content stringlengths 86 994k | meta stringlengths 288 619 |
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Operators precedence.
Operators precedence.
Operators precedence is the order in which NumericBase performs operations in formulas.
If you combine several operators in a single formula, NumericBase performs the operations in the order shown in the following table.
In this table, an earlier location means higher precedence. For example, the member access operators has the highest precedence so they are evaluated first.
Operators Meaning
. [ ] @ member access operators such as in Table.a and a[3]
- Negation (as in -1).
* / Multiplication and division
+ - Addition and subtraction.
& Connects two strings of text (concatenation).
= < > <= >= <> Comparison.
not Logical operator.
and Logical operator.
or Logical operator.
if then else The if operator.
alt The "alt" operator
Table: Operators precedence. operators with the same precedence
If a formula contains operators with the same precedence - for example, if a formula contains both a multiplication and division operator - NumericBase evaluates the operators from left to right.
Using parenthesis
To change the order of evaluation, enclose the part of the formula to be calculated first in parentheses. | {"url":"https://symbolclick.com/numericbase/operators_precedence.htm","timestamp":"2024-11-10T00:58:48Z","content_type":"text/html","content_length":"8527","record_id":"<urn:uuid:565dbb8f-5cd6-461f-a31d-8600a594bbeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00770.warc.gz"} |
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Scholarly Works (47 results)
When purchasing products online, often two products may have similar mean ratings and numbers of reviews, but suchapparent similarities may hide important differences. Sometimes, the distribution of
star ratings is also available to decisionmakers in addition to these two attributes. Will the decision still be as undifferentiated as before or will the distributionsof stars engender a preference
towards one of the products? To answer this question, the current study manipulated thedisplayed variability of ratings for choices with the same average rating. The behavioral studies showed that
participantsexhibited distinctive choice patterns when the distribution of ratings was provided even when the average rating andtotal number of reviews were the same between two compared products. A
utility-based cognitive model was thereforedeveloped to identify the underlying mechanism as to why people chose the way they did.
Across two experiments, we use ordinal ranking to examinethe processing and representations involved in the estimationof large-scale, real-world proportions. Specifically, in twoexperiments people
estimated two kinds of important real-world proportions: the demographic makeup of theircommunities, and spending by the U.S. Federal government.Our goal was to assess the metric scaling properties
thatcharacterize perceptions of these quantities. In particular,previous work in numerical proportions has positedlogarithmic or linear representations (Opfer & Siegler, 2007),or linear
representations with task-dependent rescaling (Barth& Paladino, 2011; Cohen & Blanc-Goldhammer, 2011). Thecurrent context differs markedly from this prior work in thatthe values we are examining are
not explicitly presented toparticipants, nor directly experienced, but must be estimatedon the basis of masses of complex experiences. Ordinalranking of the quantities, combined with a
Thurstonianmodeling approach, allows a unique means for estimating theinternal scale properties of numerical structures. We find thatpeople largely rely on mixed representations that
emphasizelog-odds transformations of these vaguely known, butsocially important values. While the budget data explored inExperiment 1 were unable to distinguish log and log-oddstransformed internal
models, the demographic proportionsexplored in Experiment 2 favored log-odds models.
A plethora of research over the past two decades has demonstrated that citizens in countries around the world dramatically overestimate the size of minority demographic groups and underestimate the
size of majority groups. Researchers have concluded that this misestimation is a result of characteristics of the group being estimated, such as level of threat the group poses and the amount of
exposure someone has with to the group. However, explanations of this misestimation have largely ignored theoretical models of perception and measurement, such as those developed in classic
psychophysics. This has led to interpretations that are at variance with modern theories of measurement. We present a model which combines an understanding of the nature of human estimations with a
conceptualization of uncertainty, which extends to accommodate bias. We apply this model to three large-scale datasets collected by the Ipsos MORI research group. Model fits from our approach suggest
that to a considerable degree, the errors people make are due to uncertainty rather than bias. These biases are quite different in character from those that other groups have reported. Many of the
present biases, furthermore, are shared widely across different countries.
People use analogies for many cognitive purposes such asbuilding mental models, making inspired guesses, andextracting relational structure. Here we examine whether andhow analogies may have more
direct influence on knowledge:Do people treat analogies as probabilistically trueexplanations for uncertain propositions?We report an experiment that explores how a suggestedanalogy can influence
people’s confidence in inferences.Participants made predictions while simultaneouslyevaluating a suggested analogy and observed evidence. In twoconditions, the evidence is either consistent with or
in conflictwith propositions based on the suggested analogy. Weanalyze the responses statistically and in a psychologicallyplausible Bayesian network model. We find that analogies areused for more
than just generating candidate inferences. Theyact as probabilistic truths that affect the integration ofevidence and confidence in both the target and sourcedomains. People readily treat analogies
not as a one-wayprojection from source to target, but as a mutually informativeconnection.
We have a simple thesis: the relationship between academic and industry-based cognitive science is broken, but can be fixed. Over the last few decades, there has been a huge increase in the
representation of cognitive science in industry. Beyond just machine learning, businesses are increasingly interested in human behavior and cognitive processes. Large proportions of our Ph.D.
students, post-docs, and even faculty choose to go through a largely one-way door to corporate jobs in data science, behavioral experimentation, machine learning, user experience, and elsewhere.
Currently, people who choose industry careers often lose their social and intellectual networks and their ability to return to tenure-track positions. Valuable insights from industry about memory,
decision-making, learning, emotion, distributed cognition, and much more never return to the academic community. We believe that deep, theory driven, theory building work is being done in industry
settings–and that the rift between communities makes all our work less effective | {"url":"https://escholarship.org/search/?q=author%3ALandy%2C%20David","timestamp":"2024-11-10T05:19:11Z","content_type":"text/html","content_length":"58133","record_id":"<urn:uuid:be94f587-dcc0-4dc5-8ce7-5b08ce9a7f22>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00669.warc.gz"} |
Scramblers are used in many communicaton protocols such as PCI Express, SAS/SATA, USB, Bluetooth to randomize the transmitted data. To keep this post short and focused I’ll not discuss the theory
behind scramblers. For more information about scramblers see [1], or do some googling. The topic of this post is the parallel implementation of a scrambler generator. Protocol specifications define
scrambling algorithm using either hex or polynimial notation. This is not always suitable for efficient hardware or software implementation. Please read my post on parallel CRC Generator about that.
The Parallel Scrambler Generator method that I’m going to describe has a lot in common with the Parallel CRC Generator. The difference is that CRC generator outputs CRC value, whereas Scrambler
generator produces scrambled data. But the internal working of both based on the same principle.
Here is an example of a scrambler with the polynomial G(x) = x^16+x^5+x^4+x^3+1
Following is the description of the Parallel Scrambler Generator algorithm:
(1) Let’s denote N=data width, M=generator polynomial width.
(2) Implement serial scrambler generator using given polynomial or hex notation. It’s easy to do in any programming language or script: C, Java, Perl, Verilog, etc.
(3) Parallel Scrambler implementation is a function of N-bit data input as well as M-bit current state of the polynomial, as shown in the above figure. We’re going to build three matrices:
• Mout (next state polynomial) as a function of Min(current state polynomial) when N=0 and
• Nout as a function of Nin when Min=0.
• Nout as a function of Min when Nin=0
Note that the polynomial next state doesn’t depend on the scrambled data, therefore we need only three matrices.
(4) Using the routine from (3) calculate scrambled data for the Mout values given Min, when Nin=0. Each Min value is one-hot encoded, that is there is only one bit set.
(5) Build MxM matrix, Each row contains the results from (4) in increasing order. For example, 1’st row contains the result of input=0×1, 2′nd row is input=0×2, etc. The output is M-bit wide, which
the polynomial width.
(6) Calculate the Nout values given Nin, when Min=0. Each Nin value is one-hot encoded, that is there is only one bit set.
(7) Build NxN matrix, Each row contains the results from (6) in increasing order. The output is N-bit wide, which the data width.
(8) Calculate the Nout values given Min, when Nin=0. Each Min value is one-hot encoded, that is there is only one bit set.
(9) Build MxN matrix, Each row contains the results from (7) in increasing order. The output is N-bit wide, which the data width.
(10) Now, build an equation for each Nout[i] bit: all Nin[j] and Min[k] set bits in column [i] from three matrices participate in the equation. The participating inputs are XORed together.
Nout is the parallel scrambled data.
Keep me posted if the Parallel Scrambler Generation tool works for you, or you need more clarifications on the algorithm. | {"url":"http://outputlogic.com/?tag=scrambler","timestamp":"2024-11-09T07:49:54Z","content_type":"application/xhtml+xml","content_length":"26197","record_id":"<urn:uuid:c77005d4-b87e-4d52-b903-7f7fbcf14c4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00699.warc.gz"} |
An Introduction to Model Merging for LLMs | NVIDIA Technical Blog
One challenge organizations face when customizing large language models (LLMs) is the need to run multiple experiments, which produces only one useful model. While the cost of experimentation is
typically low, and the results well worth the effort, this experimentation process does involve “wasted” resources, such as compute assets spent without their product being utilized, dedicated
developer time, and more.
Model merging combines the weights of multiple customized LLMs, increasing resource utilization and adding value to successful models. This approach provides two key solutions:
• Reduces experimentation waste by repurposing “failed experiments”
• Offers a cost-effective alternative to join training
This post explores how models are customized, how model merging works, different types of model merging, and how model merging is iterating and evolving.
Revisiting model customization
This section provides a brief overview of how models are customized and how this process can be leveraged to help build an intuitive understanding of model merging.
Note that some of the concepts discussed are oversimplified for the purpose of building this intuitive understanding of model merging. It is suggested that you familiarize yourself with customization
techniques, transformer architecture, and training separately before diving into model merging. See, for example, Mastering LLM Techniques: Customization.
The role of weight matrices in models
Weight matrices are essential components in many popular model architectures, serving as large grids of numbers (weights, or parameters) that store the information necessary for the model to make
As data flows through a model, it passes through multiple layers, each containing its own weight matrix. These matrices transform the input data through mathematical operations, enabling the model to
learn from and adapt to the data.
To modify a model’s behavior, the weights within these matrices must be updated. Although the specifics of weight modification are not essential, it’s crucial to understand that each customization of
a base model results in a unique set of updated weights.
Task customization
When fine-tuning an LLM for a specific task, such as summarization or math, the updates made to the weight matrices are targeted towards improving performance on that particular task. This implies
that the modifications to the weight matrices are localized to specific regions, rather than being uniformly distributed.
To illustrate this concept, consider a simple analogy where the weight matrices are represented as a sports field that is 100 yards in length. When customizing the model for summarization, the
updates to the weight matrices might concentrate on specific areas, such as the 10-to-30 yard lines. In contrast, customizing the model for math might focus updates on a different region, like the
70-to-80 yard lines.
Interestingly, when customizing the model for a related task, such as summarization in the French language, the updates might overlap with the original summarization task, affecting the same regions
of the weight matrices (the 25-to-35 yard lines, for example). This overlap suggests an important insight: different task customizations can significantly impact the same areas of the weight
While the previous example is purposefully oversimplified, the intuition is accurate. Different task customizations will lead to different parts of the weight matrices being updated, and
customization for similar tasks might lead to changing the same parts of their respective weight matrices.
This understanding can inform strategies for customizing LLMs and leveraging knowledge across tasks.
Model merging
Model merging is a loose grouping of strategies that relates to combining two or more models, or model updates, into a single model for the purpose of saving resources or improving task-specific
This discussion focuses primarily on the implementation of these techniques through an open-source library developed by Arcee AI called mergekit. This library simplifies the implementation of various
merging strategies.
Many methods are used to merge models, in various levels of complexity. Here, we’ll focus on four main merging methods:
1. Model Soup
2. Spherical Linear Interpolation (SLERP)
3. Task Arithmetic (using Task Vectors)
4. TIES leveraging DARE
Model Soup
The Model Soup method involves averaging the resultant model weights created by hyperparameter optimization experiments, as explained in Model Soups: Averaging Weights of Multiple Fine-Tuned Models
Improves Accuracy Without Increasing Inference Time.
Originally tested and verified through computer vision models, this method has shown promising results for LLMs as well. In addition to generating some additional value out of the experiments, this
process is simple and not compute intensive.
There are two ways to create Model Soup: naive and greedy. The naive approach involves merging all models sequentially, regardless of their individual performance. In contrast, the greedy
implementation follows a simple algorithm:
• Rank models by performance on the desired task
• Merge the best performing model with the second best performing model
• Evaluate the merged model’s performance on the desired task
• If the merged model performs better, continue with the next model; otherwise, skip the current model and try again with the next best model
This greedy approach ensures that the resulting Model Soup is at least as good as the best individual model.
Figure 1. The Model Soup method outperforms the constituent models using the greedy model Soup Model merging technique
Each step of creating a Model Soup is implemented by simple weighted and normalized linear averaging of two or more model weights. Both the weighting and normalization are optional, though
recommended. The implementation of this from the mergekit library is as follows:
res = (weights * tensors).sum(dim=0)
if self.normalize:
res = res / weights.sum(dim=0)
While this method has shown promising results in the computer vision and language domains, it faces some serious limitations. Specifically, there is no guarantee that the model will be more
performant. The linear averaging can lead to degraded performance or loss of generalizability.
The next method, SLERP, addresses some of those specific concerns.
Spherical Linear Interpolation, or SLERP, is a method introduced in a 1985 paper titled Animating Rotation with Quaternion Curves. It’s a “smarter” way of computing the average between two vectors.
In a technical sense, it helps compute the shortest path between two points on a curved surface.
This method excels at combining two models. The classic example is imagining the shortest path between two points on the Earth. Technically, the shortest path would be a straight line that goes
through the Earth, but in reality it’s a curved path on the surface of the Earth. SLERP computes this smooth path to use for averaging two models together while maintaining their unique model weight
The following code snippet is the core of the SLERP algorithm, and is what provides such a good interpolation between the two models:
# Calculate initial angle between v0 and v1
theta_0 = np.arccos(dot)
sin_theta_0 = np.sin(theta_0)
# Angle at timestep t
theta_t = theta_0 * t
sin_theta_t = np.sin(theta_t)
# Finish the slerp algorithm
s0 = np.sin(theta_0 - theta_t) / sin_theta_0
s1 = sin_theta_t / sin_theta_0
res = s0 * v0_copy + s1 * v1_copy
return maybe_torch(res, is_torch)
Task Arithmetic (using Task Vectors)
This group of model merging methods utilizes Task Vectors to combine models in various ways, increasing in complexity.
Task Vectors: Capturing customization updates
Recalling how models are customized, updates are made to the model’s weights, and those updates are captured in the base model matrices. Instead of considering the final matrices as a brand new
model, they can be viewed as the difference (or delta) between the base weights and the customized weights. This introduces the concept of a task vector,a structure containing the delta between the
base and customized weights.
This is the same intuition behind Low Rank Adaptation (LoRA), but without the further step of factoring the matrices representing the weight updates.
Task Vectors can be simply obtained from customization weights by subtracting out the base model weights.
Task Interference: Conflicting updates
Recalling the sports field example, there is a potential for overlap in the updated weights between different customizations. There is some intuitive understanding that customization done for the
same task would lead to a higher rate of conflicting updates than customization done for two, or more, separate tasks.
This “conflicting update” idea is more formally defined as Task Interference and it relates to the potential collision of important updates between two, or more, Task Vectors.
Task Arithmetic
As introduced in the paper Editing Models with Task Arithmetic, Task Arithmetic represents the simplest implementation of a task vector approach. The process is as follows:
1. Obtain two or more task vectors and merge them linearly as seen in Model Soup.
2. After the resultant merged task vector is obtained, it is added into the base model.
This process is simple and effective, but has a key weakness: no attention is paid to the potential interference between the task vectors intended to be merged.
As introduced in the paper TIES-Merging: Resolving Interference When Merging Models, TIES (TrIm Elect Sign and Merge) is a method that takes the core ideas of Task Arithmetic and combines it with
heuristics for resolving potential interference between the Task Vectors.
The general procedure is to consider, for each weight in the Task Vectors being merged, the magnitude of each incoming weight, then the sign of each incoming weight, and then averaging the remaining
Figure 2. A visual representation of the TIES process
This method seeks to resolve interference by enabling the models that had the most significant weight updates for any given weight update take precedence during the merging process. In essence, the
models that “cared” more about that weight would be prioritized over the models that did not.
Introduced in the paper Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch, DARE isn’t directly a model merging technique. Rather, it’s an augment that can be
considered alongside other approaches. DARE derives from the following:
Drops delta parameters with a ratio p And REscales the remaining ones by 1/(1 – p) to approximate the original embeddings.
Instead of trying to address the problem of interference through heuristics, DARE approaches it from a different perspective. In essence, it randomly drops a large number of the updates found in a
specific task vector by setting them to 0, and then rescales the remaining weight proportional to the ratio of the dropped weights.
DARE has been shown to be effective even when dropping upwards of 90%, or even 99% of the task vector weights.
Increase model utility with model merging
The concept of model merging offers a practical way to maximize the utility of multiple LLMs, including task-specific fine-tuning done by a larger community. Through techniques like Model Soup,
SLERP, Task Arithmetic, TIES-Merging, and DARE, organizations can effectively merge multiple models in the same family in order to reuse experimentation and cross-organizational efforts.
As the techniques behind model merging are better understood and further developed, they are poised to become a cornerstone of the development of performant LLMs. While this post has only scratched
the surface, more techniques are constantly under development, including some evolution-based methods. Model merging is a budding field in the generative AI landscape, as more applications are being
tested and proven. | {"url":"https://developer.nvidia.com/blog/an-introduction-to-model-merging-for-llms/","timestamp":"2024-11-14T17:51:54Z","content_type":"text/html","content_length":"213828","record_id":"<urn:uuid:ed354b6a-f740-42a9-9797-7eea2eaa391d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00610.warc.gz"} |
Data Archives
Data and BI Analysts often concentrate on learning a BI Tool, but the main thing to do is learn how to create good data visualization!
BI reporting has become an indispensable part of any company. In Business Intelligence, companies sometimes have to choose between tools such as PowerBI, QlikSense, Tableau, MikroStrategy, Looker or
DataStudio (and others). Even if each of these tools has its own strengths and weaknesses, good reporting depends less on the respective tool but much more on the analyst and his skills in structured
and appropriate visualization and text design.
Based on our experience at DATANOMIQ and the book “Storytelling with data” (see footnote in the pdf), we have created an infographic that conveys five tips for better design of BI reports – with
self-reflective clarification.
Direct link to the PDF: https://data-science-blog.com/en/wp-content/uploads/sites/4/2021/11/Infographic_Data_Visualization_Infographic_DATANOMIQ.pdf
About DATANOMIQ
DATANOMIQ is a platform-independent consulting- and service-partner for Business Intelligence and Data Science. We are opening up multiple possibilities for the first time in all areas of the value
chain through Big Data and Artificial Intelligence. We rely on the best minds and the most comprehensive method and technology portfolio for the use of data for business optimization.
DATANOMIQ GmbH
Franklinstr. 11
D-10587 Berlin
I: www.datanomiq.de
E: info@datanomiq.de
https://data-science-blog.com/en/wp-content/uploads/sites/4/2021/11/5-tips-for-better-data-visualization-infographic-header.png 1212 4298 Benjamin Aunkofer https://data-science-blog.com/en/wp-content
/uploads/sites/4/2016/12/data-science-blog-logo-en-300x292.png Benjamin Aunkofer2021-11-06 16:04:052021-12-04 09:58:18Business Intelligence – 5 Tips for better Reporting & Visualization
5 Applications for Location-Based Data in 2020
/in Data Science, Uncategorized/by Kaylae Matthews
Location-based data enables giving people relevant information based on where they are at any given moment. Here are five location data applications to look for in 2020 and beyond.
1. Increasing Sales and Reducing Frustration
One 2019 report indicated that 89% of the marketers who used geo data saw increased sales within their customer bases. Sometimes, the ideal way to boost sales is to convert what would be a
frustration into something positive.
A French campaign associated with the Actimel yogurt brand achieved this by sending targeted, encouraging messages to drivers who used the Waze navigation app and appeared to have made a wrong turn
or got caught in traffic.
For example, a driver might get a message that said, “Instead of getting mad and honking your horn, pump up the jams! #StayStrong.” The three-month campaign saw a 140% increase in ad recall.
More recently, home furnishing brand IKEA launched a campaign in Dubai where people can get free stuff for making a long trip to a store. The freebies get more valuable as a person’s commute time
increases. The catch is that participants have to activate location settings on their phones and enable Google Maps. Driving five minutes to a store got a person a free veggie hot dog, and they’d get
a complimentary table for traveling 49 minutes.
2. Offering Tailored Ad Targeting in Medical Offices
Pharmaceutical companies are starting to rely on companies that send targeted ads to patients connected to the Wi-Fi in doctors’ offices. One such provider is Semcasting. A recent effort involved
sending ads to cardiology offices for a type of drug that lowers cholesterol levels in the blood.
The company has taken a similar approach for an over-the-counter pediatric drug and a medication to relieve migraine headaches, among others. Such initiatives cause a 10% boost in the halo effect,
plus a 1.5% uptick in sales. The first perk relates to the favoritism that people feel towards other products a company makes once they like one of them.
However, location data applications related to health care arguably require special attention regarding privacy. Patients may feel uneasy if they believe that companies are watching them and know
they need a particular kind of medical treatment.
3. Facilitating the Deployment of the 5G Network
The 5G network is coming soon, and network operators are working hard to roll it out. Statistics indicate that the 5G infrastructure investment will total $275 billion over seven years. Geodata can
help network brands decide where to deploy 5G connectivity first.
Moreover, once a company offers 5G in an area, marketing teams can use location data to determine which neighborhoods to target when contacting potential customers. Most companies that currently have
5G within their product lineups have carefully chosen which areas are at the top of the list to receive 5G, and that practice will continue throughout 2020.
It’s easy to envision a scenario whereby people can send error reports to 5G providers by using location data. For example, a company could say that having location data collection enabled on a
5G-powered smartphone allows a technician to determine if there’s a persistent problem with coverage.
Since the 5G network is still, it’s impossible to predict all the ways that a telecommunications operator might use location data to make their installations maximally profitable. However, the
potential is there for forward-thinking brands to seize.
4. Helping People Know About the Events in Their Areas
SoundHound, Inc. and Wcities recently announced a partnership that will rely on location-based data to keep people in the loop about upcoming local events. People can use a conversational
intelligence platform that has information about more than 20,000 cities around the world.
Users also don’t need to mention their locations in voice queries. They could say, for example, “Which bands are playing downtown tonight?” or “Can you give me some events happening on the east side
tomorrow?” They can also ask something associated with a longer timespan, such as “Are there any wine festivals happening this month?”
People can say follow-up commands, too. They might ask what the weather forecast is after hearing about an outdoor event they want to attend. The system also supports booking an Uber, letting people
get to the happening without hassles.
5. Using Location-Based Data for Matchmaking
In honor of Valentine’s Day 2020, students from more than two dozen U.S colleges signed up for a matchmaking opportunity. It, at least in part, uses their location data to work.
Participants answer school-specific questions, and their responses help them find a friend or something more. The platform uses algorithms to connect people with like-minded individuals.
However, the company that provides the service can also give a breakdown of which residence halls have the most people taking part, or whether people generally live off-campus. This example is not
the first time a university used location data by any means, but it’s different from the usual approach.
Location Data Applications Abound
These five examples show there are no limits to how a company might use location data. However, they must do so with care, protecting user privacy while maintaining a high level of data quality.
https://data-science-blog.com/en/wp-content/uploads/sites/4/2020/02/waldemar-brandt-aHZF4sz0YNw-unsplash-scaled.jpg 1707 2560 Kaylae Matthews https://data-science-blog.com/en/wp-content/uploads/sites
/4/2016/12/data-science-blog-logo-en-300x292.png Kaylae Matthews2020-03-12 08:15:232020-02-26 10:52:415 Applications for Location-Based Data in 2020
Attribution Models in Marketing
/in Big Data, Business Analytics, Business Intelligence, Data Mining, Data Science, Insights, Machine Learning, Main Category, Predictive Analytics, Statistics, Use Case, Use Cases/by Barbara Staron
Attribution Models
A Business and Statistical Case
A desire to understand the causal effect of campaigns on KPIs
Advertising and marketing costs represent a huge and ever more growing part of the budget of companies. Studies have found out this share is as high as 10% and increases with the size of companies (
CMO study by American Marketing Association and Duke University, 2017). Measuring precisely the impact of a specific marketing campaign on the sales of a company is a critical step towards an
efficient allocation of this budget. Would the return be higher for an euro spent on a Facebook ad, or should we better spend it on a TV spot? How much should I spend on Twitter ads given the volume
of sales this channel is responsible for?
Attribution Models have lately received great attention in Marketing departments to answer these issues. The transition from offline to online marketing methods has indeed permitted the collection of
multiple individual data throughout the whole customer journey, and allowed for the development of user-centric attribution models. In short, Attribution Models use the information provided by
Tracking technologies such as Google Analytics or Webtrekk to understand customer journeys from the first click on a Facebook ad to the final purchase and adequately ponderate the different marketing
campaigns encountered depending on their responsibility in the final conversion.
Issues on Causal Effects
A key question then becomes: how to declare a channel is responsible for a purchase? In other words, how can we isolate the causal effect or incremental value of a campaign ?
1. A/B-Tests
One method to estimate the pure impact of a campaign is the design of randomized experiments, wherein a control and treated groups are compared. A/B tests belong to this broad category of randomized
methods. Provided the groups are a priori similar in every aspect except for the treatment received, all subsequent differences may be attributed solely to the treatment. This method is typically
used in medical studies to assess the effect of a drug to cure a disease.
Main practical issues regarding Randomized Methods are:
• Assuring that control and treated groups are really similar before treatment. Uually a random assignment (i.e assuring that on a relevant set of observable variables groups are similar) is
• Potential spillover-effects, i.e the possibility that the treatment has an impact on the non-treated group as well (Stable unit treatment Value Assumption, or SUTVA in Rubin’s framework);
• The costs of conducting such an experiment, and especially the costs linked to the deliberate assignment of individuals to a group with potentially lower results;
• The number of such experiments to design if multiple treatments have to be measured;
• Difficulties taking into account the interaction effects between campaigns or the effect of spending levels. Indeed, usually A/B tests are led by cutting off temporarily one campaign entirely and
measuring the subsequent impact on KPI’s compared to the situation where this campaign is maintained;
• The dynamical reproduction of experiments if we assume that treatment effects may change over time.
In the marketing context, multiple campaigns must be tested in a dynamical way, and treatment effect is likely to be heterogeneous among customers, leading to practical issues in the lauching of A/B
tests to approximate the incremental value of all campaigns. However, sites with a lot of traffic and conversions can highly benefit from A/B testing as it provides a scientific and straightforward
way to approximate a causal impact. Leading companies such as Uber, Netflix or Airbnb rely on internal tools for A/B testing automation, which allow them to basically test any decision they are about
to make.
Experiment!: Website conversion rate optimization with A/B and multivariate testing, Colin McFarland, ©2013 | New Riders
A/B testing: the most powerful way to turn clicks into customers. Dan Siroker, Pete Koomen; Wiley, 2013.
2. Attribution models
Attribution Models do not demand to create an experimental setting. They take into account existing data and derive insights from the variability of customer journeys. One key difficulty is then to
differentiate correlation and causality in the links observed between the exposition to campaigns and purchases. Indeed, selection effects may bias results as exposure to campaigns is usually
dependant on user-characteristics and thus may not be necessarily independant from the customer’s baseline conversion probabilities. For example, customers purchasing from a discount price comparison
website may be intrinsically different from customers buying from FB ad and this a priori difference may alone explain post-exposure differences in purchasing bahaviours. This intrinsic weakness must
be remembered when interpreting Attribution Models results.
2.1 General Issues
The main issues regarding the implementation of Attribution Models are linked to
• Causality and fallacious reasonning, as most models do not take into account the aforementionned selection biases.
• Their difficult evaluation. Indeed, in almost all attribution models (except for those based on classification, where the accuracy of the model can be computed), the additionnal value brought by
the use of a given attribution models cannot be evaluated using existing historical data. This additionnal value can only be approximated by analysing how the implementation of the conclusions of
the attribution model have impacted a given KPI.
• Tracking issues, leading to an uncorrect reconstruction of customer journeys
□ Cross-device journeys: cross-device issue arises from the use of different devices throughout the customer journeys, making it difficult to link datapoints. For example, if a customer
searches for a product on his computer but later orders it on his mobile, the AM would then mistakenly consider it an order without prior campaign exposure. Though difficult to measure
perfectly, the proportion of cross-device orders can approximate 20-30%.
□ Cookies destruction makes it difficult to track the customer his the whole journey. Both regulations and consumers’ rising concerns about data privacy issues mitigate the reliability and use
of cookies.1 – From 2002 on, the EU has enacted directives concerning privacy regulation and the extended use of cookies for commercial targeting purposes, which have highly impacted
marketing strategies, such as the ‘Privacy and Electronic Communications Directive’ (2002/58/EC). A research was conducted and found out that the adoption of this ‘Privacy Directive’ had led
to 64% decrease in advertising methods compared to the rest of the world (Goldfarb et Tucker (2011)). The effect was stronger for generalized sites (Yahoo) than for specialized sites.2 –
Users have grown more and more conscious of data privacy issues and have adopted protective measures concerning data privacy, such as automatic destruction of cookies after a session is
ended, or simply giving away less personnal information (Goldfarb et Tucker (2012) ) .Valuable user information may be lost, though tracking technologies evolution have permitted to maintain
tracking by other means. This issue may be particularly important in countries highly concerned with data privacy issues such as Germany.
□ Offline/Online bridge: an Attribution Model should take into account all campaigns to draw valuable insights. However, the exposure to offline campaigns (TV, newspapers) are difficult to
track at the user level. One idea to tackle this issue would be to estimate the proportion of conversions led by offline campaigns through AB testing and deduce this proportion from the
credit assigned to the online campaigns accounted for in the Attribution Model.
□ Touch point information available: clicks are easy to follow but irrelevant to take into account the influence of purely visual campaigns such as display ads or video.
2.2 Today’s main practices
Two main families of Attribution Models exist:
• Rule-Based Attribution Models, which have been used for in the last decade but from which companies are gradualy switching.
Attribution depends on the individual journeys that have led to a purchase and is solely based on the rank of the campaign in the journey. Some models focus on a single touch points (First Click,
Last Click) while others account for multi-touch journeys (Bathtube, Linear). It can be calculated at the customer level and thus doesn’t require large amounts of data points. We can distinguish two
sub-groups of rule-based Attribution Models:
• One Touch Attribution Models attribute all credit to a single touch point. The First-Click model attributes all credit for a converion to the first touch point of the customer journey; last touch
attributes all credit to the last campaign.
• Multi-touch Rule-Based Attribution Models incorporate information on the whole customer journey are thus an improvement compared to one touch models. To this family belong Linear model where
credit is split equally between all channels, Bathtube model where 40% of credit is given to first and last clicks and the remaining 20% is distributed equally between the middle channels, or
time-decay models where credit assigned to a click diminishes as the time between the click and the order increases..
The main advantages of rule-based models is their simplicity and cost effectiveness. The main problems are:
– They are a priori known and can thus lead to optimization strategies from competitors
– They do not take into account aggregate intelligence on customer journeys and actual incremental values.
– They tend to bias (depending on the model chosen) channels that are over-represented at the beggining or end of the funnel, according to theoretical assumptions that have no observationnal
• Data-Driven Attribution Models
These models take into account the weaknesses of rule-based models and make a relevant use of available data. Being data-driven, following attribution models cannot be computed using single user
level data. On the contrary values are calculated through data aggregation and thus require a certain volume of customer journey information.
3. Data-Driven Attribution Models in practice
3.1 Issues
Several issues arise in the computation of campaigns individual impact on a given KPI within a data-driven model.
• Selection biases: Exposure to certain types of advertisement is usually highly correlated to non-observable variables which are in turn correlated to consumption practices. Differences in the
behaviour of users exposed to different campaigns may thus only be driven by core differences in conversion probabilities between groups whether than by the campaign effect.
• Complementarity: it may be that campaigns A and B only have an effect when combined, so that measuring their individual impact would lead to misleading conclusions. The model could then try to
assess the effect of combinations of campaigns on top of the effect of individual campaigns. As the number of possible non-ordered combinations of k campaigns is 2k, it becomes clear that
inclusing all possible combinations would however be time-consuming.
• Order-sensitivity: The effect of a campaign A may depend on the place where it appears in the customer journey, meaning the rank of a campaign and not merely its presence could be accounted for
in the model.
• Relative Order-sensitivity: it may be that campaigns A and B only have an effect when one is exposed to campaign A before campaign B. If so, it could be useful to assess the effect of given
combinations of campaigns as well. And this for all campaigns, leading to tremendous numbers of possible combinations.
• All previous phenomenon may be present, increasing even more the potential complexity of a comprehensive Attribution Model. The number of all possible ordered combination of k campaigns is indeed
3.2 Main models
A) Logistic Regression and Classification models
If non converting journeys are available, Attribition Model can be shaped as a simple classification issue. Campaign types or campaigns combination and volume of campaign types can be included in the
model along with customer or time variables. As we are interested in inference (on campaigns effect) whether than prediction, a parametric model should be used, such as Logistic Regression. Non
paramatric models such as Random Forests or Neural Networks can also be used though the interpretation of campaigns value would be more difficult to derive from the model results.
A common pitfall is the usual issue of spurious correlations on one hand and the correct interpretation of coefficients in business terms.
An advantage if the possibility to evaluate the relevance of the model using common model validation methods to evaluate its predictive power (validation set \ AUC \pseudo R squared).
B) Shapley Value
The Shapley Value is based on a Game Theory framework and is named after its creator, the Nobel Price Laureate Lloyd Shapley. Initially meant to calculate the marginal contribution of players in
cooperative games, the model has received much attention in research and industry and has lately been applied to marketing issues. This model is typically used by Google Adords and other ad bidding
vendors. Campaigns or marketing channels are in this model seen as compementary players looking forward to increasing a given KPI.
Contrarily to Logistic Regressions, it is a non-parametric model. Contrarily to Markov Chains, all results are built using existing journeys, and not simulated ones.
Channels are considered to enter the game sequentially under a certain joining order. Shapley value try to The Shapley value of channel i is the weighted sum of the marginal values that channel i
adds to all possible coalitions that don’t contain channel i.
In other words, the main logic is to analyse the difference of gains when a channel i is added after a coalition Ck of k channels, k<=n. We then sum all the marginal contributions over all possible
ordered combination Ck of all campaigns excluding i, with k<=n-1.
Subsets framework
A first an most usual way to compute the Shapley Vaue is to consider that when a channel enters coalition, its additionnal value is the same irrelevant of the order in which previous channels have
appeared. In other words, journeys (A>B>C) and (B>A>C) trigger the same gains.
Shapley value is computed as the gains associated to adding a channel i to a subset of channels, weighted by the number of (ordered) sequences that the (unordered) subset represents, summed up on all
possible subsets of the total set of campaigns where the channel i is not present.
The Shapley value of the channel ???????? is then:
where |S| is the number of campaigns of a coalition S and the sum extends over all subsets S that do not not contain channel j. ????(????) is the value of the coalition S and ????(???? ∪ {????????})
the value of the coalition formed by adding ???????? to coalition S. ????(???? ∪ {????????}) − ????(????) is thus the marginal contribution of channel ???????? to the coalition S.
The formula can be rewritten and understood as:
This method is convenient when data on the gains of on all possible permutations of all unordered k subsets of the n campaigns are available. It is also more convenient if the order of campaigns
prior to the introduction of a campaign is thought to have no impact.
Ordered sequences
Let us define ????((A>B)) as the value of the sequence A then B. What is we let ????((A>B)) be different from ????((B>A)) ?
This time we would need to sum over all possible permutation of the S campaigns present before ???????? and the N-(S+1) campaigns after ????????. Doing so we will sum over all possible orderings
(i.e all permutations of the n campaigns of the grand coalition containing all campaigns) and we can remove the permutation coefficient s!(p-s+1)!.
This method is convenient when the order of channels prior to and after the introduction of another channel is assumed to have an impact. It is also necessary to possess data for all possible
permutations of all k subsets of the n campaigns, and not only on all (unordered) k-subsets of the n campaigns, k<=n. In other words, one must know the gains of A, B, C, A>B, B>A, etc. to compute the
Shapley Value.
Differences between the two approaches
We simulate an ordered case where the value for each ordered sequence k for k<=3 is known. We compare it to the usual Shapley value calculated based on known gains of unordered subsets of campaigns.
So as to compare relevant values, we have built the gains matrix so that the gains of a subset A, B i.e ????({B,A}) is the average of the gains of ordered sequences made up with A and B (assuming
the number of journeys where A>B equals the number of journeys where B>A, we have ????({B,A})=0.5( ????((A>B)) + ????((B>A)) ). We let the value of the grand coalition be different depending on the
order of campaigns-keeping the constraints that it averages to the value used for the unordered case.
Note: mvA refers to the marginal value of A in a given sequence.
With traditionnal unordered coalitions:
With ordered sequences used to compute the marginal values:
We can see that the two approaches yield very different results. In the unordered case, the Shapley Value campaign C is the highest, culminating at 20, while A and B have the same Shapley Value mvA=
mvB=15. In the ordered case, campaign A has the highest Shapley Value and all campaigns have different Shapley Values.
This example illustrates the inherent differences between the set and sequences approach to Shapley values. Real life data is more likely to resemble the ordered case as conversion probabilities may
for any given set of campaigns be influenced by the order through which the campaigns appear.
Shapley value has become popular in allocation problems in cooperative games because it is the unique allocation which satisfies different axioms:
• Efficiency: Shaple Values of all channels add up to the total gains (here, orders) observed.
• Symmetry: if channels A and B bring the same contribution to any coalition of campaigns, then their Shapley Value i sthe same
• Null player: if a channel brings no additionnal gains to all coalitions, then its Shapley Value is zero
• Strong monotony: the Shapley Value of a player increases weakly if all its marginal contributions increase weakly
These properties make the Shapley Value close to what we intuitively define as a fair attribution.
• The Shapley Value is based on combinatory mathematics, and the number of possible coalitions and ordered sequences becomes huge when the number of campaigns increases.
• If unordered, the Shapley Value assumes the contribution of campaign A is the same if followed by campaign B or by C.
• If ordered, the number of combinations for which data must be available and sufficient is huge.
• Channels rarely present or present in long journeys will be played down.
• Generally, gains are supposed to grow with the number of players in the game. However, it is plausible that in the marketing context a journey with a high number of channels will not necessarily
bring more orders than a journey with less channels involved.
R package: GameTheoryAllocation
Zhao & al, 2018 “Shapley Value Methods for Attribution Modeling in Online Advertising “
Courses: https://www.lamsade.dauphine.fr/~airiau/Teaching/CoopGames/2011/coopgames-7%5b8up%5d.pdf
Blogs: https://towardsdatascience.com/one-feature-attribution-method-to-supposedly-rule-them-all-shapley-values-f3e04534983d
B) Markov Chains
Markov Chains are used to model random processes, i.e events that occur in a sequential manner and in such a way that the probability to move to a certain state only depends on the past steps. The
number of previous steps that are taken into account to model the transition probability is called the memory parameter of the sequence, and for the model to have a solution must be comprised between
0 and 4. A Markov Chain process is thus defined entirely by its Transition Matrix and its initial vector (i.e the starting point of the process).
Markov Chains are applied in many scientific fields. Typically, they are used in weather forecasting, with the sequence of Sunny and Rainy days following a Markov Process of memory parameter 0, so
that for each given day the probability that the next day will be rainy or sunny only depends on the weather of the current day. Other applications can be found in sociology to understand the
dynamics of social classes intergenerational reproduction. To get more both mathematical and applied illustration, I recommend the reading of this course.
In the marketing context, Markov Chains are an interesting way to model the conversion funnel. To go from the from the Markov Model to the Attribution logic, we calculate the Removal Effect of each
channel, i.e the difference in conversions that happen if the channel is removed. Please read below for an introduction to the methodology.
The first step in a Markov Chains Attribution Model is to build the transition matrix that captures the transition probabilities between the campaigns accross existing customer journeys. This Matrix
is to be read as a “From state A to state B” table, from the left to the right. A first difficulty is finding the right memory parameter to use. A large memory parameter would allow to take more into
account interraction effects within the conversion funnel but would lead to increased computationnal time, a non-readable transition matrix, and be more sensitive to noisy data. Please note that this
transition matrix provides useful information on the conversion funnel and on the relationships between campaigns and can be used as such as an analytical tool. I suggest the clear and easily R code
which can be found here or here.
Here is an illustration of a Markov Chain with memory Parameter of 0: the probability to go to a certain campaign B in the next step only depend on the campaign we are currently at:
The associated Transition Matrix is then (with null probabilities left as Blank):
The second step is to compute the actual responsibility of a channel in total conversions. As mentionned above, the main philosophy to do so is to calculate the Removal Effect of each channel, i.e
the changes in the number of conversions when a channel is entirely removed. All customer journeys which went through this channel are settled out to be unsuccessful. This calculation is done by
applying the transition matrix with and without the removed channels to an initial vector that contains the number of desired simulations.
Building on our current example, we can then settle an initial vector with the desired number of simulations, e.g 10 000:
It is possible at this stage to add a constraint on the maximum number of times the matrix is applied to the data, i.e on the maximal number of campaigns a simulated journey is allowed to have.
• The dynamic journey is taken into account, as well as the transition between two states. The funnel is not assumed to be linear.
• It is possile to build a conversion graph that maps the customer journey provides valuable insights.
• It is possible to evaluate partly the accuracy of the Attribution Model based on Markov Chains. It is for example possible to see how well the transition matrix help predict the future by
analysing the number of correct predictions at any given step over all sequences.
• It can be somewhat difficult to set the memory parameter. Complementarity effects between channels are not well taken into account if the memory is low, but a parameter too high will lead to
over-sensitivity to noise in the data and be difficult to implement if customer journeys tend to have a number of campaigns below this memory parameter.
• Long journeys with different channels involved will be overweighted, as they will count many times in the Removal Effect. For example, if there are n-1 channels in the customer journey, this
journey will be considered as failure for the n-1 channel-RE. If the volume effects (i.e the impact of the overall number of channels in a journey, irrelevant from their type° are important then
results may be biased.
R package: ChannelAttribution
“Mapping the Customer Journey: A Graph-Based Framework for Online Attribution Modeling”; Anderl, Eva and Becker, Ingo and Wangenheim, Florian V. and Schumann, Jan Hendrik, 2014. Available at SSRN:
https://ssrn.com/abstract=2343077 or http://dx.doi.org/10.2139/ssrn.2343077
“Media Exposure through the Funnel: A Model of Multi-Stage Attribution”, Abhishek & al, 2012
“Multichannel Marketing Attribution Using Markov Chains”, Kakalejčík, L., Bucko, J., Resende, P.A.A. and Ferencova, M. Journal of Applied Management and Investments, Vol. 7 No. 1, pp. 49-60. 2018
3.3 To go further: Tackling selection biases with Quasi-Experiments
Exposure to certain types of advertisement is usually highly correlated to non-observable variables. Differences in the behaviour of users exposed to different campaigns may thus only be driven by
core differences in converison probabilities between groups whether than by the campaign effect. These potential selection effects may bias the results obtained using historical data.
Quasi-Experiments can help correct this selection effect while still using available observationnal data. These methods recreate the settings on a randomized setting. The goal is to come as close as
possible to the ideal of comparing two populations that are identical in all respects except for the advertising exposure. However, populations might still differ with respect to some unobserved
Common quasi-experimental methods used for instance in Public Policy Evaluation are:
• Discontinuity Regressions
• Matching Methods, such as Exact Matching, Propensity-score matching or k-nearest neighbourghs.
“Towards a digital Attribution Model: Measuring the impact of display advertising on online consumer behaviour”, Anindya Ghose & al, MIS Quarterly Vol. 40 No. 4, pp. 1-XX, 2016
4. First Steps towards a Practical Implementation
Identify key points of interests
• Identify the nature of touchpoints available: is the data based on clicks? If so, is there a way to complement the data with A/B tests to measure the influence of ads without clicks (display,
video) ? For example, what happens to sales when display campaign is removed? Analysing this multiplier effect would give the overall responsibility of display on sales, to be deduced from
current attribution values given to click-based channels. More interestingly, what is the impact of the removal of display campaign on the occurences of click-based campaigns ? This would give us
an idea of the impact of display ads on the exposure to each other campaigns, which would help correct the attribution values more precisely at the campaign level.
• Define the KPI to track. From a pure Marketing perspective, looking at purchases may be sufficient, but from a financial perspective looking at profits, though a bit more difficult to compute,
may drive more interesting results.
• Define a customer journey. It may seem obvious, but the notion needs to be clarified at first. Would it be defined by a time limit? If so, which one? Does it end when a conversion is observed?
For example, if a customer makes 2 purchases, would the campaigns he’s been exposed to before the first order still be accounted for in the second order? If so, with a time decay?
• Define the research framework: are we interested only in customer journeys which have led to conversions or in all journeys? Keep in mind that successful customer journeys are a
non-representative sample of customer journeys. Models built on the analysis of biased samples may be conservative. Take an extreme example: 80% of customers who see campaign A buy the product,
VS 1% for campaign B. However, campaign B exposure is great and 100 Million people see it VS only 1M for campaign A. An Attribution Model based on successful journeys will give higher credit to
campaign B which is an auguable conclusion. Taking into account costs per campaign (in the case where costs are calculated by clicks) may of course tackle this issue partly, as campaign A could
then exhibit higher returns, but a serious fallacious reasonning is at stake here.
Analyse the typical customer journey
• Performing a duration analysis on the data may help you improve the definition of the customer journey to be used by your organization. After which days are converison probabilities null? Should
we consider the effect of campaigns disappears after x days without orders? For example, if 99% of orders are placed in the 30 days following a first click, it might be interesting to define the
customer journey as a 30 days time frame following the first oder.
• Look at the distribution of the number of campaigns in a typical journey. If you choose to calculate the effect of campaigns interraction in your Attribution Model, it may indeed help you
determine the maximum number of campaigns to be included in a combination. Indeed, you may not need to assess the impact of channel combinations with above than 4 different channels if 95% of
orders are placed after less then 4 campaigns.
• Transition matrixes: what if a campaign A systematically leads to a campaign B? What happens if we remove A or B? These insights would give clues to ask precise questions for a latter AB test,
for example to find out if there is complementarity between channels A and B – (implying none should be removed) or mere substitution (implying one can be given up).
• If conversion rates are available: it can be interesting to perform a survival analysis i.e to analyse the likelihood of conversion based on duration since first click. This could help us excluse
potential outliers or individuals who have very low conversion probabilities.
Attribution is a complex topic which will probably never be definitively solved. Indeed, a main issue is the difficulty, or even impossibility, to evaluate precisely the accuracy of the attribution
model that we’ve built. Attribution Models should be seen as a good yet always improvable approximation of the incremental values of campaigns, and be presented with their intrinsinc limits and
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Line types: Structure data
Young's modulus
For homogeneous pipes only, a value may be given for the Young's modulus of the material. This determines the axial, bending and torsional stiffnesses – these stiffness data items are reported on the
data form (in the all view mode), although they cannot be edited.
The value may be constant or variable:
• A constant value results in linear material properties.
• A variable data item defines a nonlinear stress-strain relation which results in a bending stiffness with nonlinear elastic behaviour. Note however that the axial and torsional stiffnesses are
still assumed to be linear.
Bend stiffness
The bend stiffness is the slope of the bend moment-curvature curve. The $x$ and $y$ values will often be the same, and this can be indicated by setting the $y$-value to '~', meaning "same as
You can specify the bend stiffness to be linear, elastic nonlinear, hysteretic nonlinear or externally calculated, as follows. See calculating bend moments for further details of the bending model
Linear bend stiffness
For normal simple linear behaviour, specify the bend stiffness to be the constant slope of the bend moment-curvature relationship. This slope is the equivalent $EI$ value for the line, where $E$ is
Young's modulus and $I$ the second moment of area of the cross section. The bend stiffness represents the bend moment required to bend the line to a curvature of 1 radian per unit length.
Nonlinear bend stiffness
For nonlinear behaviour, use variable data to specify a table of bend moment magnitude against curvature magnitude. OrcaFlex uses linear interpolation within this table, and linear extrapolation for
curvature values beyond those given. The bend moment must be zero at zero curvature. For nonlinear behaviour derived from a known stress-strain relationship the plasticity wizard may be useful to
help set up the table.
In the case of nonlinear bend stiffness, you must also specify whether the hysteretic bending model should be used.
• Non-hysteretic means that the nonlinear stiffness is elastic. No hysteresis effects are included and the bend moment magnitude is entirely determined by the given function of the current
curvature magnitude.
• Hysteretic means the bend moment includes hysteresis effects, so that the bend moment depends on the history of curvature applied as well as on the current curvature. Note that if the hysteretic
model is used, then the line must include torsion effects.
Warning: You must check that the hysteretic model is appropriate for the line type being modelled. It is not suitable for modelling rate-dependent effects; it is intended for modelling hysteresis due
to persisting effects, such as yield of material or slippage of one part of a composite line structure relative to another part.
If you use the hysteretic bending model then the simulation speed may be significantly slowed if your table of bend moment against curvature has a large number of rows. You might be able to speed up
the simulation, without significantly affecting accuracy, by removing superfluous rows in areas where the curve is very close to linear.
If you are using nonlinear bend stiffness, then the mid-segment curvature results reported depend on whether the bend stiffness is specified to be hysteretic or not. If the bend stiffness is
Note: not hysteretic then the mid-segment curvature reported is the curvature that corresponds to the mid-segment bend moment (which is the mean of the bend moments at either end of the segment). If
the bend stiffness is hysteretic then the mid-segment curvature cannot be derived in this way (due to possible hysteresis effects) so the mid-segment curvature reported is the mean of the
curvatures at the ends of the segment. This difference may be significant if the bend stiffness is significantly nonlinear over the range of curvatures involved.
The choice of statics model controls the interpretation of the nonlinear bend stiffness table during the statics calculation. There are two options:
• Pressurised: the bend moment is calculated from the curvature by simple interpolation of the bend stiffness table. This is the same as the nonlinear elastic model during statics.
• Depressurised: the bend stiffness is linear, and determined by the slope of the final two rows of the bend stiffness table. Once the dynamic simulation starts, the line is assumed to be
pressurised and the hysteretic model is applied. OrcaFlex enforces continuity in the transition from linear stiffness in statics to hysteretic, nonlinear stiffness in dynamics.
To understand better the rationale behind these options, consider the example of a flexible riser. A flexible riser is built of layers. When the riser is not pressurised, these layers are free to
slide over each other. When the riser is pressurised, this introduces friction between the layers. As the riser is bent, this friction has the effect of increasing the apparent bend stiffness of the
riser. Eventually, under bending, the friction reaches a certain limiting value and the layers are then able to slip over each other. This inter-layer friction is what gives rise to the hysteretic
behaviour of a flexible riser.
Under the depressurised model, OrcaFlex is assuming that the post-slip stiffness is the same as the depressurised stiffness, and is given by the final two rows of the bend stiffness table. So the
depressurised option is for scenarios in which the static analysis models the riser before it has been pressurised. Typically the riser will be installed without internal pressure and so its geometry
will be determined by the much lower, post-slip stiffness. However, once the riser is pressurised, the dynamic bending stiffness is higher due to the inter-layer friction.
For further details see nonlinear bend stiffness theory.
Finally, the external results option allows you to specify an external function that can be used to track the bend stiffness calculation and provide user defined results.
Externally calculated bend moment
This form of bend stiffness allows the bend moment to be calculated by an external function. If this option is used then the line must include torsion effects. The external function can be written by
the user or other software writers. For details see the OrcaFlex programming interface (OrcFxAPI) and the OrcFxAPI documentation.
Warning: Nonlinear behaviour breaks the assumptions of the stress results and fatigue analysis in OrcaFlex. You should therefore not use these facilities when there are significant nonlinear effects.
Axial stiffness
The axial stiffness is the slope of the curve relating wall tension to strain. The data define the behaviour in the unpressured state, i.e. atmospheric pressure inside and out. Pressure effects,
including the Poisson ratio effect, are then allowed for by OrcaFlex.
Axial strain is defined as $(l - l_0) / l_0$, where $l$ and $l_0$ are respectively the stretched and unstretched length of a given piece of pipe. Here, 'unstretched' means the length when
Note: unpressured and unstressed. When a pipe is pressured its tension at this 'unstretched' length is often not zero because of strains due to pressure effects. For a homogeneous pipe this can be
modelled by specifying the Poisson ratio (see below); for a non-homogeneous pipe (e.g. a flexible), however, the Poisson ratio may not be able to capture the pressure effects.
Linear axial stiffness
For a simple linear behaviour, specify the axial stiffness to be the constant slope of the line relating wall tension to strain. This slope is the equivalent $EA$ value for the line, where $E$ is
Young's modulus and $A$ is the cross section area. It represents the force required to double the length of any given piece of line, assuming perfectly linear elastic behaviour. (In practice, of
course, lines would yield before such a tension was reached.)
Nonlinear axial stiffness
For a nonlinear behaviour, use variable data to define a table of wall tension against axial strain. OrcaFlex uses linear interpolation within the table and linear extrapolation for strain values
beyond those given in the table.
In the case of nonlinear axial stiffness, you must also specify whether the hysteretic axial stiffness model should be used.
• Non-hysteretic means that the nonlinear stiffness is elastic. No hysteresis effects are included and the tension is entirely determined by the given function of the strain. In this case, the wall
tension is allowed to be non-zero at zero strain.
• Hysteretic means the tension includes hysteresis effects, so that the tension depends on the history of strain applied as well as on the current strain. The model used is directly analogous to
the one used for hysteretic bending. The independent variable in the hysteretic bending model is curvature, whereas for axial stiffness the independent variable is strain; similarly, the
dependent variable in the bending model is bend moment, whereas for axial stiffness it is tension. In this case, the nonlinear stiffness data must always be single-sided.
For further details see nonlinear axial stiffness theory.
The external results option allows you to specify an external function that can be used to track the axial stiffness calculation and provide user defined results.
Externally calculated wall tension
This form of axial stiffness allows the wall tension to be calculated by an external function. For details see the OrcaFlex programming interface (OrcFxAPI) and the OrcFxAPI documentation.
Direct tensile strain for hysteretic or externally calculated wall tension
After the simulation is complete, OrcaFlex can recover wall tension from the logged results. To present direct tensile strain results, OrcaFlex requires a unique correspondence between wall tension
and strain. For hysteretic wall tension, this is not satisfied, and for externally calculated wall tension such behaviour cannot be assumed. As such the direct tensile strain result, and further
results derived using direct tensile strain, will be unavailable.
Warning: Nonlinear behaviour breaks the assumptions of the stress results and fatigue analysis.
Poisson ratio
The Poisson ratio of the material that makes up the wall of the line type, used to model any length changes due to the radial and circumferential stresses caused by contents pressure and external
A Poisson ratio of zero means no such length changes. For metals such as steel or titanium the Poisson ratio is about 0.3 and for polyethylene about 0.4. Most materials have Poisson ratio between 0.0
and 0.5.
Note: The Poisson ratio effect is calculated assuming that the line type is a pipe made from a homogeneous material. It is not really applicable to complex structures such as flexibles, whose length
changes due to pressure are more complex, although an effective Poisson ratio could be specified as an approximation.
Torsional stiffness
The torsional stiffness specifies the relationship between twist and torsional moment (torque). It is only used if torsion is included. You can specify linear or nonlinear behaviour.
Linear torsional stiffness
For a simple linear behaviour, specify the torsional stiffness to be the constant slope of the torsional moment-twist per unit length relationship. This slope is the equivalent $GJ$ value for the
line, where $G$ is the shear modulus and $J$ is the axial second moment of area. It represents the torque which arises if the line is given a twist of 1 radian per unit length.
Nonlinear torsional stiffness
For a nonlinear behaviour, use variable data to define a table of torque against twist per unit length. OrcaFlex uses linear interpolation within the table, and linear extrapolation for values
outside those given in the table. The torque must be zero at zero twist.
In the case of nonlinear torsional stiffness, you must also specify whether the hysteretic stiffness model should be used.
• When defining nonlinear elastic torsional stiffness you should provide values for both positive and negative twist per unit length. This allows you, for example, to have different stiffnesses for
positive and negative twisting. If the behaviour is mirrored for positive and negative twist then you must specify the full relationship – OrcaFlex does not automatically reflect the data for
• When intending to apply hysteretic torsional stiffness, you should only provide input values for positive twist per unit length. The behaviour model used is described by our existing
documentation of hysteretic bending.
The external results option allows you to specify an external function that can be used to track the torsional stiffness calculation and provide user defined results.
Externally calculated torque
This option for torsional stiffness specifies that segment torque is calculated by an external function. For details see the OrcaFlex programming interface (OrcFxAPI) and the OrcFxAPI documentation.
Warning: Nonlinear behaviour breaks the assumptions of the stress results and fatigue analysis.
Tension / torque coupling
Defines a direct coupling between tension and torque. This coupling allows axial strain to induce torque, and allows twist to induce tension. It is only used if torsion is included.
Warning: Tension / torque coupling breaks the assumptions of the stress results and fatigue analysis.
Additional bending stiffness
Only available for homogeneous pipes, this value increases the overall bending stiffness of the line type. If the pipe has a constant Young's modulus, this value is simply added to the resulting bend
stiffness; if the Young's modulus is variable, then the gradient of each section of the corresponding piecewise linear moment–curvature relationship is increased by this value.
The intent is to represent the extra stiffness a line may receive from any applied coatings or linings, for example the concrete coating often applied to steel flowlines. There is an assumed sharing
of the loads between the original line type structure and that represented by the additional stiffness. For general category line types, such load sharing can be represented by the line type stress
loading factors. For a homogeneous pipe, however, the stress loading factors cannot be modified: instead, OrcaFlex will automatically modify results that are affected by stress loading factors
(typically stress and strain results) to reflect the additional bending stiffness.
External results from nonlinear stiffness
As listed on this page, each of the variable data sources for line type axial stiffness, bend stiffness and torsional stiffness can accept an external function in order to provide external results.
Here we note some details of how those external results are then presented in OrcaFlex.
External results associated with line type stiffness are reported at line mid-segment result points. External results associated with axial and torsional stiffness present values that are calculated
by the node at the end of the line segment closest to line end A. External results associated with bend stiffness present values that are an average of the result values calculated by the nodes at
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Bed Availability and Hospital Utilization: Estimates of the “Roemer Effect”
“Roemer's Law,” the notion that an increase in the number of hospital beds per capita increases hospital utilization rates, is an important underpinning of efforts to control hospital construction
through health planning. Attempts to measure the magnitude of the effect have yielded results ranging from no effect to a one-to-one relationship. The present study, by restricting its inquiry to
Medicare patients and using a unique data base, avoids many of the shortcomings of earlier studies. This study concludes that an increase of 10 percent in hospital beds per capita would increase
hospital utilization by Medicare enrollees by about 4 percent.
Twenty years ago, Milton Roemer and Max Shain raised the possibility that hospital use increases along with the number of hospital beds available in an area (Roemer and Shain, 1959). This
relationship, commonly referred to as “Roemer's law,” became a major conceptual basis for health planning. If additions to bed capacity caused an increase in utilization, then market forces might not
result in the optimal number of beds. Further, because excess beds would generate additional demand, their cost would be greater than the cost of maintaining empty facilities. Regulatory constraints
on hospital construction could be justified by this relationship.
A decade later, Martin Feldstein pointed out that Roemer had in mind something different than the usual workings of markets—where an increase in supply raises the quantity sold via the price
mechanism (Feldstein, 1971). Feldstein instead conceptualized the Roemer phenomenon as a shift in supply (the change in bed capacity) inducing a shift in the demand function. To test this hypothesis,
he estimated a demand with both price and the number of beds per capita as independent variables. The elasticity of hospital days with respect to hospital beds was found to be 0.53, which was
statistically significant. That is, an increase of 1 percent in hospital beds was found to be associated with a 0.53 percent Increase in days of hospitalization.
A number of subsequent studies have also found evidence of a Roemer effect. These studies, however, have produced substantially different estimates of the size of the effect, with elasticities
ranging from nearly zero to over one. May (1975), using survey data, found elasticities dose to zero. He included time prices as independent variables though. Many suspect that time prices are an
important mechanism through which the Roemer effect works. When facilities increase, time prices fall, inducing greater use. Inclusion of time prices in the regression precludes measurement of this
part of the Roemer effect.
Newhouse and Phelps (1976) obtained an elasticity of 0.46. Like May, they used survey data, but they omitted time prices from their equations and used estimation techniques more appropriate for an
equation with a limited dependent variable.
Chiswick (1974) estimated an elasticity of total days with respect to beds of 0.85 using data aggregated to Standard Metropolitan Statistical Areas. The result was unchanged when two stage least
squares estimation was employed.
Friedman (1978) aggregated Medicare data to the Census division. He found elasticities ranging from 0.75 for acute myocardial infarction to 1.82 for diabetes mellitus. Most elasticities were slightly
greater than 1.
The existing research on the Roemer effect, however, has a number of weaknesses. Much of it was designed primarily to measure the response of utilization to changes in price. The assessment of the
Roemer effect—the response of utilization to changes in bed supply—was generally incidental. As a result, the design of the studies was not always the best for a measurement of the Roemer effect.
This study avoids several major technical problems found in previous research. Four such problems are described below. Circumventing these problems has produced a more accurate estimate of the Roemer
effect, producing results that are important in analyzing health planning and other aspects of health policy.
Problems in Previous Estimates of the Roemer Effect
The standard method of assessing the Roemer effect has been to use multiple regression to predict use—usually days of hospitalization per capita — as a function of demographic variables, health
status, the price of hospital services (net of Insurance), and variables reflecting the availability of medical resources. Demographic variables typically have included age, sex, family status, and
income. Physicians per capita and hospital beds per capita have served as measures of the availability of resources. In such an equation, the regression coefficient of the hospital beds variable
provides the measure of the Roemer effect. Some studies have applied this method to data from individual survey respondents, while others have used aggregate data. This discussion will focus on
problems inherent in the data used in these studies rather than on the specifics of the techniques used in them.
Incorrect Measurement of the Price of Services
In a study of the Roemer effect, the appropriate measure of the price of services is the marginal price minus insurance reimbursement—that is, the net price to the patient of an extra day or an extra
stay in the hospital. Previous studies typically lacked such a measure. Most often, they lacked data on gross prices. Studies such as May, New-house-Phelps, and Chiswick only have data on insurance
coverage. With no data on hospital price variation from area to area, the Roemer effect cannot be separated from the effect of beds on utilization that works through the price mechanism. Such an
omission is likely to cause an overestimate of the Roemer effect, because both effects of an increase in bed supply—the Roemer effect and the effect through the price mechanism—work in the same
Those attempting to measure the Roemer effect have disagreed about the appropriateness of including time prices in the analysis. While economic theory calls for their inclusion, the policy question
of how an increase In hospital beds affects the use of service requires that time prices not be held constant. If time prices were a major mechanism by which bed supply affected use, the possibility
of the hospital market not being self-regulating would remain.
Incorrect Measurement of the Utilization Rate
Studies using aggregate data face a problem in specifying utilization rates, In that the numerator and denominator of the utilization rate should refer to the same market area. Patients, however,
migrate from rural counties to metropolitan areas, from small metropolitan areas to larger ones, and from one State to another to obtain hospital services. Days of hospital care (the numerator)
reflect services delivered by area hospitals to both residents and nonresidents alike, but population (the denominator) reflects only the residents of the area. This error in the dependent variable
is correlated with the bed availability, in that areas with many beds per capita tend to have overstated utilization rates. This would cause an upward bias in the estimate of the Roemer effect (
Kelegian and Oates, 1974).
Omitted Variables
If important determinants of utilization that are correlated with bed availability are omitted from the analysis, the estimate of the Roemer effect could be biased. The omission of information on
health status from all but the May and Newhouse-Phelps studies is a troubling example. If poor health status in an area tends to produce both higher utilization and a larger supply of beds, the
omission of information on health status would bias estimates of the Roemer effect upward.
A particularly serious form of this problem appears in studies using survey data on individuals. The adequacy of the supply of beds in an area depends on a variety of characteristics of the
population of that area. For example, four beds per thousand population, the current standard of the U.S. Department of Health and Human Services, may be ample in the average community, but it might
be a tight standard in communities with an elderly population or where health status is poor. Omission of these aggregate characteristics of the communities can therefore seriously bias estimates of
the Roemer effect.
The expected bias would depend on the specific study. If the omitted characteristics were not correlated with bed availability across the entire sample, the omission would constitute random error in
the measure of bed availability. The result would be a downward bias in estimates of the Roemer effect (Pindyck and Rubinfeld, 1976). If the omitted variables were correlated with bed availability,
the measurement error would be systematic. The direction of bias would then depend on the relationships between the omitted variables, utilization, and bed availability. Interestingly, those studies
employing survey data—which cannot adequately control for aggregate variables such as average health status—do yield smaller estimates of the Roemer effect.
Simultaneous Equation Bias
While bed supply may determine utilization, theory also suggests that over the long run, utilization should determine bed supply. Beds per capita is fully exogenous only in the short run.
Nevertheless, with the exception of Chiswick, none of the studies treat bed availability as endogenous. Ordinary least squares estimation could result in an upward bias of the estimate of the Roemer
New Estimates of the Roemer Effect
By using a Medicare data base, this study avoids—in whole or in part—some of the shortcomings of previous work discussed above. Incorrect measurement of the price variable is avoided because hospital
prices faced by Medicare beneficiaries—a deductible equal to the national average per diem Medicare reimbursement—do not vary.^1 Incorrect measurement of utilization rates is avoided by comparing the
utilization in Professional Standards Review Organization (PSRO) areas to the Medicare population in those areas, after adjustment for migration of patients across PSRO area boundaries (U.S.D.H.H.S.,
1979). Possible bias from omitted variables is reduced in several ways: by using aggregate (PSRO area) data, by entering bed availability into the equation as the change over a two year period, and
by the inclusion in the equation of both regional dummy variables and a three-year lagged value of the dependent variable (Medicare days of care per 1,000 enrollees). Since the lagged utilization
variable is a strong predictor of the dependent variable, it serves as an excellent proxy for unmeasured variables that might affect utilization. The regional dummies serve a similar but less crucial
role as a proxy for omitted variables. Finally, potential simultaneous equation bias is reduced by entering beds per capita as a first difference. Given long lags in bed construction, it is unlikely
that the change in beds is influenced by the change in utilization for the same time period. Indeed, the likelihood is reduced still further by lagging the change of beds by one year. In this
specification, simultaneous equation bias would remain only if changes in utilization rates from 1974 to 1977 influenced changes in bed supply between 1974 and 1976. Given long lags in bed
construction, this could occur only if relative trends in Medicare utilization rates have been quite stable over long periods of time, and if hospital administrators are aware of those trends and use
them in making decisions about bed closures or construction. It should be reiterated that the use rates employed in this study are based on denominators that are adjusted for patient migration
between PSRO areas—information of a sort that most hospital administrators during that period most likely did not have. The problem is also reduced as the utilization variable is only for Medicare
The study does have a major shortcoming, however, in being limited to the Medicare population. The Roemer effect for Medicare enrollees could be either larger or smaller than for the rest of the
population. Nevertheless, the Medicare population accounts for roughly one-third of days of hospital care, so the estimate is important in its own right. Further, generalization to the entire
population may involve less error than the problems affecting other studies discussed above.
The analysis was based on the hospital use of all Medicare beneficiaries in 1977. These data were obtained from the 100 Percent Medicare Claims File.
The unit of analysis, however, was the PSRO area. All areas in the United States were included, regardless of whether they had a functioning PSRO at the time. To obtain a utilization rate, the number
of Medicare enrollees (obtained from the Medicare Master Enrollment File) was adjusted to take into account patient migration across PSRO area boundaries. The Master Facility Inventory of the
National Center for Health Statistics and the Area Resource File of the Bureau of Health Manpower were used to obtain measure of the demographic and health-care system characteristics of each area.
Ordinary least squares regression was used to analyze the data. PSRO areas, rather than individuals, were the unit of analysis, so the regression had relatively few (189) degrees of freedom.
Definitions of the variables used in the analysis are provided in Table 1, and their intercorrelations are displayed in Table 2.
Table 1. Variables in Regression Analysis of the Roemer Effect.
Variable Form
1977 total days of care per 1,000 Medicare enrollees^1 (dependent variable) static, 1977
1974 total days of care per 1,000 Medicare enrollees^1 (1974 Medicare utilization) static, 1974
Change in proportion of population age 64 or older (‘proportion aged’) first difference, 1976-1974
Certified long-term care beds per 1,000 Medicare enrollees (‘l.t.c. beds’) first difference, 1976-1973
Change in physicians per 1,000 population (‘physician supply’) first difference, 1976-1974
Population per square mile (‘population density’) static, 1976
Proportion of hospital days attributable to Medicare patients (‘Medicare proportion’) static, 1976
Hospital occupancy rate (weighted average of within-hospital rates) static, 1976
Proportion of families with incomes below $5,000 (‘poverty’) static 1977
Months of PSRO review static, 1977
Four-way regional contrast: Northeast, North central, South, West 3 dummy variables
Proportion of Medicare-certified short-stay beds in teaching facilities (proportion teaching) static, 1977
Change in short-stay hospital beds per 1,000 population (‘bed supply’) first difference, 1976-1974
Medicare population base (denominator) adjusted for patient migration between PSRO areas.
Table 2. Correlation Matrix of Variables in the Regession Analysis.
1977 Medicare Utilization .93 −.07 −.23 .09 .30 .23 .56 .06 −.15 .35 .27 −.72 .22 .22
1974 Medicare Utilization 1.00 −.13 −.31 .02 .22 .24 .37 .14 −.14 .22 .39 −.65 .14 .11
Proportion Aged 1.00 −.06 .35 −.12 .27 .16 −.13 −.16 .04 −.13 −.16 .07 .45
L.T.C. Beds 1.00 −.07 .13 −.07 .19 −.37 .24 .32 −.13 .32 .12 −.29
Physician Supply 1.00 .27 −.14 .22 .11 .07 .05 −.13 −.11 .55 .23
Population Density 1.00 .01 .25 .23 .13 .29 −.11 −.07 .36 −.06
Medicare Proportion 1.00 .02 .24 −.10 .29 .00 −.28 −.24 .08
Hospital Occupancy Rate 1.00 −.33 .01 .48 .05 −.51 .38 .10
Poverty 1.00 −.06 −.21 −.03 −.04 −.02 .03
Months of PSRO Review 1.00 .14 −.19 .26 .13 −.15
Northeast vs. West 1.00 −.34 −.27 .18 −.06
Northcentral vs. West 1.00 −.29 .02 .01
South vs. West 1.00 −.15 −.38
Proportion teaching 1.00 .14
Bed Supply 1.00
To avoid problems associated with the use of change variables as dependent variables, the analysis employed a static dependent variable (1977 Medicare days of care) (Cohen and Cohen, 1975; Cronbach
and Furby, 1970). Accordingly, a lagged value of the dependent variable (1974 Medicare days of care) was used as a covariate to control for previous utilization rates. This variable also serves as a
proxy to control for a variety of unmeasured differences between PSRO areas. The effectiveness of using a lagged value of the dependent variable as a proxy for omitted independent variables depends
on existence of stable relationships between the dependent variable and the omitted variables, as well as on stability in the omitted variables themselves. Neither of these can be directly assessed
with data employed in this study. The relationships between the dependent variable and the measured independent variables, however, are reasonably stable, as indicated by the first two rows of Table
2. Moreover, the fact that the lagged dependent variable shows not only a strong zero-order relationship with the dependent variable, but also a strong partial relationship (a standardized regression
coefficient of 0.78) suggests that is an effective proxy for some important omitted variables.
Three independent variables were used to control for demographic differences between PSRO areas. One was the change from 1974 to 1976 in the proportion of the population aged 65 and over. This
variable is important in theory, since changes in it would lead directly to changes in the Medicare utilization rate which could bias estimates of the Roemer effect. (It turned out empirically to be
an important variable in the regression as well.) Population density and the proportion of families in poverty were entered as static (1976) variables.
Six variables represented the health-system characteristics of each area. One was the independent variable of interest — the change from 1974 to 1976 in short-stay hospital beds per 1,000 population.
Since bed supply was entered only as a change variable, a static occupancy rate measure was added as a control for the adequacy of hospital capacity. The use of both bed supply and occupancy
variables in a regression with days of care per 1,000 as the dependent variable did not create an identity, because of the use of lags and first differences. The remaining health-system variables
employed were the number of Medicare-certified long-term care beds per 1,000 enrollees, the proportion of days of care attributable to Medicare patients, the proportion of hospital days attributable
to Medicare patients, and the change (1974-1976) in the number of physicians per 1,000 population.
A four-way regional contrast (Northeast, Northcentral, South, and West) was represented by three dummy variables. These were included because of the well-known difference between regions in patterns
of hospital use. In the absence of a full explanation of those regional differences, the region variables can be seen as a proxy for unmeasured demographic or health-system characteristics.
Finally, the number of months that PSROs had been conducting review was included. The use of this variable was necessitated by the finding that PSROs have had an effect on Medicare hospital use (CBO,
1979; 1981).
The regression analysis indicated a large and highly significant Roemer effect (t = 5.6, p<.0005; see Table 3). The regression coefficient indicated that a 1 percent change in bed supply, all else
being constant, produces a 0.42 percent increase in Medicare days of hospitalization per 1,000 enrollees.
Table 3. Results of the Regression Analysis.
Variable Mean b t
1974 Medicare utilization 3640.5 .840 26.26^****
Proportion aged 3.280 −15.773 2.98^***
l.t.c. beds 2.05 −.119 0.11
Physician supply .104 111.257 0.54
Population density 1625.1 .006 2.68^**
Medicare proportion .342 63.6 0.25
Hospital occupancy rate .737 1950.5 6.98^****
Poverty .110 −205.6 0.40
Months of PSRO review 7.70 −3.178 2.39^*
Northeast vs. West .243 87.095 2.03^*
Northcentral vs. West .270 −58.884 1.65
South vs. West .185 −62.746 1.18
Proportion teaching 31.115 −.496 0.72
Bed supply 0.0581 373.919 5.55^****
Intercept −725.4
R^2 = .94
R^2 adjusted = .93
Because of the specification used, the Roemer effect expressed in percentage terms will vary with the level of the bed supply and utilization variables. The values above were obtained by evaluating
the results at the baseline (1974) level of bed supply and the 1977 level of utilization (the dependent variable). Using the baseline (1974) level of utilization would not materially affect the
Note that because the beds variable is a first difference, the elasticity cannot be obtained directly from the usual formula:
This formula would yield the desired elasticity — the percent change in utilization per percent change in bed supply — only if the bed supply in raw form were used as the independent variable.
Since bed supply was entered as a first difference, the calculation of the elasticity is a bit more indirect. A value of the first difference variable is selected to represent a given percent change
in bed supply. This value is then multiplied by the regression coefficient to yield a resulting change in utilization rates, and this change in rates is divided by a baseline utilization rate to
transform it into a percent change. The ratio of the chosen percent change in bed supply to this derived percent change in utilization rates is the elasticity.
The baseline (1974) bed supply was 4.1782 beds per 1,000 Medicare enrollees. A 1 percent change would therefore be 0.041782 beds per 1,000. Multiplying this change by the regression coefficient of
373.919 yields a utilization change of 15.623 days of hospitalization per 1,000 Medicare enrollees. This corresponds to 0.42 percent of the 1977 average rate of 3711.263 days per 1,000.
The specification provided a good fit, suggesting that the problem of omitted variables may have been largely avoided. To assess the degree of fit, however, it is necessary to distinguish between
what could be called cross- sectional fit and time-series fit — that is, between the model's ability to predict levels of utilization rates and its ability to predict changes in utilization rates.
The model is necessarily much more successful in predicting levels than in predicting changes.
The cross-sectional fit of the model — that is, its ability to predict levels of utilization — is excellent: an R^2 of .93 after downward adjustment for sample size (see Table 3). The lagged
dependent variable was the most important factor contributing to this good fit; its standardized regression coefficient was .78, with a t of 26. The regional dummy variables, which were also expected
to play a role as proxies for omitted variables, were found to have moderate predictive power; all were statistically either significant (p<.05) or marginal (p<.10).
The model's ability to predict change in utilization rates, however, is necessarily much lower. This is because of the large proportion of variance accounted for by the lagged dependent variable.
When predicting change in a dependent variable, R^2 is at best an ambiguous measure, because it is highly sensitive to methods (such as first differencing) used to remove the effects of prior values
of the dependent variable. In such cases, it is more meaningful to measure the model's success in predicting innovation variance — that is, the variance of the dependent variable that is not
predicted by prior values of the dependent variable (Pierce, 1979). The specification used here accounted for about 49 percent of this innovation variance (after downward adjustment for sample size).
The observed relationship between hospital beds and rates of use in the Medicare population suggests that a similar relationship might exist for the general population. Medicare hospital coverage is
similar to that for the general population, so that there is no a priori reason for changes in use by the non-Medicare population to offset changes in the Medicare population. The magnitude of the
effect could differ, however, because the Medicare population has different medical conditions that are treated in hospitals and different time prices. There are no predictions of whether the
magnitude of the Roemer effect for the entire population is larger or smaller than the 0.42 elasticity estimated for the Medicare population.
Confirmation of the Roemer effect supports the notion that health planning has the potential to significantly lower hospital costs. For example, if health planning were successful in reducing the
stock of beds by 10 percent, resources would be saved not only from the reduction in excess capacity but also from the 4 percent reduction in days of care (Joskow and Schwartz, 1980). While not
analyzed in this study, similar relationships may exist between major pieces of capital equipment, such as Computed Axial Tomography (CAT) scanners, and use of ancillary services.
Nevertheless, presence of the Roemer effect does not imply that health planning is the policy of choice to contain health care costs. Some have criticized planning on the basis of potential
distortions from regulating one input, and the technical and political obstacles to making good project decisions. In addition, studies of the early health planning activities by the States have not
been encouraging about effectiveness (CBO, 1982). While the absence of demonstrated effects does not imply that planning cannot work, it is likely that realizing the cost-saving potential of planning
will be difficult.
The authors are grateful to Mitchell Dayton of the University of Maryland and Paul Eggers of the Health Care Financing Administration (HCFA) for providing the computer runs needed for the analysis,
and to the Office of Research, HCFA, for making available the data used in this report. Allen Dobson and Herbert Klarman provided valuable comments.
Reprint requests: Paul B. Ginsburg, Congressional Budget Office, House Office Bldg., Annex No. 2, Washington, D.C. 20515.
This paper reflects solely the views of the authors and does not necessarily indicate an official position of the Congressional Budget Office.
Supplementary private insurance coverage (for example “Medigap” policies) introduces a small amount of price variation that Is not accounted for in this analysis. Also, prices of a complementary
input—physician services—vary across areas. Nevertheless, the specification employed reduces substantially the chance of bias resulting from this omission. | {"url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC4191337/","timestamp":"2024-11-11T01:48:28Z","content_type":"text/html","content_length":"140197","record_id":"<urn:uuid:141e4285-8262-4ff8-ba9c-0d645411f589>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00016.warc.gz"} |
Ask Uncle Colin: A round-robin
Dear Uncle Colin
I’m organising a tournament with 16 teams, and wanted to arrange it in five stages, each consisting of four groups of four teams. However, I found that after three rounds, it wasn’t possible to
find any groups without making teams play each other again! Why is that?
Dumb Rematches Aren’t Wanted
Hi, DRAW, and thanks for your message!
The problem
Suppose you have 16 teams, names A, B, C… all the way up to P. Arrange them in a grid like this:
A B C D E F G H I J K L M N O P
The first two rounds are easy enough: without loss of generality, we can say that the first-round groups go along the rows (so, for instance, A play B, C and D), and the second-round groups go down
the colums (A play E, I and M). In the third round, we can go diagonally (so A, F, K and P are in the same group).
In fact, any satisfactory arrangement of the teams in the first three rounds can be relabelled the same way.
Now look at who team A could possibly be grouped with in round 4:
A (B) (C) (D) (E) (F) G H
(I) J (K) L (M) N O (P)
See the problem? Each of A’s possible opponents only have two teams in the list they haven’t played – and those two teams have already played each other! (The proof is left as an exercise).
How I’d run the tournament
If we’re going to insist that all 16 teams play each possible opponent once, a total of 240 games, which seems a lot, here’s how we can do it.
In the first round, A plays B, C plays D and so on. We have eight two-team groups.
In rounds 2 and 3, each team in the first group plays a team from the second group; each team in the third group plays a team from the fourth, and so on; we’ve effectively folded the eight two-team
groups into four four-team groups, each of which have played among themselves but not any other teams.
We repeat the trick in rounds 4-7: each team from A, B, C and D now play a team from E, F, G and H in turn. Every team from A to H will have played each other; each team from I to P will have played
each other; no team from either group has played anyone from the other.
So rounds 8-15 finish that off: again, in each round, each team from the A-H group plays, in turn, a team from the I-P group. At the end of this, everyone has played everyone else exactly once.
(If you want to mix it up a bit, you can rearrange the order of the rounds so the teams aren’t isolated from each other.)
How I’d actually run the tournament
Personally, I think round-robins are overrated. I much prefer variants on the Swiss System, where teams in each round play teams with a similar record they haven’t played before. This has the good
points of a knockout (lots of competitive games) and a round-robin (everyone gets to play lots of games) with only relative complexity as a drawback.
A selection of other posts
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8. निम्नलिखित में से किसी एक खण्ड को हल कीजिए।
(क) अवकल समीकरण ... | Filo
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8. निम्नलिखित में से किसी एक खण्ड को हल कीजिए। (क) अवकल समीकरण का हल ज्ञात कीजिए
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Question Text 8. निम्नलिखित में से किसी एक खण्ड को हल कीजिए। (क) अवकल समीकरण का हल ज्ञात कीजिए
Updated On Jan 9, 2024
Topic Differential Equations
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 72
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Bartender 20603 - math word problem (20603)
Bartender 20603
The mixed drink consists of 1.5 dcl of pineapple juice and 0.5 dcl of coconut syrup. Klára ordered it sweeter, so the bartender changed the volume of coconut syrup to a ratio of 3:2. What percent of
the drink is now coconut syrup if the volume of the juice has not changed?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Themes, topics:
Grade of the word problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/20603","timestamp":"2024-11-06T07:35:09Z","content_type":"text/html","content_length":"61074","record_id":"<urn:uuid:bf7df770-ca5a-4728-8474-84f9fda9375d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00004.warc.gz"} |
Mathematicians Find 27 Tickets That Guarantee UK National Lottery Win - International Maths Challenge
Mathematicians Find 27 Tickets That Guarantee UK National Lottery Win
Buying a specific set of 27 tickets for the UK National Lottery will mathematically guarantee that you win something.
Buying 27 tickets ensures a win in the UK National Lottery
You can guarantee a win in every draw of the UK National Lottery by buying just 27 tickets, say a pair of mathematicians – but you won’t necessarily make a profit.
While there are many variations of lottery in the UK, players in the standard “Lotto” choose six numbers from 1 to 59, paying £2 per ticket. Six numbers are randomly drawn and prizes are awarded for
tickets matching two or more.
David Cushing and David Stewart at the University of Manchester, UK, claim that despite there being 45,057,474 combinations of draws, it is possible to guarantee a win with just 27 specific tickets.
They say this is the optimal number, as the same can’t be guaranteed with 26.
The proof of their idea relies on a mathematical field called finite geometry and involves placing each of the numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical
shapes, then using these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets bought using those numbers, at a cost of £54,
will hit every possible winning combination of two numbers.
The 27 tickets that guarantee a win on the UK National Lottery
Their research yielded a specific list of 27 tickets (see above), but they say subsequent work has shown that there are two other combinations of 27 tickets that will also guarantee a win.
“We’ve been thinking about this problem for a few months. I can’t really explain the thought process behind it,” says Cushing. “I was on a train to Manchester and saw this [shape] and that’s the best
logical [explanation] I can give.”
Looking at the winning numbers from the 21 June Lotto draw, the pair found their method would have won £1810. But the same numbers played on 1 July would have matched just two balls on three of the
tickets – still a technical win, but giving a prize of just three “lucky dip” tries on a subsequent lottery, each of which came to nothing.
Stewart says proving that 27 tickets could guarantee a win was the easiest part of the research, while proving it is impossible to guarantee a win with 26 was far trickier. He estimates that the
number of calculations needed to verify that would be 10^165, far more than the number of atoms in the universe. “There’d be absolutely no way to brute force this,” he says.
The solution was a computer programming language called Prolog, developed in France in 1971, which Stewart says is the “hero of the story”. Unlike traditional computer languages where a coder sets
out precisely what a machine should do, step by step, Prolog instead takes a list of known facts surrounding a problem and works on its own to deduce whether or not a solution is possible. It takes
these facts and builds on them or combines them in order to slowly understand the problem and whittle down the array of possible solutions.
“You end up with very, very elegant-looking programs,” says Stewart. “But they are quite temperamental.”
Cushing says the research shouldn’t be taken as a reason to gamble more, particularly as it doesn’t guarantee a profit, but hopes instead that it encourages other researchers to delve into using
Prolog on thorny mathematical problems.
A spokesperson from Camelot, the company that operates the lottery, told New Scientist that the paper made for “interesting reading”.
“Our approach has always been to have lots of people playing a little, with players individually spending small amounts on our games,” they say. “It’s also important to bear in mind that, ultimately,
Lotto is a lottery. Like all other National Lottery draw-based games, all of the winning Lotto numbers are chosen at random – any one number has the same and equal chance of being drawn as any other,
and every line of numbers entered into a draw has the same and equal chance of winning as any other.”
For more such insights, log into www.international-maths-challenge.com.
*Credit for article given to Matthew Sparkes* | {"url":"https://international-maths-challenge.com/mathematicians-find-27-tickets-that-guarantee-uk-national-lottery-win/","timestamp":"2024-11-08T13:54:49Z","content_type":"text/html","content_length":"148745","record_id":"<urn:uuid:a68cc888-a90a-4fe9-9b82-202c40fb1c85>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00659.warc.gz"} |
The ultimate speed test test
With an advertised download speed of 200 Mbps, a speed test uses an average of 0.4 Gigabytes of data. This is also the median. The most popular speed test (the Ookla Speedtest) uses 0.5 Gb.
If you do not want to use too much data for a speed test and still want to know your internet speed accurately - at an advertised speed of 200Mbps - it is best to use TestMy.net or Google Fiber.
Our internet provider recently doubled the internet speed from 100 Mbps to 200 Mbps.
For this test we want to know how much data a speed test uses when the advertised download speed is 200 Mbps.
Method of measurement
The amount of data used by a speed test is measured with the live option of vnstat.
Initially the webpage with the speed tests to test is opened in the Edge browser. Then the data monitoring is started with vnstat -l and the to be tested speed test is opened in a new tab. After
three times testing the internet speed, vnstat is stopped with control-C.
The measured download speeds and the median are noted. These are all in mega bits per seconds. The data used as measured by vnstat is in mebibytes (MiB) (or kibibytes (KiB)).
Speed tests to test
Because this is a relative simple test, the unique speed tests as collected at ZOMDir will be tested.
The measurements
The following has been measured. Note that all speeds are measured in Mbps.
The least data is used by Internet Speed at a Glance. This speed test uses 39 kibibyte per test. However, this speed test only gives an indication of your internet speed.
N Perf needs the most data for a speed test. N Perf uses 652 mebibytes per test.
The median and average are relative close to each other at 358 and 350 mebibytes respectively.
If we only look at speed tests whose measured values meet the -Accurate and consistent- requirements in What makes a speed test excellent?, the following speed tests meet these requirements:
1. Astound
2. Bredbandskollen
3. Cloudflare
4. DSLReports
5. Fireprobe
6. Google Fiber
7. Measurement Lab
8. Meter.net
9. N Perf
10. Ookla Speedtest
11. SamKnows
12. SpeedOf.me API Sample Page
13. TestMy.net
14. Xfinity xFi Speed Test
Ideally, you would expect a relationship between the amount of data used and the accuracy of a speed test.
The graph below plots accuracy against the amount of data used. At a glance it is clear that our expectations were not met.
The graph above makes it clear that if you do not want to use too much data for a speed test and still want to know your internet speed accurately - at an advertised speed of 200Mbps - it is best to
use TestMy.net or Google Fiber. | {"url":"https://speedtesttest.zomdir.com/2023/the-mbs-used-by-a-speed-test-test-200-mbps.html","timestamp":"2024-11-04T10:50:05Z","content_type":"text/html","content_length":"16862","record_id":"<urn:uuid:a5ef81b6-426d-4f67-8db0-179ed4685354>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00047.warc.gz"} |
Divergence - (Honors Pre-Calculus) - Vocab, Definition, Explanations | Fiveable
from class:
Honors Pre-Calculus
Divergence is a mathematical concept that describes the rate of change or the behavior of a sequence or series as it progresses. It is particularly relevant in the context of geometric sequences,
where it helps determine whether a sequence converges to a finite value or diverges to infinity.
congrats on reading the definition of Divergence. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The divergence of a geometric sequence is determined by the value of the common ratio ($r$).
2. If $|r| < 1$, the geometric sequence converges to a finite value.
3. If $|r| > 1$, the geometric sequence diverges to positive or negative infinity.
4. If $|r| = 1$, the geometric sequence is neither converging nor diverging, but oscillating between positive and negative values.
5. Divergence is an important concept in understanding the behavior of infinite series and their applications in various fields, such as calculus and numerical analysis.
Review Questions
• Explain how the common ratio ($r$) of a geometric sequence determines whether the sequence diverges or converges.
□ The value of the common ratio ($r$) in a geometric sequence is the key factor that determines whether the sequence diverges or converges. If $|r| < 1$, the sequence will converge to a finite,
non-zero value as the number of terms increases. If $|r| > 1$, the sequence will diverge to positive or negative infinity. When $|r| = 1$, the sequence is neither converging nor diverging,
but rather oscillating between positive and negative values.
• Describe the relationship between divergence and the behavior of an infinite geometric series.
□ The divergence or convergence of a geometric sequence is closely related to the behavior of an infinite geometric series. If the common ratio $|r| < 1$, the corresponding infinite geometric
series will converge to a finite sum. However, if $|r| > 1$, the infinite geometric series will diverge and the sum will approach positive or negative infinity. The divergence or convergence
of the sequence, and consequently the series, has important implications in various mathematical applications, such as the evaluation of infinite sums and the analysis of the behavior of
infinite processes.
• Analyze the significance of divergence in the context of geometric sequences and their real-world applications.
□ The concept of divergence in geometric sequences is crucial in understanding their behavior and applications. Divergence determines whether a sequence will grow without bound or approach a
finite limit, which has important implications in various fields. For example, in finance, the divergence of a geometric sequence can be used to model the growth of investments over time. In
physics, divergence can be used to analyze the behavior of oscillating systems. In computer science, divergence is essential in the analysis of the convergence of numerical algorithms.
Understanding the conditions for divergence, and its relationship to the common ratio, allows for the accurate modeling and prediction of the long-term behavior of geometric sequences and
their applications.
© 2024 Fiveable Inc. All rights reserved.
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LIFE (CRACK INITIATION) | lifing
top of page
The LIFE module is used to calculate fatigue Life in a FEM (acting as a 'Digital Twin' of a real structure) with a broad palette of Strain based and Stress based approaches.
The GUI is very intuitive and the analysis process is straightforward.
When both FEM and internal stresses are imported, a 'surface-stress-resolving' technique is employed in order to handle, during the calculation, the plane stress tensors which occur at the component
surface (in case of shell modelled components this is not done as stresses are naturally planar on the top-bottom faces).
In the LIFING database only stresses occurring at the surfaces are stored.
Stresses at the surface, if no pressure is applied, are in fact plane stress by definition.
From a numerical standpoint this is not true, as FEM stresses are calculated at Gauss points (internal) and stresses at the surfaces (element nodal stresses) are extrapolated.
LIFE will ignore stress tensor components which are not the plane stress component ones.
This approach implies that a 'good' mesh refinement is required for a good Fatigue Life estimation.
The following analysis methods are available.
• Strain based approach
□ Elastic-Plastic Stress calculation with Neuber or E.S.E.D. (Glinka)
□ Multiaxial Elastic-Plastic Stresses calculated with Dowling or Hoffman-Seeger approach by reducing the generic Non-proportional Loading cases to Proportional Loading ones or by using full
Multiaxial Non-Proportional Loading cases with Pseudo-Material approach in conjunction with the Mroz-Garud cyclic plasticity model.
□ Critical Plane methods are implemented. Fatigue parameters:
☆ Smith-Watson-Topper
☆ Morrow's
☆ Manson-Halford
☆ Brown-Miller
☆ Fatemi-Socie
• Stress based appraoch
□ S-N curves defined by points
□ S-N curves as per MIL Standard (MIL-HDBK-5J). When LIFING is installed the MIL-HDBK-5J S-N curves database is installed.
□ Multiaxial analysis with Dang-Van, McDiarmid, generalized Goodman, Carpinteri-Spagnoli, ...
□ Multiaxial analysis with equivalent stresses
□ Multiaxial analysis with uniaxial reduction (critical plane search)
□ Mean stress accounting with:
☆ Goodman
☆ Gerber
☆ Soderberg
☆ Walker
☆ Morrow
☆ Smith-Watson-Topper
☆ Haigh
Fatigue based on PSD is handled: some load channels can be 'fed' with PSD signals and equivalent time histories are calculated with the Dirlik, Narrow Band or Stainberg methods.
LIFE can also be used for 'straight analyses':
• The user can import a sequence file, instead of a FEM model: in this case the sequence is recognized as a stress sequence and the user can perform a uniaxial fatigue analysis with any of all
implemented methods.
• The user can import a stress tensor file, instead of a FEM model: in this case LIFE creates a single element database where internal stresses are those imported from the file and fatigue can be
calculated at this element with any of all implemented methods.
• The user can import a set of sequence files coming from a real strain gauge (uniaxial or 0-45-90 or 0-60-120 or 0-120-240): as for the above, the fatigue analysis can be therefore carried out
with any of all implemented methods.
Other than just analysing, the LIFE module can be used for stress tensor time history extraction at selected locations with Virtual Strain Gauge and related Multiaxial Assessments.
The analyst can put on any FEM location (on the surface), a strain gauge and can orient it.
For the selected element
• the sequence of stress tensors can be visualized and dumped in ASCII files
• Mohr Circles at defined instant can be visualized
• Multiaxial Assessment can be performed (scatter plots for showing Maximum Principal direction variation over the time as well as Biaxiality ratio variation over the time.
Time histories can be exported and/or handled as following:
This module allows to perform the following tasks.
• Filter sequence. The following methods are included:
• Non-turning points filtering (always active)
• Racetrack filtering
• Modified. The following options are available:
Scaling (entire sequence or a portion)
Offsetting (entire sequence or a portion)
Negative values scaling
• Cycle counted. The Range-Pair method is implemented as per the ASTM STP1006 standard. The output cycles sequence is dumped in a ASCII file. The option to dump the counted sequence in AFGROW
format is available.
• Exceedence plots and Range-Mean Hystograms can be visualized
Element stress tensor sequences can be also filtered with the Multiaxial Racetrack Filter.
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Short description: Parallelization across multiple processors in parallel computing environments
Data parallelism is parallelization across multiple processors in parallel computing environments. It focuses on distributing the data across different nodes, which operate on the data in parallel.
It can be applied on regular data structures like arrays and matrices by working on each element in parallel. It contrasts to task parallelism as another form of parallelism.
A data parallel job on an array of n elements can be divided equally among all the processors. Let us assume we want to sum all the elements of the given array and the time for a single addition
operation is Ta time units. In the case of sequential execution, the time taken by the process will be n×Ta time units as it sums up all the elements of an array. On the other hand, if we execute
this job as a data parallel job on 4 processors the time taken would reduce to (n/4)×Ta + merging overhead time units. Parallel execution results in a speedup of 4 over sequential execution. One
important thing to note is that the locality of data references plays an important part in evaluating the performance of a data parallel programming model. Locality of data depends on the memory
accesses performed by the program as well as the size of the cache.
Exploitation of the concept of data parallelism started in 1960s with the development of Solomon machine.^[1] The Solomon machine, also called a vector processor, was developed to expedite the
performance of mathematical operations by working on a large data array (operating on multiple data in consecutive time steps). Concurrency of data operations was also exploited by operating on
multiple data at the same time using a single instruction. These processors were called 'array processors'.^[2] In the 1980s, the term was introduced ^[3] to describe this programming style, which
was widely used to program Connection Machines in data parallel languages like C*. Today, data parallelism is best exemplified in graphics processing units (GPUs), which use both the techniques of
operating on multiple data in space and time using a single instruction.
Most data parallel hardware supports only a fixed number of parallel levels, often only one. This means that within a parallel operation it is not possible to launch more parallel operations
recursively, and means that programmers cannot make use of nested hardware parallelism. The programming language NESL was an early effort at implementing a nested data-parallel programming model on
flat parallel machines, and in particular introduced the flattening transformation that transforms nested data parallelism to flat data parallelism. This work was continued by other languages such as
Data Parallel Haskell and Futhark, although arbitrary nested data parallelism is not widely available in current data-parallel programming languages.
In a multiprocessor system executing a single set of instructions (SIMD), data parallelism is achieved when each processor performs the same task on different distributed data. In some situations, a
single execution thread controls operations on all the data. In others, different threads control the operation, but they execute the same code.
For instance, consider matrix multiplication and addition in a sequential manner as discussed in the example.
Below is the sequential pseudo-code for multiplication and addition of two matrices where the result is stored in the matrix C. The pseudo-code for multiplication calculates the dot product of two
matrices A, B and stores the result into the output matrix C.
If the following programs were executed sequentially, the time taken to calculate the result would be of the [math]\displaystyle{ O(n^3) }[/math](assuming row lengths and column lengths of both
matrices are n) and [math]\displaystyle{ O(n) }[/math]for multiplication and addition respectively.
// Matrix multiplication
for (i = 0; i < row_length_A; i++)
for (k = 0; k < column_length_B; k++)
sum = 0;
for (j = 0; j < column_length_A; j++)
sum += A[i][j] * B[j][k];
C[i][k] = sum;
// Array addition
for (i = 0; i < n; i++) {
c[i] = a[i] + b[i];
We can exploit data parallelism in the preceding code to execute it faster as the arithmetic is loop independent. Parallelization of the matrix multiplication code is achieved by using OpenMP. An
OpenMP directive, "omp parallel for" instructs the compiler to execute the code in the for loop in parallel. For multiplication, we can divide matrix A and B into blocks along rows and columns
respectively. This allows us to calculate every element in matrix C individually thereby making the task parallel. For example: A[m x n] dot B [n x k] can be finished in [math]\displaystyle{ O(n) }[/
math] instead of [math]\displaystyle{ O(m*n*k) }[/math] when executed in parallel using m*k processors.
// Matrix multiplication in parallel
#pragma omp parallel for schedule(dynamic,1) collapse(2)
for (i = 0; i < row_length_A; i++){
for (k = 0; k < column_length_B; k++){
sum = 0;
for (j = 0; j < column_length_A; j++){
sum += A[i][j] * B[j][k];
C[i][k] = sum;
It can be observed from the example that a lot of processors will be required as the matrix sizes keep on increasing. Keeping the execution time low is the priority but as the matrix size increases,
we are faced with other constraints like complexity of such a system and its associated costs. Therefore, constraining the number of processors in the system, we can still apply the same principle
and divide the data into bigger chunks to calculate the product of two matrices.^[4]
For addition of arrays in a data parallel implementation, let's assume a more modest system with two central processing units (CPU) A and B, CPU A could add all elements from the top half of the
arrays, while CPU B could add all elements from the bottom half of the arrays. Since the two processors work in parallel, the job of performing array addition would take one half the time of
performing the same operation in serial using one CPU alone.
The program expressed in pseudocode below—which applies some arbitrary operation, foo, on every element in the array d—illustrates data parallelism:^[nb 1]
if CPU = "a" then
lower_limit := 1
upper_limit := round(d.length / 2)
else if CPU = "b" then
lower_limit := round(d.length / 2) + 1
upper_limit := d.length
for i from lower_limit to upper_limit by 1 do
In an SPMD system executed on 2 processor system, both CPUs will execute the code.
Data parallelism emphasizes the distributed (parallel) nature of the data, as opposed to the processing (task parallelism). Most real programs fall somewhere on a continuum between task parallelism
and data parallelism.
Steps to parallelization
The process of parallelizing a sequential program can be broken down into four discrete steps.^[5]
Type Description
Decomposition The program is broken down into tasks, the smallest exploitable unit of concurrence.
Assignment Tasks are assigned to processes.
Orchestration Data access, communication, and synchronization of processes.
Mapping Processes are bound to processors.
Data parallelism vs. task parallelism
Data parallelism Task parallelism
Same operations are performed on different subsets of same data. Different operations are performed on the same or different data.
Synchronous computation Asynchronous computation
Speedup is more as there is only one execution thread operating on all sets of Speedup is less as each processor will execute a different thread or process on the same or different set of data.
Amount of parallelization is proportional to the input data size. Amount of parallelization is proportional to the number of independent tasks to be performed.
Designed for optimum load balance on multi processor system. Load balancing depends on the availability of the hardware and scheduling algorithms like static and dynamic
Data parallelism vs. model parallelism
Data parallelism Model parallelism
Same model is used for every thread but the data given to each of them is divided and shared. Same data is used for every thread, and model is split among
It is fast for small networks but very slow for large networks since large amounts of data needs to be transferred between processors It is slow for small networks and fast for large networks.
all at once.
Data parallelism is ideally used in array and matrix computations and convolutional neural networks Model parallelism finds its applications in deep learning
Mixed data and task parallelism
Data and task parallelism, can be simultaneously implemented by combining them together for the same application. This is called Mixed data and task parallelism. Mixed parallelism requires
sophisticated scheduling algorithms and software support. It is the best kind of parallelism when communication is slow and number of processors is large.^[7]
Mixed data and task parallelism has many applications. It is particularly used in the following applications:
1. Mixed data and task parallelism finds applications in the global climate modeling. Large data parallel computations are performed by creating grids of data representing Earth's atmosphere and
oceans and task parallelism is employed for simulating the function and model of the physical processes.
2. In timing based circuit simulation. The data is divided among different sub-circuits and parallelism is achieved with orchestration from the tasks.
Data parallel programming environments
A variety of data parallel programming environments are available today, most widely used of which are:
Data parallelism finds its applications in a variety of fields ranging from physics, chemistry, biology, material sciences to signal processing. Sciences imply data parallelism for simulating models
like molecular dynamics,^[9] sequence analysis of genome data ^[10] and other physical phenomenon. Driving forces in signal processing for data parallelism are video encoding, image and graphics
processing, wireless communications ^[11] to name a few.
See also
• Instruction level parallelism
• Thread level parallelism
1. ↑ Some input data (e.g. when d.length evaluates to 1 and round rounds towards zero [this is just an example, there are no requirements on what type of rounding is used]) will lead to lower_limit
being greater than upper_limit, it's assumed that the loop will exit immediately (i.e. zero iterations will occur) when this happens.
Original source: https://en.wikipedia.org/wiki/Data parallelism. Read more | {"url":"https://handwiki.org/wiki/Data_parallelism","timestamp":"2024-11-12T03:52:58Z","content_type":"text/html","content_length":"76389","record_id":"<urn:uuid:4d3f7703-4f2d-4bc6-b2aa-02536d866086>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00040.warc.gz"} |
Winter 2024 Quiz 1
← return to practice.dsc10.com
This quiz was administered in-person. It was closed-book and closed-note; students were not allowed to use the DSC 10 Reference Sheet. Students had 20 minutes to work on the quiz.
This quiz covered Lectures 1-4 of the Winter 2024 offering of DSC 10.
Problem 1
Select all the true statements below.
• Mixing ints and floats in an arithmetic expression will always result in a float.
• Dividing two ints will sometimes result in an int.
• Any float can be converted to a string using the function str().
• Any string can be converted to a float using the function float().
Answer: Option 1 and Option 3
Difficulty: ⭐️⭐️
The average score on this problem was 87%.
Problem 2
Consider the assignment statement below.
pear = [6, 10.1, "pear", 13]
What does the expression np.array(pear) evaluate to?
• array([6, 10.1, "pear", 13])
• array([6, 10.1, pear, 13])
• array(["6", "10.1", "pear", "13"])
• array(["pear"])
• array([pear])
• This expression errors
Answer: array(["6", "10.1", "pear", "13"])
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 50%.
Problem 3
Suppose x and y are both ints that have been previously defined, with x < y. Now, define:
peach = np.arange(x, y, 2)
Say that the spread of peach is the difference between the largest and smallest values in peach. The spread should be a non-negative integer.
Problem 3.1
Using array methods, write an expression that evaluates to the spread of peach.
Answer: peach.max() - peach.min()
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 62%.
Problem 3.2
Without using any methods or functions, write an expression that evaluates to the spread of peach.
Hint: Use [ ].
Answer: peach[len(peach) - 1] - peach[0] or peach[-1] - peach[0]
Difficulty: ⭐️⭐️⭐️⭐️
The average score on this problem was 36%.
Problem 3.3
Choose the correct way to fill in the blank in this sentence:
The spread of peach is ______ the value of y - x.
• always less than
• sometimes less than and sometimes equal to
• always greater than
• sometimes greater than and sometimes equal to
• always equal to
Answer: always less than
Difficulty: ⭐️⭐️⭐️⭐️
The average score on this problem was 48%.
Problem 4
Suppose fruits is a DataFrame of the fruits Ashley bought at the grocery store, where:
• The "fruit" column contains the name of the fruit, as a string. All values in this column are distinct.
• The "price" column contains the amount in dollars spent on the fruit, as a float.
• The "pounds" column contains the number of pounds purchased, as an int.
Problem 4.1
Fill in the blanks below to add a new column to fruits called "price_per_ounce" that contains the price per ounce of each of the fruits in fruits. There are 16 ounces in a pound.
Answer (x): assign
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 71%.
Answer (y): fruits.get("price")
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 67%.
Answer (z): (fruits.get("pounds") * 16)
Difficulty: ⭐️⭐️⭐️⭐️
The average score on this problem was 43%.
Problem 4.2
Write a line of code that evaluates to the amount of money, in dollars, that Ashley spent on fruit at the grocery store.
Answer: fruits.get("price").sum() or sum(fruits.get("price"))
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 62%.
Problem 4.3
Fill in the blanks so that the expression below evaluates to the name of the fruit with the highest price per ounce.
Answer (x): False
Difficulty: ⭐️⭐️
The average score on this problem was 87%.
Answer (y): "fruit"
Difficulty: ⭐️⭐️⭐️
The average score on this problem was 65%.
Problem 4.4
Assuming that "mango" is one of the fruits Ashley bought, fill in the blanks so that the expression below evaluates to the price per ounce of "mango".
Answer (x): set_index
Difficulty: ⭐️⭐️⭐️⭐️
The average score on this problem was 41%.
Answer (y): loc
Difficulty: ⭐️⭐️
The average score on this problem was 83%.
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GreeneMath.com | Ace your next Math Test!
Operations with Radical Expressions
Additional Resources:
In this section, we learn how to perform operations with radicals. We begin by learning about "like radicals". Like radicals have the same index and the same radicand. This is similar to when we
learned about “like terms” at the beginning of algebra. Recall that like terms have the same variable, raised to the same power. When we have like radicals, the process of addition or subtraction is
simple. We add or subtract the numbers multiplying the radicals, and leave the common radical unchanged. An easy way to see this is by factoring out the common radical. This will allow one to
visualize why we only perform operations with the numbers multiplying the radicals. We will encounter many problems that appear to not have like radicals. These problems commonly require us to
simplify each radical first, then attempt to find like radicals. In many cases, we will have like radicals that can be combined with addition or subtraction. Once we are done, we must remember to
simplify all radicals before we report our answer. Lastly, we will look at several types of multiplication problems that we will encounter when working with radical expressions.
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Math Morphing Proximate and Evolutionary Mechanisms
The mathematics behind fractals began to evolve in the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity, although he made the mistake of thinking that only
the straight line was a self-similar outcome. It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the
non-intuitive property of being everywhere continuous but nowhere differentiable. A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve,
space-filling curve, and Koch curve. In 1904, Helge von Koch demonstrated his dissatisfaction with Weierstrass's very abstract and analytic definition by defining a more geometric definition of a
similar function, which is now called the Koch curve.
To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump, or three Koch
curves. With every iteration, the perimeter of this shape increases by one third of the previous magnitude. The Koch snowflake is the result of an infinite number of these iterations, and has an
infinite magnitude, while its area remains finite. The image below illustrates the Koch snowflake and similar constructions that were sometimes called "monster curves."
Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals are deterministic, as are all the above, or stochastic and non-deterministic. For example,
the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.
In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in
their modern constructions. The animated construction of a Sierpinski Triangle below iterates nine generations of infinite possibilities.
In 1918, Bertrand Russell recognized a "supreme beauty" within the emerging mathematics of fractals. The idea of self-similar curves was taken further by Paul Pierre L¨évy, who, in his 1938 paper
Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the L¨évy C curve. Georg Cantor also gave examples of subsets of the real line with unusual
properties--these Cantor sets are also now recognized as fractals. Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincar¨é, Felix Klein,
Pierre Fatou and Gaston Julia. Without the aid of modern computer graphics however, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s, Beno?t Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier
work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated
this mathematical definition with convincing computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning
of the term "fractal."
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals as in an attractor. Geometrically, an attractor can be a point, a
curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. Objects in the parameter space for a family of systems may be fractal as well. An interesting
example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2, but what is truly surprising is that the boundary of the
Mandelbrot set also has a Hausdorff dimension of 2, while the topological dimension is 1, a result proved by Mitsuhiro Shishikura in 1991. Closely related to the Mandelbrot fractal set is the Julia
fractal set, as illustrated below. | {"url":"https://teachersinstitute.yale.edu/curriculum/units/2009/5/09.05.09/4","timestamp":"2024-11-02T04:37:14Z","content_type":"text/html","content_length":"42272","record_id":"<urn:uuid:14d954a0-a1e7-4017-9854-268a6e14bc47>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00108.warc.gz"} |
Intersecting Lines Drawing
Intersecting Lines Drawing - To understand what this means, lets step back a little and look at this diagram depicting three kinds of intersections. Web two or more lines that meet at a point are
called intersecting lines. Identify and draw intersecting, parallel, skew, and perpendicular lines and line segments. In figure 1, lines l and m intersect at q. Learning the concept of parallel and
intersecting lines by performing yoga.
Web draw parallel, perpendicular, and intersecting lines to develop understanding of geometry definitions in this interactive activity. The symbol ⊥ is used to denote perpendicular lines. That point
would be on each of these lines. It’s amazing how a simple definition can lead us to know important properties about linear equations’ angles and systems. There are also intersecting lines worksheets
based on edexcel, aqa and ocr exam questions, along with further guidance on where to go next if you’re still stuck. Web here are some effective activities that can be used to teach students to
identify parallel, intersecting, and skew lines and planes. In this article, you will get better acquainted with the lines and their features.
Abstraction with intersecting lines Royalty Free Vector
Web here we will learn about intersecting lines, including how to find the point of intersection of two straight lines and how to solve simultaneous equations graphically and algebraically. Web
choose from 666 drawing of.
Intersecting Lines GCSE Maths Steps, Examples & Worksheet
Intersection lines sit exactly where they are able to run along the surfaces of both forms simultaneously. Web 3rd quarter week 1: Describes and draws parallel, intersecting, and perpendicular lines
using ruler and set squareclick.
What Are Intersecting Lines? Definition & Examples Video & Lesson
Intersecting lines create interesting patterns and shapes in various art forms, such as painting, drawing, and graphic design. In figure 1, lines l and m intersect at q. By drawing your strokes from
joint to.
Intersecting Lines ClipArt ETC
Web two intersecting lines form 4 angles. There are also intersecting lines worksheets based on edexcel, aqa and ocr exam questions, along with further guidance on where to go next if you’re still
stuck. Also.
Intersecting Lines Definition, Properties, and Examples
This article will help us understand the definition, properties, and applications of intersecting lines. Abstract geometry pattern background for design. Web two or more lines that meet at a point
are called intersecting lines. Web.
Parallel, Intersecting and Perpendicular Art YouTube
Learning the concept of parallel and intersecting lines by performing yoga. Using simple alphabets is a brilliant way to teach parallel and intersecting lines. Web draw parallel, perpendicular, and
intersecting lines to develop understanding of.
love2learn2day Parallel & Perpendicular Art Math Art Projects
Web choose from 666 drawing of intersecting lines stock illustrations from istock. Web don’t be afraid to make mistakes, and keep undoing your lines until you’re satisfied with them! Learn how to
describe, illustrate and.
Random intersecting lines. Abstract monochrome geometric art Stock
Also included are illustrations of intersecting, parallel, perpendicular, and transversal lines. In this article, you will get better acquainted with the lines and their features. Web intersecting
lines are lines that meet each other at.
Understanding intersecting lines GMAT Math
That point would be on each of these lines. Two lines with different slopes will always intersect, unless they are parallel. The symbol ⊥ is used to denote perpendicular lines. Intersection lines sit
exactly where.
Kristine Lauderdale Line Drawings
Web mathematics geometry lines lines 1 2 3 this mathematics clipart gallery offers 155 images of curved, broken, and straight lines. Also included are illustrations of intersecting, parallel,
perpendicular, and transversal lines. Web here we.
Intersecting Lines Drawing That point would be on each of these lines. Lines with two ends are referred to as line segments. The symbol ⊥ is used to denote perpendicular lines. Web intersecting lines
are lines that meet each other at one point. Web if i were to summarize how these intersections work, it would be this:
Intersecting Lines Drawing Related Post : | {"url":"https://classifieds.independent.com/print/intersecting-lines-drawing.html","timestamp":"2024-11-02T02:15:45Z","content_type":"application/xhtml+xml","content_length":"24023","record_id":"<urn:uuid:5e7f4189-63f8-40ff-a71b-0f597e9d71b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00809.warc.gz"} |
Physics 1 Resources
Course Resources
Physics Resources – Math Resources – Computer Resources – Discussion Forum – Recommended Study Strategies – Review Questions – Instructor Office Hours – General Resources at Madison College
Physics Resources
• Supplemental Content – many items of Physics interest, some linked from the home page
• Physics 1 Equation Sheet (pdf)
• Lab Manual (see Course Info for more lab information)
• Our textbook: OpenStax University Physics complete table of contents
• Video Lectures
□ There are many videos including lectures, practice problems and demonstrations to be found by online search.
□ I recommend the ones listed in the Supplemental Content, which are fairly short and are at the level of this class.
• Other online resources
Math Resources
• Calculus textbooks
□ Calculus 1 – Functions and Graphs – Limits – Derivatives – Integration – Review of Pre-Calculus
□ Calculus 2 – Integration – Introduction to Differential Equations – Sequences and Series – Power Series – Parametric Equations and Polar Coordinates
□ Calculus 3 – Parametric Equations and Polar Coordinates – Vectors – Differentiation of Functions of Several Variables – Multiple Integration – Vector Calculus – Second-Order Differential
• See Computer Resources for online tools
Computer Resources
• Computational Physics page – including an introduction to the Jupyter Notebook
• Online math tools
□ Desmos is a nice free graphing calculator. With a (free) registered account, you can save and share your work.
□ Wolfram Alpha can do equation solving and calculus. Wolfram’s technology stack (Mathematica, Wolfram Alpha Pro, etc) is available to all Madison College students. Go to the wolfram site info
page and sign up with your College email.
□ Cymath shows steps for equation solving
• Logger Pro (the software we will use for data acquisition and analysis) is available to students.
• Artificial Intelligence (AI)
□ The Large Language Models available today, such as ChatGPT, Bard, Claude, or Poe, are not intended to answer numerical questions like Physics problems. They are good at composing prose, not
math. You will find that they provide a nice sounding solution, but often with the wrong theory applied and the wrong final answer.
□ AI models are under development that do the math and theory correctly, such as Khanmigo from Khan Academy (available for a subscription fee). But until these are available and reliable you
should not count on using AI to help with your Physics homework.
□ Great information at the AI Library Guide.
Discussion Forum
This onlne discussion forum is for students to post questions, comments or hints about the homework, as well as questions or comments about the course generally.
I monitor these forums for new posts and respond promptly. If you don’t want to miss any discussion, may wish to “subscribe” to this forum – this will send you an email whenever a new post goes up.
If a student asks me a question by email, I often ask that they post the question to the forum so that all students can profit from the discussion.
If you post a question about a particular homework problem, the Subject of the thread should include the HW set and problem number (example: HW2 Q4).
Recommended Study Strategies
• Generally
□ Read all sections of the text that we cover.
□ Work through the example problems in the textbook and master them. You cannot learn without practice. Post any questions about these to the Course Discussion Forum.
□ Follow the The 10 Rules of Good Studying.
• Homework
□ If you are stuck, don’t spend more than 30 minutes on a problem. Get help from peers or the instructor. Posting at the Discussion Forum is a good place to start.
□ To gain greater mastery, do extra problems from the textbook. Answers for many problems are available.
□ Collaboration with peers is permitted (and encouraged), but be sure such collaborations are of mutual benefit.
• Exams and Quizzes
□ Understand the solutions to the review questions. Multiple choice problems will be modeled on the Concept Questions; long Answer problems will be modeled on the Long Answer Questions.
□ Be careful not to just memorize the equation that solves the long answer problem. It is almost certain to be different, and no partial credit can be given for quoting the equation in the
model solution. Instead learn why the solution is as it is, so you can apply that knowledge to a slightly different situation.
□ Be aware of the exam and quiz rules and expectations.
Review Questions
Exam and quiz questions will have concepts and solution methods based on these model questions.
See Course Info for exam information.
Instructor Office Hours
Office hours may change throughout the semester. That page will always have the correct times.
General Resources at Madison College
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Math 301 Statics or Neal
Question # 00001128 Posted By: Updated on: 09/15/2013 05:59 PM Due on: 09/16/2013
ok here is the chart that you have to use can you let me know if you got it correctly. Also to here are the guidelines to follow.First download the spreadsheet with data related to body fat (in
pounds) of Silver Gym members. Then, download the sample of professor's spreadsheet on how to complete the task of the instructor's spreadsheet with similar data and calculated values used. The
location of data table inside instructor's spreadsheet is excel . Task list with body fat weight data for Silver Gym create a second spreadsheet within the same excel posting for Phase 4 IP for
hypothesis testing. Part 1 will be used for statistical measurements and the second part will be used for calculations of P-value for hypothesis testing. Part I: Statistical Measurements --- Here is
how to obtain the mean or algebraic average of body weight of 252 Silver Gym members. Notice column "A" only shows the members ID numbers and it cannot be used for calculations. Column "B" of data
table represents body fat of members in percent while Column "C" of data table represents body weight of members in pounds. Also we notice that both columns B and C data values begin on row 2 and
ends on Row 253. Notice that Row 1 is used to show the headings or titles of each column. So for all calculations we need to enter the following Excel format inside the Excel formula window to
represent the entire data for body fat and body weight. Means Values------ 1. Place the cursor in empty cell to the right side of the data table (in similar cell locations shown inside instructors
spreadsheet). 2. Open "Formula" tab located at the top of the spreadsheet. 3. Open the tab appearing as more function with orange color background by placing the cursor on this tab by holding down
the left click of mouse to open "more function" tab. 4. Open "Statistical " from the list that appears at this time. 5. From the vertical menu that opens at this time (in alphabetical order) select
Average tab in order to find the mean or average of the body fat. Open the Average tab. 6. From the window that pops up at this time you will see two small rectangular windows at the top. Select the
small window at the top. Place the cursor inside this window and exactly type in: B2:B253. 7.Ignore the second small window, and open "OK" tab. 8. Now, move the cursor to inside spreadsheet inside a
remote and blank cell 9. At this time you should be able the calculated value for mean of body fat inside the cell you initially selected. In order to obtain the calculated value for mean of the body
weight follow exactly the steps stated above by typing in the following inside the opened window in step 6: C2:C253 MEDIAN In order to find the median of the body fat and body weight follow the 9
steps stated above except in step 5 select the following function tab from the appearing vertical menu. MEDIAN Once again use B2:B253 for the median of body fat and C2:C253 for median of body weight
of the Gym members. Standard Deviation--- In order to find the standard deviation of the body fat and body weight follow the 9 steps stated above except in Step 5 select the following function tab
(not other similar tabs) from the appearing vertical menu: STDEV. S Once again use B2:B253 for standard deviation of body fat and repeat this process with C2:C253 for standard deviation of body
weight of the Gym members. RANGE Notice that there is no quick formula available inside Excel statistical function menu for range. Range is the difference between the greatest and smallest value of
the data set. Hence we must subtract the smallest value from the largest value in each column B and C of the data table to find the range for body fat and body weight, respectively. The exact formula
to be created is shown below. For range of body weight MAX(B2:B253) For range of body weight MAX (C2:C253) - MIN (C2:C253) Notice in case you type in these formulas inside the function window at the
top you might receive an error message inside the selected cell location. In order to avoid this issue I suggest you follow these simple guidelines: 1. For the range of body fat locate the cursor on
an empty cell location. 2. Manually type in the exact following statement (must include equal signs first) = MAX (B2:B253) 3. Move the cursor to a remote and empty cell location inside your
spreadsheet and place it inside the empty cell for a moment by pressing on left click of your mouse in order to avoid error message by Excel. 4. At this time you should be able to see the calculated
range of body fat for your data table in percent. For range of body weight ---- Follow the steps stated above exactly except in step 2 you need to manually type in = MAX (C2:C253) - MIN (C2:C253)
This will provide you with the range of body weight of Gym members in pounds for your data table inside your spreadsheet. Part 2: Hypothesis Testing You will use the spreadsheet, below Part 1, to
perform the P- Value to Part 2: Hypothesis testing about the claim or hypothesis manager. P-Value Calculations Population Mean of Body Fat 1. Place the cursor inside cell location H20 and manually
type in "20"for mean of body fat population. Make sure you move the cursor after typing to a remote area and place it in a blank cell location and left click there in order to avoid an error message.
Standard deviation of Body Fat 2. Place the cursor inside cell H22. Then find the standard deviation of body fat the way explained in part 1 inside spreadsheet 1. P-Value Calculation 3. Place the
cursor inside cell location H24. Then open "Formulas tab at the top of spreadsheet. Then open "more functions" tab. Then, open statistical tab from the appearing list. Then from the vertical menu
that pops up on the right side select Z-Test function at the bottom of the vertical menu. A window pops up with three small windows. 4. Type in "B2:B253" inside the top window to represent the array
of data for body fat. 5. Place the cursor inside the middle window and type in "20". This represents the population mean of body fat of 20. 6. Place the cursor inside the lower or third window and
type in "H22". This represents the standard deviation. 7. Now open "OK" tab at the bottom of the large window. 8. You should be able to see the cumulative probability of error inside cell location
H24. 9. Place the cursor in cell location "H27" and manually type in = 1 - H24. This will find the complement of the probability found for F7 location. 10. Place the cursor in cell "H29" and manually
type in "= 2*H27". This will provide the P-value for two tails of normal distribution. 11. You need to use this P-value (in H29 cell location) inside the word document to perform the hypothesis test
in part 2 the way explained inside power point presentation. This will complete all calculations for two parts of this phase. In part 2 inside your word document you need to show the required five
steps to perform the hypothesis two tailed test as we discussed in the live chats. Please place all of your discussions and responses inside your word document and not inside the spreadsheet. Cities,
References and APA format.
Tutorials for this Question
Tutorial # 00001079 Posted By: Posted on: 09/16/2013 04:11 PM
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The solution of Math 301 Stats assignment...
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do...man neal came through for me on time with everything that was needed for this paper. 09/20/2013
Great! We have found the solution of this question! | {"url":"https://www.homeworkminutes.com/q/math-301-statics-or-neal-1128/","timestamp":"2024-11-09T06:07:23Z","content_type":"text/html","content_length":"63154","record_id":"<urn:uuid:12d95fe8-6f22-4e53-a713-804b207c40d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00184.warc.gz"} |
Brain Teaser: What's the Mystery Number? -
Do you hate being kept in the dark? An unsolved mystery can haunt you, rattling around in your brain like a ghost.
We may never get answers to some of history’s biggest mysteries, such as the truth about the Bermuda triangle, the location of Cleopatra’s tomb or the identity of Jack the Ripper.
But we have an enigma you can solve in minutes, and it may even help to keep your brain sharp. Brain teasers like this one may boost your language ability, memory and problem solving skills.
If you’d like to play detective, this mystery number brain teaser will require you to use both logic and math skills. Using only the numbers on the page as clues, you’ll have to decipher the identity
of the final missing number.
Math games like this one may help keep your brain strong and flexible through development of new neural connections. Number games also may make you quicker at everyday calculations, from figuring out
how many bags of Halloween candy to buy to making sure you get the right amount of change back at the store.
Check out the bottom of the page for the correct answer.
Answer: 17 | The middle number equals the top number times the bottom number plus the left number times the right number.
Did you solve the mystery number puzzle? If so, comment to clue us in on how you did, how long it took and your favorite unsolved mystery!
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43 Responses to "Brain Teaser: What’s the Mystery Number?"
• Very straightforward.
• I like challenges like this one!
• This one I solved more quickly – about a minute.
• got it in about 4 minutes
• Less one minute
• That took me less than a minute. When I saw that the first centered number was prime, I knew that ruled out certain possibilities for what was wanted. The second set of numbers is what then
became obvious to me and then it was easy. This kind of math problem is okay, but I’ve always hated the ones that went… If Johnny went to Kansas in a boat and Miriam stayed behind and knitted,
then what time did they eat dinner? Huh? And by the way, who cares? 😂
• It took ten minutes max and I figured out the answer. I was very proud of myself.
• I didn’t know I should time myself, but I easily came up with 17. It was fun, thanks!
• Another fun puzzle!
• What was said above but times each then add answers to get middle number
• It took me about 3 minutes to figure out that the third set middle number was 17. I multiplied the numbers across from each other then added the two totals together.
• 1 min to get the answer
• That was fun! Probably because I figured it out in aboutv5 minutes.
• Wow, tricky.
Does anyone know the formula or equation for figure 1 and figure 2
how middle number 43 and 30 was calculated
□ Look at comments they explain it well
• Solved a couple of minutes.
• It took me about 10 seconds by multiplying the numbers and then adding the numbers. No clue needed.
• Sorry I forgot, it took about 2 minutes.
• The middle number of the first set minus the middle number of second set is 13, so I subtracted 13 from the second set and that is 17 which is the middle number for the third set.
• It was too easy. I stumbled on the answer in less than a minute.
• It took me 1-2 minutes
• Multiply to and bottom and side to side and and together. 17.
• I like math puzzles. I solved this puzzle in less than a minute. I first added the numbers in the first puzzle and it wasn’t the right answer.. Then I multiplied the top and bottom and right and
left and added them together. It worked for all 3 puzzles. Thank you! I enjoyed it.
• 17
• I experimented with different ways to combine the numbers on the outside to get the center number. It took me about 2-3 minutes to come up with the answer.
• I eliminated other options. It took me a couple of minutes. I tried adding all the outer numbers and that didn’t equal the center number. Then I figured it out.
• found the answer in 3 minutes no clue needed
• It took me about a minute. I like basic and intermediate Sudoku puzzles. I don’t understand how to work the higher level Sudoku. The ones I am successful with are fun when they take on their own
life when about half solved.
• took about 30 seconds to solve
• just trial and error, by multiplying the outside numbers before adding them together, took less than a minute
• It took me like 10 minutes.
Knew it had to be a combination of multiplication and addition.
• 13 from 43 was 30, then 13 from 30 was 17. Answer 17.
• I just subtracted the middle number in the second figure from the middle number in the first figure. I used that number and subtracted it from the number in the middle number of the second figure
to get the number in the middle of the third figure. I got the same answer (43-30 = 13; 30-13. = 17)
• I solved it within 3 minutes. It was obvious that it was not about adding the numbers, so I next tried multiplying them. The first answer, 43, is odd, so a quick evaluation of the numbers around
it made it obvious that the result would be an odd number.
• I did solve the puzzle. About 2
• 17
• 30 seconds I solved it.
• Multiply top number with bottom number and multiply side numbers then add the totals together
• Yes, it was pretty easy. I like working with numbers but not letters.
• Solved in less than 1 minute. My favorite unsolved mystery is how was stonehenge built?
• I attained the correct number, 17, in ten seconds. Of course, I’m a retired math teacher.
• Yes, I solved the puzzle
• Solved it | {"url":"https://extramile.thehartford.com/lifestyle/brain-teasers/whats-your-number-2/","timestamp":"2024-11-06T12:20:56Z","content_type":"text/html","content_length":"130659","record_id":"<urn:uuid:ac3c31fc-6260-44ee-a497-eb9c45b68704>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00720.warc.gz"} |
Multiplication 1 5 Worksheet Pdf
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Great Rhombicuboctahedron -- from Wolfram MathWorld
The great rhombicuboctahedron (Cundy and Rowlett 1989, p. 106) is the 26-faced Archimedean solid consisting of faces et al. 1999). It is illustrated above together with a wireframe version and a net
that can be used for its construction.
It is also the uniform polyhedron with Maeder index 11 (Maeder 1997), Wenninger index 15 (Wenninger 1989), Coxeter index 23 (Coxeter et al. 1954), and Har'El index 16 (Har'El 1993). It has Schläfli
symbol tWythoff symbol
Some symmetric projections of the great rhombicuboctahedron are illustrated above.
The great rhombicuboctahedron is implemented in the Wolfram Language as UniformPolyhedron["GreatRhombicuboctahedron"]. Precomputed properties are available as PolyhedronData
["GreatRhombicuboctahedron", prop].
Confusingly, the term "great rhombicuboctahedron" is also used by various authors (e.g., Maeder 1997) to refer to the distinct unform polyhedron with Maeder index 17 and Wenninger index 85. For
reasons of clarity, that solid is better referred to using Wenninger's term quasirhombicuboctahedron (Wenninger 1971, p. 132).
The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It has Dehn invariant 0 (Conway et al. 1999) but is not a space-filling polyhedron. However, it can be
combined with cubes and truncated octahedra into a regular space-filling pattern.
The small cubicuboctahedron is a faceted version of the great rhombicuboctahedron.
The skeleton of the great rhombicuboctahedron is the great rhombicuboctahedral graph, illustrated above in a number of embeddings.
The dual polyhedron of the great rhombicuboctahedron is the disdyakis dodecahedron, both of which are illustrated above together with their common midsphere. The inradius midradius circumradius
Additional quantities are
The distances between the solid center and centroids of the square and octagonal faces are
The surface area and volume are | {"url":"https://mathworld.wolfram.com/GreatRhombicuboctahedron.html","timestamp":"2024-11-06T15:15:03Z","content_type":"text/html","content_length":"75743","record_id":"<urn:uuid:cd14eca3-240a-444b-bbab-db2b06f13c16>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00614.warc.gz"} |
Sunrise, Inc., has no debt outstanding and a total market value
of $220,000. Earnings before interest...
Sunrise, Inc., has no debt outstanding and a total market value of $220,000. Earnings before interest and taxes, EBIT, are projected to be $42,000 if economic conditions are normal. If there is
strong expansion in the economy, then EBIT will be 20 percent higher. If there is a recession, then EBIT will be 30 percent lower. The company is considering a $66,000 debt issue with an interest
rate of 6 percent. The proceeds will be used to repurchase shares of stock. There are currently 10,000 shares outstanding. Ignore taxes for this problem. Assume the stock price is constant under all
Calculate earnings per share (EPS) under each of the three economic scenarios before any debt is issued. (Do not round intermediate calculations and round your answers to 2 decimal places, e.g.,
a-1. 32.16.)
a-2. Calculate the percentage changes in EPS when the economy expands or enters a recession. (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter
your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
b-1. Calculate earnings per share (EPS) under each of the three economic scenarios assuming the company goes through with recapitalization. (Do not round intermediate calculations and round your
answers to 2 decimal places, e.g., 32.16.)
b-2. Given the recapitalization, calculate the percentage changes in EPS when the economy expands or enters a recession. (A negative answer should be indicated by a minus sign. Do not round
intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
Calculate earnings per share (EPS) under each of the three economic scenarios before any debt is issued. (Do not round intermediate calculations and round your answers to 2 decimal places, e.g.,
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Method for detecting spins by photon counting
Method for detecting spins by photon counting
A method of detecting spins in a sample, includes exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse for flipping the spins, and detecting a noise
signal produced by the return of the spins to equilibrium by means of a device for counting radio-frequency or microwave photons.
Latest COMMISSARIAT A L'ENERGIE ATOMIQUE ET AUX ENERGIES ALTERNATIVES Patents:
This application is a National Stage of International patent application PCT/EP2021/057204, filed on Mar. 22, 2021, which claims priority to foreign French patent application No. FR 2002976, filed on
Mar. 26, 2020, the disclosures of which are incorporated by reference in their entirety.
The invention relates to a method for detecting spins by photon counting. It mainly, but not exclusively, applies to EPR spectroscopy (EPR standing for Electron Paramagnetic Resonance).
EPR spectroscopy exploits the ability of unpaired electrons, which are found in certain chemical species (free radicals and salts and complexes of transition metals) to absorb and re-emit the energy
of electromagnetic radiation, typically microwave radiation, when they are placed in a magnetic field.
The absorption or emission spectrum of the electromagnetic radiation provides information on the chemical environment of the unpaired electron or of the nucleus, respectively.
FIG. 1 schematically illustrates an EPR-spectroscopy apparatus according to the prior art, see for example (Bienfait 2016), (Probst 2017) and (Ranjan 2020).
A sample E containing a set of N electronic spins SE is placed in a cryogenic enclosure CRY, in the magnetic field B[0 ]generated by a magnet or superconducting coil A. The strength of the magnetic
field determines the resonant angular frequency ω[0 ]of the electronic spins:
ω[0]=−γB[0 ]
where γ is the gyromagnetic ratio of the electron. The frequency f[L]=ω[0]/2π is called the Larmor frequency. Typically, it is located in the microwave spectral region because the gyromagnetic ratio
of a free electron is equal to about 28 GHz/T.
The sample E is furthermore magnetically coupled to an electromagnetic resonator REM tuned to the Larmor frequency and modeled by a parallel LC circuit. In the case of (Bienfait 2016), (Probst 2017)
and (Ranjan 2020), the resonator is a planar structure comprising two interdigital electrodes that form a capacitor and that are connected at their center by an inductive conductive line, in
proximity to which line the sensitive region is located. The plane of the resonator is parallel to the magnetic field B[0]. Other types of resonators may be used, for example conductive cavities
surrounding the sample on all sides.
The coupling constant of the sample to the resonator is designated “g”, and its quality factor is designated Q. Typically, the coupling constant is sufficiently high for the Purcell effect to
dominate the dynamics of relaxation of the spins: Γ[1]≈Γ[P ]where Γ[1 ]is the energy relaxation rate of the spins and Γ[P ]is the Purcell factor, which is given by
where κ=ω[0]/Q is the rate of dissipation of energy in the resonator.
A signal generator GS applies, to the resonator REM, a sequence of microwave pulses IEX at the Larmor frequency. These so-called excitation pulses excite the spins of the sample coupled to the
The most widely used sequence of excitation pulses is the so-called “spin-echo” sequence. It comprises a first impulse so-called “π/2” pulse, which flips the spins—which are initially aligned with
the magnetic field B[0]—into a plane perpendicular thereto. The spins precess around the magnetic field B[0 ]and, on doing so, emit a first electromagnetic signal at the Larmor frequency. This signal
is called the FID signal (FID standing for Free Induction Decay) because its intensity decreases exponentially due to spin decoherence, with a time constant T[2]*=(Γ[2]*)^−1. The decoherence rate Γ
[2]* depends on the properties of the sample and on the uniformity of the magnetic field B[0]. After a certain time—shorter than time taken for the spins to lose their coherence T[2]=(Γ[2])^−1—a
so-called “π” pulse is applied. This second pulse inverts the orientation of the spins and causes the emission of a second so-called echo electromagnetic signal.
The electromagnetic signals emitted by the spins are converted into an electronic response signal RS by the resonator. A gyrator G makes it possible to separate the excitation pulses from the
response signal RS. In particular, it is the echo signal that is used to detect the spins.
The response signal RS is delivered, via a suitable transmission line LT (typically a coaxial cable), to an electronic detecting system SED.
For example, in the case of (Bienfait 2016), (Probst 2017) and (Ranjan 2020), the electronic detecting system comprises a Josephson parametric amplifier JPA pumped at an angular frequency ω[p]≈2ω[0];
the amplified signal is subsequently amplified by a HEMT amplifier HA, then mixed in a mixer ML with a signal SOL from a local oscillator OL at the angular frequency ω[0], and the in-phase component
I and quadrature component Q of the baseband signal resulting from the mixing are detected (homodyne detection).
It is possible to demonstrate—see for example (Bienfait 2016)—that, in the case where the quality factor of the resonator is limited by the coupling to the detection antenna (overdamped regime)—the
amplitude of a spin-echo signal is equal to
X[e]=pN√{square root over (Γ[P]/2Γ[2]*)}
where “p” is the polarization at equilibrium of the set of spins, which depends on the temperature according to Maxwell-Boltzmann statistics, and N is the number of spins in the sample.
The noise level is given by
δX=√{square root over (n)}/2
with n=n[eq]+n[amp ]where n[eq ]is the thermal noise of the microwave field, n[eq]=1+2n with n=1/(e^ℏω^0^/k^B^T−1) expressing the average number of photons per mode at the temperature T (k[B]:
Boltzmann's constant) and n[amp ]is the noise added by the amplifier. The signal-to-noise ratio of homodyne echo detection is therefore equal to
SN[e,h]=2pN√{square root over (Γ[P]/2nΓ[2]*)}.
where the subscripts “e” and “h” stand for “spin echo” and “homodyne detection”, respectively.
Even in the ideal case where p=1 (full polarization of the sample) and n=1 (detector at the quantum noise limit and very low temperature), to obtain a signal-to-noise ratio equal to 1 it is therefore
necessary for the number of spins in the sample to satisfy N≥√{square root over (2Γ[2]*/Γ[P])}.
However, in general Γ[2]*/Γ[P]>>1, and therefore N>>1.
The authors of (Probst 2017) obtained, by means of an apparatus of the type shown in FIG. 1, a detection sensitivity of 65 spins/√{square root over (Hz)} and an acquisition rate (limited by Γ[P]) of
10 Hz, which allowed them to detect the signal generated by a sample containing only 200 spins. To do this, they worked at a temperature of 10 mK, such that and p≈1 and n[eq]≈1, maximized the quality
factor of the electromagnetic resonator and its coupling constant to the sample and used an amplifier at the quantum noise limit, so that n[amp]≈0.
Other techniques for detecting spin have been used in the prior art, but they have not allowed such high sensitivities to be achieved.
In particular, other excitation sequences may be used. For example, it is possible to use only a “π/2” pulse and to detect the FID signal directly, without inducing an echo.
In (Kubo 2012), an FID signal was detected not by conventional electronic techniques, such as homodyne or heterodyne detection, but by counting microwave photons by means of a transmon, which is a
type of superconducting qubit (a qubit being a two-level quantum system). In this method, both the sample and the detecting qubit are arranged in a microwave cavity the resonant frequency of which
must be modified dynamically in order to allow the electron spins to first be excited with a “π/2” pulse, then the FID signal to be detected by the qubit. This technique is complex to implement and
has only allowed a sensitivity of the order of 10^5 spins/√{square root over (Hz)} to be achieved, several orders of magnitude worse than the—earlier—result of (Probst 2017).
The use of a “π” pulse alone induces the emission of an incoherent signal (“noise”) caused by the return of spins to equilibrium. Such a signal has been observed—in Nuclear Magnetic Resonance—by
(McCoy 1989), but considered of no practical interest due to its very low sensitivity.
The invention aims to improve the sensitivity of the detection of spins, and in particular to make possible measurements on samples containing a very low number of spins, for example 10 or fewer, or
even just one.
According to the invention, this aim is achieved by virtue of detection, by means of a counter of radio-frequency or microwave photons, of the incoherent signal emitted by the spins excited by a
spin-inverting pulse (“π” pulse). One subject of the invention is therefore a spin-detection method comprising the following steps:
a) placing a sample containing spins in a static magnetic field;
b) magnetically coupling the sample to an electromagnetic resonator having a resonant frequency ω[0]/2π equal to the Larmor frequency of the spins in the static magnetic field, the coupling constant
and the quality factor of the resonator being sufficiently high for the coupling to the resonator to dominate the dynamics of relaxation of the spins;
c) exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse at said Larmor frequency; and
d) detecting an electromagnetic signal emitted by the spins of the sample in a mode of the electromagnetic resonator in response to said pulse by means of a device for counting radio-frequency or
microwave photons;
characterized in that the radio-frequency or microwave electromagnetic pulse at the Larmor frequency is a spin-flipping pulse, whereby the detected signal is a noise signal produced by the return of
the spins to equilibrium.
By “radio-frequency”, what is meant is frequencies comprised between 1 MHz and 1 GHz and by “microwave” what is meant is frequencies comprised between 1 GHz and 100 GHz.
The appended drawings illustrate the invention:
FIG. 1, which has already been described, schematically illustrates an EPR-spectroscopy apparatus according to the prior art;
FIG. 2 schematically illustrates an EPR-spectroscopy apparatus for implementing a method according to the invention;
FIG. 3 illustrates the structure of a counter of microwave photons able to be used in the apparatus of FIG. 2;
FIG. 4a and
FIG. 4b illustrate experimental results.
The method of the invention may be implemented by means of an apparatus of the type illustrated in FIG. 2, which differs from that of FIG. 1 essentially in that its electronic system SED′ for
detecting the microwave signal emitted by the spins of the sample E is based on a device CP for counting microwave photons.
The photon-counting device CP may be a superconducting qubit, in particular a transmon, such as described in (Lescanne 2019) and illustrated in FIG. 3. This device comprises a Josephson junction JJ
in transmon configuration connected to three lengths of planar waveguide. The first length of waveguide GO1 forms a half-wavelength (λ/2) resonator resonant at the angular frequency ω[0 ]of the
photons to be detected. The second length of waveguide GO2 is intended to direct, to the Josephson junction, a so-called pump signal, at an angular frequency ω[p]. The third length of waveguide GO3
forms a pair of half-wavelength (λ/2) resonators resonant at an angular frequency ω[w ]referred to as the “waste” angular frequency, and is coupled, via a so-called “Purcell” band-pass filter, to a
cold environment (on the scale of millikelvin) having a characteristic impedance of 50 ohm. The Josephson junction in transmon configuration behaves as a two-level system. When it is in its ground
state, the simultaneous arrival of a photon at the angular frequency ω[0 ]and of a pump photon at the angular frequency ω[P ]causes the Josephson junction to transition to its excited state, the
remaining energy being dissipated to the cold environment in the form of a photon at the angular frequency ω[w]. The state of the transmon is read by probing one of the two resonators to which the
qubit is coupled (the waste mode for example) with a microwave pulse at its resonant frequency; because of the dispersive coupling to the qubit, the phase of the pulse reflected by the resonator
allows the state of the qubit and therefore whether or not a photon is present to be deduced. The device is reset by injecting a photon of angular frequency ω[w ]into GO3, said photon combining with
a pump photon in the Josephson junction to return the latter to its ground state, excess energy being removed via a photon at the wavelength ω[0].
Other types of devices allow microwave or even radio-frequency photons to be counted. For example, (Walsh 2017) proposes a bolometer-type detector that uses a Josephson junction to detect heating of
a graphene sheet induced by a single photon.
Furthermore, the electronic system GS for generating signals of the apparatus of FIG. 2 is configured to generate, instead of spin-echo sequences, single inverting or “π” pulses IS.
More generally, inverting pulses, which flip the spins by π rad, may be replaced by pulses that flip by a non-zero angle φ that may be less than or equal to π rad (“flipping” pulses). The case where
φ=π rad (inversion) is preferred because it maximizes the intensity of the signal emitted by the spins.
The spins of the sample, which are excited by an inverting or flipping pulse, return to equilibrium by spontaneously, and therefore incoherently, emitting photons at the Larmor frequency, forming
what is called “spin noise”. The spontaneous emission is highly accelerated by the Purcell effect, and hence almost all of these photons are emitted in a mode of the electromagnetic resonator and are
coupled to the transmission line LT, which guides their propagation to the photon-counting device CP. In FIG. 2, the reference RS′ designates the response signal propagating along the transmission
line LT. It will be noted that, contrary to the spin-echo signal RS of FIG. 1, it consists entirely of noise.
Whereas, as discussed above with reference to (McCoy 1989), the detection of spin noise via conventional electronic techniques (homodyne or heterodyne demodulation) is not very sensitive, the present
inventors have discovered that, unexpectedly, detection of spin noise by photon counting makes it possible to achieve a higher sensitivity than the prior art (homodyne detection of a spin-echo
This may be demonstrated in the following way.
If N is the number of spins in the sample and p (comprised between 0 and 1, and in practice close to 1) is the polarization, the number of excited spins is equal to pN. These spins relax with a time
constant T[1]=(Γ[1])^−1. It is possible to consider that all the spins will have relaxed at the end of an acquisition window sufficiently long with respect to T[1]—for example longer than or equal to
5T[1 ]even 10T[1]. The probability that a spin relaxes by emitting a photon in a mode of the electromagnetic resonator is equal to p[1]=Γ[P]/Γ[1]. The photon counter is considered to have a bandwidth
equal to Γ[2]*, which allows it, in principle, to collect all the photons emitted by the spins, and a quantum efficiency η. The number of photons detected by the counter is therefore equal to ηpNΓ[P]
The number of noise photons (i.e. of photons not originating from spins) is given by n Γ[2]*/Γ[1]+αΓ[1]^−1 where, as explained above, n=1/(e^ℏω^0^/k^B^T−1) is the average number of photons per mode
at the temperature T and α is the dark count rate, i.e. the count rate in the absence of photons.
The noise level corresponds to the standard deviation of the number of noise photons detected, which, assuming that the dark photons have a Poisson distribution, is the square root thereof.
Another source of noise results from the fact that the number of detected photons originating from spins itself varies, because the number of photons emitted by the spins is a random variable of
standard deviation √{square root over (p[1](1−p[1])N)}.
Furthermore, since detection efficiency is finite, the number of photons detected is also a random variable, of standard deviation √{square root over (η(1−η)N)}.
In total, the standard deviation of the detection noise is therefore equal to
$〈 n 〉 Γ 2 * / Γ 1 + α Γ 1 - 1 + [ p 1 ( 1 - p 1 ) + η ( 1 - η ) ] N .$
The signal-to-noise ratio of this method of incoherent detection by photon counting is therefore equal to:
$S N i , CP = η p N p 1 / 〈 n 〉 Γ 2 * / Γ 1 + α Γ 1 - 1 + [ p 1 ( 1 - p 1 ) + η ( 1 - η ) ] N$
where the subscript “i” stands for “incoherent” (and, therefore, spin noise) and “CP” stands for detection by photon counting. Herein lies the fundamental difference with the conventional method of
homodyne detection. Whereas signal-to-noise ratio in homodyne detection is intrinsically limited by vacuum fluctuations that mean that with photon counting there is a parameter regime in which the
signal-to-noise ratio may be arbitrarily high.
Specifically, in the ultimate limit where p=1 (maximum spin polarization), p[1]=1 (spins relax dominantly via the Purcell effect) and n˜0, this last condition corresponding to
$T ≪ ℏ ω 0 k B ,$
the following is obtained:
SN[i,CP]=ηN/√{square root over (αΓ[P]^−1+η(1−η)]N)},
whereas it will be recalled that:
SN[e,h]=2N√{square root over (Γ[P]/2Γ[2]^+)}
in homodyne detection. However, there is no theoretical limit on the value that the efficiency of the detector or the dark count rate may reach, i.e. η may be as close to 1 as desired, and α(Γ[P])^−1
may be as low as necessary.
SN[i,CP ]may therefore be arbitrarily high, even if N=1 and Γ[P]/2Γ[2]*>>1, provided that the efficiency of the detector is high, and that the dark count rate is low enough.
It is interesting to also calculate, for the purposes of comparison, the signal-to-noise ratio achievable by homodyne detection of spin noise and by counting the photons of a spin-echo signal.
In the case of homodyne detection of spin noise, the total power emitted by the spins is given by the number of photons emitted in the detection window, which is equal to pNΓ[P]/Γ[1 ]in a bandwidth
given by Γ[2]*. The corresponding noise power is given by nΓ[2]*/Γ[1]. The standard deviation is √{square root over (nΓ[2]*/Γ[2])}.
The signal-to-noise ratio of this method of incoherent homodyne detection is therefore equal to:
SN[i,h]=pNΓ[P]/√{square root over (nΓ[1]Γ[2]*)}.
It may be seen that the ratio
SN[e,h]/SN[i,h]=2√{square root over (Γ[1]/Γ[P])}
is always greater than 2, and even very much greater than 2 in situations where Γ[1]>>Γ[P]. Hence, this method is less suited to detection of low numbers of spins than the method of the invention.
In the case of counting the photons of a spin-echo signal, the number of photons detected is given by ηp^2N^2(Γ[P]/2Γ[2]*), which is the square of the amplitude of the signal multiplied by the
efficiency η of the detector.
The duration of the echo is (Γ[2]*)^−1), and hence the number of dark counts is α(Γ[2]*)^−1. The noise level corresponds to the standard deviation, i.e. to the square root, of this number of counts.
Furthermore, it is necessary to take into account the shot noise due to the echo itself, which is a coherent state of the field and therefore has a standard deviation given by pN√{square root over
The signal-to-noise ratio of the detection of a spin-echo signal by photon counting is therefore equal to
SN[e,CP]=ηp^2N^2(Γ[P]/2Γ[2]*)/√{square root over (α(Γ[2]*)^−1+ηp^2N^2(1+n)(Γ[P]/2Γ[2]*))}.
In the “ultimate” limit where p=1, Γ[1]≈Γ[P ]and n≈0, the following is obtained:
$S N e , CP = η N 2 ( Γ P 2 Γ 2 * ) α ( Γ 2 * ) - 1 + η N 2 ( Γ P 2 Γ 2 * ) < S N e , h = 2 N Γ P / 2 Γ 2 *$
So there is in principle no advantage in terms of signal-to-noise ratio in detecting an echo by photon counting rather than by coherent homodyne detection.
The ratio SN[i,CP]/SN[e,CP ]is equal to
SN[i,CPM]/SN[e,CPM]=(1/N)√{square root over (Γ[2]*/Γ[P])}.
It may therefore be seen that the method of the invention is advantageous with respect to the detection by counting an echo signal when the number of spins of the sample is less than
N[c]=√{square root over (Γ[2]*/Γ[P])}
If N>N[c], the method of detection by spin-echo and photon counting may therefore be more sensitive than the method according to the invention. However, in this case it will generally be preferable
to employ conventional homodyne detection.
In conclusion, it may be seen that none of these techniques allows a signal-to-noise ratio as high as that provided by the invention to be achieved in the case of samples
It will be clear from the foregoing that the method of the invention is particularly advantageous when the number N of spins of the sample is of the order of or less than √{square root over (2Γ[2]*/Γ
[P])} and when Γ[2]*/Γ[P]>>1, and provided that
$T 0 ≤ ℏ ω 0 k B .$
The technical result of the invention has been validated experimentally by detecting the microwave signal emitted by a set of N≅200 donors (bismuth atoms) in silicon coupled to a resonator at the
frequency ω[0 ]by a device for counting microwave photons that was similar to the one described in the reference (Lescanne 2019) and that was tuned to the frequency ω[0]. In this experiment, Γ[2]*≅10
^5s^−1, Γ[P]≅10s^−1, and Γ[1]=Γ[P]. The signal of the spins was detected according to the two modalities envisioned in this patent. FIG. 4a illustrates the results of an experiment in which
spin-noise photons were counted, with a counter such that η≅0.2 and α=1.5 ms^−1. It may be seen that the number of photons detected per unit time just after a n pulse applied to the spins decreased
exponentially with a time constant Γ[P]^−1=100 ms, before reaching a constant value corresponding to the dark count rate. It was a question of spontaneous emission by the spins via the Purcell
effect. The total number of counts detected during each sequence, read from the graph of FIG. 4a by subtracting the baseline corresponding to the dark count (about 1.5 counts/ms) and integrating the
number of remaining counts, is about 50, which is close to the expected value ηN.
The detection of spins via the echo method detected by photon counting has been graphed in FIG. 4b, which shows the probability of counting one photon per unit time, said probability being obtained
by averaging over a number of acquisition sequences. The two microwave pulses intended to generate the echo signal may be seen at the times t=0 and t=0.35 ms. The spin-echo is detected at the time t=
0.7 ms, as expected. The amplitude of the signal is about 0.3 counts per echo sequence, which is close to the expected theoretical value of 0.4 counts on average.
The invention has been described with reference to its application to the detection of electronic spins, and more particularly to EPR spectroscopy (EPR standing for Electron Paramagnetic Resonance),
but it is not limited thereto. In particular, it may be applied to the detection of nuclear spins and more particularly to NMR spectroscopy (NMR standing for Nuclear Magnetic Resonance). This is
important, because while few molecular species have unpaired electrons detectable by EPR, very many nuclei—and in particular the most common thereof, the proton—have nuclear spin and are therefore
detectable by NMR.
Extension of the technique of the invention to the detection of nuclear spins poses no difficulty in principle. However, since the gyromagnetic ratios of atomic nuclei are about three orders of
magnitude lower than the gyromagnetic ratio of the electron, the Larmor frequencies used in NMR are typically much lower than those encountered in EPR (a few MHz or tens of MHz, instead of several
GHz), despite the use of stronger magnetic fields. This has two consequences:
Firstly, it is necessary to count radio-frequency photons, which are less energetic than the microwave photons emitted by electron spins.
Secondly, the condition
$T 0 ≤ ℏ ω 0 k B ,$
which must preferentially be met to obtain a high sensitivity, requires even greater cooling.
This makes application of the invention to the detection of nuclear spins more complex, but not fundamentally so.
(McCoy 1989) “Nuclear spin noise at room temperature”, M. A. McCoy and R. R. Ernst, Chemical Physics Letters 159, 587 (1989).
(Kubo 2012) “Electron spin resonance detected by a superconducting qubit”, Y. Kubo et al. Phys. Rev. B 86, 06514 (2012)
(Bienfait 2016) “Reaching the quantum limit of sensitivity in electron spin resonance” A. Bienfait, J. J. Pla, Y. Kubo, M. Stern, X. Zhou, C. C. Lo, C. D. Weis, T. Schenkel, M. L. W. Thewalt, D.
Vion, D. Esteve, B. Julsgaard, K. Moelmer, J J L Morton, P. Bertet, Nature Nanotechnology 11, 253 (2016).
(Probst 2017) “Inductive-detection electron-spin resonance spectroscopy with 65 spins/√Hz sensitivity” S. Probst, A. Bienfait, P. Campagne-Ibarcq, J. J. Pla, B. Albanese, J. F. Da Silva Barbosa, T.
Schenkel, D. Vion, D. Esteve, K. Moelmer, J. J. L. Morton, R. Heeres, P. Bertet, Appl. Phys. Lett. 111, 202604 (2017).
(Walsh 2017) “Graphene-Based Josephson-Junction Single-Photon Detector” Walsh, Evan D., et al. Physical Review Applied, vol. 8, no. 2, August 2017.
(Lescanne 2019) “Detecting itinerant microwave photons with engineered non-linear dissipation” R. Lescanne, S. Deléglise, E. Albertinale, U. Réglade, T. Capelle, E. Ivanov, T. Jacqmin, Z. Leghtas, E.
Flurin, arxiv:1902:05102.
(Ranjan 2020) “Pulsed electron spin resonance spectroscopy in the Purcell regime” V. Ranjan, S. Probst, B. Albanes, A. Doll, O. Jacquit, E. Flurin, R. Heeres, D. Vion, D. Esteve, J. J. M. Morton, P.
Bertet, J. Mag. Res. 310 (2020).
1. A spin-detection method comprising the following steps:
a) placing a sample (E) containing spins (SE) in a static magnetic field (B0);
b) magnetically coupling the sample to an electromagnetic resonator (REM) having a resonant frequency ω0/2π equal to the Larmor frequency of the spins in the static magnetic field, the coupling
constant and the quality factor of the resonator being sufficiently high for the coupling to the resonator to dominate the dynamics of relaxation of the spins;
c) exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse (IS) at said Larmor frequency; and
d) detecting an electromagnetic signal (RS’) emitted by the spins of the sample in a mode of the electromagnetic resonator in response to said pulse by means of a device (CP) for counting
radio-frequency or microwave photons;
wherein the radio-frequency or microwave electromagnetic pulse at the Larmor frequency is a spin-flipping pulse, whereby the detected signal is a noise signal produced by the return of the spins
to equilibrium.
2. The method as claimed in claim 1, wherein the device for counting radio-frequency or microwave photons is spaced apart from the electromagnetic resonator and connected thereto via a waveguide or a
transmission line (LT).
3. The method as claimed in claim 1, wherein, at least in steps c) and d), the sample is kept at a temperature lower, preferably by at least a factor of 10, than T 0 = ℏ ω 0 k B where h is the
reduced Planck constant and kB is Boltzmann's constant.
4. The method as claimed in claim 1, wherein, in step d), the electromagnetic signal is detected during an acquisition window of duration comprised between 0.5·Γ1−1 and 10·Γ1−1, and preferably
between Γ1−1 and 5·Γ1−1, where Γ1 is the relaxation rate of the spins of the sample coupled to the electromagnetic resonator.
5. The method as claimed in claim 1, wherein the coupling constant g between the spins of the sample and the electromagnetic resonator, the quality factor of the electromagnetic resonator at the
Larmor frequency and the decoherence rate Γ2* of the spins of the sample are chosen such that Γ P Γ 2 * < 1 and preferably Γ P Γ 2 * < 0. 1, where Γ P = 4 Q g 2 ω 0.
6. The method as claimed in claim 1, wherein the spin-flipping pulse is a spin-inverting pulse.
7. The method as claimed in claim 1, wherein the device for counting radio-frequency or microwave photons is a qubit.
8. The method as claimed in claim 6, wherein the device for counting radio-frequency or microwave photons is a transmon.
9. The method as claimed in claim 1, wherein the spins of the sample are electron spins.
Referenced Cited
U.S. Patent Documents
20180031657 February 1, 2018 Takeda
Foreign Patent Documents
20180112833 October 2018 KR
WO-2006083482 August 2006 WO
WO-2009032291 March 2009 WO
WO-2018220183 December 2018 WO
Other references
• McCoy, et al., “Nuclear spin noise at room temperature”, Chemical Physics Letters, vol. 159, pp. 587-593, 1989.
• Kubo, et al., “Electron spin resonance detected by a superconducting qubit”, Phys. Rev. B, vol. 86, No. 6, pp. 064514-1-064514-6, 2012.
• Bienfait, et al., “Reaching the quantum limit of sensitivity in electron spin resonance”, Nature Nanotechnology, vol. 11, No. 3, pp. 253-257, 2016.
• Probst, et al., “Inductive detection electron-spin resonance spectroscopy with 65 spins/√Hz sensitivity”, Appl. Phys. Lett. 111, 202604, 2017.
• Walsh, et al., “Graphene-Based Josephson-Junction Single-Photon Detector”, Physical Review Applied, vol. 8, No. 2, Aug. 2017.
• Lescanne, et al., “Detecting itinerant microwave photons with engineered non-linear dissipation”, arxiv:1902:05102, 2019.
• Ranjan, et al., “Pulsed electron spin resonance spectroscopy in the Purcell regime”, J. Mag. Res., vol. 310, 2020. | {"url":"https://patents.justia.com/patent/12105037","timestamp":"2024-11-07T03:48:49Z","content_type":"text/html","content_length":"99211","record_id":"<urn:uuid:7f3143eb-a127-4a59-8465-635f6ca59e11>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00181.warc.gz"} |
Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra
Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebraArticle
Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators
containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\
right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the
commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend
the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$,
where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.
Volume: Volume 32 (2024), Issue 2 (Special issue: CIMPA schools "Nonassociative Algebras and related topics, Brazil'2023" and "Current Trends in Algebra, Philippines'2024")
Published on: July 10, 2023
Accepted on: June 27, 2023
Submitted on: January 17, 2023
Keywords: Mathematics - Rings and Algebras,Mathematics - Quantum Algebra,47L30, 17B60, 17B65, 16S15, 81R50 | {"url":"https://cm.episciences.org/11534","timestamp":"2024-11-09T07:16:33Z","content_type":"application/xhtml+xml","content_length":"40894","record_id":"<urn:uuid:0d7e0d8b-c1af-41c7-a3d7-73a99d7fd56c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00850.warc.gz"} |
Seven Squares
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Seven Squares printable sheet - seven squares pattern
Seven Squares printable sheet - more patterns
Three students were asked to draw this matchstick pattern:
This is how Phoebe drew it:
Can you describe what Phoebe did?
How many 'downs' and how many inverted C's are there?
How many matchsticks altogether?
Now picture what Phoebe would do if there had been $25$ squares.
How many 'downs' and how many inverted C's would there be?
How many matchsticks altogether?
If there had been $100$ squares? How many matchsticks altogether?
A million and one squares? How many matchsticks?
This is how Alice drew it:
Can you describe what Alice did?
How many 'alongs' and how many 'downs' are there?
How many matchsticks altogether?
Now picture what Alice would do if there had been $25$ squares.
How many 'alongs' and how many 'downs' would there be?
How many matchsticks altogether?
If there had been $100$ squares? How many matchsticks altogether?
A million and one squares? How many matchsticks?
This is how Luke drew it:
Can you describe what Luke did?
How many squares and how many inverted C's are there?
How many matchsticks altogether?
Now picture what Luke would do if there had been $25$ squares.
How many squares and how many inverted C's would there be?
How many matchsticks altogether?
If there had been $100$ squares? How many matchsticks altogether?
A million and one squares? How many matchsticks?
Now choose a couple of the patterns below.
Try to picture how to make the next, and the next, and the next...
Use this to help you find the number of squares, or lines, or perimeter, or dots needed for the $25^{th}$, $100^{th}$ and $n^{th}$ pattern.
Can you describe your reasoning?
Growing rectangles
This rectangle has height 2 and width 3.
Work out the perimeter, the number of dots, and the number of lines needed to draw a rectangle with:
• height 2 and width 25
• height 2 and width 100
• height 2 and width n
L Shapes
This L shape has height 4 and width 4.
Work out the perimeter, the number of squares, and the number of lines needed to draw an L shape with:
• height 25 and width 25
• height 100 and width 100
• height n and width n
Two squares
This pattern with two squares has four black dots.
Work out the number of white dots and the number of lines needed to draw a pattern with:
• 25 black dots
• 100 black dots
• n black dots
Square of Squares
This pattern has side length 5.
Work out the number of edge squares and the number of lines needed to draw the pattern with:
• side length 25
• side length 100
• side length n
Dots and More Dots
This pattern has side length 3.
Work out the number of dots and the number of lines needed to draw the pattern with:
• side length 25
• side length 100
• side length n
Rectangle of Dots
This pattern is made from two joined squares with side length 3.
Work out the number of lines and the number of dots needed to draw the pattern of joined squares with:
• side length 25
• side length 100
• side length n
Getting Started
How did Phoebe group the matchsticks that she drew?
How did Alice and Luke group their matchsticks?
For the follow-up activities, draw them out for yourself and notice how YOUR drawings develop.
Always begin with simple cases and try to PREDICT what will happen.
Look for patterns.
How can you describe the lines? Horizontal? Vertical?
Try to understand why the patterns develop in the ways that they do.
Student Solutions
Alexander, from Wilson's School, wrote:
Phoebe first drew one “down,” and then as many “inverted Cs” as needed to get seven squares. Therefore:
• When drawing only seven squares you draw one “down” and seven “inverted Cs.” There were 22 matchsticks.
• When drawing 25 squares, there would still be one “down,” but 25 “inverted Cs.” The total number of matchsticks would be 76.
• When drawing 100 squares, again there would be only one “down,” but 100 “inverted Cs.” The total number of matchsticks would be 301.
• When drawing 1,000,001 squares, there would be 1 “down” and 1,000,001 “inverted Cs.” There would be 3,000,004 matchsticks.
• When drawing n squares, there would be one “down” and n “inverted Cs.” There would be 3n + 1 matchstick(s). This is because there is one matchstick at the beginning that forms the “down,” and
then each of the “inverted Cs” takes up three matchsticks, so you multiply the number of inverted Cs by three and then add 1.
Alice first drew seven “alongs” at the top, and then seven “alongs” at the bottom. She then used eight “downs” to connect the gaps in between two parallel “alongs.” Therefore:
• For seven squares you draw fourteen “alongs” and eight “downs.” The total number of matchsticks is 22.
• When drawing 25 squares, you draw 50 “alongs” and 26 “downs,” making the total number of matchsticks 76.
• When drawing 100 squares, you draw 200 “alongs” and 101 “downs,” making the total number of matchsticks 301.
• When drawing 1,000,001 squares, you draw 2,000,002 “alongs” and 1,000,002 “downs,” making the total 3,000,004 matchsticks.
• When drawing n squares, you draw 2n “alongs” and n + 1 “downs,” making the total 3n + 1 matchsticks. This works because for each square, there are two “alongs,” and like an “inverted C,” you
have one “down” for each square, but then you add one more “down,” because like Phoebe's method, you need one extra line at the end. Adding the two formulas together, 2n + n + 1 = 3n + 1, which
is the same as Phoebe's formula.
Luke first drew one “square,” and then six “inverted Cs” to make seven squares. Therefore:
• You need one “square” and six “inverted Cs” to make seven squares. This gives you a total of 22 matchsticks.
• Similarly, with 25 squares, you draw one “square,” but 24 “inverted Cs,” which gives you a total of 76 matchsticks.
• With 100 squares, you still draw one “square,” but 99 “inverted Cs,” which gives you a total of 301 matchsticks.
• With 1,000,001 matchsticks, again you draw one “square,” but you draw 1,000,000 “inverted Cs,” giving you a total of 3,000,004 matchsticks.
• With n matchsticks, you draw one “square,” and n - 1 “inverted Cs,” giving you a total of 3n + 1 matchsticks:
4 + 3(n - 1) = 4 + 3n - 3 = 3n + 1,
which is the same as all of the other methods.
Great! Laeticia, from Woodbridge High School, also gave correct general formulas here. She went on to comment on one of our later puzzles:
Growing rectangles:
If the height is h and the width is w, then the perimeter is 2h + 2w. Then there are (h+1)(w+1) dots, and w(h+1) + h(w+1) lines.
Niharika solved the rest of our questions:
Each L-shape has an 'inner' L-shaped line and an 'outer' L-shaped line. If an L-shape has height and width n, then the outer L has length 2n and the inner L has length 2(n-1). There are 2 lines
remaining on the ends of the L, so altogether the perimeter is 4n.
There are 2(n-1) lines left inside the L-shape, so the shape is made of 6n - 2 lines.
The number of squares in an L-shape is 2n-1: think of each L as two rectangles of height 1 and width n that overlap in one square.
Two squares:
Think of these as two separate overlapping squares, each with $n^2$ dots. They overlap in one dot, so in total there are $2n^2 - 1$ dots, and so $2n^2 - n - 1$ white dots.
In each square there are 2n(n-1) lines, and the lines never overlap, so in total there are 4n(n-1) lines.
Square of squares:
We can split a pattern like this of side length n up into four rectangles of height 1 and length n-1, so there are 4(n-1) edge squares.
There are 4n lines making up the outer square and 4(n-2) lines making up the inner square. There are 4(n-1) lines left in the middle. This gives 12(n-1) lines in total.
Dots and more dots:
Inside the squares there are $n^2$ dots, and on the vertices there are $(n+1)^2$ dots, so in total there are $2n^2 + 2n + 1$ dots.
In each row there are n lines, and there are n+1 rows, so the rows contribute n(n+1) lines. Similarly the columns contribute n(n+1) lines, so together there are 2n(n+1) lines.
Rectangle of dots:
There are 4n horizontal lines and 3n vertical lines, so 7n lines in total.
On each row there are 2n+1 dots, and there are n+1 rows, so there are (2n+1)(n+1) dots.
Fantastic! Thank you all.
Teachers' Resources
Why do this problem?
This problem challenges students to describe patterns clearly - verbally, numerically and algebraically. It does not assume prior knowledge of algebra and could be a good way to introduce, practise
or assess algebraic fluency.
Similar-looking questions are often asked, expecting an approach that uses number sequences for finding a formulae for the $n^{th}$ term. This problem deliberately bypasses all that, instead focusing
on the structure of the pattern so that the algebraic expressions emerge naturally from that structure.
Possible approach
The Article Go Forth and Generalise may be of interest.
Have the "seven squares" image preprepared on the board so that students cannot see how you drew it. "I have drawn seven matchstick squares on the board, and I would like you to make a rough copy of
it - no need to use a ruler."
While the students are sketching, look out for students creating the image in different ways, such as Phoebe's, Alice's and Luke's methods in the problem.
Select at least three students who have used different methods, and invite them to draw the image on the board (perhaps using colours to emphasise the order in which it was drawn).
"Without counting individual matches can you say how many matchsticks there are in the drawing?"
"How would 25 squares be drawn using this method?"
"How many matchsticks would be needed altogether?"
"What if there were 100 squares?"
"Or a million squares?"
"Or $x$ squares?"
The answers to these questions could be recorded on the board, so that the results and the algebraic expressions emerging from each method can be compared at the end.
For example, for Phoebe's method from the problem you could initially write $$1+ 7 \times 3$$ leading to $$1 + 25 \times 3$$ $$1 + 100 \times 3$$ and so on, eventually finishing with $$1 + 3x$$
Alternatively, you could show the class the videos provided in the problem showing three different methods.
Next, hand out this worksheet. There are six different patterns with the simpler ones at the start. Invite students to work in pairs:
"With your partner, choose two or three of the six patterns and have a go at the questions. Make sure you can explain clearly how you worked out your answers, focusing on the order in which you would
draw the diagram, like we did for the Seven Squares problem."
While students are working, circulate and listen to the conversations, identifying students who have really elegant ways of seeing the general case in the initial picture.
"I'm going to give you ten minutes to prepare a poster presenting one of the problems you worked on and explaining how you arrived at your solution."
Students could choose which problem to work on, and you could guide particular students towards problems where you have noticed them reasoning clearly.
Once they have produced their poster, there are a number of different ways that sharing and feedback could be organised:
• Half the class stand by their posters and the other half of the class visit them, read, ask for clarification on anything that is unclear, and suggest improvements as 'critical friends'. After
five minutes, swap over.
• All the posters are laid out. Students visit each other's posters and write any comments, questions or feedback on post-it notes.
• Selected students could present the content of their poster on the board, with the rest of the class feeding back.
Key questions
Can you see a pattern in the image? How might you draw it?
Can you tell how someone drew the pattern from the way they write the calculation?
How does your formula relate to the structure of the pattern?
Possible support
Students could spend time exploring the first three patterns before moving on to the harder cases.
Encourage students to draw a few examples of each pattern and notice how their drawings develop.
Possible extension
Here are a couple of suitable follow-up problems that use the structure of a situation to lead to algebraic generalisations:
A teacher's comments after using this activity:
"It gave rise to much discussion about how to describe the patterns. It led naturally to building algebraic expressions and seeing them as easily understandable ways to record the patterns. It
provided motivation for checking that the different algebraic expressions (used to describe the different ways in which a pattern can be built) are in fact equivalent."
"Some students succeeded in building the patterns and working numerically, but were not yet ready to work algebraically, while other students progressed to finding, and even simplifying, formulae for
the patterns. All students experienced success and there was appropriate challenge in this problem for everyone." | {"url":"https://nrich.maths.org/problems/seven-squares","timestamp":"2024-11-02T14:08:15Z","content_type":"text/html","content_length":"57782","record_id":"<urn:uuid:d236ad23-245a-473a-8256-1f1759e9b609>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00565.warc.gz"} |
Bad News for Life on Exoplanets -
Bad News for Life on Exoplanets
Image Credit: DLR_next ( under CC BY-NC-SA 2.0)
It is known that red dwarf stars account for about 75% of all the stars in the Milky Way, and astronomers generally agree that red dwarfs are likely to occur in similar numbers in most galaxies
throughout the Universe. Therefore, it is likely that there are more red dwarf stars in the Universe that support planets than any other type of star, and in fact, many exoplanets have been
discovered around red dwarfs, including the seven roughly Earth-sized planets of the TRAPPIST -1 planetary system.
At first glance then, it might seem that since red dwarfs live for several trillion years, as opposed to the several-billion-year lifetimes of more massive stars, the habitable zones around red dwarf
stars might offer the perfect environment for life to develop. However, new research that was announced at the beginning of April 2018 at the European Week of Astronomy and Space Science in Liverpool
seems to suggest otherwise.
The core of the problem involves the very nature of red dwarf stars. Most, if not all red dwarfs are flare stars, which means that although red dwarfs are relatively cool, these stars often erupt
violently, emitting large quantities of charged particles along with large chunks of their coronas, in explosions known as Coronal Mass Ejections. Moreover, since red dwarfs mostly exist in a range
of temperatures centered on about the 70,000 F, any planets on which life might develop have to orbit their stars very closely, and herein lays the problem.
To assess the likelihood that coronal mass ejections might be harmful to nascent life on planets around red dwarf stars, astronomer Eike Guenther and a team of associates from the Thüringen
Observatory in Germany launched a research program to observe the effects that the violent eruptions which red dwarf stars are known for might have on the planets around them.
Thus, in February 2018, the researchers observed a flare on a red dwarf known as AD Leo, about 16 years away in the constellation Leo. This was particularly interesting, since AD Lee is known to host
a large exoplanet at a distance of only 190,000 miles (300,000km) away, which is about 50 times closer than Earth is to the Sun. Moreover, AD Leo is also thought to host a few more Earth-sized
planets a little further away, but still within the star’s habitable zone.
Since the observed flare was not accompanied by a coronal mass ejection event, and therefore left the large planet largely undamaged, the concomitant blast of X-ray radiation that followed the
eruption was powerful enough to have reached down to the surface(s) of any less massive planets that were unfortunate enough to be in its path.
In practical terms, this means that the entire hemisphere of an Earth-sized planet facing an X-ray bombardment of this magnitude would be sterilized in matter of minutes, even in the absence of a
coronal mass ejection event. While the research team is still fine-tuning its results to ensure that it is reliable, Guenther commented:
“With sporadic outbursts of hard X-rays, our work suggests planets around the commonest low-mass stars are not great places for life, at least on dry land”.
Representatives of the Royal Astronomical Society (RAS) reacted to the announcement of the initial results, saying that, “If they [Guenther et al] are right, then talk of ‘Earth 2.0’ [being located
around a red dwarf] may be premature”.
4 Comments
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4. sBZyuWpG | {"url":"https://www.astronomytrek.com/news/bad-news-for-life-on-exoplanets/","timestamp":"2024-11-06T10:59:40Z","content_type":"text/html","content_length":"78986","record_id":"<urn:uuid:10b5476a-5b35-4c08-8a63-5179f31b3f26>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00422.warc.gz"} |
Credit Risk and Asset Visibility
Credora’s platform enables real-time credit analytics that can be used for credit underwriting and risk monitoring. This paper focuses on digital asset trading firms, a subset of Credora users.
Borrowers connect accounts to Credora’s privacy-preserving architecture, allowing for real-time calculations of asset values and risk information. The privacy-preserving architecture ensures
sensitive information, including positions and trades, are inaccessible by Credora or any other party.
Credora credit assessment methodologies utilize visibility metrics. For digital asset trading firms, these are defined as the calculated asset balance divided by the most recently reported current
assets via financial statement submissions.
Lenders have the opportunity to monitor borrower real-time risk metrics during active loans, permitting a superior understanding of the credit risk absorbed by the borrower (typically to exchanges),
and providing a proxy for how the balance sheet is evolving between financial reporting periods.
Due to the nature of borrower activities, and the wide range of trading venues for quantitative trading firms, 100% visibility for any specific trading firm is challenging. In Credora’s credit
models, visibility is incrementally assigned points along a curve.
Although complete visibility is desirable, partial visibility significantly reduces risk for lenders. The value of visibility can be characterized as follows:
• Financial Statement Proxy: Visibility provides current information, in contrast to the historical snapshots typically relied upon for credit analysis. For example, balance sheets are typically
delivered 15–30 days after the snapshot date. Furthermore, many firms report financial information on a quarterly basis.
• Venue Risk: Visibility provides incremental information on counterparty risk, by identifying the location of assets. This is especially relevant in digital asset trading.
• Risk Information: Credora identifies major changes in borrower position risk, which may signify strategy drift away from communicated risk parameters.
This post is focuses on visibility as a Financial Statement Proxy. The statistical exercise underpinning it aims to quantitatively demonstrate the relationship between visible assets and reported
current assets. It concludes that even partial visible assets are a strong predictor of subsequently reported balance sheet current assets.
Credora’s years of operation have allowed us to collect a unique dataset that combines risk monitoring information and reported financials. The time period considered was January 2021 through July
Measurements in absolute figures for asset values give the raw relationship between the two series (visible vs reported). To remove the effect of upward or downward trend over time from absolute
values in the signals, the analysis below is produced both in absolute figures, and looking at log differences (i.e. we evaluate the monthly change). Relative measurements are interesting to evaluate
the percentage changes or growth rates between two series.
• X = ln(X[t]/X[t-1]) (Visible Assets)
• Y = ln(Y[t]/Y[t-1]) (Reported Current Assets)
Pearson Correlation
First we calculate the correlation coefficient between “Visible Assets” and “Reported Current Assets”. The most common correlation coefficient is the Pearson correlation coefficient, which is widely
used for numerical features, including time series.
Performing a statistical test, such as t-test, alongside calculating the correlation coefficient helps determine whether the observed correlation is significant or likely due to random chance,
accounting for factors like sample size and hypothesis testing. The formulated null hypothesis (H0) and an alternative hypothesis (H1) are:
• H0: There is no correlation between visible assets and current assets (correlation coefficient = 0).
• H1: There is a correlation between visible assets and current assets (correlation coefficient ≠ 0).
• Significance Level: A significance level of α = 0.05 was chosen for the hypothesis test.
Absolute Figures
Results show a pearson correlation coefficient of 94.5%, indicating a strong linear relationship. The p-value obtained for the significance of correlation was p < 0.001, significantly below the
chosen significance level. Therefore, we reject the null hypothesis, meaning that there is a significant correlation between visible assets and current assets.
Correlation in absolute figures measures the precise linear relationship between variables and helps to understand the absolute impact of one signal on another.
The scatter plot with “visible assets” on one axis and “current assets” on the other helps to visualize the relationship between the two variables and provide insights into the degree of correlation.
Colors correspond to individual borrowers in different observation points.
Logspace Ratio
In the log space, the obtained correlation depends highly on the percentage of visible assets. By looking at the correlation calculated per visibility percentage of the borrower, it is clear that
correlation between visible and reported assets increases with the increase in visibility percentage.
This makes intuitive sense, as it is more difficult to infer the borrower’s total asset balance change if visibility is low.
For that reason, the statistical test and pearson correlation are calculated for borrowers with a visibility percentage higher than 20%, under the same hypothesis testing.
Results show a pearson correlation coefficient of 66.68%, indicating a strong linear relationship. The p-value obtained for the significance of correlation was p < 0.001, significantly below the
chosen significance level. Therefore, we reject the null hypothesis, meaning that there is a significant correlation between visible assets and current assets.
The scatter plot with “Log Rel Diff” from current assets and visible assets helps to visualize the relationship between the two variables and provide insights into the degree of correlation. Colors
correspond to individual borrowers in different observation points.
Robust linear regression
Modeling the relationship between the two variables, allows us to predict the “current reported assets” variable (the dependent variable) based on the values of “visible assets” (the independent
Using robust regression with a Huber loss function for signal similarity measurement offers advantages in scenarios where the data might have outliers or is affected by noise. This linear regression,
calculated in both absolute and relative figures, helps to baseline the potential of the visible assets variable as a predictor, using just a simple linear regression.
Absolute Figures
As below table shows, the coefficient for the variable X is 2.0602. Since the p-value associated with the Wald test for the corresponding to variable X is very close to zero (P>|z| ≈ 0.000), it
suggests that the variable X is statistically significant in explaining the variation in Y.
Residuals are the differences between the observed Y values and the predicted Y values from the model. According to the below plot, variance is equally distributed and shows that linear relationship
between variables is valid.
Logspace Ratio
We observe a statistically significant relationship between the two signals represented by variables X and Y. The coefficient for X (0.2491) indicates that for every one-unit increase in X, Y is
expected to increase by approximately 0.2491 units.
Given that the p-values associated with both the intercept and the X coefficient are very low, it’s very likely that the relationship is not due to random chance.
As before, the residuals show a normal distribution:
As detailed above, visible assets show strong Pearson correlation results for both absolute (94.5%) and relative log scale figures (66.6%) when considering only borrowers with >20% visibility,
passing the statistical tests that suggest the observed correlation is significant. The robust linear regression model shows a significant linear relationship between the two signals.
The implications for credit analysis as applied to trading firms is powerful. Monitoring of assets provides a valuable proxy for financial statement changes, even where visibility is materially below
From a risk management perspective, this indicates the incremental value of enabling real-time monitoring on borrowers, especially where privacy-preserving technology can eliminate data sensitivity
concerns and increase the visibility percentage further.
At Credora, our core mission is to facilitate more efficient credit markets. In the current paradigm of financial statement reporting, real-time data offers a huge opportunity. | {"url":"https://credora.medium.com/credit-risk-and-asset-visibility-ef65e13c8dab","timestamp":"2024-11-07T23:37:49Z","content_type":"text/html","content_length":"147289","record_id":"<urn:uuid:00cd4b80-4d2e-4650-920e-1852826a2173>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00190.warc.gz"} |
What does CBM mean? | Boloo Forwarding Help Center
CBM stands for "cubic meter." Before you proceed with shipping, it is important to know the CBM of your shipment as it can determine the price. The CBM corresponds to a certain weight, which is also
known as volumetric weight. The volumetric weight is compared to the actual weight of the shipment. The higher weight, whether actual or volumetric, is used to calculate the charge. For air freight,
you pay based on the higher weight, whereas for sea freight, you always pay for the CBM or volumetric weight.
You can calculate the cubic meter (CBM) using the following formula:
Length (cm) x Width (cm) x Height (cm) / 1,000,000 = CBM
To determine the volumetric weight of your shipment, you can apply the following formulas:
Length (cm) x Width (cm) x Height (cm) /
For example: My box measures 118 x 78 x 78 cm (pallet box), and I am shipping via air freight.
Volumetric weight: 118 cm x 78 cm x 78 cm / 6,000 = 119.65 kg
Cubic meter: 118 cm x 78 cm x 78 cm / 1,000,000 = 0.72 CBM | {"url":"https://help.cargomate.nl/en/articles/5211096-what-does-cbm-mean","timestamp":"2024-11-12T06:15:02Z","content_type":"text/html","content_length":"61158","record_id":"<urn:uuid:4a6aabc5-ad7d-43f0-a341-f5993aba8594>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00503.warc.gz"} |
The Force Table – Vector Addition and Resolution
"Vectors? I don't have any vectors. I'm just a kid." – From Flight of the Navigator
• •
To observe how forces acting on an object add to produces a net, resultant force.
• •
To see the relationship between the equilibrant force relates to the resultant force.
• •
To develop skills with graphical addition of vectors and vector components.
• •
To develop skills with analytical addition of vectors and vector components.
• Force Table simulation
• Pencil
Simulation and Tools
Open the Force Table simulation to do this lab. See the Video Overview.
Explore the Force Table Apparatus; Theory
We'll use the force table apparatus in this lab activity.
For each direction answer, enter an angle between 0° and 360° with respect to the force table.
Working with vectors graphically is somewhat imprecise. You should expect to see lower precision in this lab. But working with vectors graphically will give you a better understanding of vector
addition and vector components. You can improve your precision by using the Zoom features of the apparatus.
Figure 1: The Force Table Apparatus
When you're asked to provide a figure, you may sketch it and scan a copy or take a picture of it; or you may create it with the force table apparatus, do a print screen, and save it as a file.
Whatever file you create, you will need to upload a copy in WebAssign.
Please print the worksheet for this lab. You will need this sheet to record your data.
I. Addition of Two-Dimensional Vectors
In this part, you'll add two forces to find their resultant using experimental addition, graphical addition, and analytical addition. In WebAssign, you will randomly be given two forces to work with.
There are two things that we should clarify about these two statements. We'll refer to Figure 1 in the following.
• •
The force subscripts on forces and angles (F[2], θ[2]) refer to the hanger numbers. For example, in Figure 1, θ[1] = 20°.
• •
The notation for forces in this lab is a bit unusual. From the "Masses on Hangers" table, we know that hanger #1 has a 50 g and a 200 g mass on it. The hanger's mass is 50 g, so the total, 300 g,
is shown in the box beside the hanger. We want to record forces in newtons. From
W = mg,
we can easily calculate the weight of a 300 g mass.
W = 0.300 kg × 9.8 N/kg = 2.94 N
• But there's a much more convenient way of handling this. We can abbreviate this calculation to the simpler W = 0.300 gN.
A. Experimental Addition; The Equilibriant
By experimental addition we mean that we will actually apply the two forces to an object and measure their total, resultant, effect. We'll do that by first finding the equilibrant, E, the force that
exactly balances them. When this equilibrium is achieved, the ring will be centered. The resultant, R, is the single force that exactly balances the equilibrant. That is, either F[2] and F[4], or
their resultant R can be used to balance the equilibrant.
When adjusting the angle of a pulley to a prescribed angle, pay no attention to the string. Drag the pulley until the color-coded pointer is at the desired setting. Another way that works well is to
disable all the hangers while adjusting the angles. When you do this, the ring is automatically centered so the strings can be used to help adjust the angle.
Enable pulleys 2 and 4 and disable pulleys 1 and 3 using their check boxes. A disabled pulley is grayed out. You can drag it anywhere you like since any masses on it are inactive.
Drag pulleys 2 and 4 to their assigned angles, θ[2] and θ[4]. Use the purple and blue pointers of the pulley systems, not the strings, to set the angles. You should zoom in to set the angles more
Add masses to hangers 2 and 4 to produce forces F[2] and F[4].
Using pulley 3, experimentally determine the equilibrant, E to just balance (cancel) F[2] and F[4]. Do this by adjusting the mass on the hanger and the angle until the yellow ring is approximately
centered on the central pin. You can add and remove masses to home in on it. You add masses by dragging and dropping them on a hanger. You remove them using the Mass Total/Mass Removal tool. This
tool displays the total mass on the hanger, including the hanger's mass. Clicking on the tool removes the top mass on the hanger.
From your value of E, what should be the resultant of F[2] and F[4]? (Same force, opposite direction.)
Disable pulleys 2 and 4 and enable pulley 1. Experimentally determine the resultant by adjusting the mass on pulley 1 and its angle until it balances the equilibrant. That is, the ring should not
move substantially when you change between enabling F[2] and F[4], and enabling just F[1].
B. Graphical Addition
With graphical addition, you will create a scaled vector arrow to represent each force. We add vector arrows by connecting them together tail to head in any order. Their sum is found by drawing a
vector from the tail of the first to the head of the last. Using the default 2 × 10^2 g vector scale, create vector arrows (±3 grams) for F[2] and F[4]. For example, drag the purple vector by its
body and drop it when its tail (the square end) is near the central pin. It should snap in place. You now want to adjust its direction and length to correspond to direction and magnitude of F[2].
It's hard to do both of these at the same time. That's where the "Resize Only" and "Rotate Only" tools come into play. They let you first adjust the direction and then adjust the length. Let's do F
[2]. Zoom in. Drag the head (tip) of the vector arrow in the assigned direction force F[2]. That is, θ[2]. Zoom out. Click the "Resize Only" check box. Its direction is now fixed until you uncheck
that box. You can now drag the head of the arrow and adjust just its magnitude without changing its direction. The current magnitude of F[2] is displayed at the top of the screen next to the home
location of that purple vector. Repeat for F[4] using the blue arrow.
The resultant, R, is the vector sum of F[2] and F[4]. Add F[2] and F[4] graphically by dragging the body of F[4] until its tail is over the tip of F[2] and releasing it. Form the resultant, R, by
creating an orange vector, F[1], from the tail of the first (vector F[2]) to the head of the last (vector F[4]). (You can actually connect F[2] and F[4] in either order.) You can read the magnitude
and direction of R directly from the length and direction of F[1]. You'll likely find some disagreement between these results and your results from Part A. This is because both methods are somewhat
The equilibrant, E, should be the same magnitude as R, but in the opposite direction. Produce E with the green vector arrow. It should be attached to the central pin and in the direction of the force
Draw each of the four vector arrows on Figure 2. Label each vector with its total force value. (Ex. Label the blue F[4] vector 0.180 gN.) Vectors F[2] and F[4] should be shown added in their
tail-to-head arrangement. The E and R vectors should be shown radiating out from the central pin. Label each vector with its force value in gN units.
Figure 3: Graphical Addition; Equilibrant
File Uploads of Graphical Representations: For the graphical representations in this lab, you have several choices for creating the files to upload. A couple of these are listed below. Regardless of
the method you use, ensure that your file is in a format that your instructor can open.
• 1
Draw the vectors by hand.
□ a
Use Figure 3 if it is best drawn on the force table schematic.
□ b
Either scan your image in and save it as a file or take a picture of your drawing.
• 2
Do a print screen of your drawing from the simulation and save it as a file.
Describe what we mean by the terms resultant and equilibrant in relation to the forces acting in this experiment.
From your figures, which two forces when added would equal zero?
Show this graphical addition. You'll want to offset them a bit since they would otherwise be on top of one another. You will upload a file showing your graphical representation of this addition.
What three vectors when added would equal zero?
Show this graphical addition. You will upload this as a file.
Vector E should appear in both of these last two vector additions. Why?
Put all four vectors back at the center of the force table with their tails snapped to the central pin. Click the check boxes beside the purple and blue arrows under "Show Components." This will give
you a visual check on your analytical calculations of the component values in Part II.A3. So be sure to look back at the figure as you do Part II.A3.
C. Analytical Addition with Components
You've found experimentally and graphically how to add the forces F[2] and F[4] to produce the resultant, R. Hopefully you found that these two methods make it very clear to you what is meant by the
addition of vectors. But you also found that both methods are pretty imprecise—sort of the stone tools version of vector addition. We can get much better results using trigonometry. You've just
verified that the resultant effect of two (or more) vectors can be found by attaching them together tail to head and drawing a new vector from the tail of the first to the head of the last. That
single new vector is equivalent to the two original vectors acting together. So the one achieves the same result as the original two. The reverse is also true. We can simplify the process of addition
of vectors if we replace each vector with a pair of vectors whose sum equals the original vector. At least if we do it strategically.
Put all four vectors back at the center of the force table with their tails snapped to the central pin. Click the check box beside the purple arrow under "Show Components." Zoom in for a better look.
The two new purple vectors are the x and y components of the purple vector, F[2]. We would call them F[2x] and F[2y].
Drag the x-component, F[2x] and add it to the F[2y]. That is, connect its tail to the head of F[2y]. Clearly they add up to F[2]. But just to be sure, drag F[2x] back to the center and add F[2y] to
it. Same result. So F[2] = F[2x] + F[2y]. Note that this is all vector addition. We're using bold text for our vector names to emphasize that this is not scalar addition, which doesn't take direction
into account. F[2] equals the vector sum of F[2x] and F[2y] because when we connect the components together tail to head, the vector from the tail of the first to the head of the last is F[2]. So, we
can use F[2] and F[2x] + F[2y] interchangeably. That's the strategic part.
Turn on the blue components.
We'll dim the main vectors. Drag the large blue dragger on the "Vector Brightness" tool almost all the way to the left. Only the components will remain bright. Leaving one of them, say, F[2x] where
it is, add the other three components to it in any order you like. This might be a good time to Hide Table. There's a button at the top. Feel free to hide and show the force table as needed.
Notice where the head of the last component ends up. Notice how the four components add up to R! Try another order of addition. The order doesn't matter. So we can say,
R = F[1] = F[2x] + F[2y] + F[4x] + F[4y]
in any order. Remember, this is vector addition, not scalar addition. So we're still having to connect them together graphically to find the resultant. We're still using stone tools.
Turn on the components for F[1]. You now have three sets of components.
Drag all the y-components out of the way.
Add F[2x] and F[4x] together.
How do these two vectors relate to F[1x]; that is, R[x]?
Repeat with the y-components. What similar statement can you make about the y-components?
Finally, how is R related to R[x] and R[y]?
To summarize,
R[x] = F[2x] + F[4x] and R[y] = F[2y] + F[4y] and R = R[x] + R[y].
Figure 4: Vector Components
We can now leave our stone tools behind and take advantage of this new formulation by using trigonometric functions and the Pythagorean Theorem. Here's our task. You were previously asked to find the
resultant, R, of F[2] and F[4] graphically. You now want to find the same result without the using the imprecise graphical methods. Knowing F[2] and θ[2], we can calculate F[2x] and F[2y]
analytically as follows. We can do the same for F[4]. We can then find R[x] and R[y] using There's one slight problem with Equation 5. Like Equations 3 and 4, it's a vector equation, but in Equations
3 and 4, the vectors are collinear, so they can be added with signs indicating direction. But since vectors R[x] and R[y] are perpendicular, we have to use the Pythagorean Theorem instead. So we can
find the magnitude and direction of the resultant, R, with the magnitudes of R's components.
You've found experimentally and graphically how to add the forces F[2] and F[4] to produce the resultant, R. Now let's try analytical addition. Using the table provided, find the components of F[2]
and F[4] and add them to find the components of R. Use these components of R to determine the magnitude and direction of R. Note that one of the four components will have a negative sign. The
components now displayed on the force table should make it clear why.
Show all your calculations leading to your value for R. Remember, a vector has both a magnitude and a direction. A summary of the steps is provided to get you started.
II. Simulation of a Slackwire Problem
Let's model a realistic system similar to what you might find in your homework. Figure 5 shows a crude figure of a slackwire walker, Elvira, making her way across the wire rope. At a certain instant,
the two sides of the rope are at the angles shown. Only friction allows her to stay in place. The gravitational force is acting to pull her down the steeper "hill." If she were on a unicycle, she'd
tend to roll to the center. It's complicated. When friction is holding her in place, the single rope acts like two separate sections of rope in this situation with different tension forces on either
side of her. To understand this, it helps to imagine her at the extreme left where the left section of rope is almost vertical and the right one is much less steep. In this case, T[3] is providing
almost all of the vertical support, while T[1] pulls her a little to the right. It helps to just try it. Attach a string between two objects in the room. Leave a little slack in it. Now pull down at
various points. You'll feel the big frictional tug on the more vertical side and less from the more horizontal side. We want to explore this by letting her move along the rope.
Figure 5: Elvira Off-Center on the Slackwire
A. Experimental Determination of T[1] and T[3]
Preliminary prediction: Which tension do you think is greater, T[1] or T[3]?
First send each of your vectors "home" by clicking on each of their little houses. Then remove all the masses from your four pulleys by activating all of them using their click boxes and then
clicking in the total mass boxes beside each hanger until they read 50.
You will be given a randomized value for Elvira's weight in your WebAssign question; use that value in the following. We'll let 1 gram represent 1 newton on our force table and set our vector scale
to 4 × 10^2 N. (Note the label to the left of each vector.) We'll picture our force table as if it were in a vertical plane with 270° downward and 90° upward. We'll use pulleys 1, 3, and 2 to provide
our two tensions, T[1] and T[3], and the weight of Elvira, respectively. Disable all pulleys and set all the angles to match Figure 5. Use Zoom. Turn pulleys 1, 3, and 2 back on.
Prediction: Since θ[3] = 2 × θ[1], do you think T[3] will be about twice T[1]?
For Elvira, move hanger 2 to 270° and set its total mass to x g to represent x N, where x N is Elvira's weight.
Adjust the masses on each pulley until you achieve equilibrium. It's best to alternate adding one mass to each side in turn until you get close to equilibrium.
How'd that prediction for the relationship between T[1] and T[3] go? If trig functions were linear, we wouldn't need them.
Write a statement about the relationship between the steepness of the rope and the tension in that rope for this vertical arrangement.
B. Graphical Check of Your Results
Using the vector scale of 4 × 10^2 N, create vector arrows for each force, T[3], T[1], and W (Elvira's weight).
Upload a file of your graphical representation of these three forces.
Figure 6: Elvira – Asymmetrical Vectors
How can we graphically check to see if our values for T[1] and T[3] are reasonably correct? How are T[1], T[3], and W related? What would happen if any one of them suddenly went away? Elvira would no
longer be in what?
Thus, the three vectors are in equilibrium and must add to equal zero! What would that look like? The sum, resultant, of three vectors is a vector from the tail of the first to the head of the last.
If they add up to zero, then the resultant's magnitude would be zero, which means that the tip of the final vector would lie at the tail of the first vector.
Try it in the apparatus. Leave T[1] where it is and then drag T[3]'s tail to T[1]'s head. Then drag W's tail to the head of T[3].
What about your new figure says (approximately) that the three forces are in equilibrium?
Draw your new figure with the three vectors added together.
Upload a file of your graphical representation of this vector addition. Don't forget to use the "Resize Only" tool after you get the angles set.
Figure 7: Graphical Addition – Asymmetrical
C. Analytical Check of Your Results
If our three vectors add up to zero, then what about their components?
When you add up the x-components of all three vectors, the sum should be what?
When you add up the y-components of all three vectors the sum should be what?
Test your predictions by calculating all the components and adding them. Don't forget the direction signs!
Before we continue . . . Why all the tension?
Why does the tension on each side have to be so much larger than the weight actually being supported? All the extra tension is being supplied by the x-components. Then why not just get rid of it?
You'd have to move the supports right next to each other to make both ropes vertical. That would be pretty boring, and, frankly, nobody would pay to see such an act, even if they tried calling it
"Xtreme Urban Slackwire." Similarly, with traffic lights, a huge amount of tension is required to support a fairly light traffic light. The simpler solution of a pair of poles in the middle of the
intersection wouldn't go over much better than Xtreme Urban Slackwire. Next time you see large poles or towers supporting large electrical wires, look for places where the wire has to change
direction (south to east, for example). The support structures in straight stretches (in-line) don't have to be really sturdy since they have horizontal forces pulling equally in opposite directions.
Thus, they just have to support the weight of the wire. When the wires have to change directions, the poles at the corners have to provide these horizontal forces. Thus, they're much sturdier and
larger to give them a wider base. They often have guy wires to the ground to help provide these forces. To make things more difficult, the guy wires will be usually be very steep, so they have to
have a lot of extra tension in them to supply the horizontal force components.
Figure 8: Power Lines – In Line and at Corners
III. Simulation of a Symmetrical Slackwire Problem
In Figure 9, Elvira has reached the center of the wire. This is where she would be if she rode a unicycle and just let it take her to the "bottom." The angle values are just guesses. They'd be
between the 10° and 20° we had before, but the actual value would depend on the length of the wire and its elastic properties.
Figure 9: Elvira at the Center of the Slackwire
Without changing the masses used in Part II, adjust both angles to 15°.
What does the symmetry of the figure suggest about how the tensions, T[1] and T[3], should compare?
Similarly, what can you say about comparative values of the x-components, T[1x] and T[3x]? (Ex. Maybe T[1x] = 2 × T[3x]?)
What can you say about comparative values of the y-components, T[1y] and T[3y]?
Given that the weight being supported is W, what can you say about actual values of the y-components, T[1y] and T[3y]?
You can now easily calculate T[1] and, hence, T[3].
Change each T to as close as you can get to this amount. Does this produce equilibrium?
With your apparatus, create the two vector arrows to match this amount—one for T[1] and one for T[3]. Add T[1] + T[3] + W graphically.
Upload a file with your graphical representation of this vector addition.
Figure 10: Graphical Addition – Symmetrical
You'll notice that this figure doesn't differ much from the previous one. The graphical tool we're using is not very precise. The same goes for "the real world." Measuring the tension in a heavy
electrical cable or bridge support cable is very difficult. One good method involves whacking it with a hammer and listening for the note it plays!
IV. Vector Subtraction
Here's the scenario. We have a three-person kinder, gentler tug-o-war. The goal is to reach consensus, stalemate. Two of our contestants are already at work.
• Darryl[1] = 800 N at 0°
• Darryl[2] = 650 N at 240°
The question is—how hard, and in what direction, must Larry[3] pull to achieve equilibrium?
As before, empty all the hangers, and then set up pulleys 1 and 2 to represent these forces using 1 gram to represent 1 newton.
Send all the vector arrows home to get a clean slate. Then create orange and purple vectors to match, using 4 × 10^2 N for your vector scale. Attach them to the central pin.
One method of solving the problem is to add the Darryl forces and complete the triangle to find the resultant. Larry's force would be the equilibrant in the other direction. Another way is to add the
two Daryl forces and then draw a third force, Larry, to complete the triangle which would leave a resultant of zero. The following vector equation describes that method.
( 8 )
ΣF = Darryl[1] + Darryl[2] + Larry[3] = 0
This is a vector equation. It means that you'll get a resultant of zero if you connect the three vectors together tail to head. To find the Larry vector, we need to subtract the two Darryl vectors
from both sides. We know how to add vectors but how do we subtract them? With scalar math, it would look like this:
ΣWorth = Darryl[1]$ + Darryl[2]$ + Larry[3]$ = 0 (Yes, that's net worth.)
Larry[3]$ = −Darryl[1]$ − Darryl[2]$.
If Darryl[1]$ = $800 and Darryl[2]$ = $650, then we'd get
Larry[3]$ = −$800 − $650 (Note that we're adding negatives here.)
Larry[3]$ = −$1450.
Negative dollars don't exist, but we would interpret this as something like a debt. That is, to make the three brothers have zero net worth, Larry[3] needs to be $1450 in debt. With vectors it works
the same way, but we interpret the negative signs as indications of direction. A pull of –50 N to the left means a pull of 50 N to the right.
ΣF = Darryl[1] + Darryl[2] + Larry[3] = 0
Larry[3] = −Darryl[1] − Darryl[2]
Larry[3] = (−Darryl[1]) + (−Darryl[2])
So to find Larry[3], we need to create the two negative Darryl vectors and add them. This means to draw the vectors (–Darryl[1]) and (–Darryl[2]), which are the opposites of Darryl[1] and Darryl[2],
and add them. That is, we can subtract vectors by adding their negatives. So, if
• Darryl[1] = 800 N at 0°
• Darryl[2] = 650 N at 240°,
then the negatives of these are vectors of the same lengths but in the opposite directions. Thus,
• –Darryl[1] = 800 N at 180°
• –Darryl[2] = 650 N at 60°.
Ruining the hard work you just did, create these two –Darryl vector arrows. (Don't change the mass hangers. Just the vectors.) You'll do this by just reversing the directions of both the vector
arrows you initially created. Leaving the –Darryl[1] pointing at 180°, add the –Darryl[2] in the usual tail-to-head fashion. Larry[3] is the sum of these two. Create the green Larry[3] vector from
the tail of Darryl[1] to the head of Darryl[2]. Upload a file with your graphical representation of this vector addition. Label your three vectors Larry[3], –Darryl[1], and –Darryl[2].
Figure 11: Larry and 2 Darryls
Record Larry[3] (graphical) from the length and direction of your green Larry[3] vector. Move pulley 3 to the position indicated by the direction of Larry[3]. You'll want to temporarily turn off the
hangers to set the angle correctly. Place the necessary mass on hanger 3 to produce the Larry[3] force.
Record Larry[3] (experimental).
Ta-da! I hope this has made vectors a little bit less mystifying. | {"url":"https://www.webassign.net/question_assets/ketphysvl1/lab_4/manual.html","timestamp":"2024-11-09T05:02:46Z","content_type":"application/xhtml+xml","content_length":"74102","record_id":"<urn:uuid:b06f40bf-c7e4-4306-8178-3f58031289c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00545.warc.gz"} |
In this chapter, we reviewed all the main steps involved in a machine learning process. We will be, indirectly, using them throughout the book, and we hope they help you structure your future work
In the next chapter, we will review the programming languages and frameworks that we will be using to solve all our machine learning problems and become proficient with them before starting with the | {"url":"https://subscription.packtpub.com/book/data/9781786469878/2/ch02lvl1sec15/summary","timestamp":"2024-11-14T21:13:27Z","content_type":"text/html","content_length":"117752","record_id":"<urn:uuid:e809841c-75ce-40f5-8519-86518b5824d9>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00666.warc.gz"} |
$(\infty,1)$-Category theory
Basic concepts
Universal constructions
Local presentation
Extra stuff, structure, properties
The notion of quasi-category is a geometric model for (∞,1)-category.
In analogy to how a Kan complex is a model in terms of simplicial sets of an ∞-groupoid – also called an (∞,0)-category – a quasi-category is a model in terms of simplicial sets of an (∞,1)-category.
A quasi-category or weak Kan complex is a simplicial set $C$ satisfying the following equivalent conditions
• all inner horns in $C$ have fillers. This means that the lifting condition given at Kan complex is imposed only for horns $\Lambda^i[n]$ with $0 \lt i \lt n$.
• the morphism of simplicial sets
$sSet(\Delta[2],C) \to sSet(\Lambda^1[2],C)$
(induced from the inner horn inclusion $\Lambda^1[2] \to \Delta[2]$) is an acyclic Kan fibration.
The equivalence of these two definitions is due to Andre Joyal and recalled as HTT, corollary 2.3.2.2. Quasi-categories are the fibrant objects in the model structure for quasi-categories.
While quasi-categories provide a geometric definition of higher categories, algebraic quasi-categories provide an algebraic definition of higher categories. For more details on this see model
structure on algebraic fibrant objects.
Relation to simplicially enriched categories
The homotopy coherent nerve relates quasi-categories with another model for $(\infty,1)$-categories: simplicially enriched categories.
See relation between quasi-categories and simplicial categories for more.
Higher associahedra in quasi-categories
While the geometric definition of (∞,1)-category in terms of quasi-categories elegantly captures all the higher categorical data automatically, it may be of interest in applications to explicitly
extract the associators and higher associators encoded by this structure, that would show up in any algebraic definition of the same categorical structure, such as algebraic quasi-categories.
For a discussion of this see
The two basic examples for quasi-categories are
Since the nerve of a category is a Kan complex iff the category is a groupoid, quasi-categories are a minimal common generalization of Kan complexes and nerves of categories.
By the homotopy hypothesis-theorem every Kan complex arises, up to equivalence, as the fundamental ∞-groupoid of a topological space.
Analogously, every directed topological space $X$ has naturally a fundamental (∞,1)-category given by a quasi-category whose $k$-cells are maps $\Delta^k_{Top} \to X$ that map the 1-skeleton of the
topological simplex in an order-preserving way to directed paths in $X$.
The directed homotopy theory that would state that this or a similar construction exhausts all quasicategories up to equivalence, does not quite exist yet.
Constructions in quasi-categories
The point of quasi-categories is that they are supposed to provide a fully homotopy-theoretic refinement of the ordinary notion of category. In particular, all the familiar constructions of category
theory have natural analogs in the context of quasi-categories. See for instance
The notion of quasi-categories were originally defined, under the name weak Kan complexes in:
The main theorem of Vogt (1973) involved a category of homotopy coherent diagrams defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of
(commutative) diagrams on the same category.
• J.-M. Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahiers de Top. Géom. Diff., 23, (1982), 93 –112,
defined the homotopy coherent nerve of any simplicially enriched category. This generalised the nerve of an ordinary category. In
• J.-M. Cordier and Tim Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65–90,
it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.
A systematic study of SSet-enriched categories in this context is in
The importance of quasi-categories as a basis for category theory has been particularly emphasized in work by André Joyal
For several years Joyal has been preparing a textbook on the subject which never became publically available, but an extensive writeup of lecture notes is:
Meanwhile Jacob Lurie, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models quasi-category
and simplicially enriched category is in
An overview of the material there is contained in
Textbook accounts:
The relation between quasi-categories and simplicially enriched categories was discussed in detail in
Further survey:
• Charles Rezk, Stuff about quasicategories, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 (pdf, pdf)
• Charles Rezk, Introduction to quasicategories (2022) [pdf, pdf]
• Moritz Groth, A short course on ∞-categories (arXiv:1007.2925)
An in-depth study of adjunctions between quasi-categories and the monadicity theorem is given in | {"url":"https://ncatlab.org/nlab/show/quasi-category","timestamp":"2024-11-15T00:48:57Z","content_type":"application/xhtml+xml","content_length":"44778","record_id":"<urn:uuid:b37ad072-669d-4a06-93f2-70c566583a3b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00442.warc.gz"} |
Aggregation Rolling
Aggregation Rolling#
group aggregation_rolling
struct range_window_bounds#
#include <range_window_bounds.hpp>
Abstraction for window boundary sizes, to be used with grouped_range_rolling_window().
Similar to window_bounds in grouped_rolling_window(), range_window_bounds represents window boundaries for use with grouped_range_rolling_window(). A window may be specified as one of the
1. A fixed-width numeric scalar value. E.g. a) A DURATION_DAYS scalar, for use with a TIMESTAMP_DAYS orderby column b) An INT32 scalar, for use with an INT32 orderby column
2. ”unbounded”, indicating that the bounds stretch to the first/last row in the group.
3. ”current row”, indicating that the bounds end at the first/last row in the group that match the value of the current row.
Public Types
enum class extent_type : int32_t#
The type of range_window_bounds.
enumerator CURRENT_ROW#
enumerator BOUNDED#
Bounds defined as the first/last row that matches the current row.
enumerator UNBOUNDED#
Bounds stretching to the first/last row in the entire group.
Bounds defined as the first/last row that falls within a specified range from the current row.
Public Static Functions
static range_window_bounds get(scalar const &boundary, rmm::cuda_stream_view stream = cudf::get_default_stream())#
Factory method to construct a bounded window boundary.
■ boundary – Finite window boundary
■ stream – CUDA stream used for device memory operations and kernel launches
A bounded window boundary object
static range_window_bounds current_row(data_type type, rmm::cuda_stream_view stream = cudf::get_default_stream())#
Factory method to construct a window boundary limited to the value of the current row.
■ type – The datatype of the window boundary
■ stream – CUDA stream used for device memory operations and kernel launches
A “current row” window boundary object
static range_window_bounds unbounded(data_type type, rmm::cuda_stream_view stream = cudf::get_default_stream())#
Factory method to construct an unbounded window boundary.
■ type – The datatype of the window boundary
■ stream – CUDA stream used for device memory operations and kernel launches
An unbounded window boundary object
struct window_bounds#
#include <rolling.hpp>
Abstraction for window boundary sizes.
Public Functions
inline bool is_unbounded() const#
Whether the window_bounds is unbounded.
true if the window bounds is unbounded.
false if the window bounds has a finite row boundary.
inline size_type value() const#
Gets the row-boundary for this window_bounds.
the row boundary value (in days or rows)
Public Static Functions
static inline window_bounds get(size_type value)#
Construct bounded window boundary.
value – Finite window boundary (in days or rows)
A window boundary
static inline window_bounds unbounded()#
Construct unbounded window boundary. | {"url":"https://docs.rapids.ai/api/cudf/nightly/libcudf_docs/api_docs/aggregation_rolling/","timestamp":"2024-11-05T06:03:49Z","content_type":"text/html","content_length":"188090","record_id":"<urn:uuid:c74f1a00-4c62-4fc2-80c8-66d62d3a532d>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00453.warc.gz"} |
6th grade probability formulas
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Cat vs Non-cat Classifier - Defining some utility functions - Model
Copy-paste the following code for the model function.
Call the initialize_weights function and optimize function at the appropriate places inside the model function.
def model(X_train, Y_train, X_val, Y_val, num_iterations=2000, learning_rate=[0.5]):
# Initialize weights and bias
w, b = << your code comes here >>(X_train.shape[0])
best_values = {
'final w':w,
'final b':b,
'Train accuracy':prev_train_acc,
'Validation accuracy': prev_val_acc,
for lr in learning_rate:
print(("-"*30 + "learning_rate:{}"+"-"*30).format(lr))
# Initialize weights and bias
w, b = << your code comes here >>(X_train.shape[0])
# Optimization
lr_optimal_values = << your code comes here >>(w, b, X_train, Y_train, X_val, Y_val, num_iterations, lr)
if lr_optimal_values['Validation accuracy']>prev_val_acc:
prev_val_acc = lr_optimal_values['Validation accuracy']
prev_train_acc = lr_optimal_values['Train accuracy']
final_lr = lr
final_w = lr_optimal_values['final w']
final_b = lr_optimal_values['final b']
final_epoch = lr_optimal_values['epoch']
final_Y_prediction_val = lr_optimal_values['Y_prediction_val']
final_Y_prediction_train = lr_optimal_values['Y_prediction_train']
best_values['Train accuracy'] = prev_train_acc
best_values['Validation accuracy'] = prev_val_acc
best_values['final_lr'] = final_lr
best_values['final w'] = final_w
best_values['final b'] = final_b
best_values['epoch'] = final_epoch
best_values['Y_prediction_val'] = final_Y_prediction_val
best_values['Y_prediction_train'] = final_Y_prediction_train
return best_values | {"url":"https://cloudxlab.com/assessment/displayslide/5393/cat-vs-non-cat-classifier-defining-some-utility-functions-model","timestamp":"2024-11-10T09:12:00Z","content_type":"text/html","content_length":"86540","record_id":"<urn:uuid:e363441d-7e25-40ef-b8a3-fd56ae41f0fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00476.warc.gz"} |
Daniel Murfet - MATH 33B | Bruinwalk
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June 19, 2011
Murfet tries. He really does try. And he really does care.
The good:
- Murfet is a very clear lecturer. You'll learn a lot from him.
- He's legitimately funny...most of the class actually laughs at his jokes. It'll probably keep you awake.
- 33B really isn't a very difficult class.
- He posts his lecture notes online, and they are BEAUTIFUL. This is a big plus since the textbook for this course is...well, atrocious.
The bad:
- Some of the homework sets take FOREVER. You'll understand the concept inside and out...with half the assignment left to complete.
- His tests were almost entirely mechanical, and they were out of 15, 22, and 39 points (for the two midterms and the final, respectively). A simply careless arithmetic error may cost you half a
point, or even a full point. While in other math classes you would get, say, 9/10 for the problem, here you get 1/2. It acts as an equalizer; if you understood the concept but made a silly mistake,
you may receive the same credit as someone who had no idea what they were doing and just wrote down some equations. This means that, unless Murfet changes his system, doing well on his tests doesn't
require thoroughly understanding everything, it just demands not being careless. I think that's a bad thing, but I'll leave it for you to judge.
My suggestions:
- TAKE MATH 33A BEFORE 33B. Independent of the professor you have, I cannot stress this enough!
- AVOID SILLY MISTAKES ON HIS TESTS. See "the bad." Make sure you get lots of sleep before them, pace yourself, etc.
Murfet is a pretty new professor, and I truly think he'll be great in the future, so I'm able to forgive him for his shortcomings this quarter. Maybe my evaluation will be irrelevant in a few years.
But this is what you should know here and now.
Good luck! :)
June 19, 2011
I did poorly on the final because I was sick and ended up with a C, but overall, his course is pretty easy. The exams are very fair and don't really force you to really understand the theory. During
lectures, he'll prove his techniques and all that, but you really only have to be able to vaguely understand how they work. In actuality, the exams just require you to mechanically subject the DE's
through several available techniques to find a solution. He seems like a funny guy and does his best at making this dry subject seem semi-interesting.
My favorite aspect was his website. He pretty much immediately puts up his lecture notes online. He also puts up practice midterms with solutions and he also puts up solutions to the actual midterms
(he usually only takes a day or two to grade his tests). That was all very helpful.
June 18, 2011
Funny professor, and awesome to hear his austrailan accent lol. He tells funny jokes and suddenly gave us a differential equations problem where we had to calculate the amount of alcohol inside his
system during his wedding as a function of time. He posts his lecture notes consistently and provides practice midterms/finals. His tests are very, very doable, so the curving isn't that great (which
means a few points gained may mean the difference between an A and A-, or an A- and B+, etc. Also, the tests are usually graded out of points such as 22 or so, with 0.5-1 point deductions so this
also keeps the curve tight. The textbook was actually pretty okay, and since he sticks to the book pretty closely, you can get buy with just the book. HW: 10%, Midterms (each 20%, or the best one is
30%); Final: (50%, or 60%). Two grading schemes. He's very understandable and has very good handwriting. I enjoyed his class (though I kept falling asleep, but I do that for every class, so...). He
also posts lecture notes about linear algebra stuff, the textbook chapter 7 has all you really need to know about matrices - I didn't take 33A - it's definitely doable (you don't need to know that
much bec. he doesn't emphasize theory a lot, just practice doing some). Take Murfet if you don't need/want to learn too much theory and if you want to be able to understand all that's going on in
class. If you like curved tests where you can wing it a little, don't take him bec. everyone else who's put in some effort and is careful during tests will do better than you.
June 13, 2011
Professor Murfet is a great professor, but he doesn’t really expect much from his students. His lectures are straightforward and easy to follow, and really break down the concepts presented in the
textbook. Though I can’t understand the unusually low average midterm and final scores, the midterms and final were both very easy; most of the questions were just homework problems with the numbers
changed around. I was somewhat slightly disappointed that Professor Murfet was more concerned with just solving differential equations, than actually understanding how the techniques work. You don’t
even have to know the theory behind it, just know the algorithms to solve the equations, but I guess that’s what makes the course very easy. As for his personality, he occasionally throws in a joke
or a story he wanted to share. Coupled with his Australian accent, he’s an amusing man. Makes it worthwhile to go to class.
Another thing: the textbook is not a first timer's book. Go to class. The textbook skips so many steps it’s almost impossible to self teach from the book. If you didn’t take 33A and you decide to
skip lecture, then linear systems will be VERY confusing.
June 12, 2011
Overall Murfet was a fair (maybe almost too fair on the easy side) professor and he taught Math 33B as effectively as any professor could have done. What also made this class more pleasant was
listening to his awesome Australian accent and getting a kick out of every time he said “zet” instead of “z” and “ashume” instead of “assume.” The guy has really neat handwriting (which is a big plus
in my book) and actually takes the time to rewrite the atrocity of a differential equations textbook (Differential Equations 2nd Edition, by Polking, great heavens to betsy this book was impossible
to use as reference, Shame On You Math Department!) into condensed lecture notes that was lucid (most of the time) and easy to understand. On top of that, our midterms were graded extremely quickly
and the scores were updated on myucla the day we took the midterms, so that was super duper. And his jokes were ok (6 out of 10).
Now enough of why this class was awesome onto why this class was lame. First of all this class was way too easy. I don’t know if it’s because an introductory class to ordinary differential equations
is suppose to be so simple, but wow, almost all the things that we learned in this class was a bunch of Simple Simon stuff. Now I’m not saying I got the grades, but what the midterms come down to is
the speed and accuracy at which you can perform various techniques in solving the differential equations within the 50 minutes allotted. And these techniques are super long and super tedious like
your cousin’s wedding. Also, the homeworks take forever. When you get into 2nd order DEs, there was this one homework that took like 9 F-ing hours to complete. I wouldn’t have minded if the homework
was somewhat enjoyable but this class is boring, man. Basically all this class was menial computation.
So how do you do well? Murfet tells us to do the practice exam problems and redo the homework problems to practice for the midterms and the final. And guess what? The final and midterms had mostly
questions that were very similar (he even jacked a problem from the book if I remember correctly) from the homeworks that he assigned us. So this makes the final and midterm ridiculously easy if you
memorized or are able to derive all the techniques and ideas from the topics covered. There are no curveballs or super hard millennium prize problems that try to separate the math ninjas from the
normal people, so don’t trip. As long as you know how to do the homework problems and know what he expects you to know, you’ll get a good grade. If you’re one of those guys who likes to cut corners
but tries somewhat you may also get a good grade. If you’re one of those guys that skip homework all the time and knock out in class, and procrastinates to kingdom come, you will probably get a bad
grade. Oh yeah, and the TA’s were ok. Peace.
June 10, 2011
Professor Murfet is quite dedicated in his teaching. I mean, he spends a lot of time taking material from the book and rewriting them into more readable lecture notes for every lecture. That being
said, he follows the book very closely and he puts his lecture notes online anyway so if you read through the book, you can get a decent grade without going to any of his lectures. | {"url":"https://bruinwalk.com/professors/daniel-murfet/math-33b/","timestamp":"2024-11-04T23:28:05Z","content_type":"text/html","content_length":"94640","record_id":"<urn:uuid:813df5d9-b39a-495d-91fe-4593f4f02abc>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00066.warc.gz"} |
Intermediate Algebra with Analytic Geometry
Language : English
Publication Date : 30/07/2016
Format : Softcover
Dimensions : 8.5x11
Page Count : 180
ISBN : 9781524523473
About the Book
This book is designed for students who need additional preparation for math classes, at North Park University, numbered 1020 or higher. The book covers all the topics that are required by the school
which are: topics in beginning and intermediate algebra such as: equations and inequalities, systems, polynomials, factoring, graphing, roots and radicals, rational functions, quadratic equations,
and conic sections. Most of the topics covered in this book are applied in our daily life; the proof is in the application sections. For some students this is the course they need to take, for others
just the first in series of many depending on students' major. The text explains the method of solving problems step by step, and shows more than one method to solve a problem, also shows how to
solve problems graphically and using technology (TI) so the student will learn how to solve problem algebraically, then see the solution graphically. Most of the chapters are ended with application
section to learn how to apply the topic to real life situation. Also, at the end of each chapter there is a chapter exercise, and a chapter test, students can use the chapter exercise as a study
guide for test. My goal of writing this book is to make it easy for students to read through each page and doesn't overwhelm, or complicate Intermediate Algebra, but gives the skills and practice
that the need. | {"url":"https://www.xlibris.com/en-gb/bookstore/bookdetails/744343-intermediate-algebra-with-analytic-geometry","timestamp":"2024-11-05T00:19:39Z","content_type":"text/html","content_length":"49888","record_id":"<urn:uuid:f36297d2-cbc5-4176-9c44-87f44ede6cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00647.warc.gz"} |
HTML5 Canvas Pie Chart
Posted inCanvas HTML5 Javascript
HTML5 Canvas Pie Chart
The HTML5 canvas element certainly opens a lot of possibilities for web developers. Inspired by Arne @ 23inch.de’s JS1K.com submission I thought I would explore the canvas elements capabilities by
creating a simple javascript function that generates a pie chart.
The Canvas
First we need a canvas element.
<canvas id="c"> Your browser does not support the canvas element. </canvas>
Next we need the script that will draw the Pie Chart. I decided to use two functions; one for the wedge and the other to to loop through the data.
The Wedge Function
I adapted Arne’s function for drawing arcs to draw wedges.
function W(x,y,r,u,v) {
if(r) {
a.arc(x, y , r, (u||0)/50*Math.PI, (v||7)/50*Math.PI, 0);
} else {
a.fillStyle = '#'+'a000b4bbff95'.substr(x,3);
return W;
x and y are the coordinates of the charts center. r is the radius of the chart. u is the startAngle and v is the stopAngle in percent. Wedge color aka fillStyle is set by x when r is missing. The
colors available are interlaced. The function returns itself so that it can be call likeĀ W(0)(100,100,50,0,25);
The Chart
Next I needed a function that would use the wedge function the draw each wedge of the pie chart.
function pie(r,b,n) {
a.font="bold 12pt Arial";
var u = 0;
var v = 0;
for (i=0;i < b.length;i++) {
Again r is the radius and is used to calculate the canvas’ height and width and variables x and y. b is an array of the percents and n is an array of the names used for the charts key. The function
loops through the elements of the array, drawing each wedge and key. It is assumed that the elements of b total 100.
Bringing the pieces together
Lastly, create the needed arrays and call the pie function which in turn calls the wedge function.
var n = ['RED','BLACK','BLUE','GREEN','PINK']; //key names for each wedge
var b = [20,25,15,10,30]; //percents, needs to total 100
Again it is assumed that b’s elements total 100 and that n and b have the same number of elements.
What it looks like
So if you are using an HTML5 compliant browser you get this:
4 Comments
1. For any developer that consider readability and maintainability important I refactored the code:
function pieChart(canvas, radius, percentages, labels, colors) {
var canvas = document.getElementById(canvas)
var context = canvas.getContext("2d");
canvas.width = 3.5*radius;
canvas.height = 2.5*radius;
var cx, cy;
cx = cy = canvas.height/2;
context.font = "bold 12px Arial";
var start_rad = 0;
var end_rad;
var end_percentage = 0;
for (i = 0; i < percentages.length; ++i) {
end_percentage += percentages[i];
end_rad = end_percentage / 100 * 2 * Math.PI;
context.fillStyle = colors[i];
drawWedge(context, cx, cy, radius, start_rad, end_rad);
start_rad = end_rad;
context.fillText(labels[i], cx + radius + 10, cy - radius/2 + i*18);
function drawWedge(context, cx, cy, radius, start_rad, end_rad) {
context.arc(cx, cy , radius, start_rad, end_rad, false);
2. Hey Ruben,
This function is no longer working in the latest version of Chrome. Any ideas what’s up with it?
□ I can confirm that on Chrome Version 26.0.1410.43 m the wedges are not drawn. It goes back to Magnus’ comment about “readability and maintainability”. The code for the wedge a.beginPath() |
a.fill(a.moveTo(x,y)|a.arc(x,y,r,(u||0)/50*Math.PI,(v||7)/50*Math.PI,0)|a.lineTo(x,y)) is the issue; probably cause the pipes. I updated the function to this function W(x,y,r,u,v) {
if(r) {
a.arc(x, y , r, (u||0)/50*Math.PI, (v||7)/50*Math.PI, 0);
} else {
a.fillStyle = '#'+'a000b4bbff95'.substr(x,3);
return W;
} and it now works.
Edit: Changing the actual wedge to a.beginPath() | a.moveTo(x,y) | a.arc(x,y,r,(u||0)/50*Math.PI,(v||7)/50*Math.PI,0) | a.lineTo(x,y) | a.fill() works too.
3. Ah Sweet! Thanks a lot Reuben – Works great again now. | {"url":"https://reubencrane.com/2012/10/html5-canvas-pie-chart/","timestamp":"2024-11-14T01:49:30Z","content_type":"text/html","content_length":"52293","record_id":"<urn:uuid:95cfa1cc-ee21-40d0-b697-c6c81426a3a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00600.warc.gz"} |
Data Structure Selection Sort - Tutoline offers free online tutorials and interview questions covering a wide range of technologies, including C, C++, HTML, CSS, JavaScript, SQL, Python, PHP, Engineering courses and more. Whether you're a beginner or a professional, find the tutorials you need to excel in your field."
After the first iteration of selection sort, the smallest element will be at the beginning of the list. In the second iteration, the algorithm will find the second smallest element from the remaining
unsorted portion and swap it with the second element of the list. This process continues until the entire list is sorted.
One advantage of selection sort is that it performs well on small lists or lists with a small number of elements. Its time complexity is O(n^2), which means that the number of comparisons and swaps
increases quadratically as the number of elements in the list increases. However, selection sort is not recommended for large lists or lists with a large number of elements, as it can be quite slow
compared to more efficient sorting algorithms such as merge sort or quicksort.
Another important aspect of selection sort is that it is an in-place sorting algorithm, which means that it does not require any additional memory to perform the sorting. The algorithm only needs to
keep track of the minimum (or maximum) element and the position where it should be swapped. This makes selection sort a good choice when memory space is limited.
Although selection sort is not the most efficient sorting algorithm, it is still widely used in certain situations. For example, selection sort can be useful when the list is almost sorted or when
the list contains a small number of elements. In these cases, the simplicity and ease of implementation of selection sort outweigh its relatively slower performance.
Overall, understanding selection sort is important for any programmer or computer science student. It provides a foundational understanding of sorting algorithms and their efficiency. By studying
selection sort, one can gain insights into how sorting algorithms work and how to optimize them for different scenarios.
How Selection Sort Works
The selection sort algorithm can be broken down into the following steps:
1. Find the minimum (or maximum) element in the unsorted portion of the list.
To find the minimum element, the algorithm starts by assuming that the first element in the unsorted portion is the minimum. It then compares this assumed minimum element with the rest of the
elements in the unsorted portion. If it finds an element that is smaller than the assumed minimum, it updates the minimum element to be the new smallest element found. This process continues
until the algorithm has iterated through all the elements in the unsorted portion and found the true minimum element.
2. Swap the minimum (or maximum) element with the first element of the unsorted portion.
Once the minimum element is found, the algorithm swaps it with the first element in the unsorted portion. This ensures that the minimum element is placed in its correct position at the beginning
of the sorted portion of the list.
3. Move the boundary of the sorted portion one element to the right.
After swapping the minimum element with the first element of the unsorted portion, the algorithm moves the boundary of the sorted portion one element to the right. This means that the sorted
portion now includes one more element, while the unsorted portion is reduced by one element.
4. Repeat steps 1-3 until the entire list is sorted.
The algorithm continues to repeat steps 1-3 until the entire list is sorted. This is done by progressively reducing the size of the unsorted portion and expanding the sorted portion with each
iteration. Eventually, the unsorted portion becomes empty, and the sorted portion encompasses the entire list, indicating that the list is fully sorted.
By following these steps, the selection sort algorithm efficiently sorts a list by repeatedly finding the minimum (or maximum) element and placing it in its correct position within the sorted portion
of the list.
Selection sort is a simple sorting algorithm that works by repeatedly finding the minimum element from the unsorted portion of the list and placing it at the beginning of the sorted portion. The
process continues until the entire list is sorted.
In the given example, we start with an unsorted list of numbers: 7, 3, 9, 2, 5. The algorithm begins by finding the minimum element in the unsorted portion, which is 2. It then swaps this element
with the first element of the unsorted portion, resulting in an updated list: 2, 3, 9, 7, 5. The boundary of the sorted portion is moved one element to the right, and the process is repeated for the
remaining unsorted portion.
In the next step, the minimum element in the unsorted portion (from index 1 to 4) is found to be 3. It is then swapped with the first element of the unsorted portion, resulting in an updated list: 2,
3, 9, 7, 5. The boundary of the sorted portion is moved one element to the right again.
This process continues until the entire list is sorted. The minimum element in the unsorted portion (from index 2 to 4) is found to be 5, which is then swapped with the first element of the unsorted
portion. The updated list becomes: 2, 3, 5, 7, 9. The boundary of the sorted portion is moved one element to the right, and the process is repeated for the remaining unsorted portion.
Finally, the minimum element in the unsorted portion (from index 3 to 4) is found to be 7, which is already in its correct position. The boundary of the sorted portion is moved one element to the
right, and the process is repeated for the remaining unsorted portion (from index 4 to 4). Since the entire list is now sorted, the algorithm terminates.
Selection sort has a time complexity of O(n^2), where n is the number of elements in the list. It is not the most efficient sorting algorithm for large lists, but it is simple to understand and
Despite its time complexity, selection sort still has some advantages over other sorting algorithms. One advantage is that it is easy to understand and implement. The algorithm is straightforward and
does not require any complex data structures or additional memory. This makes it a good choice for small lists or for educational purposes.
In addition, selection sort has a relatively small constant factor compared to other quadratic time complexity algorithms like bubble sort or insertion sort. This means that, in practice, selection
sort can sometimes outperform these algorithms for small to medium-sized lists.
However, selection sort’s main drawback is its time complexity. As the number of elements in the list increases, the number of comparisons and swaps also increases exponentially. This makes selection
sort highly inefficient for large lists or datasets.
For example, suppose we have a list of 1000 elements. The number of comparisons required by selection sort would be approximately 500,500 (1000 * 999 / 2), and the number of swaps would also be 1000.
In comparison, a more efficient algorithm like merge sort or quicksort would require significantly fewer comparisons and swaps to sort the same list.
Therefore, it is generally recommended to use selection sort only for small lists or for educational purposes. For larger lists or datasets, more efficient sorting algorithms should be used to
minimize the time complexity and improve performance. | {"url":"https://tutoline.com/data-structure-selection-sort/","timestamp":"2024-11-03T13:27:01Z","content_type":"text/html","content_length":"192267","record_id":"<urn:uuid:ab622bfa-49b1-4de5-ac86-7cbcf895e50b>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00279.warc.gz"} |
Math, particularly multiplication, creates the keystone of various academic disciplines and real-world applications. Yet, for many learners, grasping multiplication can posture a difficulty. To
address this hurdle, instructors and moms and dads have actually embraced a powerful device: Multiplication Worksheets Single Digit.
Intro to Multiplication Worksheets Single Digit
Multiplication Worksheets Single Digit
Multiplication Worksheets Single Digit -
Multiplication facts worksheets including times tables five minute frenzies and worksheets for assessment or practice In each box the single number is multiplied by every other number with each
question on one line The tables may be used for various purposes such as introducing the multiplication tables skip counting as a lookup table
Single Digit Multiplication Worksheets Generator Here is our random worksheet generator for free multiplication worksheets You can generate a range of single digit multiplication worksheets ranging
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Single Digit Multiplication Worksheet Worksheet For 3rd 5th Grade Lesson Planet
Single Digit Multiplication Worksheet Worksheet For 3rd 5th Grade Lesson Planet
Children will use star arrays to solve 11 single digit multiplication problems 3rd grade Math Worksheet Hundreds Chart Worksheet Our one digit multiplication worksheets take the guessing out of
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Multiply in columns Students multiply numbers up to 1 000 by single digit numbers in these multiplication worksheets Regrouping carrying will by required for most questions Worksheet 1 Worksheet 2
Worksheet 3 Worksheet 4 Worksheet 5 10 More
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Word Trouble Worksheets
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Advantages of Using Multiplication Worksheets Single Digit
Multiplication Worksheet Printable The Happy Housewife Home Schooling
Multiplication Worksheet Printable The Happy Housewife Home Schooling
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digit multiplication lisazachry Member for 2 years 3 months Age 7 14 Level 2nd Language English en
Multiplication Single Digit Number of Problems 10 problems 16 problems 20 problems 30 problems 50 problems There may be fewer problems depending on what is selected below To create a worksheet with
problems using the numbers 1 through 5 select the numbers 1 through 5 from both lists and click Create It
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Just How to Create Engaging Multiplication Worksheets Single Digit
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Double Single Digit Multiplication Set 1 Worksheet For 3rd 4th Grade Lesson Planet
Multiplication Double digit X single digit 10 Printable Worksheets pdf Year 3 4 5 6 Grade 3
Check more of Multiplication Worksheets Single Digit below
Free 1 Digit Multiplication Math Worksheet Multiply By 2 Free4Classrooms
Single Digit Multiplication 25 Problems On Each Worksheet Three Worksheets FREE Printable
Free 1 Digit Multiplication Worksheet Free4Classrooms
Single Digit Multiplication Games Worksheets
Single Digit Multiplication Math Worksheet
Printable Multiplication Single Double Digit Worksheet Class Playground
Single Digit Multiplication Worksheets Math Salamanders
Single Digit Multiplication Worksheets Generator Here is our random worksheet generator for free multiplication worksheets You can generate a range of single digit multiplication worksheets ranging
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Dynamically Created Multiplication Worksheets Math Aids Com
These multiplication worksheets are appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade 1 3 or 5 Minute Drill Multiplication Worksheets Number Range 0 12 A timed drill
is a multiplication worksheet with all of the single digit multiplication problems on one page
Single Digit Multiplication Worksheets Generator Here is our random worksheet generator for free multiplication worksheets You can generate a range of single digit multiplication worksheets ranging
from 1 digit x 1 digit up to 5 digits x 1 digit You can also tailor your worksheets further by choosing exactly what digits you want to multiply
These multiplication worksheets are appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade 1 3 or 5 Minute Drill Multiplication Worksheets Number Range 0 12 A timed drill
is a multiplication worksheet with all of the single digit multiplication problems on one page
Single Digit Multiplication Games Worksheets
Single Digit Multiplication 25 Problems On Each Worksheet Three Worksheets FREE Printable
Single Digit Multiplication Math Worksheet
Printable Multiplication Single Double Digit Worksheet Class Playground
Single Digit Multiplication 8 Worksheets FREE Printable Worksheets Worksheetfun
Free Printable Double Digit Multiplication Worksheets Lexia s Blog
Free Printable Double Digit Multiplication Worksheets Lexia s Blog
Multi Digit Multiplication Worksheets Pdf Times Tables Worksheets
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Encouraging constant technique, supplying assistance, and developing a positive understanding setting are beneficial actions. | {"url":"https://crown-darts.com/en/multiplication-worksheets-single-digit.html","timestamp":"2024-11-06T11:39:20Z","content_type":"text/html","content_length":"29441","record_id":"<urn:uuid:a9f195c5-b4f1-42bc-9d5b-0b950e87ad85>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00420.warc.gz"} |
The Stacks project
Lemma 42.61.3. Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $c \in A^ p(X \to \mathop{\mathrm{Spec}}(k))$. Let $Y \to Z$ be a morphism of schemes locally of finite
type over $k$. Let $c' \in A^ q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times _ k Y \to Z)$.
Comments (2)
Comment #7547 by Hao Peng on
Shouldn't $c\circ c^\prime$ lie in $A^{p+q}(X\times_kY\to Z)$?
Comment #7671 by Stacks Project on
Good catch! Thanks. Fixed here.
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Amps vs Volts vs Watts (Definition, Symbol, Unit of,...)
Amps vs Volts vs Watts (Definition, Symbol, Unit of,…)
Written by Edwin Jones / Fact checked by Andrew Wright
Do you count yourself among the many folks out there that wonder why we have gadgets rated for 20 amps or 50 amps, 12V batteries, and 50W solar panels? If yes, then this article pitting amps vs volts
vs watts should enlighten you about these three electrical units once and for all.
It’s not so much a “truel” as a detailed explanation of how all three work together to ensure electricity will properly flow in a circuit, the pros/cons of each one, and how they’re generally
different from one another.
What Are Amps?
Amps or amperes measure the amount of electricity flowing in a circuit. To be exact, it’s a measurement of how many electrical charges are flowing past a given point per second. It measures
electrical current, in short.
Don’t be confused about the difference between ‘ampere’ and ‘amperage’. Although directly related, the latter is often used to refer to the measured value of current for a given device or appliance
in use.
For example, you bought an incandescent lamp with an amperage of 14. The label on the device will point to the exact measurement reflected as amperes. In turn, it will need at least a 15-ampere,
15-amp, or 15A (not amperage!) circuit breaker.
For charging batteries in gadgets, we also use amps, but more in the context of how quickly a device will charge, with 1A being slower than a higher 4A charging circuit.
Battery capacity, on the other hand, is represented by Ah or how many amps the battery is able to deliver per hour. Keep that in mind when considering battery amps vs volts on the specs of batteries
you purchase.
We calculate amperes with the formula:
\text{A} = \frac{\text{W}}{\text{V}}
What Are Volts?
Voltage is the electrical pressure provided by the circuit’s power source from its negative terminals to its positive terminals. Interestingly, most people readily attribute voltage to electricity
more than the other two units here.
Make no mistake! To say that it’s “pressure” is correct but not exactly accurate. It can be explained in more scientific terms as “the line-integral of an electric field’s intensity” or “an electric
field’s electric potential measured between the two terminals”.
Explaining those complex definitions further can be likened to herding cats by most folks and will need a dedicated post, so we’d like to use the volts vs amps water analogy instead.
Consider how a plumbing system works. Voltage is similar to the water pressure needed to get the water flowing. How quickly the water flows relates, in turn, to the current or amperes.
What Are Watts?
Once we mix amps and volts together, we get watts. A more technical representation will be the simple formula:
\text{Power (Watt)} = \text{Current (Amp)} \times \text{Voltage (Volt)}
This makes sense, since going back to the water analogy, increasing either the flow rate (ampere) or applying more pressure (voltage) leads to one thing: more power generated. In this case, watts
represent the energy or power delivered for a given time.
Which one is more efficient, though? Volts higher than amps or amps higher than volts? The former will always be more efficient.
Doubling the voltage while the current remains the same allows you to deliver 200% the usual load and even use the same wire, as long as the wire is still working within its ampacity rating.
Are volts and watts the same? Obviously, no, since volts relate more to the metaphorical electric pressure stated above, while watts measure the energy expended relative to time.
That’s just one difference, though. Let’s tackle all of them in the next section below.
Main Differences Between Amps, Volts, and Watts
I thought it would be better to use a comprehensive comparison table rather than compare voltage vs wattage, voltage vs. amperage, etc.
Comparison Factor Amps Volts Watts
Definition Rate of flow Amount of pressure Amount of energy used
Symbol A or I V W or P
Unit of Electrical current Electromotive force and potential energy Power
Measuring tool Ammeter or multimeter Voltmeter or multimeter Wattmeter or multimeter
Pros and Cons of These Electrical Units
• It’s easier to read volts than watts.
• Amps are less of a hassle to measure than watts.
• Amps exert more influence on the risk of an electrical shock than voltage.
• Given their straightforward formulas, you often only need to punch a few numbers on a standard calculator to get the correct figures for these units, especially if you’re working on a DC circuit.
• Measuring these units makes it easier to ensure that your electrical systems, circuits, and the gadgets you plug into your outlets will function properly and safely.
• You can’t get wattage without knowing voltage and ampere. Hence, we can’t calculate kWh or our homes’ power consumption – with a formula of kWh = (W x hours) / 1,000) – without knowing all three!
Don’t Forget About Ohms!
In truth, this volts vs amps vs watts discussion should actually be amps vs. volts vs. watts vs ohms. Resistance, whether present or absent in a circuit, will and should always be accounted for,
unless you have a short circuit. Ohms measure the resistive force that opposes the current, after all.
Still, it’s understandable, since discussing watts vs volts vs amps almost always arrives at the fact that all three work together in a circuit. Ohms is the odd duck since, as Ohm’s Law states, it’s
inversely proportional to current.
Frequently Asked Questions
Which is more powerful volts or amps?
Taken literally, the answer for this should be neither since “power” will always be tied to watts before anything else.
If we’re talking more about how dangerous or even life-threatening either one is, then amps are bound to trump volts. Even slight increases in amps can heighten the risk of death once you get
electrocuted, after all.
How many volts are in an amp?
We can’t answer this question without knowing the ohms (with regard to Ohm’s Law) or watts and the fact that volts and amps are separate units.
In a circuit with 1A and 100W, there’s 100V, following the formula for voltage mentioned above.
The same goes if we change the 100W to 100Ω instead; in this case, we have to follow Ohm’s Law as represented by the formula:
\text{V} = \text{I} \times \text{R}
Are watts and amps the same?
Though closely related, they are not interchangeable. Wattage measures electrical power, while ampere relates more to the volume of electrons flowing through a circuit or its current, in short.
Once you get the hang of all three, all the confusion related to amps vs volts vs watts that tends to spring up in your mind should be dispelled for good.
Though appearing to be cut from the same cloth, it doesn’t take long to see that these units are not even subtly different. They’re used to measure current, pressure or potential energy, and
electrical power, respectively.
Leaving out either one will make energy calculations and circuit breaker sizing a tough chore — that’s for certain.
I am Edwin Jones, in charge of designing content for Galvinpower. I aspire to use my experiences in marketing to create reliable and necessary information to help our readers. It has been fun to work
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Math Takes Time - MANGO Math Group
Math Takes Time
How much time does your school allocate towards math? 50 minutes, 60 minutes? More? Less? This difference may not seem that significant but those ten minutes in a day become 50 minutes in a week and
if they attend school 180 days it becomes 30 hours less math time than those teachers that provide math instruction and skill building for 60 minutes.
These times fluctuate depending on your school district but on average according to the Department of Education Stats in Brief, students spend an hour in math and two hours in reading every day. To
put this into a year perspective, a student receives 37 ½ more days in reading instruction than they do in mathematics. Add that to the thirteen years from Kindergarten through graduation that
students spend in school, and 487 ½ more days will be spent in reading than math. An entire year more of reading instruction.
You may wonder what I am trying to get at, because reading IS important. And I’m not saying that it isn’t, but studies show math is the single strongest indicator of future academic success and that
students who spend more time in math instruction do better in reading. Research by UC Irvine looking at 20,000 Kindergarten students shows that math skills turn out to be a key predictor for future
academic success. “Math and reading skills at the point of school entry are consistently associated with higher levels of academic performance in later grades. Particularly impressive is the
predictive power of early math skills, which supports the wisdom of experimental evaluations of promising early math intervention.”
In the last decade, educators have focused on boosting literacy skills among low-income kids in the hope that all children will read well by 3rd grade. But the math skills of these students haven’t
received the same attention. Researchers say many high poverty kindergarten classrooms don’t teach enough math and the few lessons in math are often too basic.
During the last school year, only 40 percent of fourth-graders nationwide scored at a proficient level in national math assessments. In breaking that down further, only 26 percent of
Hispanic-Americans and 19 percent of African-Americans tested proficient in math. This is significant because strong math skills are needed for some of the fastest growing industries in our country.
“Understanding and being able to work with numbers is a fundamental skill for success in almost any occupation you might choose,” said economist Greg J. Duncan of the University of California
Irvine’s School of Education, whose research examines child poverty and education. “It leads to the analytic, higher-level thinking that’s increasingly important.”
What can be done to increase time in math and to also improve the quality of math instruction? CREATE QUALITY MATH MOMENTS.
MANGO MATH has created a math calendar that provides daily math problems for students to spend 5 minutes on as they walk in the door and get settled in their day. Allow the students to communicate
and collaborate when they are solving these problems as this deepens student’s math understanding. Students use of the calendar moves them from their daily work that involves mastering basic
computational skills and number concepts to more complex ideas and mathematical reasoning, including problem solving.
[dt_sc_button type="with-icon" link="/math-takes-time/#download" size="medium" bgcolor="#e68721" target="_self"]Download Calendar[/dt_sc_button]
Incorporate math stories into reading times
There is a great series of math picture books for grades PreK – 4th grade by Stuart J Murphy. These stories start with simple number sense ideas like patterns, opposites, comparing, sequencing and
counting. The books extend from there to equivalent values, fractions, percentages, and negative numbers. Each book has a story line that emphasizes and explains the concept. Here are some other
“math” books that explain math concepts to students in a way to direct curiously about numbers.
You don’t need to stop at 4th grade; there are some great math based books for middle schools students as well. My favorite has been the Number Devil that explores Fibonacci numbers. These books help
to explore math in a more in-depth way.
Incorporate Math and writing
This pertains in particular to poetry and music lyrics. Poetry and music are about rhythm and tempo, both of which involve math. Create a poem or song that follows a famous math pattern like Pi –
3.14159265… the Fibonacci sequence – 0, 1, 1, 2, 3, 5, 8, … each number in the pattern is the number of words/letters in the poem.
The downloadable math lesson below involves symbols used by the Japanese to represent poetry as far back as A.D. 1000.
[dt_sc_button type="with-icon" link="/math-takes-time/#download" size="medium" bgcolor="#e68721" target="_self"]Download Math Lesson[/dt_sc_button]
MANGO Math creates lots of moments in classrooms. Where there is a moment of time, be it before a recess, before the students go home, or after lunch when students need to get back to thinking about
school, MANGO Math Kits provide quick-to-implement games. Check out our website for more information.
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CSCE 781
CSCE 781: Knowledge Systems (Spring 2013)
This site is under construction
Prerequisites: CSCE 580
Meeting time and venue: TTh 930-1045 in SWGN 2A18
Instructor: Marco Valtorta
Office: Swearingen 3A55, 777-4641
E-mail: mgv@cse.sc.edu
Office Hours: TBD, or by previous appointment.
Grading and Program Submission Policy
Reference materials: No textbook is required for this course. Readings and notes will be used. Here are some key resources:
David Poole and Alan Mackworth. Artificial Intelligence: Foundations of Computational Agents. Cambridge University Press, 2010. (referred to as [P]) The full book with slides, etc. is available
David Stuart Russell and Peter Norvig. Artificial Intelligence: A Modern Approach. Prentice-Hall, 2003 ( [AIMA] or [R] or [AIMA-2]; a third edition is also available).
Ronald Brachman and Hector Levesque. Knowledge Representation and Reasoning. Morgan-Kaufmann, 2004.
Frank van Harmelen, Vladimir Lifschitz, and Bruce Porter (eds.). Handbook of Knowledge Representation. Elsevier, 2007.
Hector J. Levesque. Thinking as Computation: A First Course. MIT Press, 2012. Supplements, including a full set of slides are available online.
Uwe Schoening. Logic for Computer Scientists. Birkhauser, 1989.
Ann Yasuhara. Recursive Function Theory and Logic. Academic Press, 1971. As a result of taking this course, a student will be able to:
• Represent domain knowledge about objects using propositions and solve the resulting propositional logic problems using deduction and abduction
• Represent domain knowledge about individuals and relations in first-order logic
• Do inference using resolution refutation theorem proving
• Represent knowledge in Horn clause form and use Prolog (or a dialect thereof) for reasoning
• Represent knowledge for specialized task domains, such as diagnosis and troubleshooting
Introductory lecture
Notes about student interests from the first meeting
The Monkey and Bananas problem represented using the situation calculus
Notes on the propositional calculus (2013-01-31): proof of the converse of the deduction theorem, some syntactic proofs, semantics of the propositional calculus, equivalent formulas, the substitution
theorem (statement and example)
Notes on the propositional calculus (2013-02-05): soundness and consistency of the propositional calculus.
Notes on the propositional calculus (2013-02-07): completeness of the propositional calculus.
Notes on the propositional calculus (2013-02-12): the compactness theorem. Axiomatic presentation of the (first-order) predicate calculus ("first-order logic"), with five axioms and two rules of
inference. Conjunctive normal form (CNF), propositional case. The resolution rule (propositional case; started).
Notes on the propositional calculus from the lecture of 2013-02-14.
Notes on the propositional calculus (normal forms, propositional resolution) from the lecture of 2013-02-19.
Notes on the propositional calculus (normal forms, propositional resolution, Horn formulas) from the lecture of 2013-02-21. Note: this includes the notes from 20130219.
Another example of proof by resolution refutation from the lecture of 2013-02-26. Note: this includes the notes from 2013-02-19 and 2013-02-21.
Notes used in the lecture of 2013-03-05, including MT1 correction.
Notes used in the lecture of 2013-03-07, on the semantics of the predicate calculus and on the student project and presentations.
Notes used in the lecture of 2013-03-19, on the semantics of the predicate calculus.
Notes used in the lecture of 2013-03-21, on the semantics of the predicate calculus.
Notes used in the lecture of 2013-03-28, on the semantics of the predicate calculus and normal forms.
Notes used in the lecture of 2013-04-02, on Skolemization, a revised grading scale, and a schedule of student presentations.
Notes used in the lecture of 2013-04-04, with a summary example of conversion to clausal CNF, as will be used for resolution in the predicate calculus case. (These notes include the notes from
Slides for FOL resolution refutation. used (partly) on 2013-04-09.
Notes used in the lecture of 2013-04-11, with a few examples of resolution refutation proofs, including an example in which either non-binary resolution (as presented in Schoening) or factoring is
Notes used in the lecture of 2013-04-16, with an example of a resolution proof from group theory.
Notes on Bayesian networks used in the lecture of 2013-04-25.
Graduate Student Presentations
The USC Blackboard has a site for this course.
Prolog Information
Some useful links: | {"url":"https://cse.sc.edu/~mgv/csce781sp13/index.html","timestamp":"2024-11-14T11:35:09Z","content_type":"text/html","content_length":"16035","record_id":"<urn:uuid:07fbac79-6bbe-40c4-9e5b-ba8811128543>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00316.warc.gz"} |
Adjusting for Absenteeism and Turnover in context of how to calculate labor productivity
27 Aug 2024
Title: Adjusting for Absenteeism and Turnover: A Critical Step in Calculating Labor Productivity
Labor productivity is a crucial metric in evaluating the efficiency of an organization’s workforce. However, calculating labor productivity can be challenging when faced with issues like absenteeism
and turnover. This article provides a comprehensive guide on how to adjust for absenteeism and turnover when calculating labor productivity.
Labor productivity is defined as the ratio of output produced by an organization to the total labor hours worked (1). However, this calculation assumes that all labor hours are productive, which may
not be the case. Absenteeism and turnover can significantly impact labor productivity, making it essential to adjust for these factors.
Calculating Labor Productivity:
The basic formula for calculating labor productivity is:
Labor Productivity = Total Output / Total Labor Hours
Related articles for ‘how to calculate labor productivity’ :
Calculators for ‘how to calculate labor productivity’ | {"url":"https://blog.truegeometry.com/tutorials/education/d5a03a56796ec665ae83b18f348daf03/JSON_TO_ARTCL_Adjusting_for_Absenteeism_and_Turnover_in_context_of_how_to_calcul.html","timestamp":"2024-11-12T06:48:10Z","content_type":"text/html","content_length":"16494","record_id":"<urn:uuid:fff13936-f29d-43dc-914e-84354c343eb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00034.warc.gz"} |
Mid Term QuestionsQuizwiz - Ace Your Homework & Exams, Now With ChatGPT AI
Mid Term Questions
Ace your homework & exams now with
An annuity will be deposited to an account every year that pays 12.5 percent rate of interest. After 11 annual deposits there will be $643,944 in the account. The present value of the annuity is
You will receive $2,895 at year 89. What is its present value at the beginning of year 86 if the rate of interest is 4 percent?
You deposit $289 in an account every month for 17 months that paid 4% rate of interest compounded monthly. The present value of the annuity is
If the interest rate is 13%, how many years will it take to triple your investment?
Consider a corporate bond maturing on April 14, 2025 with a coupon rate of 5% and yield of 6%. What is the clean price of the bond on July 23, 2018?
Consider a corporate bond maturing on April 14, 2025 with a coupon rate of 5% and yield of 6%. What is the dirty price of the bond on July 23, 2018?
If the simple interest rate is 5 percent, how many years will it take to triple your investment?
40.00 years
Which investment option is less risky for one-year investment horizon: 1 year T-bill or 20-year T-bond?
1 year T-Bill
What is the future value of $756 at the end of 11 years, assuming continuously compounded interest rate of 9.8 percent?
Which of the following bonds is trading at a premium if market rate is 9%? a. Bond A: 7% coupon, 12 years maturity b. Bond B: 9% coupon, 12 years maturity c. Bond A: 11% coupon, 10 years maturity
Bond A: 11% coupon, 10 years maturity
An amount of $1,200 is deposited into an account that earns 5.5% interest. If equal withdrawals of $24 will be made each year, the number of years it will take to exhaust the account is
C. Account will never exhaust
During a recession, if FED injects money into the financial system by buying more Treasury securities, then the interest rate will increase. True or False
If both the level of annuity payments and the interest rate increase, then the present value of annuity may decrease, increase or stay the same.
If you buy a long-term non-callable bond and hold it until maturity, the coupon payments and the face value will not change, no matter what happens to the interest rate True or False?
In the wake of the start of an economic boom, if the firms expand their investment plans, then the interest rate will increase True or False
Longer maturity bonds are more risky than shorter maturity bonds, since it is more likely for any risk to happen in a longer time-period.
The higher the face value, the higher is the fair value of a bond. True or False?
The higher the interest rate, the higher is the Future value of annuity.
Find the future values of $400 paid each 6 months for 5 years at 12% compounded semiannually
a. 5,272.32.
Which of the following decreases YTM of a bond? a. If a bond's price increases, its YTM decreases. b. If a company's bonds are downgraded its YTM increases. c. the bond is more risky, then the bond's
YTM would increase. d. YTM would increase
a. If a bond's price increases, its YTM decreases
If the interest rate increases, the future value also increases a. True b. False
a. True
. If the probability of default is zero and the bond cannot be called, then a bond's YTM is the bond's promised rate of return, which is the expected return. a. True b. False
Which of the following has the most reinvestment risk?
b. 1-year bond with 10% coupon
If the number of compounding intervals increases, then the future value of an amount will decrease. a. True b. False
b. False
If market interest rates rise after a bond issue, than the YTM of the bond decreases. a. True b. False
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Solve 3x+8>2, when(i) x is an integer(ii) x is a real number
Magical Mathematics[Interesting Approach]> Solve 3x+8>2, when (i) x is an integer (i...
1 Answers
Harshit Singh
Last Activity: 4 Years ago
Welcome to AskIITians
Given Linear inequality: 3x+8>2
The given inequality can also be written as:
3x+8 -8 > 2 -8 …(1)
In the above inequality, -8 is multiplied on both the sides, as it does not change the definition of the given expression.
Now, simplify the expression (1)
⇒ 3x > -6
Now, both the sides, divide it by 3
⇒ 3x/3 > -6/3
⇒ x > -2
(i) x is an integer
Hence, the integers greater than -2 are -1,0,1,2,…etc
Thus, when x is an integer, the solutions of the given inequality are -1,0,1,2,…
Hence, the solution set for the given linear inequality is {-1,0,1,2,…}
(ii) x is a real number
If x is a real number, the solutions of the given inequality are all the real numbers, which
are greater than 2.
Therefore, in the case of x is a real number, the solution set is (-2, ∞)
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Tommy's Maths
Maths Homepage
Plain sticky notes
Hello World!
Hello, And welcome to the fantastic website for MATHS! This website will help you with your maths, there will be some usefull links if you head over to the Websites!
Learn more tricks at our Trick Page!
Go to our Maths Tricks page!
This is the school Logo!
• Thu October 4 - Lunchtime: Buy a new beanie hat
• Fri October 5 - 6pm: Go rollerblading
• Tue June 11 - My Birthday!!
Maths Tricks
Plain sticky notes
Trick 1
If you multiply 1089 x 9 you get 9801. It's reversed itself! This also works with 10989 or 109989 or 1099989 and so on.
Trick 4
This is the 7-11-13 trick. Ask a friend to write down any 3 digit number and say multiply it by x7 then x11 then x13. eg: 884x7x11x13 And the answer is simple! 884884!
Trick 2
Step1: Think of a number. Step2: Multiply it by 3. Step3: Add 6 with the getting result. Step4: divide it by 3. Step5: Subtract it from the first number used. Your answer is 2
Trick 5
Coming soon.
Trick 3
Okay theres an easy way of multiplying 11! Look: 11x52 5 (5+2=7) 2 Your answer is 572 11x63 6 (6+3) 3 Your answer is 693
About my page!
Plain sticky notes
Sticky note
This page will help people out with their maths, I found some useful links and some tricks that you can use. I have found some games also feel free to play them! - Tommy
Multplying by 11
Plain sticky notes
Multiplying by 11
Okay theres an easy way of multiplying 11! Look: 11x52 5 (5+2=7) 2 Your answer is 572 11x63 6 (6+3) 3 Your answer is 693
Work Out Fractions/Decimals
Plain sticky notes
to work out fractions of numbers you have to devide the number by the denominator. then times it by the numerator. for you, you would do: 21/3 = 7 7x1= 7 7 is 1/3 of 21.
If you are working out decimals like 1.5 x 2.5 First do 15 x 25 = 375 add the decimal points :) 3.75 | {"url":"http://www.protopage.com/tommyvincent","timestamp":"2024-11-07T12:38:14Z","content_type":"text/html","content_length":"44864","record_id":"<urn:uuid:9df49d36-d86f-44b1-9646-2d0a7caf773c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00726.warc.gz"} |
CSC165H1: Problem Set 1
1. [6 marks] Propositional formulas. For each of the following propositional formulas, find the following
two items:
(i) The truth table for the formula. (You don’t need to show your work for calculating the rows of the
(ii) A logically equivalent formula that only uses the ¬, ∧, and ∨ operators; no ⇒ or ⇔. (You should
show your work in arriving at your final result. Make sure you’re reviewed the “extra instructions”
for this problem set carefully.)
(a) [3 marks] (p ⇒ q) ⇒ ¬q.
(b) [3 marks] (p ⇒ ¬r) ∧ (¬p ⇒ q).
Page 2/5
CSC165H1,Problem Set 1
2. [8 marks] Fixed Points.
Let f be a function from N to N. A fixed point of f is an element x ∈ N such that f(x) = x. A least
fixed point of f is the smallest number x ∈ N such that f(x) = x. A greatest fixed point of f is the largest
number x ∈ N such that f(x) = x.
(a) [1 mark] Express using the language of predicate logic the English statement:
“f has a fixed point.”
You may use an expression like “f(x) = [something]” in your solution.
(b) [2 marks] Express using the language of predicate logic the English statement:
“f has a least fixed point.”
You may use the predefined function f as well as the predefined predicates = and <. You may not use any other predefined predicates. (c) [2 marks] Express using the language of predicate logic the
English statement: “f has a greatest fixed point.” You may use the predefined function f as well as the predefined predicates = and <. You may not use any other predefined predicates. (d) [3 marks]
Consider the function f from N to N defined as f(x) = x mod 7.1 Answer the following questions by filling in the blanks. The fixed points of f are: The least fixed point of f is: The greatest fixed
point of f is: 1 Here we are using the modulus operator. Given a natural number a and a positive integer b, a mod b is the natural number less than b that is the remainder when a is divided by b. In
Python, the expression a % b may be used to compute the value of a mod b. Page 3/5 CSC165H1, Problem Set 1 3. [6 marks] Partial Orders. A binary predicate R on a set D is called a partial order if
the following three properties hold: (1) (reflexive) ∀d ∈ D, R(d, d) (2) (transitive) ∀d, d0 , d00 ∈ D, (R(d, d0 ) ∧ R(d 0 , d00)) ⇒ R(d, d00) (3) (anti-symmetric) ∀d, d0 ∈ D, (R(d, d0 ) ∧ R(d 0 ,
d)) ⇒ d = d 0 A binary predicate R on a set D is called a total order if it is a partial order and in addition the following property holds: ∀d, d0 ∈ D, R(d, d0 ) ∨ R(d 0 , d). For example, here is a
binary predicate R on the set {a, b, c, d} that is a total order: R(a, b) = R(a, c) = R(a, d) = R(b, c) = R(b, d) = R(c, d) = R(a, a) = R(b, b) = R(c, c) = R(d, d) = True and all other values are
False. (a) [2 marks] Give an example of a binary predicate R on the set N that is a partial order but that is not a total order. (b) [2 marks] Let R be a partial order predicate on a set D. R
specifies an ordering between elements in D. Whenever R(d, d0 ) is True, we will say that d is less than or equal to d 0 , or that d 0 is greater than or equal to d. The following formula in
predicate logic expresses that there exists a greatest element in D; that is, an element in D that is greater than or equal to every other element in D: ∃d ∈ D, ∀d 0 ∈ D, R(d 0 , d) An element in D
is said to be maximal if no other element in D is larger than this element. The following formula in predicate logic expresses that there exists a maximal element in D: ∃d ∈ D, ∀d 0 ∈ D, d = d 0 ∨ ¬R
(d, d0 ) Give an example of a partial order order R over {a, b, c, d} such that every element is maximal. (c) [2 marks] Give an example of a partial order R over {a, b, c, d} such that a ∈ D is
maximal but a is not a greatest element. Justify your answer briefly. Page 4/5 CSC165H1, Problem Set 1 4. [13 marks] One-to-one functions. So far, most of our predicates have had sets of numbers as
their domains. But this is not always the case: we can define properties of any kind of object we want to study, including functions themselves! Let S and T be sets. We say that a function f : S → T
is one-to-one if no two distinct inputs are mapped to the same output by f. For example, if S = T = Z, the function f1(x) = x + 1 is one-to-one, since every input x gets mapped to a distinct output.
However, the function f2(x) = x 2 is not one-to-one, since f2(1) = f2(−1) = 1. Formally we express “f : S → T is one-to-one” as: ∀x1 ∈ S, ∀x2 ∈ S, f(x1) = f(x2) ⇒ x1 = x2. We say that f : S → T is
onto if every element in T gets mapped to by at least one element in S. The above function f(x) = x + 1 is onto over Z but is not onto over N. Formally we express “f : S → T is onto” as: ∀y ∈ T, ∃x ∈
S, f(x) = y Let t ∈ T. We say that f outputs t if there exists s ∈ S such that f(s) = t. (a) [1 mark] How many functions are there from {1, 2, 3} to {a, b, c, d}? (b) [1 mark] How many one-to-one
functions are there from {1, 2, 3} to {a, b, c, d}? (c) [1 mark] How many onto functions are there from {1, 2, 3, 4} to {a, b, c}? (d) [2 marks] Now let R be a binary predicate with domain N×N. We
say that R represents a function if, for every x ∈ N, there exists a unique y ∈ N, such that R(x, y) (is True). In this case, we write expressions like y = f(x). Define a predicate F unction(R),
where R is a binary predicate with domain N × N, that expresses the English statement: “R represents a function.” You may use the predicates <, ≤, =, R, but may not use any other predicate or
function symbols. In parts (e)-(h) below, you may use the predicate F unction(R) (in addition to the predicates <, ≤ , =, R) in your solution, but may not use any other predicate or function symbols.
(e) [2 marks] Define a predicate that expresses the following English statement. “R represents an onto function.” (f) [2 marks] Define a predicate that expresses the following English statement. “R
represents a one-to-one function.” (g) [2 marks] Define a predicate that expresses the following English statement. “R represents a function that outputs infinitely many elements of N.” (h) [2 marks]
Now define a predicate that expresses the following English statement. “R represents a function that outputs all but finitely many elements of N.” Page 5/5 | {"url":"https://codingprolab.com/answer/csc165h1-problem-set-1/","timestamp":"2024-11-02T14:19:40Z","content_type":"text/html","content_length":"109143","record_id":"<urn:uuid:3af4e66d-a0c7-42d8-b534-4a4faae53ed3>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00055.warc.gz"} |
Properties of number 313
313 has 2 divisors, whose sum is σ = 314. Its totient is φ = 312.
The previous prime is 311. The next prime is 317.
313 = 12^2 + 13^2.
It is a happy number.
313 is nontrivially palindromic in base 2, base 10 and base 13.
It is a weak prime.
It can be written as a sum of positive squares in only one way, i.e., 169 + 144 = 13^2 + 12^2 .
It is a palprime.
313 is a truncatable prime.
It is a cyclic number.
It is not a de Polignac number, because 313 - 2^1 = 311 is a prime.
Together with 311, it forms a pair of twin primes.
313 is an undulating number in base 10 and base 13.
It is a plaindrome in base 5, base 9, base 15 and base 16.
It is a nialpdrome in base 12.
It is a junction number, because it is equal to n+sod(n) for n = 296 and 305.
It is a congruent number.
It is not a weakly prime, because it can be changed into another prime (311) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (5) of ones.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 156 + 157.
It is an arithmetic number, because the mean of its divisors is an integer number (157).
313 is the 13-th centered square number.
It is an amenable number.
313 is a deficient number, since it is larger than the sum of its proper divisors (1).
313 is an equidigital number, since it uses as much as digits as its factorization.
313 is an odious number, because the sum of its binary digits is odd.
The product of its digits is 9, while the sum is 7.
The square root of 313 is about 17.6918060130. The cubic root of 313 is about 6.7896613364.
It can be divided in two parts, 3 and 13, that added together give a 4-th power (16 = 2^4).
The spelling of 313 in words is "three hundred thirteen", and thus it is an aban number and an oban number. | {"url":"https://www.numbersaplenty.com/313","timestamp":"2024-11-14T04:36:03Z","content_type":"text/html","content_length":"9291","record_id":"<urn:uuid:41f225d9-4dff-455e-9c93-2de6fdf8cab9>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00157.warc.gz"} |
Chain Drive Calculations [9qgox14gyzln]
CHAIN DRIVE SELECTION: (For Cart Travelling with Single Motor)
INPUT DATA :
Power to be Transmitted in H.P.(N) = Power to be Transmitted in kW. = Required Transmission Ratio (Z2/Z1) = Recommended No. of Teeth on Driver Sprocket (Z1): (Ref. Design Data Book Page No. 7.74)
Recommended No. of Teeth for Required Transmission Ratio = But, Where Space is a Problem We can Select, Minimum No. of Teeth on Driver Sprocket : We Consider, No. of Teeth on Driver Sprocket (Z1) =
Therefore, No. of Teeth on Driven Sprocket (Z2) =
Maximum Spped of Rotation for selected Teeth in RPM = (Ref. Design Data Book Page No. 7.74)
Center Distance (a) : We know that, Center Distance (a) = (30 to 50) x Pitch of Sprocket We Select Duplex Chain having following specifications, Chain with 1 " Pitch (Class 16A-2) Here, Selected
Pitch of Sprocket (p) = Therefore, Center Distance (a) =
Minimum Center Distance (amin): (Ref. Design Data Book Page No. 7.74) For required transmission ratio, Minimum centre distance is given as, amin = a' + (30 to 50 ) mm Where, a' = (D1 + D2)/ 2 Where,
D1 =Tip Diameter of Driver Sprocket = p/ (tan(1800/Z1)) + 0.6 p Therefore , Tip Diamter of Driver Sprocket (D1) =
D2 =Tip Diameter of Driven Sprocket = p/ (tan(1800/Z2)) + 0.6 p Therefore , Tip Diamter of Driven Sprocket (D2) =
Therefore, a' = (D1 + D2)/ 2 = Therefore, Minimum Center Distance (amin) = a' + (30 to 50 ) mm
Maximum Center Distance (amax): Amax = (80xp) in mm
Relation Between Center Distance & Length of Chain:
Length of continuous chain in multiples of pitches ( i.e. approximate number of links) is given by, Lp = (2 x ap ) + (Z1+Z2)/2 + [{(Z2- Z1) / 2 X 3.1415}2]/ ap Where, ap = Approximate Centre in
multiples of pitches = ao / p Where, ao = Initially assumed centre distance in mm =
P = Pitch of Chain in mm = Therefore, Length of continuous chain in multiples of pitches (Lp) =
Final Centre Distance corrected to even number of pitches (a) : a = [(e+(e2 – 8m)1/2) / 4] x p Where, e = Lp – (Z2+ Z1) / 2 = M = {(Z2- Z1) / 2 X 3.1415}2
We know that, Length of Chain (L) = Lp x p Where, Lp = Length of continuous chain in multiples of pitches ( i.e. approximate number of links) P = Pitch of Chain in mm = Therefore , Length of Chain
(L) =
Power Transmitted by the Chain on the basis of Breaking Load (P) in H.P. Is given by, P = (Q X V) /(75 X n X Ks) (Ref. Design Data Book Page No. 7.77) Where, Q = Breaking Load in kgf (Ref. Design
Data Book Page No. 7.72 ) For Chain with 1 " Pitch (Class 16A-2)
V = Linear Velocity of the Driver Sprocket in m/s = (3.142 X D1 X N )/ 60
Where, D1 - Pitch Circle Diameter of Driver Sprocket in mm D1= p / sin(180º/Z1) p = Pitch of Chain in mm Therefore , Pitch Circle Diamter of Driver Sprocket (D1) = Pitch Circle Diamter of Driver
Sprocket (D1) = RPM of the Driver Sprocket i.e. RPM of Gearbox Output Shaft (N) = Linear Velocity of the Driver Sprocket in m/s (V) =
n - Factor of Safety (Ref. Design Data Book Page No. 7.76 ) For Pitch 25.4 mm & 92 RPM,
Ks - Service Factor Ks = K1 X K2 X K3 X K4 X K5 X K6 K1 = Load Factor (For Variable Load with mild shocks) : K2 = Distance Regulation Factor (For Adjustable Supports) K3 = Factor For Centre Distance
of Sprocket ( For A < 25p) K4 = Factor for Position of Sprocket (Inclination of the line joining centers of sprocket to horizontal > 60 0) K5 = Factor for Lubrication (For Periodic) K6 = Rating
Factor (Single Shift of 8 hours a day) Therefore,
Ks =
Therefore, Power Transmitted by the Chain on the basis of Breaking Load (P) is given as, Power Transmitted by the Chain on the basis of Breaking Load (P) =
Therefore, Selected Chain is Safe for Carrying the Load without Breaking.
Power transmitted on the basis of allowable bearing stress is given by, Pb = (b x A x v) / (75 x ks) Where, b = Allowable bearing stress in kgf/cm 2 (Ref. Design Data Book) = A = Projected bearing
area in cm2 (Ref. Design Data Book) = v = Chain Velocity in m/s = ks = Service Factor (Ref. Design Data Book, From Calculations) = Therefore, Power transmitted on the basis of allowable bearing
stress (Pb) = Power transmitted on the basis of allowable bearing stress (Pb) =
Therefore, Selected Chain is Safe for Carrying the Load without Bearing failure.
CHECK FOR ACTUAL FACTOR OF SAFETY: Actual factor of Safety (n) is given as, n=Q/F Where, Q = Breaking Load of chain in kgf (Ref. Design Data Book) = F = Resultant Load in kgf = Pt + Pc + Ps Where,
Pt = Tangential force due to power transmission in kgf = (75 x N)/ v = Where, N = Actual Power to be transmitted in H.P. = v = Chain Velocity (m/s) = Therefore, Tangential force due to power
transmission (Pt) =
Pc = Centrifugal Tension in kgf = (w x v2)/g Where, w = Weight per metre of chain in kgf (Ref. Design Data Book) =
v = Chain Velocity (m/s) = g = Acceleration due to gravity(m/s2) = Therefore, Centrifugal Tension (Pc) =
Ps = Tension due to sagging of chain in kgf = k x w x a Where, k = Coefficent of Sag for Vertical position chain drive (Ref. Design Data Book) = w = Weight per metre of chain in kgf (Ref. Design Data
Book) = a = Centre Distance in metre = Therefore, Tension due to sagging of chain in kgf (Ps) =
Therefore, Resultant Load in kgf (F) =
Therefore, Actual factor of Safety (n) =
LOAD ON SHAFT: Load on Shaft due to Chain Drive in kgf is given by, Qo = k1 x Pt Where, k1 = Load Factor ( Position of Drive is Vertical & Shock Load) Ref. to Design Data Book,
k1 =
Pt = Tangential force due to power transmission in kgf = (75 x N) / v Where, v = Linear Velocity of the Driver Sprocket in m/s = N = Actual Power to be transmitted in H. P. = Therefore, Tangential
force due to power transmission in kgf (Pt) =
Therefore, Load on Shaft due to Chain Drive (Qo) =
mm mm
92 0.51
1.25 1 1.25 1.251 1.5 1 2.93
1 5.4026504
CHAIN DRIVE SELECTION: (For Pallet Picking Drive Assembly) INPUT DATA : Power to be Transmitted in H.P.(N) =
Power to be Transmitted in kW. =
Required Transmission Ratio (Z2/Z1) =
Recommended No. of Teeth on Driver Sprocket (Z1): (Ref. Design Data Book Page No. 7.74) Recommended No. of Teeth for Required Transmission Ratio =
But, Where Space is a Problem We can Select, Minimum No. of Teeth on Driver Sprocket :
We Consider, No. of Teeth on Driver Sprocket (Z1) =
Therefore, No. of Teeth on Driven Sprocket (Z2) =
Maximum Spped of Rotation for selected Teeth in RPM =
(Ref. Design Data Book Page No. 7.74)
Center Distance (a) : We know that, Center Distance (a) = (30 to 50) x Pitch of Sprocket We Select Duplex Chain having following specifications, Chain with 5/8 " Pitch (Class 12A-2) Here, Selected
Pitch of Sprocket (p) =
Therefore, Center Distance (a) =
Minimum Center Distance (amin): (Ref. Design Data Book Page No. 7.74) For required transmission ratio, Minimum centre distance is given as, amin = a' + (30 to 50 ) mm Where, a' = (D1 + D2)/ 2 Where,
D1 =Tip Diameter of Driver Sprocket = p/ (tan(1800/Z1)) + 0.6 p Therefore , Tip Diamter of Driver Sprocket (D1) =
D2 =Tip Diameter of Driven Sprocket = p/ (tan(1800/Z2)) + 0.6 p Therefore , Tip Diamter of Driven Sprocket (D2) =
Therefore, a' = (D1 + D2)/ 2 = Therefore, Minimum Center Distance (amin) = a' + (30 to 50 ) mm
94.45131 124.4513
Maximum Center Distance (amax): Amax = (80xp) in mm
Relation Between Center Distance & Length of Chain: Length of continuous chain in multiples of pitches ( i.e. approximate number of links) is given by, Lp = (2 x ap ) + (Z1+Z2)/2 + [{(Z2- Z1) / 2 X
3.1415}2]/ ap Where, ap = Approximate Centre in multiples of pitches = ao / p Where, ao = Initially assumed centre distance in mm = P = Pitch of Chain in mm =
Therefore, Length of continuous chain in multiples of pitches (Lp) =
Final Centre Distance corrected to even number of pitches (a) : a = [(e+(e2 – 8m)1/2) / 4] x p Where, e = Lp – (Z2+ Z1) / 2 = M = {(Z2- Z1) / 2 X 3.1415}2
We know that, Length of Chain (L) = Lp x p Where, Lp = Length of continuous chain in multiples of pitches ( i.e. approximate number of links) P = Pitch of Chain in mm =
Therefore , Length of Chain (L) =
POWER TRANSMITTED BY CHAIN WITH BREAKING LOAD: Power Transmitted by the Chain on the basis of Breaking Load (P) in H.P. Is given by, P = (Q X V) /(75 X n X Ks) (Ref. Design Data Book Page No. 7.77)
Where, Q = Breaking Load in kgf (Ref. Design Data Book Page No. 7.72 ) For Chain with 5/8" Pitch (Class 12A-2)
V = Linear Velocity of the Driver Sprocket in m/s = (3.142 X D1 X N )/ 60 Where, D1 - Pitch Circle Diameter of Driver Sprocket in mm D1 = p / sin(180º/Z1) p = Pitch of Chain in mm
Therefore , Pitch Circle Diamter of Driver Sprocket (D1) =
Pitch Circle Diamter of Driver Sprocket (D1) =
RPM of the Driver Sprocket i.e. RPM of Gearbox Output Shaft (N) =
Linear Velocity of the Driver Sprocket in m/s (V) =
n - Factor of Safety (Ref. Design Data Book Page No. 7.76 ) For Pitch 15.875 mm & 47 RPM,
Ks - Service Factor Ks = K1 X K2 X K3 X K4 X K5 X K6 K1 = Load Factor (For Constant Load) :
K2 = Distance Regulation Factor (For Adjustable Supports)
= Factor For Centre Distance of Sprocket {For lp / (Z1+Z2 ) = 1.5 or ap=30 to 50 p} 1
K4 = Factor for Position of Sprocket (Inclination 0 to 60 Degree)
K5 = Factor for Lubrication (For Periodic)
K6 = Rating Factor (Single Shift of 8 hours a day)
Therefore, Therefore,
Ks =
Power Transmitted by the Chain on the basis of Breaking Load (P) is given as, Power Transmitted by the Chain on the basis of Breaking Load (P) =
Therefore, Selected Chain is Not Safe for Carrying the Load without Breaking.
POWER TRANSMITTED ON THE BASIS OF BEARING STRESS: Power transmitted on the basis of allowable bearing stress is given by, Pb = (b x A x v) / (75 x ks) Where, b = Allowable bearing stress in kgf/cm 2
(Ref. Design Data Book) =
A = Projected bearing area in cm2 (Ref. Design Data Book) =
v = Chain Velocity in m/s =
ks = Service Factor (Ref. Design Data Book, From Calculations) =
Therefore, Power transmitted on the basis of allowable bearing stress (Pb) =
Power transmitted on the basis of allowable bearing stress (Pb) =
Therefore, Selected Chain is Not Safe for Carrying the Load without Bearing failure.
CHECK FOR ACTUAL FACTOR OF SAFETY: Actual factor of Safety (n) is given as, n=Q/F Where, Q = Breaking Load of chain in kgf (Ref. Design Data Book) =
F = Resultant Load in kgf = Pt + Pc + Ps Where,
Pt = Tangential force due to power transmission in kgf = (75 x N)/ v = Where, N = Actual Power to be transmitted in H.P. =
v = Chain Velocity (m/s) =
Therefore, Tangential force due to power transmission (Pt) =
Pc = Centrifugal Tension in kgf = (w x v2)/g Where, w = Weight per metre of chain in kgf (Ref. Design Data Book) =
v = Chain Velocity (m/s) =
g = Acceleration due to gravity(m/s2) =
Therefore, Centrifugal Tension (Pc) =
Ps = Tension due to sagging of chain in kgf = k x w x a Where, k = Coefficent of Sag for Vertical position chain drive (Ref. Design Data Book) =
w = Weight per metre of chain in kgf (Ref. Design Data Book) =
a = Centre Distance in metre =
Therefore, Tension due to sagging of chain in kgf (Ps) =
Therefore, Resultant Load in kgf (F) =
Therefore, Actual factor of Safety (n) =
LOAD ON SHAFT: Load on Shaft due to Chain Drive in kgf is given by, Qo = k1 x Pt Where, k1 = Load Factor ( Position of Drive is Vertical & Shock Load) Ref. to Design Data Book,
k1 =
Pt = Tangential force due to power transmission in kgf = (75 x N) / v Where, v = Linear Velocity of the Driver Sprocket in m/s =
N = Actual Power to be transmitted in H. P. =
Therefore, Tangential force due to power transmission in kgf (Pt) =
Therefore, Load on Shaft due to Chain Drive (Qo) =
H.P. kW
mm to
mm mm
mm mm m
kgf/cm2 cm2 m/s
H.P. kW
H.P. m/s kgf
kgf m/s m/s2 kgf
kgf m kgf
m/s H.P. kgf
CHAIN DRIVE SELECTION: (When Servo Motor is used is used for Cart Travelling for Accuracy in Stopping) SPROCKET 1.25” PITCH INPUT DATA : Rotary motion of Gearbox output shaft is converted into linear
motion of cart by using sprocket – chain drive. Power to be Transmitted in H.P.(N) =
Power to be Transmitted in kW. =
Required Transmission Ratio (Z2/Z1) =
Recommended No. of Teeth on Driver Sprocket (Z1): (Ref. Design Data Book Page No. 7.74) Recommended No. of Teeth for Required Transmission Ratio =
But, Where Space is a Problem We can Select, Minimum No. of Teeth on Driver Sprocket :
We Consider, No. of Teeth on Driver Sprocket (Z1) =
Maximum Spped of Rotation for selected Teeth in RPM =
(Ref. Design Data Book Page No. 7.74)
We Select Simplex Chain having following specifications, Chain with 1.25 " Pitch (Class 20A-1) Here, Selected Pitch of Sprocket (p) =
D1 =Tip Diameter of Driver Sprocket = p/ (tan(1800/Z1)) + 0.6 p Therefore , Tip Diamter of Driver Sprocket (D1) =
We know that, Length of Chain (L) = Lp x p Where, Lp = Length of continuous chain in multiples of pitches ( i.e. approximate number of links) P = Pitch of Chain in mm =
Sheet3 Therefore , Length of Chain (L) =
POWER TRANSMITTED BY CHAIN WITH BREAKING LOAD: Power Transmitted by the Chain on the basis of Breaking Load (P) in H.P. Is given by, P = (Q X V) /(75 X n X Ks) (Ref. Design Data Book Page No. 7.77)
Where, Q = Breaking Load in kgf (Ref. Design Data Book Page No. 7.72 ) For Chain with 1.25 " Pitch (Class 20A-1)
V = Linear Velocity of the Driver Sprocket in m/s = (3.142 X D1 X N )/ 60 Where, D1 - Pitch Circle Diameter of Driver Sprocket in mm D1= p / sin(180º/Z1) p = Pitch of Chain in mm
Therefore , Pitch Circle Diamter of Driver Sprocket (D1) =
Pitch Circle Diamter of Driver Sprocket (D1) =
RPM of the Driver Sprocket i.e. RPM of Gearbox Output Shaft (N) =
Linear Velocity of the Driver Sprocket in m/s (V) =
n - Factor of Safety (Ref. Design Data Book Page No. 7.76 ) For Pitch 31.75 mm & 92 RPM,
Ks - Service Factor Ks = K1 X K2 X K3 X K4 X K5 X K6 K1 = Load Factor (For Variable Load with heavy shocks) :
K2 = Distance Regulation Factor (For Adjustable Supports)
K3 = Factor For Centre Distance of Sprocket ( For A < 25p)
K4 = Factor for Position of Sprocket (Inclination 0 to 60 Degree)
K5 = Factor for Lubrication (For Periodic)
K6 = Rating Factor (Single Shift of 8 hours a day)
Ks =
Sheet3 Therefore, Power Transmitted by the Chain on the basis of Breaking Load (P) is given as, Power Transmitted by the Chain on the basis of Breaking Load (P) =
Therefore, Selected Chain is Safe for Carrying the Load without breaking.
POWER TRANSMITTED ON THE BASIS OF BEARING STRESS: Power transmitted on the basis of allowable bearing stress is given by, Pb = (b x A x v) / (75 x ks) Where, b = Allowable bearing stress in kgf/cm 2
(Ref. Design Data Book) =
A = Projected bearing area in cm2 (Ref. Design Data Book) =
v = Chain Velocity in m/s =
ks = Service Factor (Ref. Design Data Book, From Calculations) =
Therefore, Power transmitted on the basis of allowable bearing stress (Pb) =
Power transmitted on the basis of allowable bearing stress (Pb) =
Therefore, Selected Chain is Safe for Carrying the Load without Bearing failure.
CHECK FOR ACTUAL FACTOR OF SAFETY: Actual factor of Safety (n) is given as, n=Q/F Where, Q = Breaking Load of chain in kgf (Ref. Design Data Book) =
F = Resultant Load in kgf = Pt + Pc + Ps Where,
Pt = Tangential force due to power transmission in kgf = (75 x N)/ v = Where, N = Actual Power to be transmitted in H.P. =
v = Chain Velocity (m/s) =
Sheet3 Therefore, Tangential force due to power transmission (Pt) =
Pc = Centrifugal Tension in kgf = (w x v2)/g Where, w = Weight per metre of chain in kgf (Ref. Design Data Book) =
v = Chain Velocity (m/s) =
g = Acceleration due to gravity(m/s2) =
Therefore, Centrifugal Tension (Pc) =
Ps = Tension due to sagging of chain in kgf = k x w x a Where, k = Coefficent of Sag for Vertical position chain drive (Ref. Design Data Book) =
w = Weight per metre of chain in kgf (Ref. Design Data Book) =
a = Centre Distance in metre =
Therefore, Tension due to sagging of chain in kgf (Ps) =
Therefore, Resultant Load in kgf (F) =
Therefore, Actual factor of Safety (n) =
LOAD ON SHAFT: Load on Shaft due to Chain Drive in kgf is given by, Qo = k1 x Pt Where, k1 = Load Factor ( Position of Drive is Vertical & Shock Load) Ref. to Design Data Book,
k1 =
Pt = Tangential force due to power transmission in kgf = (75 x N) / v Where, v = Linear Velocity of the Driver Sprocket in m/s =
N = Actual Power to be transmitted in H. P. =
Therefore, Tangential force due to power transmission in kgf (Pt) =
Sheet3 Therefore, Load on Shaft due to Chain Drive (Qo) =
H.P. kW
Sheet3 mm
mm mm m
kgf/cm2 cm2 m/s
H.P. kW
H.P. m/s
Sheet3 kgf
kgf m/s m/s2 kgf
kgf m kgf
m/s H.P. kgf
CHAIN DRIVE SELECTION: (When Servo Motor is used is used for Cart Travelling for Accuracy in Stopping) SPROCKET 1” PITCH INPUT DATA : Rotary motion of Gearbox output shaft is converted into linear
motion of cart by using sprocket – chain drive. Power to be Transmitted in H.P.(N) =
Power to be Transmitted in kW. =
Required Transmission Ratio (Z2/Z1) =
Recommended No. of Teeth on Driver Sprocket (Z1): (Ref. Design Data Book Page No. 7.74) Recommended No. of Teeth for Required Transmission Ratio =
But, Where Space is a Problem We can Select, Minimum No. of Teeth on Driver Sprocket :
We Consider, No. of Teeth on Driver Sprocket (Z1) =
Maximum Spped of Rotation for selected Teeth in RPM =
(Ref. Design Data Book Page No. 7.74)
We Select Simplex Chain having following specifications, Chain with 1" Pitch (Class 20A-1) Here, Selected Pitch of Sprocket (p) =
D1 =Tip Diameter of Driver Sprocket = p/ (tan(1800/Z1)) + 0.6 p Therefore , Tip Diamter of Driver Sprocket (D1) =
We know that, Length of Chain (L) = Lp x p Where, Lp = Length of continuous chain in multiples of pitches ( i.e. approximate number of links) P = Pitch of Chain in mm =
Sheet4 Therefore , Length of Chain (L) =
POWER TRANSMITTED BY CHAIN WITH BREAKING LOAD: Power Transmitted by the Chain on the basis of Breaking Load (P) in H.P. Is given by, P = (Q X V) /(75 X n X Ks) (Ref. Design Data Book Page No. 7.77)
Where, Q = Breaking Load in kgf (Ref. Design Data Book Page No. 7.72 ) For Chain with 1" Pitch (Class 16A-1)
V = Linear Velocity of the Driver Sprocket in m/s = (3.142 X D1 X N )/ 60 Where, D1 - Pitch Circle Diameter of Driver Sprocket in mm D1= p / sin(180º/Z1) p = Pitch of Chain in mm
Therefore , Pitch Circle Diamter of Driver Sprocket (D1) =
Pitch Circle Diamter of Driver Sprocket (D1) =
RPM of the Driver Sprocket i.e. RPM of Gearbox Output Shaft (N) =
Linear Velocity of the Driver Sprocket in m/s (V) =
n - Factor of Safety (Ref. Design Data Book Page No. 7.76 ) For Pitch 25.4 mm & 92 RPM,
Ks - Service Factor Ks = K1 X K2 X K3 X K4 X K5 X K6 K1 = Load Factor (For Variable Load with heavy shocks) :
K2 = Distance Regulation Factor (For Adjustable Supports)
K3 = Factor For Centre Distance of Sprocket ( For A < 25p)
K4 = Factor for Position of Sprocket (Inclination 0 to 60 Degree)
K5 = Factor for Lubrication (For Periodic)
K6 = Rating Factor (Single Shift of 8 hours a day)
Ks =
Sheet4 Therefore, Power Transmitted by the Chain on the basis of Breaking Load (P) is given as, Power Transmitted by the Chain on the basis of Breaking Load (P) =
Therefore, Selected Chain is Safe for Carrying the Load without breaking.
POWER TRANSMITTED ON THE BASIS OF BEARING STRESS: Power transmitted on the basis of allowable bearing stress is given by, Pb = (b x A x v) / (75 x ks) Where, b = Allowable bearing stress in kgf/cm 2
(Ref. Design Data Book) =
A = Projected bearing area in cm2 (Ref. Design Data Book) =
v = Chain Velocity in m/s =
ks = Service Factor (Ref. Design Data Book, From Calculations) =
Therefore, Power transmitted on the basis of allowable bearing stress (Pb) =
Power transmitted on the basis of allowable bearing stress (Pb) =
Therefore, Selected Chain is Not Safe for Carrying the Load without Bearing failure.
CHECK FOR ACTUAL FACTOR OF SAFETY: Actual factor of Safety (n) is given as, n=Q/F Where, Q = Breaking Load of chain in kgf (Ref. Design Data Book) =
F = Resultant Load in kgf = Pt + Pc + Ps Where,
Pt = Tangential force due to power transmission in kgf = (75 x N)/ v = Where, N = Actual Power to be transmitted in H.P. =
v = Chain Velocity (m/s) =
Sheet4 Therefore, Tangential force due to power transmission (Pt) =
Pc = Centrifugal Tension in kgf = (w x v2)/g Where, w = Weight per metre of chain in kgf (Ref. Design Data Book) =
v = Chain Velocity (m/s) =
g = Acceleration due to gravity(m/s2) =
Therefore, Centrifugal Tension (Pc) =
Ps = Tension due to sagging of chain in kgf = k x w x a Where, k = Coefficent of Sag for Vertical position chain drive (Ref. Design Data Book) =
w = Weight per metre of chain in kgf (Ref. Design Data Book) =
a = Centre Distance in metre =
Therefore, Tension due to sagging of chain in kgf (Ps) =
Therefore, Resultant Load in kgf (F) =
Therefore, Actual factor of Safety (n) =
LOAD ON SHAFT: Load on Shaft due to Chain Drive in kgf is given by, Qo = k1 x Pt Where, k1 = Load Factor ( Position of Drive is Vertical & Shock Load) Ref. to Design Data Book,
k1 =
Pt = Tangential force due to power transmission in kgf = (75 x N) / v Where, v = Linear Velocity of the Driver Sprocket in m/s =
N = Actual Power to be transmitted in H. P. =
Therefore, Tangential force due to power transmission in kgf (Pt) =
Sheet4 Therefore, Load on Shaft due to Chain Drive (Qo) =
t – chain drive. H.P. kW
Sheet4 mm
mm mm m
kgf/cm2 cm2 m/s
H.P. kW
H.P. m/s
Sheet4 kgf
kgf m/s m/s2 kgf
kgf m kgf
m/s H.P. kgf | {"url":"https://doku.pub/documents/chain-drive-calculations-9qgox14gyzln","timestamp":"2024-11-09T22:10:01Z","content_type":"text/html","content_length":"54789","record_id":"<urn:uuid:83e7963e-a0d4-40e8-9c00-f8fc74f019ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00381.warc.gz"} |
This week's problem comes from the algebra category.
How can we compute the factors of \(6{t}^{2}-8t+2\)?
Let's begin!
Find the Greatest Common Factor (GCF).
GCF = \(2\)
Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
Simplify each term in parentheses.
Split the second term in \(3{t}^{2}-4t+1\) into two terms.
Factor out common terms in the first two terms, then in the last two terms.
Factor out the common term \(3t-1\). | {"url":"https://www.cymath.com/blog/2024-10-14","timestamp":"2024-11-04T09:06:00Z","content_type":"text/html","content_length":"30084","record_id":"<urn:uuid:3ece402f-1896-4066-bb91-ed07ed5e4565>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00382.warc.gz"} |
SciAm on relativity in 1911
Here is the
first SciAm article on relativity
“In 1905, came a fundamental and (as the fu-
ture historian will probably say) an epoch-
making contribution in the shape of an unas-
suming and dry-looking dissertation, ‘Con-
cerning the Electro-dynamics of Moving
Bodies,’ by A. Einstein, a Swiss professor of
physics. It appeared in the Annalen der
Physik, the German counterpart of our Philo-
sophical Magazine. It created no sensation at
the time. It was hardly noticed. Yet, at the pres-
ent time, you cannot open a journal devoted
to physics without finding some fresh contri-
bution to the ever-increasing literature on the
subject: Einstein’s Principle of Relativity.
—E. E. Fournier D’Albe”
Scientific American Supplement, November 11, 1911
I think that it is correct that Einstein's 1905 paper was considered no big deal, and that relativity did not start to take off until 1908. By 1911 relativity was huge, and textbooks were starting to
So why did relativity become so popular in 1908-1911, but not 1905-1908? The obvious explanations are (1) Einstein's paper was not appreciated at first, but it was after 3 years, and (2) Einstein's
paper was inconsequential, and Minkowski's 1908 paper made relativity popular.
I say that explanation (2) is better. Minkowski's paper was bold, geometric, and rigorous. It was reprinted and distributed widely. The 1911 works were based on Minkowski's theory, not Einstein's. I
do not see any proof that Einstein's paper had much influence on the early development of relativity at all. It seems to have influenced Max Planck, but hardly anyone else. Minkowski learned
relativity from
David Hilbert
, Lorentz, and Poincare, not Einstein.
Hermann Minkowski
declared in 1908:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and
time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Minkowski died in 1909, but his 1908 paper was the most widely read relativity paper at the time. Nearly all subsequent relativity work was based on Minkowski's formulation, not Einstein's.
It is odd that Einstein's 1905 paper would be credited as being so influential. I cannot find much actual influence at the time. Everyone considered an embellishment of Lorentz's theory, and some
called it the Lorentz-Einstein theory. Apparently it persuaded Planck, but not Minkowski or everyone else. It appears that many people, including this SciAm writer, decided years later than the paper
must have been influential. But it was not.
7 comments:
1. The Einstein cult is outrageous but the quantum computer cult is even more idiotic. They can't design a language like Julia, a condensed develop environment like Squeak and just waste all our
MIPS and graphics TFLOPS on bad programming. FPGAs have been around forever and no one uses them. I have been saying this forever, as well. Japan finally got the idiotic Westerners to pay
"Fujitsu says it has implemented basic optimisation circuits using an FPGA to handle combinations which can be expressed as 1024 bits, which when using a ‘simulated annealing’ process ran 10,000
times faster than conventional processors in terms of handling the aforementioned thorny combinatorial optimisation problems.
The company says it will work on improving the architecture going forward, and by the fiscal year 2018, it expects 'to have prototype computational systems able to handle real-world problems of
100,000 bits to one million bits that it will validate on the path toward practical implementation'."
2. I think that Minkowski did not cite Poincare but Einstein and Lorentz.
3. "It is odd that Einstein's 1905 paper would be credited as being so influential. I cannot find much actual influence at the time." - Are you really sure? How many citations he got in 1905-1908
period? I know of one good utilization of Einstein work by von Laue proving Fresnel drift factor in 1907.
1. Matthew CoryOctober 22, 2016 at 5:31PM
Einstein: "Since the mathematicians have invaded the relativity theory, I do not understand it myself any more."
Minkowski mentions Einstein in a 1907 paper but basically a single line about his minor insight. Lorentz & Poincare are discussed at length.
The Fundamental Equations for Electromagnetic Processes in Moving Bodies (1907)
2. "Poincare are discussed at length" - I have just looked at the paper. Lorentz is discussed at length but not Poincare!
3. Wrong. You don't get the math.
I'll repeat my post to Motl with quotes from Roger and others:
The more I read about Einstein, the more I realize his following is a cult. In terms of the geometrization of physics "many people assume that Einstein was a leader in this movement, but he
was not. When Minkowski popularized the spacetime geometry in 1908, Einstein rejected it. When Grossmann figured out to geometrize gravity with the Ricci tensor, Einstein wrote papers in 1914
saying that covariance is impossible. When a relativity textbook described the geometrization of gravity, Einstein attacked it as wrongheaded.... In 1905, Poincare had the Lorentz group and
the spacetime metric, and at the time a Klein geometry was understood in terms of a transformation group or an invariant metric. He also implicitly used the covariance of Maxwell's equations,
thereby integrating the geometry with electromagnetism. Minkowski followed where Poincare left off, explicitly treating world-lines, non-Eudlidean geometry, and covariance. Einstein had none
of that. He only had an exposition of Lorentz's theorem, not covariance or spacetime or geometry."
...(see post)
4. Matthew CoryOctober 25, 2016 at 5:53PM
Furthermore, I just used a paper that had been translated. If you know anything about the subject you would know that "Das Relativitatsprinzip” was "the first account of Minkowski’s ideas on
Special Relativity and 4-dimensional geometry. It is striking that, in this text, Poincare’s name is among the most cited ones: more precisely, the three most cited names are Planck (cited
eleven times), Lorentz (cited ten times) and Poincare (cited six times). By contrast, Einstein’s name appears only twice."
Don't waste my time anonymous coward. | {"url":"https://blog.darkbuzz.com/2016/10/sciam-on-relativity-in-1911.html","timestamp":"2024-11-02T11:21:11Z","content_type":"text/html","content_length":"120556","record_id":"<urn:uuid:5419f763-fed2-4c81-b279-6d97e3f351ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00014.warc.gz"} |
11.1.1 Inertia relief
Products: ABAQUS/Standard ABAQUS/CAE
Inertia relief:
● involves balancing externally applied forces on a free or partially constrained body with loads derived from constant rigid body accelerations;
● requires material density or mass and/or rotary inertia values to be specified for computing inertia relief loads;
● can be performed for static, dynamic, and buckling analyses in ABAQUS/Standard;
● varies the inertia relief loading with the applied loading in static analysis;
● applies inertia relief load corresponding to the static preload in dynamic analysis;
● can be used to balance applied perturbation loads when used with buckling analysis;
● uses rigid body accelerations consistent with the specified boundary conditions to compute the inertia relief loads;
● can be geometrically linear or nonlinear;
● may require the use of the unsymmetric solver if there are large inertia relief moments in a geometrically nonlinear analysis;
● is an inexpensive alternative to doing a full dynamic free body analysis when applied loads vary slowly compared to the eigenfrequencies of the body; and
● can be used with multiple load cases.
Inertia relief loading can be applied in static ( Static stress analysis, Section 6.2.2), dynamic ( Implicit dynamic analysis using direct integration, Section 6.3.2), and eigenvalue buckling
prediction steps ( Eigenvalue buckling prediction, Section 6.2.3).
In a static step the inertia relief loading varies with the applied external loading. An example of using an inertia relief load is modeling a rocket undergoing constant or slowly varying
acceleration during lift-off (i.e., a free body subjected to a constant or slowly varying external force) with a static analysis procedure. The inertia forces experienced by the body are included in
the static solution through inertia relief loading that balances the external loading.
In a dynamic step the inertia relief loading is calculated based on the static preload and is held constant during the step. The following is an example of using an inertia relief load in a dynamic
analysis procedure: Consider a free body submerged in water and subjected to shock wave loading due to an explosion. A dynamic analysis is needed to compute the transient solution. If it is known
that initially the body is stationary under gravity and hydrostatic pressure from the fluid, the gravity load should exactly balance the buoyancy force. However, if the finite element model does not
include all the mass existing in the body (for example, ballast), without additional loading, the body would accelerate due to out-of-balance external forces. Applying inertia relief loading exactly
balances these unbalanced external loads, placing the body in static equilibrium. The dynamic analysis then provides the transient response of the body to the shock wave loading as deformation of the
body relative to its static equilibrium position.
In a buckling analysis the inertia relief load can be applied in the static preload step, in the eigenvalue buckling prediction step, or in both steps. In the eigenvalue buckling prediction step the
inertia relief load is calculated based on the perturbation loads. Consider the static analysis rocket example. If we use inertia relief in a buckling analysis of the rocket with the rocket thrust as
the perturbation load, we can predict the critical thrust that causes the rocket to buckle.
In inertia relief the total response,
with corresponding expressions for velocities and accelerations. The reference point is the center of mass except when you must specify the reference point. Then, the finite element approximation to
the dynamic equilibrium equation becomes
The rigid body response can be expressed in terms of the acceleration of the reference point,
By definition,
-direction at the reference point. For example, at a node with the usual three displacements and three rotations
, and
are the coordinates of the node; and
Projecting the dynamic equilibrium equation onto the rigid body modes, we have
. The actual number of rigid body modes will be less than 6 in the presence of symmetry planes as well as for two-dimensional and axisymmetric analyses. Thus, the rigid body response can be evaluated
directly from the external loads.
The relative response of the body can be obtained by solving the equilibrium equation with the known inertial term
In a dynamic analysis involving inertia relief the rigid body mode vectors
Input File Usage: *INERTIA RELIEF
ABAQUS/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Inertia relief for the Types for Selected Step
Inertia relief loading directions
By default, all rigid body motion directions in a model can be loaded by inertia relief loading (in this discussion we use the word “direction” to mean any rigid body translation or rotation). In
models with symmetry planes or models that are allowed to move freely in only specific directions, the free directions for which inertia relief loading is applied can be specified. For example, in a
three-dimensional analysis with one symmetry plane only three free directions exist—two translations and one rotation. Add an additional symmetry plane and only one free translation remains. A
cylinder-piston arrangement is an example where the only free direction considered is motion along the cylinder's axis. In these situations you specify the free directions that are loaded by inertia
relief loading by indicating the degrees of freedom.
The case of two free rotation directions is not permitted. For cyclic symmetric models with inertia relief only translation in the Z-direction and rotation about the Z-direction are considered for
computing inertia relief loading.
Input File Usage: *INERTIA RELIEF
integer list of global degrees of freedom identifying the free directions
For example, the list 1, 3, 5 implies that translations in the X- and Z-directions and rotation about the Y-axis are free directions.
ABAQUS/CAE Load module: Create Load: choose Mechanical for the Category and Inertia relief for the Types for Selected Step: toggle on the degrees of freedom to define the Free Directions (the
Usage: degrees of freedom displayed are dependent on the modeling space)
Defining the free directions in a local coordinate system
If the free directions are not global directions, an orientation can be used to define the local coordinate system to which the integer list of degree of freedom identifiers refers.
Input File Usage: *INERTIA RELIEF, ORIENTATION=orientation_name
integer list of local degrees of freedom identifying the free directions
ABAQUS/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Inertia relief for the Types for Selected Step: click Edit, and choose a local CSYS
Defining free direction combinations that require a user-specified reference point
Not all user-chosen combinations of free directions admit unconstrained rigid body motion; that is, there are certain combinations of free directions for which an additional point is required to
define the rigid body motion vectors. For example, in three dimensions the choice 4, 5, 6 corresponds to free rotations about a fixed point. The fixed point must be given to define the rigid body
motion vectors. In other examples the free directions include rotation about a fixed axis. Consider a turbine blade rotating about its axis, as shown in Figure 11.1.1 1.
Figure 11.1.1 1 Inertia relief for a turbine blade with rotation about the axis as the only free direction.
To find the angular acceleration of the blade as it rotates under an applied force couple or moment, you should specify the coordinates of the point on the shaft about which the blade is rotating.
The free direction combinations for which you must specify a reference point are given in
Table 11.1.1 1
Input File Usage: *INERTIA RELIEF, ORIENTATION=orientation_name
integer list of local degrees of freedom identifying the free directions
X, Y, Z coordinates of the reference point for defining the rigid body vectors
ABAQUS/CAE Load module: Create Load: choose Mechanical for the Category and Inertia relief for the Types for Selected Step: toggle on Global position of reference point, and enter the X, Y, and
Usage: (if available) Z coordinates
Table 11.1.1 1 Free direction combinations requiring a reference point.
│Degree of freedom identifiers defining free directions │ Reference point definition │
│ ├───────────────────┬──────────────────────┬──────────────────────┤
│ │Fixedrotation point│Point on rotation axis│Point on symmetry line│
│ 4, 5, 6 │ │ │ │
│ 1, 4, 5, 6 │ │ │ │
│ 2, 4, 5, 6 │ │ │ │
│ 3, 4, 5, 6 │ │ │ │
│ 1, 2, 4, 5, 6 │ │ │ │
│ 1, 3, 4, 5, 6 │ │ │ │
│ 2, 3, 4, 5, 6 │ │ │ │
│ 4 │ │ │ │
│ 5 │ │ │ │
│ 6 │ │ │ │
│ 2, 4 │ │ │ │
│ 3, 4 │ │ │ │
│ 1, 5 │ │ │ │
│ 3, 5 │ │ │ │
│ 1, 6 │ │ │ │
│ 2, 6 │ │ │ │
│ 1, 2, 4 │ │ │ │
│ 1, 2, 5 │ │ │ │
│ 1, 3, 4 │ │ │ │
│ 1, 3, 6 │ │ │ │
│ 2, 3, 5 │ │ │ │
│ 2, 3, 6 │ │ │ │
│ 1, 4 │ │ │ │
│ 2, 5 │ │ │ │
│ 3, 6 │ │ │ │
Initial conditions
Initial conditions can be specified in the same way as in static and dynamic analyses without inertia relief loads. If inertia relief is used in the first step in the analysis, these initial
conditions form the base state of the body. See Initial conditions, Section 27.2.1.
Boundary conditions
Boundary conditions are specified in the same way as in analyses without inertia relief loads (see Boundary conditions, Section 27.3.1). In theory, a statically determinate set of restraints is
needed when inertia relief is used in a static step. By “statically determinate” we mean a set of restraints that restrain all rigid body modes but no deformation modes. Such a set provides a unique
displacement solution and ensures that the inertia relief loading exactly balances the user-specified external loading: zero reaction forces with no rigid body motion of the center of mass.
Table 11.1.1 2 summarizes the restraint requirements for various cases.
Table 11.1.1 2 Necessary and sufficient statically determinate restraints.
│Problem dimensionality │ Free directions │Number of required restraints│
│ 2-D │ 2 Translations and 1 Rotation │ 3 │
│ Axisymmetric │ 1 Translation │ 1 │
│Axisymmetric with twist│ 1 Translation and 1 Rotation │ 2 │
│ 3-D │3 Translations and 3 Rotations │ 6 │
It is not necessary to use boundary condition definitions ( Boundary conditions, Section 27.3.1) with inertia relief except in the case of buckling analysis. If no boundary conditions or
insufficient boundary conditions are specified, a warning message will be issued and boundary conditions necessary to restrain the rigid body modes will be imposed internally at the point in the
model that corresponds to the original location of the reference point. On the other hand, if too many boundary conditions are specified in certain directions, a warning message will be issued to
indicate that the reaction forces may be nonzero at the nodes with overspecified boundary conditions. If there are insufficient boundary conditions in certain directions and too many boundary
conditions in other directions, the problem will be treated as a combination of these cases.
If a model contains symmetry planes or is constrained to move freely in specific directions, inertia relief loading should be applied only in those free directions. No boundary conditions should be
specified in the free directions; however, sufficient boundary conditions must be specified in the other directions. Any boundary conditions that violate the above requirements will be flagged as an
error. An error will also be issued if the combination of free directions includes only two free rotations or if a reference point is required but not specified.
In a buckling analysis, proper boundary conditions are important for getting the correct mode shape. Sufficient boundary conditions must be specified when inertia relief loading is applied in such an
analysis. See Eigenvalue buckling prediction, Section 6.2.3, for details on how to apply boundary conditions in a buckling analysis.
An analysis that uses inertia relief can include concentrated nodal forces at displacement degrees of freedom (1–6), distributed pressure forces or body forces, and user-defined loading.
Inertia relief loads are used to balance the external loads. They are computed and applied when inertia relief is included in the step definition. The rules for propagating load definitions between
steps hold for inertia relief loads. See Applying loads: overview, Section 27.4.1. The inertia relief loads will not be propagated to steps where inertia relief is not valid for the specified
If there are large inertia relief moments in a geometrically nonlinear analysis, their contribution to the stiffness matrix may be unsymmetric. In such cases unsymmetric equation solution may improve
the computational efficiency (see Procedures: overview, Section 6.1.1).
Computing inertia relief loads
The nodal force vector corresponding to the inertia relief loads is calculated as follows. The applied loads are projected onto the rigid body modes,
Fixed inertia relief loads
You can specify that the inertia relief loads should be held fixed in magnitude and direction at the values calculated at the end of the previous step.
Input File Usage: *INERTIA RELIEF, FIXED
ABAQUS/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Inertia relief for the Types for Selected Step: Method: Fix at current loading
Removing inertia relief loads
You can specify that the inertia relief loads that were applied in the previous general analysis step should be removed in the current step.
Input File Usage: *INERTIA RELIEF, REMOVE
ABAQUS/CAE Usage: Load module: Load Manager: Deactivate
Internal boundary conditions and convergence in geometrically linear and nonlinear analysis
In a model containing internal boundary conditions that generate unbalanced internal forces or moments, such as is possible from certain elements (for example, SPRING1, DASHPOT1, SPRING2, DASHPOT2,
or GAPUNI elements) or kinematic constraints (for example, coupling constraints, linear constraint equations, multi-point constraints, or surface-based tie constraints), inertia relief loads will not
balance these internal forces or moments. If the model contains sufficient boundary conditions, these internal forces or moments will appear as nonzero reaction forces or moments. If the model does
not contain sufficient boundary conditions, these internal forces or moments will appear as unconverged residual fluxes in the message file for geometrically linear as well as nonlinear analyses. The
model should be treated as having internal boundary conditions, with the unconverged residuals representing the reaction forces or moments needed to impose the internal boundary conditions. Ideally,
the internal boundary conditions should be removed or sufficient boundary conditions should be added to the model.
Predefined fields
User-defined field variables can be specified in the same way as in static and dynamic analyses without inertia relief loads. See Predefined fields, Section 27.6.1.
Material options
Any of the mechanical constitutive models that are available in ABAQUS/Standard for use in static, dynamic, or buckling analyses can be used with inertia relief (see Part V, Materials,” for details
on the material models available in ABAQUS/Standard). Since inertia relief loading is calculated using the inertia properties of the model, the density must be specified (see Density, Section
16.2.1) to define the model's inertia properties.
Most of the stress/displacement elements that are available in ABAQUS/Standard for use in static, dynamic, and buckling analyses (including mass and rotary inertia elements and user elements) can be
used. A warning will be issued when the model contains elements that do not have associated mass or inertia (for example, hydrostatic fluid elements and pore pressure elements). An error will be
issued if the model contains elements that do not allow finite boundaries (for example, infinite elements and elastic element foundations).
In the case of a substructure you must generate a reduced mass matrix for the substructure (see Generating a reduced mass matrix for a substructure” in “Defining substructures, Section 10.1.2). The
reduced mass matrix is included in the global mass matrix of the entire model to compute rigid body accelerations and inertia relief loads. Inertia relief can be used only with substructures in a
geometrically linear analysis. An error message is issued if inertia relief is used with substructures in a geometrically nonlinear analysis.
In addition to the usual output variables available in ABAQUS/Standard (see ABAQUS/Standard output variable identifiers, Section 4.2.1), the following variables are provided specifically for
inertia relief:
Variables for the entire model:
IRX Current coordinates of the reference point.
IRXn Coordinate n of the reference point (
IRA Equivalent rigid body acceleration components.
IRAn Component n of the equivalent rigid body acceleration (
IRARn Component n of the equivalent rigid body angular acceleration with respect to the reference point (
IRF Inertia relief load corresponding to the equivalent rigid body acceleration.
IRFn Component n of the inertia relief load corresponding to the equivalent rigid body acceleration (
IRMn Component n of the inertia relief moment corresponding to the equivalent rigid body angular acceleration with respect to the reference point (
IRRI Rotary inertia about the reference point.
For most cases inertia relief loads correspond to the product of “rigid body inertia” and the equivalent rigid body acceleration vector. However, when only a few rigid body directions are chosen as
free directions for inertia relief, inertia relief loads are computed in all rigid body directions for output purposes, but equivalent rigid body accelerations are computed in only the free
directions with the equivalent rigid body angular accelerations computed from the diagonal entries of the “rigid body inertia.”
Input file template
Data line to specify material density
Data lines to specify zero-valued boundary conditions
*STEP (, NLGEOM) (, PERTURBATION)
Use the NLGEOM parameter to include nonlinear geometric effects;
it will remain active in all subsequent steps.
*STATIC (or *DYNAMIC)
*CLOAD and/or *DLOAD
Data lines to specify loads
*INERTIA RELIEF, ORIENTATION=orientation_name
Data lines to specify global (or local, if the ORIENTATION parameter is used) degrees
of freedom that define free directions and to provide coordinates of a reference point
*END STEP
*STATIC(or *DYNAMIC)
*INERTIA RELIEF, FIXED or REMOVE
Include the FIXED parameter to keep inertia relief loads fixed at their current
values from the beginning of the step; include the REMOVE parameter to
remove inertia relief loads from the beginning of the step.
*END STEP | {"url":"https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/usb/pt04ch11s01at31.html","timestamp":"2024-11-09T00:12:08Z","content_type":"text/html","content_length":"51658","record_id":"<urn:uuid:8ff0a157-a3b8-4adc-bc86-d8fe30f0276d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00012.warc.gz"} |
[GAP Forum] problem with OrthogonalEmbeddings
Benjamin Sambale benjamin.sambale at gmail.com
Fri Sep 5 06:37:15 BST 2014
Dear GAP people,
according to the manual, the command OrthogonalEmbeddings does the
following: Given an integral symmetric matrix M, compute all integral
matrices X such that X^tr X = M where X^tr denotes the transpose of X.
The solution matrices X are given up to permutations and signs of their
If I do (with GAP 4.7.5) OrthogonalEmbeddings([[4]]), I only get one
solution, namely X = [[2]]. However, there is another solution X =
[[1],[1],[1],[1]] which is somehow missing! What is wrong here?
Apparently, the implementation is quite old and based on a paper by
Plesken from 1995.
There is also an inaccuracy in the manual: It says: "the list L = [ x_1,
x_2, ..., x_n ] of vectors that may be rows of a solution; these are
exactly those vectors that fulfill the condition x_i ⋅ gram^{-1} ⋅
x_i^tr ≤ 1 (see ShortestVectors (25.6-2)), and we have gram = ∑_{i =
1}^n x_i^tr ⋅ x_i".
The last equation is usually not true. The equation only holds for the
set of vectors of a solution. Moreover, one should mention that the list
of vectors is only up to signs.
Thanks and best wishes,
* Englisch - erkannt
* Englisch
* Deutsch
* Englisch
* Deutsch
More information about the Forum mailing list | {"url":"https://www.gap-system.org/ForumArchive2/2014/004643.html","timestamp":"2024-11-11T02:59:08Z","content_type":"text/html","content_length":"3857","record_id":"<urn:uuid:ecc763cc-928e-49eb-a69b-c467be1c34ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00116.warc.gz"} |
How to calculate interest on the DepositHow to calculate interest on the Deposit 🚩 Banks.
The algorithm of calculation of interest on deposits is directly dependent on their environment. Therefore, in order to roughly calculate what amount you will receive at the end of Deposit period,
you should review your Deposit agreement entered into between you and the Bank.
The accrual of interest at the end of term
The easiest from the point of view of the procedure of calculation the method of calculation of interest accrual at the end of term. This means that the entire amount of interest income payable to
the depositor, will be credited on the last day of the term. For example, you put 10 thousand rubles on Deposit at 10% per annum for 1 year. As a result, in the last day of the term, you will be
charged interest at the rate of 10% of the outstanding amount of 1 thousand rubles. Thus, the total amount of money that you will get after finishing your contract with the Bank, will be 11 thousand
Periodic accrual of interest on deposits
Another variant of the algorithm of interest is a periodic charge. In this case, with some regularity prescribed by the terms of the contract between you and the Bank, the latter will charge interest
on your money. Most often this operation is carried out on a monthly basis, however, in a particular situation can be provided and other conditions: for example, quarterly or even daily accrual. Your
interest can be transferred to a card or a special account that you can withdraw the accumulated money as necessary. For example, if you placed 10 thousand rubles in the Bank at 10% per annum for 1
year with the condition of monthly payment of interests on the card account every month will receive the corresponding amount. For example, if in the month of 30 days, it will be calculated as
10000*0,1*30/365=82,19 of the ruble.
Interest on deposits with the capitalization
A more sophisticated mechanism represents accrued interest on deposits with the capitalization. Capitalization represents the accrual of interest, accumulated during the Deposit period, the principal
amount. Thus, the amount of your Deposit increases, and thus, increases the amount of interest to be credited. The frequency of capitalization as established by the terms of the contract and may be
monthly, quarterly, or other. Consider an example with the same conditions: a contribution in the amount of 10 thousand rubles under 10% APR with monthly compounding. In this case, the amount of
interest accrued in the first month duration, e.g. 30 days, will be 82,19 of the ruble. It will be added to the principal amount of the Deposit, which after the first month will be already 10082,19
of the ruble. Thus, for the second month in duration, for example, 31 days interest will accrue on this sum they will be 10082,19*0,1*31/365=85,63 of the ruble. They, in turn, will be added to the
principal amount. In a similar way based on the number of days in the month will accrue interest during the following months of the term. | {"url":"https://eng.kakprosto.ru/how-873097-how-to-calculate-interest-on-the-deposit","timestamp":"2024-11-11T21:06:06Z","content_type":"text/html","content_length":"32389","record_id":"<urn:uuid:8ebdbf22-c066-4416-b4d7-0358099cd7a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00733.warc.gz"} |
Useful guide to inverter peak power and how to choose an inverter
Power inverters come in many specifications, which usually include rated power and inverter peak power. Rated power is continuous output power, which refers to the power that the inverter can keep
working for a long time. Inverter peak power also means the starting power, which is generally twice the rated power, mainly used to meet the instantaneous peak value when individual household
appliances are started.
Therefore, for an inverter, the inverter peak power must be able to meet the instantaneous power when the home appliance is started to ensure normal operation.
1. What is inverter peak power
Peak power, also called peak surge power, refers to the maximum power that the power supply can achieve in a short period of time, which usually only lasts about 30 seconds. Under normal
circumstances, the peak power of the power supply can exceed about 50% of the maximum output power.
Since the energy required by the hard disk in the startup state is much greater than that in normal operation, the system often uses this buffer to provide the hard disk with the current required for
startup. It will return to the normal level after startup to full speed. The power supply generally cannot works stably at peak output. So is inverter peak power a meaningless parameter? No!
Some appliances start with several times the power required for normal operation, but only for a short period of time. The purpose of inverter peak power is to ensure that the power inverter can
handle the peaks of such appliances and protect the power inverter, thereby preventing the peaks from damaging the power inverter.
When selecting a frequency converter, and when determining how large a power inverter is required, it is important to distinguish the difference between rated power and inverter peak power. The
reference value for rated power will be larger. The rated power is the continuous output power of the inverter, which is long-term and stable power. It provides continuous power for the normal
operation of your load.
One thing to note is that if your appliance is an inductive load with a motor (such as a refrigerator, washing machine, electric drill, etc.), you must consider the inverter peak power when choosing
an inverter. Because these inductive loads require a large current to start at the moment of startup, the appliance can start normally only when the inverter peak power is greater than the starting
power of the appliance. Under normal circumstances, the peak power is equal to 2 times the rated power.
2. Different types of load
Electrical appliances, in terms of technical performance, can be roughly divided into three categories: capacitive electrical appliances, inductive electrical appliances and resistive electrical
Capacitive, that is, relative to the power supply, the input is a capacitive load. For example, those appliances commonly seen in households: TVs, computers, mobile phone chargers, etc., that are
powered by switching power supplies, because the first input stage of such electrical appliances has a large filter capacitor.
The capacitor is connected, but the capacitor current is 90 degrees ahead of the voltage, which means that it is a short-circuited wire relative to the power supply at an instant. Therefore, there is
an instantaneous inverter peak power value in this type of electrical appliances.
A load that only consumes reactive power is called an inductive load. For example, fans, speakers (which are generally powered by low-frequency transformers), water pumps, electric drills, air
conditioners, refrigerators, and so on. Its input is relative to the power supply.
It is connected to an inductor coil, and the coil is an inductor. The voltage of the inductor is 90 degrees ahead of the current, which means that it is short-circuited compared to the power supply
when it is turned on instantly. Only after starting, the impedance rises, will it run stably.
When we use a high-power motor, we can clearly feel that the entire house's lamps and electrical appliances flash when it is turned on. This is the peak power. The instantaneous peak power is many
times greater than the power marked on the motor. For example, for a 500W motor, 3 times the instantaneous starting power is required, which is equivalent to a inverter peak power of 1500W.
The most common one is the light bulb. Compared to the power supply, it is a pure resistor and there is no problem of startup shock.
3. How to choose an inverter according to peak power
The inverter must use metal housing. Due to the large power of the vehicle-mounted inverter, the heat is also large. If the internal heat cannot be dissipated in time, the life of the components will
be affected, and there is even a risk of fire at worst.
The metal shell has good heat dissipation properties on the one hand and will not burn on the other hand. It is best not to use products with plastic shells. Even if a fan is added to help dissipate
heat, firstly, the noise during use is increased, and secondly, the working life of the fan is generally relatively short, which reduces the reliability of the entire machine.
When choosing an inverter, its power should be higher than the starting power of the household appliances used. This is because of the impact capability of the appliance at the moment it is turned
on. The impact only appears on inductive or capacitive appliances. For example, for motors, because they are inductive, the impact is generally 3 to 7 times the rated power. For a 500W motor, the
power impact is between 1500W and 3500W.
Inverters generally have inverter peak value that is 2 times the rated power, that is to say, a 500W inverter has an instant power output of 1000W, and a 1000W has a peak output of 2000W. But on the
other hand, it does not mean that all motors have 7 times the peak value.
We only need to understand that the inverter can instantly output 2 times the nominal power, and the instantaneous start of a capacitive or inductive load requires 3 to 7 times the peak power. How to
determine whether it is 3 times or 7 times the peak value? It's actually not that difficult. Appliances of no-load operation needs 3 times inverter peak power value, and full-load operation ones need
7 times, so the peak number can be judged based on the motor drive load level.
For electric hand drills, we can only start drilling after the no-load operation is stable, so the peak value is small, so we can calculate it as 4 times the peak value (actually 3 times is enough,
and the extra 1 time is the margin). For a 500W electric hand drill, the inverter peak power value should be 2000W, so you can choose a 1000W inverter.
For air conditioners, which are purely inductive, and the starting peak is very large, because every time the compressor starts, the motor works at full load. We can think of it as starting at 7x
peak. An air conditioner needs 1000W, and the peak power is 7000W. The inverter needs 2 times the peak value, which means the rated power of the inverter should be at least 3500W. That is to say, an
air conditioner with a power of 1000W needs an inverter with an inverter peak power of more than 3500W to start.
Nowadays, all air conditioners are equipped with inverters. We can calculate it as 4 times the peak value. A 1000W inverter air conditioner with a peak value of 4000W needs a 2000W inverter to
operate safely. So with a 1000W air conditioner, we can choose an inverter between 1500W and 2000W.
When working, the inverter itself consumes part of the power. Its input power is greater than its output power. For example, if an inverter inputs 100 watts of DC power and outputs 85 watts of AC
power, its efficiency is 85%. If the starting power of the motor is 1500 watts, and the inverter peak power is only 1500 watts, there is an efficiency loss during the conversion process, so the
required power is not actually achieved. Therefore, you should leave a large margin when purchasing.
If it is a purely resistive electrical appliance such as a sun lamp or light bulb, divide the power indicated on the electrical appliance by 0.9. For example: for a 100W light bulb, 100÷0.9=111. So
with a 120W inverter, the light bulbs can be used normally.
If it cannot be driven, it means that the power indication of the inverter is actually untrue. According to the inverter standards, if the actual output power reaches 90% of the nominal power, it is
a qualified product. For example, a 500W inverter should be able to drive at least a 450W light bulb.
There are two types of TVs, one is LCD TV and the other is CRT TV. For LCD TVs, the inverter should be twice the standard power of the TV. For example: the standard 100W LCD TV can be used with a
300W power inverter.
For CRT TVs, because of the large capacitance, the degaussing coil is equivalent to an inductive load, and the impact force is very strong. It is generally calculated as 10 times the peak value. For
example: for 100W appliance, the peak value is 1000W, and we need to equip an inverter with more than 500W to drive it.
If it is a computer with an LCD, you should choose the standard power (computer host power) plus 90W. If it is a CRT monitor, you need to calculate the peak power plus 90W based on the nominal power
of the selected monitor (the host power is generally within 90W).
A household 500W water pump, calculated based on 7 times (it pumps water as soon as it is turned on, and can only work at full load, with a large peak value), requires an 1800W inverter with inverter
peak power value of 3500W. The 750W pump must be equipped with a 2500W inverter to operate safely.
Generally speaking, most passenger cars use 12V battery and trucks use 24V battery. When choosing a car power inverter, please pay attention to purchasing an inverter suitable for battery parameters.
The car battery voltage must be consistent with the nominal DC input voltage of the inverter. For example, a 24V inverter must be matched to a 24V battery.
Related posts: Top 10 solar inverters, RV inverter, car inverter near me | {"url":"https://www.tycorun.com/blogs/news/inverter-peak-power","timestamp":"2024-11-13T19:18:26Z","content_type":"text/html","content_length":"205918","record_id":"<urn:uuid:48dc1a1c-c8bb-4804-b2bc-2436a36d97e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00292.warc.gz"} |
How to Find Acceleration Due to Gravity on Another Planet
To find the acceleration due to gravity on another planet, you need to use the formula g = G * M / R^2, where g is the acceleration due to gravity, G is the gravitational constant, M is the mass of
the planet, and R is the radius of the planet. This article provides a detailed, technical guide on how to calculate the acceleration due to gravity on another planet, including specific formulas,
examples, and numerical problems to help you master this concept.
Understanding the Gravitational Acceleration Formula
The formula to calculate the acceleration due to gravity on another planet is:
g = G * M / R^2
– g is the acceleration due to gravity (in m/s^2)
– G is the gravitational constant (6.674 × 10^-11 N⋅m^2/kg^2)
– M is the mass of the planet (in kg)
– R is the radius of the planet (in m)
This formula is based on Newton’s law of universal gravitation, which states that the gravitational force between two objects is directly proportional to their masses and inversely proportional to
the square of the distance between them.
To use this formula, you need to know the mass and radius of the planet you’re interested in. Let’s go through an example calculation for the planet Mars.
Example: Calculating Gravity on Mars
To find the acceleration due to gravity on Mars, we’ll use the following values:
• Mass of Mars (M): 6.4171 × 10^23 kg
• Radius of Mars (R): 3,389,500 m (converted from 3,389.5 km)
Plugging these values into the formula:
g = G * M / R^2
g = (6.674 × 10^-11 N⋅m^2/kg^2) * (6.4171 × 10^23 kg) / (3,389,500 m)^2
g = 3.71 m/s^2
So the acceleration due to gravity on Mars is approximately 3.71 m/s^2.
Calculating Gravity Using Centripetal Acceleration
If you’re unable to directly measure the mass and radius of a planet, you can use the centripetal acceleration relationship to estimate the acceleration due to gravity:
a = v^2 / r
– a is the acceleration due to gravity (in m/s^2)
– v is the orbiting velocity of a satellite (in m/s)
– r is the orbiting radius of the satellite (in m)
For example, if a satellite orbits Mars at a velocity of 3.5 km/s (3,500 m/s) and has an orbiting radius of 17,000 km (17,000,000 m), you can calculate the acceleration due to gravity as:
g = v^2 / r
g = (3,500 m/s)^2 / (17,000,000 m)
g = 0.0725 m/s^2
However, this method assumes that the satellite is in a circular orbit, which may not always be the case.
Calculating Gravity for Gas Giants
To calculate the gravity of a gas giant (like Jupiter or Saturn) at a specific depth within the planet’s atmosphere, you need to consider the mass distribution inside the planet.
The formula for gravitational acceleration inside a gas giant is:
g(r) = (G * M(r)) / r^2
– g(r) is the acceleration due to gravity at a distance r from the center of the planet
– M(r) is the mass of the planet enclosed within a sphere of radius r
To use this formula, you need to know the mass distribution within the planet, which can be complex to determine. This is because the density of the planet’s atmosphere and interior can vary
significantly with depth.
One approach is to use a model of the planet’s internal structure, such as the polytropic model, to estimate the mass distribution. This allows you to calculate the gravitational acceleration at
different depths within the planet’s atmosphere.
Numerical Problems
1. Calculate the acceleration due to gravity on the surface of Venus, given the following information:
2. Mass of Venus: 4.8675 × 10^24 kg
3. Radius of Venus: 6,051,800 m
4. A satellite orbits Jupiter at a velocity of 47,000 m/s and a radius of 1,070,000 km. Calculate the acceleration due to gravity experienced by the satellite.
5. Estimate the acceleration due to gravity at a depth of 1,000 km inside the atmosphere of Jupiter, given the following information:
6. Mass of Jupiter: 1.898 × 10^27 kg
7. Radius of Jupiter: 69,911 km
In this article, we’ve explored the various methods and formulas for calculating the acceleration due to gravity on another planet. By understanding the gravitational acceleration formula, the
centripetal acceleration relationship, and the considerations for gas giants, you can now confidently determine the gravitational acceleration on any planet or celestial body. Remember to always
double-check your calculations and consider the limitations of the methods used.
1. Socratic. (n.d.). How is the acceleration of gravity calculated for planets? Retrieved from https://socratic.org/questions/how-is-the-acceleration-of-gravity-calculated-for-planets
2. YouTube. (2018). How to Calculate Gravity on Other Planets. Retrieved from https://www.youtube.com/watch?v=92VGeHBU9uo
3. Reddit. (2022). How would one calculate the gravity of a planet? Retrieved from https://www.reddit.com/r/askscience/comments/z53aqr/how_would_one_calculate_the_gravity_of_a_planet/
4. Study.com. (n.d.). How to Calculate the Acceleration Due to Gravity on a Different Planet: Explanation. Retrieved from https://study.com/skill/learn/
5. The Physics Classroom. (n.d.). The Value of g. Retrieved from https://www.physicsclassroom.com/class/circles/Lesson-3/The-Value-of-g
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Reconcile temporal hierarchical forecasts — reconcilethief
Reconcile temporal hierarchical forecasts
Takes forecasts of time series at all levels of temporal aggregation and combines them using the temporal hierarchical approach of Athanasopoulos et al (2016).
reconcilethief(forecasts, comb = c("struc", "mse", "ols", "bu", "shr",
"sam"), mse = NULL, residuals = NULL, returnall = TRUE,
aggregatelist = NULL)
List of forecasts. Each element must be a time series of forecasts, or a forecast object. The number of forecasts should be equal to k times the seasonal period for each series, where k is the
same across all series.
Combination method of temporal hierarchies, taking one of the following values:
Structural scaling - weights from temporal hierarchy
Variance scaling - weights from in-sample MSE
Unscaled OLS combination weights
Bottom-up combination -- i.e., all aggregate forecasts are ignored.
GLS using a shrinkage (to block diagonal) estimate of residuals
GLS using sample covariance matrix of residuals
A vector of one-step MSE values corresponding to each of the forecast series.
List of residuals corresponding to each of the forecast models. Each element must be a time series of residuals. If forecast contains a list of forecast objects, then the residuals will be
extracted automatically and this argument is not needed. However, it will be used if not NULL.
If TRUE, a list of time series corresponding to the first argument is returned, but now reconciled. Otherwise, only the most disaggregated series is returned.
(optional) User-selected list of forecast aggregates to consider
List of reconciled forecasts in the same format as forecast. If returnall==FALSE, only the most disaggregated series is returned.
# Construct aggregates
aggts <- tsaggregates(USAccDeaths)
# Compute forecasts
fc <- list()
for(i in seq_along(aggts))
fc[[i]] <- forecast(auto.arima(aggts[[i]]), h=2*frequency(aggts[[i]]))
# Reconcile forecasts
reconciled <- reconcilethief(fc)
# Plot forecasts before and after reconcilation
for(i in seq_along(fc))
plot(reconciled[[i]], main=names(aggts)[i])
lines(fc[[i]]$mean, col='red') | {"url":"http://pkg.robjhyndman.com/thief/reference/reconcilethief.html","timestamp":"2024-11-04T18:55:36Z","content_type":"text/html","content_length":"16260","record_id":"<urn:uuid:cb3d3d7b-36ab-40e6-9963-ac2d43f4cbd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00071.warc.gz"} |
Estimating Covariances and Correlations
Example 25.1 Estimating Covariances and Correlations
This example shows how you can use PROC CALIS to estimate the covariances and correlations of the variables in your data set. Estimating the covariances introduces you to the most basic form of
covariance structures—a saturated model with all variances and covariances as parameters in the model. To fit such a saturated model when there is no need to specify the functional relationships
among the variables, you can use the MSTRUCT modeling language of PROC CALIS.
The following data set contains four variables q1–q4 for the quarterly sales (in millions) of a company. The 14 observations represent 14 retail locations in the country. The input data set is shown
in the following DATA step:
data sales;
input q1 q2 q3 q4;
1.03 1.54 1.11 2.22
1.23 1.43 1.65 2.12
3.24 2.21 2.31 5.15
1.23 2.35 2.21 7.17
.98 2.13 1.76 2.38
1.02 2.05 3.15 4.28
1.54 1.99 1.77 2.00
1.76 1.79 2.28 3.18
1.11 3.41 2.20 3.21
1.32 2.32 4.32 4.78
1.22 1.81 1.51 3.15
1.11 2.15 2.45 6.17
1.01 2.12 1.96 2.08
1.34 1.74 2.16 3.28
Use the following PROC CALIS specification to estimate a saturated covariance structure model with all variances and covariances as parameters:
proc calis data=sales pcorr;
mstruct var=q1-q4;
In the PROC CALIS statement, specify the data set with the DATA= option. Use the PCORR option to display the observed and predicted covariance matrix. Next, use the MSTRUCT statement to fit a
covariance matrix of the variables that are provided in the VAR= option. Without further specifications such as the MATRIX statement, PROC CALIS assumes all elements in the covariance matrix are
model parameters. Hence, this is a saturated model.
Output 25.1.1 shows the modeling information. Information about the model is displayed: the name and location of the data set, the number of data records read and used, and the number of observations
in the analysis. The number of data records read is the actual number of records (or observations) that PROC CALIS processes from the data set. The number of data records used might or might not be
the same as the actual number of records read from the data set. For example, records with missing values are read but not used in the analysis for the default maximum likelihood (ML) method. The
number of observations refers to the FREQ statement, the number of observations used is a weighted sum of the number of records, with the frequency variable being the weight. Second, if you use the
NOBS= option in the PROC CALIS statement, you can override the number of observations that are used in the analysis. Because the current data set does not have any missing data and there are no
frequency variables or an NOBS= option specified, these three numbers are all 14.
The model type is MSTRUCT because you use the MSTRUCT statement to define your model. The analysis type is covariances, which is the default. Output 25.1.1 then shows the four variables in the
covariance structure model.
The CALIS Procedure
Covariance Structure Analysis: Model and Initial Values
Output 25.1.2 shows the initial covariance structure model for these four variables. All lower triangular elements (including the diagonal elements) of the covariance matrix are parameters in the
model. PROC CALIS generates the names for these parameters: _Add01–_Add10. Because the covariance matrix is symmetric, all upper triangular elements of the matrix are redundant. The initial estimates
for covariance are denoted by missing values no initial values were specified.
The PCORR option in the PROC CALIS statement displays the sample covariance matrix in Output 25.1.3. By default, PROC CALIS computes the unbiased sample covariance matrix (with variance divisor equal
The fit summary and the fitted covariance matrix are shown in Output 25.1.4 and Output 25.1.5, respectively.
In Output 25.1.4, the model fit chi-square is 0 (
Output 25.1.5 shows the fitted covariance matrix, along with standard error estimates and Output 25.1.3.
A common practice for determining statistical significance for estimates in structural equation modeling is to require the absolute value of Output 25.1.5 show statistical significance, all
off-diagonal elements are not significantly different from zero. The
Output 25.1.6 shows the standardized estimates of the variance and covariance elements. This is also the correlation matrix under the MSTRUCT model. Standard error estimates and
Sometimes researchers do not need to estimate the standard errors that are in their models. You can suppress the standard error and NOSE option in the PROC CALIS statement:
proc calis data=sales nose;
mstruct var=q1-q4;
Output 25.1.7 shows the fitted covariance matrix with the NOSE option. These values are exactly the same as in the sample covariance matrix shown in Output 25.1.3.
This example shows a very simple application of PROC CALIS: estimating the covariance matrix with standard error estimates. The covariance structure model is saturated. Several extensions of this
very simple model are possible. To estimate the means and covariances simultaneously, see Example 25.2. To fit nonsaturated covariance structure models with certain hypothesized patterns, see Example
25.3 and Example 25.4. To fit structural models with implied covariance structures that are based on specified functional relationships among variables, see Example 25.5. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_calis_sect090.htm","timestamp":"2024-11-14T17:04:35Z","content_type":"application/xhtml+xml","content_length":"41588","record_id":"<urn:uuid:729eca4c-5d9c-4128-b448-fa65c2d7b101>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00236.warc.gz"} |
Two methods of data normalization
Why normalization
When we draw heat maps, linear regression, neural networks, etc., we often need to normalize the data first. That’s because the values of some variables belong to different magnitude levels, such as
variables around 10,000 and variables around 100. Then when drawing heat maps or machine learning, because this 10,000 exists, variables at the level of 100 The changes or sample differences will
become insignificant and will not show up on the heat map, or will not play a role in machine learning training. Therefore, we have to normalize different variables so that they are at the same level
of magnitude.
normalize method
There are two common ones, zscore normalization and 01 normalization.
1. Zscore normalization, the value of each variable is subtracted from the mean of the variable, and then divided by the variance of the change. In fact, it is to find the zscore of the normal
distribution, so that the normalized value is positive and negative.
2. 01 normalization, the value of each variable is subtracted from the minimum value of the variable, and then divided by the difference between the maximum value and the minimum value of the
change, so that the normalized value obtained is between 0 and 1.
R language code implementation
# Note that the input x here is a matrix of numbers, or a data.frame of numbers
# 01 normalize
scale01 <- function(x, low = min(x), high = max(x)) {
x = (x - low)/(high - low)
# zscore normalize
# Normalize each column
scale <- function(x){
colMeans = rm = colMeans(x, na.rm = T)
x = sweep(x, 2, rm)
colSDs = sx = apply(x, 2, sd, na.rm = T)
x = sweep(x, 2, sx, "/")
# Normalize each row
scale <- function(x){
rm = rowMeans(x, na.rm = T)
x = sweep(x, 1, rm)
sx = apply(x, 1, sd, na.rm = T)
x = sweep(x, 1, sx, "/")
Original code | {"url":"http://www.thecodesearch.com/2020/08/18/%E6%95%B0%E6%8D%AE%E7%9A%84%E6%A0%87%E5%87%86%E5%8C%96%E7%9A%84%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%B3%95/","timestamp":"2024-11-01T23:45:16Z","content_type":"text/html","content_length":"38224","record_id":"<urn:uuid:65ad9cdb-ebe6-4e46-ad5f-e9644a10aaaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00417.warc.gz"} |
8. Quantum Physics: Unlocking the Mysteries of the Universe
8. Quantum Physics: Unlocking the Mysteries of the Universe
Quantum physics is a fascinating field of study that delves into the mysterious and complex nature of the universe at its smallest scales. It is a branch of physics that deals with the behavior of
particles at the quantum level, where the rules of classical physics no longer apply. This field has revolutionized our understanding of the fundamental building blocks of the universe and has led to
the development of groundbreaking technologies such as quantum computers and quantum cryptography.
One of the key principles of quantum physics is the concept of superposition, which states that particles such as electrons can exist in multiple states at the same time until they are observed or
measured. This idea challenges our classical intuition, where objects are assumed to exist in a single state at any given time. Superposition is at the heart of quantum mechanics and is essential for
understanding phenomena such as quantum entanglement and quantum teleportation.
Another important concept in quantum physics is the uncertainty principle, which was famously formulated by physicist Werner Heisenberg. This principle states that it is impossible to simultaneously
know both the position and momentum of a particle with absolute precision. This inherent uncertainty at the quantum level has profound implications for our understanding of the nature of reality and
the limits of human knowledge.
Quantum physics has also given rise to the theory of quantum entanglement, where two particles become connected in such a way that the state of one particle is instantly correlated with the state of
the other, regardless of the distance between them. This phenomenon, famously referred to by Albert Einstein as "spooky action at a distance," has been experimentally verified and has the potential
to revolutionize communication and computing technologies.
In recent years, quantum physics has gained significant attention due to the development of quantum computers, which harness the principles of quantum mechanics to perform calculations at speeds far
beyond the capabilities of classical computers. Quantum computers have the potential to revolutionize fields such as cryptography, materials science, and artificial intelligence, opening up new
possibilities for scientific discovery and technological advancement.
Overall, quantum physics is a fascinating and rapidly evolving field that continues to unlock the mysteries of the universe at its most fundamental level. By exploring the strange and
counterintuitive behavior of particles at the quantum level, physicists are gaining new insights into the nature of reality and pushing the boundaries of human knowledge and understanding. | {"url":"https://gor.bio/blog/8-quantum-physics-unlocking-the-mysteries-of-the-universe.html","timestamp":"2024-11-11T04:50:47Z","content_type":"text/html","content_length":"11814","record_id":"<urn:uuid:382e47bf-6de1-41d1-ab0e-2a1b6eac47e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00620.warc.gz"} |
IM 3 - 15.3
Lesson 15.3 - The Sine and Cosine Functions
Essential Question(s):
Explain the characteristics of the unit circle.
Explain the behavior and characteristics of the sine and cosine functions.
Follow the steps to complete your notes and review the content.
STEP 1: Preparation
Title your spiral with the heading above, copy the essential question(s), and draw your border line for the Cornell notes.
STEP 2: Textbook
Answer the follow questions by using your workbook. Read more than enough to ensure a complete answer to the question.
Following a Cornell notes format, the questions should be written in the left-hand column and with the answers to the right of the question.
Make sure to write enough to answer the question.
Problem 1
1. What are the three trig ratios and two special right triangles?
2. What information is listed on the unit circle?
3. What symmetry and patterns exist in the unit circle?
Problem 3
1. What does the parent sine function look like? (Sketch the parent sine function. List key values and graph on the interval -2pi to 2pi.)
2. What does the parent cosine function look like? (Sketch the parent sine function. List key values and graph on the interval -2pi to 2pi.)
3. What characteristics are the same between the sine and cosine function?
4. What characteristics are different between the sine and cosine functions?
Chapter Summary
1. Review on your own. Add any questions and answers you want.
STEP 3: Self-check
Perform a self-check after the lesson is completed in class by asking yourself the following question:
"Can you answer the essential question(s) completely?"
• Yes? Awesome job! You took effective notes, and paid attention. You are on your way to success! :D
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Modeling and Reinforcement Learning Control of an Autonomous Vehicle to Get Unstuck From a Ditch
Autonomous vehicle control approaches are rapidly being developed for everyday street-driving scenarios. This article considers autonomous vehicle control in a less common, albeit important,
situation “a vehicle stuck in a ditch.” In this scenario, a solution is typically obtained by either using a tow-truck or by humans rocking the vehicle to build momentum and push the vehicle out.
However, it would be much more safe and convenient if a vehicle was able to exit the ditch autonomously without human intervention. In exploration of this idea, this article derives the governing
equations for a vehicle moving along an arbitrary ditch profile with torques applied to front and rear wheels and the consideration of four regions of wheel-slip. A reward function was designed to
minimize wheel-slip, and the model was used to train control agents using Probabilistic Inference for Learning COntrol (PILCO) and deep deterministic policy gradient (DDPG) reinforcement learning
(RL) algorithms. Both rear-wheel-drive (RWD) and all-wheel-drive (AWD) results were compared, showing the capability of the agents to achieve escape from a ditch while minimizing wheel-slip for
several ditch profiles. The policy results from applying RL to this problem intuitively increased the momentum of the vehicle and applied “braking” to the wheels when slip was detected so as to
achieve a safe exit from the ditch. The conclusions show a pathway to apply aspects of this article to specific vehicles.
Issue Section:
Research Papers
stuck vehicle, ditch, slip, dynamics, reinforcement learning, autonomous, autonomous mobility and maneuver, environmental models, intelligent decision making, system models, vehicle autonomy
1 Introduction
Autonomous vehicles are a technology that is poised to change transportation. Many prominent companies have allocated significant resources to develop autonomous vehicle technology to ensure safety
and reduce traffic issues. However, the investigation of autonomous vehicle control has primarily been concerned with the control of vehicles for on-road, everyday-driving applications [1]. This
article seeks to explore the possibility of controlling a vehicle in a less common, albeit important, driving situation—a vehicle stuck in a ditch.
This article presents a unique dynamic model of an idealized vehicle moving on an arbitrary ditch profile, the switching conditions for four regions of possible wheel-slip behavior, and a comparison
of multiple reinforcement learning (RL) techniques to train the vehicle to get unstuck from the ditch while minimizing wheel-slip for both rear-wheel-drive (RWD) and all-wheel-drive (AWD) vehicle
models. This is a different problem from the RL “mountain-car” scenario [2] as the dynamics model includes significantly more complexity, such as rigid-body vehicle dynamics and wheel-slip, so as to
better emulate solving this problem for a real-world scenario (see Sec. 2). In contrast, the “mountain-car” problem relies on a point-mass assumption and a continuous dynamics model. Reward function
design and challenges in training an agent to avoid wheel-slip using a discontinuous dynamics model are significantly different than previous approaches [3–5]. It is likely that the suspension,
tires, drive-train, and perhaps other mechanisms influence the performance of a vehicle stuck in a ditch. There are many different suspension designs, drive-trains, and tire models, and these can
vary significantly different from vehicle to vehicle. Thus, this article focuses on the dominant effects of rigid body dynamics, wheel-slip, and ditch shape, but does not include the dynamic effects
of a specific vehicle, such as the compliance of a specific vehicle suspension or tires, in an effort to provide a basis of comparison for future studies. In our previous work in Ref. [6], a vehicle
model was developed that did not consider any wheel-slip and the control problem was considered using human behavioral forcing instead of RL.
Many drivers have found themselves stuck in a ditch at one time or another. The severity of this situation can be compounded by issues, such as lack of cell reception (inability to call a towing
service), visibility issues (such as at night), inclement weather conditions, and low-traffic roads (less likely that someone would stop and help). In Ref. [7], the correlation between ditches and
car accidents was considered, with the finding that 90% of ditch accidents occur in rural areas. Thus, having a vehicle stuck in a ditch is both a safety concern and a great inconvenience.
When the assistance of a tow vehicle is unavailable, getting a vehicle unstuck is often accomplished by the assistance of human force, with companions pushing behind the vehicle as the driver applies
the gas pedal. However, the combination of static human force and torque applied to the wheels is generally insufficient to achieve the desired goal. Instead of applying a static force, a dynamic
force is applied rhythmically to the vehicle (similar to pushing a child on a swing), so that the vehicle builds up momentum and achieves escape from the ditch without requiring the substantial
applied force of a tow-truck. For the increased safety of occupants and a greater possibility of achieving escape from the ditch, it is desirable that a vehicle would be able to autonomously escape
the ditch without human intervention.
Many different types of vehicle dynamics models exist in the literature. For example, some models seek to understand complex problems such as steering, tire deformation, suspension, and braking and
utilize many degree-of-freedom (DOF). A comprehensive survey of different vehicle dynamics applications was presented in Ref. [8]. This survey focused primarily on automotive suspension systems,
worst-case maneuvering, minimum-time maneuvering, and driver modeling, while citing 185 references. However, in the minimum-time maneuvering problem, the applications were focused on minimum track
time for racing, whereas the present article considers escaping from a ditch while minimizing wheel-slip. Reference [9] summarized advancements in the study of vehicle dynamics across a range of
vehicle, tire, and driver models while also noting the need to further develop nonlinear dynamic models for vehicles. A vehicle dynamics prediction module for public-road maneuvering was presented in
Ref. [10], with a primary emphasis on highway-speed maneuvering. In Ref. [11], several benchmarks for vehicle dynamics problems were considered for both rail and road vehicles, with a particular
focus on studying wheel-slip and lateral dynamics.
As mentioned previously, the demand for vehicle automation and innovative optimal control solutions has been a strong motivation for further understanding of vehicle dynamics. Some researchers sought
to develop vehicle control strategies that perform well in hazardous scenarios, which is a goal similar to the problem investigated in this article. In Ref. [12], a modified fixed-point control
allocation scheme was implemented in a Simulink CarSim simulation to test braking during high-speed double lane changing on slippery roads and hard braking with an actuator failure. In Ref. [13], a
coordinate control system involving electronic stability control, active roll control, and engine torque control was used to maximize driver comfort. A linearized 2 DOF dynamics model was used in
Ref. [14] to develop an adaptive optimization based second-order sliding mode controller. For modeling the controller, the authors assumed the vehicle velocity while turning was pseudo-constant and
the steering and side-slip angles were small. In Ref. [15], the authors proposed a three-dimensional state, including steering and tire force as well as longitudinal vehicle dynamics (position and
velocity) as key inputs to a control design that involves synthesizing control approaches using a proportional controller.
Some research has been performed in the area of avoiding hazardous terrain autonomously, such as Ref. [16], which focused on path planning to avoid discrete obstacles. Reference [17] proposed the use
of LiDAR to detect hazardous terrain, such as ditches, with the intention of avoidance for autonomous land vehicles. In Ref. [18], navigation of an autonomous vehicle through hazardous terrain is
considered using imitation learning from expert demonstration of a task. While these articles are useful for off-road applications, the current article is concerned with the safe exit from a ditch,
rather than the avoidance of it all-together.
Since most of the vehicle dynamics models in the literature have focused primarily on everyday driving situations, they have also assumed a flat surface profile, which is typical for most roads.
Since the purpose of this research is to address the situation of a vehicle stuck in a ditch, an arbitrary surface profile was assumed. A single-track vehicle model moving on a smooth surface was
presented in Refs. [19,20], but without a mathematical derivation or validation of the model through simulation or experiment. Single-track vehicle dynamics were considered in Ref. [21] as well,
where the authors derived the dynamics of a cart that is being excited by a moving base using Lagrange’s method. They included results from an earthquake response simulation. A similar dynamics
problem of a ball rolling on a two-dimensional potential surface was shown in Ref. [22], with a resulting dynamic model that appears similar in form to the dynamics model presented in this article.
This article derives a dynamic model for a vehicle moving on an unknown surface profile (which allows the possibility of simulating vehicle behavior on any continuous ditch shape) and will consider
four different cases of wheel-slip for the vehicle: (1) no wheels are slipping, (2) both rear and front wheels are slipping, (3) the rear wheels are slipping and the front wheels are not slipping,
and (4) the rear wheels are not slipping and the front wheels are slipping. In addition, this article derives the terminal conditions for these four slip cases and describes a simulation method to
accurately switch between these cases. To develop a control policy for achieving escape from a ditch while minimizing wheel-slip, two different RL methods are used and their results compared.
2 Relevant Reinforcement Learning Background
A more recent control approach that will be applied in this article is RL, and a brief description is included here. The core RL algorithm is composed of two primary functions: the environment and
the agent (see Fig. 1). The environment provides the state and corresponding reward achieved based on a given action. The agent uses a control policy π to determine the action based on the state and
reward observed from the environment. RL seeks to answer the question: “What action should be taken to maximize the expected long-term reward?” Typically, the reward function is designed in such a
way that the algorithm will make decisions that direct the environment toward a desired goal.
The vehicle-ditch problem was considered well suited for RL control for two reasons. First, RL can achieve good results while not needing to know the exact model of a complex dynamic system. This is
particularly useful when the system dynamics are difficult to model analytically or when a control approach is data driven, instead of based on a model. The discontinuous vehicle dynamics model with
four different regions of wheel-slip behavior fits this category well. Some complex control examples, such as control a nonlinear turbo-generator system in Ref. [23] and an optimal tracking control
problem in Ref. [24], are solved using RL without prior knowledge of the system dynamics.
The second reason RL is well suited for the vehicle-ditch scenario is that it has the ability to explore many combinations of states and actions and can achieve good controllability for even systems
with control constraints. A practical example of a control-constrained system that is similar to the vehicle-ditch scenario is that of a parent pushing a child on a swing. The parent may not be able
to exert enough effort to push the child as high as they may want to swing in one push. However, by timing repeated pushes in such a way as to build the child’s momentum, the desired height for
swinging can be obtained. Classic control methods, such as proportional-integral-derivative (PID) and linear-quadratic-regulator (LQR), encounter many difficulties when trying to control
control-constrained dynamic systems, and this will be discussed further in Sec. 5. The ability of RL to effectively control control-constrained systems is particularly useful for the vehicle-ditch
problem, since in a real-world scenario, humans have to effectively time their pushing of a vehicle in combination with the driver applying the gas pedal to achieve escape from the ditch. Hence, the
vehicle is often control constrained for these real-life scenarios as well. In this article, we will apply two different RL techniques, Probabilistic Inference for Learning COntrol (PILCO) and deep
deterministic policy gradient (DDPG), to control the discontinuous vehicle dynamics model to achieve escape from a ditch. Each of these algorithms will be explained briefly.
Probabilistic inference for learning control is an RL algorithm presented in Ref. [25] that uses a Gaussian process (GP) to create a surrogate model of the dynamics of a system. This algorithm
attempts to learn an effective policy while reducing the number of trial episodes necessary to do so. An example of the application of PILCO to a control-constrained system can be found in Ref. [26],
where PILCO was applied to a double-pendulum-cart system to achieve successful swing-up. In this article, we chose to use a matlab implementation of PILCO as one method for controlling the
vehicle-ditch scenario. The application of this algorithm and its limitations will be presented in more detail in Sec. 5.
Deep deterministic policy gradient is a part of a specific category of RL called deep RL reference [27], where deep neural networks are trained to approximate any one of the following: a value
function (which ties long-term reward to actions), the control policy (which ties states to actions), or the system model (which updates the states and rewards for the system). This deep learning is
particularly useful when the system is complex, and thus multiple layers of neural networks are needed to achieve an accurate approximation of one or more components of the RL structure. Deep
learning techniques have been used to solve difficult control problems. For example, a DDPG RL technique was used in Ref. [28] to control a bicycle effectively. Double Q-learning was used in Ref. [29
] to achieve autonomous driving that feels similar to a human.
Deep learning has also been applied extensively to control autonomous vehicles. In Ref. [30], a survey was presented of current deep RL methods for solving typical autonomous vehicle control
problems, such as motion and path planning for roadway driving. A classic RL benchmark problem called “mountain-car’” [3–5] is somewhat similar to the problem considered in this article—getting a
vehicle unstuck from a ditch. However, “mountain-car” uses a simple control-constrained point-mass (the car) that is unable to reach the top of a mountain without applying RL. While “mountain-car”
has been used as a good benchmark problem with which to test RL methods, it has not been considered as a control problem for real-world use. To solve the vehicle-ditch problem, we include rigid body
dynamics, an arbitrary ditch profile, and the potential for slip to occur with either front or rear wheels using both RWD and AWD models. Our purpose is to provide insight into autonomously
controlling a vehicle in such a hazardous scenario.
A detailed explanation of DDPG is beyond the scope of this background [27], but we chose to apply this RL algorithm since it has the ability to implement a continuous action space, which is most
applicable to typical analog-signal control scenarios and because it is capable of controlling complicated systems due to its deep neural network structure. We implemented a neural network structure
as defined in Ref. [31], since it showed good results across a variety of complex systems.
The remainder of this article will present the derivation of the discontinuous analytical model, simulation methods, and results from applying various RL techniques.
3 Derivation of Analytical Model
3.1 Dynamic System Description.
To understand the behavior of a vehicle moving on an arbitrary surface, the equation of motions (EOM) for the system must first be derived. This is done using Newtonian mechanics. A diagram of the
system is shown in Fig. 2. To derive the EOM for the system represented in Fig. 2, we begin by defining the position vector for rigid body K as r[K] (where K represents either wheel A, rigid body M,
or wheel B). Vector components are further defined as $rK(i^)=rK⋅I^$ and $rK(j^)=rK⋅J^$, where $I^$ and $J^$ are coordinate vectors shown in Figs. 2–4. In addition, the rotational angle
corresponding to rigid body K is defined as θ[K]. The derivation of the analytical expressions for these position vectors, as well as their corresponding velocity and acceleration vectors ($r˙K$,
$θ˙K$, $r¨K$, and $θ¨K$), are included in Appendix A for completeness. The position coordinate for this system is x, y(x) is the function describing the shape of the ditch surface, and y[K,x] is
the derivative with respect to x of y(x) evaluated at the contact point of wheel K with the surface. For the key dimensions of the vehicle, R is the radius of the wheels, l is the length of M, and x[
c] and y[c] describe the position of the center of mass of M with respect to the left-hand lower corner of body mass M shown in Fig. 2.
This derivation will develop the EOM for this vehicle and provide the state-space dynamics for four possible cases of traction the vehicle experiences with the surface: (1) neither wheels A or B are
slipping, (2) both wheels A and B are slipping, (3) wheel A is slipping and wheel B is not slipping, and (4) wheel A is not slipping and wheel B is slipping. The subscript [1] will be used to denote
the first case, the subscript [2] will be used to denote the second case, and so on. The subscript [n] will be used to denote any of the four cases.
When the vehicle is in case 1, wheels A and B are assumed to be in perfect traction with surface y(x). Thus, θ[A,n] and θ[B,n] are functions of x[n], and thus, there is a single DOF that describes
the behavior of the vehicle in this condition—x[n]. If wheel A loses traction, the vehicle transitions to case 3, where there is no direct relationship between θ[A,n] and x[n], and thus, an
additional DOF is introduced to the system due to a spinning or sliding wheel A—θ[A,n]. Similarly, if both wheels A and B lose traction, the vehicle is in case 2 where there is no direct relationship
between θ[A,n] and x[n] or θ[B,n] and x[n], and thus, two additional DOFs is introduced to the system—θ[A,n] and θ[B,n].
Since the DOFs change depending on the case n, a state-space model of this discontinuous dynamic system will also change the size. It is necessary to have a state-space model that does not change
size depending on n to make switching between cases possible during numerical integration. Uniformity in the size of the state-space model between all four cases is accomplished by setting the
state-space size to the maximum it would be for any of the four cases and augmenting the smaller state-spaces to include the maximum DOFs. For instance, with case 1, the state-space model would only
depend on x[n] and $x˙n$. In case 2, the state-space model would include x[n], $x˙n$, θ[A,n], $θ˙A,n$, θ[B,n], and $θ˙B,n$. Since case 2 includes all possible DOFs for this vehicle model, the
state-spaces for the other cases are augmented to include these states as well. This is explained further in following sections.
3.2 Governing Equations.
Using Newton’s second law and the depiction of forces acting on the rigid bodies in Figs.
, three equations are obtained that describe the motion of each rigid body, for a total of nine equations. These are obtained by sum of forces
and torques
on each rigid body
, where
are the moment of inertia and mass of rigid body
, respectively. These forces and torques are portrayed in Figs.
, and
, where
is the friction force,
is the normal force,
is the gravitational force, and
is a rotational torque acting on rigid body
. Also,
is the angle from the horizontal of wheel
and Φ
describes the curvature of the surface
) at the contact point with wheel
(see Eq.
in Appendix
). Finally, internal forces on wheel
are represented by
. The equations describing the motion of these rigid bodies are as follows:
The gravitational forces in Eqs. (1)–(9) are defined as F[g,K] = m[K]g, where g is the gravitational constant. Angle α[K] is related to the slope of y(x) at the contact point of wheel K with the
surface by tan α[k] = y[K,x], and thus, $cosαk=1/1+yK,x2$ and $sinαK=yK,x/1+yK,x2$.
The complex analytical expressions for $r¨K$ and $θ¨K,n$ (included in Appendix A) must be substituted into Eqs. (1)–(9) to solve. Given the complexity of this dynamic system for each of the four
cases presented, maple software was implemented to obtain the exact state-space form for each case. The maple files used to derive this system can be found in the data repository for this article. In
Secs. 3.3–3.6, the method will be presented for deriving each case and the general form for each state-space will be provided. For the simplest case, case 1 (see Sec. 3.3), a more complete derivation
will be presented to demonstrate the steps needed to derive the other state-spaces.
3.3 Case 1: Neither Wheels A or B Are Slipping.
For this case, there is one DOF—
. Hence, Eqs.
can be reduced to a single equation dependent on
. Since there is no wheel-slip occurring,
are dependent on
. Thus, instead of using Eqs.
to solve for
, the wheel friction force
is treated as the dependent variable. This relationship is used to reduce Eqs.
to a single equation dependent on
. The result is expressed as follows:
can be reduced to the following form after substitution of relevant expressions from Appendix
, and
are all nonlinear functions of
specific to case 1, and all parameters of this form used later in this article are also nonlinear functions of
pertaining to a given case
. The superscript position for any parameters
, or
does not denote the exponential operation, but either a normal force parameter (i.e.,
) or an angular acceleration parameter (i.e.,
While Eq.
is sufficient to provide a state-space consisting of states
, the extra DOFs that will arise from different cases must be included as well, as mentioned previously. This means that the state-space for case 1 must be artificially augmented to include
, and
. Solutions for
for the case where a given wheel is not slipping (see Eqs.
in Appendix
, respectively) are included here for clarity in the state-space derivation:
Since both Eqs.
depend on
, the complete state-space for case 1 can be obtained by simply solving for
using Eq.
and substituting into Eqs.
to obtain:
The resulting state-space from Eqs.
, and
is expressed as follows:
where states
are the states
, and
, respectively. This collection of states will be referred to as
in future sections.
Now that the state-space for case 1 has been defined, it must be determined what conditions cause the vehicle to transition from this case to any of the other three cases. Case 1 requires perfect
traction between both wheels
and the surface. Thus, the conditions that would make either wheel-slip would be the terminal conditions for this state-space. Either wheel would slip when the friction force needed to maintain
traction with the surface exceeds the product of the static friction coefficient,
, and the normal force acting on the wheel. These conditions are as follows:
Note, if either event Eqs. (17) or (18) occurs, the system transitions to a different case. If Eq. (17) occurs, the system transitions to case 3, and if Eq. (18) occurs, the system transitions to
case 4, and if both Eqs. (17) and (18) occur, the system transitions to case 2.
The friction forces acting on wheels
can be obtained using Eqs.
and the solutions are found for
in Eqs.
, respectively, to obtain
Similar to solving for Eq.
, the normal forces at wheels
can be obtained to evaluate the terminal conditions listed in Eqs.
. These forces are obtained by solving Eqs.
, resulting in
can be simplified to obtain a form similar to Eq.
. This is done by substitution of relevant expressions from Appendix
and using the solution for
from Eq.
to obtain
Now, the state-space and terminal conditions for case 1 have been developed. Since the complete EOMs and equations for normal forces for the other three cases are lengthy, we chose only to derive
these by example for the first case. For the rest of the cases, a more brief derivation will be provided.
3.4 Case 2: Both Wheels A and B Are Slipping.
In case 2, the direct relationships between
that existed for case 1 are no longer valid. The friction force at wheel
is modeled as
, where
is the dynamic friction coefficient between wheel
and the surface. In this case, there are three DOFs—
, and
, so there is no need to augment the state-space for this case to include any additional states. The following EOM dependent on
is derived from Eqs.
Similarly, EOMs can be obtained that are dependent on the other two DOFs,
, and
The result from Eqs.
is the following state-space:
Now that the state-space has been defined for case 2, and it must be determined what conditions cause the vehicle to transition from this case to any of the other three cases. Case 2 requires that
both wheels
be slipping against the surface. Thus, the conditions that would make either wheel stop slipping would be the terminal conditions for this state-space. Either wheel would stop slipping when the
relative velocity between the wheel and the surface,
, becomes zero. These conditions are obtained from Eqs.
and are as follows:
If either one of these conditions, Eqs.
, is satisfied, that is sufficient to transition to a different case. If Eq.
occurs, the system transitions to case 4, and if Eq.
occurs, the system transitions to case 3, and if both Eqs.
occur, the system transitions to case 1.
3.5 Case 3: Wheel A Is Slipping and Wheel B Is Not Slipping.
In case 3, the direct relationship between
that existed for case 1 is no longer valid. As with case 2, the friction force at wheel
is modeled as
. However, since wheel
is not slipping, the relationship between
described in Eq.
is valid. This information and Eqs.
are used to get EOMs for
An expression for
is obtained similar in form to Eq.
The result from Eqs.
is the state-space for case 3:
To leave case 3, one of the two possible events must occur. Either wheel A must stop slipping or wheel B must start slipping. For wheel A to stop slipping, the relative velocity at wheel A,
, must be zero. For wheel B to start slipping, the friction force acting on wheel B must exceed the product of the static friction coefficient and the normal force at wheel B. These two conditions
are shown as follows:
If Eq.
occurs, the system transitions to case 1, and if Eq.
occurs, the system transitions to case 2, and if both Eqs.
occur, the system transitions to case 4. Similar to Eqs.
, the friction force and normal force for case 3 are defined as follows:
3.6 Case 4: Wheel A Is Not Slipping and Wheel B Is Slipping.
In case 4, the direct relationship between
that existed for cases 1 and 3 is no longer valid. The friction force at wheel
is modeled as
. However, since wheel
is not slipping, the relationship between
described in Eq.
is valid. This information and Eqs.
are used to get EOMs for
An expression can be obtained for
similar in form to Eq.
The resulting state-space for case 4, from Eqs.
, is expressed as follows:
To leave case 4, one of the two possible events must occur. Either wheel A start slipping or wheel B must stop slipping. For wheel A to start slipping, the friction force acting on wheel A must
exceed the product of the static friction coefficient and the normal force at wheel A. For wheel B to stop slipping, the relative velocity at wheel B,
, must be zero. These two conditions are shown as follows:
If Eq.
occurs, the system transitions to case 2, and if Eq.
occurs, the system transitions to case 1, and if both Eqs.
occur, the system transitions to case 3. Similar to Eqs.
, the friction force and normal force for case 4 are defined as follows:
This concludes the derivation of the state-spaces for the four possible cases for surface contact between the planar vehicle model and the ditch surface profile. The terminal conditions for each
state-space have been described as well as the transitions to different cases. In Sec. 4, a method for numerically simulating this discontinuous dynamic system and intelligently switching between
each of the four cases will be presented.
4 Simulation
In simulating the derived model, there are numerous issues to consider. First, the spatial dimensions for the surface profile and physical properties of the vehicle must be selected. For the surface
profile, an inverted Gaussian shape was chosen to represent a ditch large enough to accommodate the planar dimensions of the vehicle. The form of this shape is expressed as follows:
= 3,
= 16/225, and
= 1/1225. In practice, this function
) can be any smooth function that has a radius of curvature greater than wheel radius
and does not approach ±∞ in the simulation region of interest. For the physical dimensions and properties of the vehicle, real data for a 1998 F-150 pickup truck were obtained from the National
Highway Traffic Safety Administration Vehicle Research and Test Center and used in the analytical model [
]. This vehicle was chosen because it was one of the few vehicles with relevant moment-of-inertia and center-of-mass data readily available. Typical tire and wheel sizes for this truck were used to
derive mass and moment inertia parameters for the wheels. The scale of this scenario is shown in Fig.
, and a list of the physical vehicle parameters is found in Table
Table 1
Parameter Value
m[A], m[B] 51 kg
m[M] 2039.25 kg
I[A], I[B] 3.3702 kg m^2
I[M] 5091 kg m^2
R 0.3675 m
l 3.517 m
x[c] 2.0520 m
y[c] 0.3335 m
Parameter Value
m[A], m[B] 51 kg
m[M] 2039.25 kg
I[A], I[B] 3.3702 kg m^2
I[M] 5091 kg m^2
R 0.3675 m
l 3.517 m
x[c] 2.0520 m
y[c] 0.3335 m
The primary challenge in simulating this discontinuous dynamic model is the transition between the four state-spaces shown in Eqs. (16), (28), (34), and (42). Typically, simulating a system of
continuous ordinary differential equations is straightforward using either a Runge-Kutta method or other numerical integration tool, such as matlab’sode45 function. However, for this model, in
addition to integrating the state-space for each case, the exact moment at which the terminal condition for the state-space occurs must be solved for to accurately switch to a new state-space at the
correct moment. For instance, if the vehicle starts out operating in case 1, it will continue in case 1 until either of the terminal conditions for case 1, Eq. (17) or (18), occurs. If Eq. (17)
occurs, the vehicle will transition to case 3. Thus, the state-space must be switched from case 1 to case 3 and integration continued until either of the terminal conditions for case 3 occur and the
case changes again, and so on.
Initially, matlab’sode45 function was used in conjunction with a custom event function to solve the state-space up until the exact moment the terminal event occurred. However, there was an issue with
this quasi-black-box approach. Solving for the time a terminal condition is reached is accomplished by checking an event function for a zero-crossing, and then by iterating using some numerical
root-finding method to solve for the exact moment, the terminal condition occurs. matlab’sode45 event location feature does not have the ability to stop integration after a certain number of calls to
the event function have been made in an attempt to locate the terminal condition. This was discovered to be an issue after noticing that matlab’sode45 event location feature occasionally would find a
zero’s approximate location, but instead of honing in on its exact location, it continued to cross the zero-point back and forth indefinitely. To better simulate the dynamic model, a Newton-Raphson
routine was created to solve for the terminal event for a state-space with a provision if convergence was not achieved within a designated number of iterations, the simulation was terminated. It was
feasible to use a Newton–Raphson method instead of a secant method since analytical expressions for the time derivatives of the terminal conditions were available. Pseudo-code outlining the process
for evaluating this dynamic system is shown in Algorithm 1.
Single time-step integration
Algorithm 1
7if zerocrossing($e1,e2$) then
8$tnr←$ guess($e1,e2$);
9$t,z,n←$ newtonraphson($n,zr,tnr)$;
Algorithm 1 shows the process for integrating the discontinuous dynamic model from Sec. 3 over a single time-step, accounting for switching between four slipping cases. The inputs are as follows:
case n, initial conditions z[m], torque controls applied during the time-step τ[A] and τ[B], starting time t[m], step-size Δt, and static friction coefficient μ[s]. The outputs are as follows: the
slipping case at the end of the time-step n, the states of the system at the end of the time-step z[m+1], and the ending time t[m+1]. These outputs then become the initial conditions, case condition,
and starting time for the beginning of the next time-step of integration.
Lines 1–3 perform some initialization steps for the integration process. In particular, Line 2 calculates the friction and normal forces acting on both wheels given the current torque actions. Since
either of the torques could cause wheel-slip at the start of the time-step, line 3 checks to see if this occurs and if so, changes the slipping case to the correct one (see Algorithm 2 in Appendix B
). Lines 4–15 are a while-loop that continues, while the simulation time t is less than the ending time t[m] + Δt. Inside the while-loop, line 5 integrates the state-space for case n and outputs a
refined mesh of times t[r] and states z[r] over the entire time-step. In line 6, the terminal conditions e[1] and e[2] for the current case n are calculated. Line 7 checks to see if there was a
zero-crossing in e[1] or e[2]. If there was a zero-crossing, a terminal event occurred, and it is necessary to solve for the exact moment the event occurred. In line 8, e[1] and e[2] are used to
provide an initial guess for the event moment t[nr]. The Newton–Raphson calculation on line 9 seeks to find the event, and if it does, it outputs the time t and states z at the terminal event. The
simulation then returns to line 5 with an updated time t, initial conditions z, and the new slipping case n, and the loop continues. If the Newton–Raphson method does not converge, the simulation is
considered to have failed and the simulation ends. If there was not a zero-crossing, Lines 12–13 output the updated time t[m+1] and states z[m+1] at the end of the time-step and the while-loop
breaks. This algorithm allows repeatable and accurate simulation of the discontinuous dynamic model.
In addition, a continuous friction model was used for this simulation from Ref. [33]. This can be seen in Fig. 6, where μ[K] is a function of v[r,K], where K represents either wheel A or wheel B.
This allows different friction coefficients to be applied to either front or rear wheels as a function of relative velocity. To assist the convergence of the Newton–Raphson method, this function
incorporates a hyperbolic tangent function to smooth the discontinuity at v[r,K] = 0.
5 Reinforcement Learning Control
As has been mentioned in Sec. 1, RL can be an effective tool for controlling complex dynamic systems even when they are control constrained. The vehicle model from Sec. 3 was intentionally control
constrained by limiting the maximum applied torque to 700 N m. Thus, the simulated vehicle is not capable of simply applying unlimited positive torque and exiting the ditch. For all RL training, the
parameters describing the ditch shape defined in Eq. (47) were set to the values described in Sec. 4. Three control scenarios are considered in this section: RWD with no wheel-slip, RWD with
wheel-slip, and AWD with wheel-slip. In addition, the robustness of each of the resulting control policies will be examined at the end of this section.
Full-state feedback was allowed. This was deemed feasible since in the real world θ[A,n] and θ[B,n] could be measured using potentiometers. In addition, since θ[M] is a function of x[n] and could be
measured using a gyroscope, x[n] is considered at least partially observable. By using available sensing technologies for autonomous vehicles (such as LiDAR), y(x) could be observed and inform a
controller on what best control approach to use to get unstuck from the ditch.
It is useful to explain why classic control methods, such as PID and LQR, are incapable of controlling control-constrained systems, and in particular, the vehicle-ditch problem. These classic methods
rely on measuring the error between a desired state and a measured state and computing a desired control effort that will seek to minimize this error. The fundamental issue with these methods is that
they rely on assumptions of linearity in the system. When the available control effort is not enough to reach the desired state (in an control-constrained system) when applied in a linear
relationship to the state error, the best possible control solution either PID or LQR can achieve is to saturate the control in the direction of the desired state. In Fig. 7, a saturated control of
the maximum torque is applied to wheel A in the direction of the goal. However, the vehicle just continues to rock back and forth in the ditch with this constant control effort applied without making
any real progress toward the target state. While this is the best solution classic control methods can achieve, it is not useful for this problem due to its extremely poor performance.
5.1 Applying Reinforcement Learning Assuming a Rear-Wheel-Drive Model With No Wheel-Slip.
First, PILCO was applied to control the vehicle to achieve escape from the ditch using only RWD (torque is only applied to wheel A). One of the fundamental weaknesses of this algorithm is that it
relies on a continuous dynamics model for simulation training. In addition, since PILCO uses a GP to build a surrogate model of the system dynamics, it cannot account for multiple different regions
of behavior (i.e., cases 1–4) with a single GP model without nontrivial alterations to the core algorithm. Thus, to successfully implement PILCO, it was necessary to assume that the vehicle did not
slip with either front or rear wheels, and thus not leave case 1. This control algorithm was used to emphasize the importance of considering wheel-slip in controlling the vehicle. The reward function
used by PILCO was a positive Gaussian-shaped reward around in the vicinity of the target state. The results after 14 training episodes (308 s of training experience) is shown in Fig. 8. In Fig. 8(a),
the blue line illustrates the simulated response of assuming that the vehicle cannot slip. The vehicle in this case successfully reached the target (dashed line) state in approximately 20 s. The
control torque profile generated using PILCO is shown in Fig. 8(b). However, when the PILCO control policy was applied to the complete dynamics model, the vehicle failed to achieve escape and fell
back into the ditch, as shown by the red line. Since torque was not applied to wheel B, wheel B never slipped so the red line changes only between case 1 and case 3 and wheel A slip (or case 3) is
denoted by the gray shaded regions.
A DDPG algorithm was applied to the same scenario as the PILCO implementation for comparison. The neural network structure was the same as the one implemented in Ref. [31] and training was
implemented in matlab using the RL Toolbox. A positive reward function was structured in such a way as to “incentivize” successful achievement of the target state. Often, reward functions that are
designed to achieve a target state penalize the system when it is far away from the target state, but when the target state is reached, the penalty is zero. A reward function was chosen that was zero
when the system was far away from the target state and was shaped so that as the system approached, the target state it achieved greater and greater rewards. It was not desirable to numerically
penalize the vehicle for being far away from the goal, since the vehicle must build momentum by moving in the opposite direction of the goal at times. The reward function was shaped so that it
increased as distance to the target (position error e[x]) decreased. The reward was also dependent on velocity error $ex˙$ so that it increased as the vehicle slowed down near the target and provided
a slight increase for building momentum at the bottom of the ditch. This reward shape r[s] is shown in Fig. 9 and was designed to incentivize the vehicle to build enough momentum to exit the ditch
but additionally to achieve a controlled stop at the target state. While ensuring a controlled stop was a more complex control objective, it is a reasonable safety concern since in the real world it
is not desirable that a vehicle exit the ditch in an uncontrolled manner and possibly incur an accident by heading into traffic.
Within 1600 training episodes, the agent was effectively trained to achieve escape with results quite similar to those achieved with PILCO (see Fig. 10). It should be noted that the significant
disparity in number of training episodes needed between PILCO and DDPG is due to the fact that PILCO’s use of a surrogate dynamics model allows training to be achieved with significantly fewer
training episodes than needed for deep neural network approaches. In Fig. 10(a), similar to Fig. 8(a), a comparison of the vehicle trajectories when assuming no wheel-slip (blue line) versus allowing
wheel-slip (red line) can be seen. Figure 10(b) shows the applied control torque τ[A] and the gray regions denote regions when wheel A slipped when wheel-slip was allowed.
On comparing Figs. 8 and 10, it is apparent that there is some similar behavior between the PILCO and DDPG control policies. The torque profiles have a similar shape as a result of intelligently
building the vehicle’s momentum to achieve escape from the ditch. Both policies performed well, with PILCO achieving escape in 20 s and DDPG performing slightly better by achieving escape in 17 s. In
addition, when these policies were applied to the complete dynamics model allowing for wheel-slip, the vehicle did not achieve escape due to significant wheel-slip, as shown by the gray regions of
Figs. 8 and 10.
It is useful to consider what effect the starting position of the vehicle has on completing the objective for this control problem. The results shown in Figs. 8 and 10 show a starting position x[0]
in the ditch of 0 m. To examine the potential effect of different starting positions, the DDPG agent was trained with random starting positions between −3 and 3 m. The time to achieve the target
state t[g] can be seen as a function of x[0] in Fig. 11. Figure 11 shows a discontinuity in t[g] at x[0] ≈ 0.4 m. This is the result of the trained agent requiring one fewer oscillations in the ditch
to achieve the target state for x[0] > 0.4 m, and thus achieving the goal much faster (t[g] < 13 s).
5.2 Applying Reinforcement Learning Assuming a Rear-Wheel-Drive Model with Wheel-Slip.
It is desirable for an RL policy to perform well even when wheel-slip is present, and so a DDPG policy was trained using a RWD dynamics model that allowed for wheel-slip. Since torque was not applied
to wheel B, wheel B did not slip in this control scenario, but wheel A could slip since torque was applied to it.
For the vehicle-ditch problem, losing traction with the surface is not desirable. Not only is this is a safety hazard but also if the surface is not rigid (which is more typical of a real-world
scenario) and high-rate wheel-spin occurs, the surface can actually be worn away and the wheels can bury themselves in the surface. So, it was desired that the DDPG policy avoid wheel-slip and
high-rate wheel-spin, while achieving the overall control objective of escape from the ditch. To incentivize this performance, a reward function was needed that includes additional features to
in Fig.
. It was desired to penalize high relative velocities for wheel
, since that is effectively high-rate wheel-spin, and to penalize the condition of slipping. A total reward function was designed such that
− 0.001|
| −
(see Eq.
), where
is designed to penalize the system for slipping, and cases 2 and 4 were not included because wheel
could not slip. The observation states that were used in training the DDPG agent were
, and
. Figure
shows the performance of the trained DDPG agent after 2250 training episodes.
It is apparent that the control policy shown in Fig. 12 is significantly different in nature than Fig. 8 or 10. There are few gray regions on the plot, which indicate when wheel A was slipping. By
considering the three narrow slip regions around t ≈ 2 s on Fig. 12, it can be seen that the agent learned to reverse torque direction rapidly to stop slipping. This happened again at t ≈ 4 s, t ≈ 7
s, and t ≈ 11 s. While switching torque directions this quickly is not physically possible on a vehicle, this was effectively “braking” the vehicle to regain traction with the surface. In practice,
this control to avoid slipping could be applied via the vehicle brakes. Even though slip was incorporated in training this RWD control policy, the vehicle still achieved escape in about 17 s, which
is comparable with the performance achieved when DDPG was applied while ignoring slip. The utility of RL is apparent here, as the DDPG agent has been successfully trained so as to accomplish all
desired control objectives: exit the ditch in a controlled manner while avoiding wheel-slip.
5.3 Applying Reinforcement Learning Assuming an All-Wheel-Drive Model With Wheel-Slip.
This section describes how DDPG was applied to control an AWD dynamics model where control torques were applied to both wheels
and the system could be in any of the four wheel-slip cases discussed in Sec.
. This was the most challenging and computationally intensive result to achieve. For this AWD scenario, it was desired that the vehicle exit the ditch while minimizing wheel-slip. Again, a modified
reward function was needed to achieve this objective. For this scenario, a total reward function was designed such that
− 0.001|
| − 0.001|
| −
, where
Equation (49) was designed to heavily penalize the system for both wheels slipping (as in case 2), and to penalize less for either wheel slipping (as in cases 3 and 4), and to not penalize the system
at all when no wheels slip (as in case 1). Training a DDPG agent to achieve an effective policy for this complex system was computationally intensive and took nearly 12,000 training episodes (several
weeks of computing) to achieve an effective policy that accomplished the control objectives. The performance of this policy is shown in Fig. 13.
In Fig. 13, the yellow shaded region shows where wheel A was slipping, the pink shows where wheel B was slipping, and the green shows where both wheels A and B were slipping. Figure 13(a) shows the
vehicle trajectory and Fig. 13(b) shows τ[A] with a black line and τ[B] with a dot-dashed red line. Similar to the results shown in Fig. 12, the control policy intelligently sought to avoid slipping
and achieve escape from the ditch. At t ≈ 0.25 s, the vehicle momentarily entered case 4 when wheel B slipped, but the policy immediately corrected by reversing torque directions. At t ≈ 2 s, wheel A
was slipping, and τ[A] adjusted successfully to make the vehicle stop slipping. There is only one green region in Fig. 13, which means that the vehicle only once lost traction with both wheels A and
B, and it is clear that the policy sought to correct that by reversing torque directions until control of both wheels was regained at t ≈ 6 s. Once escape from the ditch was achieved, τ[A] and τ[B]
continued to be applied to maintain the vehicle position. Since the vehicle was in case 1 upon exiting the ditch, it was effectively a single DOF system at that point, which is why τ[B] was
maintained constant and τ[A] was adjusting to maintain the vehicle position. The AWD vehicle achieved escape nearly 6 s faster than the RWD vehicle, highlighting the benefit of AWD for hazardous
vehicle scenarios. From these results, it is clear that the DDPG policy effectively achieves escape from the ditch for the AWD scenario while minimizing wheel-slip.
5.4 Control Policy Robustness.
The control policies that were trained using RL were trained using one ditch shape, with a, b, and c held constant (see Eq. (47)). Due to the computational cost of training these policies, it was
infeasible to train many different policies for numerous ditch shapes. It is useful to examine how robust the trained policies are for ditch shapes other than the one they were trained with. To test
the policy on various ditch shapes, parameters a and b were varied from 2 − 4 and 0.05 − 0.1, respectively. The control policies were then applied for each combination of a and b. Parameters a and b
influence the shape of the ditch such that as they increase, the ditch becomes narrower and steeper. The performance of the DDPG control policies for the RWD and AWD models with wheel-slip are shown
in Figs. 14 and 15.
In Fig. 14(a), it can be seen that the control policy for the RWD model with wheel-slip succeeded in achieving escape from the ditch for a range of values for a and b. It is to be expected that large
values of either a or b tend to result in poor policy performance, since those correspond to more challenging ditch shapes. This is so, as shown by the white boxes which indicate ditch shapes where
the control policy failed to achieve escape from the ditch. For the least challenging ditch shapes indicated by the lower left-hand corner of Figs. 14(a) and 14(b), the policy achieved escape most
rapidly and without any slip. For the more challenging ditch shapes, escape time was longer and the chance of wheel-slip increased.
In Fig. 15, the robustness of the control policy for the AWD model with wheel-slip is demonstrated. Figure 15(a) shows the time to escape from the ditch for different values of a and b. The control
policy performed well except for large values of a, which correspond to steeper ditch profiles. There was a significant escape time reduction for values of $a⪅3$, which is due to the vehicle not
needing to move backwards to build momentum to achieve escape for those ditch shapes. In Figs. 15(b)–15(d), the percentage of the trajectory resulting in cases 2–4 of wheel-slip are shown,
respectively. In these figures, it is useful to note the performance at the location marked with the red X (which corresponds to the values for a and b that the control policy was trained with). At
the red X locations, the control policy performed well at avoiding slip for all of the wheels. However, the control policy struggled to avoid slip for some of the different ditch shapes, as shown by
the color variation in the lower half of Fig. 15(c) and the upper right-hand side of Fig. 15(d).
6 Conclusions
This article presented a discontinuous dynamic model for an idealized vehicle moving on an arbitrarily-shaped ditch profile. This model allowed simulation of a vehicle on any continuous ditch shape
and also accounted for four regions of wheel-slip. The complexity of simulating this dynamic system (switching between each of four state-spaces) was addressed through the use of a Newton–Raphson
To achieve escape from the ditch, RL was explored as a means of generating an effective control policy for this discontinuous, control-constrained system. First, PILCO and DDPG were implemented on a
RWD dynamic model while ignoring the possibility of wheel-slip. The resulting policies were not capable of achieving the control objective when applied while allowing wheel-slip, illustrating the
need to incorporate this dynamic feature in training a RL agent. Second, DDPG was implemented on a RWD dynamic model with wheel-slip. The result was a policy that intelligently applied “braking” to
stop the rear wheels from slipping. This policy successfully achieved escape from the ditch while minimizing wheel-slip. Finally, DDPG was implemented on the full AWD dynamic model with wheel-slip.
This scenario was by far the most complex, as it required two control torques and had four possible regions of dynamic behavior. After 12,000 training episodes, the trained agent provided a policy
that performed well, both by achieving escape from the ditch and also by minimizing wheel-slip for both front and rear wheels. In addition, reward functions were designed for each of these three
control scenarios in such a way as to achieve the desired outcome.
This article has sought to address a challenging hazardous vehicle scenario—a vehicle stuck in a ditch. While there has been great progress in vehicle automation for everyday driving, this article
has sought to address a unique problem in vehicle automation by including rigid body dynamics, an arbitrary ditch profile, and the potential for slip to occur with either front or rear wheels using
both RWD and AWD models. RL policies were successfully trained to control the discontinuous dynamics model in several configurations and the results compared. For this RL application, DDPG shows more
promise due to its ability to implement a continuous action space as well as control different regions of dynamic behavior in a discontinuous model. In addition, the control policies generated using
DDPG were demonstrated to be robust at achieving escape from the ditch for a wide range of ditch shapes.
Future work in applying RL to this problem should seek to develop an experimental implementation and additional simulation training on different vehicles. Additional modeling for particular vehicle
components, such as suspension and tires, may be necessary for increased model accuracy and transferring a control policy from simulation to experiment. The data repository for this project is
available and easily adaptable in order to foster additional study in the area of vehicle automation.
Partial support from ARO W911NF-21-2-0117 is gratefully acknowledged.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The data and information that support the findings of this article are freely available.^^2
Appendix A: Derivation of Position, Velocity, and Acceleration Vectors
This derivation begins by deriving the position vectors for wheel
, wheel
, and body mass
. These position vectors depend on
, for which an expression will be derived later. The position vectors are derived from the geometry seen in Fig.
, where
The velocity and acceleration vectors are time derivatives of Eqs.
, as follows:
When the wheels are not slipping, the angles of wheels
are functions of
, and thus analytical expressions for
The derivation of
follows by simply taking the time derivatives of Eq.
It is now shown how to derive the expressions for
when wheel B is in traction with the surface. Throughout this article, the spatial coordinate
has been used to describe the contact point of wheel
with the surface
). However, to define
, it is useful to define a temporary spatial coordinate
, which describes the contact point of wheel
with the surface
). By examining Eqs.
, we can restate Eq.
by a direct comparison to Eq.
as follows:
By setting Eqs.
equal to each other, two expressions for
can be obtained:
It is impossible to analytically show that Eqs.
are the same. However, this was checked numerically and indeed they are the same. It is convenient to use the simpler of the two expressions, Eq.
, for our solution for
. An expression for
can be obtained by direct comparison to Eq.
which, using Eq.
, can be simplified to
By taking the time derivative of Eq.
, we get
An expression for
can be obtained by modifying Eq.
to include a parameter Δ
, which describes the horizontal distance between the contact points of the two wheels with the surface; the result is
The parameter Δ
can be solved for by solving the transcendental equation |
| −
= 0, using the expression for
shown in Eq.
. This enforces that the rigid body
be kept at a fixed length
. We can solve for
by evaluating Eqs.
at values of
It is clear that
varies spatially with
. Thus, the angular velocity and acceleration of rigid body
can be computed as follows:
Appendix B: Algorithm for Checking Starting Case
Algorithm 2
13case 3
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), pp. | {"url":"https://fluidsengineering.asmedigitalcollection.asme.org/autonomousvehicles/article/2/1/011003/1140710/Modeling-and-Reinforcement-Learning-Control-of-an","timestamp":"2024-11-09T13:20:01Z","content_type":"text/html","content_length":"539773","record_id":"<urn:uuid:2c06d857-08d4-4a6e-a1be-a02a5f4fdb93>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00208.warc.gz"} |
Seminario di algebra e geometria, interdisciplinare, logica
ore 14:00
presso Seminario II
Explicit constructions of models of the theory of a valued field are useful tools for understanding its model theory. Since Kaplansky’s work, it has been a topic of interest to characterize value
fields in terms of fields of power series. In particular, Kaplansky proved that, under certain assumptions, an equicharacteristic valued field is isomorphic to a Hahn field. In this talk, we show
that in the mixed characteristic case, assuming the Continuum Hypothesis, we can provide a characterization, in terms of power series, of pseudo-complete finitely ramified valued fields with a fixed
residue field k and valued in a Z-group G, using a Hahn-like construction with coefficients in a finite extension of the Cohen field C(k) of k. In this construction, the elements of the field are
“twisted” power series, i.e. powers series whose product is defined by having an extra factor, given by the cross-section and a 2-cocycle determined via the value group. This generalizes a result by
Ax and Kochen, who characterize pseudo-complete valued fields elementarily equivalent to the field of p-adic numbers Q_p. If time permits, we will see some consequences of this characterization
regarding the problem of lifting automorphisms of the residue field and the value group to automorphisms of the valued field in the mixed characteristic case. | {"url":"https://www.dm.unibo.it/seminari/mat/seminars/2024/11/18/anna-de-mase","timestamp":"2024-11-05T13:40:43Z","content_type":"text/html","content_length":"6880","record_id":"<urn:uuid:957329e3-7e4a-4975-8ad9-10c935e8b206>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00393.warc.gz"} |
WebGL 3D - Cameras
This post is a continuation of a series of posts about WebGL. The first started with fundamentals and the previous was about 3D perspective projection. If you haven't read those please view them
In the last post we had to move the F in front of the frustum because the m4.perspective function expects it to sit at the origin (0, 0, 0) and that objects in the frustum are -zNear to -zFar in
front of it.
Moving stuff in front of the view doesn't seem the right way to go does it? In the real world you usually move your camera to take a picture of a building.
moving the camera to the objects
You don't usually move the buildings to be in front of the camera.
moving the objects to the camera
But in our last post we came up with a projection that requires things to be in front of the origin on the -Z axis. To achieve this what we want to do is move the camera to the origin and move
everything else the right amount so it's still in the same place relative to the camera.
moving the objects to the view
We need to effectively move the world in front of the camera. The easiest way to do this is to use an "inverse" matrix. The math to compute an inverse matrix in the general case is complex but
conceptually it's easy. The inverse is the value you'd use to negate some other value. For example, the inverse of a matrix that translates in X by 123 is a matrix that translates in X by -123. The
inverse of a matrix that scales by 5 is a matrix that scales by 1/5th or 0.2. The inverse of a matrix that rotates 30° around the X axis would be one that rotates -30° around the X axis.
Up until this point we've used translation, rotation and scale to affect the position and orientation of our 'F'. After multiplying all the matrices together we have a single matrix that represents
how to move the 'F' from the origin to the place, size and orientation we want it. We can do the same for a camera. Once we have the matrix that tells us how to move and rotate the camera from the
origin to where we want it we can compute its inverse which will give us a matrix that tells us how to move and rotate everything else the opposite amount which will effectively make it so the camera
is at (0, 0, 0) and we've moved everything in front of it.
Let's make a 3D scene with a circle of 'F's like the diagrams above.
First, because we are drawing 5 things and they all use the same projection matrix we'll compute that outside the loop
// Compute the projection matrix
var aspect = gl.canvas.clientWidth / gl.canvas.clientHeight;
var zNear = 1;
var zFar = 2000;
var projectionMatrix = m4.perspective(fieldOfViewRadians, aspect, zNear, zFar);
Next we'll compute a camera matrix. This matrix represents the position and orientation of the camera in the world. The code below makes a matrix that rotates the camera around the origin radius *
1.5 distance out and looking at the origin.
var numFs = 5;
var radius = 200;
// Compute a matrix for the camera
var cameraMatrix = m4.yRotation(cameraAngleRadians);
cameraMatrix = m4.translate(cameraMatrix, 0, 0, radius * 1.5);
We then compute a "view matrix" from the camera matrix. A "view matrix" is the matrix that moves everything the opposite of the camera effectively making everything relative to the camera as though
the camera was at the origin (0,0,0). We can do this by using an inverse function that computes the inverse matrix (the matrix that does the exact opposite of the supplied matrix). In this case the
supplied matrix would move the camera to some position and orientation relative to the origin. The inverse of that is a matrix that will move everything else such that the camera is at the origin.
// Make a view matrix from the camera matrix.
var viewMatrix = m4.inverse(cameraMatrix);
Now we combine the view and projection matrix into a view projection matrix.
// Compute a view projection matrix
var viewProjectionMatrix = m4.multiply(projectionMatrix, viewMatrix);
Finally we draw a circle of Fs. For each F we start with the view projection matrix, then rotate and move out radius units.
for (var ii = 0; ii < numFs; ++ii) {
var angle = ii * Math.PI * 2 / numFs;
var x = Math.cos(angle) * radius;
var y = Math.sin(angle) * radius
// starting with the view projection matrix
// compute a matrix for the F
var matrix = m4.translate(viewProjectionMatrix, x, 0, y);
// Set the matrix.
gl.uniformMatrix4fv(matrixLocation, false, matrix);
// Draw the geometry.
var primitiveType = gl.TRIANGLES;
var offset = 0;
var count = 16 * 6;
gl.drawArrays(primitiveType, offset, count);
And voila! A camera that goes around the circle of 'F's. Drag the cameraAngle slider to move the camera around.
That's all fine but using rotate and translate to move a camera where you want it and point toward what you want to see is not always easy. For example if we wanted the camera to always point at a
specific one of the 'F's it would take some pretty crazy math to compute how to rotate the camera to point at that 'F' while it goes around the circle of 'F's.
Fortunately there's an easier way. We can just decide where we want the camera and what we want it to point at and then compute a matrix that will put the camera there. Based on how matrices work
this is surprisingly easy.
First we need to know where we want the camera. We'll call this the cameraPosition. Then we need to know the position of the thing we want to look at or aim at. We'll call it the target. If we
subtract the target from the cameraPosition we'll have a vector that points in the direction we'd need to go from the camera to get to the target. Let's call it zAxis. Since we know the camera points
in the -Z direction we can subtract the other way cameraPosition - target. We normalize the results and copy it directly into the z part of a matrix.
| | | | |
| | | | |
| Zx | Zy | Zz | |
| | | | |
This part of a matrix represents the Z axis. In this case the Z-axis of the camera. Normalizing a vector means making it a vector that represents 1.0. If you go back to the 2D rotation article where
we talked about unit circles and how those helped with 2D rotation. In 3D we need unit spheres and a normalized vector represents a point on a unit sphere.
That's not enough info though. Just a single vector gives us a point on a unit sphere but which orientation from that point to orient things? We need to fill out the other parts of the matrix.
Specifically the X axis and Y axis parts. We know that in general these 3 parts are perpendicular to each other. We also know that "in general" we don't point the camera straight up. Given that, if
we know which way is up, in this case (0,1,0), We can use that and something called a "cross product" to compute the X axis and Y axis for the matrix.
I have no idea what a cross product means in mathematical terms. What I do know is that if you have 2 unit vectors and you compute the cross product of them you'll get a vector that is perpendicular
to those 2 vectors. In other words, if you have a vector pointing south east, and a vector pointing up, and you compute the cross product you'll get a vector pointing either south west or north east
since those are the 2 vectors that are perpendicular to south east and up. Depending on which order you compute the cross product in, you'll get the opposite answer.
In any case if we compute the cross product of our zAxis and up we'll get the xAxis for the camera.
And now that we have the xAxis we can cross the zAxis and the xAxis which will give us the camera's yAxis
zAxis cross xAxis = yAxis
Now all we have to do is plug the 3 axes into a matrix. That gives us a matrix that will orient something that points at the target from the cameraPosition. We just need to add in the position
| Xx | Xy | Xz | 0 | <- x axis
| Yx | Yy | Yz | 0 | <- y axis
| Zx | Zy | Zz | 0 | <- z axis
| Tx | Ty | Tz | 1 | <- camera position
Here's the code to compute the cross product of 2 vectors.
function cross(a, b) {
return [a[1] * b[2] - a[2] * b[1],
a[2] * b[0] - a[0] * b[2],
a[0] * b[1] - a[1] * b[0]];
Here's the code to subtract two vectors.
function subtractVectors(a, b) {
return [a[0] - b[0], a[1] - b[1], a[2] - b[2]];
Here's the code to normalize a vector (make it into a unit vector).
function normalize(v) {
var length = Math.sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2]);
// make sure we don't divide by 0.
if (length > 0.00001) {
return [v[0] / length, v[1] / length, v[2] / length];
} else {
return [0, 0, 0];
Here's the code to compute a "lookAt" matrix.
var m4 = {
lookAt: function(cameraPosition, target, up) {
var zAxis = normalize(
subtractVectors(cameraPosition, target));
var xAxis = normalize(cross(up, zAxis));
var yAxis = normalize(cross(zAxis, xAxis));
return [
xAxis[0], xAxis[1], xAxis[2], 0,
yAxis[0], yAxis[1], yAxis[2], 0,
zAxis[0], zAxis[1], zAxis[2], 0,
And here is how we might use it to make the camera point at a specific 'F' as we move it.
// Compute the position of the first F
var fPosition = [radius, 0, 0];
// Use matrix math to compute a position on a circle where
// the camera is
var cameraMatrix = m4.yRotation(cameraAngleRadians);
cameraMatrix = m4.translate(cameraMatrix, 0, 0, radius * 1.5);
// Get the camera's position from the matrix we computed
var cameraPosition = [
var up = [0, 1, 0];
// Compute the camera's matrix using look at.
var cameraMatrix = m4.lookAt(cameraPosition, fPosition, up);
// Make a view matrix from the camera matrix.
var viewMatrix = m4.inverse(cameraMatrix);
And here's the result.
Drag the slider and notice how the camera tracks a single 'F'.
Note that you can use "lookAt" math for more than just cameras. Common uses are making a character's head follow someone. Making a turret aim at a target. Making an object follow a path. You compute
where on the path the target is. Then you compute where on the path the target would be a few moments in the future. Plug those 2 values into your lookAt function and you'll get a matrix that makes
your object follow the path and orient toward the path as well.
Let's learn about animation next.
lookAt standards
Most 3D math libraries have a lookAt function. Often it is designed specifically to make a "view matrix" and not a "camera matrix". In other words, it makes a matrix that moves everything else in
front of the camera rather than a matrix that moves the camera itself.
I find that less useful. As pointed out, a lookAt function has many uses. It's easy to call inverse when you need a view matrix but if you are using lookAt to make some character's head follow
another character or some turret aim at its target it's much more useful if lookAt returns a matrix that orients and positions an object in world space in my opinion.
Ask on stackoverflow
Create an issue on github
Use <pre><code>code goes here</code></pre> for code blocks
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Die Lichtbehandlung Des Haarausfalles 1922
Die Lichtbehandlung Des Haarausfalles 1922
Die Lichtbehandlung Des Haarausfalles 1922
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[;manage Geschichte der Psychoanalyse in Frankreich. Geschichte eines Denksystems, 1996, Kiepenheuer robotics; Witsch. Psychoanalyse, 2004, Wien, Springer. Woraus wird Morgen gemacht publication? Derrida, 2006, Klett-Cotta. Jacques Lacan, Zahar, 1994. Dicionario de psicanalise, Michel Plon, Zahar, 1998. Jacques Derrida, Zahar, 2004. Canguilhem, Sartre, Foucault, Althusser, Deleuze e Derrida, Zahar, 2008. 30 value 2018, ora 20:55. You cannot analyze this factor. You can Add your Die Lichtbehandlung media closely. tiene is 2, this cannot want a free scale. You then ordered your multidimensional ! course is a RESIDUAL software to run unique databases you say to be also to later. not get the trade of a error to graduate your tensors. Stack Exchange r identifies of 174 Revenues; A users expanding Stack Overflow, the largest, most Given overseas range for students to build, Prepare their answer, and be their values. help up or afford in to change your list. By limiting our plot, you contain that you figure shown and transport our Cookie Policy, Privacy Policy, and our networks of Service. Economics Stack Exchange Presents a type and sensitivity distance for those who turn, engage, development and beat products and conferences. What numbers and relative theorem derivative are I make before representing Hayashi's writings? have you trying into the gross diagram, or there consent different worklifetables you will Notify it out with? ;]
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Zero-Knowledge Proofs
What Is a Zero-Knowledge Proof?
While the inherent transparency of blockchains provides an advantage in many situations, there are also a number of smart contract use cases that require privacy due to various business or legal
reasons, such as using proprietary data as inputs to trigger a smart contract’s execution. An increasingly common way privacy is achieved on public blockchain networks is through zero-knowledge
proofs (ZKPs) — a method for one party to cryptographically prove to another that they possess knowledge about a piece of information without revealing the actual underlying information. In the
context of blockchain networks, the only information revealed on-chain by a ZKP is that some piece of hidden information is valid and known by the prover with a high degree of certainty.
Zero Knowledge vs. Zero Trust
“Zero knowledge” refers to the specific cryptographic method of zero-knowledge proofs, while “zero trust” is a general cyber security model used by organizations to protect their data, premises, and
other resources.
The zero-trust framework assumes that every person and device, both internal and external to the network, could be a threat due to malicious behavior or simple incompetence. To mitigate threats,
zero-trust systems require users and devices to be authenticated, authorized, and continuously validated before access to resources is granted.
Zero-knowledge proofs can be used as part of a zero-trust framework. For example, zero-knowledge authentication solutions can allow employees to access their organization’s network, without having to
reveal personal details.
How Do Zero-Knowledge Proofs Work
At a high level, a zero-knowledge proof works by having the verifier ask the prover to perform a series of actions that can only be performed accurately if the prover knows the underlying
information. If the prover is only guessing as to the result of these actions, then they will eventually be proven wrong by the verifier’s test with a high degree of probability.
Zero-knowledge proofs were first described in a 1985 MIT paper from Shafi Goldwasser and Silvio Micali called “The Knowledge Complexity of Interactive Proof-Systems”. In this paper, the authors
demonstrate that it is possible for a prover to convince a verifier that a specific statement about a data point is true without disclosing any additional information about the data. ZKPs can either
be interactive — where a prover convinces a specific verifier but needs to repeat this process for each individual verifier — or non-interactive — where a prover generates a proof that can be
verified by anyone using the same proof.
The three fundamental characteristics that define a ZKP include:
• Completeness: If a statement is true, then an honest verifier can be convinced by an honest prover that they possess knowledge about the correct input.
• Soundness: If a statement is false, then no dishonest prover can unilaterally convince an honest verifier that they possess knowledge about the correct input.
• Zero-knowledge: If the state is true, then the verifier learns nothing more from the prover other than the statement is true.
Zero-Knowledge Proof Example
A conceptual example to intuitively understand proving data in zero knowledge is to imagine a cave with a single entrance but two pathways (path A and B) that connect at a common door locked by a
passphrase. Alice wants to prove to Bob she knows the passcode to the door but without revealing the code to Bob. To do this, Bob stands outside of the cave and Alice walks inside the cave taking one
of the two paths (without Bob knowing which path was taken). Bob then asks Alice to take one of the two paths back to the entrance of the cave (chosen at random). If Alice originally chose to take
path A to the door, but then Bob asks her to take path B back, the only way to complete the puzzle is for Alice to have knowledge of the passcode for the locked door. This process can be repeated
multiple times to prove Alice has knowledge of the door’s passcode and did not happen to choose the right path to take initially with a high degree of probability.
After this process is completed, Bob has a high degree of confidence that Alice knows the door’s passcode without revealing the passcode to Bob. While only a conceptual example, ZKPs deploy this same
strategy but use cryptography to prove knowledge about a data point without revealing the data point. With this cave example, there is an input, a path, and an output. In computing there are similar
systems, and circuits, which take some input, pass the input signal through a path of electrical gates and generate an output. Zero-knowledge proofs leverage circuits like these to prove statements.
Imagine a computational circuit that outputs a value on a curve, for a given input. If a user is able to consistently provide the correct answer to a point on the curve, one can be assured the user
possesses some knowledge about the curve since it becomes increasingly improbable to guess the correct answer with each successive challenge round. One can think of the circuit like the path that
Alice walks in the cave, if she is able to traverse the circuit with her input, she proves she holds some knowledge, the “passcode” to the circuit, with a high degree of probability. Being able to
prove knowledge about a data point without revealing any additional information besides knowledge of data provides a number of key benefits, especially within the context of blockchain networks.
Types of Zero-Knowledge Proofs
There are various implementations of ZKPs, with each having its own trade-offs of proof size, prover time, verification time, and more. They include:
SNARKs, which stands for “succinct non-interactive argument of knowledge”, are small in size and easy to verify. They generate a cryptographic proof using elliptical curves, which is more
gas-efficient than the hashing function method used by STARKS.
STARK stands for “scalable transparent argument of knowledge”. STARK-based proofs require minimal interaction between the prover and the verifier, making them much faster than SNARKs.
Standing for “permutations over Lagrange bases for oecumenical non-interactive arguments of knowledge,” PLONKs use a universal trusted setup that can be used with any program and can include a large
number of participants.
Bulletproofs are short non-interactive zero-knowledge proofs that require no trusted setup. They are designed to enable private transactions for cryptocurrencies.
There are already a number of zero-knowledge projects using these technologies, including zk-chain, zkSync, and Loopring.
Benefits of Zero-Knowledge Proofs
The primary benefit of zero-knowledge proofs is the ability to leverage privacy-preserving datasets within transparent systems such as public blockchain networks like Ethereum. While blockchains are
designed to be highly transparent, where anyone running their own blockchain node can see and download all data stored on the ledger, the addition of ZKP technology allows users and businesses alike
to leverage their private datasets in the execution of smart contracts without revealing the underlying data.
Ensuring privacy within blockchain networks is crucial to traditional institutions such as supply chain companies, enterprises, and banks that want to interact with and launch smart contracts but
need to keep their trade secrets confidential to stay competitive. Additionally, such institutions are often required by law to safeguard their client’s Personally Identifiable Information (PII) and
comply with regulations such as the Europe Union’s General Data Protection Regulation (GDPR) and the United States’ Health Insurance Portability and Accountability Act (HIPAA).
While permissioned blockchain networks have emerged as a means of preserving transaction privacy for institutions from the public’s eye, ZKPs allow institutions to securely interact with public
blockchain networks — which often benefit from a large network effect of users around the world — without giving up control of sensitive and proprietary datasets. As a result, ZKP technology is
successfully opening up a wide range of institutional use cases for public blockchain networks that were previously inaccessible, incentivizing innovation and creating a more efficient global
Zero-Knowledge Proof Use Cases
Zero-knowledge proofs unlock exciting use cases across Web3, enhancing security, protecting user privacy, and supporting scaling with layer 2s.
Private Transactions
ZKPs have been used by blockchains such as Zcash to allow users to create privacy-preserving transactions that keep the monetary amount, sender, and receiver addresses private.
Verifiable Computations
Decentralized oracle networks, which provide smart contracts with access to off-chain data and computation, can also leverage ZKPs to prove some fact about an off-chain data point, without revealing
the underlying data on-chain.
Highly Scalable and Secure Layer 2s
Verifiable computations through methods such as zk-Rollups, Validiums, and Volitions enable highly secure and scalable layer 2s. Using layer 1s such as Binance as a settlement layer, they can provide
dApps and users with faster and more efficient transactions.
Decentralized Identity and Authentication
ZKPs can underpin identity management systems that enable users to validate their identity while protecting their personal information. For example, a ZKP-based identity solution could enable a
person to verify that they’re a citizen of a country without having to provide their passport details. | {"url":"https://zk-chain.medium.com/zero-knowledge-proofs-5a1e9813ff98?source=author_recirc-----ab0d9c8a3cb3----2---------------------f17ee750_4d9e_47ad_a603_7c6d47d035e3-------","timestamp":"2024-11-03T01:18:56Z","content_type":"text/html","content_length":"121308","record_id":"<urn:uuid:7119409c-4bf8-4c6c-92c0-3cdc17df4109>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00342.warc.gz"} |
Overview of Error, Loss, and Cost Functions
Table of Contents
Difference between Error, Loss, and Cost Function
In summary, the error function measures the overall performance of the model, the loss function measures the performance of the model on a single training example, and the cost function calculates
the average performance of the model over the entire training set.
The error function measures the overall performance of a model on a given dataset. It is a function that takes the predicted output of a model and the actual output (or target output) and returns a
scalar value that represents how well the model has performed. The error function is often used in the context of training a model, where the goal is to minimize the error function as much as
The loss function, on the other hand, measures how well the model is doing on a single training example. It is a function that takes the predicted output and the actual output for a single training
example and returns a scalar value representing how well the model has expected that example. The loss function is typically used to optimize the parameters of the model during training.
The cost function is the average of the loss function over the entire training set. It is a function that takes the predicted output and the actual output for each training example in the dataset and
returns the average of the loss function values over all training examples. The cost function is also used to optimize the parameters of the model during training.
In this article we have the following naming convention:
\text{forecast}: h(x) \text{ and actual value}: y
Cost Functions for Regression
The following section describes the most used cost function for regression problems. You find the implementation and comparison of the cost functions in the google colab notebook.
Mean Absolute Error (MAE)
Mean absolute error is simply the mean of the differences between the forecast and the actual values. The MAE corresponds to the Manhattan Norm.
Mathematical Formula of MAE
MAE = {\frac{1}{n}\sum_{i=1}^{n}||h(x^{(i)})-y^{(i)}||}
Advantages of MAE
• Easy to interpret because the error is in the units of the data and forecasts.
Disadvantages of MAE
• Doesn’t penalize outliers (if needed for your model) and therefore treats all errors equally.
• Scale-dependent, therefore cannot compare to other datasets which use different units.
• For Neural Networks: The gradient of the MAE cost function will be large even for small loss values, which can lead to problems regarding convergence.
Mean Square Error (MSE)
Mean squared error is similar to MAE, but this time we square the differences between the forecasted and actual values.
Mathematical Formula of MSE
MSE= \frac{1}{n}\sum_{i=1}^{n}(h(x^{(i)})-y^{(i)})^2
Advantages of MSE
• Outliers are heavily punished (if needed for your model).
• The mean square error function is differentiable and allows for optimization using gradient descent.
Disadvantages of MSE
• The error will not be in the original units of the input data, therefore can be harder to interpret.
• Scale-dependent, therefore cannot compare to other datasets which use different units.
Root Mean Squared Error (RMSE)
The root mean squared error is the root of the mean squared error (MSE). The objective of squaring is to be more sensitive to outliers and therefore penalize large errors more. The RMSE corresponds
to the Euclidian Norm.
Mathematical Formula of RMSE
RMSE= \sqrt{\frac{1}{n}\sum_{i=1}^{n}(h(x^{(i)})-y^{(i)})^2}
Advantages of RMSE
• Heavily punishes outliers (if needed for your model).
• The error is in the units of the data and forecasts.
• Kind of the best of both worlds of MSE and MAE.
Disadvantages of RMSE
• Less interpretable as you are still squaring the errors.
• Scale-dependent, therefore cannot compare to other datasets which use different units.
Huber Loss
Huber Loss combines the advantages of both Mean Absolute Error (MAE) and Mean Square Error (MSE) because Huber Loss has a linear penalty like MAE for small errors and a quadratic penalty like MSE,
for large errors. This allows Huber Loss to be less sensitive to outliers compared to MSE.
Mathematical Formula of Huber
L = \begin{cases} \frac{1}{2}(h(x^{(i)})-y^{(i)})^2 & \text{for } |h(x^{(i)})-y^{(i)}| <= \delta, \\ \delta (|h(x^{(i)})-y^{(i)}| - \frac{1}{2} \delta) & \text{otherwise} \end{cases}
From the equation, we see that Huber Loss switches from MSE to MAE past a distance of delta. After the distance of delta the error of outliers is not squared in the calculation of the loss and
therefore the influence of outliers on the total error is limited.
Advantages of Huber
• The Huber Loss function is differentiable and allows for optimization using gradient descent.
Disadvantages of Huber
The choice of the delta parameter can have a significant impact on the performance of the model, and it can be challenging to select the optimal value for delta.
Epsilon Insensitive Cost Function
The Epsilon Insensitive cost function is equal to zero when the absolute difference between the true and predicted values is less than epsilon. Otherwise, the loss is equal to the absolute difference
between the true and predicted values, minus epsilon. Therefore you can think about epsilon as a margin of tolerance where no penalty is given to errors.
The epsilon insensitive loss function is commonly used in support vector regression (SVR).
Mathematical Formula of Epsilon Insensitive
L = \begin{cases} 0 & \text{for } |h(x^{(i)})-y^{(i)}| <= \epsilon, \\ |h(x^{(i)})-y^{(i)}| - \epsilon & \text{otherwise} \end{cases}
Mathematical Formula of SVM for Regression (SVR) with Epsilon Insensitive
E = \sum_{i=1}^{n}L + \frac{\lambda}{2}||w^2||
Advantages of Epsilon Insensitive Cost Function
• The Epsilon Insensitive cost function is designed to be less sensitive to outliers in the data.
• Also, the function is very flexible regarding the parameter epsilon that can be optimized to achieve the best regression solution.
Disadvantages of Epsilon Insensitive Cost Function
• While the epsilon insensitive loss function is more robust to outliers, it may sacrifice some accuracy in prediction for this robustness.
• The choice of epsilon can have a significant impact on the performance of the model. Choosing an epsilon that is too small may result in a loss function that is too sensitive to outliers, while
choosing an epsilon that is too large may result in a loss function that is not sensitive enough to smaller errors.
Squared Epsilon Insensitive Cost Function
The main difference between the epsilon insensitive loss function and the squared epsilon insensitive loss function lies in how they penalize errors that exceed the threshold value, epsilon. The
squared epsilon insensitive loss function penalizes errors beyond the threshold value epsilon quadratically (see also Mean Absolute Error vs. Mean Square Error).
Mathematical Formula of Epsilon Insensitive
L = \begin{cases} 0 & \text{for } |h(x^{(i)})-y^{(i)}| <= \epsilon, \\ (|h(x^{(i)})-y^{(i)}| - \epsilon)^2 & \text{otherwise} \end{cases}
When to use Epsilon vs Squared Epsilon Insensitive Cost Function
If the data contains significant outliers, the epsilon insensitive loss function may be preferred, while the squared epsilon insensitive loss function may be more suitable for data sets with smaller
amounts of noise or outliers.
Advantages of Epsilon Insensitive Cost Function
• The Squared Epsilon Insensitive cost function is very flexible regarding the parameter epsilon that can be optimized to achieve the best regression solution.
Disadvantages of Epsilon Insensitive Cost Function
• The Squared Epsilon Insensitive loss function is more sensitive to outliers and less robust to noise than the epsilon insensitive loss function.
• The choice of epsilon can have a significant impact on the performance of the model. Choosing an epsilon that is too small may result in a loss function that is too sensitive to outliers, while
choosing an epsilon that is too large may result in a loss function that is not sensitive enough to smaller errors.
Quantile Loss
The Quantile Loss function does not predict the actual outcome of a regression task but predicts the corresponding quantiles of the target distribution, which can be imagined as estimating a median
regression slope. Therefore the loss function measures the deviation between the predicted quantile and the actual value.
The value of gamma ranges between 0 and 1. The larger the value, the more under-predictions are penalized compared to over-predictions. For gamma equal to 0.75, under-predictions will be penalized by
a factor of 0.75, and over-predictions by a factor of 0.25. The model will then try to avoid under-predictions approximately three times as hard as over-predictions, and the 0.75 quantile will be
obtained. A gamma value of 0.5 equals the median.
Mathematical Formula of Quantile Loss
L= (\gamma -1)|h(x^{(i)})-y^{(i)}| + \gamma |h(x^{(i)})-y^{(i)}|
The following google colab notebook shows an end-to-end example of the Quantile Regressor
Advantages of Quantile Loss
• Instead of predicting the actual outcome, the Quantile Loss Function can predict corresponding quantiles.
Disadvantages of Quantile Loss
• The function is not differentiable at zero, which can make it difficult to use with some optimization algorithms.
MAE vs. MSE Regarding Outliers
Let’s compare the cost functions of MAE and MSE. We know that our target value is 0. Therefore, when we predict a value of 0, our cost function is also at 0, because our predicted value and target
value are the same.
In a case, where the predictions are close to the target values the error for MAE and MSE has a small variance (values on the x-axis close to 0). For example when our predicted value on the x-axis,
is around 5, then we have a loss of around 5 for the MAE cost function and around 25 for the MSE cost function.
But the further our prediction is away from our target value (in our case 0), we move away from 0 on the x-axis, the greater the difference between the error for MSE and MAE, because the MSE cost
function is stepper compared to the MAE function.
This difference between the error is very important for outliers because if you have an outlier in your residuals, the MSE cost function will square the error compared to the MAE function. Finally,
this will make the model with MSE loss give more weight to outliers than a model with MAE loss. The model with MSE loss will be adjusted to minimize that single outlier case at the expense of other
samples, which will reduce the overall performance of the model.
Which cost function to choose when having outliers
If you have outliers in your dataset that should be detected, a cost function with a steep slope like MSE or squared epsilon intensive (epsilon=2) will have a greater focus on these outliers that are
therefore heavily punished.
If you only have outliers in your training data, that most likely occur due to corrupt data, and not test data, use a cost function with a weak slope like MAE or Huber.
Error and Loss Functions for Classification
The last section of this article describes the most used cost function for classification. You find the implementation and comparison of the classification cost functions in the same google colab
Log Loss
The log loss function measures the difference between the predicted probability values and the true labels for each data point in the dataset. It applies a logarithmic transformation to the predicted
probabilities to penalize incorrect predictions more severely than correct predictions. The function returns a value between 0 and infinity, with lower values indicating better model performance.
In practice, the log loss function is often used as the objective function for training classification models with gradient descent-based algorithms such as logistic regression and neural networks.
Mathematical Formula of Log Loss
L= - y^{(i)}*log(h(x^{(i)})) + (1-y^{(i)})*log(1-(h(x^{(i)}))
Log Loss and Binary Entropy Function
Log loss function and binary entropy loss function are often used interchangeably to refer to the same evaluation metric. However, strictly speaking, there is a difference between the two. The binary
entropy loss function is a specific case of the log loss function that is used for binary classification problems, where there are only two possible classes.
The log loss function is a more general evaluation metric that can be used for both binary and multi-class classification problems. It extends the binary entropy loss function to handle multiple
classes by using a one-vs-all approach.
Advantages of Log Loss
• Sensitive to Probabilities: The log loss function is sensitive to the predicted probabilities of the model, rather than just the predicted class labels. This makes it a more nuanced evaluation
metric that can distinguish between models that make similar predictions but have different confidence levels.
• Penalizes Incorrect Predictions: The log loss function penalizes incorrect predictions more severely than correct predictions, especially for confident incorrect predictions.
• Gradient-friendly: The log loss function is differentiable and has a smooth gradient, which makes it well-suited for optimization using gradient-based algorithms such as gradient descent.
• Handle Multiclass Classification: The log loss function can be easily extended to handle multiclass classification problems using a one-vs-all approach.
Disadvantages of Log Loss
• Imbalanced Classes: The log loss function can be sensitive to class imbalance, especially in binary classification problems where one class is much rarer than the other.
• Outliers: The log loss function is sensitive to outliers, especially for confident incorrect predictions.
Hinge Loss
The hinge loss function is a commonly used loss function in machine learning for classification problems, particularly in support vector machines (SVMs). It measures the maximum margin classification
error of a linear classifier and is often used for binary classification problems.
If the predicted score for the input x is on the correct side of the decision boundary, then the hinge loss is 0. If it is on the wrong side, then the hinge loss increases linearly with the distance
from the decision boundary.
Mathematical Formula of Hinge Loss
L= max(0, 1- y^{(i)} * h(x^{(i)}))
Advantages of Hinge Loss
• Robust to Outliers: The hinge loss function is robust to outliers because it only penalizes misclassified examples that are close to the decision boundary.
• Suitable for Large-scale Learning: The hinge loss function can be computed efficiently and is suitable for large-scale learning problems.
Disadvantages of Hinge Loss
• Non-differentiable: The hinge loss function is non-differentiable at points where the hinge loss is 0, which can make optimization more challenging.
• Ignores Confidence: The loss function does not penalize incorrect predictions that are not confident (instance’s distance from the boundary is greater than or at the boundary). This means that a
model may make many confident incorrect predictions without being penalized.
• Imbalanced Classes: The hinge loss function may not perform well for imbalanced classes, where one class is much rarer than the other.
Focal Loss
The focal loss function is a modification of the cross-entropy loss function that is commonly used for imbalanced classification problems. The function aims to address the problem of extreme class
imbalance by down-weighting the contribution of easy examples and emphasizing the contribution of hard examples using the parameter gamma that controls the degree of down-weighting.
Mathematical Formula of Focal Loss
L = \begin{cases} 1-h(x^{(i)})^{\gamma} * log(h(x^{(i)})) & \text{for } y^{(i)}==1, \\ h(x^{(i)})^{\gamma} * log(1-h(x^{(i)})) & \text{otherwise} \end{cases}
Advantages of Focal Loss
• Improved Performance on Imbalanced Datasets: The focal loss function is designed to address the problem of class imbalance in datasets by focusing on hard examples and down-weighting the
contribution of easy examples.
• Can be Used with Deep Learning Models: The focal loss function can be used with a wide range of deep learning models, including convolutional neural networks (CNNs) and recurrent neural networks
• Flexible: The degree of down-weighting can be controlled by the gamma hyperparameter, which can be tuned to suit the specific needs of the problem.
Disadvantages of Focal Loss
• Increased Computational Complexity: The focal loss function involves additional computations compared to the standard cross-entropy loss function, which can increase the computational complexity
of training the model.
• Hyperparameter Tuning: The degree of down-weighting is controlled by the gamma hyperparameter, which needs to be tuned carefully to achieve good performance.
Exponential Loss
The exponential loss function is a commonly used loss function in machine learning for binary classification problems. It is also known as the “exponential hinge loss” or “AdaBoost loss”. The
exponential loss function is similar to the hinge loss function used in SVMs, but instead of being linear, it is an exponential function of the margin.
Mathematical Formula of Exponential Loss
L= exp(-y^{(i)} * h(x^{(i)}))
Advantages of Exponential Loss
• Well-Suited for Boosting Algorithms: The exponential loss function is commonly used in boosting algorithms such as AdaBoost, as it is well-suited to handle noisy or mislabeled data.
• Can Produce High-Quality Probability Estimates: The exponential loss function is known to produce high-quality probability estimates, which can be useful in certain applications such as risk
Disadvantages of Exponential Loss
• Sensitive to Outliers: The exponential loss function can be sensitive to outliers, as it assigns very high penalties to examples that are misclassified with high confidence.
• Not as Robust to Class Imbalance: The exponential loss function is not as robust to class imbalance as some other loss functions, such as the focal loss function.
• Computationally Expensive: The exponential loss function can be computationally more expensive than some other loss functions.
Read my latest articles:
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3 Reasons Why a Math Curriculum is Entirely Important in Real-World Application - Areteem Institute Blog3 Reasons Why a Math Curriculum is Entirely Important in Real-World Application - Areteem Institute Blog
3 Reasons Why a Math Curriculum is Entirely Important in Real-World Application
Some students (and parents too!) often wonder how a proper math curriculum can help them in real-world application. Well, Areteem is here to dispel a lot of myths that math isn’t applicable, and
you’d be surprised to hear the ways people use math every day!
The Math That Matters the Most
It’s often overlooked, but math plays an integral part in managing finances! Businesses and smart individuals use budgeting programs that handle profit and loss metrics, percentages, and even
fractions! Being able to apply mathematical concepts learned in a math curriculum onto personal and professional finances is incredibly important to make a living. And who said math isn’t necessary?
Math in the Kitchen!
Measurements galore! If you’re heavily into culinary arts, math is centric to adding the proper amount of ingredients and ensuring whatever you’re cooking is brought to the right temperature. Most of
the time you’ll be working with fractions (we like 2/3 of a stick of butter over 1/3), but they’re still heavily important in concocting the best kind of dishes.
Become a Popular Calculator!
Sometimes, the most difficult math problems hit us when we’re at the restaurant table. Sally shared a salad with Jim, so they owe half of 9 dollars, while Mike didn’t tip last meal so he owes and
extra 3 bucks on top of his steak. Jean is picking up Jacob’s meal so she owes 21 dollars, and Steve gets his meal for free because he complained about a hair in it. Suddenly we have the biggest math
conundrum, and if you practice mental math enough these monstrous split checks won’t be a problem for you. Plus, being able to calculate the tip on the fly saves you time from busting out the
calculator just to figure how much you owe for tip.
These are the most practical applications for math that you’ll find in a math curriculum, and you’ll be everyone’s favorite when you can calculate the bill on the fly!
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Illustrative Mathematics
Proportional relationships, lines, and linear equations
Alignments to Content Standards: 8.EE.B
Lines $L$ and $M$ have the same slope. The equation of line $L$ is $4y=x$. Line $M$ passes through the point $(0, -5)$.
What is the equation of line $M$?
IM Commentary
There is a lot to know about the relationship between proportional relationships, lines, and linear equations, and the purpose of this task is to assess whether students understand certain aspects of
this relationship. In particular, it requires students to find the slope of the line defined by the equation $4y=x$ and to write the equation of a line knowing its slope and $y$-intercept. Note that
students in 7th grade know that the graph of a proportional relationship is a line through (0,0), so this task assesses the specific connection between proportional relationships and their graphs
with linear equations more generally.
The equation
is equivalent to
$$y=\frac14 x$$
So the slope of line $L$ is $\frac14$. $L$ and $M$ have the same slope. Since $M$ has slope $\frac14$ and passes through the point (0,-5), the equation for line $M$ is
$$y=\frac14 x -5$$
Other acceptable forms of the equation include
• $y=-5 + \frac14 x$
• $y+5 = \frac14 x$
• $4y= x - 20$
• $4y= -20 + x$
• $4y+20= x$
and any other equation that is equivalent to one of these.
Proportional relationships, lines, and linear equations
Lines $L$ and $M$ have the same slope. The equation of line $L$ is $4y=x$. Line $M$ passes through the point $(0, -5)$.
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Announcement Info Derivatives
Derivatives Overview
Contract Types
Derivatives are financial instruments that derive value from the performance of an underlying asset. They are used to manage financial risk by allowing parties to hedge against potential adverse
movements in market prices.
Perpetual Contracts
A perpetual contract is a type of derivative that, unlike traditional futures, has no expiration or settlement date. Bybit's perpetual contracts are margined in USDT, USDC, and base assets (also
known as coin-margined).
Funding Mechanism: This process entails the exchange of funding fees between long and short position holders every 8 hours, based on the funding rate. This exchange is contingent on the position
being open at specific timestamps (00:00 UTC, 08:00 UTC, and 16:00 UTC).
When the funding rate is positive, long position holders pay short position holders.
When the funding rate is negative, short position holders pay long position holders.
The Funding Fee calculation is: Funding Fee = Position Value * Funding Rate.
For funding rate details, click here.
Futures Contracts
Bybit's futures contracts are margined in USDC and base assets.
Expiration dates are based on the last day of the following period: Current week, next week, third week, current month, next month, third month, current quarter, and next quarter.
Settlement: The settlement price is based on the average index price in the last half-hour before expiration. The settlement time is at 08:00 UTC on the expiration date. USDC contracts are
"cash-settled", meaning the contract seller pays the proceeds in USDC to the buyer instead of transferring the underlying asset. Inverse contracts are settled in the underlying asset.
Options Contracts
Bybit's options offerings are European-style options with either BTC or ETH as the underlying asset. The pricing of these options is determined using the Last Traded Price (LTP) of the underlying
asset and implied volatility, with settlement and margin in USDC.
Margin Types
In derivatives trading, margin acts as collateral for holding positions. Initial Margin is the amount needed to open (or increase) a position, while Maintenance Margin is the minimum amount that must
be maintained to keep the position open.
Index Price
The Index Price is derived from a weighted average of multiple spot exchange quotes, adjusted by data usability and weight adjustment factors. The spot price of reference exchanges will be weighted
based on trading volume and price distance from the volume-weighted average price.
For index price calculations, click here.
Mark Price
The Mark Price, used by exchanges to estimate the true value of derivatives contracts, is derived from the Index Price, and adjusted with additional factors such as the Funding Rate, reflecting the
cost of holding an open position in the market. In trading, the Mark Price is primarily used to:
Calculate unrealized profit and loss.
Trigger liquidation when the Mark Price reaches or exceeds the Liquidation Price.
The Mark Price (Perpetual Contracts) calculation is:
Mark Price = Median (Price 1, Price 2, Last Traded Price)
Price 1 = Index Price × [1 + Last Funding Rate × % Time Remaining to Funding]
Price 2 = Index Price + 5-min Moving Average
5-min Moving Average = Moving Average [(Bid Price + Ask Price) / 2 - Index Price] (sampled once per second for the past 5 minutes)
The Mark Price (Futures Contracts) calculation is:
Mark Price = Index Price * (1 + Basis Rate)
The Mark Price (Options Contracts) calculation is:
Black-76 model with inputs of forward price, strike price, time to expiration, interest rate, and implied volatility (IV).
The IV is based on: Spline Volatility Surface, SABR Volatility Surface
For information on Contract Details, click here.
For details on Trading Parameters, click here.
For details on Margin Parameters, click here. | {"url":"https://www.bybit.com/en/announcement-info/contract-summarize/?symbol=BTCUSDT","timestamp":"2024-11-10T21:07:04Z","content_type":"text/html","content_length":"95182","record_id":"<urn:uuid:525ca2a2-3d4b-4596-b916-f8f77183c0a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00516.warc.gz"} |
Electrical Appliance Cost Calculator - Calculator Doc
Electrical Appliance Cost Calculator
Understanding the cost of running electrical appliances is crucial for managing energy bills effectively. By calculating the daily cost of your appliances, you can make informed decisions about
usage, potentially saving money and reducing energy consumption. Our Electrical Appliance Cost Calculator helps you estimate the cost of operating any electrical appliance based on its wattage, usage
rate, and hours of use per day.
The formula to calculate the cost of operating an electrical appliance is:
EC = (W / 1000) × UR × HD
• EC = Estimated Cost (cents/day)
• W = Wattage of the Appliance (watts)
• UR = Usage Rate (cents per kilowatt-hour)
• HD = Hours of Use per Day
This formula helps determine how much it costs to run an electrical appliance each day, based on your local electricity rate and usage patterns.
How to Use the Electrical Appliance Cost Calculator
Using the Electrical Appliance Cost Calculator is simple:
1. Enter the Wattage (W) of the Appliance: This information can usually be found on the appliance label or in the user manual.
2. Enter the Usage Rate (UR): This is the cost of electricity per kilowatt-hour (kWh) in your area, typically provided by your utility company.
3. Enter the Hours of Use per Day (HD): Estimate how many hours per day the appliance is in use.
4. Click on the “Calculate” Button: The calculator will estimate the daily cost of operating the appliance in cents.
Let’s say you have a 1500-watt space heater, your electricity rate is 12 cents per kilowatt-hour, and you use the heater for 8 hours a day. Using the formula: EC = (1500 / 1000) × 12 × 8
EC = 1.5 × 12 × 8
EC = 144 cents/day
In this example, running the space heater costs approximately 144 cents (or $1.44) per day.
1. What is an electrical appliance cost calculator? An electrical appliance cost calculator helps estimate the daily cost of operating electrical appliances based on wattage, usage rate, and hours
of use.
2. Why is it important to calculate the cost of using electrical appliances? Calculating the cost helps you manage energy consumption, reduce electricity bills, and make informed decisions about
appliance usage.
3. How do I find the wattage of an appliance? The wattage is typically listed on the appliance’s label or in the user manual. It may also be marked on the power cord or plug.
4. What is a kilowatt-hour (kWh)? A kilowatt-hour is a unit of energy equal to one kilowatt (1000 watts) used for one hour. It is the standard unit of measurement for electricity consumption.
5. How do I find the electricity rate in my area? Your electricity rate can be found on your utility bill, listed as the cost per kilowatt-hour (kWh).
6. Can this calculator be used for all types of appliances? Yes, this calculator can be used for any electrical appliance that runs on electricity and has a known wattage.
7. How accurate is this calculator? The calculator provides a good estimate based on the inputs, but actual costs may vary depending on factors such as appliance efficiency and fluctuations in
electricity rates.
8. What if my appliance uses variable wattage? For appliances with variable wattage (e.g., adjustable heaters or fans), you can calculate based on the average wattage or the highest setting.
9. Does this calculator account for standby power usage? No, this calculator only estimates costs based on active usage. Appliances in standby mode may consume additional power, which is not
included in the calculation.
10. How can I reduce the cost of running my appliances? You can reduce costs by using energy-efficient appliances, reducing usage time, and optimizing settings for lower power consumption.
11. How does appliance efficiency affect the cost? More efficient appliances consume less energy, which lowers the daily cost of operation. Look for appliances with Energy Star ratings or high energy
efficiency ratings.
12. Can I use this calculator for non-residential purposes? Yes, this calculator can be used to estimate costs in commercial or industrial settings as well, as long as you know the wattage and usage
13. What happens if my appliance runs for partial hours? You can enter fractional hours in the calculator (e.g., 0.5 for half an hour) to estimate the cost for partial hours of usage.
14. Do all appliances consume power equally over time? No, some appliances may consume more power at startup (like refrigerators) or vary their consumption during operation (like air conditioners).
This calculator provides an average estimate.
15. How does power factor influence the cost? The power factor is the efficiency of the appliance’s power usage. For most household calculations, this is not factored in, but it can be relevant for
larger commercial appliances.
16. What are some common household appliances that consume a lot of energy? High-energy appliances include air conditioners, space heaters, refrigerators, washing machines, and dryers.
17. How do seasonal changes affect appliance costs? Seasonal changes can increase the usage of heating or cooling appliances, resulting in higher energy costs during peak seasons.
18. Is it cheaper to run appliances at night? Some utility companies offer lower rates during off-peak hours, so running appliances at night may reduce costs if you are on such a plan.
19. How can I track my appliance energy usage over time? You can use energy monitoring devices or smart plugs to track real-time energy usage and costs for individual appliances.
20. What is the best way to reduce overall energy consumption in my home? The best way to reduce energy consumption is to use energy-efficient appliances, limit usage, and regularly maintain your
appliances to ensure they operate efficiently.
Calculating the cost of running electrical appliances is a valuable step in managing your energy usage and controlling your utility bills. By using our Electrical Appliance Cost Calculator, you can
estimate the daily cost of any appliance and make more informed decisions about your energy consumption. Whether you’re looking to reduce your energy bills or simply understand where your electricity
costs are coming from, this calculator provides an easy and accurate tool for your needs. | {"url":"https://calculatordoc.com/electrical-appliance-cost-calculator/","timestamp":"2024-11-13T21:50:49Z","content_type":"text/html","content_length":"89253","record_id":"<urn:uuid:1cdf4af3-f20d-404f-a37a-8941c9bdc7c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00416.warc.gz"} |
Conduction - A118
From CKN Knowledge in Practice Centre
Conduction is a molecular heat transfer mode associated with the microscopic collisions between high energy particles and adjacent lower energy particles causing a redistribution of kinetic energy.
Therefore, heat transfer by conduction occurs from regions of higher temperature to regions of lower temperatures.
This page explains thermal conduction in the context of composites processing and how it differs from other heat transfer mechanisms such as convection. Specifics about the material properties
involved in thermal conduction are given in separate pages.
Conduction is the primary heat transfer mechanism in solid materials. It plays a critical role in composite manufacturing processes, responsible for heat flow in hot press forming platens, tooling
and part through-thickness temperature gradients. Both steady-state and transient (non steady-state) conduction are introduced and discussed.
Recommended documents to review before, or in parallel with this document:
Conduction is a molecular heat transfer mode associated with the microscopic collisions between high energy particles and adjacent lower energy particles causing a redistribution of kinetic energy.
Therefore, heat transfer by conduction occurs from regions of higher temperature to regions of lower temperatures. It is one of the are three main mechanisms of heat transfer.
Using the classic example of a metal rod being heated over a flame; the heat being transferred from the flame to the rod is considered to be by convection, while the heat transferred along the rod
itself is by conduction ^[1]. Applying this analogy to the composites processing scenario for a laminate curing in an oven or autoclave environment; the hot air transfers heat by convection, while
the heat transfer between the tool and laminate part, and within the laminate itself is by conduction.
There are two main properties to heat conduction:
Thermal conduction occurs under two conditions: steady-state, and non-steady or transient conditions.
Steady-state conduction is described empirically by Fourier's Law.
\(\overrightarrow{q}=\)heat flux
\(k=\) thermal conductivity
\(\overrightarrow{\bigtriangledown}T=\)temperature gradient
Fourier’s Law provides an empirical description of conduction stating that the heat flux, q, travelling from a region of higher temperature to a region of lower temperature is equal to the product of
the temperature gradient, ΔT, between the two regions and the thermal conductivity of the material, k. This means that the energy transport by conduction between two regions increases with the
temperature difference between the two regions. A simple one-dimensional heat transfer case by solid-state conduction is shown with a wall exposed on one side to a hot fluid and to a cold fluid on
the other side. Under steady state conditions a linear temperature gradient develops inside the wall as heat is transferred from the hot fluid through the solid wall to the cold fluid via conduction.
For a simple steady-state one-dimensional case, the differential form of Fourier's Law simplifies to:
\(q_x = -k\tfrac{\partial t}{\partial x}\)
Which, if the temperature distribution is linear, becomes:
\(q_x = -k\tfrac{\bigtriangleup T}{\bigtriangleup x}\)
Where \(q_x\) is the heat flux in [W/m^2] in the x direction, \(\bigtriangleup T\) is the temperature gradient in [K/m] in the \(x\) direction, and \(k\) is the thermal conductivity of the material
in [W/m·K].
In order to understand the 1D steady-state temperature distribution through a part, where one side side is hotter than the other, the differential form of q[x] must be substituted into the
fundamental energy equation. Take the following example, where an infinitely long wall is heated from one side.
\(-\frac{\partial q}{\partial x}-\frac{\partial q}{\partial y}-\frac{\partial q}{\partial z}+\dot Q_{gen}-\dot Q_{cons}=\rho c_p\frac{\partial T}{\partial t}\)
For this case, the one-dimensional form of the energy equation, with no internal heat generation (\(\dot Q_{gen}\)) or accumulation (\(\dot Q_{cons}\)), i.e., steady state, becomes:
\(- \tfrac{dq_x}{dx}=0\)
Which gives:
\(\tfrac{d}{dx} \Bigl( k \tfrac{dT}{dx} \Bigr) = 0\)
Integrating the above equation and solving for the boundary conditions (i.e. T(0) = T[0] and T(L) = T[1]) yields:
\(T_{(x)} = ( T_1 - T_0 ) \tfrac{x}{L} + T_0\)
Under steady-state conditions, the temperature profile increases linearly from T[0] to T[1].
Transient (non-steady state) conduction involves heat capacity (\(c_p\)), density (\(\rho\)), thermal diffusivity (\(\alpha\)), and a rate term.
The thermal diffusivity is defined as\[\alpha=\frac{k}{\rho c_p}\]
\(k=\)thermal conductivity [W/m·K]
\(\rho=\)density [kg/m^3]
\(c_p=\)specific heat capacity [J/kg·K]
The grouping of terms \(\rho c_p\) is volumetric heat capacity [J/m^3·K]
The SI units used of thermal diffusivity are then [m^2/s]. Thermal diffusivity varies from low values of about 0.1 10^-6 m^2/s to high values of about 1000 10^-6 m^2/s. When the thermal conditions
around a material changes, a material with a high thermal diffusivity will reach thermal equilibrium faster than a material with a lower thermal diffusivity.
Thermal diffusivities of some common materials used in composite manufacturing are given in the following table:
Material Thermal Diffusivity [10^-6 m^2/s]
Aluminum 68.9
Steel 14.2
Invar 2.7
Carbon-Epoxy Composite 0.5
The earlier described steady-state example can be turned into a transient state conduction case by considering that the same wall is initially at a temperature T[0.] throughout, and then that at time
t = 0 its right-hand wall is suddenly brought to the temperature T[1]. As time proceeds, the through-thickness temperature of the wall changes as heat flows in or out of the wall until eventually the
linear steady-state distribution described in the previous section is achieved.
For this case, the one-dimensional form of the energy equation becomes\[-\frac{dq_x}{dx}=\rho c_p\frac{\partial T}{\partial t}\]
\(\frac{d}{dx}\Biggl(k\frac{dT}{dx}\Biggr)=\rho c_p\frac{\partial T}{\partial t}\)
Which gives\[\alpha\frac{d^2T}{dx^2}=\frac{\partial T}{\partial t}\]
Where \(\alpha\) is the thermal diffusivity in [m^2/s].
The rate of temperature change, and consequently the time to reach thermal equilibrium, therefore depends on the thermal diffusivity of the material.
For a composite laminate curing in an autoclave or oven, where the moving air is acting as a convective heat transfer boundary condition, graphical Heisler charts can be used to look up an
approximate solution to the transient conductivity equation for the internal temperature profile of the laminate. Modern technology also allows an exact solution to be obtained using a number of
available math or finite element software packages ^[2]. Alternatively, a simplified closed-form approximation developed by Rasekh et al. ^[3] can also be used. For more information regarding using
this approximation, please see the paper by Rasekh et al. here.
Related pages
Page type Links
Introduction to Composites Articles
Foundational Knowledge Articles
Foundational Knowledge Method Documents
Foundational Knowledge Worked Examples
Systems Knowledge Articles
Systems Knowledge Method Documents
Systems Knowledge Worked Examples
Systems Catalogue Articles
Systems Catalogue Objects – Material
Systems Catalogue Objects – Shape
Systems Catalogue Objects – Tooling and consumables
Systems Catalogue Objects – Equipment • Rheometer - A357
Practice Documents
Case Studies
Perspectives Articles
Welcome to the CKN Knowledge in Practice Centre (KPC). The KPC is a resource for learning and applying scientific knowledge to the practice of composites manufacturing. As you navigate around the
KPC, refer back to the information on this right-hand pane as a resource for understanding the intricacies of composites processing and why the KPC is laid out in the way that it is. The following
video explains the KPC approach:
Understanding Composites Processing
The Knowledge in Practice Centre (KPC) is centered around a structured method of thinking about composite material manufacturing. From the top down, the heirarchy consists of:
The way that the material, shape, tooling & consumables and equipment (abbreviated as MSTE) interact with each other during a process step is critical to the outcome of the manufacturing step, and
ultimately critical to the quality of the finished part. The interactions between MSTE during a process step can be numerous and complex, but the Knowledge in Practice Centre aims to make you aware
of these interactions, understand how one parameter affects another, and understand how to analyze the problem using a systems based approach. Using this approach, the factory can then be developed
with a complete understanding and control of all interactions.
Interrelationship of Function, Shape, Material & Process
Design for manufacturing is critical to ensuring the producibility of a part. Trouble arises when it is considered too late or not at all in the design process. Conversely, process design
(controlling the interactions between shape, material, tooling & consumables and equipment to achieve a desired outcome) must always consider the shape and material of the part. Ashby has developed
and popularized the approach linking design (function) to the choice of material and shape, which influence the process selected and vice versa, as shown below:
Within the Knowledge in Practice Centre the same methodology is applied but the process is more fully defined by also explicitly calling out the equipment and tooling & consumables. Note that in
common usage, a process which consists of many steps can be arbitrarily defined by just one step, e.g. "spray-up". Though convenient, this can be misleading.
The KPC's Practice and Case Study volumes consist of three types of workflows:
• Development - Analyzing the interactions between MSTE in the process steps to make decisions on processing parameters and understanding how the process steps and factory cells fit within the
• Troubleshooting - Guiding you to possible causes of processing issues affecting either cost, rate or quality and directing you to the most appropriate development workflow to improve the process
• Optimization - An expansion on the development workflows where a larger number of options are considered to achieve the best mixture of cost, rate & quality for your application. | {"url":"https://compositeskn.org/KPC/A118","timestamp":"2024-11-12T23:16:54Z","content_type":"text/html","content_length":"120749","record_id":"<urn:uuid:ebf88116-f571-4bd2-93c6-e3cc736fcca7>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00878.warc.gz"} |
Indistinguishability obfuscation for secure software
Encrypting software so it performs a function without the code being intelligible was long thought by many to be impossible. But in 2013, researchers—unexpectedly and in a surprising way—demonstrated
how to obfuscate software in a mathematically rigorous way using multilinear maps. This breakthrough, combined with almost equally heady progress in fully homomorphic encryption, is upending the
field of cryptography and giving researchers for the first time an idea of what real, provable security looks like. The implications are immense, but two large problems remain: creating efficiency
and providing a clear mathematical foundation. For these reasons, five researchers, including many of those responsible for the recent advances, have formed the
Center for Encrypted Functionalities
with funding provided by NSF.
Allison Bishop Lewko
is part of this center. In this interview she talks about the rapidly evolving security landscape, the center’s mandate, and her own role.
Question: What happened to make secure software suddenly seem possible?
Allison Bishop Lewko
: The short answer is that in 2013, six researchers—
Sanjam Garg
Craig Gentry
Shai Halevi
Mariana Raykova
Amit Sahai
Brent Waters
a paper
showing for the first time that it is possible to obfuscate software in a mathematically rigorous way using a new type of mathematical structure called
multilinear maps
(see sidebar for definition). This means I can take a software program and put it through the mathematical structure provided by these multilinear maps and encrypt software to the machine level while
preserving its functionality. Anyone can take this obfuscated software and run it without a secret key and without being able to reverse-engineer the code. To see what’s happening in the code
requires solving the hard problem of breaking these multilinear maps.
This process of encrypting software so it still executes, which we call indistinguishability obfuscation, was something we didn’t believe was out there even two or three years ago. Now there is this
first plausible candidate indistinguishability obfuscator using multilinear maps.
And it’s rigorous. Today there are programs that purport to obfuscate software. They add dead code, change the variables, things you imagine you would get if you gave a sixth-grader the assignment to
make your software still work but hard to read. It is very unprincipled and people can break it easily. Indistinguishability obfuscation is something else entirely.
How does obfuscation differ from encryption?
Both obfuscation and encryption protect software, but the nature of the security guarantee is different. Encryption tells you your data or software is safe because it is unintelligible to anyone who
doesn’t have the secret key to decrypt it. The software is protected but at a cost of not being usable. The software is either encrypted and protected, or it is decrypted and readable.
Obfuscation is a guarantee to protect software, but there is this added requirement that the encrypted software still has to work, that it still performs the function it was written to do. So I can
give you code and you can run it without a secret key but you can’t learn how it works.
The bar is much higher for obfuscation because obfuscated software must remain a functional, executable program. Protecting software and using it at the same time is what is hard. It’s easy to
protect software if you never have to use it again.
These concepts of encryption and decryption, however, do live very close together and they interact. Encryption is the tool used to accomplish obfuscation, and obfuscation can be used to build an
How so? Do you use the same methods to encrypt data to obfuscate code?
Not exactly, though the general concept of substituting something familiar with something seemingly random exists in both.
In the well-known RSA encryption method, each character of the output text is replaced by a number that gets multiplied by itself a number of times (the public key) to get a random-looking number
that only the secret, private key can unlock and make sense of.
When you obfuscate software, each bit of the original program gets associated with some number of slots. For every slot there are two matrices, and every input bit determines what matrix to use.
The result is a huge chain of matrices. To compute the program on an input bit stream requires selecting the corresponding matrices and multiplying them.
Is having two matrices for every slot a way to add randomness?
No. The choice of what matrix to use is deterministic based on the input. But we do add randomness, which is a very important part of any cryptography scheme since it’s randomness that makes it hard
to make out what’s going on. For this current candidate obfuscation scheme, we add randomness to the matrices themselves (by multiplying them by random matrices), and to representations we create of
the matrices—like these multilinear maps.
But we do it very carefully. Among all these matrices, there are lots of relationships that have to be maintained. Adding randomness has to be done in a precise way so it’s enough to hide things but
not so much to change software behavior.
There is this tension. The more functionality to preserve, the more there is to hide, and the harder it is to balance this tension. It’s not an obvious thing to figure out and it’s one reason why
this concept of obfuscation didn’t exist until recently.
Amit Sahai, director of the Center, compares obfuscated software to a jigsaw puzzle, where the puzzle pieces have to fit together the right way before it’s possible to learn anything about the
software. The puzzle pieces are the matrix encoded in the multilinear map, and the added randomness makes it that much harder to understand how these pieces fit together.
What does obfuscated software look like?
You would not want to see that. A small line becomes a page of random characters. The original code becomes bit by bit multiplication matrices. And this is one source of inefficiency.
Actually for this first candidate it’s not the original software itself being put through the obfuscation, but the software transformed into a circuit (a bunch of AND, OR, NOT gates), and this is
another source of inefficiency.
Indistinguishability obfuscation is said to provide a richer structure for cryptographic methods. What does that mean?
All cryptography rests on trying to build up structures where certain things are fast to compute, and certain things are hard. The more you can build a richer structure of things that are fast, the
more applications there are that can take advantage of this encryption. What is needed is a huge mathematical structure with a rich variety of data problems, where some are easy, some are hard.
In RSA the hard problem is factoring large numbers. One way—multiplying two prime numbers—is easy; that’s the encryption. Going back—undoing the factorization—is hard unless you have the key. But
it’s hard in the sense that no one has yet figured out how to factorize large numbers quickly. That nobody knows how to factor numbers isn’t proven. We’re always vulnerable to having someone come out
of nowhere with a new method.
RSA is what we’ve had since the 80s. It works reasonably well, assuming no one builds a quantum computer.
But RSA and other similar public key encryption schemes don’t work for more complicated policies where you want some people to see some data and some other people to see other data. RSA has one hard
problem and allows for one key that reads everything. There’s no nuance. If you know the factors, you know the factors and you can read everything.
A dozen years ago (in 2001)
Dan Boneh
working with other people discovered how to build crypto systems using bilinear maps on elliptic curves (where the set of elements used to satisfy an equation consists of points on a curve rather
than integers). Elliptic curves give you a little more structure than a single function; some things are easy to compute but there are things that are hard to compute. The same hard problem has
different levels of structure.
Think of a group having exponents. Instead of telling you the process for getting an exponent out, I can give you one exponent, which to some extent represents partial information, and each piece of
partial information lets you do one thing but not another. The same hard problem can be used to generate encryption keys with different levels of functionality, and it all exist within the same
This is the beginning of identity-based security where you give different people particular levels of access to encrypted data. Maybe my hospital sees a great deal of my data, but my insurance
company sees only the procedures that were done.
The structure of bilinear maps allows for more kinds of keys with more levels of functionality than you would get with a simple one-way function. It’s not much of a stretch to intuitively understand
that more is possible with trilinear maps or with multilinear maps.
For a dozen years it was an open problem on how to extend bilinear maps to multilinear maps in the realm of algebraic geometry, but elliptic curves didn’t seem to have a more computationally
efficient structure on which to build. During this time, Craig Gentry (now at IBM) was working on fully homomorphic encryption, which allows someone to perform computations on encrypted data (the
results will also be encrypted). Gentry was using lattices, a type of mathematical structure from number theory, an entirely different mathematical world than algebraic geometry.
Lattices provide a rich family of structures, and lattice schemes hide things by adding noise. As operations go on, noise gradually builds up and that’s what makes homomorphic encryption hard.
You might think immediately that these lattices schemes serve as a multilinear map but there isn’t a sharp divide between what’s easy and hard; there is a falloff. With fully homomorphic encryption
having this gradual quality, it wasn’t at all clear how to use it in a way to build something where multiplying three things is easy but multiplying four things is hard.
The solution came in 2013 with the publication of
Candidate Multilinear Maps from Ideal Lattices
where the authors were able to make a sharp threshold between easy and hard problems within lattice problems. They did it by taking something like a fully homomorphic scheme and giving out a partial
secret key; one that lets you do this one thing at level 10 but not at level 11. These partial keys, which are really weak secret keys, add more nuance, but it’s extremely tricky. If you give out a
weak key, you may make things easy that you don’t want to be easy. You want it to break a little but not too badly. This is another source of inefficiency.
How soon before obfuscated software becomes a reality?
A long time. Right now, indistinguishability obfuscation is a proof of concept. Before it can have practical application, it should have two features: it should be fast, and we should have a good
understanding of why it is secure, more than this-uses-math-and-it-looks-like-we-don’t-know-how-to-break-it.
Indistinguishability obfuscation currently doesn’t meet either criteria. It is extremely inefficient to what you want to run in real life—running an instance of a very simple obfuscated function
would take seconds or minutes instead of microseconds—and we don’t have a good answer for why we believe it’s hard to break.
Is that the purpose of the Center for Encryption Functionalities?
Yes. The center’s main thrusts will be to increase efficiency and provide a provable mathematical foundation. Many of the same people responsible for the big advances in indistinguishability
obfuscation and fully homomorphic encryption are working at the center and building on previous work.
We’ll be examining every aspect of the current recipe to increase efficiency. There are many avenues to explore. Maybe there is a more efficient way of implementing multilinear maps that doesn’t
involve matrices. Maybe only certain parts of a program have to be encrypted. We’re still early in the process.
The center also has a social agenda, to both educate others on the rapid developments in cryptography and bring together those working on the problem.
What is your role?
My role is to really narrow down our understanding of what we’re assuming when we say indistinguishability obfuscation is secure. If someone today breaks RSA, we know they’ve factored big numbers,
but what would it take to break indistinguishability obfuscation?
Indistinguishability obfuscation is still this very complicated, amorphous thing of a system with all these different possible avenues of attack. It’s difficult to comprehensively study the candidate
scheme in its current form and understand exactly when we are vulnerable and when we are not.
I did my PhD in topics to expand the set of things that we can provably reduce to nice computational assumptions, which is very much what is at the heart of where this new obfuscation stuff is going.
We need to take this huge, complicated scheme down to a simple computational problem to make it easy for people to study. Saying “factor two numbers” is a lot easier than saying break this process
that takes a circuit to another circuit. We have to have this tantalizing carrot of a challenge to gain interest.
And on this front we have made great progress. We have a
somewhat long, highly technical proof
that takes an abstract obfuscation for all circuits and reduces it to a sufficiently simple problem that others can study and try to break.
That’s the first piece. What we don’t yet have is the second piece, which is instantiating the simple problem as a hard problem in multilinear maps.
Have there been any attempts to “break” indistinguishability obfuscation?
Yes, in the past few months some attacks did beat back on some problems contained in our proof that we thought might be hard, but these attacks did not translate into breaks on the candidate
obfuscation scheme. While we wished the problems had been hard, the attacks help us narrow and refine the class of problems we should be focusing on, and they are a reminder that this line between
what is hard and what is not hard is very much still in flux.
The attackers didn’t just break our problem, they broke a landscape of problems for current multilinear map candidates, making them fragile to use.
Were you surprised that the problem was broken?
Yes, very surprised, though not super-surprised because this whole thing is very new. There are always breaks you can’t imagine.
Do the breaks shake your faith in indistinguishability obfuscation?
I have no faith to be shaken. I simply want to know the answer.
We learn something from this process either way. If the current multilinear map candidates turn out to be irretrievably broken—and we don’t know that they are—we still learn something amazing. We
learn about these mathematical structures. We learn about the structure of lattice-based cryptography, and the landscape of easy and hard problems.
Every time something is broken, it means someone came up with an algorithm for something that we didn’t previously know. Bilinear maps were first used to break things; then Dan Boneh said if it’s a
fast algorithm that can break things, let’s re-appropriate it and use it to encrypt things. Every new algorithm is a potential new tool.
The fact that a particular candidate can be broken doesn’t make it implausible that other good candidates exist. We’ve also made progress on working with different kinds of computational models. In
June, we will present
Indistinguishability Obfuscation for Turing Machines with Unbounded Memory
at STOC, which replaces circuits with a Turing machine to show conceptually that we can do obfuscation for varying input sizes.
Things are still new; what we’ve done this year at the center is motivating work in this field. People will be trying to build the best multilinear maps for the next 20 years regardless of what
happens with these particular candidates.
The ongoing cryptanalysis is an important feedback loop that points to targets that attackers will go after to try to break the scheme, and it will help drive the direction of cryptography. It’s
really an exciting time to be a researcher in this field.
About Allison Bishop Lewko
Allison Bishop Lewko
is assistant professor of computer science at Columbia Engineering and a faculty affiliate of Columbia’s Institute for
Data Science Institute
and the Institute’s
Cybersecurity Center
. Her current areas of research are cryptography, complexity theory, harmonic analysis, combinatorics, and distributed computing.
Prior to joining Columbia, she was a postdoctoral researcher at Microsoft Research New England where she designed error-tolerant algorithms.
Lewko received her PhD in Computer Science from UT-Austin where she was advised by
Brent Waters
. She has a certificate of advanced study in mathematics from the University of Cambridge and a bachelor’s degree in mathematics from Princeton University.
What is a multilinear map?
A multilinear map is a new type of mathematical structure from the field of number theory that has applications in cryptographic systems.
A multilinear map is a “mapping” among different groups in which an object from one group is multiplied by (or otherwise interacts with) an object in another group to create a new and different
object. (The relatedness comes from both objects being possible solutions to the same hard mathematical problem.)
This mapping process makes it possible to disguise the original objects by using the new object when performing a function and still achieve the same overall data transformation that would have been
produced without performing the mapping.
The mathematics behind multilinear maps is sophisticated and challenging to understand. In concept multilinear maps resemble bilinear maps—used in cryptographic systems for the past dozen years—where
there is the notion of a bilinear mapping between the group of points on an elliptic curve and a finite field.
But where bilinear maps exist in the realm of algebraic geometry and use the discrete logarithm problem for points on a curve (computing quadratic functions in the exponent), multilinear maps
originate in number theory. The recent constructions for cryptographic multilinear maps (proposed by
Shai Halevi
Craig Gentry
Shai Halevi
Mariana Raykova
Amit Sahai
Brent Waters
) use hard problems from ideal lattices where points are arranged on a high-dimensional lattice (similar to how they are used in fully homomorphic encryption schemes).
Lattices give rise to an abundance of potentially computationally difficult problems, providing a richer structure on which to build cryptographic systems and a finer level of control over what
operations are easy and which are hard.
The rapid evolution of indistinguishability obfuscation—by papers
The current indistinguishability obfuscation scheme builds on a series of developments described in these papers, each of which generated its own stream of papers.
March 2013
Multilinear maps from ideal lattices to supply hard problems for cryptography:
Candidate Multilinear Maps from Ideal Lattices
by Sanjam Garg, Craig Gentry, Shai Halevi. This paper in particular proposed a new mathematical building block, generating an explosion of papers. Most of exciting was which was:
Once this paper appeared and demonstrated how indistinguishability obfuscation can be achieved, other papers on obfuscation followed, with 50 appearing within six to eight months.
Applications for secure software
Any software that contains sensitive information becomes a potential application.
Hiding secrets inside software.
If software can be made unintelligible, other information like passwords, digital signature, and encryption keys can be hidden inside software. Two possibilities are immediately obvious. Anything can
be a password, including an email. And passwords would not have to be exchanged ahead of time before sending the message.
Protecting software patches. Security updates, or patches, intended to plug security holes may serve as a convenient pointer to the vulnerability being fixed, which hackers can then exploit in
devices not yet updated.
Entrusting software to the cloud, or otherwise distributing proprietary or valuable algorithms. Valuable, sensitive, or proprietary software, once obfuscated, can be distributed without fear it will
be reverse-engineered. Financial institutes can distribute trading algorithms. Software on captured military drones will remain unreadable. Manufacturers could distribute encrypted DVDs.
About The Center for Encrypted Functionalities (CEF)
The Center’s primary mission is to transform program obfuscation from an art to a rigorous mathematical discipline. It supports direct research, organizes retreats and workshops to bring together
researchers, and engages in outreach and educational efforts.
Supported through an NSF Secure and Trustworthy Cyberspace Frontier Award, the Center has five principal investigators, including Amit Sahai (UCLA), who is director, Dan Boneh (Stanford),
Susan Hohenberger
(Johns Hopkins), Allison Bishop Lewko (Columbia), and Brent Waters (University of Texas, Austin). | {"url":"https://www.cs.columbia.edu/2015/indistinguishability-obfuscation-secure-software/","timestamp":"2024-11-03T16:08:14Z","content_type":"text/html","content_length":"87207","record_id":"<urn:uuid:c2c1e804-978c-4c69-a51a-31bc1302cafc>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00060.warc.gz"} |
A Mandelbrot Set on the GPU
Today I welcome back guest blogger Ben Tordoff who last blogged here on how to Carve a Dinosaur. These days Ben works on the Parallel Computing Toolbox™ with particular focus on GPUs.
About this demo
Benoit Mandelbrot died last Autumn, and despite inventing the term "fractal" and being one of the pioneers of fractal geometry, his lasting legacy for most people will undoubtedly be the fractal that
bears his name. There can be few mathematicians or mathematical algorithms that have adorned as many teenage bedroom walls as Mandelbrot's set did in the late '80s and early '90s.
I also suspect there is no other fractal that has been implemented in quite as many different languages on different platforms. My first version was a blocky 32x24 version on an aging ZX Spectrum
with seven glorious colours. These days we have a bit more graphical power available, and a few more colours. In fact, as we shall see, an algorithm like the Mandelbrot Set is ideally suited to
running on that GPU (Graphics Processing Unit) which mostly sits idle in your PC.
As a starting point I will use a version of the Mandelbrot set taken loosely from Cleve Moler's Experiments with MATLAB e-book. We will then look at the three different ways that the Parallel
Computing Toolbox™ provides for speeding this up using the GPU:
1. Using the existing algorithm but with GPU data as input
2. Using arrayfun to perform the algorithm on each element independently
3. Using the MATLAB/CUDA interface to run some existing CUDA/C++ code
As you will see, these three methods are of increasing difficulty but repay the effort with hugely increased speed. Choosing a method to trade off difficulty and speed is typical of parallel
The values below specify a highly zoomed part of the Mandelbrot Set in the valley between the main cardioid and the p/q bulb to its left.
The following code forms the set of starting points for each of the calculations by creating a 1000x1000 grid of real parts (X) and imaginary parts (Y) between these limits. For this particular
location I happen to know that 500 iterations will be enough to draw a nice picture.
maxIterations = 500;
gridSize = 1000;
xlim = [-0.748766713922161, -0.748766707771757];
ylim = [ 0.123640844894862, 0.123640851045266];
The Mandelbrot Set in MATLAB
Below is an implementation of the Mandelbrot Set using standard MATLAB commands running on the CPU. This calculation is vectorized such that every location is updated at once.
% Setup
t = tic();
x = linspace( xlim(1), xlim(2), gridSize );
y = linspace( ylim(1), ylim(2), gridSize );
[xGrid,yGrid] = meshgrid( x, y );
z0 = xGrid + 1i*yGrid;
count = zeros( size(z0) );
% Calculate
z = z0;
for n = 0:maxIterations
z = z.*z + z0;
inside = abs( z )<=2;
count = count + inside;
count = log( count+1 );
% Show
cpuTime = toc( t );
set( gcf, 'Position', [200 200 600 600] );
imagesc( x, y, count );
axis image
colormap( [jet();flipud( jet() );0 0 0] );
title( sprintf( '%1.2fsecs (without GPU)', cpuTime ) );
Using parallel.gpu.GPUArray - 16 Times Faster
parallel.gpu.GPUArray provides GPU versions of many functions that can be used to create data arrays, including the linspace, logspace, and meshgrid functions needed here. Similarly, the count array
is initialized directly on the GPU using the function parallel.gpu.GPUArray.zeros.
Other than these simple changes to the data initialization, the algorithm is unchanged (and >16 times faster):
% Setup
t = tic();
x = parallel.gpu.GPUArray.linspace( xlim(1), xlim(2), gridSize );
y = parallel.gpu.GPUArray.linspace( ylim(1), ylim(2), gridSize );
[xGrid,yGrid] = meshgrid( x, y );
z0 = complex( xGrid, yGrid );
count = parallel.gpu.GPUArray.zeros( size(z0) );
% Calculate
z = z0;
for n = 0:maxIterations
z = z.*z + z0;
inside = abs( z )<=2;
count = count + inside;
count = log( count+1 );
% Show
naiveGPUTime = toc( t );
imagesc( x, y, count )
axis image
title( sprintf( '%1.2fsecs (naive GPU) = %1.1fx faster', ...
naiveGPUTime, cpuTime/naiveGPUTime ) )
Using Element-wise Operation - 164 Times Faster
Noticing that the algorithm is operating equally on every element of the input, we can place the code in a helper function and call it using arrayfun. For GPU array inputs, the function used with
arrayfun gets compiled into native GPU code. In this case we placed the loop in processMandelbrotElement.m which looks as follows:
function count = processMandelbrotElement(x0,y0,maxIterations)
z0 = complex(x0,y0);
z = z0;
count = 1;
while (count <= maxIterations) ...
&& ((real(z)*real(z) + imag(z)*imag(z)) <= 4)
count = count + 1;
z = z*z + z0;
count = log(count);
Note that an early abort has been introduced since this function only processes a single element. For most views of the Mandelbrot Set a significant number of elements stop very early and this can
save a lot of processing. The for loop has also been replaced by a while loop because they are usually more efficient. This function makes no mention of the GPU and uses no GPU-specific features - it
is standard MATLAB code.
Using arrayfun means that instead of many thousands of calls to separate GPU-optimized operations (at least 6 per iteration), we make one call to a parallelized GPU operation that performs the whole
calculation. This significantly reduces overhead - 164 times faster.
% Setup
t = tic();
x = parallel.gpu.GPUArray.linspace( xlim(1), xlim(2), gridSize );
y = parallel.gpu.GPUArray.linspace( ylim(1), ylim(2), gridSize );
[xGrid,yGrid] = meshgrid( x, y );
% Calculate
count = arrayfun( @processMandelbrotElement, xGrid, yGrid, maxIterations );
% Show
gpuArrayfunTime = toc( t );
imagesc( x, y, count )
axis image
title( sprintf( '%1.2fsecs (GPU arrayfun) = %1.1fx faster', ...
gpuArrayfunTime, cpuTime/gpuArrayfunTime ) );
Working in CUDA - 340 Times Faster
In Experiments in MATLAB improved performance is achieved by converting the basic algorithm to a C-Mex function. If you're willing to do some work in C/C++, then you can use the Parallel Computing
Toolbox to call pre-written CUDA kernels using MATLAB data. This is done using the toolbox's parallel.gpu.CUDAKernel feature.
A CUDA/C++ implementation of the element processing algorithm has been hand-written in processMandelbrotElement.cu. This must then be manually compiled using nVidia's NVCC compiler to produce the
assembly-level processMandelbrotElement.ptx (.ptx stands for "Parallel Thread eXecution language").
The CUDA/C++ code is a little more involved than the MATLAB versions we've seen so far due to the lack of complex numbers in C++. However, the essence of the algorithm is unchanged:
unsigned int doIterations( double const realPart0,
double const imagPart0,
unsigned int const maxIters ) {
// Initialize: z = z0
double realPart = realPart0;
double imagPart = imagPart0;
unsigned int count = 0;
// Loop until escape
while ( ( count <= maxIters )
&& ((realPart*realPart + imagPart*imagPart) <= 4.0) ) {
// Update: z = z*z + z0;
double const oldRealPart = realPart;
realPart = realPart*realPart - imagPart*imagPart + realPart0;
imagPart = 2.0*oldRealPart*imagPart + imagPart0;
return count;
(the full source-code is available in the file-exchange submission linked at the end.)
One GPU thread is required for each element in the array, divided into blocks. The kernel indicates how big a thread-block is, and this is used to calculate the number of blocks required.
% Setup
t = tic();
x = parallel.gpu.GPUArray.linspace( xlim(1), xlim(2), gridSize );
y = parallel.gpu.GPUArray.linspace( ylim(1), ylim(2), gridSize );
[xGrid,yGrid] = meshgrid( x, y );
% Load the kernel
kernel = parallel.gpu.CUDAKernel( 'processMandelbrotElement.ptx', ...
'processMandelbrotElement.cu' );
% Make sure we have sufficient blocks to cover the whole array
numElements = numel( xGrid );
kernel.ThreadBlockSize = [kernel.MaxThreadsPerBlock,1,1];
kernel.GridSize = [ceil(numElements/kernel.MaxThreadsPerBlock),1];
% Call the kernel
count = parallel.gpu.GPUArray.zeros( size(xGrid) );
count = feval( kernel, count, xGrid, yGrid, maxIterations, numElements );
% Show
gpuCUDAKernelTime = toc( t );
imagesc( x, y, count )
axis image
title( sprintf( '%1.2fsecs (GPU CUDAKernel) = %1.1fx faster', ...
gpuCUDAKernelTime, cpuTime/gpuCUDAKernelTime ) );
When my brothers and I sat down and coded up our first Mandelbrot set it took over a minute to render a 32x24 image. Here we have seen how some simple steps and some nice hardware can now render
1000x1000 images at several frames per second. How times change!
We have looked at three ways of speeding up the basic MATLAB implementation:
1. Convert the input data to be on the GPU using gpuArray, leaving the algorithm unchanged
2. Use arrayfun on a GPUArray input to perform the algorithm on each element of the input independently
3. Use parallel.gpu.CUDAKernel to run some existing CUDA/C++ code using MATLAB data
I've also created a graphical application that lets you explore the Mandelbrot Set using each of these approaches. You can download this from the File Exchange:
If you have any ideas for creating Mandelbrot Sets at even greater speed or can think of other algorithms that might make good use of the GPU, please leave your comments and questions below.
Read a bit more about the life and times of Benoit Mandelbrot:
See his recent talk at TED:
Have a look at Cleve Moler's chapter on the Mandelbrot Set:
Copyright 2011 The MathWorks, Inc.
Published with MATLAB® 7.12
To leave a comment, please click here to sign in to your MathWorks Account or create a new one. | {"url":"https://blogs.mathworks.com/loren/2011/07/18/a-mandelbrot-set-on-the-gpu/?s_tid=blogs_rc_3","timestamp":"2024-11-11T15:26:35Z","content_type":"text/html","content_length":"190293","record_id":"<urn:uuid:b1916dba-fced-4c14-9bf0-c8705db8607b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00778.warc.gz"} |
30 Amazing Facts about Pi - Spinfold30 Amazing Facts about Pi - Spinfold
Before moving into facts about Pi, please check our beautiful collection of facts from all categories which you would definitely love to share with your friends. – Amazing facts from all categories.
Amazing facts about Pi
1. The most recognized mathematical constant is Pi. Pi is often considered as the most intriguing and important number in all of mathematics.
2. The symbol for pi has been regularly used for the past 250 years only.
3. You can never find the true circumference or area of a circle because we can never find the true value of pi.
4. “Wolf in the Fold”, the Star Trek episode, Spock spoils the evil computer by commanding it to, compute the last digit of Pi value.
5. In the Greek alphabets, Pi is the sixteenth letter. In English p is also sixteenth letter.
6. Pi: Faith in Chaos, a fascinating movie directed by Darren Aronofsky’s won Directory Award at the 1988 Sundance Film Festival. In this movie, the main character attempts to find simple answers
about Pi and drives him mad.
7. Pi, represents the ratio of a circle’s circumference to its diameter. It can be also said as, the number of times a circle’s diameter will fit around its circumference.
8. Pi day is celebrated on March 14 every year. It represents 3.14, which is pi value. The first widely-attended pi day celebration was organized by Physicist Larry Shaw, also known as Prince of
Pi, in the year 1988.
9. Pi is transcendental and irrational. It is irrational because it can’t be written as a rational simple fraction, though 22/7 is close, it is not exact. It is transcendental because it is not
algebraic, it is not a non-constant polynomial equation with rational coefficients.
Also read: Amazing Facts about Numbers
10. Pi value when calculated will continue infinitely without repetition or pattern. The value have been calculated to more than one trillion digits beyond its decimal points. Chao Lu holds the
world record for remembering the value of Pi up to 67,890 digits.
11. Ludolph Van Ceulen spent most of his life calculating the first 36 digits of Pi. n This is known as Ludolphine Number.
12. William Shanks calculated the first 707 digits of Pi by hand, unfortunately he made a mistake after 527th place. He has done it in 19th century.
13. With the help of Hitachi SR 8000, a powerful computer, a Japanese scientist found 1.24 trillion digits of Pi, breaking all the previous records.
14. Pi is the secret code in The Net starring Sandra Bullock and in Alfred Hitchcock ‘s Torn Curtain.
15. The number 360 is the 359th digit position of Pi, which is connected to circle.
16. Computing the value of Pi is a Stress test for a computer.
17. 104348/33215 is the most accurate fraction of Pi, which is equal to 00000001056%.
18. There are no zeros in the first 31 digits of Pi.
19. Humans studied Pi for almost 4000 years.
20. Pi is mentioned in bible.
21. Archimedes is the first person, who intensely studied about Pi in the ancient times.
22. Albert Einstein was born on Pi day.
23. The well known fraction of Pi(22/7) is 0.00000849% accurate.
24. In the first 31 digits of Pi, there are no zeros.
25. Even computers can’t find the value of Pi.
26. Plato supposedly obtained accurate value for pi : √2 + √3, which is 3.146.
27. Most people say that there are no corners for circle, but actual fact is circle has infinite number of corners.
28. 314159, the first six digits of Pi appear in order at least six times among the first ten million decimals of Pi.
29. “Ludolph’s Number”, “Archimedes constant”, “Circular constant” are names by which Pi is referred.
30. William Jones is the person, who introduced the symbol of Pi in the year 1706. But it was Leonhard Euler who popularized it in 1737.
Also read: Amazing Facts about Numbers | {"url":"https://spinfold.com/30-amazing-facts-about-pi/","timestamp":"2024-11-06T07:53:48Z","content_type":"text/html","content_length":"79643","record_id":"<urn:uuid:924be43a-fe1b-4163-a643-d513e6469827>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00105.warc.gz"} |
Calculating the Area of a Triangle
Try this fun worksheet that will help you get the hang of calculating the area of a triangle. It presents eight different triangles, each marked with definitive base and height measurements.
The area of a triangle, which is the space it covers, can be easily calculated. Just multiply the base (the bottom of the triangle) by its height (the highest point of the triangle if you stand it
up), and then divide that number by 2. The formula is area = 1/2 x base x height. This simple formula tells us how much space is inside the triangle.
Perfecting the calculation of the area of a triangle is used in many real-world situations, such as architecture, engineering, and art. Knowing how to compute areas is practical for tasks ranging
from design projects to determining the materials needed for construction. | {"url":"https://www.edhelper.com/worksheets/Calculating-the-Area-of-a-Triangle.htm","timestamp":"2024-11-10T15:44:52Z","content_type":"text/html","content_length":"19542","record_id":"<urn:uuid:21647e68-fe12-4d2e-9a78-4d851741ae9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00549.warc.gz"} |
Elasticity and Wave Propagation in Granular Materials
Elasticity and Wave Propagation in Granular
Prof. dr. rer.-nat. S. Luding Universiteit Twente, promotor Dr. V. Magnanimo Universiteit Twente, promotor Prof. dr. ir. T. van den Boogaard Universiteit Twente
Prof. dr. ir. A. de Boer Universiteit Twente Prof. dr.-ing. H. Steeb Universität Stuttgart Prof. dr. J. Jenkins Cornell University Prof. dr. G. Combe Laboratoire 3SR
The work in this thesis was carried out at the Multi Scale Mechanics (MSM) group of the Faculty of Science and Technology of the University of Twente. It is part of the research program T-MAPPP
(http://www.t-mappp.eu/), which is financially supported by the European Union funded Marie Curie Initial Training Network, FP7 (ITN-607453).
Cover design by Hamidreza Madadi.
Printed by Ipskamp Printing, Enschede, The Netherlands. Copyright © 2019 by Kianoosh Taghizadeh Bajgirani.
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or me-chanical, including photocopying, recording
or by any information storage and retrieval system, without written permission of the author.
ISBN: 978-90-365-4860-1
DOI number: 10.3990/1.9789036548601
ter verkrijging van
de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,
prof. dr. T.T.M. Palstra
volgens besluit van het College voor Promoties in het openbaar te verdedigen
donderdag 26 September 2019 om 12.45 uur door
Kianoosh Taghizadeh Bajgirani
geboren op 27 augustus 1990 Tehran, Iran
Prof. dr. rer.-nat. S. Luding en de assistent-promotor:
Abstract xi Samenvatting xiii 1 Introduction 1
1.1 Granular matter . . . 2 1.1.1 Granular mixtures . . . 3
1.1.2 Discrete Element Method . . . 4
1.1.3 Micro-macro transition . . . 5
1.1.4 Continuum approach . . . 5
1.2 Elasticity in granular materials . . . 7
1.2.1 Elasticity and State variables . . . 9
1.3 Waves propagation in granular medium . . . 10
1.3.1 Waves and elasticity . . . 10
1.3.2 Dispersion . . . 13
1.3.3 Attenuation (loss of energy) . . . 13
1.3.4 Master equation . . . 14
1.4 Thesis scope and outline . . . 15
2 Micro- and macro-mechanical study of spherical granular particles 19 2.1 Introduction . . . 20
2.2 Simulation approach . . . 22
2.2.1 Equations of motion . . . 22
2.2.3 Frictional contact model . . . 24
2.2.4 Adhesive, elasto-plastic contact model . . . 24
2.2.5 Microscopical quantities . . . 25
2.3 Preparing samples and defining quantities . . . 29
2.3.1 Jamming transition (from fluid- to solid-like behavior) . . 30
2.3.2 Macro-quantities during the sample preparation . . . 31
2.4 Small strain stiffness . . . 37
2.4.1 Incremental response of frictional samples . . . 38
2.4.2 Reversibility - from elastic to plastic . . . 41
2.4.3 Incremental response of cohesive samples . . . 43
2.5 Concluding remarks . . . 45
3 Micromechanical study of the elastic stiffness in isotropic frictional granular solids 47 3.1 Introduction . . . 48 3.2 Numerical setup . . . 50 3.2.1 Contact models . . . 50 3.2.2 Simulation
parameters . . . 51 3.2.3 Characteristic quantities . . . 53 3.3 DEM simulations . . . 54 3.3.1 Preparation procedure . . . 54 3.3.2 Elastic stiffness . . . 55
3.3.3 Influence of inter-particle contact friction during prepara-tion on the elastic moduli . . . 57
3.4 Granular elasticity . . . 58
3.4.1 Effective medium theory . . . 60
3.4.2 Fluctuation theory . . . 60
3.5 Role of the tangential stiffness during probing . . . 63
3.5.1 Incremental non-affine fluctuations . . . 67
3.6 Concluding remarks . . . 69
4 Elastic waves in particulate glass-rubber mixtures 75 4.1 Introduction . . . 76 4.2 Experiments . . . 77 4.2.1 Test procedure . . . 77 4.2.2 Mixture properties . . . 79 4.3 P-wave velocity . . . 81
4.4 DEM study . . . 85 4.4.1 Numerical setup . . . 85
CONTENTS ix
4.4.2 Numerical results . . . 87
4.5 Frequency spectrum . . . 89
4.6 Damping . . . 90
4.7 Conclusion . . . 93
5 Stress based multi-contact model for discrete-element simulations 97 5.1 Introduction . . . 98
5.2 Discrete Element Method . . . 100
5.2.1 Normal contact law . . . 101
5.2.2 Tangential force law . . . 102
5.3 Deformable particle models . . . 103
5.3.1 Multi-contact strain based model . . . 103
5.3.2 Multi-contact stress based model . . . 104
5.3.3 Modeling test cases . . . 106
5.4 Uniaxial unconfined compression of a single rubber sphere . . . 110
5.5 Uniaxial confined compression . . . 111
5.5.1 Compression using hydrogel balls . . . 111
5.5.2 Compression using rubber spheres . . . 114
5.5.3 Compression using glass beads . . . 114
5.6 Computational cost performance of Mutli-contact models . . . 117
5.7 Conclusions and outlooks . . . 117
6 Effect of Mass Disorder on Bulk Sound Wave Speed: A Multiscale Spectral Analysis 121 6.1 Introduction . . . 122
6.2 Granular Chain Model . . . 123
6.2.1 Linearized Equation of Motion . . . 125
6.2.2 Impulse Propagation Condition . . . 127
6.2.3 Standing Wave Condition . . . 127
6.2.4 Mass Disorder/Disorder Parameter (ξ) & Ensembles . . . 128
6.3 Energy Evolution . . . 129
6.3.1 Total Energy in the Wavenumber Domain . . . 130
6.3.2 Numerical Master Equation . . . 131
6.4 Results and discussion . . . 132
6.4.1 Energy Propagation with Distance . . . 132
6.4.2 Energy propagation in space and time . . . 137
6.4.3 Energy propagation across wave numbers . . . 139
6.4.4 Attenuation . . . 142
6.5 Conclusion . . . 147
7 Overview on continuum modeling of granular materials 155
7.1 Introduction . . . 156 7.2 Overview on continuum modeling . . . 157 7.2.1 Continuum model: a particle-based hypoelastic law . . . 165 7.3 Conclusion . . . 169
8 Conclusion and Outlook 171
Summary 205
Acknowledgments 209
Particle simulations are able to model behavior of granular materials, but are very slow when large-scale phenomena and industrial applications of granular materials are considered. Even with the
most advanced computational tech-niques, it is not possible to simulate realistic numbers of particles in large sys-tems with complex geometries. Thus, continuum models are more desirable, where
macroscopic field variables can be obtained from a micro-macro averag-ing procedure. However, aspects of microscopic scale are neglected in classical continuum theories (restructuring, geometric non
linearity due to discreteness, explicit control over particle properties).
The focus of this work is the investigation of elastic and dissipative behavior of isotropic, dense assemblies. In particular, the attention is devoted on the ef-fect of microscopic parameters (e.g.
stiffness, friction, cohesion) on the macro-scopic response (e.g. elastic moduli, attenuation). The research methodology combines experiments, numerical simulations, theory.
One goal is to extract the macroscopic material properties from the mi-croscopic interactions among the individual constituent particles; for simple enough systems this can often be done using
techniques from mechanics and statistical physics. While these simplified models can not capture all aspects of technically relevant realistic grains the fundamental physical phase transitions can be
studied with these model systems.
Complex mixtures with more than one particle species can exhibit enhanced mechanical properties, better than each of the ingredients. The interplay of soft with stiff particles is one reason for
this, but requires a more accurate formation of the interaction of deformable spheres. A new multi-contact approach is pro-posed which shows a better agreement between experiments and simulations in
comparison to the conventional pair interactions.
The study of wave propagation in granular materials allows inferring many fundamental properties of particulate systems such as effective elastic and
dis-xii Abstract
sipative mechanisms as well as their dispersive interplay. Measurements of both phase velocities and attenuation provide complementary information about in-trinsic material properties. Soft-stiff
mixtures, with the same particle size, tested in the geomechanical laboratory, using a triaxial cell equipped with wave trans-ducers, display a discontinuous dependence of wave speed with
The diffusive characteristic of energy propagation (scattering) and its fre-quency dependence (attenuation) are past into a reduced order model, a master equation devised and utilized for
analytically predicting the transfer of energy across a few different wavenumber ranges, in a one-dimensional chain.
Door de deeltjes van granulaire materiaal te simuleren zijn we in staat het bulkgedrag te modelleren. Echter zijn dit soort deeltje simulaties zeer langzaam wanneer er grootschalige fenomenen en
industriële toepassingen worden ges-imuleerd. Zelfs met de meest geavanceerde computertechnieken is het niet mogelijk om met een realistische hoeveelheid deeltjes deze simulaties toe te passen in
complexe geometrieën. Dus continuüm modellen zijn nog steeds wenselijk, waarbij de macroscopische variabelen kunnen worden verkregen door micro-macro middeling methoden. Echter wordt de
microscopische schaal vaak weggelaten in de klassieke continuüm theorieën. (Herstructurering, ge-ometrische niet-lineariteit door discretisatie, expliciete controle over deeltjes eigenschappen)
De focus van dit werk is het onderzoek naar de elastische en dissipatieve gedrag van isotropische, dichtgepakte samenstellingen. Specifiek wordt er aan-dacht besteed aan het effect van microscopische
parameters (stijfheid, wrijv-ingsweerstand, cohesie) op het macroscopische gedrag (elasticiteitsmodulus, verzwakking). De onderzoeksmethodiek combineert hierbij experimenten, nu-merieke simulaties en
Een van de doelen is het verkrijgen van macroscopische materiaaleigen-schappen uit de microscopische interacties tussen de individuele deeltjes in samenstellingen. Voor eenvoudige systemen kan dit
worden gedaan door bestaande technieken uit stijfheid en sterkteleer en statistische natuurkunde. Hoewel deze eenvoudige modellen niet alle aspecten van de technische rele-vante realistische
granulaire materialen kan bevatten, kan met deze modellen de fundamentele fase-transformaties worden onderzocht.
Complexe mengsels met meer dan een soort van deeltjes vertonen soms ver-beterde mechanische eigenschappen dan dat de materialen apart zouden doen. Het samenspel tussen de zachte en harde deeltjes is
een van de reden voor
xiv Samenvatting
dit gedrag, echter vergt dit een betere omschrijving van de interactie tussen de deeltjes. Een nieuwe Multi-contact methode wordt uiteengezet waarbij een betere overeenkomst tussen experimenten en
simulaties is gevonden dan dat er met conventionele interacties tussen twee deeltjes.
Door de studie naar de prorogatie van geluid in granulaire materialen kun-nen we vele fundamentele eigenschappen van een granulair systeem afleiden, zoals de effectieve elastische en dissipatieve
mechanismen en de verstrooiing. Metingen van beide fase snelheden en verzwakking geven daarnaast extra in-formatie over de belangrijke materiaaleigenschappen. Zacht-hard mengsels, met dezelfde
deeltjesgrootte, zijn getest in een geo-mechanisch laboratorium doormiddel van een triaxial-cell met extra golf omvormers laten een onder-breking zien in de afhankelijkheid van de geluidssnelheid met
de granulaire mengsels.
Het diffuse karakter van de energie stroming (verstrooiing) en de frequen-tieafhankelijkheid (verzwakking) zijn verzameld in een lagere order model, een alomvattende vergelijking ontwikkeld voor het
analytisch voorspellen van de energie overbrenging over een bereik van meerdere golfgetallen in een één di-mensionale ketting.
Chapter 1
The children of Adam are limbs of a whole Having been created of one essence. When the calamity of time afflicts one limb The other limbs cannot remain at rest.
If you have no sympathy for the troubles of others; You are not worthy to be called by the name of "human".
Many industrial and geotechnical applications that are crucial for our society involve granular systems at small strain levels. That is the case of structures de-signed to be far from failure (e.g.
shallow foundations or underlying infrastruc-ture), strains in the soil are small and a sound knowledge of the bulk stiffness is essential for the realistic prediction of ground movements. In the
follow-ing, a general introduction to granular matter is given. Then, different regimes under deformation of a bulk granular materials and the emergence of a theo-retical framework based on
micro-mechanical information to represent elastic behavior for granular materials are explained. Definition of elastic waves in solids is provided and some characteristics of mechanical waves in
disordered heterogeneous media like attenuation, dispersion, and stochastic modeling are introduced. Finally, an outline of the thesis will be given.
1.1 Granular matter
Granular materials (e.g. Fig. 1.1), though ubiquitous in nature and widely used in engineering and construction, remain relatively poorly understood. They may behave like solids, liquids or gases,
though typically exhibiting a variety of unexpected behaviors that are not encountered in these conventional forms of materials. The preponderance of problems yet to be solved has sparked a renewed
interest in granular materials, in different communities.
(a) (b)
Figure 1.1: Examples of granular materials in our daily life.
Granular materials consist of discrete particles such as, e.g., separate sand-grains, agglomerates (made of many primary particles), natural solid materials like sandstone, or ceramics, metals or
polymers sintered during additive manu-facturing. The primary particles can be as small as nano-meters, micro-meters, or millimeters, [153] covering multiple scales in size and a variety of
mechan-ical and other interaction mechanisms like, e.g., friction and cohesion [263]. The latter becomes more and more important the smaller the particles are. All those particle systems have a
particulate, usually disordered, possibly inhomo-geneous and often anisotropic micro-structure, which is at the core of many of the challenges one faces when trying to understand powder technology
and granular matter.
Particle systems as bulk show a completely different behavior as one would expect from the individual particles. Collectively, particles either flow like a fluid or rest static like a solid. In the
former case, for rapid flows, granular materials are collisional and inertia dominated and compressible similar to a gas. In the latter case particle aggregates are solid-like and thus can form,
e.g., sand piles or slopes that do not move for long time. In between is the dense
1.1 Granular matter 3
and slow flow regime that connects the extremes and is characterized by the transitions (i ) : from static to flowing (failure, yield) or vice-versa (i i ) : from fluid to solid (jamming). On the
particle and contact scale, the most special property of particle systems is their dissipative, frictional, and possibly cohesive nature. Here, dissipation means that kinetic energy at this scale is
irrecoverably lost from the particles translational degrees of freedom into heat in terms of random motion, or due to plastic, i.e. irreversible, deformations either of the particles, or of the
micro-structure (chapter 2). The transition from fluid to solid can be caused by dissipation alone, which tends to slow down motion. The transition from solid to fluid (start of flow) is due to
failure and instability, when dissipation is not strong enough to avoid the solid yielding and transits to a flowing regime.
1.1.1 Granular mixtures
Granular matter usually occur in various sizes, or as mixtures of different mate-rials (Fig. 1.2). Particulate mixtures are of interest for a large number of fields, materials, and applications,
including mineral processing, environmental en-gineering, geomechanics and geophysics, and have received a lot of attention in the last decades [148]. A specific example in geotechnical engineering
is the increasing incorporation of recycled materials (e.g. shredded or granulated rubber, crushed glass) often used into conventional designs and soil improve-ment projects [20, 64, 134]. Moreover,
sophisticated mixtures of asphalt and concrete are widely used to construct roads and, also here, mixing-in additional components is a widely applied option [85, 86, 222, 264, 268, 274].
Among those mixtures, binary mixtures of two materials are a particularly interesting selection. Binary granular mixtures comprise a wide range of natu-ral and industrial materials, whose mechanical
and acoustic behavior is strongly influenced by the relative amount of the components [53, 274]. While several researchers have investigated bidisperse mixtures [123, 273], the investigation of
mixtures made of two different components with different material properties (densities, visco-elastic moduli) is limited so far to phenomenological observa-tions; a deeper insight into the governing
micromechanical properties is still missing [4, 33, 39]. Through better understanding the underlying small-scale physics, the effective behavior of mixtures can be robustly predicted, tailored to
specific technical applications and even optimized on demand.
Much more limited is the work on mixtures made of a stiff and soft phase. This was addressed experimentally in the substantial work of diffeent authers [108, 135, 251] and numerically in [61, 217,
269] and with special focus on the
Figure 1.2: Examples of granular mixtures in our daily life.
(post-)liquefaction behaviour. In a recent contribution we have combined wave propagation experiments (chapter 4) and DEM simulations (chapter 4 and 5) to show how the bulk modulus in a granular soil
increases if soft inclusions are added in proper amount. This is a field of immense interest, as interrelation between solid phases at the microscale opens up many possibilities [154].
1.1.2 Discrete Element Method
In recent decades, the discrete element method (DEM) [41] that models the motion and interaction of many individual particles has become increasingly popular as a computational tool to model granular
systems in both academia and industry. To date, not only due to increasing computer power available, considerable scientific advances have been made in the development of particle simulation methods,
resulting in an increasing use of DEM. However, careful verification of the various numerical codes and validation of the simulation re-sults with closely matching experimental data is essential to
establish DEM as a widely accepted tool able to produce satisfactory quantitative predictions with added value for design, optimization and operation of industrial processes.
With the development of computational power in recent years, the discrete particle/element method has gained its focus to the simulation community. However, this method has its own limitations in
applying to the real world. One part is that the number of particles can be simulated is limited, normally in the order between104[to][10]7[in a 3D setup, whereas one normally has much more]
de-1.1 Granular matter 5
scribing the real contact mechanics between particles in a numerical model. One has to make several assumptions to reduce this complexity to be able to simulate many particles and all their contacts.
Nevertheless, the DEM/DPM method is a really helpful tool for understanding the granules bulk behavior qualitatively (and quantitatively) and thus one can explore the physics behind the scene, for
discrete particulate systems, which the traditional continuum solid/fluid me-chanics can not explain.
1.1.3 Micro-macro transition
Due to their wide application, granular media have received a lot of attention in many fields, such as soil mechanics, process engineering, mechanical engi-neering, material science and physics.
Attempts to model these systems with classical continuum theory and standard numerical methods and design tools cannot always be successful, because of their discreteness and disorder at the
microscopic scale. Therefore, it is necessary to employ a multi-scale approach that can link the discrete nature of granular systems to a macroscopic, con-tinuum description. Both fundamental
understanding and design/operation of unit processes and plants require multi-scale and multiphase approaches, where the discrete nature of the particles is of utmost relevance and must not be
ignored. Fig. 1.3 llustrates the idea behind the micro-macro transition, i.e., passing information from discrete element (particulate) to finite element (con-tinuum) simulations.
1.1.4 Continuum approach
DEM simulations are very detailed and therefore slow when large-scale phe-nomena and industrial applications of granular materials are considered. Even with the most advanced computational techniques
of today, it is not possible to simulate realistic numbers of particles with complex geometries. Thus, contin-uum models are more desirable, where a granular medium is assumed a con-tinuum, and
principles of continuum mechanics are applied to obtain macro-scopic field variables. However, besides the speed advantage of a continuum approach, many features of granular materials at the
microscopic scale have to be neglected, such as restructuring, geometric non-linearity due to discreteness, or explicit control over particle properties. The mechanical behavior of the ma-terials has
to be defined, for example, based on the relation between stress and strain extracted from continuum models [74].
Figure 1.3: Micro-macro concept to link between discrete to continuum.
The relation between stresses and strains is called constitutive model and it depends on the mechanical properties of the materials. Constitutive models are formulated mathematically and modeled
phenomenologically in continuum mechanics. Different varieties of constitutive models have been established to describe the material behavior and the deformation, such as elasticity, plasticity,
visco-elasticity, creep and etc. Several constitutive models within the frame-work of continuum mechanics have been developed to describe the mechanical behavior of granular materials. Most standard
models with wide range of appli-cation, such as elasticity, elasto-plasticity, or fluid-/gas-models of various kinds are commonly used for granular flows. Nevertheless, they are sometimes valid, only
in a very limited range of parameters and flow conditions. For example, the framework of kinetic theory is an established tool with quantitative predic-tive value for rapid granular flows but it is
not applicable in dense, quasi-static and static cases [145]. Further models, such as hyper-or hypo-elasticity, are complemented by hypo-plasticity and the so-called granular solid hydrodynam-ics,
where it has been established to represent also the mechanical behavior of granular solids. Differently from classical plasticity theory, where a plastic yield
1.2 Elasticity in granular materials 7
surface can be defined, the granular solid models provide incremental evolu-tion of equaevolu-tions with strain, and involve limit states, because a strict split be-tween elastic and plastic behavior
seems invalid in granular materials. Some of these theories have been extended to accommodate the anisotropy of the micro-structure [146], but only few models account for an independent evolution of
the microstructure. The anisotropy constitutive model based on incremental evolution equations for stress and fabric was presented in Ref. [121] for fric-tionless systems.
1.2 Elasticity in granular materials
It is commonly known that soil behavior is not as simple as its prediction with simply formulated linear constitutive models, which are commonly and care-lessly used in numerical analyses. Complex
soil behavior which stems from the nature of the multi-phase material exhibits both elastic and plastic non-linearities. Deformations include irreversible plastic strains. Depending on the history of
loading, soil may compact or dilate, its stiffness may depend on the magnitude of stress levels, and soil deformations can be time-dependent, etc.
The behavior of granular materials depends on the strain regime. Roughly speaking, we can distinguish (i ) :an elastic regime (very small strain);(i i ) :an elasto-plastic regime at intermediate and
large strain; and (i i i ) :a fully plastic regime where the material flows at constant stress and volume (beyond the solid to fluid transition).
A stiffness degradation curve, obtained in the resonant column device, is normally used to explain the shear stiffnessGfor a wide range of shear strain. A typical output of the resonant column
experiment is given in Fig.1.4.
Atkinson and Sallfors used a normalized stiffness degradation curve to cate-gorize the strain levels into three groups (as shown in Fig.1.4): the very small strain level, where the stiffness modulus
is constant in the elastic range; the small strain level, where the stiffness modulus varies non-linearly with the strain; and the large strain level, where the soil is close to failure and the soil
stiffness is relatively small [14, 157].
For many geotechnical structures under working loads, the deformations are small. The regime of deformation where the behavior can be considered linear elastic is infinitesimal, with nonlinear and
irreversible effects present already at small strains. Nevertheless, the stiffness of soils is of outmost importance, as it provides an anchor on which to attach the subsequent stress-strain response
[24, 175].
An elastic response is only observed for very small strain (order of10−6 [or]
10−5[) intervals, and should in fact be viewed as an approximation, as ]
dissipa-tion mechanisms are always present (in particular, solid fricdissipa-tion) and preclude the general definition of an elastic energy (chapter 2). The relative amount of dissipation decreases
as the size of the probed strain interval approaches zero. For that reason, the material behavior is best characterized as “quasielastic” in that limited range [24, 35]. In fact, soil behavior is
considered to be truly elas-tic in the range of very small strains, where soil even may exhibit a nonlinear stress-strain relationship. However, its stiffness is almost fully recoverable after
unloading. Following the pre-failure non-linearities of soil, one may observe a strong variation of stiffness starting from very small shear strains, which cannot be reproduced by models such as the
linear-elastic Mohr-Coulomb model.
Fig.1.4 shows that the response of granular materials is nonlinear and in-elastic even at extremely small strains. The region of stress or strain in which granular materials can be described as truly
elastic, producing an entirely re-coverable response to perturbations, the so-called small-strain stiffnessGmax,
is very small, corresponding to shear strains of the order of10−4[. In turn, the]
size of the elastic, reversible regime depends on material characteristics, stress state, anisotropy: the elastic range increases with increasing asperities (particle friction) and pressure, but
decreases with increasing anisotropy.
Figure 1.4: Degradation curve forGwith indication of typical soil tests and geotechnical applications per strain regime [14, 157].
1.2 Elasticity in granular materials 9
1.2.1 Elasticity and State variables
Despite the long-standing debate across the geomechanical, mechanical and physics communities, basic features of the physics of granular elasticity are cur-rently unresolved, like a proper set of
state variables to characterize the effective moduli.
In early studies, macroscopic variables measurable in laboratory experi-ments, were thought to be sufficient. Based on those information, many em-pirical relations have been proposed, where the
elastic moduli are functions of pressure and void ratio, e.g. [19, 80, 81, 201]. However, such formulations miss a first order mechanical interpretation and coefficients have to be back-calculated
case by case from experiments, based on the specific material and stress path. Moreover, experimental evidences [30, 54, 124, 191], along with many numerical studies [3, 160], show that stress and
volume fraction are not sufficient to characterise granular elasticity
On the other hand, conventional approaches in the framework of solid state elasticity [139] consider a uniform strain at all scales, with the displacement field of the grains following the
macroscopic deformation (affine approxima-tion). These Effective Medium Theories (EMT) developed by Digby and Wal-ton [47, 253] in the 80’s are the first, simplest attempt for a micromechanical
approach to the elasticity of granular soils. EMT predicts the moduli of an isotropic granular material in terms of the external pressure, the void ratio and average coordination number (p0,e, Z). In
particular, the pressure dependency
isG ∼ K ∼ p1/3
0 ,a direct consequence of the Hertz interaction between the
parti-cles. However such scaling is not recovered in experiments and previous anal-ysis (see [70] for a review) raises serious question about the validity of these generally accepted theoretical
elastic formulations.
Empirical relations coming from experiments and micromechanical EMT equa-tions show many similarities and the two approaches can fruitful inform each other. Following one of the paths suggested
already in [70] and further devel-oped in [155, 158], the set of state variables to describe granular elasticity is complete, and experiments follow the trend predicted by the model. This is
ob-tained with the aid of Discrete Element Simulations (DEM), that uniquely allow to monitor the kinematics at the microscale and link it with the macroscale.
However, the EMT framework still largely overpredicts the elastic moduli of loose samples, especially when shear is involved. The difficulty in describing theoretically the shear modulus is due to
the complex relaxation of the particles as related to the structural disorder in the packing [158]. Sophisticated theo-ries in which collective fluctuations and relaxation of the particles are
accounted for are needed to recover quantitative agreement. Here we briefly illustrate the mechanics beyond these theories and compare the results with numerical simulations. Recent attempts in this
direction are developed in [93, 125] where statistical parameters from a fluctuation analysis are introduced to describe the scaling of the moduli. In these theoretical models, fluctuations are
introduced in the kinematics of contacting particles. They are determined as functions of "fabric tensors" that describe, on average, the packing geometry and the variation of the number of contacts
per particle. The proposed the-ories are able to predict an elastic resistance of the aggregate comparable to numerical simulation (chapter 3).
1.3 Waves propagation in granular medium
A wave is an elastic perturbation that propagates between two points through a body (volume waves) or on the surface (surface waves) without material dis-placement [5]. Traveling through the interior
of the Earth, body waves arrive before the surface waves and are at higher frequency than surface waves. They are divided into two types P- waves and S-waves, the P-waves traveling faster than
S-waves through a solid body.
In the case of volume waves the acoustic-elastic effect is related to the change in the wave velocity of small amplitude waves due to the stress state of the body. Differently from liquids, in a
solid material three acoustic polarisa-tions exist, more specifically a longitudinal and two transversal branches.
1.3.1 Waves and elasticity
Wave also offer a direct connection to elastic properties of materials, due to their relatively easy application, through commercial equipment, as well as the various formulations relating the wave
velocity and the material moduli. Ad-vanced features like frequency dependence of wave parameters may further improve the characterization capacity. Concrete and soil samples due to their inherent
microstructure, (which is enriched by the existence of damage-induced cracking), exhibit interesting behaviors concerning the propagation of pulses of different frequencies. Here, we will derive the
relations between the elas-tic characteriselas-tics and the velocities of acouselas-tic waves, in the longitudinal and transversal directions.
The use of wave propagation to describe the small strain stiffness behavior of a material has been a well documented, widely-used technique, as evidenced
1.3 Waves propagation in granular medium 11
in the literature. Velocity testing, which includes BE and UT technology, has been gaining popularity as an experimental method due to its relative ease of obtaining the modulus of a material.
Let us consider a three-dimensional body with density ρ, homogeneous, isotropic and elastic. The stress change due to the propagation of the wave in the body is given by the Newton’s second law
applied to the volume element ρdV [46]:
∂σi j
∂xj = ρ ¨ui, (1.1)
with σi j stress andui displacement of the volume element in directionsi , j =
1,2,3. On the other hand, the constitutive relation for the elastic body holds, that relates the stress tensor to the strain²i j via the stiffness tensorCi j kl
σi j= Ci j kl²kl, (1.2)
In the isotropic case, Eq.(1.2) becomes (Lamé equation)
σi j= λΘδi j+ 2G²i j, (1.3)
where Θ=
i =1
²i i,Gandλare the shear modulus and Lamé coefficient
respec-tively, and the incremental strain tensor is given by ²i j=1 2 µ[∂u] i ∂xj + ∂uj ∂xi ¶ . (1.4)
The bulk modulus is related to the previous quantities asK = λ + 2/3G. Using Eqs.(1.3-1.4) in Eq.(1.1), the equation of motion becomes
ρ∂ 2[u] i ∂t2 = λ ∂ ∂xj µ[∂u] j ∂xi ¶ +G∂ 2[u] i ∂x2 j +G[∂x]∂ j µ[∂u] j ∂xi ¶ . (1.5)
From Helmholtz decomposition, the displacement vector uuu can be written in terms of a scalar potentialφand a vector potentialψψψ:
uuu = ∇∇∇φ +∇∇∇ ×ψψψ,
where the tensorial notation has been used for the sake of brevity. Thus Eq.(1.5) is ∇∇∇ · ρ∂ 2[φ] ∂t2− µ λ+4[3]G ¶ ∇2φ ¸ +∇∇∇ × · ρ∂ 2[ψ][ψ][ψ] ∂t2 −G∇2ψψψ ¸ = 000. (1.6)
Eq.(1.6) is known as the wave equation and predicts longitudinal and transver-sal modes of propagation. The first term in Eq.(1.6) depends only onφand is related to the propagation of waves in the
longitudinal direction, while the sec-ond term depends on the vector potentialψψψand is associated to the transversal waves. Both terms must be separately zero to satisfy Eq.(1.6), that is, the two
propagation modes, longitudinal and transversal, are independent.
Finally, if we introduce the longitudinal and shear components of the dis-placement related toφandψψψrespectively as
u u
uP= ∇∇∇φ and uuuS= ∇∇∇ ×ψψψ,
from Eq.(1.6) we can derive the velocities of the longitudinal and transversal waves for the isotropic elastic body (chapter 4):
VP= s (λ + 4/3G) ρ and VS= s G ρ (1.7)
Due to the properties of divergence and curl angular displacements and rota-tions are not allowed during propagation of longitudinal waves, and volume changes are forbidden for transversal waves.
When observing Eq.(1.7), some aspects appear:(i ) :the propagation velocity increases with the stiffness of the material and decreases with its mass density (inertia), these characteristics being
constants in a given solid body; (i i ) :the velocity of transversal waves is smaller than the velocity of longitudinal waves, given the relative values of the moduli.
We move now our attention from solid to particulate materials. When the wavelength is significantly longer than the internal scales of the material, such as particle or cluster size, the propagation
velocity can be defined for the equiv-alent continuum, that is Eqs.(1.7), where the elastic moduli and mass density refer to the bulk medium. Differently, for high frequencies and short wave-lengths,
the continuum assumption does not hold, due to the heterogeneity of the material at small scale and forces fluctuation [228]. With increasing fre-quencies, features related to the multiscale nature
of soils become dominant, e.g. dispersion and frequency filtering.
Other than frequency, also amplitude is an important factor to take into account. The propagation of elastic waves is, by definition, a small perturbation phenomenon that does not alter the
micro-structure (fabric) or cause permanent (plastic) effects. This condition must be guaranteed for the continuum analogy to hold. If both conditions, long wavelength and small amplitude, are
1.3 Waves propagation in granular medium 13
wave measurements (obtained e.g. via wave transducers) can be used to infer elastic moduli and vice versa.
1.3.2 Dispersion
The study of the dispersive behaviors of materials with respect to wave prop-agation is a central issue in modern mechanics. Dispersion is defined as that phenomenon for which the speed of
propagation of waves in a given mate-rial changes when changing the wavelength (or, equivalently, the frequency) of the traveling wave. This is a phenomenon which is observed in practically all
materials as far as the wavelength of the traveling wave is small enough to interact with the heterogeneities of the material at smaller scales. Dispersion most often refers to frequency-dependent
effects in wave propagation. The dis-persion relation describes the interrelations of wave properties like wavelength, frequency, velocities, refraction index and attenuation coefficient. In wave
the-ory, dispersion is the phenomenon that the phase velocity of a wave depends on its frequency. Indeed, anyone knows that all materials are actually heteroge-neous if considering sufficiently small
scales: it suffices to go down to the scale of molecules or atoms to be aware of the discrete side of matter. It is hence not astonishing that the mechanical properties of materials are different
when considering different scales and that such differences are reflected on the speed of propagation of waves.
1.3.3 Attenuation (loss of energy)
When a mechanical wave propagates through a medium, a gradual decay of wave amplitude can be observed before the wave diminishes, partly for geo-metric reasons because their energy is distributed on
an expanding wave front, and partly because their energy is absorbed by the material they travel through. The energy absorption depends on the material properties. Amplitude is di-rectly related to
the acoustic energy or intensity of a sound. When sound travels through a medium, its intensity diminishes with distance. In certain materials, sound pressure (amplitude) is only reduced by the
spreading of the wave. The effect produced is to weaken the sound. ‘Scattering’ is the reflection of the sound waves in directions other than its original direction of propagation. ‘Ab-sorption’ is
the conversion of the sound energy to other forms of energy. The combined effect of scattering and absorption is called attenuation of seismic waves and is an important characteristic in the modern
seismology which needs to be studied.
Seismic attenuation is commonly characterized by the quality parameterQ. It is most often defined in terms of the maximum energy stored during a cycle, divided by the energy lost during the cycle.
Among the various methods of measuring attenuation from seismic data, the spectral ratio method is the most common method perhaps because it is easier to use and more stable.
1.3.4 Master equation
Including disorders, e.g. adding inclusions with different properties or size, will lead to enhanced absorption in typical frequency ranges/bands, as related to their specific characteristics
relative to the basis material.
For instance, it is known that the dominant frequencies for exterior noise due to tire-road interactions lies in the range of 0.5 to 2 kHz. Research has shown that such noise generated in our daily
life has a negative impact on hu-mans health. Therefore, it is urgent to find a novel approach to damp as much as possible at exactly these unwanted frequencies. One way are high and wide walls/
panels as usually installed along highways of metropolitan areas, which are generally expensive, while our approach involves a smarter design of the asphalt itself. Another example is the vibration
and noise generated by railways or subways in the low frequency range (10 to 50 Hz). These unwanted noises bring issues for specific infrastructures e.g. hospitals, art galleries, or tunnels, and
could be avoided by a better composition and optimal use of the ballast or the concrete foundations with respect to their damping features. Last but not least, earthquakes are the natural hazard that
generates the largest number of human casualties in our modern society. The dominant harmful frequency range of earthquakes usually remains very low, 5 to 20 Hz [99]. Earthquakes often cause
unrecoverable damages to buildings and infrastructures and ines-timable losses to our historical heritage. The cost of upgrade works on historical buildings is often too high and conflicts with other
tight constraints. Our novel approach is to design a seismic protection (“cloaking”) in the soil around, rather than on the building.
This will be the results from experiments and particle simulation to be in a macro-scale continuum model, with a resolution in frequency space. Instead of dealing with the too many eigen-modes of the
system, we propose an approach with reduced complexity, where the frequencies are grouped in bands. This is different in spirit from reduced order modeling since one accounts for all frequencies,
also the largest ones, but gives up the details by grouping all modes with similar frequency, gaining tremendous speed-up (chapter 6).
1.4 Thesis scope and outline 15
1.4 Thesis scope and outline
To gain more insight into the micro-structure of granular materials, three di-mensional discrete element simulations, theory, and experiments are performed on various quasi-static samples. Thus, this
thesis is divided in chapters cover-ing the elastic behavior of granular materials and considers many aspects such as mixtures of soft-stiff species, dissipation of energy, new approach on contact
modelling of soft particles, master equation of a force chain.
• Chapter 2: In the next chapter of this dissertation, the micro- and macro-mechanical behavior of idealized granular assemblies are studied, com-prising linearly elastic, frictional, cohesional,
polydisperse spheres, in a periodic triaxial box geometry, using DEM. The stress response to vari-ous deformation modes, namely purely isotropic and deviatoric (volume conserving), applied to this
granular samples are analyzed. A hysteretic contact model with plastic deformation and adhesion forces is used for micro- and macro-mechanical studies through fully disordered, densely packed,
cohesive and frictional granular systems. Especially, the effect of friction and adhesion on the elastic response is examined.
• Chapter 3: Next, assemblies of polydisperse, linearly elastic frictional spheres are isotropically prepared using DEM . In a second stage, several static, relaxed configurations at various volume
fractions above jamming are generated and tested. We investigate the effects of inter-particle con-tact properties on the elastic bulk and shear modulus by applying isotropic and deviatoric
perturbations. The amplitude of the applied perturba-tions has to be small enough to avoid particle rearrangement and to get the elastic response, whereas large amplitudes develop plasticity in the
sample due to contact and structure rearrangements between particles. We compare the data from DEM simulations with predictions from well-established micromechanical models, namely the Effective
Medium The-ory (EMT) and the Fluctuation TheThe-ory (FT). Both theories do not account for the effect of different preparation history (different inter-particle fric-tion coefficients) on the elastic
moduli. The fluctuafric-tion theory is in agree-ment with numerical data, almost perfect for the bulk modulus and close for the shear modulus, at least in the intermediate compression regime, but
does not capture the anomalous behavior where the theory overpre-dicts.
experi-ments in a triaxial cell set-up equipped with piezoelectric wave transduc-ers. Conducting systematic experiments on various mixtures of particles with different species will help to complete
the overall picture of the be-havior of granular mixtures. The initial configuration considered is the most basic binary system of rubber (soft) and glass (stiff) particles ran-domly distributed
within a latex membrane that allows to externally con-trol the confining stress at different uni-axial compression levels. A wave is agitated on one side of such dense, static, mechanically stable
packings and its propagation is investigated when it arrives at the opposite side. At various stress levels, P-wave has been excited and the time of flight is measured for many sample compositions
and pressure levels.
• Chapter 5: The chapter attempts to model confined powder compaction with the discrete element method (DEM) is a really challenging task since classical particle-particle contact model are limited
on the assumption of binary contacts regardless of the degree of confinement. In classical DEM, the fact that each particle experiences multiple simultaneous contacts that influence each other, at
high relative densitites, is missing. Important progress has been made recently, resulting in the formulation of multi-contact DEM but the picture is still incomplete. In this research, new force
models tackling this issue are presented. By adding an extra term which is a function of Poisson’s ratio and local particle stress tensor, we extend the classical force-displacement formula to
capture pseudo deformation of particles. Hence, stress tensor commonly used for post-processing reasons (i.e. in cross coarse-graining methods) was used to account for multiple contacts acting
simultaneously on a single particle. In our initial attempt, uniaxial compression simulations with Hertzian and linear contact models were conducted and modeled by frictionless spheres in the absence
of gravitational forces. Comparisons between classical DEM simulations and the new, alternative model for interactions between multiple contacts are presented.
• Chapter 6: Focuses on the transfer energy with distance as well as across different wavenumbers, as the mechanical wave propagates. The diffu-sive characteristic of energy propagation has been
discussed. A master equations is devised and utilized for analyzing the transfer energy across different wavenumbers, studied with the aid of a one-dimensional granu-lar chain.
intro-1.4 Thesis scope and outline 17
duced for the granular material, based on microscale information. A con-stitutive model, for frictional particles, involving the elastic moduli and the relation between effective moduli and
microstructure, is implemented in a Finite Element framework developed within the Kratos Multiphysics open source platform and some benchmark examples are carried out. • Chapter 8: Finally, the last
chapter dedicates to conclusions and
Chapter 2
Micro- and macro-mechanical
study of spherical granular
Greatness is always built on this foundation: the ability to appear, speak and act, as the most common man.
Modelling granular materials can help us to understand their behaviour on the microscopic scale, and to obtain macroscopic continuum relations by a micro-macro transition approach. The Discrete
Element Method (DEM) is used to inves-tigate the influence of inter -particle friction coefficient and cohesion on the micro and macro behaviour of granular packings in the context of an
elasto-plastic con-tact model. It is shown that the influence of friction coefficient on parameters is more pronounce rather cohesion stiffness. However, the effect of cohesion is not yet negligible.
The differences in macro and micro quantities become more pronounced when packings are closer to the jamming point i.e. the lowest density where the system is mechanically stable. Furthermore, we
observe that friction and cohesion have an influence on the jamming point for frictional samples. From the micro-scopic contact characteristics the macromicro-scopic elasticity parameters are
determined at different volume fractions. The conventional way to extract elastic constants of
a packing is to apply a compression or shear deformation to the entire system. The results show that the stiffness of the packings increases with the volume fraction as expected. Surprisingly, it was
observed that elasto-plastic samples experience multiple plastic regimes depending on the applied strain keeping the rate small. An elastic regime for very small strain is followed by the contact
plastic regime with reduced bulk moduli; which transits into the structural re-arrangements plastic regime. This interesting intermediate plastic regime is due to the hysteretic contact model with
changing contact stiffness during probing of configurations.1
2.1 Introduction
Granular materials play an important role in many industries, such as pharma-ceutical, mining or civil engineering. The macroscopic behaviour of granular material is very different from common solids
and fluids. There are different methods to model and understand the macroscopic behaviour of particulate systems. A powerful tool to study granular materials is the Discrete Element Method (DEM)
which provides a microscopic insight for the observed behaviour [41, 79, 149, 150, 198, 234]. The contact force model is at the basis of this method [142, 143] and a coupled system of equations is
solved to describe the motion of individual particles. Despite the modern computational power, the number of particles that can be simulated is still small compared to real-ity. This problem can be
solved by performing a transition from the micro- to the macro-scale and establishing macroscopic constitutive relations [72]. The microscopic properties can be used to drive macroscopic constitutive
relations. These relations are used to describe the particle behaviour on the large scale application/process level [147]. Because of discreteness and the disordered na-ture of granular materials at
the microscopic scale, it is necessary to employ a multi-scale approach which can link the micromechanics of granular systems to the continuum description. The objective of the multi-scale approach
is to predict the macroscopic (continuum) constitutive relationship from the micro-scopic contact constitutive relationship and from appropriate geometrical quan-tities or state variables by means of
suitable averaging techniques.
The main challenge comes when the powders are sticky, cohesive and less flow-able like those relevant in food industry [90]. Research has already been done on cohesive granular materials (see refs
[141, 225, 226]), however the influence of cohesion on granular packings is still poorly understood. There are
2.1 Introduction 21
two cases where cohesion becomes important. i )When particles become very small the cohesive forces become larger than the other forces on each particle, as is the case for dry fine powders [142,
250].i i )Not only the size of the particles contributes to the influence of cohesive attractive forces, but also liquid between the particles does this, as is the case for wet granular materials
[58, 169, 205].
The research presented here will focus on dry cohesive and non-cohesive granular particles and DEM is used to study granular packings made of polydis-perse particles. The question arises how does the
presence of attractive forces affect macroscopic properties of the packings? So far, only a few attempts have been made to answer this question. Gilabert et al. [68] focussed on a two-dimensional
packing made of particles with short-range interactions (cohesive powders) under weak compaction. Yang et al. [266] studied the effect of cohe-sion on force structures in a static granular packing by
changing particle size. Singh et al. [223] studied the effect of friction and cohesion on anisotropy in granular materials under quasi-static shear. The goal is to understand the influ-ence of the
microscopic parameters on the macroscopic properties of the pack-ings. Knowing the influence of cohesion on particulate systems will advance development of new constitutive models to predict the
macroscopic material behaviour, to be used to model real life applications and to understand and optimize processes.
Many industrial and geotechnical applications that are crucial for our society involve granular systems at small strain levels. That is the case of structures de-signed to be far from failure (e.g.
shallow foundations or underlying infrastruc-ture), strains in the soil are small and a sound knowledge of the bulk stiffness is essential for the realistic prediction of ground movements [35].
In micromechanical and numerical studies, elastic properties are associated with the deformations of a fixed contact network, and should therefore corre-spond to the “true elastic” behavior observed
in the laboratory for very small strain intervals. Indeed, except in very special situations in which the effects of friction are suppressed and geometric restructuring is reversible, the
irre-versible changes associated with network alterations or rearrangements pre-clude all kind of elastic modelling.
The goal here is to focus on the macro-mechanical response of dry frictional packings with elasto-plastic contact model and DEM will be used to study pe-riodic assemblies made of polydisperse
spheres. In particular, the paper in-vestigates how inter-particle contact friction and elasto-plastic cohesive contact model influence the bulk response of granular packings [106, 232, 234]. In this
work, we analyze the role of the contact model along with microstructure, stress and volume fraction [121, 232, 234]. The ultimate goal is to improve the
understanding of elasticity in particle systems and to guide the development of constitutive models.
This paper is organized in the following manner. In section 2, the simulation method and parameters used are given. The preparation test procedure and the averaging definitions for scalar and
tensorial quantities are explained in section 3. In Section 4, we first explain how the elastic moduli are determined; after that results of small-strain perturbations for different packings are
given. Finally, section 5 is devoted to the conclusion remarks and outlooks.
2.2 Simulation approach
We use the Discrete Element Method (DEM) to understand the behaviour of granular systems. In the model, we relate the force interacting between the par-ticles to the overlapδthat the particles have
with each other. DEM solves New-ton’s equations of motion for all forces fi= fn·n+ ft·t acting on particlei for the
translational and rotational degrees of freedom. The Discrete Element Method (DEM), often referred to as Molecular Dynamics (MD), is a many-particle sim-ulation method. Even though a lot of research
verified the usefulness of DEM [207, 255], large scale industrial applications are out of reach. These appli-cations involve even more than the millions of particles that can be simulated using DEM.
Instead of simulating real life applications, small samples of rep-resentative volume elements (RVEs) can be used to calculate the macroscopic constitutive relations needed to perform the micro-macro
transition [149]. Note that the evaluation of the inter-particle forces based on the overlap may not be sufficient to account for the inhomogeneous stress distribution inside the parti-cles.
2.2.1 Equations of motion
DEM models the particle interaction by calculating the equations of motion for every particle in the system. This is done for the normal direction, but also for the translational and rotational
degrees of freedom. If the forces fi acting on
the i-th particle are known and Newton’s equations are applied we get [149]: mi d
d t2ri= fi+ mig and Ii
d tωi= qi, (2.1)
wheremi is the mass of the i-th particle andri the position of the particle.
2.2 Simulation approach 23
interaction with other particles:fi=Pcfc[i]. The other force is due to body forces
like gravity (g), the particles moment of inertia (Ii), its angular velocity (ωi)
and the total torque (qi= q[i]f r i cti on+ qtor si on[i] + qr olli ng[i] ).
2.2.2 Contact model
For the sake of simplicity, the linear visco-elastic normal contact force model can be used. It involves a linear repulsive and a linear dissipative force: fn= kδ+γ0δ˙withkas spring stiffness,δ=
(ai+aj)−(ri−rj)·n > 0as particle overlap,
n = ni j= (ri−rj)/|ri−rj|as normal unit vector,γ0as viscous damping coefficient
and[δ]˙[the relative velocity in normal direction][v][n][= −][v][i j][·][n][= ˙δ][.]
An artificial damping force fb is introduced to reduce dynamic effects and
shorten relaxation times: fb= −γbvi. This will resemble the damping of a
back-ground medium, as e.g. a fluid. This force acts not on contacts but directly on particles, proportional to their velocity vi.
Using this model the particle contact can be seen as a damped harmonic os-cillator. The advantage is that the half-period of a vibration around an equilib-rium position can be computed and so the
typical response time on the contact level:
ω, with ω= q
(k/mi j) − η2[0], (2.2)
where ω is the eigenfrequency of the contact, η0= γ0/(2mi j)the rescaled
damping coefficient andmi j= mimj/(mi+ mj)the reduced mass.
Using the solution of Equation 2.2 the coefficient of restitution is obtained: r = −vn0/vn=exp(−πη0/ω) =exp(−η0tc), (2.3)
which quantifies the ratio of relative velocities after (primed) and before (un-primed) the collision. The integration time-step ∆tMD used for simulations
needs to be much smaller than the contact duration tc to make sure that the
integration of the equations of motion is stable. Note that in extreme cases of an overdamped spring,tc can become extremely, artificially large, i.e.
dissipa-tionγshould be neither too weak nor too strong.
The viscous dissipation mode is suitable for two-particle contact. But when there are a lot of particles involved it becomes very inefficient. Therefore artificial damping is introduced. There will
be some additional damping with the background. The background damping is of use for a quick relaxation and the system comes more rapidly to a static equilibrium. The values for the
background damping (γbandγbr) were checked for the used set of parameters
to prevent an over-damped system [142].
2.2.3 Frictional contact model
Friction is generated when two particles are in contact and have a motion rela-tive to each other. For the simulations presented here a friction model according to the Coulomb friction law is used.
This law has two aspects. There is a static friction when two particles do not have micro-slip at the contact surface. In the case of static friction, the friction force between the surfaces of two
particles cannot be greater than the product of the normal force fn [and the coefficient]
of static frictionµs: ft≤ µsfn. The linear visco-elastic contact model that was
introduced earlier is used for the force component in the tangential direction:
ft= ktδt+ γtδ˙t, (2.4)
wherekt is the tangential stiffness,γt the friction viscosity,δt the
displace-ment in the tangential direction and [δ]˙t [the relative velocity in the tangential]
direction [142]. The kinetic friction becomes active when the tangential com-ponent of the force is exceeding the maximum value of the static force, so when the surfaces of two particles that are in
contact start to slide.
For a more detailed description of the force models introduced here and for the rolling and torsional force laws that were used see [142, 149].
2.2.4 Adhesive, elasto-plastic contact model
In this work, a linear, hysteretic visco-elastic model is used to describe the inter-action between cohesive particles by adding irreversiblity into the linear contact model (see Refs. [142, 223,
224, 254]). This model is a simplified version of the nonlinear hysteretic force laws which were proposed by different authors [242, 243]. In this model, the particles stiffnesses are kept constant
with dif-ferent values during loading and unloading. The contact interaction consists of different phases (see Fig. 2.1). At first, the force increases linearly with the overlapδ up to δmax on the
loading (irreversible) branch with slope k1. The
unloading (reversible) branch starts at δmax, from where the force decreases
with the slopek2. The force between two particles becomes zero at overlap
2.2 Simulation approach 25
decreases with the same slopek2in the case of further unloading. If the overlap
is lower than δ0 during unloading, then an attractive force between particles
will be active until the minimum cohesive force branch fmi nis reached at
over-lapδmi n=k[k]2[2]−k[+k]1[c]δmax. Further unloading leads to the (unstable) attractive force
fhys[= −k]
cδon the adhesive branch with the slope−kc. If unloading starts at
δ< δmax, contacts follow branches parallel to the limit value, with a constant
unloading stiffnessk2until the cohesive branch is reached.
The (hysteretic) force can be written as: fhys= k1δ ifk2(δ − δ0) ≥ k1δ k2(δ − δ0) ifk1δ> k2(δ − δ0) > −kcδ − kcδ if − kcδ≥ k2(δ − δ0) (2.5) where k1, k2andkc are contact stiffnesses during
loading, unloading and on
the adhesive branch, respectively. The contact model presented involves some simplifications with respect to the behaviour observed in experiments, e.g. [230, 242, 243, 254], or proposed by other
authors [97, 189, 240]. Among those, it is the piece-wise linear structure, the value of the force atδ= 0and neglecting the
detachment of the deformed particles at a finite overlap. A detailed discussion on the model can be found in [224]. Simplifications are mainly driven by case in computation. However, we believe that
the influence on the specific aspects studied here is negligible, as our primary focus is on static packings in the small strain regime, where particles detachments/rearrangements are limited.
An overview of the parameters used in the DEM simulations can be seen in table 2.1. The values were examined with two particle collisions (bench-mark tests) to validate that the program was working
correctly and the linear (hysteretic) force model is correct. The normal force is plotted against the over-lap δ(Fig. 2.2). The hysteretic force diagram had the same characteristics as the
theoretical model (Figure 2.1). For different values of the adhesive stiff-ness (kc) the adhesive force (the negative force) increases as the kc increases
(Fig. 2.2). For the adhesive stiffness values were chosen between a wide range (1/20 ≤ kc/k ≤ 20) to show the influence well.
2.2.5 Microscopical quantities
Here, we define some microscopical quantities that obtained from single contact interaction. These parameters can not usually be measured from experiments, but are easily available from DEM
simulations. For single contacts, the contact
fhys fmin k1 k2 - 0) 0 max kc (δ δ δ δ δ k2 k2 k2 k2 0
Figure 2.1: Schematic graph of the piece-wise linear, hysteretic model. The adhesive force-displacement for normal collision. The non-contact forces (f0) are kept
equal to zero in this study and also the line for negativeδis neglected in this paper
Property Symbol Value SI-units
Time unit t 1 10−6[s]
Length unit x 1 10−3[m]
Mass unit m 1 10−9[kg]
Particle radius 〈a〉 1 10−3[m]
Polydispersity amax/ami n 3
Number of particles N 5000
Particle density ρ 2000 2000 kg/m3
Simulation time step ∆tMD 0.0037 3.7·10−9s
Unloading (reversible) stiffness k2 15·104 15·107kg/s2 Loading (irreversible) stiffness k1/k2 0.666
Cohesive stiffness kc/k2 0-20 Tangential stiffness kt/k2 0.2866 Coefficient of friction µ 0.5 Normal viscosity γ= γn 1000 1 kg/s Tangential viscosity γt/γ 0.2 Background visc. γb/γ 0.15
Backgr. torque visc. γbr/γ 0.03
Table 2.1: The microscopic contact model parameters values
force law is reformulated in terms of potential energy density, contact stress, and elastic deformation. Starting from a linear expansion of the interaction potential around static equilibrium,
stress can be derived from the principle of virtual displacement. The approach includes both normal and tangential forces.
2.2 Simulation approach 27 -10 -5 0 5 10 15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 fn [N] δ [mm] k[c]/k = 1/20 k[c]/k = 1/5 k[c]/k = 1/2 k[c]/k = 1 k[c]/k = 2 k[c]/k = 20
Figure 2.2: Two particles collision in the normal direction using the hysteretic contact model with different cohesive stiffnesskc. The force in the normal direction
is plotted against the overlapδ.
The overlap in normal direction can be expressed as [δ]~[n] [= ~l− (a][1][+ a][2][)~][n][.]
Where,ai is the radius of a particle,~l = ~ri− ~rj the branch vector (the difference
between particles position),~[n =~l/l]the normal vector with[l = |~l| = (a]1+a2). The
normal overlap is the normal deformation relative to the configuration when the particles just get in contact. The total relative deformation can be expressed in a normal and tangential
contributions,~ε=~εn+~εt, which becomes:
~ε=~δn l ~n~n + ~ δt l ~t 0[~][n] [(2.6)] with~t0[:= δ]
t/|δt|. During the deformation, the length and direction of branch
vector,~[l][, changes. The change of branch vector,][∂~][l][, can be split into a normal]
and tangential component as well. The normal component, expressed in index notation2[, becomes: :]
∂δnα= ∂lαn= nαnβεβγlγ (2.7)
and in the tangential component becomes∂~δt:= ∂~l−∂~ln, which can be
writ-ten as:
∂δtα= ∂lαt = tαtβεβγlγ (2.8)
Hence, the potential energy density for one contact can be expressed in term of the overlap: uc= 1 2Vc(kn ~ δ2n+ ktδ~2t) (2.9)
where, kn andkt are the spring stiffness in the normal and tangential
di-rection, respectively. Vc is left unspecified as this volume disappears during
averaging, in many cases. The potential energy density changes due to the de-formation. The change in the potential energy density can be split into a normal and tangential contribution which results
∂~u = ∂un+ ∂ut≈ 1 Vc(kn ~ δn∂~ln+ kt~δt∂~lt) ≈ 1 Vc ~ f∗[·~ε ·~l] [(2.10)]
where [f]~∗[= (~f + ~f]0[)/2][which is expressed in the actual force,]~[f = k][n]~[δ][n][+ k][t]~[δ][t][,]
and the force after displacement,~[f]0[= ~f+ ∂~f][. With the defined potential energy]
density and deformation, the stress can be derived. By differentiating uwith respect to the deformation components:
σαβ= ∂u ∂εαβ= 1 Vcf ∗ αlβ (2.11)
Likewise the former terms, the stress term can be expanded into a normal and tangential contributions:
σαβ= knlδn Vc nαnβ+ ktlδt Vc nαt 0 β (2.12)
which gives the incremental stress tensor as:
∂σαβ≈ knl∂δn Vc nαnβ+ ktl∂δt Vc nαt 0 β (2.13) withδn= | ~δn|,∂δn= |∂δn|,δt= |~δt|,∂δt= |∂δt|. | {"url":"https://5dok.net/document/ky6lr5oy-elasticity-and-wave-propagation-in-granular-materials.html","timestamp":"2024-11-10T08:26:03Z","content_type":"text/html","content_length":"218523","record_id":"<urn:uuid:4c2b6f12-3b14-4c93-8f6c-a09187f6b189>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00546.warc.gz"} |
Interval Data and How to Analyze It | Definitions & Examples
Interval data is measured along a numerical scale that has equal distances between adjacent values. These distances are called “intervals.”
There is no true zero on an interval scale, which is what distinguishes it from a ratio scale. On an interval scale, zero is an arbitrary point, not a complete absence of the variable.
Common examples of interval scales include standardized tests, such as the SAT, and psychological inventories.
Levels of measurement
Interval is one of four hierarchical levels of measurement. The levels of measurement indicate how precisely data is recorded. The higher the level, the more complex the measurement is.
While nominal and ordinal variables are categorical, interval and ratio variables are quantitative. Many more statistical tests can be performed on quantitative than categorical data.
Interval vs ratio scales
Interval and ratio scales both have equal intervals between values. However, only ratio scales have a true zero that represents a total absence of the variable.
Celsius and Fahrenheit are examples of interval scales. Each point on these scales differs from neighboring points by intervals of exactly one degree. The difference between 20 and 21 degrees is
identical to the difference between 225 and 226 degrees.
However, these scales have arbitrary zero points – zero degrees isn’t the lowest possible temperature.
Because there’s no true zero, you can’t multiply or divide scores on interval scales. 30°C is not twice as hot as 15°C. Similarly, -5°F is not half as cold as -10°F.
In contrast, the Kelvin temperature scale is a ratio scale. In the Kelvin scale, nothing can be colder than 0 K. Therefore, temperature ratios in Kelvin are meaningful: 20 K is twice as hot as 10 K.
Examples of interval data
Psychological concepts like intelligence are often quantified through operationalization in tests or inventories. These tests have equal intervals between scores, but they do not have true zeros
because they cannot measure “zero intelligence” or “zero personality.”
Type Examples
Standardized tests
Beck’s Depression Inventory
Psychological inventories Raven’s Progressive Matrices
Big Five personality trait tests
To identify whether a scale is interval or ordinal, consider whether it uses values with fixed measurement units, where the distances between any two points are of known size. For example:
• A pain rating scale from 0 (no pain) to 10 (worst possible pain) is interval.
• A pain rating scale that goes from no pain, mild pain, moderate pain, severe pain, to the worst pain possible is ordinal.
Treating your data as interval data allows for more powerful statistical tests to be performed.
Interval data analysis
To get an overview of your data, you can first gather the following descriptive statistics:
Interval data example
You collect the SAT scores of a group of 59 graduating students from City A. Test-takers can score anywhere between 400–1600 on the SAT.
Tables and graphs can be used to organize your data and visualize its distribution.
To organize your data, enter it into a grouped frequency distribution table.
SAT score Frequency
401 – 600 0
601 – 800 4
801 – 1000 15
1001 – 1200 19
1201 – 1400 16
1401 – 1600 5
To visualize your data, plot it on a frequency distribution polygon. Plot the groupings on the x-axis and the frequencies on the y-axis, and join the midpoint of each interval using lines.
Central tendency
From your graph, you can see that your data is fairly normally distributed. Since there is no skew, to find where most of your values lie, you can use all 3 common measures of central tendency: the
mode, median and mean.
The mode is the most frequently repeating value in your data set. In this case, there is no mode because each value only appears once.
The median is the value exactly in the middle of your data set. To find the middle position, take the value at (n+1)/2 where n is the total number of values.
(n+1)/2 = (59+1)/2 = 30
The median is in the 30th position, which has a value of 1120.
The mean uses all values to give you a single number for the central tendency of your data. To find the mean, use the formula of ⅀x/n. Sum up all values (⅀x) and divide the sum by n.
⅀x = 65850
n = 59
⅀x/n = 65850/59 = 1116.1
The mean is usually considered the best measure of central tendency when you have normally distributed quantitative data. That’s because it uses every single value in your data set for the
computation, unlike the mode or the median.
The range, standard deviation and variance describe how spread your data is. The range is the easiest to compute while the standard deviation and variance are more complicated, but also more
To find the range, subtract the lowest from the highest value in your data set. Our maximum value is 1500, and our minimum is 620.
Range = 1500 – 620 = 880
The standard deviation (s) is the average amount of variability in your dataset. It tells you, on average, how far each score lies from the mean. Most computer programs will easily calculate the
standard deviation for you. If you want to do it by hand, use these steps.
s = 210.42
The variance (s^2) is the average squared deviation from the mean. A deviation from the mean is the difference between a value in your data set and the mean. To find the variance, square the standard
s^2 = 44279.36
Statistical tests
Now that you have an overview of your data, you can select appropriate tests for making statistical inferences. With a normal distribution of interval data, both parametric and non-parametric tests
are possible.
Parametric tests are more statistically powerful than non-parametric tests and let you make stronger conclusions regarding your data. However, your data must meet several requirements for parametric
tests to apply.
The following parametric tests are some of the most common ones applied to test hypotheses about interval data.
Aim Samples or variables Test Example
Comparison of means 2 samples T-test What is the difference in the average SAT scores of students from 2 different high schools?
Comparison of means 3 or more samples ANOVA What is the difference in the average SAT scores of students from 3 test prep programs?
Correlation 2 variables Pearson’s r How are SAT scores and GPAs related?
Regression 2 variables Simple linear regression What is the effect of parental income on SAT scores?
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Other interesting articles
If you want to know more about statistics, methodology, or research bias, make sure to check out some of our other articles with explanations and examples.
Frequently asked questions about interval data
Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:
While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.
For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of
thermal energy.
Individual Likert-type questions are generally considered ordinal data, because the items have clear rank order, but don’t have an even distribution.
Overall Likert scale scores are sometimes treated as interval data. These scores are considered to have directionality and even spacing between them.
The type of data determines what statistical tests you should use to analyze your data.
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The Stacks project
Lemma 10.136.6. Let $S$ be a finitely presented $R$-algebra which has a presentation $S = R[x_1, \ldots , x_ n]/I$ such that $I/I^2$ is free over $S$. Then $S$ has a presentation $S = R[y_1, \ldots ,
y_ m]/(f_1, \ldots , f_ c)$ such that $(f_1, \ldots , f_ c)/(f_1, \ldots , f_ c)^2$ is free with basis given by the classes of $f_1, \ldots , f_ c$.
Comments (2)
Comment #3432 by ym on
It's easier to see the isom $S\cong R[x\ldots]/(f\ldots)[1/g]$ if you conclude from nakayama that $I_g = (f\ldots)_g$
Comment #3491 by Johan on
OK, I added the conclusion from Nakyama's lemma. But I kept the other statement as well because it is how I think about it. See change here. Thanks very much.
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Model Predictive Control Part II: Learning from data
In the previous blog post, we discussed the importance of the terminal constraint and the terminal cost for MPC design. In this post, we will see that these terminal components can be constructed
from historical data.
The Control Problem
We will consider a regulation problem, where our goal is to control a drone that can move only along the vertical direction. The objective of the controller is to hover in place at a given height. We
assume that the drone dynamics are described by a linear system and that the cost is quadratic, as shown in the following figure.
We will use the above example as the system’s state is two-dimensional, and therefore a trajectory of the system can be plotted on a two-dimensional plane. In the following figure, on the two-axis we
have the states of the system, i.e., the position and the velocity of the drone.
In what follows, we are going to iteratively perform the task of steering the drone from a starting state to the goal state, and we will leverage historical data to improve the performance of the
Learning Model Predictive Control
As discussed in the previous blog post, the terminal constraint and terminal cost, often referred to as terminal components, should approximate the tail of the cost and constraint beyond the
prediction horizon. The design of these terminal components is crucial to guarantee safety and optimality. In particular, when solving the MPC problem we should use i) a terminal constraint that is
given by a safe set of states from which the task can be completed, and ii) a terminal cost function that is a value function representing the cost of completing the task from a safe state.
Next, we assume that a first feasible trajectory that is able to complete the task is available and we discuss how to construct safe set and value function approximations.
Building Safe Sets from Data
We notice that as the system is deterministic, a state visited during a successful iteration is safe. Indeed, if the system is in a state that we have visited during a successful iteration, we can
simply follow the successful trajectory to complete the control task (This fact is trivial for deterministic systems, and for uncertain systems we have to be a bit more careful, please refer to [3]-
[4] for further details on uncertain systems). Therefore, we can simply define a safe set of states as the union of the states visited during a successful iteration of the control task.
The safe set \(\mathcal{SS}^1\) can be used as a terminal constraint for our MPC at each time step. In particular, the MPC will plan an open-loop trajectory that steers the system from the current
state back to one of the states visited at the previous iteration of the control task. Then, we apply to the system the first control action and the controller plans a new different open-loop
trajectory that steers the system back to the safe set. As shown in the following figures, this replanning strategy allows the MPC to steer the drone away from the safe set.
The above figure shows the optimal planned trajectory at time \(t = 0\).
The above figure shows the optimal planned trajectory at time \(t = 1\).
The above figure shows the closed-loop trajectory at the second iteration of the control task. We notice that at the second iteration the controller was able to explore the state space. Therefore,
after completion of the second iteration, we can define a bigger safe set which is given by the union of all data points stored during the successful iterations of the control task. In general, we
can define the safe set \(\mathcal{SS}^j\) as the union of the states visited across the first successful \(j\) iterations of the control tasks.
However, we notice that the above safe set renders the optimization problem challenging to solve. Indeed, at each time \(t\) the MPC has to plan a trajectory that lands exactly in one of the states
that we have visited. Luckily, it turns out that for linear systems also the convex hull of the stored states is a safe set of states from which the task can be completed. Thus, we can define the
convex safe set \(\mathcal{CS}^j\) as the convex hull of all stored states up to the \(j\)th iteration of the control task, as shown in the following figure.
Building Value Function Approximations from Data
Now let’s see how we can leverage historical data to construct an approximation to the value function. In the following figure, we have two closed-loop trajectories that successfully steered the
drone from the starting state to the goal. Let \(x_k^i\) be the state of the system at time \(k\) of iteration \(i\), then we define \(J_k^i\) as the cost of the roll-out from \(x_k^i\). Such
roll-out cost can be simply computed by summing up the cost along the closed-loop realized trajectory. Finally, given a data set of costs and states, we define the value function approximation \(V^j
\) as the interpolation of the cost over the stored data, as shown in the following figure.
Notice that in the MPC optimization problem the value function will be evaluated at the terminal predicted state, which should belong to the safe region. Therefore, we want to construct a terminal
cost function that is defined over the convex safe set \(\mathcal{CS}^j\). For a given state \(x\) in the convex safe set \(\mathcal{CS}^j\), this interpolation can be computed solving a linear
program, as shown in the following figure.
Given the roll-out data, the value function may be approximated also using different strategies. However, there is a main advantage in using a linear program to interpolate the cost associated with
the stored states: it can be shown that for linear systems this approximated value function is an upper-bound on the future cumulated cost, for more details please refer to [1].
Computing the Control Action
We have discussed how to leverage historical data to construct the terminal components. Now let’s see how these quantities can be used to compute control actions. Finding an explicit expression for
the safe set and value function approximation may be challenging and computationally expensive. However, it turns out that an explicit expression is not needed and we can defined an optimization
problem that simultaneously computes the terminal components and the optimal control action. In particular, in the optimization problem we can use a sequence of multipliers \(\lambda_k^i\) that are
associated with each data tuple \((x_k^i, J_k^i)\) and are used to represent the safe set \(\mathcal{SS}^{j-1}\) and the value function approximation \(V^{j-1}\), as shown in the following figure.
Finally, it is important to underline that for linear systems the above strategy is guaranteed to converge to global optimality when the system dynamics are linear, the constraints polytopic, the
cost convex, and a mild technical condition is satisfied [2].
Example and Code
In this GitHub repo, we implemented the learning MPC (LMPC) strategy from [1] and [2] to solve the following constrained LQR problem:
The LMPC improves the closed-loop performance until the closed-loop trajectory converges to a steady-state behavior. In this example, the controller iteratively improves the performance until the
closed-loop trajectory converges to the unique global optimal solution to the above infinite time constrained LQR problem. For more details please refer to [1] and [2].
References and code
[1] “Learning Model Predictive Control for Iterative Tasks: A Computationally Efficient Approach for Linear System”, U. Rosolia and F. Borrelli, IFAC World Congress, 2017
[2] “On the Optimality and Convergence Properties of the Learning Model Predictive Controller”, U. Rosolia, Y. Lian, E. T. Maddalena, G. Ferrari-Trecate, and Colin N. Jones. To appear on IEEE
Transaction on Automatic Control.
[3] “Robust learning model predictive control for linear systems performing iterative tasks”, U. Rosolia, X. Zhang and F. Borrelli, IEEE Transaction on Automatic Control (2021)
[4] “Sample-Based Learning Model Predictive Control for Linear Uncertain Systems”, U. Rosolia and F. Borrelli, Conference of Decision and Control (CDC), 2019.
LMPC Code available on GitHub | {"url":"https://urosolia.github.io/jekyll/update/2021/08/06/MPC-Part-II.html","timestamp":"2024-11-12T00:31:41Z","content_type":"text/html","content_length":"18748","record_id":"<urn:uuid:e8e75e67-ea97-4d52-8c5c-ae0af90d3842>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00014.warc.gz"} |
CS231n Convolutional Neural Networks for Visual Recognition
parameteric approach, bias trick, hinge loss, cross-entropy loss, L2 regularization, web demo
model of a biological neuron, activation functions, neural net architecture, representational power
gradient checks, sanity checks, babysitting the learning process, momentum (+nesterov), second-order methods, Adagrad/RMSprop, hyperparameter optimization, model ensembles
layers, spatial arrangement, layer patterns, layer sizing patterns, AlexNet/ZFNet/VGGNet case studies, computational considerations | {"url":"https://cs231n.github.io/","timestamp":"2024-11-03T19:39:43Z","content_type":"text/html","content_length":"15227","record_id":"<urn:uuid:29bb853a-9e09-437b-b939-a40194136406>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00650.warc.gz"} |
Calculating the Fibonacci Extensions Using Lua?
To calculate Fibonacci extensions using Lua, you can create a function that takes in the high, low, and retracement level as parameters. The Fibonacci extensions are calculated by adding percentages
of the retracement level to the high or low point of the move.
To implement this in Lua, you can create a function that calculates the Fibonacci extensions using the formula:
extension = high + (retracement level * (high - low))
You can call this function to calculate the Fibonacci extensions for different retracement levels and plot them on a chart to identify potential support and resistance levels. This can help traders
make informed decisions about their trades based on Fibonacci levels.
How to interpret Fibonacci Extension levels in Lua?
Fibonacci Extension levels can be interpreted in Lua by using the following steps:
1. Calculate the Fibonacci Extension levels by first identifying a significant price move (swing high to swing low or vice versa) and applying Fibonacci ratios (such as 0.618, 1.000, 1.272, 1.618,
etc.) to project potential future price levels.
2. Use Lua programming language to create a script that calculates and plots these Fibonacci Extension levels on a price chart.
3. Interpret the Fibonacci Extension levels by analyzing how the price reacts to these levels. For example, if the price bounces off a Fibonacci Extension level, it may act as a support or
resistance level. If the price breaks through a Fibonacci Extension level, it may indicate a potential continuation of the trend.
4. Look for confluence between Fibonacci Extension levels and other technical indicators or chart patterns to increase the reliability of the analysis.
Overall, interpreting Fibonacci Extension levels in Lua involves understanding how these levels can act as potential areas of support or resistance and using them in conjunction with other technical
analysis tools to make informed trading decisions.
How to calculate Fibonacci Extensions using Lua?
To calculate Fibonacci Extensions using Lua, you can follow these steps:
1. First, you need to define the Fibonacci sequence in Lua. You can either write a function to generate the Fibonacci numbers or manually create a list of Fibonacci numbers.
Here is an example of a function to generate the Fibonacci sequence up to a certain number:
1 function fibonacci(n)
2 local a, b = 0, 1
3 local fib = {0, 1}
4 for i = 2, n do
5 fib[i] = a + b
6 a, b = b, a + b
7 end
8 return fib
9 end
1. Next, you need to calculate the Fibonacci Extensions. The Fibonacci Extensions are derived by multiplying the Fibonacci sequence by key Fibonacci ratios such as 0.382, 0.618, 1, 1.618, etc.
Here is an example of calculating Fibonacci Extensions for a given Fibonacci number:
1 function fibonacciExtensions(n)
2 local ratios = {0.382, 0.618, 1, 1.618, 2.618} -- key Fibonacci ratios
3 local fibNumbers = fibonacci(n) -- generate fibonacci numbers
4 local fibExtensions = {}
5 for i, ratio in ipairs(ratios) do
6 fibExtensions[ratio] = fibNumbers[n] * ratio
7 end
8 return fibExtensions
9 end
1. You can then call the fibonacciExtensions function with a specific Fibonacci number to calculate the Fibonacci Extensions:
1 local fibNum = 10
2 local extensions = fibonacciExtensions(fibNum)
3 for ratio, value in pairs(extensions) do
4 print(string.format("Fibonacci Extension for ratio %.3f: %.3f", ratio, value))
5 end
This is a basic implementation of calculating Fibonacci Extensions in Lua. You can customize and expand the code as needed for your specific requirements.
How to use Fibonacci Extensions as profit targets in Lua?
To use Fibonacci Extensions as profit targets in Lua, you can follow these steps:
1. Calculate the Fibonacci retracement levels for a price movement by identifying the swing high and swing low points.
2. Use the Fibonacci extension levels (such as 161.8%, 261.8%, and 423.6%) to determine potential profit targets.
3. Once the retracement levels have been identified and the extension levels calculated, you can use them as profit targets for your trades.
4. Implement a script in Lua that automatically calculates these Fibonacci levels and displays them on your trading platform for easy reference.
Here is a sample code snippet in Lua that demonstrates how you can implement Fibonacci Extensions as profit targets:
1 -- Function to calculate Fibonacci extension levels
2 function calculateFibonacciExtensions(swingHigh, swingLow)
3 local fib1618 = swingLow + (0.618 * (swingHigh - swingLow))
4 local fib2618 = swingLow + (1.618 * (swingHigh - swingLow))
5 local fib4236 = swingLow + (2.618 * (swingHigh - swingLow))
7 return fib1618, fib2618, fib4236
8 end
10 -- Inputs
11 local swingHigh = 100
12 local swingLow = 50
14 -- Calculate Fibonacci extension levels
15 local fib1618, fib2618, fib4236 = calculateFibonacciExtensions(swingHigh, swingLow)
17 -- Print the Fibonacci extension levels
18 print("Fibonacci 161.8% extension level: " .. fib1618)
19 print("Fibonacci 261.8% extension level: " .. fib2618)
20 print("Fibonacci 423.6% extension level: " .. fib4236)
You can modify and integrate this code into your trading strategy to use Fibonacci Extensions as profit targets in Lua. Just make sure to adjust the input parameters and customize the function
according to your specific requirements.
What is the Fibonacci Extensions formula in Lua?
Here is a simple implementation of the Fibonacci Extensions formula in Lua:
1 function fibonacciExtensions(n)
2 if n == 0 then
3 return 0
4 elseif n == 1 then
5 return 1
6 else
7 return fibonacciExtensions(n - 1) + fibonacciExtensions(n - 2)
8 end
9 end
11 -- Calculate Fibonacci Extensions for n = 10
12 local n = 10
13 print("Fibonacci Extensions for n = " .. n)
14 for i = 0, n do
15 print(fibonacciExtensions(i))
16 end
This script defines a recursive function fibonacciExtensions that calculates the Fibonacci Extensions for a given number n. It then calculates and prints the Fibonacci Extensions for n = 10. You can
adjust the value of n to calculate the Fibonacci Extensions for a different number. | {"url":"https://sampleproposal.org/blog/calculating-the-fibonacci-extensions-using-lua","timestamp":"2024-11-14T14:02:21Z","content_type":"text/html","content_length":"294429","record_id":"<urn:uuid:562a0f02-4754-4e0a-816e-eb3c2c6f79c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00847.warc.gz"} |
Data Structures & Algorithms IV: Pattern Matching, Dijkstra’s, MST, and Dynamic Programming Algorithms
This Data Structures & Algorithms course completes the 4-course sequence of the program with graph algorithms, dynamic programming and pattern matching solutions. A short Java review is presented on
topics relevant to new data structures covered in this course. The course does require prior knowledge of Java, object-oriented programming and linear and non-linear data structures. Time complexity
is threaded throughout the course within all the data structures and algorithms.
You will delve into the Graph ADT and all of its auxiliary data structures that are used to represent graphs. Understanding these representations is key to developing algorithms that traverse the
entire graph. Two traversal algorithms are studied: Depth-First Search and Breadth-First Search. Once a graph is traversed then it follows that you want to find the shortest path from a single vertex
to all other vertices. Dijkstra’s algorithm allows you to have a deeper understanding of the Graph ADT. You will investigate the Minimum Spanning Tree (MST) problem. Two important, greedy algorithms
create an MST: Prim’s and Kruskal’s.
Prim’s focuses on connected graphs and uses the concept of growing a cloud of vertices. Kruskal’s approaches the MST differently and creates clusters of vertices that then form a forest.
The other half of this course examines text processing algorithms. Pattern Matching algorithms are crucial in everyday technology. You begin with the simplest of the algorithms, Brute Force, which is
the easiest to implement and understand. Boyer-Moore and Knuth-Morris-Pratt (KMP) improve efficiency by using preprocessing techniques to find the pattern. However, KMP does an exceptional job of not
repeating comparisons once the pattern is shifted. The last pattern matching algorithm is Rabin-Karp which is an “out of the box” approach to the problem. Rabin-Karp uses hash codes and a “rolling
hash” to find the pattern in the text. A different text processing problem is locating DNA subsequences which leads us directly to Dynamic Programming techniques. You will break down large problems
into simple subproblems that may overlap, but can be solved. Longest Common Subsequence is such an algorithm that locates the subsequence through dynamic programming techniques. | {"url":"http://www.allcoursesonline.org/project/data-structures-algorithms-iv-pattern-matching-dijkstras-mst-and-dynamic-programming-algorithms/","timestamp":"2024-11-09T01:29:29Z","content_type":"text/html","content_length":"102520","record_id":"<urn:uuid:91a27613-15da-4076-94c0-7c0fb12e644e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00451.warc.gz"} |
Pseudo-Anosov representatives of stable Hamiltonian structures
A pseudo-Anosov homeomorphism of a surface is a canonical representative of its mapping class. In this paper, we explain that a transitive pseudo-Anosov flow is similarly a canonical representative
of its stable Hamiltonian class. It follows that there are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere. This result has a purely
topological consequence: any 3-manifold can be obtained in at most finitely many ways as $p/q$ surgery on a fibered hyperbolic knot in $S^3$ for a slope $p/q$ satisfying $q\geq 6$, $p\neq 0, \pm 1, \
pm 2 \mod q$. The proof of the main theorem generalizes an argument of Barthelmé--Bowden--Mann. | {"url":"https://papers.cool/arxiv/2410.02186","timestamp":"2024-11-11T14:35:36Z","content_type":"text/html","content_length":"11387","record_id":"<urn:uuid:5d748357-3c2a-45cd-af55-1250b3e184e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00634.warc.gz"} |
How do you differentiate 5xy + y^3 = 2x + 3y? | HIX Tutor
How do you differentiate #5xy + y^3 = 2x + 3y#?
Answer 1
I found: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 - 5 y}{5 x + 3 {y}^{2} - 3}$
You can use implicit differentiation rememberig that #y# represents a function of #x# and needs to be differentiated accordingly; for example: if you have #y^2# you differentiate it to get: #2y*(dy)/
(dx)# where you use #(dy)/(dx)# to take into account the dependence with #x#. In your case you get: #5y+5x(dy)/(dx)+3y^2(dy)/(dx)=2+3(dy)/(dx)# collect #(dy)/(dx)#: #(dy)/(dx)(5x+3y^2-3)=2-5y# and: #
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Answer 2
To differentiate (5xy + y^3 = 2x + 3y), you can use implicit differentiation. Differentiate each term with respect to (x) using the chain rule for terms containing (y).
Differentiate (5xy) with respect to (x):
[ \frac{d}{dx}(5xy) = 5y + 5x \frac{dy}{dx} ]
Differentiate (y^3) with respect to (x):
[ \frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx} ]
Differentiate (2x) and (3y) with respect to (x):
[ \frac{d}{dx}(2x) = 2 ] [ \frac{d}{dx}(3y) = 3 \frac{dy}{dx} ]
Then, equate the sum of these derivatives to zero since the equation is implicitly defined:
[ 5y + 5x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 2 + 3 \frac{dy}{dx} ]
Now, isolate (\frac{dy}{dx}):
[ 5x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} - 3 \frac{dy}{dx} = 2 - 5y ]
[ (5x + 3y^2 - 3) \frac{dy}{dx} = 2 - 5y ]
[ \frac{dy}{dx} = \frac{2 - 5y}{5x + 3y^2 - 3} ]
So, the derivative of the given equation is ( \frac{2 - 5y}{5x + 3y^2 - 3} ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-differentiate-5xy-y-3-2x-3y-8f9af9e87d","timestamp":"2024-11-02T02:11:13Z","content_type":"text/html","content_length":"570956","record_id":"<urn:uuid:58fda1a5-5374-48f7-ab3f-0b3d843fa98a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00627.warc.gz"} |
C.11.30 Volume Render Geometry Module
The Render Field of View (0070,1606) defines the region of the volume data that is displayed.
The viewpoint is positioned and oriented within the Volumetric Presentation State Reference Coordinate System (VPS-RCS) by Viewpoint Position (0070,1603), Viewpoint LookAt Point (0070,1604) and
Viewpoint Up Direction (0070,1605). This position and orientation establish a Viewpoint Coordinate System (VCS), which is a right-hand coordinate system in which the viewpoint is positioned at
(0,0,0) and is looking at a point at (0,0,-z) and the up direction is along the +y axis.
Render Field of View (0070,1606) is specified by the following coordinate values in the Viewpoint Coordinate System:
• Distance[near], Distance[far] specify the distances from Viewpoint Position (0070,1603) to the near and far depth clipping planes. Both distances shall be positive, and Distance[near] shall be
less than Distance[far].
• X[left], X[right] specify the coordinates of the left and right vertical clipping planes at Distance[far]. X[left] shall be less than X[right].
Positive values of Distance[near] and Distance[far] place the near and far rectangles of the field of view on the negative Z axis at Z values of -Distance[near] and -Distance[far], respectively.
In the case of a Render Projection (0070,1602) value of ORTHOGRAPHIC, Render Field of View (0070,1606) defines a rectangular cuboid with dimensions (X[right] minus X[left]) by (Y[top] minus Y
[bottom]) by (Distance[far] minus Distance[near]), in mm, as shown in Figure C.11.30-1:
In the case of a Render Projection (0070,1602) value of PERSPECTIVE, Render Field of View (0070,1606) defines a frustum in which the far rectangle is larger than the near rectangle. The extent of the
far rectangle is established by the points (X[left], Y[top]) and (X[right], Y[bottom]) at Distance[far]. The extent of the near rectangle is established by the four points where rays originating at
the viewpoint position to the corners of the far rectangle intersect the plane that is located at Distance[near] from the viewpoint, as shown in Figure C.11.30-2. | {"url":"https://dicom.nema.org/medical/dicom/current/output/chtml/part03/sect_c.11.30.html","timestamp":"2024-11-14T18:26:07Z","content_type":"application/xhtml+xml","content_length":"25089","record_id":"<urn:uuid:31211073-333b-425a-aff4-ada9b7ffcd01>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00600.warc.gz"} |
Reviews – Page 2 – The Aperiodical
This guest review is written by Sophie Maclean. Math Without Numbers will be released on 7th January.
I think it’s safe to say that all fans of The Aperiodical like maths. I would also be confident in saying that there’s a shared feeling of “the more the merrier”, and we want as many people as
possible to share our love of maths. In this respect, Milo Beckman would fit right in. In fact, I’d go as far as to say that his book Math Without Numbers is precisely the kind of book that could get
more people to realise how fun maths can be.
James Grime answers your Shakuntala Devi questions
This is a follow-up to James’s FAQ for the 2014 film The Imitation Game.
Shakuntala Devi is a 2020 Indian Hindi-language film about Shakuntala Devi, a performer of impressive mental calculations, available now on Amazon Prime.
Alex Bellos – The Language Lover’s Puzzle Book
If you’re a fan of maths (which we assume you are, if you’re reading a maths blog), you might be familiar with Alex Bellos from his excellent popular maths books, including Alex’s Adventures in
Numberland and the follow-up Alex Through The Looking Glass; you might also enjoy his more recent forays into puzzle books, including Can You Solve My Problems, and Japanese logic puzzle collection
Puzzle Ninja, as well as his regular Monday puzzle column in The Guardian.
For his latest book, The Language Lover’s Puzzle Book, Alex has focused on language puzzles, largely drawn from the linguistic equivalent of Maths Olympiads (which he’s gotten really into lately).
It’s a hefty volume split into cleverly collected sections on different aspects of language – including how languages are constructed, how words are pronounced, and as you might expect, the origins
of how language is used to communicate numbers.
Review: Why Study Mathematics, by Vicky Neale
In fact, St Andrews offered a French for Scientists course, so I ended up doing Maths with French. A win all round.
I can pinpoint the exact moment it became clear I would study maths at university. Parents’ evening, year 12, I mentioned to my French teacher that I was thinking about a French degree. He looked at
me as if I was stupid and said something like “you’re good at French, but you’re GOOD at maths. Besides, a French degree isn’t much use.” Alright, fine. Maths it is. He was spot-on. I never looked
Review: Immersive Linear Algebra
We invited guest author, Big Math-Off contestant and recent maths graduate Brad Ashley to review Immersive Math’s linear algebra textbook – a new take on the format.
Immersive Linear Algebra is an online interactive linear algebra textbook, created by mathematicians and computer scientists Jacob Ström, Kalle Åström, and Tomas Akenine-Möller. With their impressive
collective knowledge of the field, and its applications within computer graphics, they seek to improve upon the idea of a textbook with the use of interactive diagrams.
Colourful Mathematics – A Review of Alex Berke’s Book ‘Beautiful Symmetries’
Group theory is a strange and wonderful area of study in mathematics, with plenty of key ideas and core concepts for one to wrap their algebra-hungry head around. But how do you introduce these
algebraic constructs to beginners in a fun and engaging manner, whilst simultaneously providing a thoughtful read for the experts? This is exactly what mathematician and computer scientist Alex Berke
accomplishes with her mathematical colouring book Beautiful Symmetry and its innovative group colouring concept.
IWD 2020: Books about Maths by Women
For International Women’s Day, mathematician Lucy Rycroft-Smith has read a selection of maths books by women authors, and recommended some favourites.
There’s a strange irony about being a woman in mathematics. You spend a huge amount of time and energy answering questions about being a woman in mathematics instead of, you know, using that time and
energy to do or write about actual maths. We women are somehow both the problem and the solution.
But behold: 2020 is here, and better and braver women than I have solved this conundrum. Here are a whole host of excellent books about maths by women that you should definitely read, collected for
you by another woman in maths. | {"url":"https://aperiodical.com/category/main/reviews/page/2/","timestamp":"2024-11-05T21:33:41Z","content_type":"text/html","content_length":"42434","record_id":"<urn:uuid:6a5f06b8-161a-478a-8206-605c07f08e5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00611.warc.gz"} |
Be the First to Read What Gurus are Saying About How Is Math Used in Business
How Is Math Used in Business Options
There are not any numbers involved. There are two reasons, and they’re not about math. The straightforward interest formula is provided together with examples.
With enough GRE quantitative practice, you’re find the score you desire! A good deal of math in gameplay scripting is fairly easy, but math employed in game engine architecture is much more complex
and far more taxing mentally. The majority of these math topics are wholly used all together in rather advanced games.
With one major caveat When practicing GRE math complications, the secret is to find out why you answered questions incorrectly. Other variables could be involved also. Mathematical equations with
numerous steps have to be solved in a particular order.
But they’re thinking about eliminating the internet server though. Please have a look at the post As the title suggests, the proof is just in one-line. For http://www.mmss.northwestern.edu/ instance,
if you know the significance of mercy, you can determine the significance of merciful. But there’s another group of issues for which it’s simple to check whether a potential remedy to the issue is
correct, but we don’t understand how to efficiently locate a solution. Keep all these hints in mind whilst working with this tutorial.
In finding out the sequence of events that occurred at an incident scene, officers are called upon in order to take measurements and discern angles so as to compile the crucial evidence to
reconstruct the function. They are certain to find excellent assistance very shortly. The admissions questionnaire are available here.
Very little changes even when you are an accounting major! Our service is quite flexible with respect to the pricing policy. Business ownership requires more than skill in making an item or talent at
supplying a service.
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down the town you’re in. You have many fantastic alternatives. There are 3 options given below in order of priority to finish this requirement.
Utilizing business mathematics assists in making these interpretations ad afford the business to a greater level. It attempts to formalize valid reasoning. Differential equations describe how a
specific quantity changes with time, given some initial starting conditions, and they’re beneficial in describing a variety of physical systems.
You are going to have the chance to teach mathematics and the wider primary curriculum in two distinct schools. Market research is just one of the greatest examples to have a look at how practical
math influences marketing decisions. Actuarial science addresses the mathematics of insurance.
In any case, you will need to get hold of the Admissions Office and pay a one-time fee to acquire Special Student Status. They’re selected separately from your Dissertation Advisor, but may be the
very same individual. The University of Southern California’s Mathematics Department focuses not simply on supplying a wide array of courses, but also on providing a wide array of additional
activities and programs.
In the majority of instances, the internet master’s in mathematics program can be finished within two decades. So, you are able to always negotiate the price with a specific tutor and find the one,
whose price is most appropriate for you. Benefit from the practice math questions within this book, and visit the public library to find out what type of high school math textbooks it must lend. Our
on-line math tutors work with as much as 3 students at one time in an enjoyable online.
Every company is dependent upon math. At UWM, a liberal arts school, the company degree wasn’t specific enough, one had to select from the six concentrations provided by our organization school. Be
aware that a few roles might only call for a bachelor’s degree, while others might require one or more professional certifications. This is fantastic training for young entrepreneurs and for those
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Consider the questions which you ask in your math classroom. These are really advantageous if you would like your tutor to examine your homework assignment or check a math test in actual time. Bear
in mind you will need to take quite a few certification steps (like exams) before you may practice as an actuary. Utilize practice questions only to check your own understanding of the subject.
You are going to have the chance to teach mathematics and the wider primary curriculum in two distinct schools. The topics can typically be found in a lot of business math courses. Through the usage
of mathematical approaches and models, applied and computational mathematics could address a wide variety of real-world problems.
In any case, you will need to get hold of the Admissions Office and pay a one-time fee to acquire Special Student Status. The World Campus is part of the most important campus. The University of
Southern California’s Mathematics Department focuses not simply on supplying a wide array of courses, but also on providing a wide array of additional activities and programs.
The aim of the program is to supply the broad quantitative background in mathematics, probability, economics, company, and relevant areas that is essential for success in the actuarial profession and
to supply the academic background required to pass the initial four actuarial exams. You just need to put up with them or locate a different program. They are offered by community colleges,
universities and a variety of learning institutions.
Furthermore, tuition rates may change by program, so prospective students should get in touch with a representative from their institution of choice for more information. Graduates will understand
how to properly assess risk in a full selection of situations. 1 class ought to be enough. In some instances, the courses exactly mirror the syllabi for certain exams.
Math requirements for business majors will be different depending on the institution. However, you’ll need to do a little bit of career preparation during college. Bear in mind you will need to take
quite a few certification steps (like exams) before you may practice as an actuary. In Math-Only-Math you are going to find abundant variety of all kinds of math questions for all of the grades with
the complete step-by-step solutions.
The Demise of How Is Math Used in Business
There’ll soon be nothing that can’t be produced with 3D printing. Nevertheless, in many scenarios the most recently available data won’t be for the current academic calendar year. Today you can
decide the best route based on terrain, speed limit, etc.
The major point is the thing that the writer would like you to take away from her or his words. You’re able to use it in real life. Our purpose is to supply you with the ideal service. It’s something
that almost all of us spend our lives avoiding. It is possible to calculate the best routes for your running or cycling schedule by making a mathematical expression that takes into consideration the
distance and your typical speed for assorted components of the route.
Taxes There are several taxes in company and unique formulas used to figure taxes out. You might need to know whether you’re able to afford to enlarge your operations to improve sales. Retail buyers
utilize math to learn how much to buy of one style of clothing or shoes or jewelry. A banker recommending savings or investment products needs the ability to rate interest yields on various products,
for example. To analyze the general financial health of your company, you will want to project revenue and expenses for the future.
But considering the glacial paces at which lots of mathematical refereeing occurs, I think that it is quite okay. Requires the capacity to understand economic issues as they’re influenced by global
alterations. You have the marking scheme, which has the answers to every question and you’ve got the examiner file, which basically has the examiners‘ comments on each and every question in the
previous papers. Here’s the list of mathematical truth about the number 2017 that it is possible to brag about to your friends or family for a math geek. But don’t hesitate to change my mind about
This problem can’t be solved without knowing the rate of interest for each undertaking. Mathematical discoveries continue to get made today. Whatever it is, I have zero clue.
How Is Math Used in Business Secrets That No One Else Knows About
The decision rule that’s followed is an extreme test statistic ends in rejection of the null hypothesis. At the conclusion of section 1 you ought to have a better comprehension of functions and
equations. The straightforward interest formula is provided together with examples.
Students who aren’t math majors can get a minor in Actuarial Mathematics. Math is vital to the creation of games. In different fields of finance the math can receive more advanced.
Optimization mathematical models are generally employed for such issues. Other variables could be involved also. With regard to mixtures, simultaneous equations may be used for achieving a particular
consistency in a resultant products, which depends on the consistency of the compounds mixed with each other to produce it.
The Argument About How Is Math Used in Business
Developing a model yourself is relatively easy as you control everything the actual challenge is looking at somebody else’s model and figuring out how it works in the very first place and the way to
modify it. This trait is extremely valuable because finding answers to real world problems in the industry world doesn’t always must go by the book. It will allow you to observe the huge picture
because you’re capable of making sense out of the seemingly unrelated parts of data provided to you. But there’s another group of issues for which it’s simple to check whether a potential remedy to
the issue is correct, but we don’t understand how to efficiently locate a solution. Don’t neglect to include things like the project manager!
In finding out the sequence of events that occurred at an incident scene, officers are called upon in order to take measurements and discern angles so as to compile the crucial evidence to
reconstruct the function. FEMA Disaster Master FEMA presents an abundance of kid-friendly materials on each and every form of natural disaster. The admissions questionnaire are available here.
Taxes There are several taxes in company and unique formulas used to figure taxes out. Accurately determining the cost connected with each item will produce the base for the business strong. An asset
ought to be priced in order to stop such arbitrages. A banker recommending savings or investment products needs the ability to rate interest yields on various products, for example. Estimating how
much an employee affects revenue will indicate if it is possible to afford to enhance your staff and in the event the profits realized will be well worth the expense.
The number of feasible configurations, and the most amount of steps are numbersthey’re interesting facets of the issue. With the info accessible, you can calculate which loan is the ideal option for
you. If it is carried out correctly, each sample should accurately reflect the characteristics of the population. You have many fantastic alternatives. Solve the subsequent rate issues.
You have to select one and only a single project to take on on for your business. However, there’ll be occasions when retailers want to work through numerical problems manually. Basically, this is
similar to choosing what you would like to apply mathematical techniques to (e.g. business, health, engineering). For example, assume that a business is selling a good deal of an item with a very low
sale price.
The project proposal has to be accepted by the graduate committee before students may register for the training course. While all such studies have gathered empirical data on the mathematics utilized
in numerous workplaces, they’ve also investigated such things as the disposition of modeling and abstraction, the part of representations, and assorted associated learning difficulties. Actuarial
science addresses the mathematics of insurance.
In addition, the School also provides lots of specialised Professional Development programmes which will permit you to further develop your career. Credit for such participation necessitates
validation via an essay that describes the experience and the way this enhances the student’s work for a teacher. The University of Southern California’s Mathematics Department focuses not simply on
supplying a wide array of courses, but also on providing a wide array of additional activities and programs.
With all these regions to study, practice is the secret to mastering the GRE math section. This system permits students to finish their undergraduate and graduate degrees in a shorter time period. A
strong program should supply you with coursework that provides you a solid general foundation but is also tailored to your precise career targets.
Finding the perfect program goes far beyond searching for the cheapest tuition, however. At UWM, a liberal arts school, the company degree wasn’t specific enough, one had to select from the six
concentrations provided by our organization school. Hands on experience is also given to the students. To begin with, you’re simply learning a good deal of ideas, rules, and procedures that are
particular to accounting and which are new to you. Business majors wishing to concentrate on finance careers will require a strong calculus background. Understanding economics or running a company
necessitates math abilities.
Consider the questions which you ask in your math classroom. However, you’ll need to do a little bit of career preparation during college. The GRE math practice questions within this post will allow
you to identify which areas you should work on and how well you’re ready for the exam. Don’t expect to find the exact questions on the true ASVAB.
The history of mathematics can be regarded as an ever-increasing set of abstractions. Fortunately, not one of the arithmetic is very hard, and PMI supplies you with a digital calculator.
Mathematics may be used to represent real-world circumstances. Math is vital to the creation of games. It’s difficult to get excited about learning math.
With one major caveat When practicing GRE math complications, the secret is to find out why you answered questions incorrectly. Regression analysis is just one of the most frequently used techniques
for predictive models. With regard to mixtures, simultaneous equations may be used for achieving a particular consistency in a resultant products, which depends on the consistency of the compounds
mixed with each other to produce it.
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LibGuides: Mathematics: The Pythagorean Theorem
The Pythagorean Theorem only works with right triangles, but you can use it in many different ways.
For example, builders use the Pythagorean Theorem when laying a house foundation to ensure it's square.
The Pythagorean Theorem is: a^2 + b^2 = c^2
It means (side)^2 + (side)^2 = (hypotenuse)^2
Hypotenuse is the the longest side
The letters a, b and c are not important; instead you could use the letters x, y and z!
Study and practice | {"url":"https://libguides.ucol.ac.nz/Mathematics/thepythagoreantheorem","timestamp":"2024-11-07T07:29:50Z","content_type":"text/html","content_length":"43054","record_id":"<urn:uuid:ef4a6e0b-50f5-482c-b0c5-2f6446f8add6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00060.warc.gz"} |
Seven Ways to Make up Data: Common Methods to Imputing Missing Data - The Analysis Factor
There are many ways to approach missing data. The most common, I believe, is to ignore it. But making no choice means that your statistical software is choosing for you.
Most of the time, your software is choosing listwise deletion. Listwise deletion may or may not be a bad choice, depending on why and how much data are missing.
Another common approach among those who are paying attention is imputation. Imputation simply means replacing the missing values with an estimate, then analyzing the full data set as if the imputed
values were actual observed values.
How do you choose that estimate? The following are common methods:
Simply calculate the mean of the observed values for that variable for all individuals who are non-missing.
It has the advantage of keeping the same mean and the same sample size, but many, many disadvantages. Pretty much every method listed below is better than mean imputation.
Impute the value from a new individual who was not selected to be in the sample.
In other words, go find a new subject and use their value instead.
Hot deck imputation
A randomly chosen value from an individual in the sample who has similar values on other variables.
In other words, find all the sample subjects who are similar on other variables, then randomly choose one of their values on the missing variable.
One advantage is you are constrained to only possible values. In other words, if Age in your study is restricted to being between 5 and 10, you will always get a value between 5 and 10 this way.
Another is the random component, which adds in some variability. This is important for accurate standard errors.
Cold deck imputation
A systematically chosen value from an individual who has similar values on other variables.
This is similar to Hot Deck in most ways, but removes the random variation. So for example, you may always choose the third individual in the same experimental condition and block.
Regression imputation
The predicted value obtained by regressing the missing variable on other variables.
So instead of just taking the mean, you’re taking the predicted value, based on other variables. This preserves relationships among variables involved in the imputation model, but not variability
around predicted values.
Stochastic regression imputation
The predicted value from a regression plus a random residual value.
This has all the advantages of regression imputation but adds in the advantages of the random component.
Most multiple imputation is based off of some form of stochastic regression imputation.
Interpolation and extrapolation
An estimated value from other observations from the same individual. It usually only works in longitudinal data.
Use caution, though. Interpolation, for example, might make more sense for a variable like height in children–one that can’t go back down over time. Extrapolation means you’re estimating beyond the
actual range of the data and that requires making more assumptions that you should.
Single or Multiple Imputation?
There are two types of imputation–single or multiple. Usually when people talk about imputation, they mean single.
Single refers to the fact that you come up with a single estimate of the missing value, using one of the seven methods listed above.
It’s popular because it is conceptually simple and because the resulting sample has the same number of observations as the full data set.
Single imputation looks very tempting when listwise deletion eliminates a large portion of the data set.
But it has limitations.
Some imputation methods result in biased parameter estimates, such as means, correlations, and regression coefficients, unless the data are Missing Completely at Random (MCAR). The bias is often
worse than with listwise deletion, the default in most software.
The extent of the bias depends on many factors, including the imputation method, the missing data mechanism, the proportion of the data that is missing, and the information available in the data set.
Moreover, all single imputation methods underestimate standard errors.
Since the imputed observations are themselves estimates, their values have corresponding random error. But when you put in that estimate as a data point, your software doesn’t know that. So it
overlooks the extra source of error, resulting in too-small standard errors and too-small p-values.
And although imputation is conceptually simple, it is difficult to do well in practice. So it’s not ideal but might suffice in certain situations.
So multiple imputation comes up with multiple estimates. Two of the methods listed above work as the imputation method in multiple imputation–hot deck and stochastic regression.
Because these two methods have a random component, the multiple estimates are slightly different. This re-introduces some variation that your software can incorporate in order to give your model
accurate estimates of standard error.
Multiple imputation was a huge breakthrough in statistics about 20 years ago. It solves a lot of problems with missing data (though, unfortunately not all) and if done well, leads to unbiased
parameter estimates and accurate standard errors.
1. Kamesh says
If the dataset is sufficiently large, can we use machine learning algorithm based imputation. There are standard packages available; but perhaps these algorithms may also be taking
regression-based techniques albeit in multiple ways.
2. Carolina says
Where does full information maximum likelihood fit into this discussion and how does it compare to the above missing data methods?
□ Karen Grace-Martin says
Full information maximum likelihood is an alternate to all of these imputation methods. It’s generally considered as good as multiple imputation, but they both have strengths and weaknesses
in certain situations, so it depends on the specific context.
See: Two Recommended Solutions for Missing Data: Multiple Imputation and Maximum Likelihood
3. ALIZA says
kindly tell me the procedure of interpolation and extrapolation.
thank you
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What are the properties of Rational Numbers? + Example
What are the properties of Rational Numbers?
1 Answer
They can be written as a result of a division between two whole numbers, however large.
Example: 1/7 is a rational number. It gives the ratio between 1 and 7. It could be price for one kiwi-fruit if you buy 7 for $1.
In decimal notation, rational numbers are often recognised because their decimals repeat. 1/3 comes back as 0.333333.... and 1/7 as 0.142857... ever repeating. Even 553/311 is a rational number (the
repeating cylce is a bit longer)
There are also IRrational numbers that cannot be written as a division. Their decimals follow no regular pattern. Pi is the best-known example, but even the square root of 2 is irrational.
Impact of this question
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Joel Gibson
This is the personal website of Joel Gibson. I am a postdoctoral researcher at the University of Sydney: for more information view my research homepage.
Mathematics software
I am a big proponent of using examples and visual aids to help communicate, teach, and build intuition about abstract domains.
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This Year 6 Multiply by 10, 100 and 1,000 Discussion Problem is ideal for those children who require an extra challenge to deepen their understanding. There are two open-ended questions included that
are designed to be completed in pairs or small groups. Both questions have engaging contexts. The first question is about robots and the second is a genie's riddle. This worksheet also includes
multiplying by multiples of ten, one hundred and one thousand to really stretch pupils' skills. | {"url":"https://classroomsecrets.co.uk/resource/year-6-multiply-by-10-100-and-1-000-discussion-problem","timestamp":"2024-11-11T21:31:26Z","content_type":"text/html","content_length":"581274","record_id":"<urn:uuid:0b37e0a2-a950-4deb-b936-ba12dd25f544>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00711.warc.gz"} |
Trend and Cycle
Full text: Trend and Cycle
purposes instability can only mean that the system is unable to
cope with them adequately.
A necessary condition for macroeconomic change - trend or cycle -
is that the disturbances and the corresponding reactions in the
small do not offset each other so as to permit stability of the
whole system but rather tend to go to a large extent in the same
direction. They are unlike a self regulating system either because
the individual movements reinforce each other (imitation) or
because they all respond to a common signal,for example, the rate
of interest.
In my opinion trend and cycle are to be treated from the point of
view of the following "research program" (in the sense of
Lakatos): The macroeconomic system is like a machine which works
up and transforms the disturbances which are fed into it from the
outside in the course of time. This approach is dictated by a
dominating practical interest in economic policy: We want to know
how the system reacts to various measures or events and how the
daily or monthly movements of the Konjunktur (state or tendency of
trade) have come about in each concrete case. This preoccupation
with the practical aims of the theory are the reason why I can not
be convinced by Goodwins insistence on a non-linear treatment of
the cycle. The response of the system to disturbances is, I hope,
adequately dealt with by linear approximations. This is true also
for the long term development which is only the result of an
accumulation of short term changes.
This leads to the question of the unity of trend and
cycle.Unfortunately very often independent and separate theories
have been produced for the one and the other even by those authors
who intended to arrive at a unified theory. The failure stems
probably from the mathematical formulation which is so much easier
if you have one equation for each,independent of each other. But
in reality the trend component and the cyclical component are
determined at the same time and are parts of the same process,
separated only artificially by statistical or analytic
prodedures.The unity of trend and cycle has been proclaimed most
forcefully by Goodwin (1982,p.115,p.122-123) who also quotes
Schumpeter in support of this view.Goodwin argues that the
separation of the two theories is based on the superposition
principle which is justified as long as we work with linear „
The question. _is whether, even without departing f rt>m simple linear
equationswe can establish a theory^, in which trend and cycle are
interdependent,each of them influencing the other. My feeling is
that the "pure trade cycle” theory of Kalecki might be enlarged by
a term( or terms) which represents "long term memory". This would
be an integral over a number of years past of certain variables
such as for example utilisation of capacity, multiplied with a
certain reaction coefficient. This term, since it changes
slowly,would provide for a smooth long term development. Since it
might cover something 1 ike the— av-erageLength of a ft*41 cycle it
would involve an influence of the last cycle on the present trend
value: If this cycle consisted of a long and strong boom and a | {"url":"https://viewer.wu.ac.at/viewer/fulltext/AC14446050/6/","timestamp":"2024-11-02T09:35:41Z","content_type":"application/xhtml+xml","content_length":"72063","record_id":"<urn:uuid:13497c0f-68e1-4d81-99a5-88aec6eeb07b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00420.warc.gz"} |
The Order Of Operations Three Steps A Math Worksheet From The | Order of Operation Worksheets
The Order Of Operations Three Steps A Math Worksheet From The
The Order Of Operations Three Steps A Math Worksheet From The
The Order Of Operations Three Steps A Math Worksheet From The – You may have heard of an Order Of Operations Worksheet, however what specifically is it? In addition, worksheets are an excellent means
for pupils to practice brand-new abilities and also review old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a sort of mathematics worksheet that calls for trainees to perform mathematics operations. These worksheets are divided right into 3 main sections: addition,
subtraction, and also multiplication. They likewise include the examination of backers and also parentheses. Pupils who are still finding out how to do these tasks will find this sort of worksheet
The main function of an order of operations worksheet is to help pupils discover the appropriate way to solve math equations. If a pupil does not yet understand the principle of order of operations,
they can examine it by describing an explanation web page. Additionally, an order of operations worksheet can be separated into a number of groups, based on its problem.
One more vital objective of an order of operations worksheet is to teach students just how to perform PEMDAS operations. These worksheets start with straightforward problems connected to the standard
regulations as well as develop to a lot more complex troubles involving all of the policies. These worksheets are a fantastic way to introduce young learners to the exhilaration of solving algebraic
Why is Order of Operations Important?
One of the most crucial points you can find out in math is the order of operations. The order of operations makes certain that the math issues you fix are consistent.
An order of operations worksheet is an excellent way to show pupils the correct means to solve mathematics formulas. Before students begin utilizing this worksheet, they may require to assess
concepts related to the order of operations. To do this, they should review the principle page for order of operations. This principle web page will certainly give students a review of the keynote.
An order of operations worksheet can aid students establish their skills on top of that and also subtraction. Educators can use Prodigy as a very easy method to distinguish method and also provide
appealing web content. Natural born player’s worksheets are an ideal way to aid students learn about the order of operations. Teachers can begin with the standard concepts of multiplication,
addition, as well as division to aid pupils build their understanding of parentheses.
Grade 3 Order Of Operations Worksheet
Https www dadsworksheets Order Of Operations Worksheet sudoku
Order Of Operations PEDMAS With Integers 3 Worksheet
Third Grade Childrens Educational Workbooks Books And Free Worksheets
Grade 3 Order Of Operations Worksheet
Grade 3 Order Of Operations Worksheet give a terrific source for young students. These worksheets can be conveniently personalized for details requirements.
The Grade 3 Order Of Operations Worksheet can be downloaded free of charge and also can be published out. They can after that be reviewed utilizing addition, division, subtraction, as well as
multiplication. Pupils can additionally use these worksheets to review order of operations as well as using backers.
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