibility: A university wants to establish a control system model that represents the student population as an output, with the desired student population as an input. The administration determines the rate of admissions by comparing the current and desired student populations. The admissions office then uses this rate to admit students. Draw a functional block diagram showing the administration and the admissions office as blocks of the system. Also show the following signals; the desired student population, the actual student population, the desired student rate as determined by the administration, the actual student rate as generated by the admissions office, the dropout rate, and the net rate of influx.
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Article | Open
Human protein secretory pathway genes are expressed in a tissue-specific pattern to match processing demands of the secretome
• npj Systems Biology and Applications 3, Article number: 22 (2017)
• doi:10.1038/s41540-017-0021-4
Revised:
Accepted:
Published online:
Abstract
Protein secretory pathway in eukaryal cells is responsible for delivering functional secretory proteins. The dysfunction of this pathway causes a range of important human diseases from congenital disorders to cancer. Despite the piled-up knowledge on the molecular biology and biochemistry level, the tissue-specific expression of the secretory pathway genes has not been analyzed on the transcriptome level. Based on the recent RNA-sequencing studies, the largest fraction of tissue-specific transcriptome encodes for the secretome (secretory proteins). Here, the question arises that if the expression levels of the secretory pathway genes have a tissue-specific tuning. In this study, we tackled this question by performing a meta-analysis of the recently published transcriptome data on human tissues. As a result, we detected 68 as called “extreme genes” which show an unusual expression pattern in specific gene families of the secretory pathway. We also inspected the potential functional link between detected extreme genes and the corresponding tissues enriched secretome. As a result, the detected extreme genes showed correlation with the enrichment of the nature and number of specific post-translational modifications in each tissue’s secretome. Our findings conciliate both the housekeeping and tissue-specific nature of the protein secretory pathway, which we attribute to a fine-tuned regulation of defined gene families to support the diversity of secreted proteins and their modifications.
Introduction
In eukarya, the protein secretory pathway is an essential, efficient, and accurate molecular machinery for preparing and exporting proteins to expose the extracellular environment. This machinery includes various functional modules which are compartmentalized along the endoplasmic reticulum (ER) and Golgi apparatus. These modules are responsible for folding, processing of the post-translational modifications (PTMs), and trafficking of the proteins routed to the membrane of extracellular space.1, 2 In human, a functioning secretory pathway is essential for the body physiology. The majority of the hormones, peptidases, receptors/channels, extracellular matrix components, coagulation factors, transporters are all clients of this machinery.2 Unsurprisingly, dysfunction of the secretory pathway is the cause of a variety of systemic or developmental diseases, like cancer, diabetes, Parkinson’s disease, and congenital neurodegenerative disorders.3–7 The molecular biology and biochemistry of this pivotal pathway are well-studied for its core components.8,9,10 However, the knowledge how these components are expressed across tissues is lacking. Although, primary transcription is a key player in defining which genes has specific expression in certain tissue(s), yet, until recent advances in sequencing technologies it was not possible to measure the precise quantity of the RNA expression level in the genome scale.11 The recent studies based on RNA-sequencing (RNA-seq) have shown that human tissues exhibit unique transcriptional signatures that show stability even in postmortem sampls.12 The Genotype-Tissue Expression Project (GTEx),13 and the Human Protein Atlas (HPA)5 has been recently published as two independent and comprehensive RNA-seq data sources on 30 human tissues. In HPA study, one of the major conclusion of the paper was that the largest fraction of the tissue enriched transcriptome codes for the secretory proteins (secretome). The secretory pathway has evolved to process specific PTMs encoded in secretory protein. Among the PTMs, glycosylation, sulfation and adding GPI-anchored (glycosylphosphatidylinositol) are the major modifications. Each secretory protein has its composition regarding the PTMs type and number of the sites. Therefore, tissue-specific secretome implies in each tissue a different set of proteins with specific PTMs form enter to the secretory pathway. This further means, in each tissue, functional modules which are responsible for the processing of the PTMs types are faced with the particular load of the sites to processes which is different from other tissue. Borrowed from manufacturing world, if there is an input pressure on a particular operating module in a production pipeline, to release the pressure more processing units needs to be used in that specific modules. In the context of the secretory pathway, the response to the tissue-specific pressure on processing specific PTMs can be a fine tuning of the components expression in a particular functional module. In this study, we performed a meta-analysis approach utilizing the transcriptome data to detect such adjustment (Fig .1a). Also, we also examined whether the genes coding secretory pathway components indicate a change in their expression level in connection with the explained processing pressure. The results of this study advance the fundamental understanding of the tissue-specific function of the secretion pathway in human tissues. The findings can also possibly aid surpassing a long time standing challenges in biopharmaceutical protein production, since the current bottleneck in the production of human proteins is the functional difference between the host (e.g., CHO cells) and parent secretion system.14, 15
Overall assessment of the expression level of secretory pathway genes in different tissues
Secretory pathway processes proteins in a stepwise manner. These steps include several functional modules such as translocation, folding and glycosylation. Each module involves a set of cooperative proteins, in most cases, encoded by the genes which belong to the same gene family. We previously have defined 169 components (proteins) of the secretory pathway in yeast, and we mapped them into the subsystems representing distinct functional modules16. Using a similar approach, we allocated our defined 575 core components of the human general secretory pathway into 13 subsystems (Fig. 1b, Table EV1). As earlier mentioned, the major focus of this study was to dissect the extent to which the expression levels of genes encoding the components of the secretory pathway are tuned over different tissues. Therefore, using available transcriptome data, we investigated the tissue-wise variations in the mRNA levels of these 575 genes. We used GTEx13 and HPA5 as the two independent and comprehensive RNA-seq datasets on 30 intersected human tissues. Both datasets have provided an unprecedented resolution on RNA levels in the tissues, and the correlation between their measurements has shown to be significant.12 We chose to analyze the GTEx data as the primary dataset because he it benefits from a careful experimental design with more tissues samples. The expression levels are normalized for different confounding parameters and variation sources such as individual, sex, and age. Although in the paper the authors have reported some variation depending on the individual, sex or age, however, we did not found for any of our analyzed gene lists from secretory pathway significant dependency on the individual, sex or age expression variation (Fig. S3). We used HPA data as a control.
Before analyzing the variations, we performed a descriptive analysis of the expression levels of the genes both in the secretory pathway and secretome based on the HPA gene expression categories. Therefore, we assigned the genes to the groups such as expressed in all, tissue-elevated and tissue-enriched.5, 12 Of all 575 secretory pathway components, ~75% (n = 435) belonged to the expressed in all category, while ~25% (n = 140) were in tissue-specific categories (such as tissue-elevated and tissue-enriched) (see the methods for the definitions, Table EV1). The distributions of the 435 genes expression (expressed in all) are similar in different tissues (in log10 FPKM) based on both GTEx and HPA data (median approximately equal to 10 FPKM (Fragments Per Kilobase Million)). The pancreas, skeletal muscle, heart, and liver slightly lower median expression (Fig. S1).
On the other hand, 10–20% of the transcriptome in human tissues (70% in the pancreas and salivary glands) translates into the secreted or the cell-membrane proteins. It has been shown that secretome holds the largest fraction of the tissue-specific proteome(Fig. 2a).5 In human proteome, 3328 proteins were predicted having an N-terminal signal peptide that dictates their entrance into the secretory pathway.5 From this group, 1218 are secreted proteins, and 1607 are cell membrane proteins.5 Contrary to the secretory pathway components, most of these proteins were assigned to the tissue-specific categories (e.g., tissue enriched) (Fig. 2a). This simple descriptive analysis indicates most of the secretory pathway genes are expressed in all tissues, while the secretome is tissue specific. Considering this, we contended that if the secretory pathway genes also follow any tissue-specific expression adjustment, despite its ubiquitous expression.
Analyzing the expression variations, as a preliminary assessment, we first checked for within-tissue variation of the secretory pathway genes as a whole machinery. Therefore, we performed a correlation analysis of the expression profiles of the genes encoding secretory pathway components across 30 tissues. As a result, interestingly, tissues were separated into the two groups. One group includes ~70% of all tissues (n = 23) showing medium to high correlation scores (ρ = 0.83 to 0.98). On the other hand, eight tissues including the pancreas, blood, kidney, skeletal muscle, heart, testis, and brain (cerebellum, and cerebrum) as the second group, showed low to medium correlation scores with the tissues in the first group (median coefficient ρ = 0.57 ± 0.17, permutation test p < 0.05). We repeated the analysis using HPA data and observed a similar clustering pattern (Fig. S2A). With a negligible effect of including the secretory pathway genes, the pancreas and the blood showed low cross-tissue correlation scores also at the whole transcriptome level. The weak correlation merits a potential confounding effect due to a deviance from their expression profile in these tissues (Fig. S2B). But, collectively, these results strengthen the idea of that opposite to the ubiquitous expression of the secretory pathway, at least in eight tissue there is a possible adjustment in the genes expression levels.
Finding tissue-specific fine-tuning in secretory pathway gene families
Most of the subsystems in secretory pathway are comprised of several gene families. We showed earlier that eight tissues cluster away from other tissues because of the variations in the expression level of their secretory pathway’s genes (Fig. 2b). Therefore, to trace the differences causing these tissues to cluster away, we intended to reanalyze the correlations in the gene family level. Analyzing the variation in the gene family and subsystem level helped us to interpret the results in proper biological context. We identified 30 gene families with the size range between 4–44 gene members. These gene families are spread over different subsystems (summed up to 348 genes, see EV2 for detail). For example, post-Golgi trafficking (the largest subsystem) includes nine gene families including RAB family as the largest family (n = 72). RAB family genes encode for several different GTPase (diverged from the same ancestral origin17, 18) which are involved in vesicles trafficking from the ER to the Golgi and further down into the extracellular space. Reminding that most of the secretory pathway genes were assigned to the category expressed in all category (86%), the greatest fraction of all subsystems and corresponding gene families also assigned to this category (Fig. S4A, EV2). For instance, genes families in translocation subsystem all have their genes in expressed in all category. Interestingly, among 12 genes that have tissue-enriched, 11 genes are testis-specific proteins, and one gene (CRYAA) is a kidney-specific chaperone. The testis-specific genes spread to over ERAD (Endoplasmic-reticulum-associated protein degradation) (5 genes), protein folding (2 genes), Golgi glycosylation (2 genes) and trafficking regulation (2 genes) (Fig. S4B). For testis, as one the eight outlier tissues, these genes represent a specific expression for the secretory pathway. However, for the other seven tissues, there were no genes assigned to the tissue-enriched category Fig. S4B. We designed a 1110 pair-wise correlation analysis using the expression levels of each gene family expression profile across tissues. As a result, we observed each gene family showing specific correlations pattern across tissues.
To give an example, we take the expression levels of the five gene families in the pancreas, as one of the least correlated tissues (median ρ = 0.49 ± 0.11, Fig. 3). As you can see in Fig. 3, each of these gene families has their correlation pattern. For instance, the correlation scores for the SRP (signal receptor protein) and DNAJ (chaperone) gene families (involved in translocation and protein folding in ER) are low between the pancreas and most other tissues. On the other hand, the pair-wise correlations scores in most cases are high for the RAB or the SEC gene families (involved in membrane coat formation). It is remarkable to note that while the expression profile of secretory pathway in the pancreas as a whole machinery does do not correlate with most tissues (Fig. 2b), in gene family level, some gene families show a high correlation with most tissues. This evidently highlights the tuning of the expression profiles of the secretory pathway at gene family level rather than machinery as the whole. Among all correlation scores, ARF gene family (11 genes) shows high scores for most tissue-pairs. ARF (ADP-ribosylation factor) genes belong to the trafficking regulation subsystem and are involved in vesicle budding and uncoating within the Golgi apparatus19 (Fig. S5A). Conversely, the TBC (TBC1 domain family) gene family (19 genes) in more than 50% of the tissue pairs show low correlation (R < 0.6) (Fig. S5A and SB). TBC family genes are GTPase-activating proteins and are involved in the regulation of the vesicle trafficking.20 These results indicate in some of the particular gene families; the expression levels are modulated in a tissue-specific way.
Identification of tissue-specific “extreme” genes
We showed that tissue-specific modulation in the expression levels of individual gene families could lead to the pair-wise low correlation scores for secretory pathway’s expression profile. In continue, we sought to identify the most extreme expression variations in gene families with low correlation scores. Therefore, for each gene family, we ran the Grubbs test21 to detect the outlier gene expressions, assuming that the total expression level of a gene family can vary among tissues (see Methods). We call the detected outlier genes as “extreme” genes, and we created a network of these identified genes connected to their corresponding particular tissue (Fig. 4a). This network visualizes which extreme genes from which gene families is specific or shared between any tissues (Fig. 4b). The detected extreme genes contribute the most to the low correlation scores calculated for each gene families across tissues (Fig. S5A). The tissues earlier were shown to cluster apart (Fig. 2b) have the largest set of detected extreme genes. These extreme genes are in the gene families which code the components for the subsystems like trafficking regulation, ERAD, protein folding and post-Golgi trafficking. Among the tissues, the Skeletal muscle has the largest number of the extreme genes (Fig. 4b). Noteworthy, for cross-validation, the genes with tissue-enriched category (from HPA) are also among the detected extreme genes. But, most of the extreme genes are marked as expressed in all category, therefore we instead suggest tissue-specific tuning for them.
As a separate validation, we compared our results with GTEx preferential expression analysis results. In the GTEx study, the authors performed a pair-wise differential gene expression analysis among tissue.13 The genes q > 0.99 (FDR = 0.01) and log2 fold change ≥4 in exclusive tissues were reported by them as tissue-preferential genes.13 Extracting preferential genes encoding secretory pathway components, in trustingly, we observed a large overlap with our detected extreme genes (Fig. S7). Meanwhile, we also checked for the tissue-enriched genes of the secretome to see if they are also reported as preferential genes. As a result, we found most of them are reported as preferential genes in GTEx study with top fold changes (Fig. S7). Some of these genes such as Leptin (LEP), insulin (INL), or prolactin (PRL) encode well-known secretory proteins, and their secretion has been studied for many years. This comparison made us confident on our method of detecting extreme genes that are assigned in expressed in all category.
Here we discuss some of the detected extreme genes in the three tissues with lowest correlation scores including the pancreas, skeletal muscle, and cerebrum (Fig. 4c). In the skeletal muscle and the pancreas, extreme genes that were uniquely associated with either of the two tissues, OPTN for skeletal muscle or SEL1L (involved in ERAD) for pancreas, showed an evident higher expression level (>10-fold change) comparing to the other tissues. It has been shown that OPTN plays a major role in the maintenance of the Golgi complex, in membrane trafficking and exocytosis, and it interacts with myosin VI and Rab8.22, 23 Surprisingly, RAB12, another extreme gene in the skeletal muscle, is shown to interact with OPTN.24 These findings suggest that even if secretory pathway genes were expressed rather ubiquitously in all tissues, specific tissues could spike the expression of specific genes in defined subsystems in a tissue-specific fashion. Reminding from our manufacturing example, now we could identify the units (extreme genes) that seem to be fine-tuned in a particular tissue. As next step, we, therefore, explored if these genes ultimate expression is correlated with enrichment of specific PTMs and functions in the secreted or membrane proteins specific in corresponding tissues.
Tissue-specific enrichment in secreted and membrane proteins PTMs associated with expression tuning of the secretory pathway genes
To estimate the PTMs enrichment in each tissue, first, we had to define the tissue-specific secretome and membrane proteins. Therefore, we assembled a comprehensive list of 4098 genes encoding conventional (with signal peptide) (n = 3328) and unconventional (without signal peptide) (n = 680) secreted or membrane proteins (shown in Fig. 2a). Then, we extracted the GTEx expression profiles of these genes and performed hierarchical clustering of the tissues based on their expression correlation matrix (Pearson correlation). We limited this analyses to the genes in tissue-specific categories based HPA (2047 genes). The heat map of the clustering results reveals the tissue-specific expression patterns of the secreted and membrane proteins (Fig. 5). The tissues such as pancreas, testis, brain, skeletal muscle or kidney that were clustered in the separate clade with low correlation scores (shown in Fig. 2b) also show clear and specific expression in their secretome and membrane genes (Fig. 5). Instead, rest of the tissues that clustered together in Fig. 2b with high correlation scores (e.g. colon, ovary, breast, or bladder) share a sizeable number of highly expressed secreted and membrane proteins. Also, the number of secreted or membrane proteins unique to each tissue has a broad dynamic range, which reflects the complexity of the secretory requirements differs in each tissue.
Next, to integrate the PTMs information, we obtained information from UniProt on the number of sites for N-glycosylation (NG), the disulfide bond (DS), O-glycosylation (OG), GPI-anchored (GP) for all of the defined tissue-specific secreted and membrane proteins. Then, we integrated this information with the clustering result (Fig. 5). In general, most of the tissue-specific secreted and membrane proteins are enriched with N-linked glycosylation and disulfide sites. Specifically, pancreas and pituitary secretome displayed a lower enrichment in N-linked glycosylation sites and highly enrichment in disulfide sites (Fig. 5). O-linked and GPI-anchored sites are enriched rather in specific tissues. For example, the liver secretome is enriched in O-linked sites, whereas brain sub-regions are enriched with GPI-anchored membrane proteins (Fig. 5).
Of the PTMs, we chose to explore the correlation between the disulfide sites load and the expression levels of the disulfide isomerase as processing components in each tissue. This is because of complexity of disulfide bond processing in less than other PTMs in secretory pathway regarding a number of the involved gene family and processing reactions. Therefore, to estimate the disulfide sites load on the secretory pathway in each tissue, we defined an enrichment estimator using the expression levels of proteins harboring disulfide sites as a proxy (see Methods). In brief, the estimator is a product function of the expression levels of secreted or membrane proteins and their corresponding number of the disulfide sites. We hypnotized higher estimator values to underscore a higher pressure on disulfide isomerases. Thus, higher values should correlate with the expression of the gene encoding the disulfide isomerase. Consistent with this hypothesis, we observed that the expression level of the PDI gene family, responsible for disulfide isomerase activity, linearly correlated with the disulfide enrichment estimator in each tissue (Fig. 6a). Strikingly, the expression level of ERO1LB gene, previously detected as a pancreas-specific extreme gene (Fig. 4c) was strongly correlated (p-value < 0.001) with calculated disulfide enrichment estimator in the pancreas (Fig. 6a). ERO1LB, an oxidoreductase involved in disulfide bond formation in the ER, is known to efficiently reoxidizes P4HB. P4HB is an enzyme which catalyzes the protein disulfide formation. Oxidation of P4HB by ERO1LB allow P4HB to sustain additional rounds of disulfide formation.25 We therefore also observed a correlation between the expression of P4HB in liver (a shared extreme gene by the liver and pancreas, Fig. 4c) and PDIA4 (a shared extreme gene by the liver and thyroid, Fig. 4c) to the estimator values and found a positive correlation (p-value < 0.001 for P4HB and p-value < 0.01 for PDIA4) (Fig. 6a). These observations are clear evidence which suggests the tissue-specific fine-tuning of the PDI family expression level in response to the enrichment of the disulfide sites. As an experimental validation, we found a recent report in the literature that the expression level ERO1LB is precisely regulated in the pancreas.26
Discussion
As we mentioned in the introduction, Uhlen et al. (2015)5 has recently shown that secretome is the largest fraction of the tissue-specific proteome. However, among the genes coding of the secretory pathway components which process and deliver the secretome proteins, only a small fraction (13%) found to be selectively expressed in certain tissues (mostly in testis). We reasoned that expression of secretome in tissue-specific way could put different PTMs processing pressure on secretory pathway subsystems which are responsible for processing the corresponding PTMs types. Searching for the footprints of this pressure, we detected expression spikes in individual members of gene family members in particular tissues. Detecting this kind of expression modulation in most gene-centric expression analysis such as differential expression analysis in difficult, while considering all members of a gene family expression gives us insight in the expression adjustment across various tissues. Gene expression is a dominant form of biological regulation that contributes to conferring tissue-specific functionality to diverse cell processes. It has previously shown by Kaessmann lab that purifying selection plays a key role in shaping the evolution of gene expression levels in mammalian organs27. Therefore, the tuning of the gene expression in each tissue evolutionary is independent of other tissues. The physiological and phenotypic demands have been the primary driving selection pressure on tissues. Secretory pathway has many gene families, and this indicates its function and complexity has been evolved through many gene duplications and neofunctionalization. So, it is not irrational for the cells to overexpress a specific member of a gene family to release the pressure caused by high processing load in a specific tissue. We could show this clearly in the case of the PDI gene family (Fig. 6a). Our result suggests that the expression levels in PDI gene family are tuned in respect to the processing load of disulfide sites (inside ER) in each tissue.
However, we do not assume this as the only source for the tissue-specific function of the secretory pathway, and signaling pathways, regulatory loops, and biological interactions are still important players.28, 29 On the other hand, drawing this conclusion is not trivial for other PTMs, because a large fraction of UniProt information on PTMs is based on the computational prediction; therefore, it includes a certain degree of false positives. Despite the recent advances, databases to serve tissue-specific information on N-linked or O-linked glycosylation are lacking. Also, it has been shown that even for a protein with experimentally detected glycosylation sites the glycoforms can be very heterogenic. Therefore, it is not easy to correlate the number of detected size with the processing load.30,31,32 The gene family size is another problem, for example despite PDI gene family, other genes families in secretory pathway are large, and therefore, it is more complex to link the detected extreme genes with specific processing load. For example, RAB gene family which is one of the human largest gene family has been studied comprehensively from the evolutionary and molecular point of view. However, due to their complex interaction network and complex function, more analysis and experiment design are needed to validate and understand why some members of this family have extreme expression level in an individual tissue. Although the experimental validation of our results remained to be explored, we found one external validation for TMED2, shared extremely gene between the liver and the pancreas (Fig. 4c). In a recent study, TMED2 is shown to be a pancreas-specific protein,33 and it plays a critical role in cargo detection from ER (COPII vesicle) and the regulation of exocytic trafficking from the Golgi to the plasma membrane.34,35,36 All in all, these results shed light on important fundamental cross-tissue differences in the expression levels of the genes coding the secretory pathway’s component. A key question which remains to be explored is whether tissue-specific fine-tuning is the result of tissue specialization through evolution or the presence of regulatory programs specific to each tissue to fine-tune the control of its secretory pathway. This knowledge will empower us to boost our understanding of important diseases linked to the secretory pathway function in human and, on the other hand, to design better heterologous proteins expression host for biotechnological production.
Data collection
Transcriptome
We obtained the FPKM values for the human tissues from the analysis that has been performed by Uhlén et al. between12 on comparing the recently published RNA-Seq data generated by the Genotype-Tissue Expression (GTEx) consortium13, 37 and HPA consortium.5 In these datasets cutoff of 1 FPKM is used to indicate the presence or absence of transcripts for each gene in a tissue. We also used the categories defined in their paper. All human protein-coding genes were classified into one of six categories based on the FPKM levels in 32 tissues: (1) “Not detected”: FPKM < 1 in all tissues; (2) “Tissue enriched”—at least a 5-fold higher FPKM level in one tissue compared to all other tissues; (3) “Group enriched”—5-fold higher average FPKM value in a group of 2–7 tissues compared to all other tissues; (4) “Expressed in all tissues”—detected in all 32 tissues with FPKM >1; (5) “Tissue enhanced”—at least a 5-fold higher FPKM level in one tissue compared to the average value of all 32 tissues; (6) “Mixed”—the remaining genes detected in 1–31 tissues with FPKM >1 and in none of the above categories. We used the GTEx data sets as the main expression datasets in our analysis, which its measurements are for 20344 genes across 32 human tissues. The GTEx data is based on measurements for 1641 samples from 175 individuals representing 43 sites: 29 solid organ tissues, 11 brain sub-regions, whole blood, and two cell lines: Epstein–Barr virus–transformed lymphocytes (LCL) and cultured fibroblasts from the skin.13 The data from HPA5 were used in parallel to analyze the consistency. Interactome data: For protein–protein interaction data, we used the CCSB database for humans generated by Rolland et al. (2014),38 which includes 14000 high-quality binary protein–protein interactions. Protein complexes: Protein complex information retrieved from a census of human soluble protein complex data generated by Havugimana et al. (2012),39 which is a network of 13993 high-confidence physical interactions among 3006 stably associated soluble human proteins.
Data processing, correlation analysis and visualization
We used recurrently “plyr,” “tidyr,” and “dplyr” R (https://www.r-project.org/) packages for all data processing steps and correlations analysis. The “pheatmap” and “ggplot2” packages used for visualization of the clustering results and plotting.
Detection of the extreme genes
To detect the extreme genes in each gene family we used the Grubbs test21 using “outliers” package in R and GTEx as genes expression level source. The core formula of the calculated G-statistic for Grubbs test for each gene families is:
$G= max X - X ̄ s$
(1)
Where with $X ̄$ and s denotes the sample mean and standard deviation, respectively. The Grubbs’ test statistic is the largest absolute deviation from the sample mean in units of the sample standard deviation.40
The outliers (extreme genes) are collected for all the gene families across tissues by filtering them based on an inbuilt two-sided test with calculated p-values <0.05. The Grubbs test assumes the input data has a normal distribution; however, the gene expression in the gene families violate this assumption. To avoid the false positives in the detection, we repeated the run by using HPA as independent expression resource. The output converted to a binary matrix of tissues-extreme genes and visualized as a network in Cytoscape.41
Defining human secretory pathway
To collect the core components of the human secretory pathway, using the biomart package in R, first, we obtained the orthologs of 163 components of our previously reconstructed secretory pathway model in yeast.16 Also, the additional components were added up to 575, based on collecting relevant components from a comprehensive literature survey and KEGG secretion-related pathways including protein processing in the endoplasmic reticulum (ko04141) and SNARE interactions in vesicular transport (ko04130) (EV1). We defined 13 subsystems (Fig. 1a) based on the overlapping functions of the components adopting from our previously work on yeast secretory pathway genome-scale model.16 The Genes in each subsystem further classified into 30 gene families based on their nomenclature. The gene families consist of 347 genes and serve the core functional core of each subsystem. The rest of the secretory pathway genes are spread in different subsystem as functional units along with gene families (EV3, EV1).
Defining the human secretome
We parsed the human UniProt GFF file and extracted the selected seven secretory features for the human proteome, including the following: Signal Peptide, N-glycosylation sites, O-glycosylation sites, Disulfide bond, GPI-anchored, Transmembrane domain, Localization. The obtained PTMs information was used to build a protein-specific information matrix. Each column of the matrix represents a specific PTMs type and each row belongs to a specific secretory protein. To define the tissue-specific enrichment of the different PTMs types, we integrated the constructed PTMs information matrix with the correlation analysis of the expression profiles from the genes encoding the secretome and membrane proteins. Among analyzed proteins as secretory proteins, 1242 proteins were without predicted signal peptide which 680 of them predicted to be secreted by unconventional secretion (secretome P NN-score >0.6). (Bendtsen et al. 2004 and Nickel & Seedorf, 2008) (EV2). We excluded these proteins from the analysis. For the clustering and visualization of the heatmaps corresponding to the secretome expression data (Fig. 6), we used the ComplexHeatmap packages.42
Disulfide enrichment score
To be able to compare tissues for the enrichment of the disulfide sites in their secretory load we defined a disulfide enrichment estimator DS e for each tissue to be as:
$D S e =log10 ∑ i = 1 n f p k m i ⋅ d s i$
(2)
Where i is the number of the genes that are secreted or membrane proteins from 1to n, fpkm i is the FPKM expression value of gene i and ds i is the number of the disulfide sites in corresponding coded protein.
Data availability
The HPA data used in the analysis is available from the original paper supplementary files (DOI: 10.1126/science.1260419) (ref. 5) and their download section in their database (http://www.proteinatlas.org/). The GTEx data used in the analysis is available from the original paper supplementary files (DOI: 10.1126/science.aaa0355) (ref. 13). The HPA and GTEx comparison data is available at the published papers supplementary files (DOI 10.15252/msb.20155865) (ref. 12). All the PTMs data for human proteome are available at UniProt data base GFF file for human proteins (http://www.uniprot.org/).43 All the codes for the data analysis and visualization are available upon request.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
We acknowledge Swedish National Infrastructure for Computing (SNIC) at C3SE assistance in providing computational infrastructure.This work was supported by the Knut and Alice Wallenberg Foundation and the Novo Nordisk Foundation.
Affiliations
1. Department of Biology and Biological Engineering, Kemivägen 10, Chalmers University of Technology, SE41296, Gothenburg, Sweden
• Amir Feizi
• , Francesco Gatto
• & Jens Nielsen
2. Novo Nordisk Foundation Center for Biosustainability, Chalmers University of Technology, SE41296, Gothenburg, Sweden
• Amir Feizi
• , Mathias Uhlen
• & Jens Nielsen
3. Novo Nordisk Foundation Center for Biosustainability, Technical University of Denmark, DK2970, Hørsholm, Denmark
• Mathias Uhlen
• & Jens Nielsen
4. Science for Life Laboratory, Royal Institute of Technology, SE-17121, Stockholm, Sweden
• Mathias Uhlen
• & Jens Nielsen
Contributions
J.N. and A.F. designed the project. A.F. performed all the analysis steps and the interpretation of the results. A.F. made all the figures and tables. F.G. assisted with expression analysis, interpretation of the results and improving the figures. A.F., F.G., M.U., and J.N. wrote the manuscript. All the authors contribute with proof-reading of the manuscript.
Competing Interests
The authors declare that they have no competing financial interests.
Corresponding author
Correspondence to Jens Nielsen.
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
10. 10.
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# Eagle: Converting DXF circles to pads
Has anyone been able to convert shapes from DXF file to pads in Eagle? Is it possible to convert circles to Eagle pads?
I have just arranged and positioned things using Fusion360 and now I want to convert it to proper things in Eagle.
So far I have manually edited .lbr file (converted <circle> to <pad>) but I'm looking for something more convenient.
Before: <circle x="-22.86" y="10.49" radius="0.45" width="0.001" layer="17"/>
After: <pad name="AP0 x="-22.86" y="10.49" drill="0.9"/>
If somebody is interested:
Regards, Paweł
• Looks like a repetitive symbol.Would be pretty easy to make an Eagle schematic symbol & board footprint for that. I've imported DXF files, but only for board dimension information. – CrossRoads Feb 4 at 13:18
Things that make what you want to do impractical: -
• A circle is just a shape with no inherent drill information associated with it
• A circle as drawn and converted to DXF is unlikely to be to scale
• The relative positions of circles may not be to scale
• Dimension information in the DXF won't be understood readily by a PCB tool
• A DXF circle carries zero information about the pad stack-up (copper layer by layer)
I'm looking for something more convenient
• Talking about accuracy...that is why I'm using Fusion360...for accuracy. Eagle/KiCAD just sucks when we are talking about doing more complex placement and object relations. Anyway... I will just edit .lbr with text editor. I still believe this is way much better/accurate to keep X and Y the same and change only object type: Before: <circle x="-22.86" y="10.49" radius="0.45" width="0.001" layer="17"/> After: <pad name="AP0 x="-22.86" y="10.49" drill="0.9"/> Cheers, Paweł – felixd Feb 4 at 15:07
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# compactification
Let $X$ be a topological space. A (Hausdorff) compactification of $X$ is a pair $(K,h)$ where $K$ is a Hausdorff topological space and $h:X\rightarrow K$ is a continuous function such that
• $K$ is compact
• $h$ is a homeomorphism between $X$ and $h(X)$
• $\overline{h(X)}^{K}=K$ where $\overline{A}^{K}$ denotes closure in $K$ for any subset $A$ of $K$
$h$ is often considered to be the inclusion map, so that $X\subseteq K$ with $\overline{X}^{K}=K$.
Title compactification Compactification 2013-03-22 12:15:42 2013-03-22 12:15:42 Evandar (27) Evandar (27) 8 Evandar (27) Definition msc 54D35 Hausdorff compactification Compact AlexandrovOnePointCompactification
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# The difference in the measures of two complementary angles is 12°.find the measures of the angles.
let the smaller no. be x
therefore bigger no. = x +12
complementary angle = 90
therefore the 2 angles are = x + x +12 = 90
2x + 12 = 90
2x = 90 - 12 = 78
2x = 78
therefore x = 78 / 2 = 39
therefore x + 12 = 39 + 12 = 51
therefore two angles are 39 and 51
hope it helped u............
• 6
a+b=90
a-b=12
a=12+b
now 12+b+b
12+2b=90
2b=90-12
2b=78
b=78/2=39
a=12+39=51
a=51 , b=39
• 0
let the smaller angle be x
therefore bigger angle = x +12
complementary angle = 90
therefore the 2 angles are = x + x +12 = 90
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?2x + 12 = 90
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?2x = 90 - 12 = 78
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2x = 78
therefore x = 78 / 2 = 39
therefore x + 12 = 39 + 12 = 51
therefore two angles are 39 and 51
hope it helped u............ ? ? ? ? ? ??
• 0
What are you looking for?
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Introduction
2018 will be remembered by many investors as a difficult year. In sharp contrast to 2017, when S&P500 was performing spectacularly (+19.4% YoY), 2018 brought a bitter disappointment to many. Not only in terms of absolute performance (-7% YoY), driven heavily by the impressive sell-off in the last quarter of the year, but also in terms of volatility.
In this article we would like to discuss volatility and analyse arguments supporting two seemingly contradictory strategies that employ betting on future volatility. Both of them are bullish with respect to the S&P500 performance. Where they differ is the approach towards utilizing VIX futures. One of the strategies tries to hedge exposure towards a possible recession by adding a rolling hedge using long position in futures on VIX. Those securities are traded on the CBOE futures exchange since 2004. The other trading idea assumes short position on VIX in order not to hedge but to enhance the returns of the portfolio. The innovative element here is the attempt to minimize the tail risk associated with destructive volatility spikes that usually accompany market crashes. The reason for using VIX futures (for this analysis 1 month-maturity futures are used, rolled down to provide continuity) and not VIX spot prices is explained in the following paragraphs.
CBOE VIX Index
In order to start the analysis, a formal introduction to the VIX index is required. According to CBOE, VIX Index is the measure of “market expectations of near-term volatility conveyed by S&P 500 stock index option prices”. What does it mean in practice?
The VIX Index is a measure of the expected volatility of the US stock market. It is a continuous, 30-day expected volatility of the stock market, derived from real-time, mid-quote prices of S&P 500 Index call and put options. The model used for calculating daily values of VIX is described below but the simple intuition is as following: market participants express their beliefs regarding future market moves by generating demand for put and call options, This provides a market price for those instruments, which in turn can be used to calculate implied volatility using Black-Scholes model. Hence, VIX is directionally neutral, and moves in tandem with market sentiment.
History
The concept of creating an index that would measure implied volatility was first proposed by Prof. M. Brenner and Prof. Dan Galai in 1986 in their papers “New Financial Instruments for Hedging Changes in Volatility” and “Hedging Volatility in Foreign Currencies”. However, VIX was born only after the crunch of 1987 when the need to efficiently hedge volatility exposure led CBOE (Chicago Board Options Exchange) to create the CBOE Market Volatility Index, known today as VIX, in 1993. When the VIX was first introduced, it used S&P 100 at-the-money options. In 2003 the underlying asset was changed to S&P 500 options in order to better resemble broad market volatility.
However, VIX on its own is not a tradable security as it encapsulated a rolling basket of options as its underlying. To place bets on the behaviour of the index next degree derivative must have been created. Legend holds that Mark Cuban called Goldman Sachs in 2002 looking for a way to hedge his market risk. He wanted to go long on VIX as this index tended to soar when the market was plummeting. In an usual Goldman Sachs – fashion this instrument was quickly created benefiting from the wave of post-deregulation creativity. New VIX formula was proposed and implemented by CBOE. Eventually, the VIX futures were launched in 2004, followed by the VIX options in 2006.
However, it was only with the Great Recession in 2008 that the VIX saw huge growth in trading volumes. This was largely a psychological effect as the VIX hit an all-time-high of $80, when every other asset was crashing. At the same time, competing products entered the market, both ways, from the long-vol (VXX, UVXY, TVIX) to the short-vol (XIV, SVXY, ZIV) with varying levels of leverage. The aftermath of the crisis saw a long period of economic healing, boosted by low interest rates, inflation and volatility. Short-vol became the overcrowded trade of the decade, luring investors with seemingly ludicrous returns (XIV 10-year return: 1,518%). As trades picked nickels, the metaphorical steam-roller finally came crashing. On the February 5th 2018 Dow Jones plummeted by 1175 points, worst absolute daily downturn in history. The crash and following panic led to an astonishing spike in the VIX as it more than doubled from$17.31 to $37.32. The sweet dreams of the short-vol traders came to an end, with the XIV falling more than 90% in a single day. The issuer Credit Suisses came in and sterilize the product. Traders had to take a 90% haircut. Mechanics Here we would like to get deeper into the underlying model for the VIX spot price, being also the expected annualised variation of the broad market. Cboe Options Exchange calculates the VIX Index using standard and weekly SPX options that are listed for trading on Cboe Options. Only SPX options with Friday expirations are used to calculate the VIX Index (Cboe lists SPX options that expire on days other than Fridays as well, but non-Friday SPX expirations are not used to calculate the VIX Index). Only SPX options with more than 23 days and less than 37 days to the expiration are used to calculate the VIX Index. These SPX options are then weighted to yield a constant, 30-day measure of the expected volatility of the S&P 500 Index. The generalized formula used in the VIX Index calculation is: $\sigma^2=\dfrac{2}{T} \sum_i \dfrac{\Delta K_i}{K_i^2}e^{RT}Q(K_i) - \dfrac{1}{T} \left[ \dfrac{F}{K_0}-1 \right] ^2$ Where: σ VIX divided by 100 T Time to expiration F forward index level derived from index option prices K0 first strike below the forward index level, F Ki strike price of ith out-of-the-money option; a call if Ki > K0 and a put if Ki < K0; both put and call if Ki = K0 Ki interval between strike prices – half the difference between the strike on either side of Ki: $\Delta K_i= \dfrac{K_{i+1}-K_{i-1}}{2}$ (Note: K for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, K for the highest strike is the difference between the highest strike and the next lower strike.) R Risk-free interest rate to expiration Q(Ki) The midpoint of the bid-ask spread for each option with strike Ki More on mechanics of the calculation can be found in the VIX paper by CBOE. Given that implied volatility, in general, exceeds realized volatility, the VIX has a positive skew. Statistically speaking, this implies a greater chance of positive outcomes. At the same time, VIX’s delta is convex to negative returns. This means that positive movements give little or no changes to the prices, while negative movements causes larger spikes. This agrees with behavioural economics in that investors reach more to negative changes. The index measures market expectations towards volatility for the following 30 days with at-the-money options. This is backed mathematically by the Black-Scholes model pricing options, which famously states that implied volatility plays a huge role in determining the price. Interestingly, empirical data has shown VIX’s tendencies for mean reversion, abandoning extremi for the medium-term average. That said, this average fluctuates incessantly. The VIX futures term structure details the implied volatility over the next few months. Like other asset classes, VIX futures in normality is in contango, with a positive gradient as the volatility increases with maturity. However, backwardation can occur when markets are in crises, as short-term volatility exceeds long-term volatility. Also, VIX futures are different from futures with standard, tradable underlying. Since it is not possible to directly trade VIX at the spot price the price of VIX futures has much smaller predictive value. This is due to the fact that without tradable underlying, arbitrage-based futures pricing does not apply; hence the price of futures reflects future spot VIX in more relaxed fashion. Instead of assuming that VIX spot in 1 month will be trading at the level of today’s 1-month maturity future price, it is often a reverse relation. In 1 month futures will converge to today’s vix spot price. This is due to aforementioned mean reversion as well as the usual market tendency to overstate volatility. Finally, VIX level has a very straightforward interpretation. It can be well illustrated with an example. VIX level of, lets say,$20, indicates that the term structure and pricing of options on S&P500 indicate a price movement of 20% of the underlying (S&P500) during the next year. To get monthly variance VIX has to be divided by the square root of 12, and for daily changes it is a square root of 262 (average number of trading days during one year). Therefore, the VIX at $20 is a bet on monthly variance of 5.77% and daily variance of 1.23%. Given this formula we can derive that the spike in 2008 up to$80 was representing market belief that S&P500 will drop by 4.94% the next day and 23.09% during the following month.
First Strategy: Long broad market + Long VIX Futures
First approach to VIX comes from the paper by W.J. Heslinga (2012) “Tactical Asset Allocation with VIX Futures”. Proposed portfolio focuses on taking the advantage of VIX futures negative correlation with the broad market. Long position on the broad market is taken by constructing theoretical portfolio as following:
• 60.5% Equity represented by The S&P 500 Index (S&PCOMP)
• 30.5% Bonds represented by Barclays Capital US Aggregate Index (LHAGGBD)
• 1.3% High Yield Bonds represented by Barclays Capital US High Yield Index (LHYIELD)
• 1.2% Hedge Funds represented by HFRX Global Hedge Fund Index (HFRXHF$) • 0.4% Commodities represented by S&P GSCI Index (GSCITOT) • 4.5% Real Estate represented by S&P US REIT Index (SBBRUSL) • 1.6% Private Equity represented by S&P Listed Private Equity Index (SPLPEI$)
Portfolio weights were motivated by Markowitz Mean-Variance analysis. For the details we recommend checking the original paper. Analysis is conducted using data spanning from 2005 to 2011. First date is when VIX futures obtained liquidity sufficient for semi-frictionless trading. The other leg of this strategy is long position in VIX futures. For the sake of this analysis two long VIX futures positions are discussed. First of them assumes allocation of 2.5% of capital to this security, the other analyses 10% allocation. Historical prices of this securities can be traced using the ticker CVXCS00 on the Reuters DataStream.
Motivation
Biggest advantage of implementing VIX contracts for hedging purposes is their negative correlation with the broad market. Not only it is negative, it gets more negative during the times of financial stress.
From the perspective of an investor a very interesting statistical relation can be observed. Correlation between asset classes experience changes between normal, calm market and crisis, distressed market. First regime is described by numbers without brackets, second regine in brackets. For example: correlation between bonds and equity under first regime is equal to -0.29. This enables to decrease overall beta of the portfolio by diversification. However, under crisis regime this correlation decreases to -0.18. It means, that the hedging effect of diversification is decreasing precisely when it is most needed.
The opposite happens with VIX correlation. In this case, comparing VIX-equity correlation, we can see that first regime produces strongly negative correlation of -0.76. This result is favourable for hedging on its own. However, when crisis began in 2008 this negative correlation jumped to even more negative -0.87. This means, that VIX was the only security that benefited from the crisis in terms of providing hedge for the discussed portfolio.
Results
Over the analysed period our strategy produced following results:
It is very clear that adding VIX future position to analysed portfolio is very beneficial. Average return increased, variance decreased and portfolio has more positive skew. In addition, VIX futures experience smaller volatility than VIX spot itself and are more suited to provide hedge and diversification.
Criticism
This strategy has a very strong downside. It performs better than pure broad market long portfolio if and only if there is a financial stress on the market at some point. If not, longing VIX futures is an expensive process. Not only because of the negative correlation but also because of high prices attached to call options on relatively peaceful market. In other words, this strategy is expensive to implement and works well under specific circumstances.
Second Strategy: Long S&P500 + Short VIX Futures
The second strategy is originally proposed in the paper by A. Dondoni, D.M. Montagna and M. Maggi (2018) titled “Shorting volatility as a portfolio enhancing strategy”. Here the approach is radically different. Goal of the transaction is to benefit from well-performing S&P500 enhanced by continuously shorting VIX futures with 1 month to maturity.
Why 1-month maturity future?
VIX futures terms structure has specific properties. First of all, it is most of the time upward sloping so the futures are traded in contango. Because most of the time implied volatility captured by the VIX Index includes a risk premium which disappears at the time of maturity, selling those futures is, provided normal market environment, very profitable. What posed a great weakness for the first strategy is exactly the advantage of the second one.
To understand this behaviour behavioural economics might come in handy. The same as in the famous volatility smile, implied volatility of an underlying asset does not want to stay constant. Some might argue that high futures premium is due to the value of the insurance they provide against market shock. The truth lays probably in the middle with inherent risk-aversion of the investors that drives put options prices, and the VIX as a whole, up, and the tail risk that those contracts can hedge.
The strategy
Shorting VIX can be performed in 3 ways in order to enhance the gains on the overall portfolio. All 3 will be presented with the conclusion pointing at the most efficient one. All the calculations cover the period between 2005 and 2014.
All-In
First attempt is a classical “all-in” with VIX futures contract being a simple short rolling position in 1 month-maturity futures. Below are the results of this approach divided with respect of the percentage of capital allocated in the futures:
The results are positive. Adding short position on VIX futures with maturity of 1 month not only increases the return but also decreases the standard deviation of the portfolio.
Short UX1, Long UX3
Second approach tries to protect portfolio to adverse stock movements by adding long position on VIX futures with the maturity of 3 months. This way the benefit of limiting the losses is introduced and as long as term structure of the futures remains in contango, the time decay of value on the 3-month future (UX3) will be compensated by the gain on shorted UX1.
Here the results are weaker in absolute terms but variance of the portfolio decreases more providing very similar Sharpe ratios.
Short UX1 with contrarian indicator
Last strategy aspires to be the best of both worlds. In normal times it is exactly the same as the “all-in”, however it has one more parameter. The strategy monitors the difference between UX1 and VIX. This value is obtained for each day of trading. Another step is to calculate the long moving average of this parameter. Here 1-year windom was used (260 sessions). Finally, we calculate shorter moving average. Best results were obtained using 15 days MA. The signal is defined as the moment when the short moving average gets more distant from the long average by 2 standard deviation. If the signal is triggered our position on futures gets flipped for the duration of the next month (ie. if we were shorting volatility we are now longing it).
Thanks to the “flip” indicator the strategy would avoid heavy losses of 2008. Signals were send on October and November of that year producing the return of 45% instead of the loss of the same amount. However, signal was also send on November 2014 resulting in the loss of 14%.
Final results are presented below:
This strategy is by all means the most profitable and obtains very favourable results compared to the two explained before.
Conclusion
Today, more than one year after the deadly spike, shorting VIX once again is incredibly popular. The number of open contracts is now minus 150 thousand, indicating net 150 thousand short position. It seems that market participants are more optimistic about the market in general. Of course, on the 3rd of February 2018 everyone was probably equally as optimistic.
The choice looks quite simple. If you believe in the strong bull market to come in the foreseeable future, short VIX. This is what most of market participants have been doing for some time now, with the record high short-volatility position. In particular, it has been for some time now subject of study whether the hedge fund industry can be considered as a “short put on volatility” (see Ang (2014) and Jurek and Stafford (2015)). The fact that all kinds of hedge funds (from macro to event driven, from relative value to merger arbitrage) but long-short ones have correlations which are statistically significant suggest this might be the case. It is also worth remembering how occasional falls in the stock market lead to huge losses in such cases, also causing such institutional players to default. Today’s situation, in any case, seem to suggest that record-high stock valuations are here to stay according to hedge funds players. So far, they proved right: at the beginning of the year volatility expectation was higher than the one that was actually witnessed by the market and thanks to a steep VIX futures curve the second strategy proposed in the article would have outperformed the S&P500. On the contrary, if you are not sure if you will be able to stomach the afternoon that stock market crashes, hedge and go long. One thing you have to remember: today investors are looking for returns. Safety seems to be just a background concern. Only the future will show how those approaches will perform.
Previously on volatility by Bocconi Students Investment Club:
Trading Volatility Using Options: a French Case
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# 3.8. SMS meshing workflow, cyclone, Discrete Element Model (DEM)¶
This tutorial shows how to use the new meshing workflow, called SMS (Segregated Mesher/Solver). The SMS workflow breaks the CFD workflow into two major steps: Mesher and Solver modes. Each mode of operation is performed in a separate tab in the GUI.
During the Meshing step, the user focuses on generating the mesh and verifies it is appropriate before moving on to the Solver setup. Only three panes are available during Meshing: Geometry, Regions, and Mesh. Once the mesh is generated it must be accepted (after inspection) to unlock the other panes, and move to the second step (Solver mode). The solver mode allows to set all other model settings, and run the MFiX solver to obtain and visualize the simulation results.
The SMS workflow is currently available as a beta testing feature, and can be turned on from the settings menu (select SMS: Segregated Mesher/Solver (beta) from the Mesher workflow drop down list). The blank template will automatically prompt the user to use the SMS workflow and users are encouraged to try it and provide feedback to the development team.
## 3.8.1. Create a new project¶
• On the main menu click on the New button.
• Create a new project by double-clicking on “Blank” template.
• Enter a project name (say ‘cyclone_sms’ and browse to a location for the new project.
• If you have not switched to SMS workflow, you will be prompted to switch. Accept SMS mode.
We are now starting a new project in SMS workflow. There are two tabs in the GUI to setup and run the simulation. The Mesher tab is where the mesh is setup and generated. The Modeler tab is where the rest of the settings and the simulation are performed. The Modeler tab is initially locked until the mesh has been generated and accepted. Running the MFiX solver is also not allowed until the mesh has been generated. At run time, the solver will read the mesh and proceed with the CFD solution.
## 3.8.2. Import the geometry¶
We will use an STL file as the geometry input. Download the file here
Go to the Geometry pane:
• Choose STL file as the geometry input.
• Navigate to where you downloaded cyclone.stl, select it and answer Yes when prompted to copy the file to the project directory.
• Verify the STL normals are pointing in the right direction. In the model window, show the Geometry, hide the Background Mesh, Hide the Regions (visibility is toggled by clicking on each icon). Set the Geometry style to edges, opacity to 1.0, check the Show normals box, set the Scale to 0.05, set count to 10,000 (to show all normals).
The normal vectors should point towards the fluid region (here towards the inside of the cyclone). For internal flows, the tips of the arrows are generally not visible, except through openings in the STL geometry. Openings in the STL geometry are allowed as long as they are outside of the domain extents.
Note
When the normals are pointing in the wrong direction, they can be flipped in the Geometry parameters. Check the Flip normals box and notice the difference in the Model view. Uncheck the box since the original STL file was correctly oriented.
• Toggle the Region visibility back on.
• In Domain extent, click on Autosize, and adjust the following:
• Enter 0.0 for the Min Y value.
• Enter 0.9 for the Max Y value.
• Enter 0.25 for the Max Z value.
Note that the cyclone intet and outlets extend beyond the domain extents. This is intentional, as it usually provides a cleaner intersection and easier way to define boundary conditions regions when they are aligned with the domain box (planes along x=xmin, x=xmax etc.).
• Save the project by clicking the button.
## 3.8.3. Create boundary conditions regions¶
Here, we need to define the mesh boundaries. Boundaries include the cyclone wall (curved surface defined by the STL geometry), and the inlet and outlets, defined along the domain outer box (plane boundaries). A Boundary Condition type must be defined, but it will be possible to change it later (in Modeler Tab). The actual boundary conditions (like the inlet velocity, or outlet pressure) do not need to be set at this point.
Go to the Regions pane:
• There is already a Background IC that is predefined in the Blank template. We can ignore it for now.
• Click the button to create a new region to be used by pressure outflow boundary condition.
• Select Pressure outflow in the Boundary type drop down menu.
• Enter a name for the region in the Name field (“Top outlet”).
• Change the region Color to Blue.
• Adjust the From/To coordinates in the x-direction: From -0.07 to 0.07.
• Adjust the From/To coordinates in the z-direction: From -0.07 to 0.07.
• Click the button to create a new region to be used by another pressure outflow boundary condition.
• Select Pressure outflow in the Boundary type drop down menu.
• Enter a name for the region in the Name field (“Bottom outlet”).
• Change the region Color to Yellow.
• Adjust the From/To coordinates in the x-direction: From -0.07 to 0.07.
• Adjust the From/To coordinates in the z-direction: From -0.07 to 0.07.
• Click the button to create a new region to be used by mass inflow boundary condition.
• Select Mass inflow in the Boundary type drop down menu.
• Enter a name for the region in the Name field (“Front inlet”).
• Change the region Color to Red.
• Adjust the From/To coordinates in the x-direction: From xmin to -0.025.
• Adjust the From/To coordinates in the y-direction: From 0.55 to 0.80.
• click the button to create a new region that covers the entire domain to be used for the wall boundary condition.
• Select No-slip wall in the Boundary type drop down menu.
• Enter a name for the region in the Name field (“Wall”).
• Change the region Color to Green.
• Check Select facets (STL) and cyclone.stl, use all for the selection method. You should see thet there are 7554 facets selected.
Note
The pressure outlet and mass inflow boundaries are located along the domain box. They are defined as rectangular 2D regions and must overlap the actual boundary areas (intersection of the STL file with the box planes). The exact boundary area will be computed automatically when preprocessing is performed. Since there is only one Boundary condition on the Top, Bottom and Front planes, we could also have used the entire planes instead of adjusting the region coordinates.
• Save the project by clicking the button.
## 3.8.4. Setup the mesh¶
Go to the Mesh pane, Background sub-pane:
• Enter 25 for the x cell value.
• Enter 70 for the y cell value.
• Enter 33 for the z cell value.
Go to the Mesh pane, Mesher sub-pane:
• Keep all default settings and click Generate. Select the default solver, and click Run
• Look at the console output to verify the mesh generation completed successfully.
• In the Model view, hide the Geometry, Background Mesh and Regions, show the Boundary_Mesh, set Color by to bc_id, Style to edges and opacity to 1.0.
• The Boundary_Mesh should look like a closed surface with colors matching the Boundary Conditions regions.
• Color the Boundary_Mesh by small_cell. This will show small cells, as defined by the Small cell tolerance. The default value of 0.01 means a cut cell is considered small if its volume is below 0.01 times the volume of the corresponding standard, uncut cell. These small cells will be removed from the computation. Having a Small cell tolerance set to 0.0 will keep all small cells and this may lead to numerical stiffness and loss of convergence.
• Having a few small cells is usually acceptable. In some cases it is possible to reduce the number of small cells by adjusting some cut cell parameters. Increase the Snap tolerance to 0.1 in the x, y, and z-direction. Generate the mesh again, and visualize the small cells. The vertical column of small cells was eliminated.
• Mesh statistics (histogram) can ve viewed in the Mesh Stats viewport. Various quantities are available. For example the aspect ratio of cut cells is between 1 and 8, with most aspect ratios between 1 and 2. This is acceptable.
After inspection, the mesh is deemed acceptable and we can move to the second step (Solver mode). Click Accept. This will unlock the Modeler tab.
Note
Optional but recommended: To see examples of unacceptable meshes, try the following:
• Go back to the Geometry pane, and flip the normals. This will incorrectly orient the STL file. This is a common error when importing STL files. Generate the mesh and visualize the Boundary_Mesh. Having an open surface and boundaries inside-out is an indication that the normals are not properly oriented. The mesh should not be accepted.
• Uncheck the Flip normals box in the Geometry pane to get correct otientation. Go to the Region pane, and select the Front inlet region in the region table and set the Boundary type to None. Save the project and generate the mesh. The Boundary_Mesh will look like an open surface, this is an indication a Boundary condition is missing. The mesh should not be accepted.
Remember to set the Front inlet Region boundary type to Mass inflow again before continuing. Generate the mesh, inspect it and accept it before moving to the Modeler tab.
## 3.8.5. Model settings¶
Switch to the Modeler tab (second tab at the bottom left corner of the GUI).
• On the Model pane, enter a descriptive text in the Description field.
• Select “Discrete Element Model (MFiX-DEM)” in the Solver drop-down menu.
• Keep all other settings at their default values.
## 3.8.6. Regions settings¶
We already have the Background_IC region that is predefined. We will use it to initialize the flow field. We have defined all Boundary Condition regions during the meshing step. There are no other regions that need to be defined in this tutorial. We will re-use Background_IC region for the VTK output file.
## 3.8.7. Fluid settings¶
• Change Density to Ideal gas law.
• Keep all other settings at their default values.
## 3.8.8. Solids settings¶
On the Solids pane, Materials sub-pane:
• Click the button to create a new solid.
• Enter a descriptive name in the Name field (“Glass beads”).
• Verify the solids model is already set to “Discrete Element Model (MFiX-DEM)”.
• Enter the particle diameter of 0.005 m in the Diameter field.
• Enter the particle density of 2500 kg/m3 in the Density field.
• Select the Solids pane, DEM sub-pane.
• Check the Enable automatic particle generation checkbox. Although we will start with an empty cyclone, this setting is necessary so we don’t attempt to read initial particle location from partile_input.dat.
• Keep all other settings at their default values.
## 3.8.9. Initial Conditions settings¶
On the Initial conditions pane:
• There is already a pre-defined “Background IC” region. This will initialize the entire flow field with air at rest.
• Keep all settings at their default values.
## 3.8.10. Boundary Conditions settings¶
On the Boundary conditions pane:
• The Top outlet and Bottom outlet Boundary Conditions are already set to atmospheric pressure. No changes are needed.
• The Wall Boundary Conditions is already set to No-slip wall. No changes are needed.
• Select the Front Inlet region from the list.
• In the Glass beads tab: set the Volume fraction to 0.01, and Z-axial velocity to -2.0.
• In the Fluid tab, verify the Gas volume fraction was automatically set to 0.99 and set the Z-axial velocity to -2.0.
• Keep all other settings at their default values.
## 3.8.11. Numerics settings¶
On the Numerics pane:
• Set the Fluid normalization to 0.0.
• Keep all other settings at their default values.
## 3.8.12. Output settings¶
On the Output pane:
• On the Basic sub-pane, check the Write VTK output files (VTU/VTP) checkbox.
• Select the VTK sub-pane.
• Create a new output by clicking the button.
• Select “Particle Data” from the ‘Output type’ drop-down menu.
• Select the “Background IC” region from the list to save all the particle data.
• Click OK to create the output.
• Enter a base name for the *.vtu files in the Filename base field (“Particles”).
• Change the Write interval to 0.01 seconds.
• Select the Diameter and Translational Velocity check-boxes.
## 3.8.13. Run settings¶
Save the project by clicking the button.
On the Run pane:
• Set the Stop time to 1.0.
• Set the Time step to 0.01.
• Set the Maximum time step to 0.01.
• Keep all other settings at their default values.
## 3.8.14. Run the simulation¶
• Run the project by clicking the button.
• On the Run dialog, select the executable from the combo-box.
• Click the Run button to actually start the simulation.
## 3.8.15. View results¶
Results can be viewed, and plotted, while the simulation is running.
• Create a new visualization tab by pressing the next to the Model tab.
• Click the 3D view button to view the vtk output files.
• On the VTK results tab, the visibility and representation of the *.vtk files can be controlled with the menu on the side.
• Change frames with the , , , and buttons.
• Click the button to play the available vtk files.
• Change the playback speed under the section on the sidebar.
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Cut Datasets¶
Original datasets are considerably big in size. If you want to save your time when loading original datasets into the DL Workbench, cut them as described in the following sections.
ImageNet Dataset¶
Cut ImageNet Dataset¶
1. Save the script to cut datatsets to the following directory:
• Linux*, macOS*: /home/<user>/Work. Replace <user> with your username.
• Windows* : C:\Work
Note
Replace <user> with your username. Run the following command in a terminal for Linux, macOS and in the Windows PowerShell* for Windows.
python /home/<user>/Work/cut_dataset.py \
--source_archive_dir=/home/<user>/Work/imagenet.zip \
--output_size=20 \
--output_archive_dir=/home/<user>/Work/subsets \
--dataset_type=imagenet \
--first_image=10
python C:\\Work\\cut_dataset.py
--source_archive_dir=C:\\Work\\imagenet.zip
--output_size=20
--output_archive_dir=C:\\Work\\subsets
--dataset_type=imagenet
--first_image=10
This command runs the script with the following arguments:
Parameter
Explanation
--source_archive_dir
--output_size=20
Number of images to be left in a smaller dataset
--output_archive_dir
Full directory to the smaller dataset, excluding the name
--dataset_type
Type of the source dataset
--first_image
Optional . The index of the image to start cutting from. Specify if you want to split your dataset into training and validation subsets. The default value is 0.
Pascal Visual Object Classes (VOC) Dataset¶
Cut Pascal VOC Dataset¶
1. Save the script to cut datatsets to the following directory:
• Linux*, macOS*: /home/<user>/Work. Replace <user> with your username.
• Windows* : C:\Work
Note
Replace <user> with your username. Run the following command in a terminal for Linux, macOS and in the Windows PowerShell* for Windows.
python /home/<user>/Work/cut_dataset.py \
--source_archive_dir=/home/<user>/Work/voc.tar.gz \
--output_size=20 \
--output_archive_dir=/home/<user>/Work/subsets \
--dataset_type=voc \
--first_image=10
python C:\\Work\\cut_dataset.py
--source_archive_dir=C:\\Work\\voc.tar.gz
--output_size=20
--output_archive_dir=C:\\Work\\subsets
--dataset_type=voc
--first_image=10
This command runs the script with the following arguments:
Parameter
Explanation
--source_archive_dir
--output_size=20
Number of images to be left in a smaller dataset
--output_archive_dir
Full directory to the smaller dataset, excluding the name
--dataset_type
Type of the source dataset
--first_image
Optional . The index of the image to start cutting from. Specify if you want to split your dataset into training and validation subsets. The default value is 0.
Common Objects in Context (COCO) Dataset¶
Cut COCO Dataset¶
1. Save the script to cut datatsets to the following directory:
• Linux*, macOS*: /home/<user>/Work. Replace <user> with your username.
Note
Replace <user> with your username.
* Windows* : C:\Work
Note
Replace <user> with your username. Run the following command in a terminal for Linux, macOS and in the Windows PowerShell* for Windows.
python /home/<user>/Work/cut_dataset.py \
--source_images_archive_dir=/home/<user>/Work/coco_images.zip \
--source_annotations_archive_dir=/home/<user>/Work/coco_annotations_.zip \
--output_size=20 \
--output_archive_dir=/home/<user>/Work/subsets \
--first_image=10
python C:\\Work\\cut_dataset.py
--source_images_archive_dir=C:\\Work\\coco_images.zip
--source_annotations_archive_dir=C:\\Work\\coco_annotations_.zip
--output_size=20
--output_archive_dir=C:\\Work\\subsets
--first_image=10
This command runs the script with the following arguments:
Parameter
Explanation
--source_images_archive_dir
--source_annotations_archive_dir
--output_size
Number of images to be left in a smaller dataset
--output_archive_dir
Full directory to the smaller dataset excluding the name
--dataset_type
Type of the source dataset
--first_image`
Optional . The number of the image to start cutting from. Specify if you want to split your dataset into training and validation subsets. The default value is 0.
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## The For Statement
The 'for' loop is more compact than the 'while' and 'do' loops and automatically updates the loop counter at the end of each iteration. Both 'for' and 'while' loops are designed for different situations. You'll learn more about when to use each later.
### 14. Omitting the Test
Answer:Yes. (Actually, a good answer would be "sounds dangerous", because it is.)
# Omitting the Test
When the test part of a for is omitted it is as if the value true were put in its place. So,
for ( initialize ; ; change ) loopBody ;
is the same as:
for ( initialize ; true ; change ) loopBody ;
This is done for compatibility with the language C. It should not be used in newly written programs.
Several syntactic oddities were included in Java so that it would look familiar to C programmers.
Question 14:Could all three parts be omitted from a for?
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2013
11-11
# Bridge
n people wish to cross a bridge at night. A group of at most two people may cross at any time, and each group must have a flashlight. Only one flashlight is available among the n people, so some sort of shuttle arrangement must be arranged in order to return the flashlight so that more people may cross.
Each person has a different crossing speed; the speed of a group is determined by the speed of the slower member. Your job is to determine a strategy that gets all n people across the bridge in the minimum time.
The first line of input contains n, followed by n lines giving the crossing times for each of the people. There are not more than 1000 people and nobody takes more than 100 seconds to cross the bridge.
The first line of output must contain the total number of seconds required for all n people to cross the bridge. The following lines give a strategy for achieving this time. Each line contains either one or two integers, indicating which person or people form the next group to cross. (Each person is indicated by the crossing time specified in the input. Although many people may have the same crossing time the ambiguity is of no consequence.) Note that the crossings alternate directions, as it is necessary to return the flashlight so that more may cross. If more than one strategy yields the minimal time, any one will do.
4
1
2
5
10
17
1 2
1
5 10
2
1 2
//* @author: [email protected]
import java.util.*;
public class Main
{
public static void main(String[] args)
{
Scanner in=new Scanner(System.in);
int a=in.nextInt();
ArrayList< Integer> t=new ArrayList< Integer>();
for(int i=0;i< a;i++)
if(a==1){
System.out.println(t.get(0));
System.out.println(t.get(0));
}
else{
int total=0;
Collections.sort(t);
ArrayList< Integer> t2=new ArrayList< Integer>();
ArrayList< Integer> t3=new ArrayList< Integer>();
int m1=t.get(0);
int m2=t.get(1);
int w=a;
while(w>3)
{
total+=t.get(w-1)+m1+Math.min(m2*2,m1+t.get(w-2));
if(m2*2>(m1+t.get(w-2)))
{
}
else
{
}
w-=2;
}
if(w==2){
total+=m2;
}
else {
total+=m1+m2+t.get(2);
}
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# Solution: 2014-05 Nonnegative determinant
Let $$n$$, $$k$$ be positive integers and let $$A_1,A_2,\ldots,A_n$$ be $$k\times k$$ real matrices. Prove or disprove that $\det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.$ (Here, $$A^t$$ denotes the transpose of the matrix $$A$$.)
The best (most elementary) solution was submitted by 김정민. Congratulations!
Alternative solutions were submitted by 조준영 (+3), 채석주 (+3), 이영민 (+3), 심병수 (+3), 박훈민 (+3), 장기정 (+3), 정성진 (+3), 황성호 (+3), 이종원 (+3), 김일희 (+2), 남재현 (+3), 박경호 (+3).
GD Star Rating
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# Immersed Spheres of Finite Total Curvature into Manifolds
created by mondino on 18 Sep 2012
modified on 26 Jun 2013
[BibTeX]
Accepted Paper
Inserted: 18 sep 2012
Last Updated: 26 jun 2013
We prove that a sequence of, possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence stays within a class $\gamma$ of $\pi_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.
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### Home > CC2MN > Chapter 5 > Lesson 5.2.2 > Problem5-40
5-40.
The Kennedy High School cross-country running team ran the following distances in recent practices:
$3.5$ miles, $2.5$ miles, $4$ miles, $3.25$ miles, $3$ miles, $4$ miles, and $6$ miles.
Find the mean and median of the team’s distances.
To find a mean, add together all of the values and divide the sum by the total number of values.
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# sksurv.linear_model.IPCRidge#
class sksurv.linear_model.IPCRidge(alpha=1.0, *, fit_intercept=True, copy_X=True, max_iter=None, tol=0.001, solver='auto', positive=False, random_state=None)[source]#
Accelerated failure time model with inverse probability of censoring weights.
This model assumes a regression model of the form
$\log y = \beta_0 + \mathbf{X} \beta + \epsilon$
L2-shrinkage is applied to the coefficients $$\beta$$ and each sample is weighted by the inverse probability of censoring to account for right censoring (under the assumption that censoring is independent of the features, i.e., random censoring).
See 1 for further description.
Parameters
• alpha (float, optional, default: 1.0) –
Small positive values of alpha improve the conditioning of the problem and reduce the variance of the estimates. alpha must be a non-negative float i.e. in [0, inf).
For numerical reasons, using alpha = 0 is not advised.
• fit_intercept (bool, default: True) – Whether to fit the intercept for this model. If set to false, no intercept will be used in calculations (i.e. X and y are expected to be centered).
• copy_X (bool, default: True) – If True, X will be copied; else, it may be overwritten.
• max_iter (int, default: None) – Maximum number of iterations for conjugate gradient solver. For ‘sparse_cg’ and ‘lsqr’ solvers, the default value is determined by scipy.sparse.linalg. For ‘sag’ solver, the default value is 1000. For ‘lbfgs’ solver, the default value is 15000.
• tol (float, default: 1e-4) – Precision of the solution. Note that tol has no effect for solvers ‘svd’ and ‘cholesky’.
• solver ({'auto', 'svd', 'cholesky', 'lsqr', 'sparse_cg', 'sag', 'saga', 'lbfgs'}, default: 'auto') –
Solver to use in the computational routines:
• ’auto’ chooses the solver automatically based on the type of data.
• ’svd’ uses a Singular Value Decomposition of X to compute the Ridge coefficients. It is the most stable solver, in particular more stable for singular matrices than ‘cholesky’ at the cost of being slower.
• ’cholesky’ uses the standard scipy.linalg.solve function to obtain a closed-form solution.
• ’sparse_cg’ uses the conjugate gradient solver as found in scipy.sparse.linalg.cg. As an iterative algorithm, this solver is more appropriate than ‘cholesky’ for large-scale data (possibility to set tol and max_iter).
• ’lsqr’ uses the dedicated regularized least-squares routine scipy.sparse.linalg.lsqr. It is the fastest and uses an iterative procedure.
• ’sag’ uses a Stochastic Average Gradient descent, and ‘saga’ uses its improved, unbiased version named SAGA. Both methods also use an iterative procedure, and are often faster than other solvers when both n_samples and n_features are large. Note that ‘sag’ and ‘saga’ fast convergence is only guaranteed on features with approximately the same scale. You can preprocess the data with a scaler from sklearn.preprocessing.
• ’lbfgs’ uses L-BFGS-B algorithm implemented in scipy.optimize.minimize. It can be used only when positive is True.
All solvers except ‘svd’ support both dense and sparse data. However, only ‘lsqr’, ‘sag’, ‘sparse_cg’, and ‘lbfgs’ support sparse input when fit_intercept is True.
• positive (bool, default: False) – When set to True, forces the coefficients to be positive. Only ‘lbfgs’ solver is supported in this case.
• random_state (int, RandomState instance, default: None) – Used when solver == ‘sag’ or ‘saga’ to shuffle the data.
coef_#
Weight vector.
Type
ndarray, shape = (n_features,)
intercept_#
Independent term in decision function. Set to 0.0 if fit_intercept = False.
Type
float or ndarray of shape (n_targets,)
n_iter_#
Actual number of iterations for each target. Available only for sag and lsqr solvers. Other solvers will return None.
Type
None or ndarray of shape (n_targets,)
n_features_in_#
Number of features seen during fit.
Type
int
feature_names_in_#
Names of features seen during fit. Defined only when X has feature names that are all strings.
Type
ndarray of shape (n_features_in_,)
References
1
W. Stute, “Consistent estimation under random censorship when covariables are present”, Journal of Multivariate Analysis, vol. 45, no. 1, pp. 89-103, 1993. doi:10.1006/jmva.1993.1028.
__init__(alpha=1.0, *, fit_intercept=True, copy_X=True, max_iter=None, tol=0.001, solver='auto', positive=False, random_state=None)[source]#
Methods
__init__([alpha, fit_intercept, copy_X, ...]) fit(X, y) Build an accelerated failure time model. get_params([deep]) Get parameters for this estimator. Predict using the linear accelerated failure time model. score(X, y[, sample_weight]) Return the coefficient of determination of the prediction. set_params(**params) Set the parameters of this estimator.
fit(X, y)[source]#
Build an accelerated failure time model.
Parameters
• X (array-like, shape = (n_samples, n_features)) – Data matrix.
• y (structured array, shape = (n_samples,)) – A structured array containing the binary event indicator as first field, and time of event or time of censoring as second field.
Return type
self
get_params(deep=True)#
Get parameters for this estimator.
Parameters
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
Returns
params – Parameter names mapped to their values.
Return type
dict
predict(X)[source]#
Predict using the linear accelerated failure time model.
Parameters
X ({array-like, sparse matrix}, shape = (n_samples, n_features)) – Samples.
Returns
C – Returns predicted values on original scale (NOT log scale).
Return type
array, shape = (n_samples,)
score(X, y, sample_weight=None)[source]#
Return the coefficient of determination of the prediction.
The coefficient of determination $$R^2$$ is defined as $$(1 - \frac{u}{v})$$, where $$u$$ is the residual sum of squares ((y_true - y_pred)** 2).sum() and $$v$$ is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a $$R^2$$ score of 0.0.
Parameters
• X (array-like of shape (n_samples, n_features)) – Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
• y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True values for X.
• sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
Returns
score$$R^2$$ of self.predict(X) wrt. y.
Return type
float
Notes
The $$R^2$$ score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score(). This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).
set_params(**params)#
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.
Parameters
**params (dict) – Estimator parameters.
Returns
self – Estimator instance.
Return type
estimator instance
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## Calculus 10th Edition
$(-3,4,5)$
The point is located three units behind the $yz$-plane (this means the $x$-component is $-3$), four units to the right of the $xz$-plane (this means the $y$-component is 4), and five units above the $xy$-plane (this means the $z$-component is 5.)
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Continuum mechanics/Sets: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
Study guide
Up to date as of January 14, 2010
From Wikiversity
A familiarity with the notation of sets is essential for the student who wants to read modern literature on finite elements. This handout gives you a brief review of set notation. More details can be found in books on advanced calculus.
Sets
A set is a well-defined collection of objects. As far as we are concerned, these objects are mainly numbers, vectors, or functions.
If an object x is a member of a set A, we write
$x \in A ~~~~~~~~(x~\text{belongs to}~A)~.$
If x is not a member of A, we write
$x \notin A ~~~~~~~~(x~\text{does not belong to}~A)~.$
An example of a finite set (of functions) is
$S = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}~.$
Another example is the set of integers greater than 5 and less than 12
$B = \{6, 7, 8, 9, 10, 11\}~.$
If we denote the set of all integers by $\mathbb{Z}$, then we can alternatively write
$B = \{n ~|~ n \in \mathbb{Z}, 5 < n < 12\}~.$
The set $\mathbb{Z}^{+}$ of positive integers is an infinite set and is written as
$\mathbb{Z}^{+} = \{n ~|~ n \in \mathbb{Z}, n > 0\}~.$
An empty (or null) set is a set with no elements. It is denoted by $\varnothing$. An example is
$\varnothing = \{n ~|~ n \in \mathbb{Z}^{+}, n < 0\}~ = \{\}~.$
Subsets
If A and B are two sets, then we say that A is a subset of B if each element of A is an element of B.
For example, if the two sets are
$A = \{\sin(x), \cos(x)\}~ B = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}~~\text{and}~~$
we write
$A \subset B ~~~~~(A~\text{is a proper subset of}~B)~.$
On the other hand, if A is a subset of B which may be the set B itself we write
$A \subseteq B ~~~~~(A~\text{is a subset of}~B)~.$
If A is not a subset of B, we write
$A \not\subset B ~~~~~(A~\text{is not a subset of}~B)~.$
Equality of sets
Two sets A and B are equal if they contain exactly the same elements. Thus,
$A = B \iff A \subseteq B ~\text{and}~ B \subseteq A ~.$
The symbol $\iff$ means if and only if.
For example, if
$A = \{x ~|~ x^2 = 4\} ~\text{and}~ B = \{2, -2\}$
then A = B.
Union, Intersection, Difference of Sets
The union of two sets A and B is the set of all elements that are in A or B.
$C = A \cup B = \{x ~|~ x \in A ~\text{or}~ x \in B\}~.$
The intersection of two sets A and B is the set of all elements that are both in A and in B.
$D = A \cap B = \{x ~|~ x \in A ~\text{and}~ x \in B\}~.$
The difference of two sets A and B is the set of all elements that are in A but not in B.
$E = A - B = \{x ~|~ x \in A ~\text{and}~ x \not\in B\}~.$
The complement of a sets A (denoted by A') is the set of all elements that are not in A but belong to a larger universal set U.
$A^{'} = U - A = \{x ~|~ x \not\in A\}~.$
Countable Sets
Suppose we have a set A. Such a set is called countable if each of its members can be labeled with an integer subscript of the form
$A = \{a_1, a_2, a_3, a_4, \dots\}~.$
Obviously, each finite set is countable. Some infinite sets are also countable. For instance, the set of integers is countable because you can label each integer with an subscript that is also an integer. However, you cannot do that with the real numbers which are uncountable.
The set of functions
$P = \{f_k ~|~ f_k(x) = x^k, k=1,2,\dots\}$
is countable.
The set of points on the real line
$A = \{x ~|~ 0 \le x \le 1\}$
is not countable because the points cannot be labeled a1, a2, $\dots$.
Cartesian Product
The Cartesian product of two sets A and B is the set of all ordered pairs (a,b), such that
$A \times B = \{(a,b) ~|~ a \in A, b \in B \}~.$
In general, $A \times B \neq B \times A$.
For example, if
$A = \{1, 5, 7\} ~\text{and}~ B = \{3, 6\}$
then
$A \times B = \{(1,3),(1,6),(5,3),(5,6),(7,3),(7,6)\}$
and
$B \times A = \{(3,1),(3,5),(3,7),(6,1),(6,5),(6,7)\} \neq A \times B~.$
The Set of Real Numbers ($\mathbb{R}$)
The set of real numbers ($\mathbb{R}^{}$) can be visualized as an infinitely long line with each real number being represented as a point on this line.
We usually deal with subsets of $\mathbb{R}^{}$, called intervals.
Let a and b be two points on $\mathbb{R}^{}$ such that $a \le b$. Then,
• The open interval (a,b) is defined as
$(a,b) = \{x ~|~ x \in \mathbb{R}, a < x < b\}~.$
• The closed interval [a,b] is defined as
$[a,b] = \{x ~|~ x \in \mathbb{R}, a \le x \le b\}~.$
• The half-open intervals (a,b] and [a,b) are defined as
$(a,b] = \{x ~|~ x \in \mathbb{R}, a < x \le b\} ~\text{and}~ [a,b) = \{x ~|~ x \in \mathbb{R}, a \le x < b\}~.$
Let $p \in \mathbb{R}^{}$ and ε > 0. Then the neighborhood of p is defined as the open interval
$\text{nbd}(p;\epsilon) := (p - \epsilon, p+\epsilon) = \{ x ~|~ p - \epsilon < x < p + \epsilon \}$
Let $X \subset \mathbb{R}^{}$. Then p is an interior point of X if if we can find a nbd(p) all of whose points belong to X.
If every point of X is an interior point, then X is called an open set. For example, the interval (a,b) is an open set. So is the real line $\mathbb{R}^{}$.
A set $X \subset \mathbb{R}^{}$ is called closed if its complement $X^{'} = \mathbb{R}^{} - X$ is open.
The closure $\bar{X}$ of a set $X \subset \mathbb{R}^{}$ is the union of the set and its boundary points (a rigorous definition of closed sets can be made using the concept of points of accumulation).
Open and Closed Sets in $\mathbb{R}^n$
The concept of the real line can be extended to higher dimensions. In two dimensions, we have $\mathbb{R}^{2}$ which is defined as
$\mathbb{R}^{2} = \mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y) ~|~ x,y \in \mathbb{R}^{}\}~.$
$\mathbb{R}^{2}$ can be thought of as a two-dimensional plane and each member of the set $\mathbf{x} = (x,y)$ represents a point on the plane.
In three dimensions, we have
$\mathbb{R}^{3} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y,z) ~|~ x,y,z \in \mathbb{R}^{}\}~.$
In n dimensions, the concept is extended to mean
$\mathbb{R}^{n} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{}\dots\times\mathbb{R}^{} = \{(x_1,x_2,x_3,\dots,x_n) ~|~ x_1,x_2,x_3,\dots,x_n \in \mathbb{R}^{}\}~.$
In the case of sets in $\mathbb{R}^{n}$ the concept of distance in $\mathbb{R}^{}$ is extended so that
$\text{nbd}(\mathbf{p};\epsilon) := \{ \mathbf{x} ~|~ \mathbf{p} \in \mathbb{R}^{n}, |\mathbf{x} - \mathbf{p}| < \epsilon \}$
where
$|\mathbf{x} - \mathbf{p}| = \sqrt{(x_1 - p_1)^2 + (x_2 - p_2)^2 + \dots + (x_n - p_n)^2}~.$
The definition of interior point also follows from the definition in $\mathbb{R}^{}$. Thus if $\Omega \subset \mathbb{R}^{n}$, then $p \int \mathbb{R}^{n}$ is an interior point if we can always find a nbd(p;ε), all of whose points belong to Ω. If every point on Ω is an interior point, then Ω is an open set. As in the real number line, a closed set is the complement of an open set. One way of creating a closed set is by taking an open set Ω and its boundary Γ. This particular closed set is called the closure $\bar\Omega$ of Ω. A rigorous definition can once again be obtained using the concept of points of accumulation.
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# Law of the wall
(Difference between revisions)
Revision as of 07:58, 6 September 2011 (view source)GeeZ (Talk | contribs) (Changed image (now is a clear png file generated from a *.svg).)← Older edit Revision as of 08:32, 7 September 2011 (view source)Peter (Talk | contribs) Newer edit → Line 24: Line 24: In the image y is replaced with the letter n. In the image y is replaced with the letter n. + + [[Image:Img_lawOfTheWall.svg]] [[Image:Img_lawOfTheWall_whiteBG.png]] [[Image:Img_lawOfTheWall_whiteBG.png]] {{stub}} {{stub}}
## Revision as of 08:32, 7 September 2011
In the log layer the velocity profile can be estimated with the log law:
$u^+ = \frac{1}{\kappa} \, ln(y^+) + B$
and close to the wall in the viscous sublayer
$u^+ = y^+$
Where:
$u^+$ Dimensionless velocity $y^+$ Dimensionless wall distance $\kappa$ von Karman's constant ($\approx 0.41$) $B$ Constant ($\approx 5.1$)
We should have a lin-log plot here of a typical turbulent boundary layer to illustrate where the log-law is valid, anyone have one handy?
In the image y is replaced with the letter n.
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#### Howdy, Stranger!
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# Question 2.4 - Uniqueness of universal objects
edited January 2020
Uniqueness of universal objects.
Recall the definitions of terminal object and product from Chapter 3 in the notes.
(a) Show that if $$t$$ and $$t'$$ are both terminal objects in a category, then $$t$$ and $$t'$$ are isomorphic.
(b) Let $$a$$ and $$b$$ be objects of a category. Show that if $$p$$ and $$p'$$ are both products of $$a$$ and $$b$$, then they are isomorphic.
(c) Discuss the similarities between your two proofs. Could the same idea be used to show that any two initial objects are isomorphic?
## Comments
• Options
1.
edited January 2020
(a) Let us define the functions $$f : t \rightarrow t'$$ and $$g : t' \rightarrow t$$ Their composition gives us $$g \circ f : t \rightarrow t$$ and $$f \circ g : t' \rightarrow t'$$ Since there is only one morphism from any $$a$$ to $$t$$ or $$t'$$ then these two compositions must necessarily be the identity function.
(b) Since $$p, p'$$ are both products of $$a, b$$ then there is a unique morphism from $$c$$ to them ( $$h, h'$$ ). If t we have two morphisms $$\alpha : p \rightarrow p'$$ and $$\beta : p' \rightarrow p$$ . Since $$h, h'$$ are unique, we have:
$$\alpha \circ h : c \rightarrow p' = h'$$ $$\beta \circ h' : c \rightarrow p = h$$ $$\alpha \circ \beta \circ h' = h'$$ $$\beta \circ \alpha \circ h = h$$ And our only choice is for $$\alpha, \beta$$ be the identity functions.
(c) both proofs used the uniqueness of morphisms and the commuting diagrams. I think we could follow the same idea for the initial objects, since there is only one morphism from $$0 \rightarrow \$$.
Comment Source:(a) Let us define the functions \$$f : t \rightarrow t' \$$ and \$$g : t' \rightarrow t \$$ Their composition gives us \$$g \circ f : t \rightarrow t \$$ and \$$f \circ g : t' \rightarrow t' \$$ Since there is only one morphism from any \$$a \$$ to \$$t \$$ or \$$t' \$$ then these two compositions must necessarily be the identity function. (b) Since \$$p, p' \$$ are both products of \$$a, b \$$ then there is a unique morphism from \$$c \$$ to them ( \$$h, h' \$$ ). If t we have two morphisms \$$\alpha : p \rightarrow p' \$$ and \$$\beta : p' \rightarrow p \$$ . Since \$$h, h' \$$ are unique, we have: $$\alpha \circ h : c \rightarrow p' = h'$$ $$\beta \circ h' : c \rightarrow p = h$$ $$\alpha \circ \beta \circ h' = h'$$ $$\beta \circ \alpha \circ h = h$$ And our only choice is for \$$\alpha, \beta \$$ be the identity functions. (c) both proofs used the uniqueness of morphisms and the commuting diagrams. I think we could follow the same idea for the initial objects, since there is only one morphism from \$$0 \rightarrow \ \$$.
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# Tag Info
7
Flavor is entirely orthogonal to color - the gluon(s) neither "know" nor "care" about flavor, but the existence of different quarks still leads to phenomena descending from the strong force that you wouldn't get without them: What remains of the strong force on the scale between nucleons is often called the residual strong force and can ...
3
Looks like the classic "catch my sloppiness" exercise on Schwartz. (My students got extra credit for those). I edited your question to drop P&S in favor of S, clearly your intention. Let's only deal with $F_\pi\sim 93$MeV, to avoid confusion. In your (1), you took his τs to be Pauli σs, when he clearly takes them to be the real SU(2) ...
2
No, "lifetime of an up quark" is utterly meaningless (at least here, but I'd be hard pressed to find legitimate contexts for it...). The lifetime discussed is that of a neutral pion, decaying by the F diagram (sorry) In words, the pion "resolves" to virtual states of its valence quarks, u or d, which then couple to two real photons, to ...
1
Just to state the result for the beta-function associated to QCD $$\beta = -\frac{g^2}{32 \pi^2} \left(\frac{11}{3}N_c - \frac{2}{3}N_f \right)$$ in which $N_c$ is the amount of colours and $N_f$ is the amount of flavours. Essentially, both terms boil down to antiscreening and screening respectively. A single quark can be ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient
Khanlar R Mamedov and Ozge Akcay*
Author Affiliations
Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, 33343, Turkey
For all author emails, please log on.
Boundary Value Problems 2014, 2014:110 doi:10.1186/1687-2770-2014-110
Received: 30 November 2013 Accepted: 25 April 2014 Published: 13 May 2014
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
### Abstract
In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.
MSC: 34A55, 34L40.
##### Keywords:
Dirac operator; inverse problem; necessary and sufficient condition
### 1 Introduction
In this paper, we consider the boundary value problem generated by the system of Dirac equations on the finite interval :
(1)
with boundary conditions
(2)
where
, are real valued functions, , , λ is a spectral parameter,
and .
The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1,2]. The reconstruction of the potential from one spectrum and norming constants was investigated in [3]. For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4-6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7,8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in [9] and discontinuity conditions inside an interval were worked out in [10,11]. The inverse problem for weighted Dirac equations was obtained in [12]. The reconstruction of the potential by the spectral function was given in [13]. For the Dirac operator with peculiarity, the inverse problem was found in [14]. Inverse nodal problems for the Dirac operator were examined in [15,16]. In the case of potentials that belong entrywise to , for some , the inverse spectral problem for the Dirac operator was studied in [17], and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method [18] was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in [19]. Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20-22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23,24] and discontinuous conditions inside an interval were obtained in [25]. In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.
In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let and () be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities () are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data () to indicate a method of determining the potential and to find necessary and sufficient conditions for () to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.
We give a brief account of the contents of this paper in the following section.
### 2 Preliminaries
Let be solution of the system (1) satisfying the initial conditions
The solution has an integral representation [26] as follows:
(3)
where
is a quadratic matrix function and is the solution of the problem
(4)
Equation (4) gives the relation between the kernel and the coefficient of (1). Let be solutions of the system (1) satisfying the initial conditions
The characteristic function of the problem (1), (2) is
(5)
where is the Wronskian of the solutions and and independent of . The zeros of the characteristic function coincide with the eigenvalues of the boundary value problem (1), (2). The functions and are eigenfunctions and there exists a sequence such that
(6)
Denote the normalizing numbers by
The following relation is valid:
(7)
where . In fact, since and are solutions of the problem (1), (2), we get
Multiplying the equations by , , , , respectively, adding them together, integrating from 0 to π and using the condition (2),
is found. From (6) as , we obtain
The following two theorems are obtained by Huseynov and Latifova in [27].
Theorem 1 (i) The boundary value problem (1), (2) has a countable set of simple eigenvalues () where
(8)
(ii) The eigen vector-functions of problem (1), (2) can be represented in the form
(iii) The normalizing numbers of problem (1), (2) have the form
(9)
Theorem 2 (i) The system of eigen vector-functions () of problem (1), (2) is complete in space.
(ii) Letbe an absolutely continuous vector-function on the segmentand. Then
(10)
moreover, the series converges uniformly with respect to.
(iii) Forseries (10) converges in; moreover, the Parseval equality holds:
(11)
From [27], the following inequality holds:
(12)
where is a positive number and this inequality is valid in the domain
where () are zeros of the function and δ is a sufficiently small number.
In Section 3, the fundamental equation
is derived by using the method by Gelfand-Levitan-Marchenko, where
and
In Section 4, we show that the fundamental equation has a unique solution and the boundary value problem (1), (2) can be uniquely determined from the spectral data. In Section 5, the result is obtained from Lemma 6 that the function defined by (3) satisfies the equation
where
where is the solution of the fundamental equation. In Lemma 7, using the fundamental equation, the Parseval equality
is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that () are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function by the spectral data ().
Note that throughout this paper, denotes the transposed matrix of ϕ.
### 3 Fundamental equation
Theorem 3For each fixedthe kernelfrom the representation (3) satisfies the following equation:
(13)
where
(14)
and
(15)
whereandare, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when.
Proof According to (3) we have
(16)
It follows from (3) and (16) that
and
Using the last two equalities, we obtain
or
(17)
where
It is easily found by using (14) and (15) that
(18)
Let . Then according to the expansion formula (10) in Theorem 2, we obtain uniformly on
(19)
From (18), we find
(20)
It follows from (3) that
(21)
Taking into account (21) and expansion formula (10) in Theorem 2, we get
Substituting , we obtain
(22)
Now, we calculate
(23)
Using (7) and the residue theorem, we get
(24)
where is oriented counter-clockwise, N is a sufficiently large number. Taking into account the asymptotic formulas as
and the relations ([20], Lemma 1.3.1)
it follows from (12) and (24) that
(25)
Thus, using (17), (19), (20), (22) (23), and (25), we find
Since can be chosen arbitrarily,
is obtained. □
### 4 Uniqueness
Lemma 4For each fixed, (13) has a unique solution.
Proof When , (13) can be rewritten as
where
(26)
Now, we shall prove that is invertible, i.e. has a bounded inverse in .
Consider the equation , . From this and (24),
We show that
In fact,
Thus, the operator is invertible in . Therefore the fundamental equation (13) is equivalent to
and is completely continuous in . Then it is sufficient to prove that the equation
(27)
has only the trivial solution . Let be a non-trivial solution of (27). Then
It follows from (14) that
Using (21), we get
Substituting into the last two integrals, we obtain
(28)
Using the Parseval equality,
it follows from (28) that
Since the system () is complete in , we have , i.e.. For invertible in , is obtained. □
Theorem 5Letandbe two boundary value problems and
Then
Proof According to (14) and (15), and . Then, from the fundamental equation (13), we have . It follows from (4) that a.e. on . □
### 5 Reconstruction by spectral data
Let the real numbers () of the form (8) and (9) be given. Using these numbers, we construct the functions and by (14) and (15) and determine from the fundamental equation (13).
Now, let us construct the function by (3) and the function by (4). From [2], and have a derivative in both variables and these derivatives belong to .
Lemma 6The following relations hold:
(29)
(30)
Proof Differentiating to x and y, (13), respectively, we get
(31)
(32)
It follows from (14) and (15) that
(33)
(34)
and using the fundamental equation (13), we obtain
(35)
Multiplying (31) on the left by B and we get
(36)
and multiplying (32) on the right by B and we have
(37)
Adding (36) and (37) and using (34), we find
(38)
From (33), we get
(39)
Integrating by parts and from (35)
(40)
is obtained. Substituting (40) into (38) and dividing by , we have
(41)
Multiplying (13) on the left by in the form of (4) and adding to (41)
(42)
is obtained. Setting
we can rewrite (42) as follows:
(43)
According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.
(44)
Differentiating (3) and multiplying on the left by B, we have
(45)
On the other hand, multiplying (3) on the left by and then integrating by parts and using (35), we find
(46)
It follows from (45) and (46) that
Taking into account (4) and (44),
is obtained. For , from (3) we get (30). □
Lemma 7For each function, the following relation holds:
(47)
Proof It follows from (3) and (21) that
(48)
Using the expression
the fundamental equation (13) is transformed into the following form:
(49)
From (48), we get
(50)
and for the kernel we have the identity
(51)
Denote
and using (48) it is transformed into the following form:
where
(52)
Similarly, in view of (50), we have
(53)
According to (52),
It follows from (49) and (51) that
(54)
From (18) and the Parseval equality we obtain
Taking into account (54), we have
whence, by (52) and (53),
is obtained, i.e., (47) is valid. □
Corollary 8For any functionand, the following relation holds:
(55)
Lemma 9The relation
(56)
is valid.
Proof (1) Let . Consider the series
(57)
where
(58)
Using Lemma 6 and integrating by parts, we get
Applying the asymptotic formulas in Theorem 1, is found. Consequently the series (57) converges absolutely and uniformly on . According to (55) and (58), we have
Since is arbitrary, is obtained, i.e.
(59)
(2) Fix and assume . Then, by virtue of (59),
where
The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □
Lemma 10For allthe equality
is valid.
Proof It is easily found that
According to (56), we get
(60)
We shall prove that for any n, . Assume the contrary, i.e. there exists m such that . Then for , it follows from (60) that . On the other hand, since as
. This contradicts the condition , . Hence, for any n. From (60), we have
Thus, we get , for any n. Since
we find , and then is obtained. □
Theorem 11For the sequences () to be the spectral data for a certain boundary value problemof the form (1), (2) with, it is necessary and sufficient that the relations (8) and (9) hold.
Proof Necessity of the problem is proved in [27]. Let us prove the sufficiency. Let the real numbers () of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers () are spectral data for the constructed boundary value problem . The theorem is proved. □
The algorithm of the construction of the function by the spectral data () follows from the proof of the theorem:
(1) By the given numbers () the functions and are constructed, respectively, by (14) and (15).
(2) The function is found from (13).
(3) is calculated by (4).
### Competing interests
The authors declare that they have no competing interests.
### Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
### Acknowledgements
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
### References
1. Gasymov, MG, Levitan, BM: The inverse problem for the Dirac system. Dokl. Akad. Nauk SSSR. 167, 967–970 (1966)
2. Gasymov, MG, Dzabiev, TT: Solution of the inverse problem by two spectra for the Dirac equation on a finite interval. Dokl. Akad. Nauk Azerb. SSR. 22(7), 3–6 (1966)
3. Dzabiev, TT: The inverse problem for the Dirac equation with a singularity. Dokl. Akad. Nauk Azerb. SSR. 22(11), 8–12 (1966)
4. Misyura, TV: Characteristics of spectrums of periodical and antiperiodical boundary value problems generated by Dirac operation. II. Teoriya funktsiy, funk. analiz i ikh prilozheiniya, pp. 102–109. Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkov (1979)
5. Nabiev, IM: Solution of a class of inverse problems for the Dirac operator. Trans. Natl. Acad. Sci. Azerb.. 21(1), 146–157 (2001)
6. Nabiev, IM: Characteristic of spectral data of Dirac operators. Trans. Natl. Acad. Sci. Azerb.. 24(7), 161–166 (2004)
7. Sakhnovich, A: Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems. Inverse Probl.. 22(6), 2083–2101 (2006). Publisher Full Text
8. Fritzsche, B, Kirstein, B, Roitberg, IY, Sakhnovich, A: Skew-self-adjoint Dirac system with a rectangular matrix potential: Weyl theory, direct and inverse problems. Integral Equ. Oper. Theory. 74(2), 163–187 (2012). Publisher Full Text
9. Latifova, AR: The inverse problem of one class of Dirac operators with discontinuous coefficients by the Weyl function. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb.. 22(30), 65–70 (2005)
10. Amirov, RK: On system of Dirac differential equations with discontinuity conditions inside an interval. Ukr. Math. J.. 57(5), 712–727 (2005). Publisher Full Text
11. Huseynov, HM, Latifova, AR: The main equation for the system of Dirac equation with discontinuity conditions interior to interval. Trans. Natl. Acad. Sci. Azerb.. 28(1), 63–76 (2008)
12. Watson, BA: Inverse spectral problems for weighted Dirac systems. Inverse Probl.. 15(3), 793–805 (1999). Publisher Full Text
13. Mamedov, SG: The inverse boundary value problem on a finite interval for Dirac’s system of equations. Azerb. Gos. Univ. Ucen. Zap. Ser. Fiz-Mat. Nauk. 5, 61–67 (1975)
14. Panakhov, ES: Some aspects inverse problem for Dirac operator with peculiarity. Trans. Natl. Acad. Sci. Azerb.. 3, 39–44 (1995)
15. Yang, CF, Huang, ZY: Reconstruction of the Dirac operator from nodal data. Integral Equ. Oper. Theory. 66, 539–551 (2010). Publisher Full Text
16. Yang, CF, Pivovarchik, VN: Inverse nodal problem for Dirac system with spectral parameter in boundary conditions. Complex Anal. Oper. Theory. 7, 1211–1230 (2013). Publisher Full Text
17. Albeverio, S, Hryniv, R, Mykytyuk, Y: Inverse spectral problems for Dirac operators with summable potentials. Russ. J. Math. Phys.. 12(14), 406–423 (2005)
18. Krein, MG: On integral equations generating differential equations of the second order. Dokl. Akad. Nauk SSSR. 97, 21–24 (1954)
19. Mamedov, KR, Çöl, A: On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci.. 35(14), 1712–1720 (2012). Publisher Full Text
20. Marchenko, VA: Sturm-Liouville Operators and Applications, Am. Math. Soc., Providence (2011)
21. Freiling, G, Yurko, V: Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York (2008)
22. Guliyev, NJ: Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl.. 21, 1315–1330 (2005). Publisher Full Text
23. Mamedov, KR, Cetinkaya, FA: Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition. Bound. Value Probl. (2013). BioMed Central Full Text
24. Akhmedova, EN, Huseynov, HM: On solution of the inverse Sturm-Liouville problem with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb.. 27(7), 33–44 (2007)
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27. Huseynov, HM, Latifova, AR: On eigenvalues and eigenfunctions of one class of Dirac operators with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb.. 24(1), 103–112 (2004)
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# Introduction to the textreuse package
#### 2020-05-15
The textreuse package provides classes and functions to detect document similarity and text reuse in text corpora. This introductory vignette provides details on the TextReuseTextDocument and TextReuseCorpus classes, as well as functions for tokenizing, hashing, and measuring similarity. See the pairwise, minhash/LSH, or alignment vignettes for details on solving text similarity problems.
vignette("textreuse-pairwise", package = "textreuse")
vignette("textreuse-minhash", package = "textreuse")
vignette("textreuse-alignment", package = "textreuse")
For these vignette we will use a small corpus of eight documents published by the American Tract Society and available from the Internet Archive. The full corpus is also available to be downloaded if you wish to test the package.
## TextReuse classes
### TextReuseTextDocument
The most basic class provided by this package is the TextReuseTextDocument class. This class contains the text of a document and its metadata. When the document is loaded, the text is also tokenized. (See the section on tokenizers below.) Those tokens are then hashed using a hash function. By default the hashes are retained and the tokens are discarded, since using only hashes results in a significant memory savings.
Here we load a file into a TextReuseTextDocument and tokenize it into shingled n-grams, adding an option to retain the tokens.
library(textreuse)
file <- system.file("extdata/ats/remember00palm.txt",
package = "textreuse")
doc <- TextReuseTextDocument(file = file, meta = list("publisher" = "ATS"),
tokenizer = tokenize_ngrams, n = 5,
keep_tokens = TRUE)
doc
## TextReuseTextDocument
## file : /private/var/folders/vn/mfg6b6pd33s8grnh8nwcf43r0000gn/T/RtmpDMTxet/Rinst5916123cb63e/textreuse/extdata/ats/remember00palm.txt
## hash_func : hash_string
## id : remember00palm
## publisher : ATS
## tokenizer : tokenize_ngrams
## content : Remember
## By
## Rat Palmer.
## Boston:
##
## THE AMERICAN TRACT SOCI]
##
## Depositories, 28 Cornhill, Boston ; and 13 Biblb House,
## Astor Place, New York.
## Entered, according to Act of Congress, in the year 1865
We can see details of the document with accessor functions. These are derived from the S3 virtual class TextDocument in the NLP package. Notice that an ID has been assigned to the document based on the filename (without the extension). The name of the tokenizer and hash functions are also saved in the metadata.
meta(doc)
## $file ## [1] "/private/var/folders/vn/mfg6b6pd33s8grnh8nwcf43r0000gn/T/RtmpDMTxet/Rinst5916123cb63e/textreuse/extdata/ats/remember00palm.txt" ## ##$hash_func
## [1] "hash_string"
##
## $id ## [1] "remember00palm" ## ##$publisher
## [1] "ATS"
##
## \$tokenizer
## [1] "tokenize_ngrams"
meta(doc, "id")
## [1] "remember00palm"
meta(doc, "date") <- 1865
head(tokens(doc))
## [1] "remember by rat palmer boston"
## [2] "by rat palmer boston the"
## [3] "rat palmer boston the american"
## [4] "palmer boston the american tract"
## [5] "boston the american tract soci"
## [6] "the american tract soci depositories"
head(hashes(doc))
## [1] -96275747 -1721204321 707361410 -626087009 -532862870 141807655
wordcount(doc)
## [1] 11399
The tokens() and hashes() function return the tokens and hashes associated with the document. The meta() function returns a named list of all the metadata fields. If that function is called with a specific ID, as in meta(doc, "myfield"), then the value for only that field is returned. You can also assign to the metadata as a whole or a specific field, as in the example above.
In addition the content() function provides the unprocessed text of the document.
The assumption is that is that you want to tokenize and hash the tokens from the start. If, however, you wish to do any of those steps yourself, you can load a document with tokenizer = NULL, then use tokenize() or rehash() to recompute the tokens and hashes.
Note that a TextReuseTextDocument can actually contain two kinds of hashes. The hashes() accessor gives you integer representations of each of the tokens in the document: if there are 100,000 tokens in the document, there will be 100,000 hashes. The minhashes() accessor gives you a signature that represents the document as a whole but not the specific tokens within it. See the minhash vignette for details: vignette("textreuse-minhash").
### TextReuseCorpus
The class TextReuseCorpus provides a list of TextReuseTextDocuments. It derives from the S3 virtual class Corpus in the tm package. It can be created from a directory of files (or by providing a vector of paths to files).
dir <- system.file("extdata/ats", package = "textreuse")
corpus <- TextReuseCorpus(dir = dir, tokenizer = tokenize_ngrams, n = 5,
progress = FALSE)
corpus
## TextReuseCorpus
## Number of documents: 8
## hash_func : hash_string
## tokenizer : tokenize_ngrams
The names of the items in a TextReuseCorpus are the IDs of the documents. You can use these IDs to subset the corpus or to retrieve specific documents.
names(corpus)
## [1] "calltounconv00baxt" "gospeltruth00whit"
## [3] "lifeofrevrichard00baxt" "memoirjamesbrai00ricegoog"
## [5] "practicalthought00nev" "remember00palm"
## [7] "remembermeorholy00palm" "thoughtsonpopery00nevi"
corpus[["remember00palm"]]
## TextReuseTextDocument
## file : /private/var/folders/vn/mfg6b6pd33s8grnh8nwcf43r0000gn/T/RtmpDMTxet/Rinst5916123cb63e/textreuse/extdata/ats/remember00palm.txt
## hash_func : hash_string
## id : remember00palm
## minhash_func :
## tokenizer : tokenize_ngrams
## content : Remember
## By
## Rat Palmer.
## Boston:
##
## THE AMERICAN TRACT SOCI]
##
## Depositories, 28 Cornhill, Boston ; and 13 Biblb House,
## Astor Place, New York.
## Entered, according to Act of Congress, in the year 1865
corpus[c("calltounconv00baxt", "lifeofrevrichard00baxt")]
## TextReuseCorpus
## Number of documents: 2
## hash_func : hash_string
## tokenizer : tokenize_ngrams
Accessor functions such as meta(), tokens(), hashes(), and wordcount() have methods that work on corpora.
wordcount(corpus)
## calltounconv00baxt gospeltruth00whit lifeofrevrichard00baxt
## 134616 16593 44283
## memoirjamesbrai00ricegoog practicalthought00nev remember00palm
## 131939 124544 11399
## remembermeorholy00palm thoughtsonpopery00nevi
## 11532 64758
Note that when creating a corpus, very short or empty documents will be skipped with a warning. A document must have enough words to create at least two n-grams. For example, if five-grams are desired, then the document must have at least six words.
## Tokenizers
One of the steps that is performed when loading a TextReuseTextDocument, either individual or in a corpus, is tokenization. Tokenization breaks up a text into pieces, often overlapping. These pieces are the features which are compared when measuring document similarity.
The textreuse package provides a number of tokenizers.
text <- "How many roads must a man walk down\nBefore you'll call him a man?"
tokenize_words(text)
## [1] "how" "many" "roads" "must" "a" "man" "walk" "down"
## [9] "before" "you'll" "call" "him" "a" "man"
tokenize_sentences(text)
## [1] "how many roads must a man walk down" "before you ll call him a man"
tokenize_ngrams(text, n = 3)
## [1] "how many roads" "many roads must" "roads must a"
## [4] "must a man" "a man walk" "man walk down"
## [7] "walk down before" "down before you'll" "before you'll call"
## [10] "you'll call him" "call him a" "him a man"
tokenize_skip_ngrams(text, n = 3, k = 2)
## [1] "how must walk" "many a down" "roads man before"
## [4] "must walk you'll" "a down call" "man before him"
## [7] "walk you'll a" "down call man" "how roads a"
## [10] "many must man" "roads a walk" "must man down"
## [13] "a walk before" "man down you'll" "walk before call"
## [16] "down you'll him" "before call a" "you'll him man"
## [22] "must a man" "a man walk" "man walk down"
## [25] "walk down before" "down before you'll" "before you'll call"
## [28] "you'll call him" "call him a" "him a man"
You can write your own tokenizers or use tokenizers from other packages. They should accept a character vector as their first argument.
As an example, we will write a tokenizer function using the package which splits a text on new lines, perhaps useful for poetry. Notice that the function takes a single string and returns a character vector with one element for each line. (A more robust tokenizer might strip blank lines and punctuation, include an option for lowercasing the text, and check for the validity of arguments.)
poem <- "Roses are red\nViolets are blue\nI like using R\nAnd you should too"
cat(poem)
## Roses are red
## Violets are blue
## I like using R
## And you should too
tokenize_lines <- function(string) {
stringr::str_split(string, "\n+")[[1]]
}
tokenize_lines(poem)
## [1] "Roses are red" "Violets are blue" "I like using R"
## [4] "And you should too"
## Hash functions
This package provides one function to hash tokens to integers, hash_string().
hash_string(tokenize_words(text))
## [1] -78131211 -909288800 -647481819 -909500956 -1640531430 -78235283
## [7] -904724921 -889252160 317438038 937035765 -890718890 -78132909
## [13] -1640531430 -78235283
You can write your own hash functions, or use those provided by the digest package.
## Comparison functions
This package provides a number of comparison functions for measuring similarity. These functions take either a set (in which each token is counted one time) or a bag (in which each token is counted as many times as it appears) and compares it to another set or bag.
a <- tokenize_words(paste("How does it feel, how does it feel?",
"To be without a home",
"Like a complete unknown, like a rolling stone"))
b <- tokenize_words(paste("How does it feel, how does it feel?",
"To be on your own, with no direction home",
"A complete unknown, like a rolling stone"))
jaccard_similarity(a, b)
## [1] 0.65
jaccard_dissimilarity(a, b)
## [1] 0.35
jaccard_bag_similarity(a, b)
## [1] 0.4
ratio_of_matches(a, b)
## [1] 0.75
See the documentation for ?similarity-functions for details on what is measured with these functions.
You can write your own similarity functions, which should accept two sets or bags, a and b, should work on both character and numeric vectors, since they are used with either tokens or hashes of tokens, and should return a single numeric score for the comparison. You will need to implement a method for the TextReuseTextDocument class.
## Parallelization
This package will use multiple cores for a few functions is an option is set. This only benefits the corpus loading and tokenizing functions, which are often the slowest parts of an analysis. This is implemented with the parallel package, and does not work on Windows machines. (Regardless of the options set, this package will never attempt to parallelize computations on Windows.)
To use the parallel option, you must specify the number of CPU cores that you wish to use:
options("mc.cores" = 4L)
If that option is set, this package will use multiple cores when possible.
You can figure out how many cores your computer has with parallel::detectCores(). See help(package = "parallel") for more details.
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# What is the value of ab^2 ?
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What is the value of ab^2 ? [#permalink] 09 Jun 2010, 05:22
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What is the value of ab^2?
(1) a = b - 1
(2) a = b^2 - 1
[Reveal] Spoiler: OA
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Re: Data sufficiency question on equations from MGMAT [#permalink] 09 Jun 2010, 05:44
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The following question is located in the Equations, Inequalities & VICs Manhattan Guide book on page 135. I just need some clarification on the last part of their working out of the answer. I'll post this below in a spoiler section, so please click to find where I am a bit stuck.
What is the value of $$ab^2$$?
(1) $$a=b-1$$
(2) $$a=b^2-1$$
[Reveal] Spoiler:
MGMAT provides the following working out:
"if we combine statements (1) and (2), we find that a and b can still have 2 different values.
$$b-1=b^2-1$$
$$b=b^2$$
$$0=b^2-b$$
$$0=b(b-1)$$
b= 0 or 1
a= -1 or 0
So $$b=0$$ when $$a=-1$$, and $$b=1$$ when $$a=0$$. However, in either case $$ab^2=0$$. Therefore statements (1) and (2) combined are sufficient.
Please explain how b can equal 0 or 1?
Could I deduce b=1 using this equation from above: $$0=b(b-1)$$. Take out the b on the outside of the brackets and solve the remaining $$0=(b-1)$$ equation, therefore $$b=1$$?
Likewise, how do I deduce that b=0 also? Is it from this equation $$0=b^2-b$$?
What I seem to find difficult is how would you know that b has two answers. THANKS IN ADVANCE!
What is the value of $$ab^2$$?
(1) $$a=b-1$$. Clearly insufficient.
(2) $$a=b^2-1$$. Clearly insufficient.
(1)+(2) $$b-1=b^2-1$$ --> $$b^2=b$$ --> $$b^2-b=0$$ --> $$b(b-1)=0$$. For the product of two numbers ($$b$$ and $$b-1$$) to be zero one of them (or both) must be zero. So either:
$$b=0$$ --> $$a=b-1=-1$$ --> $$ab^2=0$$;
or
$$b-1=0$$, $$b=1$$ --> $$a=b-1=0$$ --> $$ab^2=0$$.
The same answer for both cases. Sufficient.
Hope it helps.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 09 Jun 2010, 12:12
Thanks for the quick reply Bunuel, kudos for the explanation.
What I am slightly concerned about is that if a similar question appears on the GMAT, I may fail to spot the fact that b can have two answers. If $$b^2-b=0$$ then surely b must equal 0. Why do we then go and factorise to produce a new expression: $$b(b-1)=0$$? And when solving for b using this latter expression, why do we drop the first b and do $$b-1=0$$ therefore $$b=1$$ instead of $$b(b-1)=0$$ and solve?
Sorry if this sounds convoluted but I'm probably overlooking a rule somewhere, which essentially needs clarification.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 09 Jun 2010, 12:45
Expert's post
Thanks for the quick reply Bunuel, kudos for the explanation.
What I am slightly concerned about is that if a similar question appears on the GMAT, I may fail to spot the fact that b can have two answers. If $$b^2-b=0$$ then surely b must equal 0. Why do we then go and factorise to produce a new expression: $$b(b-1)=0$$? And when solving for b using this latter expression, why do we drop the first b and do $$b-1=0$$ therefore $$b=1$$ instead of $$b(b-1)=0$$ and solve?
Sorry if this sounds convoluted but I'm probably overlooking a rule somewhere, which essentially needs clarification.
First of all $$b^2-b=0$$ is a quadratic equation and it can have 2 solutions.
Next: $$b^2-b=0$$ means that either $$b=0$$ (0^2-0=0) or $$b=1$$ (1^2-1=0) (so it's not necessary $$b$$ to be zero). We did not drop $$b=0$$ we just found the second solution $$b=1$$.
You might need to revise basics of algebra.
Hope it helps.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 10 Jun 2010, 09:42
Ah got it now , silly mistake really. As it's been so long since I've touched algebra that I thought all quadratics looked like $$Ax^2+ Bx - C= 0$$ so the $$b(b-1)=0$$ threw me a little. I realize now that $$b(b-1)=0$$ is the same as saying $$(b) . (b-1)=0$$ which follows the concept of what a quadratic looks like after its been factorised ie $$(x+1) . (x-1)=0$$. Thus, we can solve the two values for b by taking each expression individually (b & (b-1)) and equating them each to zero.
Thanks for your help Bunuel, very kind!
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Re: Data sufficiency question on equations from MGMAT [#permalink] 12 Jun 2010, 22:50
Bunuel,
Please can you explain my confusion?
If I place the equation as b-1 = b^2-1 and solve it by thinking
b-1 = (b-1) (b+1) ==> b = -1..... which is only one solution... is this method wrong?? or it is ok? Please kindly tell me if i am making some mistake.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 13 Jun 2010, 03:55
Expert's post
amitjash wrote:
Bunuel,
Please can you explain my confusion?
If I place the equation as b-1 = b^2-1 and solve it by thinking
b-1 = (b-1) (b+1) ==> b = -1..... which is only one solution... is this method wrong?? or it is ok? Please kindly tell me if i am making some mistake.
First of all the roots of the equation $$b-1 = b^2-1$$ are $$b=0$$ and $$b=1$$ (see the solution in my post), if you got these roots you solved the equation correctly and if you got different roots you solved incorrectly.
Next, I don't understand why aren't you cancelling (-1) from $$b-1=b^2-1$$ --> $$b=b^2$$.
But anyway if you don't and we proceed the way you are doing: $$b-1=b^2-1$$ --> $$b-1=(b-1)(b+1)$$. Now how is $$b=-1$$ the root of this equation? $$b-1=-2\neq{(b-1)(b+1)}=0$$.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 13 Jun 2010, 06:29
Sorry, it was an error from my side. I mean to say if I arrive with only one solution ==> b=0, accroding to the method i mentioned. Then why it is considered wrong?
b-1 = b^2 - 1
b-1 = (b-1) (b+1)
1 = b+1 ==> b=0
Why I am not cancelling 1 on each side? I dont know... That is the first thought when i looked at the equation. I want to make a note in my mind why it is wrong so that i dont make these mistakes on the test day.
Thanks for your ever woderful support.
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Re: Data sufficiency question on equations from MGMAT [#permalink] 13 Jun 2010, 06:43
Expert's post
amitjash wrote:
Sorry, it was an error from my side. I mean to say if I arrive with only one solution ==> b=0, accroding to the method i mentioned. Then why it is considered wrong?
b-1 = b^2 - 1
b-1 = (b-1) (b+1)
1 = b+1 ==> b=0
Why I am not cancelling 1 on each side? I dont know... That is the first thought when i looked at the equation. I want to make a note in my mind why it is wrong so that i dont make these mistakes on the test day.
Thanks for your ever woderful support.
I see. When you write $$1 = b+1$$ after $$b-1 = (b-1) (b+1)$$ what you are actually doing is reducing (dividing) the equation by $$b-1$$ but we can not do that as it can equal to zero.
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.
So basically when you divide by $$b-1$$ you assume that it doesn't equal to zero, thus missing the second valid solution for $$b-1 = (b-1) (b+1)$$ which is $$b-1=0$$ --> $$b=1$$.
Again if we proceed the way you are doing $$b-1 = (b-1) (b+1)$$ --> $$(b-1)-(b-1) (b+1)=0$$ --> factor out $$b-1$$ --> $$(b-1)(1-b-1)=0$$ --> $$(b-1)b=0$$ (the type of equation you'd receive right away if you'd cancel out $$-1$$) --> $$b=0$$ or $$b=1$$.
Hope it's clear.
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What is the value of ab? [#permalink] 20 Nov 2013, 10:59
What is the value of ab?
(1) a = b - 1
(2) a = b^2 - 1
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Re: what is the value of ab? [#permalink] 20 Nov 2013, 11:06
1)Doesn't give anything about a and b.So not sufficient.Eliminate A and D
2)Doesn't give anything about a and b.So not sufficient. Eliminate B
1+2) (b-1)=b^2-1
(b-1)=(b-1)(b+1)
1=b+1
b=0
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Re: what is the value of ab? [#permalink] 20 Nov 2013, 11:12
Expert's post
vikrantgulia wrote:
1)Doesn't give anything about a and b.So not sufficient.Eliminate A and D
2)Doesn't give anything about a and b.So not sufficient. Eliminate B
1+2) (b-1)=b^2-1
(b-1)=(b-1)(b+1)
1=b+1
b=0
Just a caveat:Note the part that is highlighted,
You can't cancel out (b-1) on both the sides, as you don't know whether b=1 or not.
Thus, either b=1 or b=0. In either case, you get ab=0.
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Re: what is the value of ab? [#permalink] 20 Nov 2013, 12:36
mau5 wrote:
vikrantgulia wrote:
1)Doesn't give anything about a and b.So not sufficient.Eliminate A and D
2)Doesn't give anything about a and b.So not sufficient. Eliminate B
1+2) (b-1)=b^2-1
(b-1)=(b-1)(b+1)
1=b+1
b=0
Just a caveat:Note the part that is highlighted,
You can't cancel out (b-1) on both the sides, as you don't know whether b=1 or not.
Thus, either b=1 or b=0. In either case, you get ab=0.
Just to add for St 2 we have
a = (b-1)(b+1) => a=a(b+1) { from St 1 we have a=b-1}
a= a(b+1)
1=b+1 => b=0, so ab is 0, sufficient.
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Re: What is the value of ab? [#permalink] 21 Nov 2013, 02:10
Expert's post
vikrantgulia wrote:
What is the value of ab?
(1) a = b - 1
(2) a = b^2 - 1
Merging similar topics. Please refer to the solutions above.
Quote:
What is the value of ab?
(1) a = b + 1
(2) a^2 = b + 1
Discussed here: what-is-the-value-of-ab-143841.html
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Re: what is the value of ab? [#permalink] 21 Nov 2013, 02:11
Expert's post
viksingh15 wrote:
mau5 wrote:
vikrantgulia wrote:
1)Doesn't give anything about a and b.So not sufficient.Eliminate A and D
2)Doesn't give anything about a and b.So not sufficient. Eliminate B
1+2) (b-1)=b^2-1
(b-1)=(b-1)(b+1)
1=b+1
b=0
Just a caveat:Note the part that is highlighted,
You can't cancel out (b-1) on both the sides, as you don't know whether b=1 or not.
Thus, either b=1 or b=0. In either case, you get ab=0.
Just to add for St 2 we have
a = (b-1)(b+1) => a=a(b+1) { from St 1 we have a=b-1}
a= a(b+1)
1=b+1 => b=0,
so ab is 0, sufficient.
You cannot reduce a=a(b+1) by a and write 1=b+1 because you exclude a possible solution a=0. Check here for more: what-is-the-value-of-ab-95543.html#p737359
Hope this helps.
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Re: What is the value of ab^2 ? [#permalink] 30 Jun 2015, 22:07
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Re: What is the value of ab^2 ? [#permalink] 30 Jun 2015, 22:07
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# Figure 1
Combined particle identification in the TPC and TOF for data from Pb-Pb collisions at $\snn$ = 2.76 TeV, shown as a two-dimensional plot. The PID signals are expressed in terms of the deviation from the expected response for pions in each detector.
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# American Institute of Mathematical Sciences
October 2013, 33(10): 4549-4566. doi: 10.3934/dcds.2013.33.4549
## A Liouville theorem of degenerate elliptic equation and its application
1 School of Mathematical Science, Fudan University, Shanghai, 200433, China
Received November 2012 Revised February 2013 Published April 2013
In this paper, we apply the moving plane method to the following degenerate elliptic equation arising from isometric embedding,\begin{equation*} yu_{yy}+au_y+\Delta_x u+u^\alpha=0\text{ in } \mathbb R^{n+1}_+,n\geq 1. \end{equation*} We get a Liouville theorem for subcritical case and classify the solutions for critical case. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.
Citation: Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
##### References:
[1] A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V.,, Amer. Math. Soc. Transl.(2), 21 (1962), 412. Google Scholar [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar [3] W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [4] W.-X. Chen and C.-M. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [5] W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations,", AIMS, (2010). Google Scholar [6] S-Y A. Chang and P. C. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry,, Math. Research Letters, 4 (1997), 91. Google Scholar [7] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I(8), 5 (1956), 1. Google Scholar [8] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, Boundary problems in differential equations, (1960), 97. Google Scholar [9] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Comm. Math. Phys., 68 (1979), 209. Google Scholar [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb R^n$,, Mathematical analysis and applications, (1981), 369. Google Scholar [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar [12] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [13] J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces,, Chin. Ann. of Math., 2 (1981), 407. Google Scholar [14] J.-X. Hong and G.-G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications,, Int. Math. Res. Notices, 2012 (): 2889. Google Scholar [15] M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain,, (Russian) Dokl. Akad. Nauk SSSR, 77 (1951), 181. Google Scholar [16] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$,, Commment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar [17] C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar [18] C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301. Google Scholar [19] Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153. Google Scholar [20] O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,", Translated from the Russian by Paul C. Fife. Plenum Press, (1973). Google Scholar [21] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. Google Scholar [22] J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar [23] X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$,, Pacific J. Math., 225 (2006), 361. doi: 10.2140/pjm.2006.225.361. Google Scholar
show all references
##### References:
[1] A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V.,, Amer. Math. Soc. Transl.(2), 21 (1962), 412. Google Scholar [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar [3] W.-X. Chen, C.-M. Li and B. OU, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [4] W.-X. Chen and C.-M. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [5] W.-X. Chen and C.-M. Li, "Methods on Nonlinear Elliptic Equations,", AIMS, (2010). Google Scholar [6] S-Y A. Chang and P. C. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry,, Math. Research Letters, 4 (1997), 91. Google Scholar [7] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I(8), 5 (1956), 1. Google Scholar [8] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, Boundary problems in differential equations, (1960), 97. Google Scholar [9] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Comm. Math. Phys., 68 (1979), 209. Google Scholar [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb R^n$,, Mathematical analysis and applications, (1981), 369. Google Scholar [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar [12] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [13] J.-X. Hong, On boundary value problems for mixed equations with characteristic degenerate surfaces,, Chin. Ann. of Math., 2 (1981), 407. Google Scholar [14] J.-X. Hong and G.-G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications,, Int. Math. Res. Notices, 2012 (): 2889. Google Scholar [15] M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain,, (Russian) Dokl. Akad. Nauk SSSR, 77 (1951), 181. Google Scholar [16] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R^n$,, Commment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar [17] C.-M. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar [18] C.-M. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301. Google Scholar [19] Y.-Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153. Google Scholar [20] O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,", Translated from the Russian by Paul C. Fife. Plenum Press, (1973). Google Scholar [21] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. Google Scholar [22] J.-C. Wei and X.-W. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar [23] X.-W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$,, Pacific J. Math., 225 (2006), 361. doi: 10.2140/pjm.2006.225.361. Google Scholar
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2018 Impact Factor: 1.143
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Release - Unity
___________________________________
Unity the creation of Eddage, I have taken on the task of completing the skin where he left off. For more information you can see his original thread here.
___________________________________
watch gallery
___________________________________
The skinshortcuts addon is used to implement the main and sub menus on the home screen.
Unity is compatible with Helix and Isengard at this point, and should be installed using the repo link below.
___________________________________
I am still working through the last remaining skin script files. These should hopefully be completed soon.
Code:
------------------- Incomplete Files ------------------- script-NextAired-TVGuide.xml
Very nice concept.
Awesome! So glad you took this on. I always loved this skin. Have been dying to see it progress. Thank you!
Thanks for the Isengard updates. This has been my skin of choice for a while.
Leopold's Repository: Home of LibreELEC Dev Updater ...
Very nice.
[H]i-[d]eft [M]edia [K]een [V]ideosaurus
Kodi 3D MVC Hardware Options
Kodi A/V Samples wiki
3D: Zidoo X9S | Intel D34010WYK w/Kodi MVC build
4K: nVidia Shield TV (2nd gen)
Awesome skin. Thanks for your hardwork. I am enjoying it.
Is there a way to make the submenu appear once the parent menu is selected instead of clicking on the down arrow on the remote controller or keyboard?
Thanks again?
This skin is awesome, it should replace "confluence" in the next versions of kodi. I would like to see the android lollipop wallpaper replaced, the skin needs something more original.
(2015-09-20, 07:39)dibson Wrote: Awesome skin. Thanks for your hardwork. I am enjoying it.
Is there a way to make the submenu appear once the parent menu is selected instead of clicking on the down arrow on the remote controller or keyboard?
Thanks again?
I am still trying to figure out a way that looks good to be able to accomplish this. The problem is the dynamic size of the sub menu once skin.shortcuts started being used to allow customization of the main and sub menus. I'll be sure to update this thread if anything gets figured out.
Beutiful skin but...erm...I can't find the File Manager
Where is it hiding?
Panasonic 42G10 ♦ Onkyo TX SR507 ♦ Teufel Concept R2 ♦ TV Box (COOLEME MB1/Libreelec)
By default it is a submenu of System (once on System, hit down to bring up the submenu). If you have customized your main menu with skinshortcuts, its wherever you put it.
I had leftover settings from a previous Unity installation and they messed up my menus.
A skin reset fixed it.
Panasonic 42G10 ♦ Onkyo TX SR507 ♦ Teufel Concept R2 ♦ TV Box (COOLEME MB1/Libreelec)
This is the best skin ever!
do you still work on this?
Thanks
Yes, I am slowly finishing up the last couple of files.
While you are working on those last files, could you take a look at the following?
- When searching for subtitles, the info bar and the time\end time stay on screen and hide part of the list of subtitles and a large part of the filename
- The first and last home menu entries should be centered when selected. Now it is difficult to see if the last or the next-to-last item is selected.
- The info bar displays the info label instead of the info text. This happens when there is no tagline in the info screen (the genre of the film is displayed instead).
Now that I think of it, the problem might be in the info screen code. Is it supposed to show the tagline below the film title?
That's all and thanks for the skin
Panasonic 42G10 ♦ Onkyo TX SR507 ♦ Teufel Concept R2 ♦ TV Box (COOLEME MB1/Libreelec)
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Solution: Consider Lewis formulas A, B, and C: (a) Are A, B, and C co...
Question
Consider Lewis formulas A, B, and C:
(a) Are A, B, and C constitutional isomers, or are they resonance contributors?
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# Use of the tcl parameter for inductance change
#### Alec_t
Joined Sep 17, 2013
12,735
Out of curiosity I was trying to see how LTspice models inductance changes with temperature. According to the LTspice Help, inductors can have a temperature coefficient of inductance, specified by parameter tcl or Tcl.
However, on running a simple sim (below, using either parameter), LTS complains that neither parameter is recognised ! Also, the error message uses lower case for both parameters, so how would LTS (which is generally case-insensitive) distinguish the two parameters, even if it recognised them?
Am I missing something, or is this a bug?
#### eetech00
Joined Jun 8, 2013
3,095
Out of curiosity I was trying to see how LTspice models inductance changes with temperature. According to the LTspice Help, inductors can have a temperature coefficient of inductance, specified by parameter tcl or Tcl.
View attachment 258060
However, on running a simple sim (below, using either parameter), LTS complains that neither parameter is recognised ! Also, the error message uses lower case for both parameters, so how would LTS (which is generally case-insensitive) distinguish the two parameters, even if it recognised them?
View attachment 258059
Am I missing something, or is this a bug?
Hi
TCL should be TC1
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# Publications with P. Lücke
#### All publications by Philipp Lücke and S. Shelah
Click the Sh:-number to get to the papers' detail page (which may include pdf's of the paper).
Can't find an Sh:-number (in particular, an "F-number")? You can try your luck here.
number title
Sh:982 Lücke, P., & Shelah, S. (2012). External automorphisms of ultraproducts of finite models. Arch. Math. Logic, 51(3-4), 433–441.
Sh:1014 Lücke, P., & Shelah, S. (2014). Free groups and automorphism groups of infinite structures. Forum Math. Sigma, 2, e8, 18.
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Outlook: KAR Auction Services Inc Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Sell
Time series to forecast n: 06 Feb 2023 for (n+6 month)
Methodology : Modular Neural Network (DNN Layer)
## Abstract
KAR Auction Services Inc Common Stock prediction model is evaluated with Modular Neural Network (DNN Layer) and Stepwise Regression1,2,3,4 and it is concluded that the KAR stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell
## Key Points
1. Which neural network is best for prediction?
2. What is prediction model?
3. Why do we need predictive models?
## KAR Target Price Prediction Modeling Methodology
We consider KAR Auction Services Inc Common Stock Decision Process with Modular Neural Network (DNN Layer) where A is the set of discrete actions of KAR stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4
F(Stepwise Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (DNN Layer)) X S(n):→ (n+6 month) $∑ i = 1 n r i$
n:Time series to forecast
p:Price signals of KAR stock
j:Nash equilibria (Neural Network)
k:Dominated move
a:Best response for target price
For further technical information as per how our model work we invite you to visit the article below:
How do AC Investment Research machine learning (predictive) algorithms actually work?
## KAR Stock Forecast (Buy or Sell) for (n+6 month)
Sample Set: Neural Network
Stock/Index: KAR KAR Auction Services Inc Common Stock
Time series to forecast n: 06 Feb 2023 for (n+6 month)
According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell
X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)
Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)
Z axis (Grey to Black): *Technical Analysis%
## IFRS Reconciliation Adjustments for KAR Auction Services Inc Common Stock
1. When assessing a modified time value of money element, an entity must consider factors that could affect future contractual cash flows. For example, if an entity is assessing a bond with a five-year term and the variable interest rate is reset every six months to a five-year rate, the entity cannot conclude that the contractual cash flows are solely payments of principal and interest on the principal amount outstanding simply because the interest rate curve at the time of the assessment is such that the difference between a five-year interest rate and a six-month interest rate is not significant. Instead, the entity must also consider whether the relationship between the five-year interest rate and the six-month interest rate could change over the life of the instrument such that the contractual (undiscounted) cash flows over the life of the instrument could be significantly different from the (undiscounted) benchmark cash flows. However, an entity must consider only reasonably possible scenarios instead of every possible scenario. If an entity concludes that the contractual (undiscounted) cash flows could be significantly different from the (undiscounted) benchmark cash flows, the financial asset does not meet the condition in paragraphs 4.1.2(b) and 4.1.2A(b) and therefore cannot be measured at amortised cost or fair value through other comprehensive income.
2. A net position is eligible for hedge accounting only if an entity hedges on a net basis for risk management purposes. Whether an entity hedges in this way is a matter of fact (not merely of assertion or documentation). Hence, an entity cannot apply hedge accounting on a net basis solely to achieve a particular accounting outcome if that would not reflect its risk management approach. Net position hedging must form part of an established risk management strategy. Normally this would be approved by key management personnel as defined in IAS 24.
3. When measuring hedge ineffectiveness, an entity shall consider the time value of money. Consequently, the entity determines the value of the hedged item on a present value basis and therefore the change in the value of the hedged item also includes the effect of the time value of money.
4. At the date of initial application, an entity shall use reasonable and supportable information that is available without undue cost or effort to determine the credit risk at the date that a financial instrument was initially recognised (or for loan commitments and financial guarantee contracts at the date that the entity became a party to the irrevocable commitment in accordance with paragraph 5.5.6) and compare that to the credit risk at the date of initial application of this Standard.
*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.
## Conclusions
KAR Auction Services Inc Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating. KAR Auction Services Inc Common Stock prediction model is evaluated with Modular Neural Network (DNN Layer) and Stepwise Regression1,2,3,4 and it is concluded that the KAR stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell
### KAR KAR Auction Services Inc Common Stock Financial Analysis*
Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementBa2B2
Balance SheetBa3Baa2
Leverage RatiosBaa2B2
Cash FlowBaa2Ba3
Rates of Return and ProfitabilityBaa2Ba1
*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?
### Prediction Confidence Score
Trust metric by Neural Network: 93 out of 100 with 609 signals.
## References
1. Clements, M. P. D. F. Hendry (1995), "Forecasting in cointegrated systems," Journal of Applied Econometrics, 10, 127–146.
2. Mnih A, Hinton GE. 2007. Three new graphical models for statistical language modelling. In International Conference on Machine Learning, pp. 641–48. La Jolla, CA: Int. Mach. Learn. Soc.
3. Mikolov T, Chen K, Corrado GS, Dean J. 2013a. Efficient estimation of word representations in vector space. arXiv:1301.3781 [cs.CL]
4. L. Busoniu, R. Babuska, and B. D. Schutter. A comprehensive survey of multiagent reinforcement learning. IEEE Transactions of Systems, Man, and Cybernetics Part C: Applications and Reviews, 38(2), 2008.
5. Imai K, Ratkovic M. 2013. Estimating treatment effect heterogeneity in randomized program evaluation. Ann. Appl. Stat. 7:443–70
6. C. Wu and Y. Lin. Minimizing risk models in Markov decision processes with policies depending on target values. Journal of Mathematical Analysis and Applications, 231(1):47–67, 1999
7. M. J. Hausknecht and P. Stone. Deep recurrent Q-learning for partially observable MDPs. CoRR, abs/1507.06527, 2015
Frequently Asked QuestionsQ: What is the prediction methodology for KAR stock?
A: KAR stock prediction methodology: We evaluate the prediction models Modular Neural Network (DNN Layer) and Stepwise Regression
Q: Is KAR stock a buy or sell?
A: The dominant strategy among neural network is to Sell KAR Stock.
Q: Is KAR Auction Services Inc Common Stock stock a good investment?
A: The consensus rating for KAR Auction Services Inc Common Stock is Sell and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of KAR stock?
A: The consensus rating for KAR is Sell.
Q: What is the prediction period for KAR stock?
A: The prediction period for KAR is (n+6 month)
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# Thread: TTS voices on Android not seen/selectable in Primo 9.6.29
1. Sergeant
Join Date
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## TTS voices on Android not seen/selectable in Primo 9.6.29
Hello, I tried to get the TTS voices running, but it just won't succeed and it's driving me nuts!
I started a discussion here: TTS voices on Android not seen/selectable in Primo 9.6.29
but I was suggested to try this in the Android section (even though there are many android questions..)
Here's the problem:
I have Primo for Android 9.6.29.329069
am trying to get the TTS with Acapela running, but everything I do just won't work:
put some ZIP files with the TTS voices to content\voice, but only the natural voices show up in the voices selection in Primo.
I tried files like:
Voice_TTS-nua_dut-m1-lua-dri40-vssq5f22.zip
Voice_TTS-Pro-Dutch-Android-zervdim_v2.zip
Voice_TTS-svox_dut-f1-lua.zip
so, please tell me how to get these working, or at least how to select them in Primo!
ps: I need the Dutch voices
Strange thing is that it worked with a prior Primo version (don't know which one I had before)
btw: Yes, I did search and read A LOT, both in here as well as several other forums, but never found the solution.
Changing in the .INI file from 'Dutch' to 'dut' etc. won't work either.
I also tried to set the Android TTS engine to the default Google one and to the SVOX that I have too. I don't think that iGo identifies if the Android OS has a TTS activated.
I tried several 'sources' to get the voices from, but even though people say that they work (on their versions of iGo) the y won't work with my device, because they are not even visible!
So, can anyone help me with the TTS voices? Maybe a working Dutch version (or even an English one just to find out if iGo can work with TTS)
2. ## The Following User Says Thank You to brightlight For This Useful Post:
3. Sergeant
Join Date
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## Re: TTS voices on Android not seen/selectable in Primo 9.6.29
Anyone?
Is there no develop of iGO for Android anymore? This site is so silent...
Or has everyone moved to the 'regular' iGo section?
4. Private
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## Re: TTS voices on Android not seen/selectable in Primo 9.6.29
hi,
I use IVONA as default TTS engine with Romanian main voice
and in my sys.txt I have this on the end of the file
Code:
[tts]
automatic_direction_conversion=1
announce_exit_directions=1
announce_street_name=1
priority="HIGH"
ttspro_da_voice_enabled=1
tts_speedcam_sounddelay0=90
tts_speedcam_sounddelay1=140
tts_speedcam_sounddelay2=140
tts_speedcam_sounddelay3=140
tts_speedcam_sounddelay11=90
tts_speedcam_sounddelay17=140
And on "content/voice/Voice_TTS.zip/info.ini" file in TTS voice zip I have some lines modified like this
Code:
languagekey="Romanian"
languagename="Română"
luavoice=1
phoneme_lang="ROR"
tts_destnames=1
tts_engine="Android TTS"
tts_langname="ro"
tts_mode_mixed=0
tts_rate=100
ttsvoice=1
And for me it works on Romanian even if IVONA is set to default English ( I have English and Romanian engine installed )
5. Corporal
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## Re: TTS voices on Android not seen/selectable in Primo 9.6.29
If you are trying to use Svox then you need the Svox basic package plus an Svox voice such as Victoria.
You could also try using Loquendo.
Under 'settings\voice input & output\text-to speech settings' select preffered voice and engine. Tick always remember my settings.
6. Sergeant
Join Date
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## Re: TTS voices on Android not seen/selectable in Primo 9.6.29
I have SVOX, Acapela, Google and Pico with correctly installed voices, but neither will work with iGo.
They DO work on EVERY other application, so it definately is a problem with iGO!
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# How do you complete the square to solve x^2 – 5x + 8=0?
May 22, 2015
Given ${x}^{2} - 5 x + 8 = 0$
Completing the square
${x}^{2} - 5 x + {\left(\frac{5}{2}\right)}^{2} + 8 - {\left(\frac{5}{2}\right)}^{2} = 0$
${\left(x - \frac{5}{2}\right)}^{2} = \frac{25}{4} - 8 = - \frac{7}{8}$
$x - \frac{5}{2} = \pm \sqrt{- \frac{7}{8}}$
Note: there are no Real roots
$x = \frac{5}{2} \pm \sqrt{\frac{7}{8}} i$
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# Problem 36 LC Diff Eq
I have given the differential equations describing the system for Problem 36. This is called a state-space form, and it is useful because it expresses the time derivatives of the state variables in terms of the state variables themselves. In other words, if you know what the state of the system is, you also know how the state is changing. I would pair this with a numerical solution to derive results. I don't know why it is displaying sideways, but one can download and rotate within the picture viewer.
Note by Steven Chase
10 months, 3 weeks ago
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Sort by:
@Lil Doug Here it is. I'm going to bed now
- 10 months, 3 weeks ago
@Steven Chase can't we solve it analytically?
- 10 months, 3 weeks ago
@Steven Chase Your handwriting is far better than Latex.
- 10 months, 3 weeks ago
Here is the rotated form
- 10 months, 3 weeks ago
@Steven Chase Hello ,have a look in this problem please
- 10 months, 3 weeks ago
@Steven Chase How to solve 31st problem?
- 10 months, 3 weeks ago
@Steven Chase Did u have attempted my all latest 3-4 problems.?
There are 2-3-4 and many attempts showing of each problem.
- 10 months, 3 weeks ago
No, I haven't. I was going to try the pulley one, but then I noticed that the masses were the same.
- 10 months, 3 weeks ago
@Steven Chase so what happened if masses are same,
- 10 months, 3 weeks ago
@Steven Chase Other than bead problems, which type of problems u like the most??
- 10 months, 3 weeks ago
@Steven Chase ohhww I just realized that $M>m$ , I am doing it right now.
Don't forget to attend. :)
- 10 months, 3 weeks ago
@Steven Chase Now it's reposted, No need to worry, Go ahead and rock.
And try to solve using pen.
- 10 months, 3 weeks ago
@Steven Chase in the latest problem there are 2 attempt.
Did u have tried??
I think Brilliant should add a new feature in which all people with viewers, attempt, solvers, their profile photo should be displayed.
- 10 months, 3 weeks ago
Yeah, I tried it. Maybe I just don't understand pulleys with friction well enough
- 10 months, 3 weeks ago
@Steven Chase Did you have forgot those concepts OR not yet learned how to solve tough problems of rotational mechanics ?
- 10 months, 3 weeks ago
I'm fine with rotational dynamics in general. But pulleys with friction, maybe not so much
- 10 months, 3 weeks ago
@Steven Chase ok No problem. I am posting a new General Rotational mechanics problem within half an hour.
Don't forget to drop a solution Bye Love you. :)
- 10 months, 3 weeks ago
Do you have the solution to that pulley one?
- 10 months, 3 weeks ago
@Steven Chase No sir , currently I don't have.
Whenever I will get, or if I able to solve it, I will definitely give you.
- 10 months, 3 weeks ago
@Steven Chase this one is also a rotational problem https://brilliant.org/problems/maximum-tension/
- 10 months, 3 weeks ago
@Steven Chase it is up now.
- 10 months, 3 weeks ago
@Steven Chase I have made a little bit change in problem it is $b≥$ instead of $b=$
- 10 months, 3 weeks ago
@Steven Chase did u have solved the latest problem?
- 10 months, 3 weeks ago
@Steven Chase I think you like bead types of problems nah!
I have posted one more right now.
I know that you didn't owe me anything, but whenever you will be free, please post its solution, it is a request.
- 10 months, 2 weeks ago
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# Loan Repayment Question From Pre Calculus Test (4 questions witihn) [closed]
Edit: A farmer borrows $80,000 to purchase new machinery. The interest is calculated monthly at the rate of 2% per month, and is compounded each month. The farmer intends to pay the loan with interest in two equal installments of$M at the end of the first and second years.
i) How much does the farmer owe at the end of the first month?
ii) Write an expression involving M for the total amount owed by the farmer after 12 months, just after the first installment of $M has been paid. iii)Find an expression for the amount owed at the end of the second year and deduce that:$M = \frac{80000(1.02)^{24}}{(1.02)^{12} + 1}$iv) What is the total interest over the two year period? Any help will be appreciated, but I was particularly stuck with (ii) and (iii) ## closed as off-topic by John Doe, Davide Giraudo, John B, Namaste, JMPJun 4 '17 at 19:11 This question appears to be off-topic. The users who voted to close gave this specific reason: • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Doe, Davide Giraudo, John B, Namaste, JMP If this question can be reworded to fit the rules in the help center, please edit the question. ## 3 Answers The original loan was$\$80000$, and you probably figured out that after a month the farmers owes $\$80000\times 1.02$. After 12 month, the total owed is$ \$80000\times 1.02^{12}$. At this point the farmers pays $M$, so the answer to (ii) is $\$80000\times 1.02^{12}-M.$At the start of the second year, the amount owed is the answer to part (ii). With interest, at the end of the second year, the amount is$ (\$80000\times 1.02^{12}-M)\times 1.02^{12}$. Remember that you were told that this is the same as $M$, since you have two equal payments. Therefore $$(80000\times 1.02^{12}-M)\times 1.02^{12}=M$$
Expanding the parenthesis, and moving the terms with $M$ on one side will yield the answer to part (iii)
Recall that the formula for compound interest is $A = P(1+i)^n$,
where:
• $A$ is the final amount
• $P$ is the initial amount
• $i$ is the interest rate (in decimals)
• $n$ is the time that has elapsed.
For question (I):
Since the initial amount is $80 000$,
$A = 80000(1+0.02)^1$
$A = 81,600$
For question (II):
$A = 80000(1+0.02)^{12} - m$
$A = 101,459 - m$
Using this information, can you see what (III) is asking?
Let $a_n$ be the money owed in the $n^{th}$ month and let $r = 1.02$ be the rate at which the money owed increases.
Notice that $$a_n = ra_{n-1} \space\text{ and } \space a_0 = \80,000 \space \text{ for } \space 0\le n \le 12$$ We can find an explicit formula from the recursion by noting the pattern (and proving it with induction if you feel the need) to get
$$a_n = a_0 r^n \space \text{ for } \space 0 \le n \le 12$$ After a year, then, $$a_{12} = a_0 r^{12} \implies a_{12} = a_0 r^{12} - M \space \text{ after paying the annual installment }$$
Now define $b_n = b_{n-1}r$ with $b_0 = a_0 r^{12} - M$
After another year and payment, we get $$b_{12} = b_0r^{12}- M = (a_0 r^{12} - M)r^{12} - M = 0$$
Solve for $M$ to get $$M = \frac {a_0r^{24}}{r^{12} + 1}$$
• Very complicated way of looking at it, but it works. Maybe the concepts aren't so complicated, but the way you've written it looks foreign to me. – user432114 Jun 4 '17 at 2:05
• @Dodsy: That's a fair perspective. Personally, I will do any problem using sequences that can be done that way. It typically is the most intuitive and comfortable approach for me. I also prefer to derive formulas from scratch rather than use preexisting ones in many cases (at least when first learning it). I despise how mathematics is handled in grade school, where they often just throw formulas at you unjustifiably. – infinitylord Jun 4 '17 at 2:15
• I think your answer is more than adequate, and agree with your comment fully and completely! – user432114 Jun 4 '17 at 2:24
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The following table describes specialized objects to store data represented in population genetics packages. Conversion between all types is possible.
Anyone developing a package for population genetic analysis is encouraged to use or build upon these data structures. If a new data structure is needed, please provide a conversion method to one or more of the classes listed below.
Class {type} (package) Strengths Weaknesses
DNAbin {S3} (ape) stores all sets of sequences (aligned or not) less compact than 2-bit coding (but by a factor 4 at most)
uses matrices (aligned) or lists so usual R’s commands (names, rownames, [, [[, \$) can be used
many as.DNAbin methods in ape (inc. from BioConductor)
efficient functions in ape (dist.dna, seg.sites, base.freq, read.FASTA) and in pegas (haplotype)
loci {S3} (pegas) low memory usage not really appropriate for some analyses (e.g., multivariate analyses)
all levels of ploidy and any number of alleles needs to improve the treatment of NA’s (especially when data are read with read.vcf()
genotypes can be phased
any kind of individual data can be associated in the data frame
efficient to compute genotype and allele frequencies
genind {S4} (adegenet) stores allelic counts; ideal for multivariate analyses requires more memory
additional slots for individual data less efficient to compute frequencies
all levels of ploidy
genpop {S4} (adegenet) equivalent to genind at group level; ideal for multivariate analysis requires more memory
genlight {S4} (adegenet) stores binary SNPs using bit-level coding; very memory efficient more computationally intensive to handle; less functionalities
additional slots for individual data and population strata
all levels of ploidy assumes bi-allelic loci
genclone {S4} (poppr) inherits genind object; gains all advantages all the same weaknesses plus slightly more memory
stores multilocus genotype/lineage definitions (@mlg slot) for clonal populations
snpclone {S4} (poppr) inherits genlight object; gains all advantages all the same weaknesses plus slightly more memory
stores multilocus genotype/lineage definitions (@mlg slot) for clonal populations
genambig {S4} (polysat) stores microsatellite data with ambiguous ploidy does not handle any other data type
exports to genpop objects cannot easily be transferred to any other object
phyDat {S3} (phangorn) very general inspired by R data.frame, factor and contrasts, can contain any discrete data type; nucleotides, amino acids and codons have some more support designed having phylogenetic analysis in mind; requires alignments, where all sequences have same length
can be converted to and from DNAbin objects (as.DNAbin / as.phyDat)
a few generic functions work on it: c, unique, subset and utility functions baseFreq, allSitePattern, etc. data are not necessarily very memory efficient (as integer + contrast matrix), but stores only unique site patterns and their weights (as double)
“efficient” maximum likelihood, maximum parsimony and distance functions in phangorn
gtype {S3} (strataG) a simple R list containing a matrix where the first column is a stratification scheme and columns afterward are either haplotypes or diploid loci. If haploid data, the gtype object can also contain a list of DNA sequences. Can likely be made more efficient in terms of storage and preprocessing for other analytical routines in package
can be converted to data.frame or matrix with appropriate as. functions.
has manipulation functions like subset which will select certain strata and/or loci, merge to combine mulitple gtypes, and summary.
can create input files for Genepop, STRUCTURE, fastsimcoal, Arlequin, MEGA, and PHASE
multiDNA {S4} (apex) stores multiple DNAbin objects from ape
multiPhyDat {S4} (apex) stores multiple phyDat objects from phangorn
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# Degenerate eigenvalues and finding normalized eigenvectors
I am trying to find the eigenvalues and the normalized eigenvectors of the matrix: $$\begin{bmatrix}0 & i & 0\\0 & 0& i\\i & 0 & 0 \end{bmatrix}$$
It is stated that the eigenvalues are : $$\lambda_1 = i\omega$$, $$\lambda_2 = i$$ and $$\lambda_3 = i\omega^2$$ and two of the normalised eigenvectors are: $$\frac{1}{\sqrt 3}\begin{bmatrix}1\\1\\1\end{bmatrix}$$ and $$\frac{1}{\sqrt 3}\begin{bmatrix}1\\\omega^2\\\omega\end{bmatrix}$$.
I'm not sure what $$\omega$$ means in this case and how to arrive to this result, given that $$i$$ is a degenerate eigenvalue.
• I suspect that $\omega$ has been used in this way previously in your study materials. – amd Apr 23 at 17:38
The matrix is equal to $$iM$$, where$$M=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$$Therefore, its eigenvalues are the eigenvalues of $$M$$ times $$i$$. But the characteristic polynomial of $$M$$ is $$1-x^3$$ and therefore its eigenvalues are $$\{1,\omega,\omega^2\}$$, where$$\omega=-\frac12\pm\frac{\sqrt3}2i$$
$$\omega$$ is used for a lot of different things in math, but in this context it's generally used to represent a non-principal cube root of $$1$$. That is, $$\omega$$ is a number such that $$\omega^3=1$$ but $$\omega \neq 1$$. While there are two numbers that satisfy that relation, they are conjugates of each other and thus indistinguishable by polynomials over the reals. (Taking different values to be $$\omega$$ will result in the expressions you have evaluating to the same set of eigenvalues, albeit in a different order.) If a particular number is needed, it's generally taken to be the first one, going counter-clockwise, from $$1$$ in the complex plane, i.e. -$$\frac12+\frac{\sqrt3}2i$$.
$$i$$ is not degenerate; $$\frac{1}{\sqrt 3}\begin{bmatrix}1\\1\\1\end{bmatrix}$$ is the only (up to scaling) eigenvalue for that eigenvalue. Both its arithmetic and geometric multiplicities are $$1$$.
• Another usual notation is $j$. – Bernard Apr 23 at 17:40
Here a direct calculation.
The characteristic polynomial of our matrix $$A=\left[\begin{array}{rrr} 0 & i & 0 \\ 0 & 0 & i \\ i & 0 & 0 \end{array}\right]$$ is given by \begin{align*} \chi_A(t) &= \det(t\cdot I_3-A) \\ &= \left|\begin{array}{rrr} t & -i & 0 \\ 0 & t & -i \\ -i & 0 & t \end{array}\right| \\ &\overset{\ast}{=}\left|\begin{array}{rrr} t - i & t - i & t - i \\ 0 & t & -i \\ -i & 0 & t \end{array}\right| \\ &= (t-i)\left|\begin{array}{rrr} 1 & 1 & 1 \\ 0 & t & -i \\ -i & 0 & t \end{array}\right| \\ &\overset{\circledast}{=}(t-i)\left|\begin{array}{rrr} 1 & 1 & 1 \\ 0 & t & -i \\ 0 & i & t + i \end{array}\right| \\ &= (t-i)\left|\begin{array}{rr} t & -i \\ i & t + i \end{array}\right| \\ &= (t-i)\cdot\{t^2+i\,t+i^2\} \\ &=(t-i)\cdot\{t^2+i\,t-1\} \\ &\overset{q}{=} (t-i)\cdot\left(t-\frac{-i+\sqrt{3}}{2}\right)\cdot \left(t-\frac{-i-\sqrt{3}}{2}\right) \end{align*} The equality marked $$\ast$$ is obtained by replacing row one with the sum of all three rows of the previous matrix. The equality marked $$\circledast$$ is obtained by adding $$i$$ times row one to row three. The equality marked $$q$$ is obtained from the quadratic formula.
Our factorization of $$\chi_A(t)$$ shows that the eigenvalues of $$A$$ are \begin{align*} \lambda_1 &= i & \lambda_2 &= \frac{-i+\sqrt{3}}{2} & \lambda_3 &= \frac{-i-\sqrt{3}}{2} \end{align*}
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# What determines the transmit frequency in this 555-based transmitter?
I'm a general class ham, and am just starting to do my own homebrewing. I was planning on basing a QRP transmitter off of this instructables project, seeing as I have a veritable plethora of 555 ics in my shack toolbox.
The problem is, this transmitter is designed to operate a bit below the commerical AM band. How is the general operating frequency being set? I'd like to get it set around the 1.8 mhz band (160 meters).
Help is much help appreciated, as I'm going to use this to teach the other kids in the ham radio club at my school how to build basic transmitters. Picture of circuit: http://www.instructables.com/image/FD38QHUGIKL42D3/Schematic.jpg">
• What are the boundaries of the 1.8 MHz band? – AndrejaKo Mar 26 '11 at 12:58
• 1.8 to 2.0 MHz. – technowizard12 Mar 26 '11 at 20:02
• I'd go with something a little better; this transmitter almost certainly has FCC issues. – Brian Carlton Sep 13 '11 at 20:37
From what I can see form the circuit, the timer is running in astable mode. The frequency is controlled by the equivalent resistor made by adding resistances of P1 and R3, resistor R1 and capacitor C1.
If you want to experiment, go to 555 calculator and take a look at bottom schematic. Your P1+R3 are its R1, your R1 is its R2 and your C1 is its C.
UPDATE: I'll try to make it a bit clearer how this transmitter gets its frequency. First, read through the whole instructable. There is a nice explanation related to harmonics in it.
This transmitter controls antenna output using the Q1 transistor. The transistor is triggered by the output of the 555 timer. Therefore there is direct relation between 555 frequency and transmission frequency.
The timer itself is controlled by two resistors and a capacitor. Timer monitors situation on the C1 capacitor. When it is $\frac {2}{3}$ full, timer will emit high output and start discharging the capacitor. When it is $\frac {1}{3}$ full, timer will start emitting low output and start charging the capacitor. When the capacitor is charging, current is going through resistors (P1+R3) and R1. They limit the charging current and modify the time it takes to charge the capacitor. When the capacitor is being discharged, current goes from C1 through resistor R1 into discharge pin which is connected to ground during discharge. This way, R1 controls the discharge time.
Now about the 1.8 MHz band. You may be able to directly transmit at that band by using proper timer settings. For example TS555 timers made by STmicroelectroncs can provide up to 2.7 MHz frequency in astable mode. To get the 1.8 MHz frequency, you can use formulas from the 555 timer. Basically, you should pick the resistors, potentiometer and capacitor so that $((R3+P1)+R1)*C1=8.05*10^{-7}$. If you for example take a 22 pF capacitor (they are commonly used for microcontroller crystals oscillators), resistors added together should be around 37 $k\Omega$. You can take for example R1 to be 8.2 $k\Omega$ and then set the P1+R3 to be 20 $k\Omega$. After that, you can calculate exactly what kind of potentiometer and resistor you need for the transmitter to work correctly using the calculator.
I recommend that you do some more research before making the circuit with the values I recommended. Capacitors usually have high tolerances, so its impact on the circuit should be minimized. Resistors with 1% can be very cheaply obtained, but precise potentiometers or rheostats may be expensive. For example at local stores here, a good multi-turn potentiometer costs between 10€ and 20€, while cheap single turn one costs about 2€.
The point of the above paragraph is that there may be other set of values which could make it much easier and cheaper to set the correct frequency and provide higher precision. I unfortunately don't have enough experience to provide a better set of values.
• Where do you figure the 8.05*10^-8? – technowizard12 Mar 28 '11 at 4:55
• @technowizard12 From the formulas on the site I linked to. $\frac{1}{\ln(2)}=1.442695...\approx1.45$. Next we have $f=\frac{1.45}{(R_1+2R_2)C}$. If we take that f=1.8 MHz, we get $\frac{1.45}{1.8*10^6}=(R_1+2R_2)C$. The left side of the equation is $\approx 8.056*10^{-7}$. I don't know how I got $10^{-8}$ back then. The rest of the calculation is correct. – AndrejaKo Mar 28 '11 at 6:00
• I'm not so sure about the rest of the calculation being correct, but the 555 calculator does give frequency of 1798201 Hz for the component values I provided. Strange... – AndrejaKo Mar 28 '11 at 7:30
This circuit is going to generate square waves into your antenna. As a licensed ham, you are responsible for transmitting a clean signal. I'm all for trying stuff, but it's best that you put this into a dummy load instead of an antenna, and see if you can hear it on your receiver. Then tune around and look for harmonics and other spurious output. As a matter of fact, the article states that the 555 is running at a fraction of the desired frequency, and you are picking up one of the harmonics on your radio.
This is not as off-the-wall as it sounds. It's a very common practice to "multiply" the frequency to get a higher frequency that the oscillating circuit can't reach by itself. It's also common to build RF amplifiers that are anything but linear. What's needed, though, is a filter to remove all but the desired signal. What you'll need is at least a tuned tank circuit before the antenna. The result won't be a very strong signal, but it may be legal.
Note that a perfect square wave has only odd harmonics. It should be interesting to tune through the bands and see if the odd harmonics are stronger than the even ones. This also suggests that you don't try to, say, run at 500 kHz trying to get 2 MHz. You'd be better off with 400 kHz or 667 kHz. Now, your square wave won't be perfect, so there will still be some energy at the even harmonics. But still, there's a lot to play around with here.
One final note. Almost everything about this circuit will be unsuitable for AM. The original article even said so. But if you get that far, I think that should be discussed in another question.
• I totally agree with your analysis (+1). The circuit (link) is more akin to a spark transmitter than an AM system. Its modulation (?) looks more like narrowband FM or PWM - using pin 4 (the reset pin) as a control input, certainly not AM as I would recognise it. The output transistor is totally redundant and will not work as an emitter follower. Fortunately the really poor aerial should limit the range to within a few feet of the offending item. Its a shame that folks put up these projects for beginners. – JIm Dearden Oct 4 '14 at 10:08
• Funny that--in my original answer, I had the sentence, "This is marginally better than a spark gap", but I removed it, as there was enough to worry about already, and the SE community isn't all that thrilled with these sorts of remarks. As to the modulation, I think the idea is that you talk into the mic, and yeah, I can tell that's my voice coming out on the radio, although it doesn't sound too good...It's kind of a proof of concept (that mostly fails). – gbarry Oct 4 '14 at 19:56
AndrejaKo, I agree with 99.99% of your answer, so my point is a bit of a nitpicker, but maybe apprpriate when instructing. I agree with you that the charging and discharging paths of the timing capacitor, C1, are different: Rcharging= R3 + P1 + R1, while Rdischarge = R1. Depending upon your timer (I'm going by the datasheet of theLM555, which I'm not sure is appropriate for 1.8MHz opn, but it shows the output to be HI when C1 charges from 1/3Vcc to 2/3Vcc, and LO when C1 discharges from 2/3Vcc to 1/3Vcc. This means Time-ON = ln(2)C1(R3+P1+R1), while Time-OFF = ln(2)C1(R1). Adding these two times together gives the total period of one cycle of the output astable wave : T = ln(2)C1(R3+P1+2R1). Since f = 1/T, we can next solve for the circuit parameters: 1.8 x 10^6 = 1/[ln(2)C1(R3+P1+2R1)] C1(R3+P1+2R1) = 8.015 x 10^-7
And then from here, without my having any first hand experience with this communication technique, I yield to Mr. AndreaJo's advice concerning choosing of the individual component values.
Ray
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## Annals of Functional Analysis
### The weak Haagerup property for $C^{*}$-algebras
Qing Meng
#### Abstract
We define and study the weak Haagerup property for $C^{*}$-algebras in this article. A $C^{*}$-algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$-algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$-algebras.
#### Article information
Source
Ann. Funct. Anal., Volume 8, Number 4 (2017), 502-511.
Dates
Accepted: 5 January 2017
First available in Project Euclid: 22 June 2017
https://projecteuclid.org/euclid.afa/1498096870
Digital Object Identifier
doi:10.1215/20088752-2017-0014
Mathematical Reviews number (MathSciNet)
MR3717172
Zentralblatt MATH identifier
06841331
#### Citation
Meng, Qing. The weak Haagerup property for $C^{*}$ -algebras. Ann. Funct. Anal. 8 (2017), no. 4, 502--511. doi:10.1215/20088752-2017-0014. https://projecteuclid.org/euclid.afa/1498096870
#### References
• [1] M. Bożejko and G. Fendler, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2, 297–302.
• [2] N. P. Brown and N. Ozawa, $C^{*}$-algebras and Finite-Dimensional Approximations, Grad. Stud. Math. 88, Amer. Math. Soc., Providence, 2008.
• [3] J. D. Canniere and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500.
• [4] Z. Dong, Haagerup property for $C^{*}$-algebras, J. Math. Anal. Appl. 377 (2011), no. 2, 631–644.
• [5] E. G. Effros and Z. J. Ruan, Operator Spaces, London Math. Soc. Monogr. (N.S.) 23, Oxford Univ. Press, New York, 2000.
• [6] U. Haagerup and S. Knudby, The weak Haagerup property, II: Examples, Int. Math. Res. Not. IMRN 2015, no. 16, 6941–6967.
• [7] P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), no. 2, 311–313.
• [8] P. Jolissaint, Haagerup approximation property for finite von Neumann algebras, J. Operator Theory 48 (2002), no. 3, 549–571.
• [9] S. Knudby, Semigroups of Herz-Schur multipliers, J. Funct. Anal. 266 (2014), no. 3, 1565–1610.
• [10] S. Knudby, The weak Haagerup property, Trans. Amer. Math. Soc. 368 (2014), no. 5, 3469–3508.
• [11] G. Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser. 294, Cambridge Univ. Press, Cambridge, 2003.
• [12] Y. Suzuki, Haagerup property for $C^{*}$-algebras and rigidity of $C^{*}$-algebras with property (T), J. Funct. Anal. 265 (2013), no. 8, 1778–1799.
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Select Page
Direction for the question: Refer the given diagram to answer the questions:
Question 1: Approximately, what was the actual profit made by the department store in the second quarter of 1999?
(a) Rs. 160 lakh
(b) Rs. 170 lakh
(c) Rs. 180 lakh
(d) Rs. 210 lakh
Question 2: In which of the following quarters, did the departmental store make the least amount of profits?
(a) Third quarter of 2000
(b) Second quarter of 1999
(c) First quarter of 1999
(d) Third quarter of 1998
Question 3: During the period 1998-2000, how many quarters exceeded the profit of Rs. 150 lakh?
(a) 6
(b) 5
(c) 4
(d) 3
Question 4: In the year 2000, total profit made by the departmental store was approximately
(a) Rs. 540 lakh
(b) Rs. 630 lakh
(c) Rs. 720 lakh
(d) Rs. 770 lakh
Question 5: The total annual profit made by the departmental store increased by approximately what per cent from 1997 to 2000?
(a) 40%
(b) 50%
(c) 88%
(d) 120%
Answer 1: (a) In the second quarter the profit is above 150 but below 170 lakhs, which means the answer should be approximately = 160 lakhs. Correct option is (a)
Answer 2: (d) From the table it can be inferred that among the options 3rd quarter of 1998 is lowest.
Answer 3: (a) It can be easily counted from the table (refer to arrow marks.) Correct option is (a)
Answer 4: (d) The 4 quarters data are: 150lakh, 180lakh, 210lakh, 230lakh. Total profit = 150 + 180 + 210 + 230 = 770 lakh Correct option is (d)
Answer 5: (c) In 1997 the total was = 135+115+ 90+70 = 410lakh. In 2000 the total was = 150 + 180 + 210 + 230 = 770 lakh Percentage increase = $\frac{770-410}{410}\times 100=\frac{360}{410}\times 100=87.8%\approx 88%.$ Correct option is (c)
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$25_r = 23_{10}$ solve for the base r
first of all this is a homework question and I don't want the solution. I just want a reference to how to solve similar questions like this. I believe it's explained my course textbook, "Computer Organization & Architecture: Themes and Variations" but I currently cannot afford the textbook.
Here's the question:
For each of the following numbers, state the base in use; that is, what are the values of r, s, and t?
a. $25_r = 23_{10}$
b. $1001_s = 19684_{10}$
c. $1011_t = 4931_{10}$
I recognize it's similar to solving an equation. I'm guessing I have to find r, s and t and they will be a specific base that matches the base ten number. I tried searching online for similar questions but I'm not sure what to search for so I have no clue where to start for solving these equations.
Any help would be appreciated.
• 1. Can you identify a more general conceptual question that you want information on? That will make this more useful to others in the future. 2. Book recommendation requests are generally off-topic for this site. (e.g., meta.cs.stackexchange.com/q/303, meta.cs.stackexchange.com/q/874/755). A request for a cheaper textbook for computer architecture is off-topic, but if you can identify a specific topic/concept, it might be on-topic. 3. What have you tried? What research have you done? Where have you looked? We expect you to make a serious effort on your own to find resources. – D.W. Jan 29 '15 at 1:17
• This question seems to be covered by Wikipedia, e.g., en.wikipedia.org/wiki/Radix and en.wikipedia.org/wiki/Positional_notation. If the answer to your question can be found on Wikipedia, that often suggests you need to put more effort into researching your question on your own before asking, and it typically means your question isn't a good fit for this site. – D.W. Jan 29 '15 at 1:20
1 Answer
Hint:
Consider $25_{10}$. The number can be written as $5 \cdot 10^0 + 2 \cdot 10^1$.
How can $25_{r}$ be written?
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# [Remember Your Squares] Something I Found
1. Feb 17, 2010
### SomeGuy121
Let x = 1.
Let n = Next Odd Number
Let y = Previous Sum
x2 = x
+3 = 4 = (1+x)2
+5 = 9 = (3+x)2
+n = n+y = (n-2 + x)2
You could make a program to list all the squares without invoking the multiplication function or squaring function using a simple loop.
C++ Example:
#include "stdafx.h"
#include <iostream>
using namespace std;
int _tmain(int argc, _TCHAR* argv[])
{
cout<<"Squares: \n\n";
int sum=1,nextOdd=1; // Sum is Starting Integer Squared, Declared X in the For Loop below
for(int x=1;x<100;x++)
{
cout<<x<<" Squared is "<<sum<<"\n";
nextOdd+=2;
sum+=nextOdd;
}
system("pause");
}
2. Feb 17, 2010
### CRGreathouse
You've discovered that
$$\sum_{k=1}^n2k-1=n^2$$.
Congratulations.
Can you transform that into a formula for $$\sum_{k=1}^nk$$? Can you find one for $$\sum_{k=1}^nk^2$$?
You can check your work afterward, and even glimpse what's beyond:
http://mathworld.wolfram.com/FaulhabersFormula.html
3. Feb 17, 2010
### SomeGuy121
Are you being sarcastic with "congratulations"?
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# 27.2 - The T-Test For One Mean
27.2 - The T-Test For One Mean
Well, geez, now why would we be revisiting the t-test for a mean $$\mu$$ when we have already studied it back in the hypothesis testing section? Well, the answer, it turns out, is that, as we'll soon see, the t-test for a mean $$\mu$$ is the likelihood ratio test! Let's take a look!
## Example 27-4
Suppose that a random sample $$X_1 , X_2 , \dots , X_n$$ arises from a normal population with unknown mean $$\mu$$ and unknown variance $$\sigma^2$$. (Yes, back to the realistic situation, in which we don't know the population variance either.) Find the size $$\alpha$$ likelihood ratio test for testing the null hypothesis $$H_0: \mu = \mu_0$$ against the two-sided alternative hypothesis $$H_A: \mu ≠ \mu_0$$.
Our unrestricted parameter space is:
$$\Omega = \left\{ (\mu, \sigma^2) : -\infty < \mu < \infty, 0 < \sigma^2 < \infty \right\}$$
Under the null hypothesis, the mean $$\mu$$ is the only parameter that is restricted. Therefore, our parameter space under the null hypothesis is:
$$\omega = \left\{(\mu, \sigma^2) : \mu =\mu_0, 0 < \sigma^2 < \infty \right\}$$
Now, first consider the case where the mean and variance are unrestricted. We showed back when we studied maximum likelihood estimation that the maximum likelihood estimates of $$\mu$$ and $$\sigma^2$$ are, respectively:
$$\hat{\mu} = \bar{x} \text{ and } \hat{\sigma}^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2$$
Therefore, the maximum of the likelihood function for the unrestricted parameter space is:
which simplifies to:
$$L(\hat{\Omega})= \left[\dfrac{ne^{-1}}{2\pi \Sigma (x_i - \bar{x})^2} \right]^{n/2}$$
Now, under the null parameter space, the maximum likelihood estimates of $$\mu$$ and $$\sigma^2$$ are, respectively:
$$\hat{\mu} = \mu_0 \text{ and } \hat{\sigma}^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \mu_0)^2$$
Therefore, the likelihood under the null hypothesis is:
which simplifies to:
$$L(\hat{\omega})= \left[\dfrac{ne^{-1}}{2\pi \Sigma (x_i - \mu_0)^2} \right]^{n/2}$$
And now taking the ratio of the two likelihoods, we get:
which reduces to:
$$\lambda = \left[ \dfrac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{\sum_{i=1}^{n}(x_i - \mu_0)^2} \right] ^{n/2}$$
Focusing only on the denominator for a minute, let's do that trick again of "adding 0" in just the right away. Adding 0 to the quantity in the parentheses, we get:
which simplifies to:
$$\sum_{i=1}^{n}(x_i - \mu_0)^2 = \sum_{i=1}^{n}(x_i - \bar{x})^2 +n(\bar{x} - \mu_0)^2$$
Then, our likelihood ratio $$\lambda$$ becomes:
$$\lambda = \left[ \dfrac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{\sum_{i=1}^{n}(x_i - \mu_0)^2} \right] ^{n/2} = \left[ \dfrac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{ \sum_{i=1}^{n}(x_i - \bar{x})^2 +n(\bar{x} - \mu_0)^2} \right] ^{n/2}$$
which, upon dividing through numerator and denominator by $$\sum_{i=1}^{n}(x_i - \bar{x})^2$$ simplifies to:
Therefore, the likelihood ratio test's critical region, which is given by the inequality $$\lambda ≤ k$$, is equivalent to:
which with some minor algebraic manipulation can be shown to be equivalent to:
So, in a nutshell, we've shown that the likelihood ratio test tells us that for this situation we should reject the null hypothesis $$H_0: \mu= \mu_0$$ in favor of the alternative hypothesis $$H_A: \mu ≠ \mu_0$$ if:
$$\dfrac{(\bar{x}-\mu_0)^2 }{s^2 / n} \ge k^{*}$$
Well, okay, so I started out this page claiming that the t-test for a mean $$\mu$$ is the likelihood ratio test. Is it? Well, the above critical region is equivalent to rejecting the null hypothesis if:
$$\dfrac{|\bar{x}-\mu_0| }{s / \sqrt{n}} \ge k^{**}$$
Does that look familiar? We previously learned that if $$X_1, X_2, \dots, X_n$$ are normally distributed with mean $$\mu$$ and variance $$\sigma^2$$, then:
$$T = \dfrac{\bar{X}-\mu}{S / \sqrt{n}}$$
follows a T distribution with n − 1 degrees of freedom. So, this tells us that we should use the T distribution to choose $$k^{**}$$. That is, set:
$$k^{**} = t_{\alpha /2, n-1}$$
and we have our size $$\alpha$$ t-test that ensures the probability of committing a Type I error is $$\alpha$$.
It turns out... we didn't know it at the time... but every hypothesis test that we derived in the hypothesis testing section is a likelihood ratio test. Back then, we derived each test using distributional results of the relevant statistic(s), but we could have alternatively, and perhaps just as easily, derived the tests using the likelihood ratio testing method.
[1] Link ↥ Has Tooltip/Popover Toggleable Visibility
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# Understanding Bayes: Updating priors via the likelihood
[Some material from this post has been incorporated into a paper to be published in AMPPS]
In a previous post I outlined the basic idea behind likelihoods and likelihood ratios. Likelihoods are relatively straightforward to understand because they are based on tangible data. Collect your data, and then the likelihood curve shows the relative support that your data lend to various simple hypotheses. Likelihoods are a key component of Bayesian inference because they are the bridge that gets us from prior to posterior.
In this post I explain how to use the likelihood to update a prior into a posterior. The simplest way to illustrate likelihoods as an updating factor is to use conjugate distribution families (Raiffa & Schlaifer, 1961). A prior and likelihood are said to be conjugate when the resulting posterior distribution is the same type of distribution as the prior. This means that if you have binomial data you can use a beta prior to obtain a beta posterior. If you had normal data you could use a normal prior and obtain a normal posterior. Conjugate priors are not required for doing bayesian updating, but they make the calculations a lot easier so they are nice to use if you can.
I’ll use some data from a recent NCAA 3-point shooting contest to illustrate how different priors can converge into highly similar posteriors.
## The data
This year’s NCAA shooting contest was a thriller that saw Cassandra Brown of the Portland Pilots win the grand prize. This means that she won the women’s contest and went on to defeat the men’s champion in a shoot-off. This got me thinking, just how good is Cassandra Brown?
What a great chance to use some real data in a toy example. She completed 4 rounds of shooting, with 25 shots in each round, for a total of 100 shots (I did the math). The data are counts, so I’ll be using the binomial distribution as a data model (i.e., the likelihood. See this previous post for details). Her results were the following:
Round 1: 13/25 Round 2: 12/25 Round 3: 14/25 Round 4: 19/25
Total: 58/100
The likelihood curve below encompasses the entirety of statistical evidence that our 3-point data provide (footnote 1). The hypothesis with the most relative support is .58, and the curve is moderately narrow since there are quite a few data points. I didn’t standardize the height of the curve in order to keep it comparable to the other curves I’ll be showing.
## The prior
Now the part that people often make a fuss about: choosing the prior. There are a few ways to choose a prior. Since I am using a binomial likelihood, I’ll be using a conjugate beta prior. A beta prior has two shape parameters that determine what it looks like, and is denoted Beta(α, β). I like to think of priors in terms of what kind of information they represent. The shape parameters α and β can be thought of as prior observations that I’ve made (or imagined).
Imagine my trusted friend caught the end of Brown’s warm-up and saw her take two shots, making one and missing the other, and she tells me this information. This would mean I could reasonably use the common Beta(1, 1) prior, which represents a uniform density over [0, 1]. In other words, all possible values for Brown’s shooting percentage are given equal weight before taking data into account, because the only thing I know about her ability is that both outcomes are possible (Lee & Wagenmakers, 2005).
Another common prior is called Jeffreys’s prior, a Beta(1/2, 1/2) which forms a wide bowl shape. This prior would be recommended if you had extremely scarce information about Brown’s ability. Is Brown so good that she makes nearly every shot, or is she so bad that she misses nearly every shot? This prior says that Brown’s shooting rate is probably near the extremes, which may not necessarily reflect a reasonable belief for someone who is a college basketball player, but it has the benefit of having less influence on the posterior estimates than the uniform prior (since it is equal to 1 prior observation instead of 2). Jeffreys’s prior is popular because it has some desirable properties, such as invariance under parameter transformation (Jaynes, 2003). So if instead of asking about Brown’s shooting percentage I instead wanted to know her shooting percentage squared or cubed, Jeffreys’s prior would remain the same shape while many other priors would drastically change shape.
Or perhaps I had another trusted friend who had arrived earlier and seen Brown take her final 13 shots in warm-up, and she saw 4 makes and 9 misses. Then I could use a Beta(4, 9) prior to characterize this prior information, which looks like a hump over .3 with density falling slowly as it moves outward in either direction. This prior has information equivalent to 13 shots, or roughly an extra 1/2 round of shooting.
These three different priors are shown below.
These are but three possible priors one could use. In your analysis you can use any prior you want, but if you want to be taken seriously you’d better give some justification for it. Bayesian inference allows many rules for prior construction.”This is my personal prior” is a technically a valid reason, but if this is your only justification then your colleagues/reviewers/editors will probably not take your results seriously.
## Updating the prior via the likelihood
Now for the easiest part. In order to obtain a posterior, simply use Bayes’s rule:
$\ Posterior \propto Likelihood \ X \ Prior$
The posterior is proportional to the likelihood multiplied by the prior. What’s nice about working with conjugate distributions is that Bayesian updating really is as simple as basic algebra. We take the formula for the binomial likelihood, which from a previous post is known to be:
$\ Likelihood \ = \ p^x \big(1-p \big)^{n-x}$
and then multiply it by the formula for the beta prior with α and β shape parameters:
$\ Prior \ = \ p^{\alpha-1} \big(1-p \big)^{\beta-1}$
to obtain the following formula for the posterior:
$\ Posterior \ = \ p^x \big(1-p \big)^{n-x} p^{\alpha-1} \big(1-p \big)^{\beta-1}$
With a little bit of algebra knowledge, you’ll know that multiplying together terms with the same base means the exponents can be added together. So the posterior formula can be rewritten as:
$\ Posterior \ = \ p^x p^{\alpha-1}\big(1-p \big)^{n-x} \big(1-p \big)^{\beta-1}$
and then by adding the exponents together the formula simplifies to:
$\ Posterior \ = \ p^{\alpha-1+x} \big(1-p \big)^{\beta-1+n-x}$
and it’s that simple! Take the prior, add the successes and failures to the different exponents, and voila. The distributional notation is even simpler. Take the prior, Beta(α, β), and add the successes from the data, x, to α and the failures, n – x, to β, and there’s your posterior, Beta(α+x, β+n-x).
Remember from the previous post that likelihoods don’t care about what order the data arrive in, it always results in the same curve. This property of likelihoods is carried over to posterior updating. The formulas above serve as another illustration of this fact. It doesn’t matter if you add a string of six single data points, 1+1+1+1+1+1+1 or a batch of +6 data points; the posterior formula in either case ends up with 6 additional points in the exponents.
## Looking at some posteriors
Back to Brown’s shooting data. She had four rounds of shooting so I’ll treat each round as a batch of new data. Her results for each round were: 13/25, 12/25, 14/25, 19/25. I’ll show how the different priors are updated with each batch of data. A neat thing about bayesian updating is that after batch 1 is added to the initial prior, its posterior is used as the prior for the next batch of data. And as the formulas above indicate, the order or frequency of additions doesn’t make a difference on the final posterior. I’ll verify this at the end of the post.
In the following plots, the prior is shown in blue (as above), the likelihood in orange (as above), and the resulting posteriors after Brown’s first 13/25 makes in purple.
In the first and second plot the likelihood is nearly invisible because the posterior sits right on top of it. When the prior has only 1 or 2 data points worth of information, it has essentially no impact on the posterior shape (footnote 2). The third plot shows how the posterior splits the difference between the likelihood and the informed prior based on the relative quantity of information in each.
The posteriors obtained from the uniform and Jeffreys’s priors suggest the best guess for Brown’s shooting percentage is around 50%, whereas the posterior obtained from the informed prior suggests it is around 40%. No surprise here since the informed prior represents another 1/2 round of shots where Brown performed poorly, which shifts the posterior towards lower values. But all three posteriors are still quite broad, and the breadth of the curves can be thought to represent the uncertainty in my estimates. More data -> tighter curves -> less uncertainty.
Now I’ll add the second round performance as a new likelihood (12/25 makes), and I’ll take the posteriors from the first round of updating as new priors for the second round of updating. So the purple posteriors from the plots above are now blue priors, the likelihood is orange again, and the new posteriors are purple.
The left two plots look nearly identical, which should be no surprise since their posteriors were essentially equivalent after only 1 round of data updates. The third plot shows a posterior still slightly shifted to the left of the others, but it is much more in line with them than before. All three posteriors are getting narrower as more data is added.
The last two rounds of updating are shown below, again with posteriors from the previous round taken as priors for the next round. At this point they’ve all converged to very similar posteriors that are much narrower, translating to less uncertainty in my estimates.
These posterior distributions look pretty similar now! Just as an illustration, I’ll show what happens when I update the initial priors with all of the data at once.
As the formulas predict, the posteriors after one big batch of data are identical to those obtained by repeatedly adding multiple smaller batches of data. It’s also a little easier to see the discrepancies between the final posteriors in this illustration because the likelihood curve acts as a visual anchor. The uniform and Jeffreys’s priors result in posteriors that essentially fall right on top of the likelihood, whereas the informed prior results in a posterior that is very slightly shifted to the left of the likelihood.
My takeaway from these posteriors is that Cassandra Brown has a pretty damn good 3-point shot! In a future post I’ll explain how to use this method of updating to make inferences using Bayes factors. It’s called the Savage-Dickey density method, and I think it’s incredibly intuitive and easy to use.
## Notes:
Footnote 1: I’m making a major assumption about the data: Any one shot is exchangeable with any other shot. This might not be defensible since the final ball on each rack is worth a bonus point, so maybe those shots differ systematically from regular shots, but it’s a toy example so I’ll ignore that possibility. There’s also the possibility of her going on a hot streak, a.k.a. having a “hot hand”, but I’m going to ignore that too because I’m the one writing this blog post and I want to keep it simple. There’s also the possibility that she gets worse throughout the competition because she gets tired, but then there’s also the possibility that she gets better as she warms up with multiple rounds. All of these things are reasonable to consider and I am going to ignore them all.
Footnote 2: There is a tendency to call any priors that have very little impact on the posterior “non-informative”, but, as I mentioned in the section on determining priors, uniform priors that seem non-informative in one context can become highly informative with parameter transformation (Zhu & Lu, 2004). Jeffreys’s prior was derived precisely with that in mind, so it carries little information no matter what transformation is applied.
## R Code
shotData<- c(1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0) #figure 1 from blog, likelihood curve for 58/100 shots x = seq(.001, .999, .001) ##Set up for creating the distributions y2 = dbeta(x, 1 + 58, 1 + 42) # data for likelihood curve, plotted as the posterior from a beta(1,1) plot(x, y2, xlim=c(0,1), ylim=c(0, 1.25 * max(y2,1.6)), type = "l", ylab= "Density", lty = 3, xlab= "Probability of success", las=1, main="Likelihood Curve for 3-pt Shots", sub= "(Binomial Data, 58/100)",lwd=2, cex.lab=1.5, cex.main=1.5, col = "darkorange", axes=FALSE) axis(1, at = seq(0,1,.2)) #adds custom x axis axis(2, las=1) # custom y axis ## Function for plotting priors, likelihoods, and posteriors for binomial data ## Output consists of a plot and various statistics ## PS and PF determine the shape of the prior distribution. ## PS = prior success, PF = prior failure for beta dist. ## PS = 1, PF = 1 corresponds to uniform(0,1) and is default. If left at default, posterior will be equivalent to likelihood ## k = number of observed successes in the data, n = total trials. If left at 0 only plots the prior dist. ## null = Is there a point-null hypothesis? null = NULL leaves it out of plots and calcs ## CI = Is there a relevant X% credibility interval? .95 is recommended and standard plot.beta <- function(PS = 1, PF = 1, k = 0, n = 0, null = NULL, CI = NULL, ymax = "auto", main = NULL) { x = seq(.001, .999, .001) ##Set up for creating the distributions y1 = dbeta(x, PS, PF) # data for prior curve y3 = dbeta(x, PS + k, PF + n - k) # data for posterior curve y2 = dbeta(x, 1 + k, 1 + n - k) # data for likelihood curve, plotted as the posterior from a beta(1,1) if(is.numeric(ymax) == T){ ##you can specify the y-axis maximum y.max = ymax } else( y.max = 1.25 * max(y1,y2,y3,1.6) ##or you can let it auto-select ) if(is.character(main) == T){ Title = main } else( Title = "Prior-to-Posterior Transformation with Binomial Data" ) plot(x, y1, xlim=c(0,1), ylim=c(0, y.max), type = "l", ylab= "Density", lty = 2, xlab= "Probability of success", las=1, main= Title,lwd=3, cex.lab=1.5, cex.main=1.5, col = "skyblue", axes=FALSE) axis(1, at = seq(0,1,.2)) #adds custom x axis axis(2, las=1) # custom y axis if(n != 0){ #if there is new data, plot likelihood and posterior lines(x, y2, type = "l", col = "darkorange", lwd = 2, lty = 3) lines(x, y3, type = "l", col = "darkorchid1", lwd = 5) legend("topleft", c("Prior", "Posterior", "Likelihood"), col = c("skyblue", "darkorchid1", "darkorange"), lty = c(2,1,3), lwd = c(3,5,2), bty = "n", y.intersp = .55, x.intersp = .1, seg.len=.7) ## adds null points on prior and posterior curve if null is specified and there is new data if(is.numeric(null) == T){ ## Adds points on the distributions at the null value if there is one and if there is new data points(null, dbeta(null, PS, PF), pch = 21, bg = "blue", cex = 1.5) points(null, dbeta(null, PS + k, PF + n - k), pch = 21, bg = "darkorchid", cex = 1.5) abline(v=null, lty = 5, lwd = 1, col = "grey73") ##lines(c(null,null),c(0,1.11*max(y1,y3,1.6))) other option for null line } } ##Specified CI% but no null? Calc and report only CI if(is.numeric(CI) == T && is.numeric(null) == F){ CI.low <- qbeta((1-CI)/2, PS + k, PF + n - k) CI.high <- qbeta(1-(1-CI)/2, PS + k, PF + n - k) SEQlow<-seq(0, CI.low, .001) SEQhigh <- seq(CI.high, 1, .001) ##Adds shaded area for x% Posterior CIs cord.x <- c(0, SEQlow, CI.low) ##set up for shading cord.y <- c(0,dbeta(SEQlow,PS + k, PF + n - k),0) ##set up for shading polygon(cord.x,cord.y,col='orchid', lty= 3) ##shade left tail cord.xx <- c(CI.high, SEQhigh,1) cord.yy <- c(0,dbeta(SEQhigh,PS + k, PF + n - k), 0) polygon(cord.xx,cord.yy,col='orchid', lty=3) ##shade right tail return( list( "Posterior CI lower" = round(CI.low,3), "Posterior CI upper" = round(CI.high,3))) } ##Specified null but not CI%? Calculate and report BF only if(is.numeric(null) == T && is.numeric(CI) == F){ null.H0 <- dbeta(null, PS, PF) null.H1 <- dbeta(null, PS + k, PF + n - k) CI.low <- qbeta((1-CI)/2, PS + k, PF + n - k) CI.high <- qbeta(1-(1-CI)/2, PS + k, PF + n - k) return( list("BF01 (in favor of H0)" = round(null.H1/null.H0,3), "BF10 (in favor of H1)" = round(null.H0/null.H1,3) )) } ##Specified both null and CI%? Calculate and report both if(is.numeric(null) == T && is.numeric(CI) == T){ null.H0 <- dbeta(null, PS, PF) null.H1 <- dbeta(null, PS + k, PF + n - k) CI.low <- qbeta((1-CI)/2, PS + k, PF + n - k) CI.high <- qbeta(1-(1-CI)/2, PS + k, PF + n - k) SEQlow<-seq(0, CI.low, .001) SEQhigh <- seq(CI.high, 1, .001) ##Adds shaded area for x% Posterior CIs cord.x <- c(0, SEQlow, CI.low) ##set up for shading cord.y <- c(0,dbeta(SEQlow,PS + k, PF + n - k),0) ##set up for shading polygon(cord.x,cord.y,col='orchid', lty= 3) ##shade left tail cord.xx <- c(CI.high, SEQhigh,1) cord.yy <- c(0,dbeta(SEQhigh,PS + k, PF + n - k), 0) polygon(cord.xx,cord.yy,col='orchid', lty=3) ##shade right tail return( list("BF01 (in favor of H0)" = round(null.H1/null.H0,3), "BF10 (in favor of H1)" = round(null.H0/null.H1,3), "Posterior CI lower" = round(CI.low,3), "Posterior CI upper" = round(CI.high,3))) } } #plot dimensions (415,550) for the blog figures #Initial Priors plot.beta(1,1,ymax=3.2,main="Uniform Prior, Beta(1,1)") plot.beta(.5,.5,ymax=3.2,main="Jeffreys's Prior, Beta(1/2,1/2)") plot.beta(4,9,ymax=3.2,main="Informed Prior, Beta(4,9)") #Posteriors after Round 1 plot.beta(1,1,13,25,main="Beta(1,1) to Beta(14,13)",ymax=10) plot.beta(.5,.5,13,25,main="Beta(1/2,1/2) to Beta(13.5,12.5)",ymax=10) plot.beta(4,9,13,25,main="Beta(4,9) to Beta(17,21)",ymax=10) #Posteriors after Round 2 plot.beta(14,13,12,25,ymax=10,main="Beta(14,13) to Beta(26,26)") plot.beta(13.5,12.5,12,25,ymax=10,main="Beta(13.5,12.5) to Beta(25.5,25.5)") plot.beta(17,21,12,25,ymax=10,main="Beta(17,21) to Beta(29,34)") #Posteriors after Round 3 plot.beta(26,26,14,25,ymax=10,main="Beta(26,26) to Beta(40,37)") plot.beta(25.5,25.5,14,25,ymax=10,main="Beta(25.5,25.5) to Beta(39.5,36.5)") plot.beta(29,34,14,25,ymax=10,main="Beta(29,34) to Beta(43,45)") #Posteriors after Round 4 plot.beta(40,37,19,25,ymax=10,main="Beta(40,37) to Beta(59,43)") plot.beta(39.5,36.5,19,25,ymax=10,main="Beta(39.5,36.5) to Beta(58.5,42.5)") plot.beta(43,45,19,25,ymax=10,main="Beta(43,45) to Beta(62,51)") #Initial Priors and final Posteriors after all rounds at once plot.beta(1,1,58,100,ymax=10,main="Beta(1,1) to Beta(59,43)") plot.beta(.5,.5,58,100,ymax=10,main="Beta(1/2,1/2) to Beta(58.5,42.5)") plot.beta(4,9,58,100,ymax=10,main="Beta(4,9) to Beta(62,51)")
view raw updating.R hosted with ❤ by GitHub
## References:
Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge University Press.
Lee, M. D., & Wagenmakers, E. J. (2005). Bayesian statistical inference in psychology: Comment on Trafimow (2003). Psychological Review, 112(3), 662-668.
Raiffa, H. & Schlaifer, R. (1961). Applied statistical decision theory. Division of Research, Graduate School of Business Administration, Harvard University.
Zhu, M., & Lu, A. Y. (2004). The counter-intuitive non-informative prior for the Bernoulli family. Journal of Statistics Education, 12(2), 1-10.
## 7 thoughts on “Understanding Bayes: Updating priors via the likelihood”
1. […] Understanding Bayes: Updating priors via the likelihood In this post I explain how to use the likelihood to update a prior into a posterior. The simplest way to illustrate likelihoods as an updating factor is to use conjugate distribution families (Raiffa & Schlaifer, 1961). A prior and likelihood are said to be conjugate when the resulting posterior distribution is the same type of distribution as the prior. This means that if you have binomial data you can use a beta prior to obtain a beta posterior. If you had normal data you could use a normal prior and obtain a normal posterior. Conjugate priors are not required for doing bayesian updating, but they make the calculations a lot easier so they are nice to use if you can. […]
2. […] shown in another post how the likelihood works as the updating factor for turning priors into posteriors for parameter […]
3. That’s a very interesting post 🙂
I’ve got a general question.
Let k be a parameter which must be estimated. It lies within the interval [a;b], a and b being finite real numbers.
Let us further assume we dispose of a series of measurements x of known standard deviations.
X is a complex function of k.
Do we dispose of some type of guarantee that Jefrreys prior imitates the shape of the likelihood in such a way that the posterior distribution delivers us a result close enough to the Maximum Likelihood Estimate?
If not, what about Bernardo’s prior?
I’m asking this question for cases where only few measurements are at hand.
In such a situation, I personally would simply use the likelihood l_x(k) in order to define the prior as f0(k) = l_x(k)/I, I being the integral of l_x over the possible values of k (which is a closed interval).
While this doubtlessly sounds very scandalous to any orthodox Bayesian, this has the great advantage of preventing an unphysical prior from erasing the influence of the experimental data on the posterior.
I’d be very interested to know, however, if either Jeffreys prior or Bernardo’s prior could do the job as well.
Cheers.
• Interesting question, lotharson. As you correctly guess, I’m not inclined to agree that one should use the likelihood to define the prior. Here’s my general reaction:
You want to know if there is any guarantee that the shape of a posterior derived from Jeffreys’s prior gives general agreement with the shape of the likelihood (and presumably if the mode closely agrees with the MLE?). This guarantee is partially provided by the fact that the Jeffreys’s prior’s shape is invariant to re-parameterization of the problem (sharing this property with Likelihoods).
In your proposed solution, to get your prior you divide the likelihood with respect the marginal likelihood (its integration over the parameter space) but I don’t see how you can do that without defining the relative weights for different parts of the parameter space (i.e., a prior). Could you clarify for me? If each part of the parameter space is weighted equally then your new prior will effectively be the posterior resulting from multiplying the likelihood to a uniform prior. If you then update this new prior with the likelihood you have essentially just counted your data twice. (If the integration over the parameter space is weighted non-uniformly then I think I’ve missed something in your example.)
Could you clarify what an unphysical prior is? (improper, or..?)
In short I’m not sure I fully understood your question. Let me know if you think I totally missed your point. And thanks for commenting!
4. JM says:
Why didn’t you continue with the example of Cassandra Brown all the way through updating the posterior? I think it would provide a much clearer explanation.
5. This post was clear and helpful!
One question for you – You mention that it would greatly complicate the analysis if the shots were not independent (i.e. If Cassandra improved or tired over time). Do you have any advice on how one would begin approaching that problem? If we assume Cassandra improved her shooting by .03 every ten shots (So her shooting percentage equaled something like 0.40+0.003t.. I realize that seems odd in this context and eventually she would be making over 100% of her shots…) But I’m thinking one would have to weight newer observations more than old ones or estimate both the starting point and the trend? Just trying to determine to what extent this is possible – thanks!
6. […] Introduction to the concept of likelihood and its applications [preprint] (which takes from some of my blog posts [1, 2]) […]
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# Find sum binomial coefficients
$$\sum_{k>=1}^{\infty} {2N \choose N-k}k$$
How to find this sum?
I know that the answer is $\frac{1}{2}N{2N \choose N}$
But it is very interesting to know the solution :)
Our first goal in dealing with this sum is to get rid of the $k$. The standard approach is to rewrite something like $\binom{n}{k} \cdot k$ as $\frac nk \binom{n-1}{k-1} \cdot k$, or $n \binom{n-1}{k-1}$. Here, the bottom index doesn't match the extra factor, but we can make it so with a little extra work: \begin{align} \binom{2N}{N-k} k &= \binom{2N}{N+k}k \\ &= \binom{2N}{N+k}(N+k) - \binom{2N}{N+k} N \\ &= \frac{2N}{N+k} \binom{2N-1}{N+k-1}(N+k) - N \binom{2N}{N+k} \\ &= 2N \binom{2N-1}{N+k-1} - N \binom{2N}{N+k}. \end{align} At this point, we have two sums that are both easier to deal with: $$\sum_{k \ge 1} \binom{2N}{N-k} k = 2N \sum_{k \ge 1} \binom{2N-1}{N+k-1} - N \sum_{k \ge 1} \binom{2N}{N+k}.$$ The sum of all binomial coefficients of the form $\binom{2N-1}{i}$ is $2^{2N-1}$, and our first sum takes only those binomial coefficients of this form where $i \ge N$. These are the second half, which by symmetry is equal to the first half, so the first sum simplifies to $2^{2N-2}$.
We're in much the same position with the second sum, except that the $\binom{2N}{i}$ coefficients also have a central coefficient $\binom{2N}{N}$, which is left out here. The sum of all coefficients that aren't the central one is $2^{2N} - \binom{2N}{N}$, and this sum is half of that.
Putting these facts together, we get \begin{align} \sum_{k \ge 1} \binom{2N}{N-k} k &= 2N \Bigg(2^{2N-2}\Bigg) - N \Bigg(2^{2N-1} - \frac12\binom{2N}{N}\Bigg) \\ &= N \cdot 2^{2N-1} - N \cdot 2^{2N-1} + \frac N2 \cdot \binom{2N}{N} \\ &= \frac N2 \cdot \binom{2N}{N}. \end{align}
Here is a slightly different variation.
We obtain \begin{align*} \sum_{k=1}^\infty\binom{2N}{N-k}k&=\sum_{k=0}^N\binom{2N}{N-k}k\tag{1}\\ &=\sum_{k=0}^N\binom{2N}{k}(N-k)\tag{2}\\ &=N\sum_{k=0}^N\binom{2N}{k}-\sum_{k=1}^N\binom{2N}{k}k\\ &=N\left(2^{2N-1}+\frac{1}{2}\binom{2N}{N}\right)-2N\sum_{k=1}^N\binom{2N-1}{k-1}\tag{3}\\ &=2^{2N-1}N+\frac{N}{2}\binom{2N}{N}-2N\sum_{k=0}^{N-1}\binom{2N-1}{k}\tag{4}\\ &=2^{2N-1}N+\frac{N}{2}\binom{2N}{N}-2N\cdot2^{2N-2}\tag{5}\\ &=\frac{N}{2}\binom{2N}{N} \end{align*}
Comment:
• In (1) we set the upper limit of the sum to $N$ since $\binom{2N}{N-k}=0$ for $k>N$. We also start with $k=0$ without changing anything since we are adding zero only.
• In (2) we change the order of summation $k\rightarrow N-k$.
• In (3) we use the symmetry of the binomial coefficients and the fact that they sum up to $2^{2N}$. We have additionally to respect the central binomial coefficient since we have an odd number of summands. We also use the binomial identity $\binom{p}{q}=\frac{p}{q}\binom{p-1}{q-1}$ for the right-hand sum.
• In (4) we shift the index of the sum by $1$.
• In (5) we again use the symmetry of the binomial coefficients and the fact that they sum up to $2^{2N-1}$. This time we have an even number of summands and need not to care for a central binomial coefficient.
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MapExpression -- the class of map expressions
Description
MapExpression is a type of Expression representing a map.
i1 : MapExpression {a,b,c} o1 = a <----- b c o1 : Expression of class MapExpression
For the programmer
The object MapExpression is , with ancestor classes Expression < BasicList < Thing.
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# Second law of thermodynamics
The second law of thermodynamics states that the entropy of an isolated system never decreases, because isolated systems always evolve toward thermodynamic equilibrium, a state with maximum entropy.
The second law is an empirically validated postulate of thermodynamics. In classical thermodynamics, the second law is a basic postulate defining the concept of thermodynamic entropy, applicable to any system involving measurable heat transfer. In statistical thermodynamics, the second law is a consequence of unitarity in quantum mechanics. In statistical mechanics information entropy is defined from information theory, known as the Shannon entropy. In the language of statistical mechanics, entropy is a measure of the number of alternative microscopic configurations corresponding to a single macroscopic state.
The second law refers to increases in entropy that can be analyzed into two varieties, due to dissipation of energy and due to dispersion of matter. One may consider a compound thermodynamic system that initially has interior walls that restrict transfers within it. The second law refers to events over time after a thermodynamic operation on the system, that allows internal heat transfers, removes or weakens the constraints imposed by its interior walls, and isolates it from the surroundings. As for dissipation of energy, the temperature becomes spatially homogeneous, regardless of the presence or absence of an externally imposed unchanging external force field. As for dispersion of matter, in the absence of an externally imposed force field, the chemical concentrations also become as spatially homogeneous as is allowed by the permeabilities of the interior walls. Such homogeneity is one of the characteristics of the state of internal thermodynamic equilibrium of a thermodynamic system.[1][2]
The second law may be expressed in many specific ways, but the first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics).
## Introduction
The first law of thermodynamics provides the basic definition of thermodynamic energy, also called internal energy, associated with all thermodynamic systems, but unknown in classical mechanics, and states the rule of conservation of energy in nature. [clarification needed]
The concept of energy in the first law does not, however, account for the observation that natural processes have a preferred direction of progress. The first law is symmetrical with respect to the initial and final states of an evolving system. But the second law asserts that a natural process runs only in one sense, and is not reversible. For example, heat always flows spontaneously from hotter to colder bodies, and never the reverse, unless external work is performed on the system. The key concept for the explanation of this phenomenon through the second law of thermodynamics is the definition of a new physical quantity, the entropy. [clarification needed]
For mathematical analysis of processes, entropy is introduced as follows. In a fictive reversible process, an infinitesimal increment in the entropy (dS) of a system results from an infinitesimal transfer of heat (δQ) to a closed system divided by the common temperature (T) of the system and the surroundings which supply the heat.[3]
$dS = \frac{\delta Q}{T} \!$
The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.[4] For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.[5][6] [clarification needed]
## Various statements of the law
The second law of thermodynamics may be expressed in many specific ways,[7] the most prominent classical statements[8] being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.[9]
### Carnot's principle
The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. It states
The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.[10][11][12][13][14][15][16]
[clarification needed]
### Clausius statement
The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.[17] His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.[18]
Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.
### Kelvin statement
Lord Kelvin expressed the second law as
It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.[19]
### Equivalence of the Clausius and the Kelvin statements
Derive Kelvin Statement from Clausius Statement
Suppose there is an engine violating the Kelvin statement: i.e.,one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the graph. The net and sole effect of this newly created engine consisting of the two engines mentioned is transferring heat $\Delta Q=Q\left(\frac{1}{\eta}-1\right)$ from the cooler reservoir to the hotter one, which violates the Clausius statement. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.
### Planck's proposition
Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.
It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.[20][21]
### Relation between Kelvin's statement and Planck's proposition
It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law. For example, see.[22] One text gives a statement that for all the world looks like Planck's proposition, but attributes it to Kelvin without mention of Planck.[23] One monograph quotes Planck's proposition as the "Kelvin-Planck" formulation, the text naming Kelvin as its author, though it correctly cites Planck in its references.[24] The reader may compare the two statements quoted just above here.
### Planck's statement
Planck stated the second law as follows.
Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.[25][26][27]
### Principle of Carathéodory
Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:[28]
In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.[29]
With this formulation he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, $\delta Q=TdS$.[30] [clarification needed]
Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.[31][32][33][34] [clarification needed]
### Planck's Principle
In 1926 Max Planck wrote an important paper on the basics of thermodynamics.[33][35] He indicated the principle
The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.[31][32]
This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."[36] Using a now obsolete form of words, Planck himself wrote: "The production of heat by friction is irreversible."[37][38]
Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.[39] Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.
The link to Kelvin's statement is illustrated by an equivalent statement by Allahverdyan & Nieuwenhuizen, which they attribute to Kelvin: "No work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source."[40][41]
### Statement for a system that has a known expression of its internal energy as a function of its extensive state variables
The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).[42][43] [clarification needed]
### Gravitational systems
In non-gravitational systems, objects always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time.
However, this is not always the case for systems in which the gravitational force is important. The most striking examples are black holes, which – according to theory – have negative heat capacity. The larger the black hole, the more energy it contains, but the lower its temperature. Thus, the supermassive black hole in the center of the Milky Way is supposed to have a temperature of 10−14 K, much lower than the cosmic microwave background temperature of 2.7K, but as it absorbs photons of the cosmic microwave background its mass is increasing so that its low temperature further decreases with time.
For this reason, gravitational systems tend towards non-even distribution of mass and energy. The universe in large scale is importantly a gravitational system, and the second law may therefore not apply to it.
## Corollaries
### Perpetual motion of the second kind
Main article: Perpetual motion
Before the establishment of the Second Law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of First Law of Thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.
### Carnot theorem
Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
• All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs.
• All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.
In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot however further postulated that some caloric is lost, not being converted to mechanical work. Hence no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.
Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.
### Clausius Inequality
The Clausius Theorem (1854) states that in a cyclic process
$\oint \frac{\delta Q}{T} \leq 0.$
The equality holds in the reversible case[44] and the '<' is in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.
### Thermodynamic temperature
For an arbitrary heat engine, the efficiency is:
$\eta = \frac {A}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)$
where A is the work done per cycle. Thus the efficiency depends only on qC/qH.
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is to say, the efficiency is the function of temperatures only: $\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).$
In addition, a reversible heat engine operating between temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 andT3. This can only be the case if
$f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).$
Now consider the case where $T_1$ is a fixed reference temperature: the temperature of the triple point of water. Then for any T2 and T3,
$f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.$
Therefore if thermodynamic temperature is defined by
$T = 273.16 \cdot f(T_1,T) \,$
then the function f, viewed as a function of thermodynamic temperature, is simply
$f(T_2,T_3) = \frac{T_3}{T_2},$
and the reference temperature T1 will have the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)
### Entropy
According to the Clausius equality, for a reversible process
$\oint \frac{\delta Q}{T}=0$
That means the line integral $\int_L \frac{\delta Q}{T}$ is path independent.
So we can define a state function S called entropy, which satisfies
$dS = \frac{\delta Q}{T} \!$
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.
For any irreversible process, since entropy is a state function, we can always connect the initial and terminal status with an imaginary reversible process and integrating on that path to calculate the difference in entropy.
Now reverse the reversible process and combine it with the said irreversible process. Applying Clausius inequality on this loop,
$-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0$
Thus,
$\Delta S \ge \int \frac{\delta Q}{T} \,\!$
where the equality holds if the transformation is reversible.
Notice that if the process is an adiabatic process, then $\delta Q=0$, so $\Delta S\ge 0$.
### Energy, available useful work
An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature TR and pressure PR — so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain TR; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain PR.
Whatever changes to dS and dSR occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy Stot of the isolated total system must not decrease:
$dS_{\mathrm{tot}}= dS + dS_R \ge 0$
According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μiRNi, so that:
$dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,$
where μiR are the chemical potentials of chemical species in the external surroundings.
Now the heat leaving the reservoir and entering the sub-system is
$\delta q = T_R (-dS_R) \le T_R dS$
where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.
It therefore follows that any net work δw done by the sub-system must obey
$\delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,$
It is useful to separate the work δw done by the subsystem into the useful work δwu that can be done by the sub-system, over and beyond the work pR dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:
$\delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,$
It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,
$E = U - T_R S + p_R V - \sum \mu_{iR} N_i$
The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,
$dE + \delta w_u \le 0 \,$
i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.
In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.
$dS_{tot} \ge 0$ Is equivalent to $dE + \delta w_u \le 0$
This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)
This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.
## History
Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the École Polytechnique.
The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.
Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).
Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.
The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.
It has been shown that not only classical systems but also quantum mechanical ones tend to maximize their entropy over time. Thus the second law follows, given initial conditions with low entropy. More precisely, it has been shown that the local von Neumann entropy is at its maximum value with a very high probability.[45] The result is valid for a large class of isolated quantum systems (e.g. a gas in a container). While the full system is pure and therefore does not have any entropy, the entanglement between gas and container gives rise to an increase of the local entropy of the gas. This result is one of the most important achievements of quantum thermodynamics.[dubious ]
Today, much effort in the field is attempting to understand why the initial conditions early in the universe were those of low entropy,[46][47] as this is seen as the origin of the second law (see below).
### Informal descriptions
The second law can be stated in various succinct ways, including:
• It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir (Kelvin, 1851).
• It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir (Clausius, 1854).
• If thermodynamic work is to be done at a finite rate, free energy must be expended. (Stoner, 2000)[48]
### Mathematical descriptions
Rudolf Clausius
In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:[49]
$\int \frac{\delta Q}{T} = -N$
where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:
The entropy of the universe tends to a maximum.
This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified version of a more extended and precise description.
In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:
$\frac{dS}{dt} \ge 0$
where
S is the entropy of the system and
t is time.
The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. An alternative way of formulating of the second law for isolated systems is:
$\frac{dS}{dt} = \dot S_{i}$ with $\dot S_{i} \ge 0$
with $\dot S_{i}$ the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature $T_{a}$ it gives the so-called dissipated energy $P_{diss}=T_{a}\dot S_{i}$.
The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:
$\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}$ with $\dot S_{i} \ge 0$
Here
$\dot Q$ is the heat flow into the system
$T$ is the temperature at the point where the heat enters the system.
If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.
For open systems (also allowing exchange of matter):
$\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}$ with $\dot S_{i} \ge 0$
Here $\dot S$ is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.
Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.
## Derivation from statistical mechanics
Further information: H-theorem
Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.[50][51][52]
Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.[citation needed] The first part of the second law, which states that the entropy of a thermally isolated system can only increase is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of $E$ is:
$S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,$
where $\Omega\left(E\right)$ is the number of quantum states in a small interval between $E$ and $E +\delta E$. Here $\delta E$ is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of $\delta E$. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on $\delta E$.
Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then $\Omega$ will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that $\Omega$ is maximized as that is the most probable situation in equilibrium.
If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that $\Omega$ is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value). Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number $\Omega$ of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of $1/\Omega$. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity H increases monotonically as a function of time during the intermediate out of equilibrium state.
### Derivation of the entropy change for reversible processes
The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:
$dS =\frac{\delta Q}{T}$
where the temperature is defined as:
$\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}$
See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.
The generalized force, X, corresponding to the external variable x is defined such that $X dx$ is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate $E_{r}$ is given by:
$X = -\frac{dE_{r}}{dx}$
Since the system can be in any energy eigenstate within an interval of $\delta E$, we define the generalized force for the system as the expectation value of the above expression:
$X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,$
To evaluate the average, we partition the $\Omega\left(E\right)$ energy eigenstates by counting how many of them have a value for $\frac{dE_{r}}{dx}$ within a range between $Y$ and $Y + \delta Y$. Calling this number $\Omega_{Y}\left(E\right)$, we have:
$\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,$
The average defining the generalized force can now be written:
$X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,$
We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x + dx. Then $\Omega\left(E\right)$ will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between $E$ and $E+\delta E$. Let's focus again on the energy eigenstates for which $\frac{dE_{r}}{dx}$ lies within the range between $Y$ and $Y + \delta Y$. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are
$N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,$
such energy eigenstates. If $Y dx\leq\delta E$, all these energy eigenstates will move into the range between $E$ and $E+\delta E$ and contribute to an increase in $\Omega$. The number of energy eigenstates that move from below $E+\delta E$ to above $E+\delta E$ is, of course, given by $N_{Y}\left(E+\delta E\right)$. The difference
$N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,$
is thus the net contribution to the increase in $\Omega$. Note that if Y dx is larger than $\delta E$ there will be the energy eigenstates that move from below E to above $E+\delta E$. They are counted in both $N_{Y}\left(E\right)$ and $N_{Y}\left(E+\delta E\right)$, therefore the above expression is also valid in that case.
Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:
$\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,$
The logarithmic derivative of $\Omega$ w.r.t. x is thus given by:
$\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,$
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:
$\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,$
Combining this with
$\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,$
Gives:
$dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,$
### Derivation for systems described by the canonical ensemble
If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:
$P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}$
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:
$S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)$
that
$dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}$
Inserting the formula for $P_{j}$ for the canonical ensemble in here gives:
$dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}$
### General derivation from unitarity of quantum mechanics
The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently, the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics.[53][54] This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the general transition matrix.
## Non-equilibrium states
It is only by convention, for the purposes of thermodynamic analysis, that any arbitrary occasion of space-time is said to be in thermodynamic equilibrium. In general, an occasion of space-time found in nature is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.[55][56] If it is assumed, for the purposes of physical analysis, that one is dealing with a system in thermodynamic equilibrium, then statistically it is possible for that system to achieve moments of non-equilibrium. In some statistically unlikely events, hot particles "steal" the energy of cold particles, enough that the cold side gets colder and the hot side gets hotter, for a very brief time.
The physics involved in such events is beyond the scope of classical equilibrium thermodynamics, and is the topic of the fluctuation theorem (not to be confused with the fluctuation-dissipation theorem). This was first proved by Bochov and Kuzovlev,[57] and later by Evans and Searles.[58][59] It gives a numerical estimate of the probability that a system away from equilibrium will have a certain change in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamical equations of motion but assumes the Axiom of Causality, which is equivalent to assuming uncorrelated initial conditions (namely, uncorrelated past). Such events have been observed at a small enough scale where the likelihood of such a thing happening is significant. Quantitative predictions of this theorem have been confirmed in laboratory experiments by use of optical tweezers apparatus.[60]
## Arrow of time
Further information: Entropy (arrow of time)
The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction.
The second law has been proposed to supply an explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).[61]
## Controversies
### Maxwell's demon
Main article: Maxwell's demon
James Clerk Maxwell
James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.
One of the most famous responses to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.
Maxwell's demon repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations, not just presiding over natural processes.
Main article: Loschmidt's paradox
Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of nearly all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behavior of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.
One proposed resolution of this paradox is as follows. The Loschmidt scenario refers to a strictly isolated system or to a strictly adiabatically isolated system. Heat and matter transfers are not allowed. The Loschmidt reversal times are fantastically long, far longer than any laboratory isolation of the required degree of perfection could be maintained in practice. In this sense, the Loschmidt scenario will never be subjected to empirical testing. Also in this sense, the second law, stated for an isolated system, will never be subjected to empirical testing. A system, supposedly perfectly isolated, in strictly perfect thermodynamic equilibrium, can be observed only once in its entire life, because the observation must break the isolation. Two observations would be needed to check empirically for a change of state, one initial and one final. When transfer of heat or matter are permitted, the requirements of perfection are not so tight. In practical laboratory reality, therefore, the second law can be tested only for systems with transfer of heat or matter, and not for isolated systems.
Due to this paradox, derivations of the second law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past or, equivalently, that the entropy in the past was lower than in the future. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately derived from the initial conditions of the Big Bang.[50][62]
### Poincaré recurrence theorem
The Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence, which is of the order of $\sim \exp\left(S/k\right)$.[63] The result applies to physical systems in which energy is conserved. The Recurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the system will never return to it. There are many possible ways to resolve this paradox, but none of them is universally accepted.[citation needed] The most reasonable argument is that for typical thermodynamical systems the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence.
## Quotations
The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
—Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)
There have been nearly as many formulations of the second law as there have been discussions of it.
—Philosopher / Physicist P.W. Bridgman, (1941)
Clausius is the author of the sybillic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.
Truesdell, C., Muncaster, R.G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN0-12-701350-4, p.17.
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2. ^ Bailyn, M. (1994), Section 71, pp. 254–256.
3. ^ Bailyn, M. (1994), p. 120.
4. ^ J. S. Dugdale (1996, 1998). Entropy and its Physical Meaning. Tayler & Francis. p. 13. ISBN 0-7484-0569-0. "This law is the basis of temperature."
5. ^ Zemansky, M.W. (1968), pp. 207–209.
6. ^ Quinn, T.J. (1983), p. 8.
7. ^ "Concept and Statements of the Second Law". web.mit.edu. Retrieved 2010-10-07.
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