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Covert - math word problem (73874)
Covert the fraction 5/8 into a decimal using long division. Round any repeating decimals to the nearest hundredth.
Correct answer:
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You need to know the following knowledge to solve this word math problem:
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Understanding Security Through Probability
This post was also authored by Min-yi Shen and Martin Lee.
Security is all about probability. There is a certain probability that something bad will happen to your networks or your systems over the next 24 hours. Hoping that nothing bad will happen is
unlikely to change that probability. Investing in security solutions will probably reduce the chance of something bad happening, but by how much? And where should resources be most profitably
Cyber security is a complex environment with many unknowns and interdependencies. TRAC data scientists research this environment to try and understand how different variables affect security.
Bayesian graph models are one of our most useful tools for understanding probabilities in security and to explore how the likelihood of outcomes can be changed.
Figure 1 – A Simple Bayesian Network
This example graph shows that C is affected by both A and B, whereas B is only affected by A. This notation of nodes and arrows is particularly useful to describe complex outcomes.
We can apply this theoretical representation to a real world scenario:
Figure 2 – “System Crash” Bayesian Network
A lightning strike during stormy weather hits a power line and causes a power failure. This in turn may cause the operating system to malfunction, causing a system crash. Alternatively a large power
spike may cause a hardware failure that may also result in a system crash. But a system crash can also happen without stormy weather. Malware may interfere with the operating system or directly cause
the system to crash.
The directed graph is useful because we can quickly deduce which variables are independent from other variables. We can see that a computer crash can be the result of a malware infection, an
operating system failure, and/or hardware failure. We can also see that malware infection is independent of the stormy weather. In other words, you can get malware regardless of the weather outside.
We can also make operational decisions from such a graph. If the weather outside is stormy, we are at heightened risk for a system crash. Hence we may wish to change our security activities. Equally,
we can also deduce that if we are aware of a power failure, we are in danger of experiencing a system crash no matter what the weather.
Once we have constructed such a graph we can begin to collect metrics and assign probabilities to the various outcomes on the graph. We can use the power of Bayesian network analysis to calculate the
likelihood of outcomes. For example, let’s say the probability of experiencing a system failure during stormy weather is 2.37%, or 2.27% if the weather is not stormy. From this, and using the
probability of stormy weather, I can calculate the most likely scenario for computer infection: a malware infection and no storm (69.3%) compared with no malware infection and a storm (2.98%).
Real world examples can be daunting for data scientists and applied researchers. We are usually faced with a scenario where there is only partial data, and no obvious network structure. Nevertheless,
we can apply probabilistic reasoning to construct such a network and deduce the outcome probabilities. Applying model parameter estimation using techniques popularized by R.A. Fisher, such as the
maximum likelihood, allows us to estimate the maximum value for probabilities, but only if we are able to observe enough outcomes.
Observing and counting enough outcomes to make these kinds of conclusions has been a major challenge to data scientists. However, the advent of modern big data platforms such as Hadoop makes
searching for, and counting, the events that we are interested in so much easier. Armed with these tools we are able to construct and populate real world models of the cyber security environment and
begin to make conclusions about what variables affect this environment. And most importantly, we can begin to measure and make predictions about how to make the Internet a safer place.
Min-yi Shen studied theoretical chemistry at the University of Chicago, spending four years simulating the dynamics of protein folding with his own accelerated 1/sqrt(x) function, which takes up 80%
of the CPU time in simulations. He subsequently moved to UCSF as a postdoc, building a statistical classifier for protein structure models, and spent a summer in Harvard with the future Nobel
Laureate Martin Karplus. He then left academia and joined Applied Biosystems to help developing their next-generation DNA sequencer. He brought his prior experience with data analysis when he joined
Cisco security in 2011, where he worked as a data scientist before this term was conceived.
Cisco Cybersecurity Viewpoints
Where security insights and innovation meet. Read the e-book, see the video, dive into the infographic and more...
Why Cisco Security?
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Lecture 14: Poisson Process - I | Probabilistic Systems Analysis and Applied Probability | Electrical Engineering and Computer Science | MIT OpenCourseWare
« Previous | Next »
Lecture Topics
• Review of Bernoulli process
• Definition of Poisson process
• PMF of number of arrivals
• Distribution of interarrival times
• Other properties of the Poisson process
Lecture Activities
Recitation Problems and Recitation Help Videos
Review the recitation problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.
Recitation Help Videos
Tutorial Problems
Review the tutorial problems in the PDF file below and try to solve them on your own.
« Previous | Next » | {"url":"https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/pages/unit-iii/lecture-14/","timestamp":"2024-11-02T03:20:44Z","content_type":"text/html","content_length":"98635","record_id":"<urn:uuid:53e3cd8b-4be4-4cfb-aab8-ad2d6cb11e0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00302.warc.gz"} |
How do I deal with (diploma) Courses with several components?
Diploma Courses
This article explains how to deal with 14-19 Diploma or other courses with several components:
Options can:
-- handle courses that span across 2 or more Option Blocks, and
-- even if there are 2 or more TeachingGroups for each component of the course, it will ensure that the students are allocated consistently to the groups.
An advanced** example to illustrate:
A school has a 14-19 Diploma course which is to span across four of the five Option Blocks.
ie. a student will choose 5 subjects, of which 4 can be for the diploma.
A student must choose all of the diploma course components, or none.
Step 1.
Suppose the 4 diploma parts are called Dip1, Dip2, Dip3, Dip4.
In this example we will complicate things by saying that there are two Teaching Groups for each of these, specified on the Subjects Screen like this:So we will need to ensure (see below) that
Options allocates the students consistently, so that each of the parallel groups remain separate and with the same membership for each of Dip1, Dip2, Dip3, Dip4.
(Note: if there is only one group for each 'subject' then everything is simpler!).
Step 2.
If a student chooses Dip1, s/he must choose Dip2 and Dip3 and Dip4.
This can be ensured by using the Rules for Students' Choices Screen, by entering every possible pairing of the 4 groups, like this: Step 3.
The Choices Screen will look like this: Step 4.
Before you can use the AutoCreate Screen to build up a Pattern, you need to tell Options that all of the students must be assigned to the groups consistently like this:Note: If there is only one
group for each Subject (ie. each component of the course) then you do not need to use this rule.
Options will then look for solutions that obey these rules, when you click on Go:
**More general or simpler cases:
In this uncommon example we have used 4 diploma components, with 2 groups for each, and with the students to be assigned consistently.
In most cases the problem will not be this complicated.
The most usual case is 'double options' (eg. BTEC1 & BTEC2, or Double Maths at A-level). | {"url":"https://timetabler.kayako.com/article/168-how-do-i-deal-with-diploma-courses-with-several-components","timestamp":"2024-11-09T12:57:41Z","content_type":"text/html","content_length":"22536","record_id":"<urn:uuid:f07e271f-ebf5-4114-9a27-2c75f3f5c7c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00703.warc.gz"} |
Data Science Tips for Beginners!
Q1. You are given a train data set having 1000 columns and 1 million rows. The data set is based on a classification problem. Your manager has asked you to reduce the dimension of this data so that
model computation time can be reduced. Your machine has memory constraints. What would you do? (You are free to make practical assumptions.) Answer: Processing a high dimensional data on a limited
memory machine is a strenuous task, your interviewer would be fully aware of that. Following are the methods you can use to tackle such situation: Since we have lower RAM, we should close all other
applications in our machine, including the web browser, so that most of the memory can be put to use. We can randomly sample the data set. This means, we can create a smaller data set, let’s say,
having 1000 variables and 300000 rows and do the computations. To reduce dimensionality, we can separate the numerical and categorical variables and remove the correlated variables. For numerical
variables, w
Data preprocessing is a step in the data mining and data analysis process that takes raw data and transforms it into a format that can be understood and analyzed by computers and machine learning.
When we talk about data , we usually think of some large datasets with some rows and columns. While that is a likely scenario, it is not always the case — data could be in so many different forms:
Structured Tables, Images, Audio files, Videos , etc. Machines don’t understand text, image, or video data as it is, they understand 1s and 0s. Real-world data also contains noises, missing values,
etc. which cannot be directly used for ML models. Hence, data preprocessing is required for cleaning the data and making it suitable for an ML model which increases the accuracy and efficiency of the
model. It involves the following steps: Getting the Dataset Importing Libraries Importing Dataset Data Quality Assessment: i) Finding and Processing Missing/Inconsistent/Duplicate Data ii) Mixed Data
List of Common Machine Learning Algorithms Here is the list of commonly used machine learning algorithms. These algorithms can be applied to almost any data problem: Linear Regression Logistic
Regression Decision Tree SVM Naive Bayes kNN K-Means Random Forest Dimensionality Reduction Algorithms Gradient Boosting algorithms GBM XGBoost LightGBM CatBoost 1. Linear Regression It is used to
estimate real values (cost of houses, number of calls, total sales etc.) based on continuous variable(s). Here, we establish relationship between independent and dependent variables by fitting a best
line. This best fit line is known as regression line and represented by a linear equation Y= a *X + b. https://www.analyticsvidhya.com/wp-content/uploads/2015/08/Linear_Regression.png 2. Logistic
Regression Don’t get confused by its name! It is a classification not a regression algorithm. It is used to estimate discrete values ( Binary values like 0/1, yes/no, true/false ) based on given set
of independen
1 . Sentiment Analyzer of Social Media This is one of the interesting and innovative machine learning projects. As, social media like Facebook, Twitter, and YouTube is the ocean of big data.
Therefore, mining these data can be beneficial in a number of ways to understand user sentiments and opinions. This project can be effective for digital marketing and branding to understand the
opinion or reaction for a product or service of a customer. 2 . Music Recommendation System Are you a lover of music? Always love to listen to your favorite one? Then, you will be glad to know about
this interesting machine learning project idea. This can also be an innovative project. The goal of this project is to recommend music based on user listening history. 3 . Credit Card Fraud Detection
Project Companies that involve a lot of transactions with the use of cards need to find anomalies in the system. The project aims to build a fraud detection model on credit cards. We will use the
transaction and th
Here are the 10 MOST IMPORTANT tips for making you a Compelling value proposition & helps create an impact. 1. From the PORTFOLIO OF PROJECTS which is actually your LIVING RESUME, choose ONE project
that you are SURE about to talk & your story SHOULD cover the following at a minimum with Demonstrable evidence
1. Pick the brain of an expert. There are myriads of ways to learn data science. You can read articles, watch videos, enroll in onli n e courses, turn up at meetups, etc. But one thing that you
cannot “learn” is the experience . That you have to gain throughout years of working in the field. There is much to learn from Data science experts, their experience in managing end-to-end machine
learning and deep learning projects, their philosophy when constructing a data science team from scratch, | {"url":"https://datahunger.blogspot.com/","timestamp":"2024-11-10T17:41:18Z","content_type":"text/html","content_length":"148424","record_id":"<urn:uuid:f91950e3-3e26-493d-a6c9-9b8a22558484>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00475.warc.gz"} |
Vol. 3 No. 3 (2011) , Article ID: 4145 , 4 pages DOI:10.4236/eng.2011.33028
Comparison of Wind-Induced Displacement Characteristics of Buildings with Different Lateral Load Resisting Systems
^1Department of Civil Engineering, The Federal University of Technology, Akure, Nigeria
^2Works Department, Ladoke Akintola University of Technology, Ogbomoso, Nigeria
E-mail: arumcnwchrist@yahoo.co.uk, engr.akinkunmi@gmail.com
Received December 25, 2010; revised January 5, 2011; accepted January 17, 2011
Keywords: Moment Frame, Shear Wall, Dual System, Inter-Storey Drift, Lateral Displacement, Wind Load
Due to excessive displacements of tall buildings occasioned by lateral loads, lateral load resisting systems are usually provided to curtail the load effect. The resistance may be offered by Frame
Action, Shear Walls, or combined Walls and Frames (also known as Dual System). In this study, finite element based software, ETABS, was used to generate and analyse three-dimensional building models
for the assessment of the relative effectiveness of the various lateral load resisting systems. Three models were used, one each for the three resisting systems. Each model consisted of three samples
representing three different building heights of 45 m, 75 m, and 99 m. Wind Design Spreadsheet complying with the appropriate British Standards was used to compute preliminary wind load coefficients
using the wind speed values from the relevant wind isopleth map of Nigeria as primary data. Lateral wind load was then applied at floor levels of each of the building samples. Each building sample
was subjected to three-dimensional analysis for the determination of both the lateral displacements of storey tops and interstorey drifts. The results of the work showed that the dual system was the
most efficient lateral-load resisting system based on deflection criterion, as they yielded the least values for lateral displacements and inter-storey drifts. The moment frame was the least stiff of
the resisting systems, yielding the highest values of both the lateral displacement and the inter-storey drift.
1. Introduction
In general, as the height of a building increases, its overall response to lateral load (such as wind and earthquake) increases. When such response becomes sufficiently great such that the effect of
lateral load must be explicitly taken into consideration in design, a multistory building is said to be tall. Tall buildings are prone to excessive displacements, necessitating the introducetion of
special measures to contain these displacements. The lateral load effects on buildings can be resisted by Frame action, Shear Walls, or Dual System. Peak interstorey drift and lateral displacement
(or sidesway) are two essential parameters used for assessing the lateral stability and stiffness of lateral force resisting systems of tall buildings.
Park, Hong & Seo [1] in their work pointed out that the efficiency of lateral load resisting system or the amount of materials required for multistorey buildings heavily depends on drift limits.
Kowalczyh [2] noted the difficulties faced by Structural Engineers in selecting strong and stiff enough deformation resisting systems that will curtail the drift within acceptable code limits. Chen &
Lui [3] extensively discussed susceptibility of multistorey buildings to sway under lateral wind loading and therefore advocated the need for good understanding of the nature of wind load and
estimation of interstorey drift. Sindel et al. [4] reiterated the importance of the knowledge of lateral displacements at the top of multistorey buildings because of its usefulness in assessing the
stability and stiffness of multistorey buildings.
In view of the foregoing, this study examined the three deflection limiting structural systems (moment frame, shear-wall and dual system) prominently utilized in tall building structures, in order to
establish their relative effectiveness.
2. Lateral Load Resisting Systems
2.1. Shear Wall System
Khajehpour [5] described Shear Walls as stiff structures with high ductility which keeps the deformations of non-ductile framing systems in the elastic range. He noted that the in-plane load
resistance is the principal strength of shear walls and that the resistance against both gravity and lateral loads can be assigned to shear walls if they are appropriately located in a building.
2.2. Moment Frame System
Moment Frame is suitable where the presence of Shear Wall is undesirable, especially in situations where architectural limitation is imposed. The results of Khajehpour’s work [5] revealed that Moment
Frame is economical up to twenty stories while Smith & Coull [6] submitted that it is economical for buildings up to twenty-five stories, above which their behaviour is costly to control.
The lateral deflection of Moment Frame is caused by two modes of deformation, namely chord drift which accounts for 20% of the total drift of structures, and frame racking which accounts for 80% of
the storey drift [5].
2.3. Dual Frame-Wall System
Gardiner, Bull, and Carr [7] in their work noted that the wall element in dual system is responsible for an increase of stiffness which is beneficial in terms of drift control. Nawy [8] observed that
if they were to act independently, the shear walls would deflect as vertical cantilevers with greater interstorey drift occurring at the top while the frames would deflect at more uniform rate or
with greater interstorey drift at the bottom. With rigid diaphragms he noted that there is a forced compatibility of frame and wall deflection at each storey and this induces interaction forces
between shear walls and frames. He showed that the pattern of these forces is such that the shear walls tend to support the frame at the lower stories and the frame tends to support the shear walls
at the upper stories.
3. Wind Load Determination
Lungu and Rackwitz [9] in their studies established that wind effects on buildings and structures depend on the general wind climate, the exposure of buildings, type of structures and their elements,
the dynamic properties, and the shape and dimensions of the building structure. BS6399-2 [10] is the widely adopted code for wind load estimation in Nigeria. For the computation of wind loads, the
Standard Method was employed. The static approach used in this study is based on a quasi-steady assumption equivalent to a structure that is dynamically displaced in its lowest frequency mode and
assumes a building to be a fixed rigid body in the wind direction. According to BS 6399-2 [10], the site wind speed is given by the following formula:
V[b] = basic wind speed;
S[a] = altitude factor;
S[d] = wind direction;
S[s] = seasonal factor; and S[p] = probability factor.
According to the Code [10], the effective wind speed is computed as follows:
S[b] = terrain and building factor.
The dynamic pressure is obtained as follows:
and the net load, P, on an area of a building element is given as the product of net pressure across the surface p, and the loaded area A, as given by Equation (4).
C[a] = size effect factor;
A basic wind speed of 56 m/s was employed in this work, the maximum value for Nigeria.
4. Storey Drifts Limitation
Sindel et al. [4] defined the storey drift as the difference of maximum lateral displacements of any two adjacent floors under the factored loads, divided by respective storey height. In terms of
elastic deflections, the storey drift, S is given by
^th floor under the factored loads.
h = storey height .
BS 8110-2 [11] restricts relative lateral deflection in any one storey under the characteristic wind load to a maximum of
5. Modeling Assumptions and Idealizations
The following assumptions apply in this work:
1) Static or equivalent static loads as recommended by BS 6399-2 [10] are considered.
2) Dead loads are assumed to be invariant with changes in member sizes.
3) The material of concrete is assumed to be linearly elastic and P-∆ effects are not considered.
4) Structural members are straight and prismatic.
5) In order to reflect actual behaviour of structures, all frames are assumed to be rigid in plane, hence they constrain the horizontal shear deflection of all vertical bents at floor levels to be
related by the horizontal translations and floor slab rotations.
6) All connections between members of all building models are assumed rigid while the buildings are fixed at the base.
7) The buildings are assumed to be office complex type meant for general use located in exposure B.
8) Member sizing is carried out based on the provision for worst combination of load forces as stipulated by BS 8110-1 [12].
9) It is assumed that window openings are installed in all shear walls at 1.0 m above floor levels with total width of 2.60 m and stretch to a height of 1.2 m while the door openings 2.10 m high by
1.2 m wide are introduced at the floor levels.
10) Major axis is taken to be the axis about which the section has the larger second moment of area.
11) The wall piers of wall-frame structures are uniform.
6. Materials and Methods
Finite Element Method of analysis was employed in the structural modeling of all three lateral load resisting systems. This was achieved by the use of both Wind Design Workbook by Buczkowski [13] and
ETABS computer analysis software package.
6.1. Generation of Building Analytical Models
Three building models were generated with the aid of ETABS, representing the three lateral load resisting systems under investigation. Each model consisted of three building samples which differed
only by the number of storeys, namely, 15, 25 and 33 storeys. Thus, nine samples altogether were generated, loaded and their threedimensional analyses were performed using ETABS software. Each sample
had four bays in the X direction and three bays in the Y direction. The storey height was fixed at 3.0 m. The bay width was 5 m centre-to-centre equal span in both X and Y directions. This brought
the overall plan dimensions of the building to 20.0 m by 15.0 m. The general arrangement drawings for typical floors for the moment frame, the shear wall system and the dual system are shown
respectively in Figures 1, 2 and 3.
In the moment resisting frame (Figure 1), every joint is an intersection of beams and columns and it is this network of beams and columns that resists the lateral load by the bending action of the
members. In the shear wall building (Figure 2), the interior of the building framework consists of beams and columns while the exterior frame of the building consists of shear walls and beams without
any columns. Finally, the dual system (Figure 3) has columns and beams both within the interior and the exterior framing of the building but in addition
Figure 1. General arrangement drawing for a typical floor for the moment frame.
Figure 2. General arrangement drawing for a typical floor for the shear wall system.
Figure 3. General arrangement drawing for a typical floor for the dual system.
it also has shear walls incorporated in the exterior framework of the building.
A typical three-dimensional model with shear wall is shown in Figure 4 for a 15-storey sample while Figure 5 shows the maximum nodal values of the wind load for the same model, on a plane elevation.
Every building model was assigned fixed bottom support condition while a rigid diaphragm constraint was allotted to both slab and deck members. Each model was loaded with gravity loading in addition
to wind load.
6.2. Loading
The basic wind speed was obtained from the map of wind speed isopleths for Nigeria. In order to fully consider local conditions, the works of Arum [14] and Auta
Figure 4. Typical 3D model with shear wall for the 15-storey building sample.
[15] were utilised. Wind Design Workbook developed by Buczkowski [13] which implements the full Standard Method of calculating wind loads for buildings in accordance with BS 6399-2 [10] was employed
to compute the preliminary wind load which were later used as secondary wind data in ETABS software. Wind load was computed for each of the four sides of each building sample. A uniform shell live
load of 2.5 kN/m^2 was assigned to the solid slab along the negative global Z-direction. A live load reduction was effected on the total distributed imposed shell loads in compliance with minimum
imposed floor load stipulated by BS 6399-1 [16] for office buildings. The dead load for both slab and filled deck were automatically generated by ETABS. In addition, a minimum imposed load of 1.5 kN/
m^2 was assigned to the deck in accordance with BS 6399 [15].
Static Load Combinations
Each of the nine samples was subjected to load combination as provided for in BS 8110-1 [12]. The load combinations included the following:
Figure 5. Maximum nodal values of the wind load for the 15-storey build-ing sample.
1) 1.4DL + 1.4WL, and
2) 1.2DL + 1.2LL + 1.2WL combinations.
DL = Dead load
WL = Wind load
LL = Live load
7. Results & Discussion
7.1. Results
The results of the lateral displacements curve for the 15-storey building sample, which is typical of all building heights considered, are presented in Figure 6 while their corresponding interstorey
drifts are shown in Figure 7.
The values of the lateral displacements and their comparisons for the three lateral load resisting systems are shown in Tables 1 to 3 respectively for the 15-storey, 25-storey and 33-storey
The values of the inter-storey drifts as well as the comparison of the values for the three lateral load systems are shown presented in Tables 4, 5 and 6 respectively for the 15-storey, 25-storey and
33-storey buildings.
Figure 6. Lateral displacements curve for the 15-storey building sample.
Figure 7. Inter-storey drifts curve for the 15-storey building sample.
Table 1. Comparison of lateral displacements for a 15-storey building with different lateral load resisting systems.
Table 2. Comparison of lateral displacements for a 25-storey building with different lateral load resisting systems.
Table 3. Comparison of lateral displacements for a 33-Storey building with different lateral load resisting systems.
7.2. Discussion
A comparison of the values for the lateral displacements as contained in Tables 1, 2 and 3 for 15-storey, 25-storey and 33-storey buildings respectively shows that the lateral displacement is
greatest at the topmost storeys for the three lateral force resisting systems, having the greatest values for the moment frame and the least for the dual system. For the 15-storey building, the
lateral displacement of the moment frame at the level of the 1^st storey is about 20 times greater than that of the shear wall system and about 24 times that of the dual system. Thus, from the
standpoint of resistance to lateral displacement, the moment frame is the worst while the dual system is the best. The shear wall lies in-between being far better than the moment frame and marginally
Table 4. Comparison of inter-storey drifts for a 15-storey building with different lateral load resisting systems.
Table 5. Comparison of inter-storey drifts for a 25-storey building with different lateral load resisting systems.
Table 6. Comparison of inter-storey drifts for a 33-storey building with different lateral load resisting systems.
worse than the dual system. In the moment resisting frame system, since the stiffness of the columns is comparable to that of the beams, the beam-column joints rotate as well as translate laterally
and the high deformation components of rotation and translation sum up to give high displacement values. In the case of the shear wall system, the shear walls act like deep beams which are much
stiffer than the columns. In this case the deformation due to rotation of the joints is significantly reduced and the major component of the deformation is due to lateral translation, resulting in
overall reduced lateral displacement values. Finally, since the dual system incorporates both columns and beams as well as shear walls, the system is the stiffest of the three and the extra
resistance offered by the periphery columns further reduces the lateral displacement prevailing in the case of the shear wall system. In practice, in order to reach a decision on which of the shear
wall and the dual system to use, the extra stiffness offered by the dual system should be weighed against the extra cost in material and construction of the periphery columns and beams, which are
absent in the shear wall system.
As shown in Tables 4 to 6, for all storey variants considered, the inter-storey drift is greatest for moment frame and least for dual system. Furthermore, for all storey variants, the greatest
inter-storey drifts for the moment frame occur at the lowest third along the building height, with the exception of the first storey which has about half the drift value of the average value of the
storeys located in the lowermost third of the building height. For the shear wall and the dual system, the drift is greatest for the storeys located within the middle third of the building height. In
addition, for the 25 and 33 storeys, whereas the drift is greater in the shear wall than for the dual system at the lower floors, from the 20^th to the 24^th storey for the 25-storey building and
from the 29^th storey to the 32^nd for the 33-storey building, the drift of the dual system is greater than for the shear wall. It should be noted that this reversal occurred only for the tall
buildings (25 and 33 storeys) and not for the 15-storey variant. The implication of this is that for tall buildings, there exists a height for every building from which greater building stiffness
would be achieved by employing shear wall system rather than dual system.
Furthermore, the lateral displacement curves of Figure 6 show that for the moment frame, the lateral displacement increases rapidly from the first storey to about two-thirds of the building height
from where the increase with height begins to dampen. The average slope of the curve within the first two-thirds of the building height is very gentle. For the shear wall and dual systems, the
relationship between building height and lateral displacement is almost linear with a steep slope.
8. Conclusions
From the results of this work the following conclusions are apt:
1) The lateral displacement in moment frames is the greatest among the three lateral load resisting systems investigated; the lateral displacement in dual frames is the least while the lateral
displacement in shear wall systems is slightly higher than that of the dual system.
2) Interstorey drift is greatest in moment frames and least in dual systems while that of the shear wall system is slightly higher than that of the dual system.
3) Among the building samples studied, the greatest interstorey drift occurred at the bottom third of the moment frames. For the shear wall and the dual system, the drift is greatest for the storeys
located within the middle third of the building height. In addition, for the 25 and 33 storeys, whereas the drift is greater in the shear wall than for the dual system at the lower floors, from the
20^th to the 24^th storey for the 25-storey building and from the 29^th storey to the 32^nd for the 33-storey building, the drift of the dual system is greater than for the shear wall.
9. REFERENCES
1. H. S. Park, K. Hong and J. Seo, “Drift Design of SteelFrame Shear-Wall Systems for Tall Buildings,” The Structural Design of Tall Buildings, Vol. 11, No. 1, 2002, pp. 35-49. doi:10.1002/tal.187
2. R. Kowalczyh, “Tall Buildings: Past, Present & Future Developments,” URE: Summer School Urban Steel Structures, Gdansk, 2005.
3. W. Chen and E. M. Lui, “Principles of Structural Design,” CRC Press Taylor & Francis Group, London, 2006.
4. Z. Sindel, R. Akbas and S. Tezcar, “Drift Control & Damage in Tall Buildings,” Engineering Structures, Vol. 18, No. 12, 1996, pp. 959-962. doi:10.1016/0141-0296(95)00215-4
5. S. Khajehpour, “Optimal Conceptual Design of HighRise Office Buildings,” Ph.D. Thesis, University of Waterloo, Ontario, 2001.
6. B. S. Smith and A. Coull, “Tall Building Structures: Analysis & Design,” John Wiley & Sons, Inc., New York, 1991.
7. D. R. Gardiner, D. K. Bull and A. J. Carr, “Internal Forces of Concrete Floor Diaphragms in Multistorey Buildings,” Department of Civil Engineering, University of Canterbury NZSEE Conference,
Christchurch, 2008.
8. E. G. Nawy, “Concrete Construction Engineering Handbook,” CRC Press, Taylor & Francis Group, London, 2008.
9. D. Lungu and R. Rackwitz, “Joint Committee on Structural Safety, Probabilistic Model Code Part 2: Loads,” Joint Committee on Structural Safety, Section 2.13 on WINDS, Tampa, 1997.
10. British Standards Institution, “BS 6399: Loading for Buildings—Part 2: Code of Practice for Wind Loads,” British Standards Institution, London, 1997.
11. British Standards Institution, “BS 8110: Structural Use of Concrete—Part 2: Code of Practice for Special Circumstances,” British Standards Institution, London, 1985.
12. British Standards Institution, “BS 8110: Structural Use of Concrete—Part 1: Code of Practice for Design & Construction,” British Standards Institution, London, 1997.
13. C. Buczkowski, “Wind Design Workbook for Microsoft Excel—Version 2.1,” 2009. http://www.structural-engineering.fsnet.co.
14. C. Arum, “Lightweight Steel Frameworks for Shed Buildings in Different Wind Regions of Nigeria,” Nigerian Journal of Industrial and Systems Studies, Vol. 1, No. 4, 2002, pp. 5-14.
15. S. M. Auta, “Wind Load Estimation on Tall Building: Part 1: Comparison of Russian & Nigerian Codes of Practice,” Asian Journal of Civil Engineering (Building & Housing), Vol. 7, No. 3, 2006, pp.
16. British Standards Institution, “BS 6399: Loading for Buildings—Part 1: Code of Practice for Dead and Imposed Loads,” British Standards Institution, London, 1996. | {"url":"https://file.scirp.org/Html/6-8101310_4145.htm","timestamp":"2024-11-05T23:44:08Z","content_type":"application/xhtml+xml","content_length":"51295","record_id":"<urn:uuid:2bed54c8-6a73-44f0-9ded-7d313204b00b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00079.warc.gz"} |
Get the Log of a Value in R - With Examples - Data Science Parichay
In this tutorial, we will look at how to compute the logarithm of a value using log functions in R with the help of some examples.
How to compute the log of a value in R?
You can use the built-in log() function to compute the log of a value in R. Pass the value for which you want to compute the log as an argument. The following is the syntax –
# log function in R
log(x, base)
The log() function takes the following arguments –
• x – The value for which you want to compute the log.
• base (optional) – The logarithmic base to use.
By default, the log() function computes the natural log of a value if you don’t specify the base. If you specify the base, it computes the log with respect to the given base.
Note that R also has direct functions to compute the logarithm with common bases such as log10() to compute log with 10 as the base and log2() to compute log with 2 as the base.
Let’s look at some examples of using the above functions in R.
Natural log in R
To get the natural log of a value in R, pass the value to the log() function without any additional arguments. For example, let’s compute the natural log of some numbers using this method –
# natural log of value
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[1] 0
[1] 0.6931472
We get the natural log of 1 as 0 and 2 as 0.693.
Log with base 10 in R
Let’s now compute the log with base 10 for a value. For this, you can either directly use the log10() function or use the log() function with base=10.
Let’s look at an example.
# log with base 10
print(log(100, 10))
[1] 2
[1] 2
Here, we compute the log with base 10 of the number 100 using the functions log10() and log() (with the base as 10). You can see that we get the same result from both methods.
Log with base 2 in R
You can similarly compute the log with base 2 for a value in R. That is, either directly use the log2() function or use the log() function with base=2.
Let’s look at an example.
# log with base 2
print(log(8, 2))
[1] 3
[1] 3
Here, we compute the log with base 2 of the number 8 using the functions log2() and log() (with the base as 2). You can see that we get the same result from both methods.
Log with custom base for a value in R
As shown above, you can use the log() function with a custom base value. For example, let’s compute the log of 9 with base 3.
# log with custom base
print(log(9, 3))
[1] 2
We get 2 as the output.
In this tutorial, we looked at how to compute the logarithm of a value in R using the function log() which computes the natural log by default but can be customized to compute the log with respect to
a different base using the base parameter. We also looked at some specific log functions such as log10() and log2() which compute the log with respect to base 10 and 2 respectively.
You might also be interested in –
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How to Add Specific Field into Form with SUBSTITUTE Formula
Hey Community,
I am trying to have my OPPTY # field prepopulated in my form using the SUBSTITUTE formula because a lot of our users tend to enter it incorrectly. Can you please help me figure out how to solve this?
I will paste a screenshot of the format below.
• @John- Michael Diedrich Is it possibly the same issue in your other post regarding the URL? I believe there was an issue with the # and it needed to be swapped out with something (maybe %23 but
could be remembering wrong).
• Thanks for the response Paul. I am assuming it is something along those line, however I am not sure what "%##" needs to be there for it to work. I tried playing with it but have had no luck.
Any help is appreciated!
• Was the recommendation in the other post not correct?
• It was not unfortunately. I think it will require a different "%##" number in order for it to work properly. That is where I struggle and need help, because there are unique codes for specific
columns that I do not know about.
• %23 is correct for the #, but you also need to replace any spaces with %20.
• Thank you for the help Paul, those code numbers are helpful. I tried re-writing the Substitute formula but had no luck. Below is how I tried writing it if you want to see:
And here is the OPPTY # column I am trying to have prepopulated on the form:
Any help is greatly appreciated!
• It looks like you are not including the Oppty # field in the URL.
Notice how you have "Work%20Order%20ID" in the URL to indicate you want to populate that field and then the following bit is what you want to populate the field with (in this case it happens to
be the cell contents).
You need to do the same for the Oppty # field. You need to list the field and then list what you want to populate it with. Right now you are basically combining the Work Order ID and the Oppty #
as a single string and dropping it into the Work Order ID field of the form.
• Sorry Paul I am trying to follow and correct the formula but I am not fully understanding. I updated the formula to the below and now the Work Order ID doesn't populate. Do you know how the
formula is supposed to be written?
• You need to string it together in the proper order. Field value field value. In the above you have field field value value.
• Understood, thank you Paul. Will I need to include the form link for each field value? Or should I use the below:
="FormLink?Work%20Order%20ID=" + SUBSTITUTE([Work Order ID]@row, "#", "%23") + ?Oppty%20# = SUBSTITUTE([Oppty #]@row, "OPPTY", "%20")
Sorry this formula is giving me a lot of trouble.
• I personally would write out the formula that puts everything together and then just wrap the entire thing in a series of SUBSTITUTE functions to make sure everything gets replaced as needed.
=SUBSTITUTE(SUBSTITUTE("FormLink?............" + [Work Order ID]@row + "?Oppty #=" + [Oppty #]@row, "#", "%23"), " ", "%20")
• Thank you for the help Paul, but unfortunately this did not return a value in the Work Order ID and Oppty # fields. I typed it out as the below:
=SUBSTITUTE(SUBSTITUTE("FormLink" + [Work Order ID]@row + "?Oppty #=" + [Oppty #]@row, "#", "%23"), " ", "%20")
Please let me know if you can spot the issue, any help is appreciated!
• Wha tis the final output of the above formula? Can you drop the URL in here?
• Here is the exact formula I used based on your input:
=SUBSTITUTE(SUBSTITUTE("FormLink" + [Work Order ID]@row + "?Oppty #=" + [Oppty #]@row, "#", "%23"), " ", "%20")
And when it is clicked the form shows the below:
Please let me know what you think when you get a chance, below is the old formula I had that would populate the Work Order ID but not the OPPTY #:
="FormLink" + SUBSTITUTE([Work Order ID]@row, "#", "%23") + SUBSTITUTE([Oppty #]@row, "OPPTY", "%20")
Thank you for the help
• If you look at the new formula, you will see that the field for Work Order ID is not specified.
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Frontiers | Estimating Attractor Reachability in Asynchronous Logical Models
• ^1Instituto Gulbenkian de Ciência, Oeiras, Portugal
• ^2Department of Computer Science and Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
• ^3Instituto de Engenharia de Sistemas e Computadores Investigação e Desenvolvimento, Lisbon, Portugal
• ^4Aix Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, Marseille, France
Logical models are well-suited to capture salient dynamical properties of regulatory networks. For networks controlling cell fate decisions, cell fates are associated with model attractors (stable
states or cyclic attractors) whose identification and reachability properties are particularly relevant. While synchronous updates assume unlikely instantaneous or identical rates associated with
component changes, the consideration of asynchronous updates is more realistic but, for large models, may hinder the analysis of the resulting non-deterministic concurrent dynamics. This complexity
hampers the study of asymptotical behaviors, and most existing approaches suffer from efficiency bottlenecks, being generally unable to handle cyclical attractors and quantify attractor reachability.
Here, we propose two algorithms providing probability estimates of attractor reachability in asynchronous dynamics. The first algorithm, named Firefront, exhaustively explores the state space from an
initial state, and provides quasi-exact evaluations of the reachability probabilities of model attractors. The algorithm progresses in breadth, propagating the probabilities of each encountered state
to its successors. Second, Avatar is an adapted Monte Carlo approach, better suited for models with large and intertwined transient and terminal cycles. Avatar iteratively explores the state space by
randomly selecting trajectories and by using these random walks to estimate the likelihood of reaching an attractor. Unlike Monte Carlo simulations, Avatar is equipped to avoid getting trapped in
transient cycles and to identify cyclic attractors. Firefront and Avatar are validated and compared to related methods, using as test cases logical models of synthetic and biological networks. Both
algorithms are implemented as new functionalities of GINsim 3.0, a well-established software tool for logical modeling, providing executable GUI, Java API, and scripting facilities.
1. Introduction
Logical modeling has been widely used to study gene regulatory and signalling networks (see e.g., Glass and Siegelmann, 2010; Saadatpour and Albert, 2012; Abou-Jaoudé et al., 2016). Briefly, in a
logical model, the evolution of the discretised level of each component depends on the current values of its regulators whose influences are dictated by logical functions. Here, we rely on the
generalized framework initially introduced by Thomas and d'Ari (1990) and implemented in our software tool GINSIM (Chaouiya et al., 2012; Naldi et al., 2018). Because precise knowledge of the
durations of underlying mechanisms is often lacking, one assumes that, when multiple components are called to change their levels, all update orders have to be considered. This corresponds to the
asynchronous updating scheme (Thomas and d'Ari, 1990; Thomas, 1991). The dynamics of these models are classically represented by State Transition Graphs (STGs) where nodes embody the model states and
edges represent the state transitions; each path in this graph accounts for a potential trajectory of the system. In contrast, synchronous updates, which amount to consider equal or negligible delays
associated to component changes, define deterministic dynamics, easier to analyse but less realistic.
Model attractors (stable states or cyclic attractors) represent long term, stable equilibria. Cyclic attractors denote stable oscillations as observed in cell cycle or circadian rythms (see e.g.,
Fauré et al., 2006; Fauré and Thieffry, 2009; Chaves and Preto, 2013), whereas stable states are associated with cell lineages or other cellular responses to external cues or perturbations (see e.g.,
Sánchez et al., 2008; Calzone et al., 2010; Naldi et al., 2010; Collombet et al., 2017). Modeling molecular networks involved in cancer has been focusing on attractors and their reachability
properties (see e.g., Huang et al., 2009; Flobak et al., 2015; Remy et al., 2015; Cho et al., 2016). Indeed, attractor likelihood may provide relevant predictions as attractors reflect cellular
responses (e.g., healthy or not). For instance, to uncover patterns of genetic alterations in bladder tumors, Remy et al. (2015) considered an asynchronous logical model and checked how model
perturbations modify the probabilities of reaching attractors related to proliferative phenotypes.
Not surprisingly, the number of states of logical models grows exponentially with the number of regulatory components. Moreover, due to the asynchronous updating scheme, the dynamics are
non-deterministic; they possibly encompass alternative trajectories toward a given state as well as transient cycles. All this turns the identification and reachability analysis of model attractors
into a difficult challenge. In this context, methods have been developed to find stable states—also referred as point attractors—and complex, oscillatory attractors (or, at least to circumscribe
their location) (Naldi et al., 2007; Garg et al., 2008; Zañudo and Albert, 2013; Klarner et al., 2015). Here, we primarily aim at efficiently determining attractors reachable from specific initial
condition(s) as well as estimating the reachability probability of each of those attractors in asynchronous dynamics.
An STG can be readily interpreted as the transition matrix of a finite Markov Chain. Generally, STGs encompass distinct attractors (or recurrent classes) and thus define absorbing chains (Grinstead
et al., 1997). However, most existing results relate to recurrent (or irreducible) chains (Prum, 2012). Moreover, we aim at avoiding the construction of the whole dynamics (or the associated
transition matrix); we thus rely on the logical rules as implicit descriptions of state transitions. Finally, we have here a specific interest on reachability properties.
Following a background section, we present two approaches to assess reachable attractors. First, the FIREFRONT algorithm is a quasi-exact method that starts from an initial state and simultaneously
follows all (concurrent) trajectories while propagating state probabilities. This algorithm follows a principle similar to those employed for infinite Markov chains (Munsky and Khammash, 2006;
Henzinger et al., 2009). To enable state space sampling and tackle models with large transient cyclic behaviors, we developed AVATAR, which is a Monte Carlo approach adapted to cope with strongly
connected components. Both methods have been implemented as new functionalities of the software tool GINSIM (Naldi et al., 2018). They are applied to a range of models, illustrating their respective
performances and specificities.
2. Methods
In this section, we first briefly introduce the basics on Logical Regulatory Graphs (LRGs), their state transition graphs (STGs), attractors as well as absorbing Markov chains. We then present the
algorithm FIREFRONT. The rest of the section focuses on AVATAR, an adaptation of the classical Monte Carlo simulation to cope with cyclical behaviors. It is worth noting that for small enough models
it is possible to explicitly construct the STGs and identify reachable attractors, but it is not straightforward to evaluate their reachability probabilities.
2.1. Background
2.1.1. Basics on Logical Models and Their Dynamics
Definition 1. A Logical Regulatory Graph (LRG) is a pair (G, K), where:
• G = {[g[i]}i = 0, …n] is the set of regulatory components. Each g[i] ∈ G is associated to a variable v^i denoting its level, which takes values in D[i] = {0, …M[i]} ⊊ ℕ; $v={\left({v}^{i}\right)}_
{i=0,\dots n}$ is a state of the system, and S = ∏[i = 0, …n]D[i] denotes the state space.
• ([K[i])i = 0, …n] denotes the logical regulatory functions (or logical rules); K[i]:S → D[i] is the function that specifies the evolution of g[i]; ∀v ∈ S, K[i](v) is the target value of g[i] that
depends on the state v.
The asynchronous dynamics of an LRG is represented by a graph as follows.
Definition 2. Given a logical regulatory graph (G, K), its asynchronous State Transition Graph (STG) is denoted (S, T), where:
• S is the state space,
• T = {(v, v′) ∈ S^2∣v′ ∈ Succ(v)}, where for each state v, Succ(v):S → 2^S is the set of successor states w, satisfying the asynchronous property (one component is updated at a time):
$∃gi∈G with{Ki(v)=vi and wi=vi+Ki(v)−vi|Ki(v)−vi|,∀gj∈G∖{gi}, wj=vj.$
Note that, from the STG defined above, one can consider the sub-graph reachable from a specific initial state v[0] or from a set of states {[v[i]}i ∈ {0, …m}] ⊆ S.
We further introduce some notation and classical notions.
Given an STG (S, T), we write v → v′ if and only if there exists a path between the states v and v′. In other words, there is a sequence of states of S such as: ${v}_{0}=v,{v}_{1},\dots {v}_{k-1},{v}
_{k}={v}^{\prime }$, and for all j ∈ {1, …k}, (v[j−1], v[j]) ∈ T. Furthermore, we denote $v\stackrel{k}{\to }{v}^{\prime }$ such a path of length k.
A Strongly Connected Component (SCC) is a maximal set of states A ⊆ S such that ∀v, v′ ∈ A with v≠v′, v → v′. This is to say, there is a path between any two states in A, and this property cannot be
preserved adding any other state to A.
Attractors of an LRG are defined as the terminal SCCs of its STG (i.e., there is no transitions leaving the SCC). If a terminal SCC is a single state we call it a stable state, otherwise it is a
complex attractor.
2.1.2. Markov Chains and Absorption
The incidence matrix of an STG (S, T) naturally translates into an |S| × |S|-transition matrix Π, which is a stochastic matrix (for all v ∈ S, $\sum _{u\in S}\Pi \left(v,u\right)=1$):
$∀v,v′∈S Π(v,v′)>0⇔(v,v′)∈T,∀v∈S Π(v,v)=1⇔Succ(v)=∅, Π(v,v)=0otherwise.$
We assume that probabilities of concurrent transitions are uniformly distributed: ∀v ∈ S, ∀v′ ∈ Succ(v), Π(v, v′) = 1/|Succ(v)|. Extension to other distributions would be rather straightforward.
A Markov chain (μ[0], Π) is defined by the finite set S, the transition matrix Π, and the initial law μ[0] (that depends on the selection – or not – of an initial condition). We want to define the
chain stopped when it reaches an attractor. For that, we consider the quotient graph of (S, T) with respect to the equivalence relation: u ~ v ⇔ u → vandv → u. In this quotient graph, each node
gathers a set of states and corresponds to a class of the Markov chain. The absorbing nodes of the quotient graph (i.e., nodes with no output arcs) form the absorbing classes of the chain (μ[0], Π),
all the other classes being transient. Note that the number of absorbing classes is the number of attractors of the corresponding STG. Let θ be this number and a[1], …a[θ] the absorbing classes.
Now, let us stop the chain (μ[0], Π) when it reaches an absorbing class: we thus define the Markov chain X on the set $\stackrel{~}{S}={T}\cup {A}$, where ${T}\subset S$ is the set of all the
transient states, and ${A}=\left\{\left\{{a}_{i}\right\},i=1,\dots \theta \right\}$ (each element a[i] being an absorbing class). The transition matrix π of X is:
$π(u,ai)=∑v∈aiΠ(u,v) ∀u∈T,∀ai∈A,π(ai,u)=0 ∀u∈T,∀ai∈A,π(ai,ai)=1 ∀ai∈A,π(ai,aj)=0 ∀ai∈A,∀aj∈A,i≠j,π(u,v)=Π(u,v) ∀u,v∈T.$
Reordering the states by considering first the transient ones, (i.e., those belonging to ${T}$) and then the absorbing classes (i.e., the elements of ${A}$), the transition matrix π is under its
canonical form:
where Q(u, v) = π(u, v) for $u,v\in {T}$, L(u, a) = π(u, a) for $u\in {T}$ and $a\in {A}$, 0 is the null matrix (no transition from an absorbing class to a transient state), and I the identity
matrix. One can easily verify that:
π^k(u, v) denotes the probability that, started in state u, the chain is in state v after k steps: ${\pi }^{k}\left(u,v\right)={ℙ}_{u}\left({X}_{k}=v\right)\stackrel{\Delta }{=}ℙ\left({X}_{k}=v|{X}_
{0}=u\right)$. Proofs of the next, well-known results can be found in [e.g., (Grinstead et al., 1997), chap. 11].
• Q^k tends to 0 when k tends to infinity, and
$limn→+∞∑k=0nQk=(I-Q)-1. (1)$
• The hitting time of ${A}$ is almost-surely finite.
• From any $u\in {T}$, the probability of X being absorbed in $a\in {A}$ is ${ℙ}_{u}\left({X}_{\infty }=a\right)={\left(Id-Q\right)}^{-1}L\left(u,a\right).$
By an abuse of terminology, we will refer to ℙ[u](X[∞] = a) as the probability to reach the attractor a from the initial state u.
2.2. Firefront
FIREFRONT is our first method to identify attractors and assess their reachability probabilities. Although simple, it is effective for restricted types of dynamics as demonstrated in section 4.
Briefly, the algorithm progresses in breadth from an initial state v[0], which is first assigned probability 1. It distributes and propagates the probability of each visited state to its successors,
according to the transition matrix Π.
At any step k, the set of states being expanded and carrying a fraction of the original probability is called firefront as it corresponds to the front line of the breadth-first exploration of the
STG: ${F}_{k}=\left\{v\in S,\exists {v}_{0}\stackrel{k}{\to }v\right\}$. Basically this procedure, called expansion, calculates at each iteration k and for each state v the probability of the Markov
chain X to be in v after k steps from state v[0]: ${ℙ}_{{v}_{0}}\left({X}_{k}=v\right)={\pi }^{k}\left({v}_{0},v\right)$. Clearly, by the definition of the set F[k], ℙ[v[0]](X[k] ∈ F[k]) = 1; the
firefront will ultimately contain only states that are stable states or members of complex attractors. In what follows, we will simply denote the firefront set F, omitting the index k. Actually,
attractors are not kept in F, they are instead stored in another set A (see below), hence F becomes ultimately empty.
In practice, to tackle efficiency bottlenecks avoiding the exploration of unlikely trajectories, we introduce a set of neglected states N. Furthermore, to ensure that the algorithm terminates
whenever the reachable attractors are all stable states, we consider the set of attractors A. In the course of the exploration the firefront F is reduced as explained below:
• if the probability associated with a state v ∈ F drops below a certain value α, then v is moved from F to N (set of neglected states). As a consequence, the immediate successors of v will not be
explored at this time. If a state v ∈ N is visited again as being the successor of a state in F, its probability is properly updated (we will say that it accumulates more probability), and if this
probability exceeds α, then v is moved from N back to F (see Figure 1, step 7);
• if a state in F has no successors, it is moved to A (set of stable states); if it is already in A, its probability increases according to this new trajectory.
FIGURE 1
Figure 1. Illustration of FIREFRONT operation, with $\alpha =\frac{1}{16}$: (1) The exploration starts from initial state v[1] in F associated with probability 1, sets A and N are empty; (2)
successors replace v[1] in F, associated with their probabilities; (3–4) states in F are replaced by their successors, but the stable state v[7] goes in A; (4) v[3], v[4], v[6] stay in F with updated
probabilities; (5) probability of v[8] in A increases as it is visited again; (6) v[5] goes to N as its probability is lower than α; (7) v[5] is removed from N and put back in F as its probability
increased when visited again from v[1]. Transitions explored in the current iteration are in blue, their sources being labeled with their probabilities. Red nodes are in A, and gray nodes are in N.
The exploration will halt when F is empty or the maximum number of iterations is reached.
At each step, the sum of the probabilities of the states in F, N, and A is 1.
Unlike forest fires, which do not revisit burnt areas, the algorithm will, in general, revisit the same state in the presence of a cycle. This invalidates our colorful metaphor unless imagining
uncannily rapid forest regeneration. The presence of cycles thus poses some difficulties because the algorithm would never terminate. To address this issue, FIREFRONT detects periodicities of the
ensemble of states entering and exiting F (i.e., states with a sustained oscillating probability); three sequential occurrences of exactly the same set F are assumed to be sufficient evidence that
the simulation is locked within a complex attractor. In this situation, all the states found in F between the second and third occurrences are used to compose the complex attractor. To do so
efficiently, FIREFRONT uses a reversible hash-function. This heuristic thus enables the identification of complex attractors from oscillating behaviors throughout expansions. Nevertheless, since
FIREFRONT progression can still become locked in large and complex cycles for a lengthy number of expansions, the user may specify a maximum depth (number of expansions) to guarantee its termination
in useful time.
When available, the algorithm can be provided with a description of the complex attractors, equipping FIREFRONT with a function called oracle that indicates whether a state belongs to a listed
complex attractor. In this case, FIREFRONT halts the exploration whenever it reaches a state recognized by the oracle, and treats all members of the corresponding attractor as a single element of A
collectively accumulating incoming probabilities.
FIREFRONT terminates when: 1) the total probability in F drops to zero or below some predefined threshold β, or 2) the predefined maximum depth is reached. Given the initial state v[0], the
probability associated to each attractor a ∈ A is a lower bound of ℙ[v[0]](X[∞] = a). An upper bound is obtained by adding to this value β and the sum of probabilities accumulated in N. An outline of
FIREFRONT is presented in Algorithm 1, and Figure 1 provides an illustration on a toy example.
2.3. Avatar
AVATAR is proposed as an alternate algorithm to identify model attractors and quantify their reachability, considering specific initial state(s) or the whole state space. AVATAR is an adaptation of
the classical Monte Carlo simulations that aims at efficiently coping with (transient and terminal) SCCs.
2.3.1. The Algorithm
When exhaustive enumeration is not feasible, Monte Carlo simulation is classically used to estimate the likelihood of an outcome. Concerning attractor reachability in logical models, this means
following random paths along the asynchronous dynamics (the STG). Each simulation halts when either a stable state (with no successor) or the maximal depth are reached. Performing a large number of
simulations allows estimating reachability probabilities of stable states. The simulation does not record past states, and thus memory requirements are minimal. However, a major drawback is that
cycles are not detected. Consequently, without restricting the number of steps, the simulation does not terminate when a trajectory enters a terminal SCC. Moreover, in the presence of a transient
cycle, it may re-visit the same states an unbounded number of times before exiting. That is why we propose an appropriate modification of this approach.
AVATAR is outlined in Algorithm 2 (further description of AVATAR and its ancillary procedures is provided in the Supplementary Material S1). It avoids repeatedly visiting states by detecting that a
previously visited state is reached, indicating the presence of a cycle in the dynamics. Having detected a cycle, the algorithm modifies the STG in order to dismantle the cycle, linking its states to
its exiting states (i.e., targets of transitions leaving the cycle). It is important, however, to associate these new transitions with appropriate probabilities; the probability of a transition from
any cycle state to a given exit must match the corresponding asymptotic probability, considering the infinitely many possible trajectories. The STG is thus rewired so as to replace all the
transitions between the cycle states by transitions from each cycle state toward each cycle exit (see Figure 2). Each rewiring creates a new so-called incarnation of the dynamics. Such an
incarnation—Sanskrit name of our algorithm—is a graph with the same states as the original STG, but with different transition probabilities. This rewiring relies on theoretical foundations that are
presented in section 2.3.2. Upon rewiring, the simulation proceeds from the current state.
FIGURE 2
Figure 2. Illustration of AVATAR operation: The transition matrix π is partitioned into the sub-matrices q for transitions between states v[1], …v[4] of the cycle to be discovered (Top Left), and r
for transitions leaving the cycle (Top Right). Exploration starts at v[1] (denoted in blue as well as its leaving transitions with their probabilities), v[2] is selected for the second iteration, and
v[1] is indicated as being already visited in red. Exploration proceeds until revisiting v[1] at the 5^th step. Having identified a cycle, the rewiring procedure is launched, removing transitions of
the cycle (dotted red) and adding transitions toward exits (green). Probabilities are computed, resulting in a new matrix π^1, with ${q}_{ij}^{1}=0$ and ${r}_{ij}^{1}={\left({\left(Id-q\right)}^{-1}r
\right)}_{ij},i=1,\dots 4$. From v[1], an exit of the cycle is chosen according to these probabilities (step 6).
Because it is generally more efficient to rewire a large transient than to iteratively rewire portions of it, upon encountering a cycle, AVATAR performs an extension step controlled by a parameter τ
that is a modified Tarjan's algorithm for SCC identification (Tarjan, 1972)—trajectories exploration is performed up to a depth of τ away from states of the original cycle. The subsequent rewiring is
then performed over the (potentially) extended cycle. In the course of a single simulation, the value of τ is doubled within each attempt to enlarge a cycle in order to speed up the identification of
large transients.
When a newly visited state v has no successor, it is a stable state. But if v was part of a cycle in a previous incarnation, v belongs to a complex attractor, which is computed as the equivalence
class containing all the cycles that included v in past incarnations.
As for FIREFRONT, the algorithm can be complemented with the previous knowledge of the attractors (oracles). This obviously improves AVATAR's performance. Moreover, AVATAR not only evaluates the
probability of the attractors being reached from an initial condition, it can also be used to assess the probability distribution of the attractors for the whole state space (i.e., considering all
possible initial states). AVATAR is also able to use the knowledge regarding the identified transient SCCs within one iteration to alleviate the cost of identifying and possibly rewiring large cycles
in upcoming iterations, thus boosting the overall efficiency of the simulation. The knowledge regarding the sizes of the transient SCCs and average depths of the found attractors can provide valuable
insights into the model dynamics.
2.3.2. Theoretical Foundations of Avatar Rewiring
The rewiring performed by AVATAR to force the simulation exiting a cycle modifies the probabilities associated to transitions. This is properly done so as to ensure a correct evaluation of the
reachability probabilities performing a (large) number of random walks over our Markov chain X. This procedure amounts to modify the chain. It is formalized below and illustrated in Figure 2.
Suppose that X[t] = c[1], and X[t+k] = c[1] for t and k two positive integers. The walk has thus traveled along the cycle C = (c[1], c[2], …c[k]) (with c[i] ∈ S and (c[i], c[i+1]) ∈ T, ∀i = 1, …k).
Note that this cycle may contain “direct shortcuts”: (c[i], c[j]) ∈ T, j ≠ i+1 (mod k). We denote by B the set of states directly reachable from C: B = {v ∈ S\C, (c[i], v) ∈ T, c[i] ∈ C}. Let q be
the k × k sub-matrix of π, for states c[1], …c[k], and r the k×|B| sub-matrix of π, defining transitions from C to B. To force the walk leaving the cycle (rather than being trapped there for a long
time), the transition matrix is modified as follows:
• remove the transitions between the states of C; the sub-matrix q is replaced by q^1 = 0, the null matrix;
• add an arc from each state of C to each state of B; the sub-matrix r is replaced by ${r}^{1}\stackrel{\Delta }{=}\sum _{j=0}^{\infty }{q}^{j}r$. By Equation (1), section 2.1.2, $\forall {c}_{i}\in
C,\forall v\in B,{r}^{1}\left({c}_{i},v\right)=\left[{\left(Id-q\right)}^{-1}r\right]\left({c}_{i},v\right).$
Y denotes this new chain. Property 1 asserts that, starting from any transient state u, X, and Y have the same asymptotical behaviors.
Property 1. $\forall u\in {T}$, $\forall a\in {A}$, ℙ[u](Y[∞] = a) = ℙ[u](X[∞] = a).
Proof. Transition matrices of X and Y are the same except around the states of the cycle C; they behave differently only when traveling along C: from c[i], entry state of C, X runs along C for l
steps (l ≥ 0), leaving C through a state v ∈ B with probability ${q}^{l}r\left({c}_{i},v\right)$, whereas Y would go directly from c[i] to v, with probability ${r}^{1}\left({c}_{i},v\right)$. Hence,
for all $u\in {T}$, $a\in {A}$ and j ≥ 0, we have ℙ[u](Y[j] = a) ≥ ℙ[u](X[j] = a) and thus,
All the terms being positive, the Property is proved. Therefore, the rewiring does not asymptotically affect the output of the simulation.
Despite the inherent simplicity and time efficiency of the rewiring step, its dependency on matrix inversions can lead to a memory bottleneck for very large cycles. As such, the current
implementation of AVATAR uses a ceiling size for a cycle to be rewired. When AVATAR finds a cycle, it still attempts to extend it as far as possible. If the extended cycle has some exits, it needs to
be rewired. However, if the extended cycle has more states than the specified ceiling, only a sub-cycle (with as much states as allowed) of the detected cycle is rewired. Furthermore, the user can
also choose an approximate strategy for rewiring that still guarantees the selection of exit states when entering a cycle without the need to perform an exact estimation of their likelihood. This is
done by assigning uniform probabilities from the states of a cycle to its exits. Although this strategy is not prone to memory bottlenecks, its approximate nature can lead to biases on the computed
reachability probabilities.
3. Implementation
Both FIREFRONT and AVATAR are implemented in the context of GINSIM, which supports the definition and analysis of logical models (Chaouiya et al., 2012; Naldi et al., 2018). Figure 3 provides a
snapshot of the desktop GUI, showing the selection of the algorithm, specification of model modifications (perturbation or reduction), initial conditions, and algorithm parameters. MONTECARLO
simulations are also available, as well as a modified version of AVATAR with the approximate strategy described above. User documentation of Firefront and Avatar is provided in the Supplementary
Material S2.
FIGURE 3
The implementations of FIREFRONT and AVATAR rely on adequate data structures—states are easily indexable through meaningful and compact hash keys, and sets of states are implemented as a map of
states for highly efficient indexations, additions and removals. Our implementation of FIREFRONT halts the STG exploration after a predefined number of expansions (10^3 by default). AVATAR
implementation includes a heuristic optimization controlled by optional parameters whose default values were found to be appropriate for the tested models. This optimization considers tradeoffs
between costly rewirings and simulations freely proceeding along cycles, as well as between memory cost of keeping state transitions after rewiring and not profiting from rewirings in previous
simulations. AVATAR further supports sampling over (portions of) the state space. In this case, iterations within a simulation start from states randomly selected over the unconstrained model
Both algorithms provide textual and visual displays of the results: attractors and their reachability probabilities, maximal size of encountered transient SCCs, and plots of the evolution of the set
contents for FIREFRONT and of the probability estimates for AVATAR (see section 4).
4. Results
To validate the proposed algorithms, we considered a number of case studies including randomly generated, synthetic and published biological models. All are briefly described below. We analyzed how
FIREFRONT and AVATAR perform on these case studies and compared, when possible, to outcomes produced by BOOLNET (Müssel et al., 2010) and MONTECARLO simulations. BOOLNET is an R package not only able
to generate random Boolean models, but also to identify attractors and to perform Markov chain simulations. We further compared AVATAR with MABOSS, a C++ software implementing a Monte Carlo kinetic
algorithm to produce time trajectories of Boolean models (Stoll et al., 2012), and with the probabilistic model checker PRISM (Kwiatkowska et al., 2011, 2017). The experiments were run using an Intel
(R) i7-7500U CPU @ 2.7GHz and 8GB of RAM.
4.1. Case Studies Description
Two sets of synthetic models were generated. First, we used BOOLNET (Müssel et al., 2010) to define random models with 10 to 15 components, each with 2 regulators and logical rules randomly selected
(uniform distribution)^1. From the resulting set of random models, three models were selected for exhibiting multi-stability (Table 1). Additionally, we constructed a “synthetic” model exhibiting a
large complex attractor and a few transient cycles. To further challenge our algorithms, we modified this last model, adding one component in such a way that the complex attractor turned into a
transient cycle with very few transitions leaving toward a stable state (see synthetic models 1 and 2 in Table 1).
TABLE 1
Table 1. Characteristics of the models used as case studies to challenge Firefront and Avatar: type of variables (Boolean vs. multi-valued), number of input components (these remain constant) and
internal components, number and type of attractors with number of states in the case of complex attractors, size of the state space with the total number of model states.
Our case studies also include published biological models. First, a Boolean model of the mammalian cell cycle control (Fauré et al., 2006), which has 10 components and exhibits one stable state
(quiescent state) and one complex attractor (cell cycle progression). These attractors arise in (two) disconnected regions of the state space, controlled by the value of the sole input component
(CycD, which stands for the presence of growth factors).
Second, Sanchez et al.'s multi-valued model of the segment polarity module—involved in early segmentation of the Drosophila embryo—defines an intra-cellular regulatory network. Instances of this
network are connected through inter-cellular signaling (Sánchez et al., 2008). Here, we consider three cases: 1) the intra-cellular network (one cell), 2) the composition of two instances (i.e., two
adjacent cells), and 3) the composition of four instances. Initial conditions are specified by the action of the pair-rule module (Wg-expressing cell for the single cell model) that operates earlier
in development (see Sánchez et al., 2008 for details).
Third, we consider the interaction network of genes frequently altered in bladder cancer as proposed in Remy et al. (2015). This model includes 4 input components leading to different responses
(EGFR, FGFR3 stimuli, Growth inhibitor, DNA damage), 23 internal components and 3 output components representing cellular responses or phenotypes (Proliferation, Apoptosis, Growth Arrest). Depending
on the input values, the model displays multistability or not, with a combination of stable states and complex attractors. This case study further demonstrates the capacity of AVATAR in assessing
large complex attractors, quantifying attractor reachability, and revealing transient dynamics.
Finally, using a model of T helper cells differentiation (Naldi et al., 2010) and a model of cell fate decision in response to death receptor engagement (Calzone et al., 2010), we provide additional
illustrations in the Supplementary Materials S4, S5.
Supplementary Material S6 provides an archive containing all the models in the GINsim format (zginml).
4.2. Firefront and Avatar in Action
Results are summarized in Table 2. Generally, FIREFRONT and AVATAR show efficiency gains against alternatives and are further able to surpass the drawbacks of BOOLNET (applicable to Boolean models
only) and MONTECARLO (unable to identify transient and terminal cycles).
TABLE 2
Considering random models 1 to 3, FIREFRONT and AVATAR are able to efficiently find the stable states and complex attractors of these models and to estimate their reachability probabilities. BOOLNET
is slower for these random models. MONTECARLO is not only less efficient but is also unable to detect the complex attractors. For instance, in random model 2, less than 8% of the simulations
For synthetic model 1, FIREFRONT takes over a minute to distribute the probability out of the large transient cycles. For synthetic model 2, FIREFRONT could not distribute more than 5% of the
probability out of the transient SCC (purposely constructed with 8 196 states and a dozen exits). The presence of multiple large transient SCCs causes FIREFRONT to accumulate a large number of states
in F, leading to some time overhead and difficulty to distribute the probabilities. States of transient SCCs are revisited until the probabilities of their incoming transitions drop below α, which
can take long. As such, the computational performance of FIREFRONT is greatly influenced by the structure of the STG (e.g., state outdegrees or sizes of transient SCCs). The Supplementary Material S3
provides illustrations of the structures of the dynamics. In contrast, AVATAR is able to adequately identify and exit transient SCCs. For this reason, AVATAR was able to escape the transient SCC
planted in synthetic model 2 thanks to its rewiring procedure, and could identify and quantify the attractors for both synthetic models. BOOLNET completed synthetic models 1 and 2, after 7 and 5
days, respectively, which highlights the need for the proposed methods to face efficiency bottlenecks for models with large and complex SCCs.
Starting in the region of the state space where the mammalian cell cycle model has a (unique) complex attractor (i.e., with the presence of CycD), AVATAR, FIREFRONT, and BOOLNET could assess its
reachability from the quiescent state; when sampling the state space, both AVATAR and BOOLNET could correctly quantify the reachability of the two attractors (FIREFRONT was not applicable as it
requires a starting initial state). Expectedly, MONTECARLO could not retrieve the complex attractor, being unable to exit it in all runs.
With regards to the segment polarity model, FIREFRONT was efficient for all cases (single, two and four cells), although its ability to distribute all the probability decreases with the increase of
model size. Since it did not reach the allowed maximum number of iterations, its stopping condition was that the total probability in F dropped below β, with all the residual probability in the
neglected set, which in the end contained approximately 140, 52 000, and 210 000 states for the models of single, two and four cells, respectively. This would suggest that α was not small enough with
respect to the number of concurrent trajectories toward the attractors (see Supplementary Material S4 for illustration). Although AVATAR's performance is constrained by the need to assess the complex
structure of the two and four cells' models (for instance the largest encountered transient SCC for sp2 has over a million states), it is adequately able to find the attractors, even those with a low
reachability probability. Given the fact that the attractors of these models are stable states, MONTECARLO was able to retrieve them, in particular those attractors reachable without the need to
visit large transient SCCs.
Figure 4 complements these results by showing, for two of our case studies: with FIREFRONT, the evolution of the cardinals of the sets F, N, and A (and their corresponding probabilities), and with
AVATAR, the convergence of the estimated reachability probabilities of the attractors.
FIGURE 4
Figure 4. Plots computed by FIREFRONT and AVATAR throughout simulations for the random3 (Top) and the sp1 (Bottom) models (see Tables 1, 2). Left plots show the numbers of states to be expanded (in F
), of neglected states (in N), and of attractors (in A). Middle plots show the cumulative probabilities of the 3 sets. Right plots show the convergence of the reachability probability of each
The application of AVATAR over the bladder tumourigenesis model—with results illustrated in Table 3—enabled the quantification of attractor reachability over the whole state space, for 8 combinations
of input values. Stable states were gathered in 3 classes, corresponding to the cell phenotypes Proliferation, Apoptosis and Growth Arrest, which are indicated by the values of the 3 output
components of the model. The model displays several complex attractors. The reachability quantification of the attractors is relevant in the cases of multi-stability, i.e., when several attractors
arise for the same input condition (compare with Table S2 in Remy et al., 2015). AVATAR discloses structural properties of the model dynamics such as the sizes of encountered transient SCCs and mean
depths of the attractors (not shown).
TABLE 3
We also performed the analysis of model perturbations to illustrate the biological relevance of assessing attractor probabilities. To this end, we considered the case of activating mutations of
fibroblast growth factor receptor 3 (FGFR3) and of the oncogene PI3K, one of the co-occurrent genetic perturbations observed in bladder tumors (see Remy et al., 2015). Figure 5 illustrates how
probabilities of the attractors are modified under those perturbations. It supports the conclusions drawn in Remy et al. (2015): mutating FGFR3 in PIK3-mutated tumors seems to be advantageous (to
increase the probability of Proliferation); a third mutation is required for uncontrolled proliferation (i.e., the loss of all the phenotypes but Proliferation).
FIGURE 5
Figure 5. Probabilities of the phenotypes for the bladder tumourigenesis model in the wild type and mutant contexts: probabilities for the double mutant FGRF3 overexpression (FGFR3 E1) and PI3K
overexpression (PI3K E1) suggest a slight advantage in mutating FGFR3 in a PI3K-mutated context (by increasing the probability of Proliferation); a third mutation of the tumor suppressor CDKN2A
(coding for p16INKa) leads to the sole phenotype Proliferation (see Remy et al., 2015).
For completeness, we also compared AVATAR with MABOSS and PRISM. For this, we used GINSIM export facilities of logical models to MABOSS and PRISM formats.
MABOSS is a related command-line tool that generalizes Boolean models by defining stochastic rates associated with component updates (Stoll et al., 2012). MABOSS primary goal is to compute temporal
evolutions of state probability distributions and to estimate stationary distributions. To this end, it relies on the Gillespie algorithm. MABOSS is thus well suited to get a quantitative view of
temporal evolutions in the form of stochastic trajectories (see e.g., Abou-Jaoudé et al., 2016). When running MABOSS on our case studies, it appeared that the tool was able to provide the
reachability probabilities of the stable states of the random models 1 to 3. However, the presence of large transient SCCs or of complex attractors hinders the evaluation of such a measure for the
synthetic models and for the cell cycle model. Table 3 includes the results obtained with MABOSS for the analysis of the bladder tumourigenesis model. Reachability probabilities obtained for the
stable states are close to those provided by AVATAR. While MABOSS is clearly faster than AVATAR, it is unable to assess complex attractors being thus applicable only when attractors are known to be
stable states.
PRISM is a model checker that supports probabilistic reachability queries (Kwiatkowska et al., 2011, 2017). To compare AVATAR and PRISM, we repeated the analysis of the segment polarity model with 2
cells. Results are provided in Table 4. Notably, PRISM is extremely efficient to evaluate the number of reachable states, a feature not provided by AVATAR. PRISM performs an exhaustive exploration to
evaluate exact reachability probabilities. However, as demonstrated with AVATAR, a restricted sample of the dynamics may provide good enough probability estimates in a much shorter time. This feature
is particularly useful for larger models. Indeed, for the sp4 model, PRISM ran out of memory and was thus unable to evaluate the number of reachable states and conclude the analysis (even when
increasing the amount of available memory to CUDD to 8Gb).
TABLE 4
5. Discussion
For models of regulatory networks controlling cell fates, it is of a real interest to identify the model attractors, as well as quantify their reachability over the whole state space or from specific
initial conditions. In particular, the impact of model perturbations (e.g., corresponding to observed mutations) on attractors and their basins of attraction has been used to better understand the
fates of tumor cells (Huang et al., 2009; Kim et al., 2017; Shah et al., 2018). Most studies rely on Boolean models under a synchronous updating scheme. However, while stable states are identical
whatever the updating scheme, it is not the case for the complex attractors, neither for the basins of all attractors. Because the synchronous scheme stems from the assumption that delays associated
with component updates are equal, asynchronous updates have been considered more realistic (Thomas, 1991; Abou-Jaoudé et al., 2016). In the context of non-deterministic asynchronous dynamics, it is
then relevant to assess the likelihood to reach an attractor and how model perturbations modifies this reachability likelihood. For example, this approach has been used to assess patterns of genetic
alterations in bladder tumourigenesis (Remy et al., 2015), or yet to highlight the synergetic roles of Notch gain-of-function and p53 loss-of-function in promoting metastasis (Cohen et al., 2015).
Attractor identification could be achieved by analysing the State Transition Graph (STG) kept in memory but, due to combinatorial explosion, this is impractical for large models. In any case, we are
still left with the problem of quantifying attractor reachability in asynchronous dynamics. As an attempt to surpass efficiency bottlenecks and quantification biases of existing methods, we have
delineated two novel strategies. FIREFRONT performs a memoryless breath-first exploration of the STG, avoiding any further exploration of states which fall bellow a given threshold α. AVATAR performs
a modified version of the Monte Carlo algorithm, avoiding the exploration of states previously visited by rewiring and appropriately associating new probabilities with state transitions. To
adequately choose the algorithm and optimal values of associated parameters, information about the structure of the dynamics would be needed, which is generally unachievable. Broadly, the breadth of
the explored STG and the structure of transient Strongly Connected Components (SCCs) clearly impact FIREFRONT's performances. AVATAR's performances are influenced by the degree of connectivity of the
SCCs. Ideally, AVATAR should avoid to rewire SCCs from which it can easily exit (low connectivity or high exit ratio). On the other hand, it should rewire SCCs from which it is hard to escape. It is
also much more efficient to rewire a whole SCC than to iteratively rewire portions of it. While sizes and structures of SCCs are not known a priori, AVATAR incorporates heuristics that evolve running
parameters to the information collected in the course of the simulation.
Results from synthetic and real biological models reveal the ability of FIREFRONT and AVATAR to efficiently assess attractor reachability. This type of analysis will permit further biological
insights into the dynamics of regulatory and signalling networks. For example, as mentioned above, how model perturbations modify the probability to reach an attractor can reveal the role of single
or combined mutations in disease progression. Usage of both algorithms is facilitated through their implementation in GINsim, which provides a convenient graphical user interface.
As future work, the consideration of non-uniform transition probabilities could be easily handled. In particular, when priority classes can be defined by classifying component updates into e.g., slow
and fast processes (Fauré et al., 2006), some trajectories are discarded thus modifying the structure of the STG, and therefore the reachability properties. Furthermore, confronting asymptotic model
dynamics against experimental time series could provide the ground for model validation.
Author Contributions
CC, PM, JC, and ER designed the research. CC supervised the work, PM supervised the computational implementations, and ER focused on the theoretical foundations. NM specified the algorithms, and
implemented the first prototypes. RH revised the algorithms to improve performances, and worked on the GINSIM implementation. NM, RH, ER, PM, and CC wrote the paper. All authors reviewed the content
of the paper and agreed to endorse it.
This work was supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal), grants PTDC/EIA-CCO/099229/2008, UID/CEC/50021/2013, IF/01333/2013, and PTDC/EEI-CTP/2914/2014.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
CC would like to thank the Fundação Calouste Gulbenkian for its continuous support.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphys.2018.01161/full#supplementary-material
GINSIM 3.0 is freely available at http://ginsim.org. The accompanying user documentation, algorithmic details and supplementary results will be made publicly available upon publication.
1. ^This process is automated in BOOLNETR2GINSIM, a small program available at https://github.com/ptgm/BoolNetR2GINsim that accepts user-defined parameters, calls BOOLNET and writes the resulting
model to a GINML file (the GINSIM format).
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Citation: Mendes ND, Henriques R, Remy E, Carneiro J, Monteiro PT and Chaouiya C (2018) Estimating Attractor Reachability in Asynchronous Logical Models. Front. Physiol. 9:1161. doi: 10.3389/
Received: 06 April 2018; Accepted: 02 August 2018;
Published: 07 September 2018.
Edited by:
Xiaogang Wu
, University of Nevada, Las Vegas, United States
Copyright © 2018 Mendes, Henriques, Remy, Carneiro, Monteiro and Chaouiya. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in
accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Pedro T. Monteiro, Pedro.Tiago.Monteiro@tecnico.ulisboa.pt
Claudine Chaouiya, chaouiya@igc.gulbenkian.pt
^†These authors have contributed equally to this work | {"url":"https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2018.01161/full","timestamp":"2024-11-06T05:14:38Z","content_type":"text/html","content_length":"623090","record_id":"<urn:uuid:fa8c5270-1fbf-47dc-903e-9c50a4b5250e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00401.warc.gz"} |
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The 1)2))is used to determine whether there is a trend in proportion for particular categories of an analysed variables (features). It is based on the data gathered in the contingency tables of 2
features. The first feature has the possible The contingency table of
This statistic asymptotically (for large expected frequencies) has the Chi-square distribution with 1 degree of freedom.
The p-value, designated on the basis of the test statistic, is compared with the significance level
The settings window with the Chi-square test for trend can be opened in Statistics menu → NonParametric tests → Chi-square, Fisher, OR/RR → Chi-square for trend.
We examine whether cigarette smoking is related to the education of residents of a village. A sample of 122 people was drawn. The data were recorded in a file. }
We assume that the relationship can be of two types i.e. the more educated people, the more often they smoke or the more educated people, the less often they smoke. Thus, we are looking for an
increasing or decreasing trend.
Before proceeding with the analysis, we need to prepare the data, i.e., we need to indicate the order in which the education categories should appear. To do this, from the properties of the Education
variable, we select Codes/Labels/Format… and assign the order by specifying consecutive natural numbers. We also assign labels.
As the graph shows, the more educated people are, the less often they smoke. However, the result obtained by people with junior high school education deviates from this trend. Since there are only
two people with lower secondary school education, it did not have much influence on the trend. Due to the very small size of this group, it was decided to repeat the analysis for the combined primary
and lower secondary education categories.
A small value was again obtained p=0.0078 and confirmation of a statistically significant trend.
Because of the decrease in people watching some particular soap opera there was carried out an opinion survey. 100 persons were asked, who has recently started watching this soap opera, and 300
persons were asked, who has watched it regularly from the beginning. They were asked about the level of preoccupation with the character's life. The results are written down in the table below:
The new viewers consist of 25\% of all the analysed viewers. This proportion is not the same for each level of commitment, but looks like this:
Cochran W.G. (1954), Some methods for strengthening the common chi-squared tests. Biometrics. 10 (4): 417–451
Armitage P. (1955), Tests for Linear Trends in Proportions and Frequencies. Biometrics. 11 (3): 375–386 | {"url":"http://manuals.pqstat.pl/en:statpqpl:porown2grpl:nparpl:trendpl","timestamp":"2024-11-06T23:39:46Z","content_type":"text/html","content_length":"61484","record_id":"<urn:uuid:82eeabec-cf8b-4a5a-b2ea-b4a7fbe9a233>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00498.warc.gz"} |
A Multi-Dimension Spatial Method for Topology Awareness and Multipath Generating
School of Computer Science and Engineering, Northeastern University, ShenYang 110169, China
Authors to whom correspondence should be addressed.
Submission received: 15 May 2019 / Revised: 7 June 2019 / Accepted: 24 June 2019 / Published: 3 July 2019
Multipath diversity significantly impacts multipath transmission quality. Enough multipath diversity would minimize the negative influence brought by an individual path, thus improving tolerance
capability of network congestion and failure. However, multipath diversity is hard to guarantee on overlay networks because of inaccurate awareness of underlay network and multipath generating
methods considering little about underlay diversity. In this paper, we design a multi-dimension spatial method for topology awareness and multipath generating. Analyzing that the complicated underlay
networks with multiple autonomous systems reduce the accuracy of network positioning for topology awareness, we decompose the underlay networks into multiple dimensions, namely intra and inter
autonomous system dimensions. We generate independent views for each autonomous system and merge views by embedding exchange points. Then, we design some spatial mechanisms to evaluate link diversity
and to constrain multipath generating. Based on the multi-dimensional view, multipath generating is conducted in inter and intra autonomous system phases. Experiments demonstrate that the proposed
method improves topology awareness accuracy and guarantees multipath diversity better and the transmission quality is improved.
1. Introduction
Various innovative internet applications are emerging thanks to the network infrastructures with ever growing quality, and, in turn, applications are demanding higher transmission capability to
achieve better quality of experience (QoE). Although underlay networks are usually of good performance (e.g., high bandwidth, short latency), it is still challenging to guarantee end-to-end QoE,
partially due to the “best-effort” layer-3 routing mechanism. According to 3GPP (3rd Generation Partnership Project), the conversational applications [
], e.g., real-time video conversation, demand not only specific end-to-end bidirectional bandwidth and delay, but also smooth procedure. The end-to-end QoE is vulnerable to link abnormality, e.g.,
outage and congestion, which happen randomly in underlay networks and are difficult to overcome using the single path transmission mode [
]. Therefore, the multipath transmission methodology has been coming up as a preventive or remedial measure to strengthen the end-to-end transmission quality.
The distinct advantage of multipath transmission is that it aggregates transmission capabilities of multiple paths. On one hand, it expands the bandwidth by concurrently exploiting redundant links.
On the other hand, it weakens the negative influence on the end-to-end transmission brought by link abnormality from a certain path. We mainly attribute this advantage to the uncorrelation of
multiple paths, and name it multipath diversity. From the perspective of which part of networks multipath transmission enhances, we classify the studies of multipath transmission into two categories,
namely, (i) on the end-host side, (ii) operating the intermediate network, and (i) some improved studies of multipath transmission control protocol (MPTCP) represent the enhancement on the end part [
], due to the fact that it focuses on optimizing the end flow control and schedule by using paths provided by multi-homed hosts. Actually, the multi-homed scenario is not common for most end devices.
(ii) In [
], multipath diversity is improved in the intermediate networks on the internet protocol (IP) layer, while some modification has to be imposed on network infrastructures, e.g., router. Alternatively,
overlay relays are adopted to redirect network traffic to multiple overlay paths as in [
]. In this paper, we mainly consider relevant two issues of generating multiple relay-based paths on overlay networks, namely (1) the topology match between overlay and underlay networks, and (2)
generating multiple paths with high path diversity.
(1) The topology match between overlay and underlay networks is a classical and still challenging issue for designing network architectures and deploying applications on overlay networks due to the
characteristics of traditional TCP/IP protocol stacks. The mismatch problem is likely to result in transmission degradation for end users and low link utilization for bearing networks. We have
analyzed the potentially serious consequences of topology mismatch for transmission in [
] including unacceptable large transmission delay and useless redirected network traffic. An efficient and widely studied methodology to solve the mismatch problem is topology awareness and a
representative method to improve topology awareness for overlay networks is network positioning [
]. Based on the measure results, the overlay nodes are positioned with network coordinates in a coordinate system. While we discover that the underlay networks are complicated thanks to the abundant
deployments of bearing networks from multiple Internet Service Providers (ISPs), the complicated bearing networks have some features of a complex network, and a main feature is that the networks
devices are deployed in a cascade manner [
]. For example, there exist various autonomous systems (AS) and they might cover overlapping geographic areas and be interconnected in several locations. The complicated bearing networks would lead
to positioning inaccuracy, mainly due to the fact that the reference nodes for overlay nodes in different autonomous systems would cause delay deviation when overlay nodes in different autonomous
systems are positioned together. Therefore, in this paper, in order to reduce topology match deviation and improve positioning accuracy, we try to simplify the complicated bearing networks and
decompose the bearing network into multi-dimensions, namely horizontal dimension for intra-AS and vertical dimension for inter-AS. Firstly, we independently generate the coordinate-based topology
view for overlay nodes in each autonomous system, i.e., the horizontal dimension. Then, we merge these views into a universal view by embedding exchange exports between autonomous systems, i.e., the
vertical dimension.
(2) Multipath diversity impacts the multipath transmission quality when capabilities of multiple concurrent paths with little correlation are aggregated. High multipath diversity helps improve the
concurrent utilization of network links and minimizes the negative influence brought by an individual path, so that tolerance capability of network congestion and failure are improved [
]. In practice, it is more challenging to guarantee enough multipath diversity on overlay networks than in underlay bearing networks, mainly due to that there exists a little mechanism for overlay
nodes in traditional networks to become aware of the precise underlay network topology. In this paper, we take advantages of abundant links existing in the complicated bearing networks and propose a
multipath generating method. The motivation is as follows: In real-world underlay networks, various Internet Service Providers (ISPs) have deployed their own infrastructures of different scopes, so
that there might exist redundant links between a pair of origin-destination hosts. An ideally typical scenario is that two paths between an origin–destination pair separately traverse two different
service providers. Thus, based on the existing underlay link diversity, we hold the opinion that links through different service providers are more likely to be independent, and multiple paths
consisting of independent links would gain high multipath diversity. On the basis of the multi-dimension topology match view in (1) above, we also design the multipath generating method from two
dimensions, namely inter-AS (among autonomous systems) and intra-AS (among relay nodes in an autonomous system). Firstly, we use autonomous systems as the basic unit and try to redirect traffic to
paths belonging to totally different autonomous systems; this is named inter-AS. While practically this ideal scenario is not always feasible, due to the fact that some geographic areas are not
covered by multiple autonomous systems, in some areas, paths have to pass through the same autonomous systems. Then, we have to minimize (maximize) the multipath correlation (multipath diversity) in
one autonomous system. We design some spatial mechanisms to improve multipath diversity, e.g., we use the cosine similarity (“Deviation Angle”) to evaluate link correlation, and use “Transmission
Surface” to constrain available nodes.
After analyzing the above issues, we make the following contributions in this paper:
(1) In order to improve network positioning accuracy for topology awareness, we decompose the bearing networks into a multi-dimension structure, namely intra-AS and inter-AS dimensions and design the
multi-dimension topology view.
(2) We design some spatial mechanisms to evaluate the correlation of first-hop links , and to constrain multipath generating.
(3) We design an inter-AS and intra-AS multipath generating method to improve multipath diversity by taking advantages of underlay diversity.
(4) We evaluate the multi-dimension spatial method on the OMNet++ simulation platform considering topologies generated with BRITE and brought from practical real-world networks.
The remaining parts of this paper are arranged as follows. In
Section 2
, we describe the background and state the problems. A multi-dimensional structure is designed to improve the accuracy of network positioning and topology awareness in
Section 3
Section 4
introduces the multi-dimension spatial method of multipath generating. The performance of the proposed method evaluated in
Section 5
Section 6
introduces related work and
Section 7
concludes this paper.
2. Background and Problem Statement
2.1. Multipath Transmission on Overlay Networks and SDSON
Multipath transmission on overlay networks has the advantages of flexible and efficient development and deployment in bearing networks compared with other multipath implementations on other network
layers, partially thanks to its no modification on network devices in bearing networks and efficient redirection on network traffic. We achieve multipath transmission on application-layer overlay
networks based on relaying network traffic. We deploy application-layer relay nodes in bearing networks to redirect networks traffic into multiple concurrent end-to-end paths. In order to achieve
optimal and intelligent control and management on relay nodes, we use the software-defined networking (SDN) paradigm to orchestrate relay nodes and conduct forwarding control and we name the
SDN-based multipath transmission framework Software-Defined Service Overlay Network (SDSON) [
The structure of SDSON is shown in
Figure 1
, which is built on the application-layer overlay networks. The bottom plane is the bearing network layer, consisting of network forwarding devices; this underlay network provides the basic
transmission for overlay networks. The middle plane is the forwarding plane of SDSON, and there exist two network entities of SDSON, namely Relay Server (RS) and User Agent (UA); RS works as the
forwarding nodes, i.e., the relay nodes that providing relayed forwarding services, and UA provides the end host with the access to SDSON. The top plane is the control plane of SDSON, where Relay
Controller (RC) conducts global control and management functions. A scenario of end-to-end multipath transmission is shown in
Figure 1
, and three multiple paths are generated on forwarding plane, and corresponding paths in underlay networks. Like other multipath transmission solutions on application-layer overlay networks, in
SDSON, there exist two challenging problems to solve, namely, (i) the topology match between overlay networks and underlay networks, and (ii) the enhancement on multipath transmission performance.
2.2. Topology Awareness
Topology awareness helps improve the topology match performance between overlay networks and bearing networks, and an effective topology awareness methodology is network positioning. However, due to
the complexity of existing bearing networks, the accuracy of network positioning is significantly impacted, thus the accuracy of topology awareness is degraded. We use an example to discuss the
potential accuracy degradation. In order to simplify the network scenario, we use autonomous systems to represent the bearing networks. A simple but representative bearing network is depicted in
Figure 2
. For simplicity, the underlay devices are not revealed, and overlay nodes are deployed in AS1 and AS2, which have an exchange point (Er) in geographic location L1.
We use the following example to discuss the positioning inaccuracy. Two overlay nodes (O1 and O2) are deployed in AS1 and one node (O3) in AS2. Overlay nodes O1, O2 and O3 are all in location L2, and
the delays between three overlay nodes and Er are D(O1,Er), D(O2,Er) and D(O3,Er). Due to the fact that the packet transmitted from O3 to O1 has to pass through Er, the delay between O3 and O1 is
estimated as D(O3,O1)=D(O3,Er)+D(O1,Er). The delay between O2 and O1 is small because they are in the same autonomous systems and close. Thus, the delay between O2 and O1 is far less than that
between O3 and O1, that is, D(O2,O1)«D(O3,O1). However, when O1 is chosen as the reference node in a network positioning method, the position of O2 would be close to O1 while O3 is distant from O1.
Then, the delay deviation between O2 and O3 comes up. Actually, they are in the same location and they should be close. The above delay deviation severely impacts the accuracy of topology awareness
when network positioning methods are adopted.
2.3. Multipath Diversity
Multipath diversity means the transmission differences of multiple paths, and higher multipath diversity indicates that there exist fewer joint nodes or links in underlay networks between paths.
Multipath diversity is an important issue that significantly influences the performance of multipath transmission—for instance, paths obtaining high multipath diversity could efficiently aggregate
different resources of multiple paths, and one individual path would bring less impact to overall paths when multipath diversity is higher. Multipath diversity is determined by many factors in
bearing networks and we should explore and exploit the diversity of transmission resources in existing underlay bearing networks and enhance multipath diversity.
2.3.1. Diversity of Underlay Network
Generally, end-to-end packets are forwarded via one or a set of autonomous systems. i.e., packets may pass via border devices, named exchange points, between autonomous systems. We decompose underlay
networks in two dimensions—intra-AS and inter-AS—as shown in
Figure 3
a; vertical layers signify three autonomous systems, and in each layer locate forwarding nodes (solid circles) and dotted rectangle boxes denote six geographic locations. The aggregated networks are
depicted in
Figure 3
b and the integrated topology in
Figure 3
c. We discuss end-to-end multipath transmission between hosts C1, C2 and C3. For C1 and C2, there exist two types of end-to-end multiple paths, i.e., completely intra-AS paths in AS1, and inter-AS
paths via AS2 or AS3. For C1 and C3, only the second type exists. Notations are listed in
Table 1
2.3.2. Enhancing Multipath Diversity
Our objective is to discover MP(i,j) to guarantee MPD(i,j) as high as possible, i.e., MP(i,j) = argmax MPD(i,j). For simplicity, we combine two paths within the two types mentioned above, and there
exist three combinations, (i) intra-AS with intra-AS, (ii) intra-AS with inter-AS, and (iii) inter-AS with inter-AS.
(i) Suppose $P 1 ∈ T 1$, $P 2 ∈ T 2$, IF $T 1 ≠ T 2$, $P 1$ and $P 2$ are disjoint, ELSE, intra-AS geometric multipath diversity is involved.
(ii) Suppose $P 1 ∈ T 1$, $P 2 ∈ T 2 ∪ T 3$ (P2 via two autonomous systems), IF $T 1 ≠ T 2$ and $T 1 ≠ T 3$, $P 1$ and $P 2$ are disjoint, then we select a proper exchange point between $T 2$ and $T
3$, ELSE, intra-AS geometric multipath diversity is involved.
(iii) Suppose $P 1 ∈ T 1 ∪ T 2$, $P 2 ∈ T 3 ∪ T 4$, (P1 and P2 both via two autonomous systems), IF $T 1 ≠ T 3$, $T 1 ≠ T 4$ and $T 2 ≠ T 3$, $T 2 ≠ T 4$, $P 1$ and $P 2$ are disjoint; then, we
select the SPs between $T 1$ and $T 2$, $T 3$ and $T 4$, ELSE, intra-AS geometric multipath diversity is involved.
3. Constructing Multi-Dimensional Topology View
3.1. Constructing Independent Views
We prefer to use centralized network positioning methods [
] to generate coordinates for RSes in SDSON, e.g., Vivaldi [
], MDS [
] and GNP [
]. We obtain required inputs by measuring [
]. The coordinate of
$R S i$
$C i$
. We use D(i,j), MR(i,j) and MM(i,j) as the distance, measure relation and metric between
$R S i$
$R S j$
3.2. Merging Topology Views
We merge independent views by relations of locations and embedding SPs. The information of SPs is hard to obtain due to privacy and security policy, so we infer SPs from MM(i,j) based on deployment
characteristics, e.g., exchange point for two autonomous systems is usually in the same location, and latency between RSes near the exchange point is relatively small. To reduce search complexity, we
only search MR() between RSes in locations of specific distance. i.e., $0 ≤ N B ( L p , L q ) ≤ D T$, and DT is the location distance threshold. In addition, only top h minimum MM() within each city
are chosen, and LT sets the exchange point latency threshold according to empirical measure, so that exchange point would exist if $M M ( ) ≤ L T$. The embedding procedure is shown in Algorithm 1.
Algorithm 1 Embedding exchange points
MM(N,N), N is the number of RSes
$T W$, W is the number of autonomous systems
DT, LT
$L M ( N , N )$, M is the number of locations (cities)
$N B M ( R S P , R S Q )$, P,Q are the numbers of neighbor RSes.
C(N,N), coordinate of exchange point, center of two RSes
1: while $T i ∈ T W and T J ∈ T W$do
2: while $L c ∈ L M and L d ∈ L M and 0 ≤ N B ( L c , L d ) ≤ D T$do
3: while $R S f ∈ T i and R S f ∈ L c and R S g ∈ T j and R S g ∈ L d$do
4: MM(f,g) ≤ LT and Get top h minimum MM(f,g)
5: $N B k ( ) ← ( R S f , R S g ) , L k ( f , g ) = 1$ (k= c or d or c-d)
6: $C ( f , g ) ← c e n t e r ( C f , C g )$
7: end while
8: end while
9: end while
4. Multipath Generating
Our proposition is that, if two links are more “distant”, they have higher diversity. The “distant” has two explanations: one is geographic “distant”, i.e., links via different locations have little
correlation, and the other is that links in different autonomous systems are independent, although in the same location.
4.1. Spatial Multipath Diversity
We introduce some spatial mechanisms to characterize and improve multipath diversity. The relevant items and descriptions are listed in
Table 2
referring to samples depicted in
Figure 4
a, where black circles denote coordinates of RSes. TD indicates the deterministic transmission trend so as to reduce aimless redirection and facilitate convergence. For instance, two paths 7→5→2→9
and 7→14→15→9 violate the principle of TD 7→9 because hop 5→2 reverses 7→9, and hop 7→14 diverges two much from 7→9. TS constrains intermediate nodes to reduce search complexity and is constructed
between two RSes with several criteria. The TS ABCD is a rectangle symmetric about 7-9. The TS ellipse EF contains the RS whose distance sum to 7 and 9 is smaller than a specific value, e.g.,1.5×D
(7,9). DA implies the direction difference of two links. Supposing that OMN is a triangle, according to the triangle geometric formula,
$2 D O M D O N cos ∠ M O N = D O M 2 + D O N 2 D M N 2$
; then, we get
$D M N = D O M 2 + D O N 2 D M N 2 − 2 D O M D O N cos ∠ M O N 2$
. When
$D O M$
$D O N$
stay constant,
$∠ M O N$
$D M N$
, i.e., within
$0 ∘$
$180 ∘$
, bigger
$∠ M O N$
brings bigger
$D M N$
. We adopt this geometry feature to characterize node diversity, so that, in
Figure 4
a, RS2 and RS12 would be more distant with bigger
$| α | + | β |$
and thus links 7→2 and 7→12 would be with more difference, which we seem as link diversity.
4.2. Exploiting Underlay Diversity
Links from different autonomous systems are usually independent, so we try to establish multiple paths via various autonomous systems. In
Figure 4
b, three paths are to be established between origin and destination ARSes from AS1 and AS5. Assuming that AS4 and AS6 are different and have SPs with AS1 and AS5; then, we hope that packets are
redirected into AS4 and AS6 earlier and out later, thus packets on two paths might pass less common links. Assuming also that AS2 and AS3 together could bridge AS1 and AS5, we try to find these
combined autonomous systems. The multipath generating procedure of a multi-dimensional spatial method is described in Algorithm 2. When MP(i,j) contains candidate multiple paths, we use DA to
evaluate the diversity of first hops and filter links with high diversity, i.e., selecting relatively bigger
$D A ( l k a , l k c )$
$D A ( l k b , l k d )$
, where
$l k a , l k b ∈ P 1 , l k c , l k d ∈ P 2 , P 1 , P 2 ∈ M P ( i , j )$
. When MP(i,j) has less than two paths, then we use above filtering constrains on Origin(i,j) and Dest(i,j) to construct paths. When there still exist no available paths, we just select RSes that
satisfy TS and DA filtering constraints.
Algorithm 2 Generating multiple paths with MDSM
$R S i ∈ L s , R S j ∈ L t , R S i ∈ T u , R S j ∈ T v$
$T N$, N is the number of autonomous systems
$L M$, M is the number of locations (cities)
$N B ( M , M ) , N B M ( P , P )$, P is the number of RSes
MP(i,j), multiple paths from $R S i$ to $R S j$
Origin(i,j), Dest(i,j), candidate RSes near two ARSes
1:while $L k ∈ L M and 0 ≤ N B ( k , s ) ≤ 1$do
2:while $T d ∈ T N and L k ( u , d ) = 1 and L k ∈ T S ( i , j )$do
3:Add(Origin(i,j), d, k)
4:while $L r ∈ L s e t and 0 ≤ N B ( r , t ) ≤ 1$do
5:while $T w ∈ T s e t and L r ( v , w ) = 1 and L r ∈ T S ( i , j )$do
6:Add(Dest(i,j), w, r)
7:if $T w = T d$then
8:Add(MP(i,j), k, r, 1)
9:else if $L b ( w , d ) = 1 and L b ∈ T S ( w , d )$then
10:Add(MP(i,j), k, b, r, 2)
11:end if
12:end while
13:end while
14:end while
15:end while
5. Evaluation and Analysis
We evaluate the performance of the Multi-Dimension Spatial Method (MDSM) in SDSON on OMNeT++ simulation platform. Based on the basic bearing networks provided by INet, we implement the function
modules of SDSON on the application-level overlay networks. The main network entities of SDSON are Relay Server, User Agent and Relay Controller and they are implemented as function modules on the
application layers. For the purpose of extensively evaluating the performance of improving the accuracy of topology awareness and enhancing PD using MDSM in SDSON, we build the underlay bearing
network with two approaches: (i) the underlay bearing networks are generated using topology tool BRITE [
] and (ii) we bring the backbone networks of the practically deployed bearing networks to the simulation platform.
(i) We use BRITE to generate the underlay bearing networks with scale-free feature. While generating the bearing network topology, we use the same connection strategy on both autonomous systems and
router levels, i.e., the scale-free feature for connections between routers and the relations between autonomous systems comply with the scale-free feature. The scale-free underlay network is shown
Figure 5
. The rectangle nodes represent the routers, and the nodes with the same colour are routers in the same autonomous systems. Relay Servers access the bearing networks by connecting to the routers in
bearing networks. For simplicity, the topology of underlay bearing networks could approximately represent the overlay network.
(ii) We bring backbone topologies of bearing networks from three Internet Service Providers in China, namely, ChinaNet, CMNet, and UNINet and to OMNet++, and the router nodes and connections are
referring to their practical deployments. The backbone topology of underlay bearing networks from the one provided is depicted in
Figure 6
a. The combination of two backbone topologies are depicted in
Figure 6
b. The exchange points of different backbone topology are connected with bold links. Relay Servers are connected to access routers.
5.1. Accuracy of Topology Awareness
To evaluate the accuracy of topology awareness, we compare MDSM with the other three positioning methods, namely MDS-MAP [
], Vivaldi [
] and GNP [
]. In MDSM, we only use MDS-MAP to generate the independent view for each autonomous system. These methods are conducted on different networks depicted in
Figure 5
. Then, we compare the relative errors among these positioning methods. In MDSM, the relative error is the average error of all independent views. Referring to [
], we use the relative error to indicate the accuracy and it is defined as
Relative Error
$| M i j − C i j | M i j ( ∀ i , j , i ≠ j , M i j ≠ 0 ) ,$
$M i j$
denotes the practical delay between the ith and jth nodes, and
$C i j$
denotes the estimated delay calculated from their coordinates of the ith and jth nodes. The relative errors of these methods are shown in
Table 3
. We observe that the relative error of MDSM is the smallest and the accuracy of topology awareness is improved.
5.2. Diversity of Multipath Generating
In this section, we evaluate the performance of MDSM in improving multipath diversity and we define the Multipath Stretch to evaluate the diversity among multiple paths. Multipath Stretch could
indicate the correlation between paths by reflecting the joint ratio of nodes between multiple paths. The definition of Multipath Stretch of k paths is shown as follows: Multipath Stretch = $∑ 1 ≤ m
, n ≤ k , m ≠ n 1 2 C k 2 ( J m n N m + J m n N n ) ,$ where $J m n$ denotes the quantity of joint router nodes between the mth and nth paths, respectively. $N m$ and $N n$ denote the quantity of
router nodes on the mth and nth paths. Multipath Stretch indicates that, when the correlation between multiple paths is higher, i.e., they have larger ratio of joint nodes, then Multipath Stretch
becomes higher, and it indicates lower multipath diversity. In order to obtain statistics of packets on the router-level, the router in OMNeT++ platform is modified with a module to parse packets so
that packets on multiple paths could be traced.
The evaluation scenario is set as, the video stream would be delivered between origin-destination end hosts using a real-time protocol. In order to get extensive evaluation results, 200 pairs of
origin–destination end hosts are selected, and using the statistics of traced packets we calculate the
Multipath Stretch
. We compare
Multipath Stretch
among the proposed MDSM and other multipath transmission methods. In the work of [
], DR-MP generates multiple paths by deflecting packets from the shortest path. In the work of [
], GA and KSP-MP generates multiple overlay paths with by combining genetic and k-shortest algorithm. In the work of [
], GMS-MP selects the multihomed paths group that has the least correlation. In MDSM, we set the
Deviation Angles
with two values,
$10 ∘$
$20 ∘$
We calculate the CDF (cumulative distribution function) of
Multipath Stretch
for different multipath generating methods. The CDF is shown in
Figure 7
and it could be observed that the ratio of relatively lower
Multipath Stretch
of MDSM with
$20 ∘$Deviation Angle
is larger than other methods; this means that MDSM could obtain higher multipath diversity. This is because MDSM directly uses some objective factors of underlay networks to characterize multipath
diversity; in contrast with other methods, they adopt some indirectly inferring approaches. For instance, DR-MP uses deflection that could not directly discover end-to-end disjoint paths, GA and
KSP-MP might discover path sharing links, and GMS-MP indirectly infers the correlation using packet transmission features and could not probe path diversity when the features do not appear.
5.3. Performance of Multipath Transmission
In this section, we use the performance of multipath transmission to indirectly evaluate MDSM in improving multipath diversity. High multipath diversity helps improve the tolerance capability of
multiple paths, so that multipath transmission should perform better with higher multipath diversity. The evaluation scenario adopts the bearing networks practically deployed as in
Figure 6
b. We still use the transmission scenario in the above
Section 5.2
. The quantity of end-to-end concurrent multiple paths is up to 3. We collect the sending rates for each origin end host and the variation of sending rate is depicted in
Figure 8
a. Background traffic with different degrees are injected into underlay bearing networks to act as the accidental occurrence of link congestion and failure. With the reference to evaluation
configuration in work [
], the evaluation scenario is configured as the total time being 32 s, the beginning time for congestion traffic or link failure follows a Weibull distribution, the duration interval congestion
traffic varies between five seconds and ten seconds following a Pareto truncated distribution, and the failure links are chosen with a Power-Law based distribution.
Figure 8
b depicts the total injected traffic bandwidth acting as congestion. We compare the aggregating rates of different multipath generating methods, and also consider the default single path (DSP for
Figure 8
c shows the aggregating rate and it reveals that generally multipath transmission achieves better performance than single path, especially encountering link congestion and traffic burst. This is due
to the fact that multipath transmission weakens congestion or failure influence of an individual path on the overall transmission and aggregate abundant transmission resources. Furthermore, MDSM
achieves better performance most of the time in comparison with others; the main reason is that it directly characterizes multipath diversity and exploits multiple paths, and higher multipath
diversity brings better tolerance capability.
6. Related Work
6.1. Topology Awareness
Topology awareness is an efficient methodology for topology match between overlay and underlay networks. P4P uses network coordinates to enhance forwarding with topology awareness [
]. ASAP search for the neighbor autonomous systems to obtain topology relations [
]. In [
], it is proved that coordinates would be sufficient for most applications requiring topology-awareness. Representative network positioning methods include GNP [
], Vivaldi [
] and MDS-MAP [
]. However, the topology-aware solutions above generate network coordinates for all nodes together in a coordinate system, and do not consider the complicated bearing networks, which actually would
reduce the accuracy of positioning and topology awareness.
6.2. Multipath Diversity
Lower path correlation contributes to higher fault-tolerance capability, so that, when abnormal nodes or links occur, they would bring less impact on whole paths. High multipath diversity indicates
low multipath correlation. The multipath diversity of MPTCP is likely to be improved between multi-homed hosts [
]; however, this scenario is not common. Path diversity is provided by deflecting packets to neighbors lying off the shortest-path in [
]. Multipath diversity is maximized using hybrid GA and k-shortest path algorithm in [
]. Multipath diversity is improved by avoiding shared bottlenecks between joint paths in [
]. In [
], partial sampling schemes are introduced to enhance path diversity for scaling overlay routing. In [
], natural diversity of wide-area networks is leveraged. The above solutions use feedback of probes or transmission to infer or estimate multipath diversity, whereas sometimes highly correlated paths
could also exhibit uncorrelated states in appearance, but paths are tightly coupled inside. In addition, the solutions do not directly exploit the diversity among autonomous systems.
7. Conclusions
Multipath diversity significantly impacts multipath transmission quality; however, multipath diversity is difficult to guarantee on overlay networks thanks to the inaccurate topology awareness of
underlay networks and the multipath generating methods considering little about underlay diversity. In this paper, we proposed a multi-dimension spatial method to improve topology awareness accuracy
and multipath diversity. The topology view generated with network positioning methods was decomposed into intra and inter autonomous system dimensions. Based on the multi-dimension topology view, we
designed some spatial mechanisms to generate multiple paths. Numerous simulations demonstrated that the proposed method improved topology awareness accuracy and multipath diversity.
The multi-dimensional spatial method could also be further considered in other networks that have similar features of existing bearing networks—for example, devices are deployed in the cascade
manner, and the multi-dimensional organization approach could simplify the complexity.
Author Contributions
Conceptualization, Y.G., W.L., W.Z., Y.Z., H.L. and S.Z.; Data Curation, Y.G.; Formal Analysis, Y.G.; Investigation, Y.G., Y.Z., H.L. and S.Z.; Methodology, Y.G., W.L. and W.Z.; Supervision, W.L. and
W.Z.; Validation, Y.G.; Writing—Original Draft Preparation, Y.G.; Writing—Review & Editing, Y.G., W.L. and W.Z..
This work was supported by the National Key Research and Development Program of China Grant No. 2018YFB1702000, the National Natural Science Foundation of China Grant No. 61671141 and the Liaoning
Provincial Natural Science Foundation of China Grant No. 20180551007.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. The framework of Software-Defined Service Overlay Network (SDSON) built on bearing networks. The bottom plane consists of underlay bearing networks. The middle plane is the forwarding
plane, which consists of Relay Servers (relay forwarding nodes). The top plane is the control plane, which consists of Relay Controller (control node).
Figure 3. The underlay bearing networks consisting of multiple autonomous systems in different locations. (a) shows the underlay networks with three different autonomous systems (represented by three
layers), and in each autonomous systems, forwarding nodes are deployed in several locations (represented by dotted rectangles); (b) shows the independent topology view of each autonomous systems and
(c) shows the merged topology view of three independent views.
Figure 4. Sample topologies consist of overlay nodes and different autonomous systems. (a) shows a coordinate-based view of Relay Serves and solid circles represent the coordinates of Relay Servers;
(b) shows the underlay bearing networks consisting of various autonomous systems deployed in different locations (L1–L6).
Figure 5. The network topology of the scale-free topology feature generated with BRITE. Nodes with different colours mean they belong to different autonomous systems.
Figure 6. The backbone topology of practically deployed bearing networks from ChinaNet, CMNet. (a) shows the topology of one underlay network and circles represent the backbone routers; (b) shows the
integrated topology of underlay networks belonging to ChinaNet and CMNet; the bold lines are exchange points between them.
Figure 7.
The CDF (cumulative distribution function) of multipath stretch calculated from different multipath generating methods. (
) shows the CDF of multipath stretch in bearing networks generated with BRITE with scale-free feature in
Figure 5
; (
) shows the CDF of multipath stretch in practically deployed bearing networks from Internet Service Providers in
Figure 6
Figure 8. Evaluation of multipath transmission performance with injected congestion traffic and link failure. (a) depicts the sending rate from origin hosts; (b) depicts the total bandwidth of
traffic injected into bearing networks as congestion; (c) depicts the aggregating rates from destination host using different transmission methods.
Notation Definition and Description
$T s e t = { T 1 ∼ T n }$ topology view set, $T i$ denotes the view of $A S i$
$T i = { R S 1 ∼ R S m }$ $T i$ has m RSes
$L s e t = { L 1 ∼ L p }$ location set, city is the basic unit of location
$R S i j ∈ L j$ $R S i$ is deployed in $L j$
$N B ( i , j )$ distance between $L i$ and $L j$, or $R S i$, and $R S j$
$L k ( p , q ) = 1$ $T p$ and $T q$ have SPs in $L k$
$M P ( i , j ) = { P 1 ∼ P q }$ q multiple paths between $R S i$ and $R S j$
$R S i ∈ P j$ $R S i$ is on $P j$
$P j ⊆ T k$ $∀ R S i ∈ P j , R S i ∈ T k$
$M P D ( i , j )$ the multipath diversity of $M P ( i , j )$
Table 2.
Geometric items and descriptions referring to
Figure 4
Item Definition and Description
Transmission Direction (TD) Vector from the coordinate of origin-ARS to that of destination-ARS, TD of 7-9 is TD(7,9)
Transmission Surface (TS) The candidate RSes are restricted within the surface, which is enclosed with constrained criteria, TS of 7-9 is TS(7,9), which includes rectangle surface ABCD, ellipse EF
Deviation Angle (DA) The angle between the link and TD, $α$ is the deviation angle of 7-2 and 7-9, $β$ the 7-12 and 7-9, Then, the angle of 7-2 and 7-12 is DA(7-2,7-12)=$| α | + | β |$
Node Quantity MDSM MDS-MAP GNP Vivaldi
30 0.163 0.215 0.231 0.230
50 0.175 0.227 0.245 0.251
70 0.183 0.232 0.255 0.257
MDSM: Multi-Dimension Spatial Method; MDS-MAP: Multi-dimension Scaling-MAP; GNP: Global Network Positioning.
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
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Guan, Y.; Lei, W.; Zhang, W.; Zhan, Y.; Li, H.; Zhang, S. A Multi-Dimension Spatial Method for Topology Awareness and Multipath Generating. Symmetry 2019, 11, 870. https://doi.org/10.3390/sym11070870
AMA Style
Guan Y, Lei W, Zhang W, Zhan Y, Li H, Zhang S. A Multi-Dimension Spatial Method for Topology Awareness and Multipath Generating. Symmetry. 2019; 11(7):870. https://doi.org/10.3390/sym11070870
Chicago/Turabian Style
Guan, Yunchong, Weimin Lei, Wei Zhang, Yuzhuo Zhan, Hao Li, and Songyang Zhang. 2019. "A Multi-Dimension Spatial Method for Topology Awareness and Multipath Generating" Symmetry 11, no. 7: 870.
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Strip Option Strategy - Sana Securities
When to use: Strip Option Strategy is used when the investor is bearish on the stock and expects volatility in the near future.
How it works: Strip option strategy use three option contracts of the same underlying stock, with the same expiry date and same strike prices. In this strategy, you buy 2 at-the-money put options and
1 at-the-money call option.
For example: On 6^th September 2013, when the share of HDFC was trading at Rs. 760.00, you decided to buy 2 at-the-money put options at a premium of Rs. 14.95 expiring 26th September 2013; and you
bought 1 at-the-money call option at a premium of Rs. 71.90, with a Strike Price of Rs. 760.00 expiring 26th September 2013.
Risk/Reward: In strip option strategy, the risk is limited to the net premium paid for the position and the maximum reward/profit is unlimited. In this strategy you can make huge profits if the
underlying stock makes a strong move either on the upside or downside on expiry (however, larger gains will be made with a downward movement).
For 1 ATM Call Option: If the price of HDFC rises above Rs. 760.00 (i.e. the strike price), you can exercise your long call option, but the price of the stock must rise above Rs. 831.90 (i.e. the
strike price + the amount of premium) for you to exercise your option and make a profit.
For 2 ATM Put Options: If the price of the share falls below Rs. 760.00 (i.e. the strike price of the long put options), you can exercise your option, but the price of the stock must fall below Rs.
730.10 (i.e. the strike price minus the premiums), for you to exercise your option and make a profit.
The table below shows the net payoff of a strip option strategy assuming different spot prices on the expiry date:
The table above allows you to easily see the break-even points, maximum profit and the loss potential at expiry in rupee terms. The calculations are presented below.
The two break-even points occur when the price of the underlying share equals Rs. 709.10 and Rs. 861.80.
First Break-even Point = Strike price – (net premium paid/2) = Rs. 709.10 (760.00 – (101.80/2))
Second Break-even Point = Strike price + net premium paid = Rs. 861.80 (760.00 + 101.80)
The profit potential is unlimited but higher gain will be made if the share price moves downward.
The maximum loss which the investor may suffer is equal to the net premium paid i.e. 101.80. Maximum loss, in this example, will be incurred if the share price remains equal the strike price (i.e.
Rs. 760.00).
How to use the Strip Option Strategy Excel calculator
Just enter your expected spot price on expiry, option strike price and the amount of premium, to estimate your net pay-off from the Strip Option Strategy.
Note: The example and calculations are based assuming a single share though in reality options are based on lots of many shares. For example HDFC’s option contract is for 250 shares. Accordingly the
net premium paid will be Rs. 25,450 for 3 lots (i.e. 101.80*250) in our example.
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Category: Feedback
So after the last post on feedback I recieved a few messages asking about how I was using the feedback sheets in lessons and how I was using them for KS4 so here it is. Part of my Search for
focus this year has been developing this process, and there are still some small aspects that I beleive need ironing out but it is currently maneageable and effective which has been my a topic
aim from the start.
The main thing that I am doing now to gauge what feedback I give is using topic assessments every 2 weeks. The examples shown below are after a 2 week stretch on simultanous equations
with a high achieving KS4 set, at the time of the assessment we had covered algebraic linear and quadratic simultaneous equations and done a small amount on graphical linear Archives
simultaneous equations. The only feedback I give now is on these assessments, after all if a student gets 20 questions wrong in their books on a particular subtopic and this has helped
lead to mastery when completing the assessments, why bother marking all those mistakes when everything is self assessed in class time and they now understand it as proven by the
assessment? I am strongly of the opinion that these periods of self assessment are incredibly powerful and I insist that all work must be marked and corrected in class so that I can
conduct my AFL during classtime.
After the class has done the assessment I get these marked and fedback on for the next upcoming lesson, this particular set took me 1 hour and 33 minutes (I was asked to time it by a
member of the department, so no.. I am not that sad), it took me slightly longer than usual as some of the methods and feedback elements were slightly more complex than basic fractions
with year 7. Again each memeber of the class recieved written feedback on their own relevent sheet, some had algebraic linear equations, some quadratic focussing on either the 'y='
method or the substitution method when using the equation of a circle, and others had graphical simultaneous equations. So feedback responses were all being targeted at indivudual
areas for improvement.
Once completed, the next lesson students have around 30 minutes to read through the feedback and complete the relevent rosponse areas, students have questions to work through focusing
on fluency, reasoning and problem for each topic. Along with this, each feedback sheet has it's own relevent 'help sheet' that you can see in the pictures below. One thing I found was
that giving 32 students a bunch of questions on a topic they couldn't do on the assessment was a recipe for a riot of 'I don't get it', as it was impossible to provide extended
feedback for 6 different sub topics beyond the written element in order for them to make a confident start on the questions. If needed students can collect the relevent help sheet and
start working straight away. I have found this particularly useful for some of my classes that I have focussed on building resilience, students are now directing themselves towards
'revision material' or 'help sheets' rather than relying on me. Some example pictures shown below:
So far this system has been working well, I have run it across all of my year groups a few times now and the quality of the responses has improved tremendously. Students have been more
engaged within the lesson and are becoming more and more motivated to improve upon areas they need to develop.
One student read their feedback the lesson after the DIRT session last week and said "Sir this has made my week, thank you!". Atleast I know it's being appreciated by some one.
14 Comments
It's not really something I am fond on putting my marking out into the public domain, so I'll start it off straight away with the reason I feel uncomfortable to do so. I know that my
marking isn't yet perfect, although I do beleive I am doing my best at this point in time. I have been constantly re-assessing my feedback this year and refining the way I approach it,
earlier in August I had just over 100 students fill in a questionnaire asking them what they thought of the feedback they received in mathematics, and asked them to write in some
qualitative feedback. This was actually really useful, and I would definitely recommend trying it out. Ultimately, this has helped lead to where I currently am with my own feedback
this new term. As teachers, marking is quite personal, especially if you have spent hours doing it! So having others pass judgement on it is always going to be a tough conversation (if
the conversations isn't a positive one anyway).
This is pretty much how I start the day, 6:45 in my classroom. I can not bring myself to take any marking home, I find that it takes me 10 minutes a book, apposed to 2 minutes when I
am in 'work mode' which doesn't tend to switch off in my classroom.. So I decided that I would get to work with enough time to put in an hours marking everyday if needed, this has
meant avoiding those long hours of book marking sitting on the sofa switching between watching Luther and making another cup of coffee.
So first of all, all of my marking tends to take place in class, ALL classwork is marked by students in green pen, there is rarely a task that we ever do in class that we don't get
instant feedback on and self assessed in class. I find that it very quickly allows me to undertake my AFL around the room when you can see ticks or crosses very clearly in the students
books. So the only part that I actually onduct a detailed mark on is end of topic assessments and teacher assessed homeworks, then finally on student responses to feedback. Fortunately
these tend to fall every 12-14 days which falls in line with the schools marking policy of two weeks. Below is a typical topic assessment, this is with a 'bottom set' year 7 class I
teach 3 times a week.
While marking the assessments I have (for this particular assessment) 5 different
feedback sheets
in front of me; simplifying fractions, finding fractions of an amount, multiplying/dividing fractions, adding/subtracting fractions, and converting improper fractions and mixed
numbers. As I am marking the assessment, I then pick up the sheet which each student needs to improve on and provide their feedback on this particular sheet. At the start of the next
lesson students then stick these sheets in on a double page spread and work on their responses on the opposite page. What I had found previously was that writing in a question was
taking me an extra few minutes per book, and then once in class it was extremely difficult to ensure that all students had completed that question as one student would finish in a few
minutes where as others would need much longer. This is why I have put these new sheets together, it gives students time to consolidate, challenge and extend the topic they struggled
on and allows them to work through reasoning and problem solving style questions in line with the new GCSE.
Some examples of books from this year 7 class are shown below:
So far, after around 9-10 DIRT sessions this term across all my classes, students have been really engaged in these feedback sessions, there is enough work to be getting on with that
it allows me to get around the class and provide verbal feedback to those who struggled more than others on particular topics, while those who are happy to work through the whole sheet
can move onto all levels of questions and get some challenge on the problem solving questions. I have been dedicating around 30 minutes of a lesson to these DIRT sessions so far and
have found this is just enough time that everyone will be able to complete the 'fluency' questions while others are completing all levels of questions.
The only written feedback I am now giving is on the
9-1 Feedback Sheets
I have been uploading. It's taking me on average 1 hour to mark a set of 30 mini assessments and provide feedback and actions for each student. And this is happening after every topic
or series of subtopics, for example my year 9's have been doing algebraic fractions and we have recently done an assessment on simplifying them. Students then had to work on either
factorising quadratics, simplifying basic algebraic fractions or simplifying algebraic fractions involving quadratics. See example image below:
Anyway that's what I'm doing! If you have been using these in a different way or have any ideas then please let me know! You can access all of these sheets on the link below:
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Inconsistent Mathematics
First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008
Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the
presence of a contradiction without turning every sentence into a theorem.
Inconsistent Mathematics began historically with foundational considerations. Set-theoretic paradoxes such as Russell's led to attempts to produce a consistent set theory as a foundation for
mathematics. But, as is well known, set theories such as ZF, NBG and the like were in various ways ad hoc. Hence, a number of people including da Costa (1974), Brady (1971), (1989), Priest, Routley,
and Norman (1989), considered it preferable to retain the full power of the natural abstraction principle (every predicate determines a set), and tolerate a degree of inconsistency in set theory.
Brady, in particular, has extended, streamlined and simplified these results on naive set theory in his book (2006); for a clear account see also Restall's review (2007)
These constructions require, of course, that one dispense with the logical principle ex contradictione quodlibet (ECQ) (from a contradiction every proposition may be deduced), as well as any
principle which leads to it, such as disjunctive syllogism (DS) (from A-or-B and not-A deduce B). But considerable debate, in Burgess (1981) and Mortensen (1983), made it clear that dispensing with
ECQ and DS was not so counter-intuitive, especially when a plausible story emerged about the special conditions under which they continue to hold.
It should also be noted that Brady's construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged
by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for
mathematics, and that naive set theory is reducible to logic via the naive comprehension schema. The only change needed is a move to an inconsistency-tolerant logic.
In addition, mathematics has a metalanguage; that is, names for mathematical statements and other parts of syntax, self-reference, proof and truth. Gödel's contribution to the philosophy of
mathematics was to show that the first three of these can be rigorously expressed in arithmetical theories, albeit in theories which are either inconsistent or incomplete. The possibility of a
well-structured example of the former alternative was not taken seriously, again because of belief in ECQ. However, in addition natural languages seem to have their own truth predicate. Combined with
self-reference this produces the Liar paradox, "This sentence is false", an inconsistency. Priest (1987) and Priest, Routley, and Norman (1989) argued that the Liar had to be regarded as a statement
both true and false, a true contradiction. This represents another argument for studying inconsistent theories, namely the claim that some contradictions are true. Kripke (1975) proposed instead to
model a truth predicate differently, in a consistent incomplete theory. We see below that incompleteness and inconsistency are closely related.
But mathematics is not its foundations. Hence there is a further independent motive, to see what mathematical structure remains when the constraint of consistency is relaxed. But it would be wrong to
regard this as in any way a loss of structure. If it is different at all, then it represents an addition to known structure.
Robert K. Meyer (1976) seems to have been the first to think of an inconsistent arithmetical theory. At this point, he was more interested in the fate of a consistent theory, his relevant arithmetic
R#. There proved to be a whole class of inconsistent arithmetical theories; see Meyer and Mortensen (1984), for example. In a parallel with the above remarks on rehabilitating logicism, Meyer argued
that these arithmetical theories provide the basis for a revived Hilbert Program. Hilbert's program was widely held to have been seriously damaged by Gödel's Second Incompleteness Theorem, according
to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer's construction was that within his arithmetic R# it was demonstrable by simple finitary
means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert's goal of conclusively demonstrating that mathematics is
trouble-free proves largely achievable.
The arithmetical models used by Meyer-Mortensen later proved to allow inconsistent representation of the truth predicate. They also permit representation of structures beyond natural number
arithmetic, such as rings and fields, including their order properties. Recently, these inconsistent arithmetical models have been completely characterised by Graham Priest; that is, Priest showed
that all such models take a certain general form. See Priest (1997), (2000).
One could hardly ignore the examples of analysis and its special case, the calculus. There prove to be many places where there are distinctive inconsistent insights; see Mortensen (1995) for example.
(1) Robinson's non-standard analysis was based on infinitesimals, quantities smaller than any real number, as well as their reciprocals, the infinite numbers. This has an inconsistent version, which
has some advantages for calculation in being able to discard higher-order infinitesimals. Interestingly, the theory of differentiation turned out to have these advantages, while the theory of
integration did not. A similar result, using a different background logic, was obtained by Da Costa (2000).(2) Another place to find applications of inconsistency in analysis is topology, where one
readily observes the practice of cutting and pasting spaces being described as "identification" of one boundary with another. One can show that this can be described in an inconsistent theory in
which the two boundaries are both identical and not identical, and it can be further argued that this is the most natural description of the practice. (3) Yet another application is the class of
inconsistent continuous functions. Not all functions which are classically discontinuous are amenable of inconsistent treatment; but some are, for example f(x)=0 for all x<0 and f(x)=1 for all x≥0.
The inconsistent extension replaces the first < by ≤, and has distinctive structural properties. These inconsistent functions may well have some application in dynamic systems in which there are
discontinuous jumps, such as quantum measurement systems. Differentiating such functions leads to the delta functions, applied by Dirac to the study of quantum measurement also. (4) Next, there is
the well-known case of inconsistent systems of linear equations, such as the system (i) x+y=1, plus (ii) x+y=2. Such systems can potentially arise within the context of automated control. Little work
has been done classically to solve such systems, but it can be shown that there are well-behaved solutions within inconsistent vector spaces. (5) Finally, one can note a further application in
topology and dynamics. Given a supposition which seems to be conceivable, namely that whatever happens or is true, happens or is true on an open set of (spacetime) points, one has that the logic of
dynamically possible paths is open set logic, that is to say intuitionist logic, which supports incomplete theories par excellence. This is because the natural account of the negation of a
proposition in such a space says that it holds on the largest open set contained in the Boolean complement of the set of points on which the original proposition held, which is in general smaller
than the Boolean complement. However, specifying a topological space by its closed sets is every bit as reasonable as specifying it by its open sets. Yet the logic of closed sets is known to be
paraconsistent, ie. supports inconsistent theories; see Goodman (1981) for example. Thus given the (alternative) supposition which also seems to be conceivable, namely that whatever is true is true
on a closed set of points, one has that inconsistent theories may well hold. This is because the natural account of the negation of a proposition, namely that it holds on the smallest closed set
containing the Boolean negation of the proposition, means that on the overlapping boundary both the proposition and its negation hold. Thus dynamical theories determine their own logic of possible
propositions, and corresponding theories which may be inconsistent, and are certainly as natural as their incomplete counterparts.
Category theory throws light on many mathematical structures. It has certainly been proposed as an alternative foundation for mathematics. Such generality inevitably runs into problems similar to
those of comprehension in set theory, see eg. Hatcher (1982, p.255-260). Hence there is the same possible application of inconsistent solutions. There is also an important collection of categorial
structures, the toposes, which support open set logic in exact parallel to the way sets support Boolean logic. This has been taken by many to be a vindication of the foundational point of view of
mathematical intuitionism. However, it can be proved that that toposes support closed set logic as readily as they support open set logic. That should not be viewed as an objection to intuitionism,
however, so much as an argument that inconsistent theories are equally reasonable as items of mathematical study.
Duality between incompleteness/intuitionism and inconsistency/paraconsistency has at least two aspects. First there is the above topological (open/closed) duality. Second there is Routley * duality.
Discovered by the Routleys (1972) as a semantical tool for relevant logics, the * operation dualises between inconsistent and incomplete theories of the large natural class of de Morgan logics. Both
kinds of duality interact as well, where the * gives distinctive duality and invariance theorems for open set and closed set arithmetical theories. On the basis of these results, it is fair to argue
that both kinds of mathematics, intuitionist and paraconsistent, are equally reasonable.
A very recent development is the application to explaining the phenomenon of inconsistent pictures. The best known of these are perhaps M.C.Escher's masterpieces Belvedere, Waterfall and Ascending
and Descending. In fact the tradition goes back millennia to Pompeii. Escher seems to have derived many of his intuitions from the Swedish artist Oscar Reutersvaard, who began in 1934. Escher also
actively collaborated with the English mathematician Roger Penrose. There have been several attempts to describe the mathematical structure of inconsistent pictures using classical consistent
mathematics, by theorists such as Cowan, Francis and Penrose. As argued in Mortensen (1997), however, no consistent mathematical theory can capture the sense that one is seeing an impossible thing.
Only an inconsistent theory can capture the content of that perception. This amounts to an appeal to cognition, that is the epistemological justification of paraconsistency as above. One can then
proceed to describe inconsistent theories which are candidates for such inconsistent contents. There is an analogy with classical mathematics on this point. Projective geometry is a mathematical
theory which is interesting because we are creatures with an eye, since it explains why it is that things look the way they do in perspective. This study has been further developed in Mortensen
(2002a), where category theory is applied to give a general description of the relationships between the various theories and their consistent cut-downs and incomplete duals. For an informal account
which highlights the difference between visual "paradoxes" and the philosophically more common paradoxes of language, such as the Liar, see Mortensen (2002b). More recently, concrete mathematical
descriptions have been obtained for one class of inconsistent figures, exemplified by the "Crazy Crate" (also found in Belvedere ), see Mortensen (2006).
Recently, an alternative technique for dealing generally with contradictions has emerged. Brown and Priest (2004) have proposed a technique they call "Chunk and Permeate", in which reasoning from
inconsistent premisses proceeds by separating the assumptions into consistent theories (chunks), deriving appropriate consequences, then passing (permeating) those consequences to a different chunk
for further consequences to be derived. They suggest that Newton's original reasoning in taking derivates in the calculus, was of this form. This is an interesting and novel approach, though it must
meet the objection that to believe a conclusion obtained on this basis, one should believe all the premisses equally; and so an argument of the more common form, appealing to all the premisses
without fragmenting them, should be eventually forthcoming. Developments are to be awaited with interest.
It should be emphasised that these constructions do not in any way challenge or repudiate existing mathematics, but extend our conception of what is mathematically possible.
• Brady, R., 1971, "The Consistency of the Axioms of Abstraction and Extensionality in a Three-Valued Logic", Notre Dame Journal of Formal Logic, 12: 447–453.
• –––, 1989, "The Nontriviality of Dialectical Set Theory", in G. Priest, R. Routley and J. Norman (eds.), Paraconsistent Logic, Munich: Philosophia Verlag.
• –––, 2006, Universal Logic, Stanford: CSLI Publications.
• Brown, B., and G. Priest, 2004, "Chunk and Permeate: A Paraconsistent Inference Strategy. Part I: The Infinitesimal Calculus", The Journal of Philosophical Logic, 33: 379–388.
• Burgess, J., 1981, "Relevance, a Fallacy?", Notre Dame Journal of Formal Logic, 22: 97–104.
• Da Costa, Newton C.A., 1974, "On the Theory of Inconsistent Formal Systems", Notre Dame Journal of Formal Logic, 15: 497–510.
• –––, 2000, "Paraconsistent Mathematics", in D. Batens et al. (eds), Frontiers of Paraconsistent Logic, Hertfordshire: Research Studies Press, 165–180.
• Goodman, N., 1981, "The Logic of Contradictions", Zeitschrift fur Mathematische Logic und Grundlagen der Arithmetik, 27: 119–126.
• Hatcher, W. S., 1982, The Logical Foundations of Mathematics, Oxford: Pergamon.
• Kripke, S., 1975, "Outline of a Theory of Truth", The Journal of Philosophy, 72: 690-716.
• Meyer, R. K., 1976, "Relevant Arithmetic", Bulletin of the Section of Logic of the Polish Academy of Sciences, 5: 133–137.
• Meyer, R. K. and C. Mortensen, 1984, "Inconsistent Models for Relevant Arithmetics", The Journal of Symbolic Logic, 49: 917–929.
• Mortensen, C., 1983, "Reply to Burgess and to Read", Notre Dame Journal of Formal Logic, 24: 35–40.
• –––, 1995, Inconsistent Mathematics, Kluwer Mathematics and Its Applications Series, Dordrecht: Kluwer. [Errata available online].
• –––, 1997, "Peeking at the Impossible", Notre Dame Journal of Formal Logic, 38: 527–534.
• –––, 2000, "Prospects for Inconsistency", in D. Batens et al. (eds.), Frontiers of Paraconsistent Logic, London: Research Studies Press, 203–208.
• –––, 2002a, "Towards a Mathematics of Impossible Pictures", in W. Carnielli, M. Coniglio and I. D'Ottaviano (eds.), Paraconsistency: The Logical Way to the Infinite, (Lecture Notes in Pure and
Applied Mathematics, Volume 228), New York: Marcel Dekker, 445–454.
• –––, 2002b, "Paradoxes Inside and Outside Language", Language and Communication, 22: 301–311.
• –––, 2006, "An Analysis of Inconsistent and Incomplete Necker Cubes", Australasian Journal of Logic, 4: 216–225.
• Priest, G., 1987, In Contradiction, Dordrecht: Nijhoff.
• –––, 1997, "Inconsistent Models for Arithmetic: I, Finite Models", The Journal of Philosophical Logic, 26: 223–235.
• –––, 2000, "Inconsistent Models for Arithmetic: II, The General Case", The Journal of Symbolic Logic, 65: 1519–29.
• Priest, G., R. Routley and J. Norman (eds.), 1989, Paraconsistent Logic, Munich: Philosophia Verlag.
• Restall, G., 2007, "Review of Brady Universal Logic", Bulletin of Symbolic Logic, 13/4: 544–547.
• Routley, R. and V. Routley, 1972, "The Semantics of First Degree Entailment", Noûs 6: 335–359.
[Please contact the author with suggestions.]
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Lesson 4
More Balanced Moves
Problem 1
Mai and Tyler work on the equation \(\frac25b+1=\text-11\) together. Mai's solution is \(b=\text-25\) and Tyler's is \(b=\text-28\). Here is their work. Do you agree with their solutions? Explain or
show your reasoning.
\(b=\text-10\boldcdot \frac52\)
\(b = \text-25\)
Problem 3
Describe what is being done in each step while solving the equation.
1. \(2(\text-3x+4)=5x+2\)
2. \(\text-6x+8=5x+2\)
3. \(8=11x+2\)
4. \(6=11x\)
5. \(x=\frac{6}{11}\)
Problem 4
Andre solved an equation, but when he checked his answer he saw his solution was incorrect. He knows he made a mistake, but he can’t find it. Where is Andre’s mistake and what is the solution to the
\(\displaystyle \text{-}2(3x-5) &= 4(x+3)+8\\\text{-}6x+10 &= 4x+12+8\\\text{-}6x+10 &= 4x+20\\ 10 &= \text{-}2x+20\\\text{-}10 &= \text{-}2x\\ 5 &= x\)
Problem 5
Choose the equation that has solutions \((5, 7)\) and \((8, 13)\).
Problem 6
A length of ribbon is cut into two pieces to use in a craft project. The graph shows the length of the second piece, \(x\), for each length of the first piece, \(y\).
1. How long is the ribbon? Explain how you know.
2. What is the slope of the line?
3. Explain what the slope of the line represents and why it fits the story. | {"url":"https://im.kendallhunt.com/MS/teachers/3/4/4/practice.html","timestamp":"2024-11-12T17:17:14Z","content_type":"text/html","content_length":"73896","record_id":"<urn:uuid:072bf0de-e3ec-457a-9978-28f78dc9d2e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00493.warc.gz"} |
I am a researcher at Inria Saclay in the MATHEXP team and a teacher in the computer science department of Ecole polytechnique. My research interests lie in computer algebra, experimental mathematics
and algebraic geometry. Concretely, this amounts to:
• designing algorithms to solve mathematical problems and explore mathematical areas;
• implementing these algorithms and studying their computational complexity.
Recently, I gave an online lecture where I introduce my research topic, check it out!
For the period 2022–2027, I received an ERC Starting Grant for a project called 10000 DIGITS. There are positions available.
Since January 2022, I am Associate Editor of SIAGA, the SIAM Journal on Applied Algebra and Geometry.
(keywords) computer algebra, experimental mathematics, algebraic geometry, symbolic integration, seminumerical algorithms
École polytechnique (teaching)
INRIA Saclay, Bât. Alan Turing
1 rue Honoré d’Estienne d’Orves
Campus de l’École polytechnique
91120 Palaiseau
48.7146, 2.2056 (map)
+33 (0)1 77 57 80 36
gpg public key
PhD students | {"url":"https://mathexp.eu/lairez/","timestamp":"2024-11-14T05:26:00Z","content_type":"text/html","content_length":"5460","record_id":"<urn:uuid:6b84f684-7c77-4932-8909-173c2cefc061>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00366.warc.gz"} |
What is 1111 in Binary? Unveiling the Secrets of a Binary Sequence - Dfahos
What is 1111 in Binary? Unveiling the Secrets of a Binary Sequence
What is 1111 in Binary? Unveiling the Secrets of a Binary Sequence
Binary the language of computers may seem like an enigma to many. However within this complex system lies a particular sequence that piques curiosity – 1111.
• Introduction
• Definition of Binary
Binary in essence is a numerical system based on two digits 0 and 1.
• Understanding Binary System
• Basics of Binary
To comprehend 1111 a grasp of binary basics is essential. Unlike the decimal system binary operates on powers of 2.
Binary digits or bits are the building blocks of this system. Each 1 or 0 represents a bit forming the language of computers.
Decoding 1111
What does 1111 translate to in the decimal system?
Exploring the operations involving 1111 showcases the versatility and applicability of this binary pattern.
Where does 1111 find practical use?
• Historical Perspective
• Origin of Binary System
The roots of binary trace back to ancient civilizations with intriguing insights into its origin and development.
How did binary evolve and when did it become a cornerstone in mathematics and computing?
• Exploring Binary Patterns
• Patterns and Sequences
Beyond 1111 binary is rich in patterns and sequences.
• Common Misconceptions
• Misinterpretations of 1111
Misconceptions surrounding the interpretation of 1111 often lead to confusion. Addressing these clears the air.
What myths surround the binary sequence 1111? Debunking these myths ensures a more accurate understanding.
Significance in Computing
• Role in Computer Programming
Computer programming heavily relies on binary including the use of 1111 in coding structures and algorithms.
• Binary in Digital Communication
Digital communication platforms utilize binary showcasing its relevance in the modern era.
Binary in Electronics
The integration of binary in electronics has revolutionized technological advancements from circuits to processors.
Contributions to Technological Advancements
How has binary including the sequence 1111 contributed to the leaps in technological innovations?
Cultural References
• Binary in Cultural Symbols
Binary finds its way into cultural symbols representing a fusion of technology and human expression.
• References in Pop Culture
Popular culture often references binary showcasing its influence on contemporary art music and literature.
Binary Beyond 1111
Exploring additional binary patterns reveals the diversity within this numerical system.
Beyond simple sequences complex binary numbers open doors to advanced mathematical concepts.
Practical Applications
Binary impact extends beyond the digital realm touching everyday life in unexpected ways.
• Utilization in Different Fields
Various fields leverage binary demonstrating its versatility and adaptability.
Educational Insights
• Teaching Binary in Schools
Insights into how binary is taught in schools shed light on educational approaches and challenges.
Educational resources play a crucial role in fostering a deeper understanding of binary among learners.
Future Prospects
• Continuation of Binary Relevance
Is binary here to stay? Assessing the future prospects of binary unveils its enduring relevance.
• Innovations and Developments
Ongoing innovations and developments in binary technology pave the way for exciting possibilities.
The Fascination of Binary
• Attraction to Binary Systems
What fascinates individuals about binary systems including the allure of 1111?
• Enthusiasts and Communities
Exploring the communities and enthusiasts dedicated to the study and appreciation of binary.
Final Word
In the binary sequence 1111 serves as a gateway to the intricate world of binary language. Its historical significance practical applications and cultural references showcase the depth and
versatility of binary in our modern live.
1. What is the decimal equivalent of 1111 in binary?
□ Understanding the numeric value of the binary sequence 1111.
2. How is binary taught in schools?
□ Insights into educational approaches and challenges in teaching binary.
3. Are there other notable binary patterns besides 1111?
□ Exploring the diversity within the binary numerical system.
4. Why is binary significant in computing and technology?
□ Unveiling the crucial role of binary in shaping the digital landscape.
5. What myths surround the interpretation of 1111 in binary?
□ Debunking common misconceptions and clarifying myths.
What is 1111 in Binary? Unveiling the Secrets of a Binary Sequence
Source of Image: https://www.pexels.com/photo/multicolored-text-on-the-screen-3872166/ | {"url":"https://dfaho.com/what-is-1111-in-binary-unveiling-the-secrets-of-a-binary-sequence/","timestamp":"2024-11-08T04:22:25Z","content_type":"text/html","content_length":"113445","record_id":"<urn:uuid:69e3263c-458c-4dae-a346-aecb0e6d447b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00595.warc.gz"} |
Guided tour of Logic I: Tools for Thinking, with sample chapters
A selection of sample chapters is introduced below, with context provided by a brief description of the program's topic sequence.
For a more detailed topic listing, you may want to view the Table of Contents
from the teacher’s manual. (The teacher's manual has all the same chapter and section titles as the student book, so the teacher's manual table of contents shows you the topic sequence for the
student book as well.)
Introduction to logic
Many users of Logic I: Tools for Thinking are new to logic. Logic is a big field, and the sheer variety of logic topics chosen for inclusion in this or that logic textbook can be a bit bewildering.
Our Introduction to Logic from the teacher's manual will be helpful for users of any logic textbook. It covers such topics as what is logic, what are the parts or branches of logic, how should
Christians view logic, and why study logic. It also features an annotated bibliography of logic programs.
Reasoning and logic
How are the students of Logic I: Tools for Thinking introduced to logic? After studying Chapter 1, which gets them thinking about reasoning, they meet a definition of logic in Chapter 2. They also
encounter the organizing metaphor for this book. Logic is presented as a toolbox, containing a wide variety of tools, each of which is used for a specific purpose on a specific kind of material.
Statements and logical operators
Chapters 4 through 13 develop tools for use on two kinds of material: statements and logical operators. Logical operator may not be a term familiar to you, but actually, you know many logical
operators already. For example, the conjunction operator is commonly expressed by the English word "and." The sentence "Parsnip is gray and Parsnip is fuzzy" contains a conjunction operator.
Chapter 10 presents a technique for telling whether a statement that contains logical operators is true or false. It employs a tool, introduced in Chapter 7, called a Parsnip tree (officially called
a parse tree, but renamed in this book in honor of our cat Parsnip Poffle-Hoover). Chapter 10 of the Teacher's Manual comments on Chapter 10 of the student textbook, and provides exercise solutions.
Moleculan, or the sentential calculus
On the foundation laid in Chapters 4 through 13, Chapters 14 through 26 present a system of symbolic logic for use on statements that contain logical operators. Because such statements are called
molecular statements, and because our other cat is named Tiger Molecule, the book refers to this system as Moleculan. (It is more commonly known as the sentential calculus, the propositional
calculus, sentence logic, or propositional logic.)
In this part of the book, the students learn the grammar of the Moleculan language, and they gain experience translating from English to Moleculan and vice versa. This focus on translation into and
out of Moleculan is a distinctive and important feature of Logic I: Tools for Thinking. Students also learn how to tell whether a given Moleculan sentence is true or false, using a truth table. In
Chapter 19, they encounter tautologies, sentences that are always true, and learn how to determine whether a certain sentence is a tautology. (Keep in mind as you look at Chapter 19 that the student
who has been working his way through all the chapters is already familiar with much in the chapter that you may find unfamiliar or perplexing at first glance.)
Arguments in Moleculan
Next the students meet arguments, a very important class of objects on which logic tools can be used. They learn what a valid argument is, and they learn to test Moleculan arguments for validity,
using truth tables. They also encounter a sizable number of common argument forms, or patterns of argumentation that one meets frequently.
Informal logic
Moleculan is an example of formal logic. Formal logic is useful in such areas as mathematics and computer programming (not to mention deciphering tax instructions!) Informal logic is more useful in
everyday life, and so in Chapters 27 through 31 the book shifts its focus to some important kinds of informal arguments. Chapter 27 connects this new focus to what has come before, by distinguishing
among deductive, inductive and conductive arguments. Chapter 28 and Chapter 29 present the appeal to authority. (And here's Chapter 28 of the Teacher's Manual.) Chapter 30 presents the argument from
analogy. In Chapter 31, students meet the inductive generalization. Then in Chapters 32-33, they begin their study of questions, which present an interesting set of logical issues all their own.
Informal arguments have a very different feel from formal arguments, and analyzing them calls upon students' wisdom in a way that formal arguments do not. But developing wisdom—rather than mere
mechanical reasoning skills, as valuable as they are in their place—is what we're aiming for, and so the extra effort is abundantly repaid.
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Scholarship 12/11613-0 - Geometria algébrica - BV FAPESP
Grant number: 12/11613-0
Support Opportunities: Scholarships in Brazil - Post-Doctoral
Effective date (Start): November 01, 2012
Effective date (End): October 31, 2015
Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator: Herivelto Martins Borges Filho
Grantee: Gary Russell Cook
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
We propose to conduct research in the area of algebraic geometry; in particular the study of plane quartic curves predominantly over finite fields. The aim of this research is primarily to address
Problems 1 and 2.Problem 1. How many rational points are there on a plane quartic curve over finite fields?Problem 2. The projective classification of plane quartic curves over finite fields.Details
of Problems 1 and 2There are several bounds on the number of points on a plane quartic curve over a finite field of order q and for some instances of q there are sharpness results for these bounds.
These bounds are given in Section 2 of the research proposal. There are no results about the existence of plane quartic curves with n rational points over a finite field of order q, where n is
between these bounds, excluding sharpness results.The plane quartic curves are projectively classified over finite fields of small order; the smallest field over which the plane quartic curves are
not projectively classified is of order eleven. There is not a classification of plane quartic curves over a general finite field as there is for plane cubic curves. Approaches to Problems 1 and 2The
following gives an outline of the approaches to be taken in tackling Problems 1 and 2. As is made explicit in the research proposal, this is subject to change, as some approaches may prove fruitless
and other approaches and techniques could arise during the research.The aim of studying the projective classification of plane quartic curves over fields of order smaller than eleven and determine if
there exist any results that can be extrapolated to more general finite fields.The study of plane quartic curves over the complex numbers may reveal results that can be translated to plane quartic
curves over finite fields.Are there any results to Problems 1 and 2 that apply over fields of specific order q; for example, q is a prime.Non-singular plane quartic curves possess a genus of 3, so
non-singular plane quartic curves may be studied through the more general study of curves of genus 3.Lines in a cubic surfaces project onto bitangents of plane quartic curves; this is described in
Section 3 of the research proposal together with a brief account of what is known about the number of lines on a cubic surface and the number of bitangents on the plane quartic curve onto which it
projects.This leads to the question: Can the problem of determining how many rational points there are on a non-singular plane quartic curve or the classification of non-singular plane quartic curves
be addressed through a study of their bitangents or through a study of the cubic surfaces.What can be determined about the classification of plane quartic curves over finite fields through a detailed
study of the classification of plane cubic curves over finite fields?
News published in Agência FAPESP Newsletter about the scholarship:
More itemsLess items
Articles published in other media outlets ( ):
More itemsLess items
VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA)
We propose to conduct research in the area of algebraic geometry; in particular the study of plane quartic curves predominantly over finite fields. The aim of this research is primarily to address
Problems 1 and 2.Problem 1. How many rational points are there on a plane quartic curve over finite fields?Problem 2. The projective classification of plane quartic curves over finite fields.Details
of Problems 1 and 2There are several bounds on the number of points on a plane quartic curve over a finite field of order q and for some instances of q there are sharpness results for these bounds.
These bounds are given in Section 2 of the research proposal. There are no results about the existence of plane quartic curves with n rational points over a finite field of order q, where n is
between these bounds, excluding sharpness results.The plane quartic curves are projectively classified over finite fields of small order; the smallest field over which the plane quartic curves are
not projectively classified is of order eleven. There is not a classification of plane quartic curves over a general finite field as there is for plane cubic curves. Approaches to Problems 1 and 2The
following gives an outline of the approaches to be taken in tackling Problems 1 and 2. As is made explicit in the research proposal, this is subject to change, as some approaches may prove fruitless
and other approaches and techniques could arise during the research.The aim of studying the projective classification of plane quartic curves over fields of order smaller than eleven and determine if
there exist any results that can be extrapolated to more general finite fields.The study of plane quartic curves over the complex numbers may reveal results that can be translated to plane quartic
curves over finite fields.Are there any results to Problems 1 and 2 that apply over fields of specific order q; for example, q is a prime.Non-singular plane quartic curves possess a genus of 3, so
non-singular plane quartic curves may be studied through the more general study of curves of genus 3.Lines in a cubic surfaces project onto bitangents of plane quartic curves; this is described in
Section 3 of the research proposal together with a brief account of what is known about the number of lines on a cubic surface and the number of bitangents on the plane quartic curve onto which it
projects.This leads to the question: Can the problem of determining how many rational points there are on a non-singular plane quartic curve or the classification of non-singular plane quartic curves
be addressed through a study of their bitangents or through a study of the cubic surfaces.What can be determined about the classification of plane quartic curves over finite fields through a detailed
study of the classification of plane cubic curves over finite fields?
News published in Agência FAPESP Newsletter about the scholarship:
More itemsLess items
Articles published in other media outlets ( ):
More itemsLess items
VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA) | {"url":"https://bvs.fapesp.br/en/bolsas/137089/quartic-curves-over-finite-fields/","timestamp":"2024-11-07T12:43:37Z","content_type":"text/html","content_length":"64400","record_id":"<urn:uuid:4f838290-2922-4104-a546-24f517d4dd15>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00147.warc.gz"} |
Machine learning algorithms could offer a glimpse into the heart of the proton
Feb. 22, 2019 Research Highlight Physics / Astronomy
Machine learning algorithms could offer a glimpse into the heart of the proton
A computer algorithm that learns as it goes could hold the key to discovering what happens inside protons and neutrons
By using cutting-edge computational techniques, RIKEN researchers have found a way to simplify the fiendishly difficult mathematics that describes the physics of fundamental particles^1.
The electrical forces that bind the atomic nucleus and electrons to each other to form an atom are very well understood. Quantum electrodynamics (QED) combines relativity and quantum theory to
describe how electrically charged particles interact in great detail. However, this theory does not apply to the individual subatomic particles themselves: a complete understanding of the physics
that binds so-called quarks to make a proton or a neutron is still being developed.
Quantum chromodynamics, or QCD, attempts to describe the interaction between quarks using a concept known as ‘color’ charge, analogous to the electrical charge used as the basis for QED. But the
mathematical tools used by particle physicists to simplify QED calculations are not useful in QCD, so alternatives need to be found.
One promising approach is the holographic model, which attempts to construct an equation of motion in a space that has an additional dimension. Just as a hologram recovers three-dimensional objects
from a two-dimensional image, this model mimics the quantum behaviors of the original system.
“One of the most important features of the holographic model is that problems in a difficult quantum field theory like QCD can be mapped to equations of motion that are relatively easy to solve,”
explains Akinori Tanaka from the RIKEN Center for Advanced Intelligence Project and the RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS).
But building a holographic model to describe QCD is an enormous task, with so many variable parameters. “So far there is no mechanical method that can find a candidate of holographic model for a
given quantum field theory,” says Tanaka. So he, along with Akio Tomiya from the RIKEN Brookhaven National Laboratory Research Center and their colleagues at Osaka University, tackled this problem
using machine learning.
Machine learning is a computational technique that uses algorithms to build a mathematical model from a sample data set, which is then progressively improved.
“When we calculated the potential energy between quarks using our machine-learned holographic model, we found it qualitatively reproduces the same behavior seen in computer-simulated QCD calculations
based on original quantum field theory,” says Tanaka.
This useful tool could help scientists solve knotty QCD problems, and hence lead to a better understanding of how nuclei behave during nuclear reactions, such as fission and fusion, and the formation
of more exotic particles, such as those created in the Large Hadron Collider at CERN in Switzerland.
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• 1. Hashimoto, K., Sugishita, S., Tanaka, A. & Tomiya, A. Deep learning and holographic QCD. Physical Review D 98, 106014 (2018). doi: 10.1103/PhysRevD.98.106014 | {"url":"https://www.riken.jp/en/news_pubs/research_news/rr/20190222_FY20180051/index.html","timestamp":"2024-11-08T09:13:29Z","content_type":"text/html","content_length":"20220","record_id":"<urn:uuid:68ac0599-af0e-431d-9076-82955e697925>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00348.warc.gz"} |
What is standard error of regression coefficient?
What is standard error of regression coefficient?
The standard error of a coefficient estimate is the estimated standard deviation of the error in measuring it. Also, the estimated height of the regression line for a given value of X has its own
standard error, which is called the standard error of the mean at X.
What is standard error in regression formula?
Standard Error of Regression Slope Formula / TI-83 Instructions. SE of regression slope = sb1 = sqrt [ Σ(yi – ŷi)2 / (n – 2) ] / sqrt [ Σ(xi – x)2 ]. The equation looks a little ugly, but the secret
is you won’t need to work the formula by hand on the test.
What does coefficient standard errors mean?
The standard error of the coefficient measures how precisely the model estimates the coefficient’s unknown value. The standard error of the coefficient is always positive. Use the standard error of
the coefficient to measure the precision of the estimate of the coefficient.
What is standard error in regression table?
The standard error (SE) is an estimate of the standard deviation of an estimated coefficient. It is often shown in parentheses next to or below the coefficient in the regression table. It can be
thought of as a measure of the precision with which the regression coefficient is estimated.
What is error in regression?
An error term appears in a statistical model, like a regression model, to indicate the uncertainty in the model. The error term is a residual variable that accounts for a lack of perfect goodness of
What is standard error in statistics?
The standard error of the mean, or simply standard error, indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were
to repeat a study using new samples from within a single population.
What is standard error in regression excel?
The standard error of the regression is the precision that the regression coefficient is measured; if the coefficient is large compared to the standard error, then the coefficient is probably
different from 0. Observations.
What is meant by standard error and what are its practical uses?
Standard error is used to estimate the efficiency, accuracy, and consistency of a sample. In other words, it measures how precisely a sampling distribution represents a population. It can be applied
in statistics and economics.
What is standard error and why is it important?
Standard error statistics measure how accurate and precise the sample is as an estimate of the population parameter. It is particularly important to use the standard error to estimate an interval
about the population parameter when an effect size statistic is not available.
What does standard error mean in regression?
The standard error of the regression is the average distance that the observed values fall from the regression line. In this case, the observed values fall an average of 4.89 units from the
regression line. If we plot the actual data points along with the regression line, we can see this more clearly:
What is the standard error of a regression model?
There are 32 pairs of dependent and independent variables: labelled (y i,x i ),where 1<=i<=32.
The SE of y i was calculated earlier by GLM,but was NOT calculated from the regression of y on x.
What is the formula for the SE of prediction of each y i,given R² y,x,the deviation of y i from the regression on x i,…
How do you interpret standard error?
In the first step,the mean must be calculated by summing all the samples and then dividing them by the total number of samples.
In the second step,the deviation for each measurement must be calculated from the mean,i.e.,subtracting the individual measurement.
In the third step,one must square every single deviation from the mean.
What is standard error interpretation?
The standard error of the mean, or simply standard error, indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were
to repeat a study using new samples from within a single population. | {"url":"https://www.resurrectionofgavinstonemovie.com/what-is-standard-error-of-regression-coefficient/","timestamp":"2024-11-05T19:31:29Z","content_type":"text/html","content_length":"36224","record_id":"<urn:uuid:dc8567e9-9bac-43ee-854b-2ab5152097be>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00578.warc.gz"} |
Fun Ways to Teach Factorials to Kids
Did you know that factorials are not just complex equations reserved for mathematicians? They can also be taught to children in fun and engaging ways! Factorials may seem daunting, but by breaking
down the concept and using creative methods, you can make learning about factorials an enjoyable experience for young minds. Let’s explore some exciting ways to explain factorials to children and
help them develop a solid understanding of this mathematical concept.
Key Takeaways:
• Factorials can be taught to kids using visual aids and interactive activities.
• Start by explaining the concept of the factorial function and how it is used to calculate products.
• Provide examples and show step-by-step calculations to help children grasp the concept.
• Show children the practical applications of factorials in everyday scenarios.
• Engage kids in hands-on activities and games to make learning about factorials fun and interactive.
What is the Factorial Function?
The factorial function is represented by an exclamation point (!) and is used to calculate the product of all whole numbers from a given number down to 1. For example, 6! (read as “6 factorial”)
means multiplying all the numbers from 6 to 1. This concept can be simplified for children by explaining that it involves counting down and multiplying the numbers in a specific order.
Understanding factorials is the key to unlocking the fascinating world of mathematics. By teaching factorials to young learners in a simplified manner, we can make this concept accessible and
enjoyable for kids of all ages.
“Factorials are like a magical math trick! They help us calculate big numbers by multiplying smaller numbers in a special way.”
Let’s explore this fascinating mathematical concept and discover fun ways to introduce factorials to children.
Factorial Calculation
1! 1
2! 2 x 1 = 2
3! 3 x 2 x 1 = 6
4! 4 x 3 x 2 x 1 = 24
Understanding Factorials through Examples
To help children understand factorials, it’s important to provide them with examples. By demonstrating step-by-step calculations, kids can grasp the concept and see the pattern in calculating
Let’s start with some simple examples:
1. Factorial of 1: 1! = 1
2. Factorial of 2: 2! = 2 x 1
3. Factorial of 3: 3! = 3 x 2 x 1
4. Factorial of 4: 4! = 4 x 3 x 2 x 1
As you can see, each factorial is calculated by multiplying the given number by the numbers below it until reaching 1. This step-by-step approach helps children understand how factorials work and how
to calculate them.
Let’s calculate 4! = 4 x 3 x 2 x 1:
1. Start with 4: 4 x
2. Multiply by the next number, 3: 4 x 3 x
3. Continue with 2: 4 x 3 x 2 x
4. End with 1: 4 x 3 x 2 x 1 = 24
Therefore, 4! is equal to 24.
By following this step-by-step process, children can learn to calculate factorials of different numbers and gain a deeper understanding of how they work.
Practical Applications of Factorials
While factorials might seem like an abstract mathematical concept, they have practical applications that can be introduced to children. Factorials are used in combinatorial analysis to calculate
combinations and permutations. By providing real-life examples, children can see the relevance and usefulness of factorials in everyday scenarios.
“Factorials are like secret codes that help us solve puzzles and figure out different ways to organize things. They may seem tricky at first, but once you understand how to use factorials, you’ll
see how they can make everyday tasks easier!”
One practical application of factorials is arranging a deck of cards. Let’s say you want to know how many different orders you can arrange the cards in a deck. The factorial function can help you
find the answer! Since a standard deck has 52 cards, you would calculate 52 factorial (52!). This means multiplying all the numbers from 52 down to 1:
52! = 52 x 51 x 50 x … x 3 x 2 x 1
Once you calculate the factorial, you’ll find that there are approximately 8 x 10^67 possible ways to arrange a deck of cards! That’s a huge number, and factorials help us understand the vast number
of possibilities in various situations.
Factorials also come in handy when organizing objects in different orders. Let’s say you have a set of books on a shelf and you want to know how many ways you can arrange them. By using factorials,
you can easily calculate the number of possible arrangements.
Number of Books Possible Arrangements (Factorial)
As you can see from the table, the number of possible arrangements increases rapidly as the number of books increases. Factorials help us understand the different ways objects can be organized and
provide a mathematical foundation for these arrangements.
So, factorials may seem like a complex concept, but they have practical applications in various scenarios. By introducing factorials to children and providing real-life examples, we can help them
understand the importance and usefulness of factorials in everyday situations. Whether it’s organizing objects or calculating combinations, factorials provide a valuable mathematical tool to solve
puzzles and explore different possibilities.
Engaging Activities and Games
Make learning about factorials fun for kids by incorporating interactive activities and games. These hands-on approaches will keep young learners engaged and make the concept of factorials more
enjoyable to explore.
Create Factorial Flashcards
One effective way to teach factorials to kids is by creating flashcards. Design flashcards with numbers on one side and their corresponding factorials on the other. For example, one card could have
the number 3 on one side and “3!” on the other. Encourage children to match the numbers to their respective factorials, helping them visualize the concept and reinforce their understanding.
Design Factorial Puzzles or Riddles
Engage children’s problem-solving skills by designing puzzles or riddles that involve calculating factorials. For instance, create a crossword puzzle where the clues require calculating the
factorials of specific numbers. Alternatively, craft riddles that challenge kids to determine the factorial of a given number as part of deciphering the answer. These activities will encourage
critical thinking and make learning factorials a playful and interactive experience.
Visualize Factorials with Manipulatives
Another way to engage kids in learning factorials is to use objects or manipulatives to visualize the concept. Provide counting cubes or colored tiles that children can arrange and multiply to
demonstrate factorials. For instance, for 4!, children can use four cubes or tiles to represent 4 x 3 x 2 x 1, physically showing the multiplication process. This hands-on approach makes factorials
more tangible and helps children grasp the concept more effectively.
Integrating these engaging activities and games into your factorial lessons will spark children’s curiosity and make learning about factorials a fun-filled adventure.
Factorials and Probability
Show children how factorials can be used to calculate probabilities. Use simple examples like flipping a coin multiple times or rolling a dice. Explain that factorials help determine the number of
possible outcomes and the likelihood of specific events occurring. By connecting factorials to probability, children can see the practical application of this mathematical concept.
Understanding probability is an important skill, and factorials provide a useful tool for calculating it. For instance, let’s take the example of flipping a coin. When a fair coin is flipped once,
there are two possible outcomes: heads (H) or tails (T). However, if we want to calculate the probability of getting heads on two consecutive flips, we need to consider the number of possible
outcomes for both flips.
This is where factorials come into play. By multiplying the number of outcomes at each step, we can determine the total number of possible outcomes. In the case of flipping a coin twice, the
factorial of 2 (2!) represents the total number of possible outcomes:
2! = 2 x 1 = 2
So, there are 2 possible outcomes when flipping a fair coin twice: HH and TT. Each outcome has an equal probability of occurring, as the coin is fair. By introducing factorials in this context,
children can grasp the concept of probability and understand how factorials help calculate it.
Let’s take another example involving rolling a six-sided dice. When rolling a dice once, there are six possible outcomes: 1, 2, 3, 4, 5, or 6. If we want to calculate the probability of rolling a
specific number twice in a row, such as rolling two 2s, we need to consider the number of possible outcomes for each roll.
The factorial of 2 (2!) represents the total number of possible outcomes:
2! = 2 x 1 = 2
So, there are 2 possible outcomes when rolling a dice twice and aiming to roll two 2s: 2-2 or any other combination. The probability of rolling two 2s in a row is 1/36 since there are 36 possible
outcomes when rolling a dice twice. By using factorials, children can understand how to calculate probabilities and how they are interconnected with the mathematics of factorials.
Using Factorials for Multiple Events
Factorials become even more valuable when calculating the probability of multiple events. Consider a scenario where you want to calculate the probability of flipping a fair coin three times and
getting heads each time.
The factorial of 3 (3!) represents the total number of possible outcomes:
3! = 3 x 2 x 1 = 6
So, there are 6 possible outcomes when flipping a coin three times: HHH, HHT, HTH, HTT, THH, or TTH. Out of these 6 outcomes, only one satisfies the condition of getting heads each time (HHH).
Therefore, the probability of flipping a fair coin three times and getting heads each time is 1/6.
Introducing factorials in probability calculations can help children understand the concept of favorable outcomes and the calculation of probabilities. This knowledge can be applied to various
real-life situations, such as games, sports, and everyday decision-making.
Number of Events Factorial Total Possible Outcomes
1 1! 2
2 2! 4
3 3! 6
4 4! 24
5 5! 120
By using factorials, children can explore the fascinating world of probability and understand how it relates to factorials. They will gain a deeper appreciation for the mathematical concept of
factorials and its practical applications beyond mere calculation.
Encouraging Further Exploration
Foster a love for math and encourage children to explore factorials further. Recommending online resources, books, or educational games can help young learners delve deeper into the topic of
One valuable online resource is the Khan Academy, which offers interactive lessons, practice exercises, and videos to help kids understand factorials. Another excellent platform is Coolmath4kids,
where kids can explore factorial concepts through games and puzzles.
Encourage children to ask questions and think critically about factorials in various contexts. By promoting continued exploration, children can develop a stronger understanding of factorials and
expand their mathematical knowledge.
Here are some books that can supplement their learning:
1. “Math for Smarty Pants” by Marilyn Burns – This book introduces factorial concepts in a fun and engaging way, perfect for young learners.
2. “The Number Devil: A Mathematical Adventure” by Hans Magnus Enzensberger – This captivating story incorporates various mathematical concepts, including factorials, to inspire curiosity and
Remember, learning about factorials doesn’t have to end in the classroom. Encouraging children to explore math beyond their textbooks can ignite a lifelong passion for the subject.
Recommended Educational Games and Books for Teaching Factorials
Resource Description
Khan Academy An online platform offering interactive lessons, practice exercises, and videos to help kids understand factorials.
Coolmath4kids A website featuring games and puzzles that engage children in factorial concepts.
Math for Smarty Pants A book by Marilyn Burns that introduces factorials in a fun and engaging way for young learners.
The Number Devil: A Mathematical Adventure A captivating story by Hans Magnus Enzensberger that incorporates factorials and other mathematical concepts to spark curiosity and exploration.
By exploring these resources, children can enhance their understanding of factorials and enjoy the journey of mathematical discovery.
Teaching factorials to children can be an exciting and engaging experience with the use of creative methods. By breaking down the concept, using visual aids, and incorporating interactive activities,
young learners can develop a solid understanding of factorials and their practical applications.
Encourage children to explore further and continue their mathematical journey of discovery. By fostering a love for math and providing opportunities for hands-on learning, factorials can become an
exciting part of a child’s education. With these fun and interactive approaches, kids can learn and appreciate the wonders of factorials.
So, whether it’s through counting cubes, solving puzzles, or exploring probability, teaching factorials to young learners can create a solid foundation for their mathematical development. Let’s
inspire their curiosity, engage their minds, and unlock the world of factorials for kids.
How do I explain factorials to a child?
Factorials can be explained to children by breaking down the concept into smaller numbers and using visual aids. Start with simple examples and gradually increase the complexity to help them
understand the pattern in calculating factorials.
What is the factorial function?
The factorial function is represented by an exclamation point (!) and is used to calculate the product of all whole numbers from a given number down to 1. For example, 6! means multiplying all the
numbers from 6 to 1.
How can children understand factorials through examples?
Children can understand factorials by using examples. Start with smaller numbers like 1 and gradually move to larger factorials such as 4!. By going step-by-step and demonstrating the multiplication
process, children can grasp the concept of factorials.
What are the practical applications of factorials?
Factorials have practical applications in combinatorial analysis, such as calculating combinations and permutations. Children can understand this concept by relating it to real-life examples like
arranging objects in different orders or organizing a deck of cards.
How can I engage children in learning about factorials?
Make learning about factorials fun by incorporating interactive activities and games. Use flashcards, puzzles, or manipulatives to visualize factorials. These hands-on activities will help keep
children engaged and make the learning experience enjoyable.
How are factorials related to probability?
Factorials can be used to calculate probabilities. By using simple examples like flipping a coin multiple times or rolling a dice, children can understand how factorials help determine the number of
possible outcomes and the likelihood of specific events occurring.
How can I encourage children to explore factorials further?
Foster a love for math by recommending online resources, books, or educational games that delve deeper into the topic of factorials. Encourage children to ask questions and think critically about
factorials in various contexts to expand their mathematical knowledge.
How can I teach factorials to young learners?
Factorials can be taught to young learners by using creative methods such as breaking down the concept into smaller numbers, using visual aids, and incorporating interactive activities. By making the
learning experience enjoyable, children can grasp the concept of factorials more effectively.
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Imagine Sarah, a young researcher, intrigued by patterns in nature. She finds a link between math and the tangled...
Knot Theory: Untangling the Loops in Mathematics | {"url":"https://www.littleexplainers.com/how-to-explain-factorials-to-a-child/","timestamp":"2024-11-06T10:27:30Z","content_type":"text/html","content_length":"273801","record_id":"<urn:uuid:530fef45-bbbf-411b-a7ff-f5778048e474>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00001.warc.gz"} |
Determining Whether a Whole Number is a Solution to an Equation
Learning Outcomes
• Determine whether a number is a solution of an equation
Determine Whether a Number is a Solution of an Equation
Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable
that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!
Solution of an Equation
A solution to an equation is a value of a variable that makes a true statement when substituted into the equation.
The process of finding the solution to an equation is called solving the equation.
To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of [latex]x+2=7?[/latex] If you said [latex]5[/latex],
you’re right! We say [latex]5[/latex] is a solution to the equation [latex]x+2=7[/latex] because when we substitute [latex]5[/latex] for [latex]x[/latex] the resulting statement is true.
[latex]\begin{array}{}\\ \hfill x+2=7\hfill \\ \hfill 5+2\stackrel{?}{=}7\hfill \\ \\ \hfill 7=7\quad\checkmark \hfill \end{array}[/latex]
Since [latex]5+2=7[/latex] is a true statement, we know that [latex]5[/latex] is indeed a solution to the equation.
The symbol [latex]\stackrel{?}{=}[/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[/latex] or not-equal sign
Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true.
□ If it is true, the number is a solution.
□ If it is not true, the number is not a solution.
Determine whether [latex]x=5[/latex] is a solution of [latex]6x - 17=16[/latex].
Substitute [latex]\color{red}{5}[/latex] for x. [latex]6\cdot\color{red}{5}-17=16[/latex]
Multiply. [latex]30-17=16[/latex]
Subtract. [latex]13\not=16[/latex]
So [latex]x=5[/latex] is not a solution to the equation [latex]6x - 17=16[/latex].
try it
Determine whether [latex]y=2[/latex] is a solution of [latex]6y - 4=5y - 2[/latex].
Show Solution
try it
In the following video we show more examples of how to verify whether an integer is a solution to a linear equation. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/determining-whether-a-whole-number-is-a-solution-to-an-equation/","timestamp":"2024-11-10T12:50:21Z","content_type":"text/html","content_length":"52955","record_id":"<urn:uuid:076c4c56-d477-4f41-8ddf-23c9f8087be2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00364.warc.gz"} |
Some Problems in Graph Coloring: Methods, Extensions and Results
The « Habilitation à Diriger des Recherches » is the occasion to look back on my research work since the end of my PhD thesis in 2006. I will not present all my results in this manuscript but a
selection of them: this will be an overview of eleven papers which have been published in international journals or are submitted and which are included in annexes. These papers have been done with
different coauthors: Marthe Bonamy, Daniel Gonçalves, Benjamin Lévêque, Amanda Montejano, Mickaël Montassier, Pascal Ochem, André Raspaud, Sagnik Sen and Éric Sopena. I would like to thanks them
without whom this work would never have been possible. I also take this opportunity to thank all my other co-authors: Luigi Addario-Berry, François Dross, Louis Esperet, Frédéric Havet, Ross Kang,
Daniel Král’, Colin McDiarmid, Michaël Rao, Jean-Sébastien Sereni and Stéphan Thomassé. Working with you is always a pleasure ! Since the beginning of my PhD, I have been interested in various fields
of graph theory, but the main topic that I work on is the graph coloring. In particular, I have studied problems such as the oriented coloring, the acyclic coloring, the signed coloring, the square
coloring, . . . It is then natural that this manuscript gathers results on graph coloring. It is divided into three chapters. Each chapter is dedicated to a method of proof that I have been led to
use for my research works and that has given results described in this manuscript. We will present each method, some extensions and the related results. The lemmas, theorems, and others which I took
part are shaded in this manuscript. # The entropy compression method. In the first chapter, we present a recent tool dubbed the entropy compression method which is based on the Lovász Local Lemma.
The Lovász Local Lemma was introduced in the 70’s to prove results on 3-chromatic hypergraphs [EL75]. It is a remarkably powerful probabilistic method to prove the existence of combinatorial objects
satisfying a set of constraints expressed as a set of bad events which must not occur. However, one of the weakness of the Lovász Local Lemma is that it does not indicate how to efficiently avoid the
bad events in practice. A recent breakthrough by Moser and Tardos [MT10] provides algorithmic version of the Lovász Local Lemma in quite general circumstances. To do so, they used a new species of
monotonicity argument dubbed the entropy compression method. This Moser and Tardos’ result was really inspiring and Grytczuk, Kozik and Micek [GKM13] adapted the technique for a problem on combi-
natorics on words. This nice adaptation seems to be applicable to coloring problems, but not only, whenever the Lovász Local Lemma is, with the benefits of providing better bounds. For example, the
entropy compression method has been used to get bounds on non-repetitive coloring [DJKW14] that improve previous results using the Lovász Local Lemma and on acyclic-edge coloring [EP13]. In this
context, we developed a general framework that can be applied to most of coloring problems. We then applied this framework and we get the best known bounds, up to now, for the acyclic chromatic
number of graphs with bounded degree, non-repetitive chromatic number of graphs with bounded degree, facial Thue chromatic index of planar graphs, ... We also applied the entropy compression method
to problems on combinatorics on words: we recently solved an old conjecture on pattern avoidance. # Graph homomorphisms and graph colorings In this chapter, we present some notions of graph colorings
from the point of view of graph homomorphisms. It is well-known that a proper k-coloring of a simple graph G corresponds to a homomorphism of G to Kk. Considering homomorphisms from a more general
context, we get a natural extension of the classical notion of coloring. We present in this chapter the notion of homomorphism of (n,m)-colored mixed graphs (graphs with arcs of n different types and
edges of m different types) and the related notions of coloring. This has been introduced by Nešetřil and Raspaud [NR00] in 2000 as a generalization of the classical notion of homomorphism. We then
present two special cases, namely homomorphisms of (1, 0)-colored mixed graphs (which are known as oriented homomorphisms) and homomorphisms of (0,2)-colored mixed graphs (which are known as signed
homomorphisms). While dealing with homomorphisms of graphs, one of the important tools is the notion of universal graphs: given a graph family F, a graph H is F-universal if each member of F admits a
homomorphism to H. When H is F-universal, then the chromatic number of any member of F is upper-bounded by the number of vertices of H. We study some well-known families of universal graphs and we
list their structural properties. Using these properties, we give some results on graph families such as bounded degree graphs, forests, partial k-trees, maximum average degree bounded graphs, planar
graphs (with given girth), outerplanar graphs (with given girth), . . . Among others, we will present the Tromp construction which defines well known families of oriented and signed universal graphs.
One of our major contributions is to study the properties of Tromp graphs and use them to get upper bounds for the oriented chromatic number and the signed chromatic number. In particular, up to now,
we get the best upper bounds for the oriented chromatic number of planar graphs with girth 4 and 5: we get these bounds by showing that every graph of these two families admits an oriented
homomorphism to some Tromp graph. We also get tight bounds for the signed chromatic number of several graph families, among which the family of partial 3-trees which admits a signed homomorphism to
some Tromp graph. # Coloring the square of graphs with bounded maximum average degree using the discharging method The discharging method was introduced in the early 20th century, and is essentially
known for being used by Appel, Haken and Kock [AH77, AHK77] in 1977 in order to prove the Four- Color-Theorem. More precisely, this technique is usually used to prove statements in structural graph
theory, and it is commonly applied in the context of planar graphs and graphs with bounded maximum average degree. The principle is the following. Suppose that, given a set S of configurations, we
want to prove that a graph G necessarily contains one of the configuration of S. We assign a charge ω to some elements of G. Using global information on the structure of G, we are able to compute the
total sum of the charges ω(G). Then, assuming G does not contain any configuration from S, the discharging method redistributes the charges following some discharging rules (the discharging process
ensures that no charge is lost and no charge is created). After the discharging process, we are able to compute the total sum of the new charges ω∗(G). We then get a contradiction by showing that ω
(G) ̸= ω∗(G). Initially, the discharging method was used as a local discharging method. This means that the discharging rules was designed so that an element redistributes its charge in its
neighborhood. However, in certain cases, the whole graph contains enough charge but this charge can be arbitrarily far away from the elements that are negative. In the last decade, the global
discharging method has been designed. This notion of global discharging was introduced by Borodin, Ivanova and Kostochka [BIK07]. A discharging method is global when we consider arbitrarily large
structures and make some charges travel arbitrarily far along those structures. In some sense, these techniques of global discharging can be viewed as the start of the “second generation” of the
discharging method, expanding its use to more difficult problems. The aim of this chapter is to present this method, in particular some progresses from the last decade, i.e. global discharging. To
illustrate these progresses, we will consider the coloring of the square of graphs with bounded maximum average degree for which we obtained new results using the global discharging method. Coloring
the square of a graph G consists to color its vertices so that two vertices at distance at most 2 get distinct colors (i.e. two adjacent vertices get distinct colors and two vertices sharing a common
neighbor get distinct colors). This clearly corresponds to a proper coloring of the square of G. This coloring is called a 2-distance coloring. It is clear that we need at least ∆ + 1 colors for any
2-distance coloring since a vertex of degree ∆ together with its ∆ neighbors form a set of ∆ + 1 vertices which must get distinct colors. We investigate this coloring notion for graphs with bounded
maximum average degree and we characterize two thresholds. We prove that, for sufficiently large ∆, graphs with maximum degree ∆ and maximum average degree less that 3 − epsilon (for any epsilon > 0)
admit a 2-distance coloring with ∆ + 1 colors. For maximum average degree less that 4 − epsilon, we prove that ∆ + C colors are enough (where C is a constant not depending on ∆). Finally, for maximum
average degree at least 4, it is already known that C′∆ colors are enough. Therefore, thresholds of 3 − epsilon and 4 − epsilon are tight. | {"url":"https://hal-lirmm.ccsd.cnrs.fr/tel-01376199v1","timestamp":"2024-11-09T23:15:37Z","content_type":"application/xhtml+xml","content_length":"122918","record_id":"<urn:uuid:913d62a5-30ed-4066-8641-589685058b69>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00356.warc.gz"} |
The module seq provides the lightweight, generic sequence container Seq for unmovable data and is embedded into the program during compile time. Elements of Seq are stacked on top of each other.
Initially a sequence is empty. A longer sequence is constructed attaching a new head to the existing sequence, representing the tail.
Multiple sequences may share the same tail, permitting memory-efficient organisation of hierarchical data.
Put this in your Cargo.toml:
## Cargo.toml file
seq = "0.5"
The "default" usage of this type as a queue is to use Empty or ConsRef to construct a queue, and head and tail to deconstruct a queue into head and remaining tail of a sequence.
Constructing two sequences seq1 as [1,0] and seq2 as [2,1,0], sharing data with seq1
use seq::Seq;
// constructing the sequence 'seq1'
const seq1: Seq<i32> = Seq::ConsRef(1, &Seq::ConsRef(0, &Seq::Empty));
// construction the sequence 'seq2' sharing data with 'seq1'
const seq2: Seq<i32> = Seq::ConsRef(2, &seq1);
Deconstructing a sequence
use seq::Seq;
fn print_head<'a>(seq: &'a Seq<i32>) {
println!("head {}", seq.head().unwrap());
Extend an existing sequence. Note the lifetime of the return type matches the one of the tail.
use seq::Seq;
fn extend<'a>(head: i32, tail: &'a Seq<i32>) -> Seq<'a, i32> {
return Seq::ConsRef(head, tail);
Extend an existing sequence with dynamic element residing in heap-memory
use seq::Seq;
fn extend_boxed<'a>(head: i32, tail: &'a Seq<i32>) -> Box<Seq<'a, i32>> {
return Box::new(Seq::ConsRef(head, tail));
Iterate a sequence
use seq::Seq;
fn sum_up(seq: &Seq<i32>) -> i32 {
return seq.into_iter().fold(0, |x, y| x + y);
seqdef The seqdef! macro defines a stack-allocated sequence variable for the speficied data list, the last data item in the list will be the top most in the sequence.
SeqIterator The sequence iterator representation
Seq A single-ended, growable, unmovable queue of data, linking constant data with dynamic data.
empty Function returns static reference to empty list | {"url":"https://docs.rs/seq/latest/seq/","timestamp":"2024-11-05T05:44:31Z","content_type":"text/html","content_length":"26159","record_id":"<urn:uuid:93253cf4-9f56-4355-8fcc-d425c36f280a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00097.warc.gz"} |
Poker Hands - rankings and probability | HotSlots
Although understanding poker hands isn’t rocket science, it comes pretty close to that for people with no poker knowledge whatsoever.
The great thing about poker hands — and, by extension, poker hand rankings — is that some practice is all that is required before a baseline understanding can be achieved.
Whether it is a matter of keeping a poker hands ranking list or a poker hand rankings FAQ by your side while playing a poker game, it won’t take long to nail down how poker hands work.
In this post, we’ll take care of explaining everything that has to do with poker hands, including poker hand rankings, how poker hands compete with one another and the probability of making each
poker hand.
Our goal is to help you identify a winning poker hand when you see one; thus mastering the art of winning the pot.
Poker hands can be made up of just one to five cards. To do this, either or both of the two hole cards must be used, as well as the five community cards.
The goal in Texas hold’em is to form the strongest poker hand depending on the cards dealt, but this concept is inverted in lowball poker games.
Here’s where things get somewhat tricky for beginners: a specific combination of cards are needed to form the different poker hands.
Have a look at the traditional poker hand rankings below (listed from strongest to weakest) and how they can be made.
Consisting of the five consecutive cards A-K-Q-J-10 of the same suit, a royal flush is the best poker hand. A royal flush beats all the other hands, and it can only tie with another royal flush.
A straight flush — consisting of five consecutive cards of the same suit — is the second-best poker hand. A straight flush beats all the other hands except for a royal flush and a higher straight
FOUR OF A KIND (QUADS)
A four of a kind, also referred to as quads, features four cards of the same rank, such as four nines. This hand beats all the other hands, except for a royal flush, a straight flush or another four
of a kind of a better rank. For example, a four of a kind made up of four jacks will beat another four of a kind made up of four nines.
A full house is made up of a three of a kind combined with a pair of a different rank. For example, three eights and two aces would make a full house.
Only a royal flush, a straight flush and a four of a kind can beat a full house. If two players have a full house, the one with the highest card rankings will win.
The best way to make a full house is when a pocket pair is dealt. Pocket pairs complement a full house pretty well, since other players won’t stand the same chance of making an identical full house
that way.
Pocket aces, pocket Kings, pocket Queens, pocket Jacks and pocket tens are very desired hands for this very reason. Such hands can form a wide range of poker combinations, so try to make the most out
of them if you’re lucky enough to have them dealt to you.
A flush consists of five cards of the same suit, but they must not be consecutive cards. For example, 10, eight, five, three and two of diamonds would make a flush.
Although desirable, a flush is far from the most powerful hand in the game since it can be beaten by a royal flush, a straight flush, a four of a kind and a full house. Having said that, a flush can
beat a straight, a three of a kind, a two pair, a pair and a high card.
A straight consists of five consecutive cards of different suits. For example, a K-Q-J-10-9 of different suits would make a straight flush. Should they be of the same suit, those cards would make a
straight flush.
Straights beat a three of a kind, a two pair, a pair and a high card. A straight of a higher rank or any hand better than that will beat a straight.
You might come across the phrase ‘Broadway straight’, which refers to the best possible straight hand of 10 through ace.
Three cards of the same rank are required to make a three of a kind, which only beats three other hands: a two pair, a pair and a high card.
A two pair consists of one pair of the same rank and another pair of another rank. For example, two jacks and two queens. A two pair beats any one pair as well as high cards.
A pair simply consists of two cards of the same rank, such as two queens. A one pair beats a high card and, at most, a one pair of a lower rank.
A high card is the worst poker hand possible. It is made up of five cards that don’t form any of the hands listed above. A high card won’t beat any made hands except for another high card of a lower
Ties can be quite frequent in poker games, so it’s important to know what happens in the event that your poker hand ties.
In poker, ties are settled by what are known as kickers or high cards. The kicker refers to the cards in a poker hand that don’t contribute to the made hand.
For example, A-A-10-J-5 and A-A-10-6-3 feature one pair of aces. The rest of the cards are the tie-breakers, that is, the kickers.
In this case, the former poker hand wins since its kicker (J) beats the other hand’s kicker (6).
If the high card is the same for both hands, the subsequent high card would then be the kicker.
For example, in a showdown between A-A-10-J-5 and A-A-10-6-3 the kicker wouldn’t be the Jack since both hands feature it.
The kicker would therefore be the best-ranking card after that, which would see the former hand win thanks to its kicker (J) that beats the other hand’s kicker (6).
In cases when all kickers are identical, the hands are considered full ties. Should this happen, the pot would be split in equal value.
This tends to happen when players’ poker hands are made up of five cards, since there are fewer kickers to act as tie-breakers.
On the other hand, kickers are more abundant when it comes to a one pair or a three of a kind.
As a disclaimer, identifying the winning poker hand can still take quite a long time to get used to. Whether it is online poker or other poker games played in real life, you’ll need quite a lot of
practice to get the hang of how poker hands rival each other.
To save you the hassle of consulting the list of poker hands every time you play a poker game, we’ve taken the time to write a few pointers about why exactly some hands beat others.
We’d advise new poker players to keep this post handy while playing, since it can really help you understand each poker hand ranking quickly.
Simply put, the royal flush is the best poker hand. “Does a royal flush beat-” Yes, yes it does.
The reason why the royal flush is the best poker hand is that it is extremely rare.
We’ll delve into the exact probability of being dealt a royal flush later on, but just to paint a clear picture, we’ll say that some professional players fail to see a single royal flush once in
their poker careers.
What happens if two royal flushes collide? The pot would simply be split.
If you’re lucky enough to be dealt a royal flush, do not bother slow playing it. It’s important to play aggressively and rack up as big of a pot as possible, since you definitely wouldn’t want your
royal flush to go to waste!
The odds of making a straight flush are pretty low. Think about it: a straight flush is simply a royal flush with lower-ranking cards!
Does a straight flush beat a straight, a full house and the rest? The answer is yes. The only hand that beats a straight flush is a royal flush.
The odds of not only forming five cards of the same suit, but also of sequential order are extremely small — second-only to the odds of a royal flush.
A full house poker hand is thought of as one of the best hands in poker, but it can get beaten by three other hands: a royal flush, straight flush and four of a kind.
With that said, a full house still beats a large number of hands. A full house poker hand beats a straight, three of a kind, two pair, one pair and high card.
As a side note, the exact name of a full house depends on the cards used to form the poker hand.
For example, a full house consisting of J-J-J-5-5 would be called ‘Jacks full of fives’.
A flush in poker is quite an average-to-good hand, since it beats five hands and loses to the remaining four.
What exactly does a flush beat? A flush can beat a straight, three of a kind, two pair, one pair and a high card. However, it loses out to a royal flush, straight flush, four of a kind and full
As a side note, the exact name of a flush in poker depends on the cards used to form the poker hand.
For example, a flush consisting of A-J-10-5-3 would be called an ‘ace high flush’.
DOES A FLUSH BEAT A FULL HOUSE?
Although a flush wins against most hands, the same doesn’t hold for a stand-off against a full house.
When starting to play poker, the misconception that a flush is rarer than a full house can pop up, but this is not the case.
Despite being somewhat similar in terms of their probability of happening, a full house beats a flush since it comes by slightly less frequently than its counterpart.
A STRAIGHT BEATS A THREE OF A KIND
Why does a straight beat a three of a kind? The reason for it is that the outcome of combining five cards of sequential rank is lower than that of combining three cards of the same rank.
A three of a kind is often the winning poker hand, especially when there aren’t many other strong hand draws.
When it comes to misconceptions about poker hand rankings, we cannot avoid mentioning the assumption that forming two pairs is harder than forming a three of a kind.
That is not the case. A three of a kind beats a two pair since it is a marginally rarer hand. Just because a hand requires more cards to be made doesn’t mean that it is rarer than hands with fewer
The scenario we’ve described in this section perfectly sums this up; a two pair (consisting of four cards) is not rarer than a three of a kind (which consists of three cards), despite the fact that
it requires one more card to be made.
The high card is the worst poker hand, and there’s little you can do if you’re dealt one. Most high card hands tend to be folded before the river since they will lose out to any possible poker hand
combination — even a pair beats a high card!
A high card is usually only worth keeping in the game if:
• you have a Jack or higher.
• you don’t need to call large bets.
Sinking money into a high card is rarely a good idea, which is why most players tend to fold this poker hand when faced with a bet or raise.
People who hate math will most likely want to skip this part, but calculating the probability of a poker hand isn’t as mind-boggling as it sounds.
Think about it: understanding the probability of each poker hand will not only sate your curiosity, but it can help you understand why certain poker hands beat others.
Let’s start off with the basis: there are 52 cards in a deck, and in the context of Texas hold’em, five cards are needed to form a poker hand.
To calculate the probability of a specific hand, we must count the number of ways said hand can occur and divide the figure by the total number of possible five-card draws — a figure that stands at
Since we’re counting combinations (C), we’re looking for n objects taken r at a time, and this number of combinations can be expressed as n! / r!(n – r)!.
52C5 = 52! / 5!(52 – 5)! = 52! / 5!47! = 2,598,960
This formula can be used to count the number of possible five-card hands and the number of ways a particular hand can be dealt. To find probability, we divide the latter by the former.
Below, you’ll find a breakdown of each hand probability, with a table compiling each nugget of information listed below.
There are only four ways to make a royal flush — through each of the four suits. In order to calculate the probability of a royal flush, we must divide four by 2,598,960, placing the exact
probability at around 0.000154%, or in case you prefer odds, 649,739:1.
There are 36 possible ways to form a straight flush — nine times that of a royal flush. If we divide 36 by 2,598,960, we’ll get the exact probability to get a straight flush: 0.00139%, or
Although you’ll find that forming a four of a kind is much more likely compared to the two hands we’ve mentioned above, this hand is still extremely rare. With 624 possible combinations for a four of
a kind, the probability of coming across one is around 0.02401%, or 4,164:1.
There are 3,744 possible combinations to make a full house, putting the probability at 0.1441%, or 693.166 :1.
There are 5,108 ways to make a flush (excluding a royal and a straight flush). This puts the probability of forming a flush at 0.1965%, or 508.8019:1.
There are a good 10,200 ways to make a straight (excluding a royal flush and a straight flush). This puts the probability of forming a straight at 0.3925% or 253.8:1.
As we move on from the rare hands, we’ll get to see how easier it is to form the more common hands. Having said that, the probability of landing one is still low, statistically speaking.
This is the case with the three of a kind, where there are 54,912 ways to form this hand, which puts the probability of forming one at 2.1128%, or 46.32955:1.
There are 123,552 ways to form two pairs, putting the probability of forming one at 4.7539%, or 20.03535:1.
There always tends to be at least one player who forms one pair during a round of poker. The reason for that is because there are 1,098,240 ways to form one pair, putting the probability of forming
one at 42.2569%, or 2.366477:1.
There are 1,302,540 ways to form a high card in five-hand poker variants, putting the probability of forming one at 50.1177%, or 0.9953015:1.
The only reason this probability isn’t higher is that the probability of forming a winning combination takes up the rest of the odds.
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The royal flush is at the top of the list when it comes to the poker hand rankings. It has no close competitor, since the rest of the poker hands have a much higher probability of occurring.
The royal flush cannot be beaten; at most, it can tie to hands of equal value. In this case, an equal-value hand would be another royal flush.
It is very uncommon to see this hand, but it is known to feature in some very high-stakes tournaments — making the action even more thrilling to spectators.
The most promising royal flush draws are ace-king suited, ace-queen suited and ace-jack suited, since the player would then only need three of the five community cards dealt to make a royal flush.
The only time is it possible for two players to have a royal flush is if it is on the board, that is, if the A-K-Q-J-10 are the community cards.
Should this be the case, the hands will tie. It would then be up to the kickers to decide who of the two players takes the pot.
Whichever high card outranks the other will win the pot.
There are 10 poker hands that you should know about. Starting from the strongest to the weakest, these are: royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind,
two pair, one pair and high card.
That list is valid for traditional Texas hold'em variants which feature five community cards and two hole cards.
Learning all the poker hand rankings isn't a matter of how hard you study them but how many poker hands you actually play.
The fastest way to learn all the poker hand rankings is to keep a list of poker hands by your side when playing poker: be it live or online poker.
In time, you'll get used to the poker hand rankings and won't need to keep the list by your side.
Another way to learn the poker hands rankings is to keep in mind the probability of them occurring.
Although you don't have to remember the exact odds of the poker hands being dealt, you can keep in mind that straight flush and four of a kind are very rare hands, and they beat a full house, a
standard flush and all the other pair hands.
In Texas hold'em, there are 1,326 starting hands in poker. With that said, many of these hands have the same value since all the suit are worth the same.
For example, an ace-Queen of spades and an ace-Queen of diamonds have the exact same value.
Therefore, there are 169 non-equivalent starting hands in Texas hold 'em, which is the total count of 13 pocket pairs, 13 × 12 / 2 = 78 suited hands and 78 unsuited hands (13 + 78 + 78 = 169).
As a side note, the 's' and 'o' that are often used when describing starting hands refer to 'suited' and 'offsuit', respectively.
For example, a description of a starting hand could look like this: KJo, which refers to a King and a Jack offsuit.
The best possible starting hand in Texas hold'em is pocket aces. The ace is the strongest card in nearly all forms of poker, and starting with two of them puts you in a favourable early position.
With that said, there are still five community cards to come, which may possibly help your opponents more than you and cause them to overtake you by the time the showdown arrives.
The next most powerful starting hand is two Kings. Other good starting hands include two Queens, an ace and a King of the same suit and an Ace and King of different suits.
Two cards of the same suit greatly improve your chances of making a flush, although this doesn't mean that you should play every suited hand you are dealt.
The worst starting hand in Texas hold'em is a seven and two offsuit, since very few hands can be formed this way.
Some players think that a three and a two offsuit is the worst starting hand in the game, but this hand can potentially form a straight if the community cards are favourable.
The seven-two hand is, therefore the worst hand in the game since it offers very few possibilities to make a promising hand.
Having said that, that hand could still form powerful hands in the right circumstances, such as a full house, four of a kind and more.
This largely depends on the rules of the game being played. In Texas hold'em, five cards are used to form a hand, but this may differ in other variants.
There are two ways to play poker: tight and loose. Playing tight means being very particular about the hands you play, thus betting on a lower range of poker hands.
On the other hand, playing loose means pursuing a very high range of hands until later streets, where they'll mostly have to fold to stronger or better-played hands.
In normal poker games, we advise playing tight for a number of reasons.
First off, not all hands are worth playing, especially considering the number of chips that a weak hand can send to waste. Playing fewer hands will not only preserve your chip stack, but it will also
disguise your strong hands in the future.
Once you learn to play tight — that is, folding relatively weak hands and only pursuing hands on the stronger side — you'll want to play them aggressively.
Doing so will render you unpredictable to your opponents, since they'll have no idea whether you're sitting on a very strong hand or just air.
Such practice is called semi-bluffing, where a relatively promising hand is played extremely aggressively. If other players decide to call your ostensible bluff, they'll be greeted with a formidable
hand that can very easily overcome theirs.
The stakes are usually much higher in tournaments compared to normal poker games. As a result, players will tend to play tight, which refers to the practice of only pursuing hands that they believe
have good showdown value.
Having said that, many poker events have been won by what most consider low-value hands. We recommend assessing each situation and playing accordingly.
Another handy tip is to widen the range of hands to bet on. This is because the goal of most poker tournaments is to rack up enough chips for the bubble, at which point chips become ever so valuable.
The best strategy you could incorporate when playing poker or other cash games is to establish a budget and stick to it no matter what. This isn't just a poker strategy; sticking to a budget must be
done when playing any cash game out there.
Before looking to play cash games, you should have an accurate idea of the size of your budget, and you should stick to it no matter what the result of your wagering is.
When you start playing poker games, it isn't a good idea to constantly look to make a profit. You are better off establishing a budget and stretching your bankroll as much as possible while learning
about the game.
One of the most common poker mistakes out there is throwing bankroll management out of the window and going all-in in one final attempt to recoup past losses. This is a very bad idea and it often
ends up losing the player more money than anticipated.
Astray to what most players might think, a three pair poker hand does not qualify as a hand combination.
This is because a poker hand can only be made up of five cards, and three pairs would make use of six. | {"url":"https://hotslots.io/blog/poker/poker-hands/","timestamp":"2024-11-14T10:24:46Z","content_type":"text/html","content_length":"101635","record_id":"<urn:uuid:ed27911e-8f0d-4825-9680-9374a7ae8b10>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00091.warc.gz"} |
What is RSA & How Does An RSA Encryption Work?
An Ultimate Encryption Guide: What is RSA & How does an RSA work?
Information exchange is the most significant aspect of modern businesses. Every company needs to provide data to users. Therefore, organizations ask to store, analyze and exchange user data.
According to The State of Email Security 2022, over 64% of companies have experienced ransomware attacks.
The rise in cyberattacks and higher possibilities of data leaks during information exchange needs reliable security measures. So, how will you ensure the security of data exchanged?
Cryptographic encryptions can help businesses secure their data. However, algorithms used for encryptions were not designed for secure data exchange without compromising the speed. For example,
Diffie–Hellman (DH) algorithm used symmetric encryptions where two parties sharing data use a mutually agreed public security key.
However, it was effective encryption for non-sensitive data; crucial information needed a more reliable encryption protocol. This is where RSA changed the entire cybersecurity scenario through
asymmetric encryptions. It has two separate keys for encryptions and decryptions.
Here, we will discuss RSA. How it works. The benefits of using it for your security systems.
Let us first discuss what is RSA!
What is RSA?
Rivest-Shamir-Adleman (RSA) is an encryption algorithm that uses asymmetric encryptions to secure data exchange between two parties.
There are two types of encryptions symmetric and asymmetric. Both of them differ in security key pair usage. Symmetric encryptions have the same key for encryption and decryptions, while asymmetric
security keys are different.
Let’s take an example of a message you want to send to your friend to maintain secrecy. Now, if your friend lives on the other side of the world, the most logical way will be to send a message
through email. However, email is not a completely secure channel.
However, if you can code sign the email, encryption will secure the message. Now, if you are using symmetric encryptions, you and your friend need to share the security key beforehand.
Such a scenario of mutually sharing the security key may not create a problem for you to secure the message. However, RSA solves a problem through two separate security key pairs. First, the message
is encrypted using a public key and can be decrypted only through a private key.
RSA has several use cases, including digital signatures, SSL certificates, and the implementation of secure connections for VPN clients. It was a cryptographic algorithm first developed by Ron
Rivest, Adi Shamir, and Leonard Adleman in 1977. The MIT-based academics created a one-way function for data exchange which was difficult for hackers to invert.
Key takeaways
• RSA encryptions are based on a mathematical approach called the trapdoor function
• It does not allow a function to be reversed without additional information
• The private key of the recipient provides that additional information allowing access to change function
• RSA enables the conversion of text, data, and images into anonymous information
• It provides encryption of emails, hard disks, online messengers, etc.
Now that we know what is RSA let us understand how it works.
How an RSA Works?
RSA is an asymmetric system where there are two security pairs generated. For example, public and private keys are two different keys generated for the encryption and decryption of data. However, a
compelling use case of such encryptions is the usage of symmetric and asymmetric systems.
Many SSL certificate providers leverage RSA encryptions to encrypt symmetric keys. This provides a dual advantaFirst, symmetric encryptions have higher data transmission speeds, and an asymmetric
system ensures better security.
But how does RSA ensure better security?
Now that we know how an RSA works, let’s understand the algorithm,
RSA Encryption Algorithm
Step 1 Key generation
It is essential to understand that security key generation needs complex mathematical calculations. So, if you want to generate security keys, the RSA algorithm will use the following approach,
• Two large prime numbers, X and Y, are chosen.
• These prime numbers need to be large enough for hackers to figure out.
• Calculate the totient function: F(n)=(X-1)(Y-1)
• Select an integer e which will act as a co-prime number to F(n) and 1
• Calculate d, which you can find through an extended euclidean algorithm
Here it is essential that (n,e) will be your public key and (n,d) will be your private key.
Step 2 Encrypting data.
Suppose the data is in a plaintext format known as P. Ciphertext that encrypts data P can be calculated as below,
C=Pe *modn
Step 3 Decrypting the data.
A data recipient needs a private key(n,d) to decrypt the data.
P=Cd *modn
RSA Algorithm Example
Here is an example of the RSA algorithm through pseudocode,
int x = 60, int y = 50;
int n = x * y;
// n = 3000.
// compute the totient, phi
int F = (x-1)*(y-1);
// F =2891
int e = findCoprime(F);
// find an 'e' which is > 1 and is a co-prime of F.
// e = 20 satisfies the current values.
// Using the extended euclidean algorithm, find 'd' which satisfies
// this equation:
d = (1 mod (F))/e;
// d = 2981 for the example values.
public_key = (e=20, n=3000);
private_key = (d=2981, n=3000);
// Given the plaintext P=123, the ciphertext C is :
C = (123^20) % 3000 = 820;
// To decrypt the cipher text C:
P = (820^2981) % 3000 = 123;
Advantages of RSA encryptions
RSA encryptions are the backbone of many security models creating a secure environment for organizations to exchange information.
• It provides more robust encryption than other algorithms
• The one-way function ensures irreversible data transmissions
• Two security keys help secure the message
• A secure channel of communication helps in preventing man-in-the-middle attacks.
• Cracking RSA encryptions require solving complex mathematical problems
• Implementing the RSA encryption is easy
• Transmitting confidential data with RSA is highly secure
Where is RSA encryption used?
RSA encrypts secure messages, protects websites from cyber attacks, and improves customer trust. For example, if you want to send a message securely, you must use mathematical calculations to create
security keys. You need to know the recipient’s public key to encrypt data and provide a private key to decrypt it.
Similarly, if you want to code sign your applications to improve trust among customers and ensure that they can verify your identity, RSA can be helpful. You need first to hash the software and alert
your recipient with a hash file to match it later. Further, code signs your app using the certificate provided by a certificate authority and append the hashed file.
Getting a code signing certificate using RSA encryptions or SSL certificates begins with generating a certificate signing request(CSR). Then, depending on the type of code signing certificate, you
can provide details of an individual or organization to CA in CSR for the vetting process.
CA will verify details and issue a code signing certificate that uses RSA encryptions to secure data. Once you code the app, users can confirm their identity before downloading and installing the
You can also use RSA encryptions to,
• Implement RSA encryptions in systems like OpenSSL, WolfCrypt, Cryptlib, and other cryptographic repositories.
• Encrypt emails with RSA for higher security
• Establish a secure channel for TLS handshakes
• Secure VPN connections
RSA Vulnerabilities
RSA public keys result from two large prime numbers generated anonymously. So, the security of the algorithm depends on the prime factorization. In plain words, data security assumes that hackers
can’t determine two random prime numbers within a specific time.
In theory, the assumption of two large prime numbers being secure does not always stand the test of time. In other words, in real-life scenarios, hackers can attack security keys through
counter-algorithms like Greatest Common Deviser(GCD). So, there is no denying that public critical vulnerabilities are prevalent and can cause issues for RSA encryption.
Difference Between AES and RSA Algorithm
Comparison between AES and RSA algorithms is the debate of symmetric vs. asymmetric systems. Advanced Encryption Standard(AES) is based on encryption depending on the symmetric key. In other words,
the same key is used for both encryption and decryption. Implementing the AES encryptions requires putting data and security keys in the software files.
Once AES is implemented, you will have encrypted, which is decrypted by putting the same key again in the software file. RSA provides a robust security system through two different security keys. So,
you need a public key to encrypt and a private key to decrypt data.
Both have their advantages and disadvantages. For example, RSA is far more secure but slow. On the other hand, AES is fast but less secure. So, the best way to ensure higher security is to use both
You can use RSA algorithms to encrypt the security key of AES encryptions and ensure faster data transmission with enhanced protection.
Encryptions are one of the most effective security measures to ensure data protection. However, your systems will be vulnerable if you don’t choose a suitable cryptographic algorithm for your
encryptions. RSA is one such algorithm that can help you improve security and protect user data.
Other algorithms like AES are popular, but the one-way function makes asymmetric systems efficient. Here we have discussed how an RSA works to protect your system, stands compared to AES, and its
benefits. However, which encryption approach suits your system will depend on specific requirements. | {"url":"https://www.clickssl.net/blog/what-is-rsa","timestamp":"2024-11-09T19:45:09Z","content_type":"text/html","content_length":"106182","record_id":"<urn:uuid:4ceb86f6-af3e-470e-94df-40002ff1399e>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00578.warc.gz"} |
Inverse of a percentage math
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Author Message
reids Posted: Tuesday 06th of Apr 11:09
Well there are just two people who can help me out at this point in time, either it has to be some math guru or it has to be the Almighty himself. I’m fed up of trying to solve problems
on inverse of a percentage math and some related topics such as perfect square trinomial and distance of points. I have my finals coming up in a a few days from now and I don’t know what
to do ? Is there anyone out there who can actually spare some time and help me with my problems ? Any sort of help would be highly appreciated .
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Vofj Posted: Wednesday 07th of Apr 10:55
Timidrov You can check out Algebrator. This software literally helps you solve questions in math very fast. You can plug in the questions and this program will go through it with you step by step
so you will be able to understand easily as you solve them. There are some demos available so you can also take a look and see how incredibly helpful the program is. I am sure your
inverse of a percentage math can be solved faster here.
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cufBlui Posted: Thursday 08th of Apr 16:30
Even I’ve been through that phase when I was trying to figure out a solution to certain type of questions pertaining to graphing function and y-intercept. But then I found this piece of
software and it was almost like I found a magic wand. In the blink of an eye it would solve even the most difficult questions for you. And the fact that it gives a detailed and elaborate
explanation makes it even more handy. It’s a must buy for every algebra student.
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Koem Posted: Saturday 10th of Apr 15:04
I remember having often faced problems with graphing parabolas, angle complements and proportions. A really great piece of algebra program is Algebrator software. By simply typing in a
problem from workbook a step by step solution would appear by a click on Solve. I have used it through many math classes – Basic Math, Algebra 2 and Pre Algebra. I greatly recommend the
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Cnetz Posted: Saturday 10th of Apr 16:15
This sounds almost incredible. Can you tell me where I can obtain the program?
From: fo2
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nedslictis Posted: Monday 12th of Apr 13:12
This one is actually quite different. I am recommending it only after trying it myself. You can find the information about the software at https://softmath.com/algebra-policy.html.
Back to top | {"url":"https://www.softmath.com/algebra-software/exponential-equations/inverse-of-a-percentage-math.html","timestamp":"2024-11-12T06:30:06Z","content_type":"text/html","content_length":"43487","record_id":"<urn:uuid:f43f137f-b6cb-4c24-b53b-c47308d1e970>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00875.warc.gz"} |
Calculus: Early Transcendentals (4th Edition) - eBook
The author’s objective for Calculus: Early Transcendentals, 4th Edition, (PDF) is that it’s well written, could be easily read by a calculus student and would stimulate them to engage in the material
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Idris: Using the properties of simple constructors for equality and inequality
Among the first question I asked anyone about idris was about proofs that end in having to prove that two things aren’t equal. Idris itself stops being particularly helpful in those cases, giving the
weird feeling that you’re just on your own. In some cases you can provide positive proof of the opposite of the required proof and save yourself. In some cases, some rewriting gives a contradiction
that idris itself is able to turn into a self evident falsehood.
I wrote a function that, for bools, inverts a proof that two things aren’t equal into a proof that the opposite of one is equal to the other, and leveraged that plenty in the idris I’ve written.
Early on I received some advice that I definitely didn’t fully understand, but came in useful as the foundation of other proof code due to being able to somewhat magically help me combine deciadable
cases on a constructor with two arguments.
I’ve been working on providing proof of more properties of this object recently and I finally understand this, and can explain it hopefully in plain language that’s helpful.
My nat-like binary numbers (ugly but functional proof of interop with nat)
data Bin = O Nat Bin | BNil
First let’s start with what I was given:
— Thanks: https://www.reddit.com/r/Idris/comments/8yv5fn/using_deceq_in_proofs_extracting_and_applying_the/e2e8a6l/?context=3
O_injective : (O a v = O b w) -> (a = b, v = w)
O_injective Refl = (Refl, Refl)total aNEBMeansNotEqual : (a : Nat) -> (b : Nat) -> (v : Bin) -> Dec (a = b) -> Dec (O a v = O b v)
aNEBMeansNotEqual a b v (Yes prf) = rewrite prf in Yes Refl
aNEBMeansNotEqual a b v (No contra) =
No $ \prf =>
let (ojAB, _) = O_injective prf in
contra ojABtotal vNEWMeansNotEqual : (v : Bin) -> (w : Bin) -> Dec (v = w) -> Dec (O Z v = O Z w)
vNEWMeansNotEqual v w (Yes prf) = rewrite prf in Yes Refl
vNEWMeansNotEqual v w (No contra) =
No $ \prf =>
let (_, ojVW) = O_injective prf in
contra ojVW
I’d like to dissect what’s going on here and really dig into how this helps with many proof obligations one might run into on Bin (and really any constructor with multiple arguments).
O_injective is relying on idris to take a single proof step, namely that because a pairs with b and v pairs with w in a proof, then it can infer that a=b and v=w. Since the proof is a function, it
conveniently returns both so we can destructure them. Without this, we’d need to use contradictions and either ‘with’ rules or induction to prove them.
The following decision rules follow from this, using the indicated destructured part of the proof to refute the clause with the given contradiction (either the shift or the tail), giving us an easy
way to turn a decision about inequality in one part of the constructor to a decision about equality in the constructor as a whole.
It didn’t occur to me at the time why the function is called ‘injective’ because my math is very rusty, but upon refreshing, I realized two things:
• Simple constructors are implicitly injective because each unique set of argument values results in a unique outcome in the space of the constructor’s type. Despite having this code for a while,
my experiences with more complex proofs suggested that idris would require me to rewrite these myself. I admittedly was weirdly blind to it for a while.
• Not only simply injective, constructors are also bijective: each unique value of the constructor fully populated also represents a unique set of inputs, and idris also allows us to write this:
O_bijective : (a = b) -> (c = d) -> (O a c = O b d)
O_bijective Refl Refl = Refl
With these, we can compose and decompose proofs about constructors without having to go through labor to show the outcome of each case. I wanted to call it out specifically because it’s very useful
and might be missed unless one knows this pattern can be used. | {"url":"https://prozacchiwawa.medium.com/idris-using-the-properties-of-simple-constructors-for-equality-and-inequality-13dea8292af8","timestamp":"2024-11-05T16:06:47Z","content_type":"text/html","content_length":"97737","record_id":"<urn:uuid:c67a6d5e-696b-405d-b076-f5338caca67c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00827.warc.gz"} |
Doubling Your Money
One of my first real articles was about the Rule of 72 (more Einstein Finance). The Rule of 72 is a simple heuristic for figuring out when your money can double in value, given a specific
(compounding) growth rate.
Let’s make sure we are clear: the model I am working from assumes:
• I am putting an amount into a savings vehicle and not adding any more (so the model is flawed already but stay with me on this)
• The rate of return stays the same throughout the period (again flawed)
When I say doubling, based on those assumptions, it is when the initial investment is now worth twice what it was initially.
I attempted to clarify my initial post with a very grainy-looking graph in Einstein: The Rule of 72 a few years later, but I think we can do better than that now.
First a simple table following the formula:
Where T is the number of period and r is the interest rate compounded in that period, and the ln() function is the Natural Log (mascot of the University of Waterloo MathSoc).
Rate (r) Period to Double (T) in years
0.50% 139.0
1.00% 69.7
1.50% 46.6
2.00% 35.0
2.50% 28.1
3.00% 23.4
3.50% 20.1
4.00% 17.7
4.50% 15.7
5.00% 14.2
5.50% 12.9
6.00% 11.9
6.50% 11.0
7.00% 10.2
7.50% 9.6
8.00% 9.0
8.50% 8.5
9.00% 8.0
9.50% 7.6
10.00% 7.3
10.50% 6.9
11.00% 6.6
11.50% 6.4
12.00% 6.1
12.50% 5.9
13.00% 5.7
13.50% 5.5
14.00% 5.3
14.50% 5.1
15.00% 5.0
12% or higher and doubling in less than 6 years.
Simple calculation, isn't it? You can see that it doesn't take long to go from taking 135 years to double your investment to 15 years to double your investment (0.5% to 4.5%), but it is easier to see
in a graph how this all works:
This is a straightforward model, given very few folks dump a load of money into a single investment and let it grow with no intervention. Still, it is worthwhile to understand that when someone talks
about getting a 4.0% growth on their investment, that means their investment will double in 18 years (or so). It is a handy model to remember.
Feel Free to Comment
1. I remember when I first learned the rule of 72. It was comparable to my fascination of ants. Seriously, a colony of ants is quite amazing. Did you know they even build a landfill for all of their
waste? Those little guys are quite intriguing…
2. The last time I was in the MathSoc office at University of Waterloo (and that was a looong time ago) the Natural Log was gone, replaced by an empty box of Tide, with a sign that said something
like “Natural Log, recycled”.
1. The office has moved I believe as well. I was a Computer Science Club member, so we looked across the hall at MathSoc a great deal. The CSC was the cooler place to be, I always thought.
4. Very clear! Thanks for facilitating my financial education.
So if I take my current investment portfolio and say I’m earning 8%, it will double in 9 years. Now I need to check if I’m getting that rate of growth. I’m still contributing also.
5. The rule of 72 is great tool for dividend growth stocks I find cause of the real returns of dividends. Great recap !!!
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Spatial derivatives of Gaussian process
Gaussian correlation derivative
The equations above work for any covariance function, but then we need to have the derivatives of the covariance function with respect to the spatial variables. Here we calculate these derivatives
for the Gaussian, or squared exponential, correlation function \(R\).
\[R(x, u) = \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell - u_{\ell})^2 \right) \]
\[\frac{\partial R(x, u)}{\partial x_i} = -2\theta_i (x_i - u_{i}) \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell - u_{\ell})^2 \right) \] The second derivative with respect to the same dimension
\[ \frac{\partial^2 R(x, u)}{\partial x_i^2} &= \left(-2\theta_i + 4\theta_i^2 (x_i - u_{i})^2 \right) \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell - u_{\ell})^2 \right) \\ &= \left(-2\theta_i +
4\theta_i^2 (x_i - u_{i})^2 \right) R(x, u) \]
The cross derivative for \(i \neq k\) is \[ \frac{\partial^2 R(x, u)}{\partial x_i \partial x_k} &= 4\theta_i \theta_k (x_i - u_{i}) (x_k - u_{k}) \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell -
u_{\ell})^2 \right) \\ &= 4\theta_i \theta_k (x_i - u_{i}) (x_k - u_{k}) R(x, u) \]
The second derivative with respect to each component, which is needed for the gradient distribution below, is the following for the same dimension \(i\):
\[ \frac{\partial^2 R(x, u)}{\partial x_i \partial u_i} &= \left(2\theta_i - 4\theta_i^2 (x_i - u_{i})^2 \right) \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell - u_{\ell})^2 \right) \\ &= \left(2\
theta_i - 4\theta_i^2 (x_i - u_{i})^2 \right) R(x, u) \]
And the following for \(i \neq k\):
\[ \frac{\partial^2 R(x, u)}{\partial x_i \partial u_k} &= -4\theta_i \theta_k (x_i - u_{i}) (x_k - u_{k}) \exp \left(-\sum_{\ell =1}^d \theta_\ell (x_\ell - u_{\ell})^2 \right) \\ &= -4\theta_i \
theta_k (x_i - u_{i}) (x_k - u_{k}) R(x, u) \]
Gradient distribution
A big problem with using the gradient of the mean function of a GP is that it doesn’t give an idea of its distribution/randomness. The mean of the gradient could be predicted to be zero in a region
where the surface is not flat simply because it has no information in that region yet.
First we want to know what type of distribution the gradient follows. Since the derivative is a linear operator, and a linear operator applied to a normal r.v. is also normal, the gradient must be a
multivariate random variable. For a more intuitive explanation, consider a \(\delta\) approximation to the gradient.
\[ \frac{\partial y(x)}{\partial x} = \lim_{\delta \rightarrow 0} \frac{1}{\delta} \left( \begin{bmatrix} y(x+\delta e_1) \\ \vdots \\ y(x+\delta e_d) \end{bmatrix} - \begin{bmatrix} y(x) \\ \vdots \
\ y(x) \end{bmatrix} \right)\]
For any finite \(\delta\), this vector’s components are a linear combination of normal random variables, and thus the vector has a multivariate distribution. We still need to show that in the limit
as \(\delta \rightarrow 0\), it retains a multivariate distribution, but I won’t do that here.
Thus the gradient follows a multivariate distribution, and now we will find its mean and covariance.
Gradient expected value
The expected value of the gradient is easily found since it equals the gradient of the expected value. This is true because both gradients and expected value and linear operators and thus can be
\[ E \left[\frac{\partial y(x)}{\partial x} \right] = \frac{\partial E[y(x)]}{\partial x} \\ = \frac{\partial \Sigma(x,X)}{\partial x_i} \Sigma_X^{-1}(Y_X - \mu_X) \]
Variance of the gradient
The variance is harder to calculate. I used this reference (McHutchon, n.d.) to get started, but the derivation presented here is much simpler.
We need to find the covariance of the gradient vector of \(y\) at a point \(x\). \[ \text{Cov}(\nabla_x y(x))\] The \((i,j)\) entry of this matrix is \[ \text{Cov}(\frac{\partial y(x)}{\partial x_i},
\frac{\partial y(x)}{\partial x_j})\] We can write this as a limit.
\[ \lim_{\delta \rightarrow 0} \text{Cov}\left(\frac{y(x+\delta e_i) - y(x)}{\delta}, \frac{y(x+\delta e_j) - y(x)}{\delta}\right)\] The covariance function is bilinear, so we can split it into four
\[ \lim_{\delta \rightarrow 0} \frac{1}{\delta^2} \left( \text{Cov}\left[y(x+\delta e_i), y(x+\delta e_j)\right] - \text{Cov}\left[y(x+\delta e_i), y(x)\right] \\ - \text{Cov}\left[y(x), y(x+\delta
e_j)\right] + \text{Cov}\left[y(x), y(x)\right] \right) \]
This can be recognized as the second order derivative of \(\text{Cov}(y(u), y(v))\) with respect to \(u_i\) and \(v_j\) evaluated at \(u=v=x\). See finite difference differentiation for more details.
We have to use \(u\) and \(v\) instead of a single \(x\) to be clear which component of the covariance function is being differentiated.
Thus we have the following. It looks obvious, so I’m not sure I even needed the previous step. \[ \text{Cov}(\frac{\partial y(x)}{\partial x_i},\frac{\partial y(x)}{\partial x_j}) = \frac{\partial \
text{Cov}(y(u), y(v))}{\partial u \partial v} |_{u=x, v=x}\] Let \(U\) be the matrix with rows \(u\) and \(v\). Recall that \[ \text{Cov}\left( \begin{bmatrix} y(u) \\ y(v) \end{bmatrix} \right) = \
Sigma_{U} - \Sigma_{UX} \Sigma_{X}^{-1} \Sigma_{XU}\] The \((1,2)\) element of this is the covariance we are looking for. \[ \text{Cov}(y(u), y(v)) = \Sigma_{u,v} - \Sigma_{u,X} \Sigma_X^{-1} \Sigma_
{X,v}\] Now we need to differentiate this with respect to \(u_i\) and \(v_j\).
\[ \text{Cov} \left( \frac{\partial y(x)}{\partial x_i},\frac{\partial y(x)}{\partial x_j} \right) = \frac{\partial^2 \Sigma_{u,v}}{\partial u_i \partial v_j} - \frac{\partial \Sigma_{u,X}}{\partial
u_i} \Sigma_X^{-1} \frac{\partial \Sigma_{X,v}}{\partial v_j} |_{u=x,v=x}\] Therefore we have found the distribution of the gradient.
\[ \frac{\partial y(x)}{\partial x} | Y_X \sim N(\frac{\partial \Sigma(x,X)}{\partial x}^T \Sigma_X^{-1} (Y_X - \mu_X), \frac{\partial^2 \Sigma(x_1, x_2)}{\partial x_1 \partial x_2} + \frac{\partial
\Sigma(x_1, X)}{\partial x_1} \Sigma_x^{-1} \frac{\partial \Sigma(X, x_2)}{\partial x_2} |_{x_1=x,x_2=x}) \]
Distribution of the gradient norm squared
Let \[g(x) = \frac{\partial y(x)}{\partial x} | Y_X \] \[g(x) = \nabla_x y(x) | Y_X \] This is a vector. We are often interested in the gradient norm, or its square, \(g(x)^T g(x) = ||g(x)||^2\).
This is the sum of correlated squared normal variables, i.e. the sum of correlated chi-squared variables. Since \(g(x)\) has a multivariate normal distribution, the square of its norm is probably
distributed according to some kind of chi-squared distribution. We can first try to find its expectation using.
Mean of gradient norm squared
\[ E \left[ ||g(x)||^2 \right] = E \left[ \sum_{i=1}^d g_i(x)^2 \right] = \sum_{i=1}^d E \left[g_i(x)^2 \right]\]
\[ E \left[ g_i(x)^2 \right] = \text{Var} \left[g_i(x) \right] + E \left[g_i(x) \right]^2\] We just found these variances and expectations, so this is a closed form equation.
Full distribution of gradient norm squared
In this section we will derive the full distribution of \(||g(x)||^2\) following the fantastic answer from this Math Stack Exchange answer (halvorsen 2015), all credit for this section goes there.
The general idea is that if we could decorrelate the chi-squared variables, then it would be a sum of chi-squared variables, which is a known distribution that is easy to work with.
General derivation
Let \(X\) be a random vector with multivariate distribution with mean \(\mu\) and covariance matrix \(\Sigma\). \[ X \sim N(\mu, \Sigma)\] Let \(Q(X)\) be a quadratic form of \(X\) defined by the
matrix \(A\). \[Q(X) = X^TAX\] Let \(Y = \Sigma^{-1/2}X\). \(Y\) is a decorrelated version of \(X\), so \(Y \sim N(\Sigma^{-1/2} \mu, I)\).
Let \(Z = Y - \Sigma^{-1/2}\mu\). \(Z\) is a version of \(Y\) with mean zero, so \(Z \sim N(0, I)\).
Now we have
\[Q(X) = X^TAX = (Z + \Sigma^{-1/2} \mu)^T \Sigma^{1/2} A \Sigma^{1/2} (Z + \Sigma^{-1/2} \mu)\]
The spectral theorem allows the middle term to be decomposed as below, where P is the orthogonal matrix of eigenvectors and \(\Lambda\) is the diagonal matrix with positive diagonal elements \(\
lambda_1, \ldots, \lambda_n\).
\[\Sigma^{1/2} A \Sigma^{1/2} = P^T \Lambda P \] Let \(U=PZ\). Since \(P\) is orthogonal and using the distribution of \(Z\), we also have that \(U \sim N(0, I)\).
Putting these together, we can change \(Q(X)\) as follows.
\[ Q(X) = X^TAX = (Z + \Sigma^{-1/2} \mu)^T \Sigma^{1/2} A \Sigma^{1/2} (Z + \Sigma^{-1/2} \mu) \\ = (Z + \Sigma^{-1/2} \mu)^T P^T \Lambda P (Z + \Sigma^{-1/2} \mu) \\ = (PZ + P\Sigma^{-1/2} \mu)^T \
Lambda (PZ + P\Sigma^{-1/2} \mu) \\ = (U + b)^T \Lambda (U + b) \\ \] Here we defined \(b= P \Sigma^{-1/2} \mu\).
Since \(\Lambda\) is diagonal, we have \[Q(X) = X^TAX = \sum_{j=1}^n \lambda_j (U_j + b_j)^2\]
The \(U_j\) have standard normal distribution and are independent of each other. \((U_j + b_j)^2\) is thus the square of normal variable with mean \(b_j\) and variance 1, meaning it has a noncentral
chi-squared distribution with mean \(b_j^2 +1\) and variance \(4b_j^2 + 2\). Thus \(Q(X)\) is distributed as a linear combination of \(n\) noncentral chi-squared variables. Since \(\lambda_j\) is
different for each, \(Q(X)\) does not have a noncentral chi-squared distribution. However, we can easily find its mean, variance, sample from it, etc.
The mean and variance of \(Q(X)\) are \[ E[Q(X)] &= \sum_{j=1}^n \lambda_j E[(U_j + b_j)^2] \\ &= \sum_{j=1}^n \lambda_j (b_j^2 + 1) \\ \]
\[ \text{Var}[Q(X)] &= \sum_{j=1}^n \lambda_j^2 \text{Var} \left[ (U_j + b_j)^2 \right] \\ &= \sum_{j=1}^n \lambda_j^2 (4b_j^2 + 2) \\ \]
Relating this back to \(||g||^2\)
For \(||g||^2\) we had \(A=I\), \(\mu=\frac{\partial \Sigma_{x,X}}{\partial x}^T \Sigma_X^{-1} (Y_X - \mu_X)\), and \(\Sigma=\frac{\partial^2 \Sigma(x_1, x_2)}{\partial x_1 \partial x_2} + \frac{\
partial \Sigma(x_1, X)}{\partial x_1} \Sigma_x^{-1} \frac{\partial \Sigma(X, x_2)}{\partial x_2}\)
Thus we need \(P\) and \(\Lambda\) from the eigendecomposition \(\Sigma = P^T \Lambda P\). \(\Lambda\) will give the \(\lambda_j\).
Then we need \(b = P \Sigma^{-1/2} \mu\). We can calculate \(\Sigma^{-1/2}\) as the square root of the matrix \(\Sigma^{-1}\) using the eigendecomposition again. We can decompose \(\Sigma^{-1} = W^
TDW\) (using different symbols from before to avoid confusion), where \(D\) is diagonal and W is orthogonal. Then \(\Sigma^{-1/2} = W^T S W\), where \(S\) is the diagonal matrix whose elements are
the square roots of the elements of \(D\). This can easily be proven as follows. \[ \Sigma^{-1/2} \Sigma^{-1/2} = W^T S W W^T S W = W^T S S W = W^T D W = \Sigma^{-1}\]
Now that we know how to calculate \(\lambda\) and \(U\), we can calculate the distribution of \(||g||^2\). | {"url":"https://cran.r-project.org/web/packages/GauPro/vignettes/surface_derivatives.html","timestamp":"2024-11-08T07:53:57Z","content_type":"text/html","content_length":"27440","record_id":"<urn:uuid:f61c12db-7cad-4795-9297-700438f7ed3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00288.warc.gz"} |
Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
We use the optimal fluctuation method to evaluate the short-time probability distribution P (H, L, t) of height at a single point, H = h (x = 0, t), of the evolving KardarParisiZhang (KPZ) interface
h (x, t) on a ring of length 2L. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards
Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of P(H). At large L/√t the faster-decaying tail has a double structure: it is L-independent, -lnP ∼ |H
|^5/2 /t^1/2, at intermediately large |H|, and L-dependent, -lnP ∼ |H|^2 L/t, at very large |H|. The transition between these two regimes is sharp and, in the large L/√t limit, behaves as a
fractional-order phase transition. The transition point H = H+ c depends on L/√t. At small L/√t, the double structure of the faster tail disappears, and only the very large-H tail, -lnP ∼ |H|^2 L/t,
is observed. The slower-decaying tail does not show any L-dependence at large L/√t, where it coincides with the slower tail of the GOE TracyWidom distribution. At small L/√t this tail also has a
double structure. The transition between the two regimes occurs at a value of height H = 1DUMMY-c which depends on L/√t. At L/√t → 0 the transition behaves as a mean-field-like second-order phase
transition. At |H| < |1DUMMYc | the slower tail behaves as -lnP - |H|^2 L/t, whereas at |H| > |Hc it coincides with the slower tail of the GOE TracyWidom distribution.
• growth processes
• large deviations in non-equilibrium systems
• macroscopic fluctuation theory
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Statistics and Probability
• Statistics, Probability and Uncertainty
Dive into the research topics of 'Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/finite-size-effects-in-the-short-time-height-distribution-of-the-","timestamp":"2024-11-09T03:33:09Z","content_type":"text/html","content_length":"59293","record_id":"<urn:uuid:fca23365-4966-4e58-b9af-c2d25187c4a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00322.warc.gz"} |
3.5.10 Operations of Fixed Point Types
Static Semantics
1 The following attributes are defined for every fixed point
2/1 S'Small — S'Small denotes the small of the type of S. The value of this attribute is of the type universal_real. {specifiable (of Small for fixed point types) [partial]} {Small clause} Small may
be specified for nonderived ordinary fixed point types via an attribute_definition_clause (see §13.3); the expression of such a clause shall be static.
3 S'Delta — S'Delta denotes the delta of the fixed point subtype S. The value of this attribute is of the type universal_real.
3.a Reason: The delta is associated with the subtype as opposed to the type, because of the possibility of an (obsolescent) delta_constraint.
4 S'Fore — S'Fore yields the minimum number of characters needed before the decimal point for the decimal representation of any value of the subtype S, assuming that the representation does not
include an exponent, but includes a one-character prefix that is either a minus sign or a space. (This minimum number does not include superfluous zeros or underlines, and is at least 2.) The value
of this attribute is of the type universal_integer.
5 S'Aft — S'Aft yields the number of decimal digits needed after the decimal point to accommodate the delta of the subtype S, unless the delta of the subtype S is greater than 0.1, in which case the
attribute yields the value one. [(S'Aft is the smallest positive integer N for which (10**N)*S'Delta is greater than or equal to one.)] The value of this attribute is of the type universal_integer.
6 The following additional attributes are defined for every decimal fixed point subtype S:
7 S'Digits — S'Digits denotes the digits of the decimal fixed point subtype S, which corresponds to the number of decimal digits that are representable in objects of the subtype. The value of this
attribute is of the type universal_integer. Its value is determined as follows: {digits (of a decimal fixed point subtype)}
Implementation Note:
Although a decimal subtype can be both
-constrained and digits-constrained, the digits
is intended to control the Size attribute of the subtype. For decimal types, Size can be important because input/output of decimal types is so common.
• 10 The digits of a base subtype is the largest integer D such that the range –(10**D–1)*delta .. +(10**D–1)*delta is included in the base range of the type.
11 S'Scale — S'Scale denotes the scale of the subtype S, defined as the value N such that S'Delta = 10.0**(–N). {scale (of a decimal fixed point subtype)} [The scale indicates the position of the
point relative to the rightmost significant digits of values of subtype S.] The value of this attribute is of the type universal_integer.
11.a Ramification: S'Scale is negative if S'Delta is greater than one. By contrast, S'Aft is always positive.
12 S'Round — S'Round denotes a function with the following specification:
function S'Round(X : universal_real)
return S'Base
14 The function returns the value obtained by rounding X (away from 0, if X is midway between two values of the type of S).
15 (39) All subtypes of a fixed point type will have the same value for the Delta attribute, in the absence of delta_constraints (see §J.3).
16 (40) S'Scale is not always the same as S'Aft for a decimal subtype; for example, if S'Delta = 1.0 then S'Aft is 1 while S'Scale is 0.
17 (41) {predefined operations (of a fixed point type) [partial]} The predefined operations of a fixed point type include the assignment operation, qualification, the membership tests, and explicit
conversion to and from other numeric types. They also include the relational operators and the following predefined arithmetic operators: the binary and unary adding operators – and +, multiplying
operators, and the unary operator abs.
18 (42) As for all types, objects of a fixed point type have Size and Address attributes (see §13.3). Other attributes of fixed point types are defined in §A.5.4.
Wording Changes from Ada 95
18.a/2 Corrigendum: Clarified that small may be specified only for ordinary fixed point types. | {"url":"https://aada.m2osw.com/alexis-ada-reference-manual/section-3-declarations-and-types/35-scalar-types/3510-operations-fixed-p","timestamp":"2024-11-14T18:54:49Z","content_type":"application/xhtml+xml","content_length":"34374","record_id":"<urn:uuid:4ab234eb-a6d5-4120-bc66-b59962d6cf24>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00374.warc.gz"} |
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TY - DATA T1 - Hybridization Toolbox for Model Predictive Control PY - 2023/09/21 AU - Leila Gharavi UR - DO - 10.4121/2a4a7bed-63b9-43d9-a4d2-192bc9163dd1.v1 KW - Hybridization Framework KW - Model
Predictive Control KW - Evasive Maneuvers KW - Vehicle Control KW - Automated Driving KW - Max-Min-Plus-Scaling Systems KW - Hybrid Systems KW - Computational Efficiency KW - Hybrid Control N2 -
This toolbox can be used to hybridize any nonlinear function given as its input argument, which can be either a nonlinear prediction model or the nonlinear function expressing the boundary of the
feasible region, i.e. the nonlinear constraints.
A grid is generated on the function domain and the toolbox returns the hybridized form of the nonlinear function. The user can select the type and form of approximation based on the problem type:
• For model approximation, the options are
1. selecting the grid type and
2. specifying the number of affine modes in the MMPS formulation.
• For constraint approximation, the options are
1. specifying the number of subregions,
2. selecting between polytopic (MMPS-based) or ellipsoidal approximation, and
3. choosing between boundary-based or region-based approximation.
ER - | {"url":"https://data.4tu.nl/export/refman/datasets/2a4a7bed-63b9-43d9-a4d2-192bc9163dd1","timestamp":"2024-11-04T18:32:15Z","content_type":"text/plain","content_length":"1709","record_id":"<urn:uuid:7f18dbbc-4b66-4753-b625-ac9de3f33bf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00045.warc.gz"} |
Why Power in Pure Inductive and Pure Capacitive Circuit is Zero?
This article describes why the power in pure inductive and capacitive circuits is zero. The inductors and capacitors are the basic building blocks of an electric circuit, and you will understand the
concept of no power drawn by these elements after reading this article. Let’s dive into the details.
Power in a Pure Inductive Circuit
The active power drawn by a pure inductive and a capacitive circuit is zero. In a pure inductive circuit, the current lags the voltage by 90° because the inductive load always opposes the rate of
change of current.
The back EMF or counter EMF is generated in the inductive load due to the rate of change of the current. The back EMF generated in the inductive circuit is given by.
The minus sign indicates that the voltage induced in the inductive circuit opposes the applied voltage. Because of this opposition, the current through the inductive load lags the voltage. In a
purely inductive circuit, the current lags the voltage by 90 degrees. The waveform of current and voltage in the purely inductive circuit is shown below.
The voltage and current in a purely inductive circuit do not rise or fall together. There is always a phase shift of 90° between voltage and current. The power factor of the purely inductive circuit
is ;
The power in an AC circuit is given by;
P = VICosΦ ———–(1)
The power factor of the purely inductive circuit is zero(lagging).
CosΦ = 0 ———–(2)
The power in a purely inductive circuit is given by,
P = VICosΦ
P = VI x 0
P = 0
Thus, the pure inductive circuit consumes zero active power, and it consumes reactive power only from the supply source.
Power in a Pure Capacitive Circuit
In a purely capacitive circuit, the current leads the voltage by 90° because the capacitive load always opposes the rate of change of voltage. The Pure capacitive circuit is given below.
The current through the capacitor leads the applied voltage by 90°in a purely capacitive circuit.
The Power factor of a pure capacitive load is zero(leading).
The power in an AC circuit is given by;
P = VICosΦ ———–(3)
The power factor of the purely capacitive circuit is zero(leading).
CosΦ = 0 ———–(4)
The power in a purely inductive circuit is given by,
P = VICosΦ
P = VI x 0
P = 0
Thus, a pure capacitive circuit consumes zero active power. The pure capacitive circuit consumes reactive power from the supply source.
Mathematical Proof:
The active power drawn by a pure capacitive load can be mathematically proved to be zero.
1 thought on “Why Power in Pure Inductive and Pure Capacitive Circuit is Zero?”
1. I really appreciate thanks for your guidance.
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5.5 The Hall Effect
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Describe the Hall effect
• Calculate the Hall emf across a current-carrying conductor
The information presented in this section supports the following AP® learning objectives and science practices:
• 3.C.3.1 The student is able to use right-hand rules to analyze a situation involving a current-carrying conductor and a moving electrically charged object to determine the direction of the
magnetic force exerted on the charged object due to the magnetic field created by the current-carrying conductor. (S.P. 1.4)
We have seen the effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. One result is the Hall effect, which has important implications and
Figure 5.19 shows what happens to charges moving through a conductor in a magnetic field. The field is perpendicular to the electron drift velocity and to the width of the conductor. Note that
conventional current is to the right in both parts of the figure. In part (a), electrons carry the current and move to the left. In part (b), positive charges carry the current and move to the right.
Moving electrons feel a magnetic force toward one side of the conductor, leaving a net positive charge on the other side. This separation of charge creates a voltage $ε,ε, size 12{ε} {}$ known as the
Hall emf, across the conductor. The creation of a voltage across a current-carrying conductor by a magnetic field is known as the Hall effect, after Edwin Hall, the American physicist who discovered
it in 1879.
One very important use of the Hall effect is to determine whether positive or negative charges carries the current. Note that in Figure 5.19(b), where positive charges carry the current, the Hall emf
has the sign opposite to when negative charges carry the current. Historically, the Hall effect was used to show that electrons carry current in metals and it also shows that positive charges carry
current in some semiconductors. The Hall effect is used today as a research tool to probe the movement of charges, their drift velocities and densities, and so on, in materials. In 1980, it was
discovered that the Hall effect is quantized, an example of quantum behavior in a macroscopic object.
The Hall effect has other uses that range from the determination of blood flow rate to precision measurement of magnetic field strength. To examine these quantitatively, we need an expression for the
Hall emf, $ε,ε,size 12{ε} {}$ across a conductor. Consider the balance of forces on a moving charge in a situation where $B,B,size 12{B} {}$$v,v,size 12{v} {}$ and $ll size 12{l} {}$ are mutually
perpendicular, such as shown in Figure 5.20. Although the magnetic force moves negative charges to one side, they cannot build up without limit. The electric field caused by their separation opposes
the magnetic force, $F=qvB,F=qvB,size 12{F= ital "qvB"} {}$ and the electric force, $Fe=qE,Fe=qE,size 12{F rSub { size 8{e} } = ital "qE"} {}$ eventually grows to equal it. That is,
5.10 $qE = qvB qE = qvB size 12{ ital "qE"= ital "qvB"} {}$
5.11 $E=vB.E=vB. size 12{E= ital "vB"} {}$
Note that the electric field $EE size 12{E} {}$ is uniform across the conductor because the magnetic field $BB size 12{B} {}$ is uniform, as is the conductor. For a uniform electric field, the
relationship between electric field and voltage is $E=ε/l,E=ε/l,size 12{E=ε/l} {}$ where $ll size 12{l} {}$ is the width of the conductor and $εε size 12{ε} {}$ is the Hall emf. Entering this into
the last expression gives
5.12 $ε l = vB . ε l = vB . size 12{ { {ε} over {l} } = ital "vB" "." } {}$
Solving this for the Hall emf yields
5.13 $ε = Blv ( B , v , and l , mutually perpendicular ) , ε = Blv ( B , v , and l , mutually perpendicular ) , size 12{ε= ital "Blv"``` \( B,`v,`"and"`l,`"mutually perpendicular" \) ,} {}$
where $εε size 12{ε} {}$ is the Hall effect voltage across a conductor of width $ll size 12{l} {}$ through which charges move at a speed $v.v.size 12{v} {}$
One of the most common uses of the Hall effect is in the measurement of magnetic field strength $B.B.size 12{B} {}$ Such devices, called Hall probes, can be made very small, allowing fine position
mapping. Hall probes can also be made very accurate, usually accomplished by careful calibration. Another application of the Hall effect is to measure fluid flow in any fluid that has free charges
(most do). (See Figure 5.21.) A magnetic field applied perpendicular to the flow direction produces a Hall emf $εε size 12{ε} {}$ as shown. Note that the sign of $εε size 12{ε} {}$ depends not on the
sign of the charges, but only on the directions of $BB size 12{B} {}$ and $v.v.size 12{v} {}$ The magnitude of the Hall emf is $ε=Blv,ε=Blv,size 12{ε= ital "Blv"} {}$ where $ll size 12{l} {}$ is the
pipe diameter, so that the average velocity $vv size 12{v} {}$ can be determined from $εε size 12{ε} {}$ providing the other factors are known.
Example 5.3 Calculating the Hall emf: Hall Effect for Blood Flow
A Hall effect flow probe is placed on an artery, applying a 0.100-T magnetic field across it, in a setup similar to that in Figure 5.21. What is the Hall emf, given the vessel’s inside diameter is
4.00 mm and the average blood velocity is 20.0 cm/s?
Because $B,B,size 12{B} {}$$v,v,size 12{v} {}$ and $ll size 12{l} {}$ are mutually perpendicular, the equation $ε=Blvε=Blv size 12{ε= ital "Blv"} {}$ can be used to find $ε.ε.size 12{ε} {}$
Entering the given values for $B,B,size 12{B} {}$$v,v,size 12{v} {}$ and $ll size 12{l} {}$ gives
5.14 ε = Blv = 0.100 T 4 . 00 × 10 − 3 m 0 .200 m/s = 80.0 μV . ε = Blv = 0.100 T 4 . 00 × 10 − 3 m 0 .200 m/s = 80.0 μV . alignl { stack { size 12{ε= ital "Blv"= left (0 "." "100"`T right ) left (4
"." "00" times "10" rSup { size 8{ - 3} } `m right ) left (0 "." "200"`"m/s" right )} {} # ="80" "." 0`"μV" {} } } {}
This is the average voltage output. Instantaneous voltage varies with pulsating blood flow. The voltage is small in this type of measurement. $εε size 12{ε} {}$ is particularly difficult to measure,
because there are voltages associated with heart action (electrocardiogram (ECG) voltages) that are on the order of millivolts. In practice, this difficulty is overcome by applying an AC magnetic
field, so that the Hall emf is AC with the same frequency. An amplifier can be very selective in picking out only the appropriate frequency, eliminating signals and noise at other frequencies. | {"url":"https://texasgateway.org/resource/55-hall-effect?book=79106&binder_id=78821","timestamp":"2024-11-07T19:25:33Z","content_type":"text/html","content_length":"68366","record_id":"<urn:uuid:252dd7ac-8179-49da-99ef-6ee33a1955c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00304.warc.gz"} |
Chan's Algorithm
This visualization shows the gift wrapping step used in Chan's Algorithm for computing the convex hull.
A set of convex polygons:
An intermediate position. The algorithm has just identified a new vertex of the resulting polygon:
The algorithm determines the tangents to all polygons from the current vertex:
The algorithm is finished: | {"url":"https://jaryard.com/projects/live-cg/chans-algorithm.html","timestamp":"2024-11-10T10:55:00Z","content_type":"text/html","content_length":"3501","record_id":"<urn:uuid:450a4198-d954-42b1-8d55-ec41083107e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00606.warc.gz"} |
DNA/RNA Graphs Package
Skip to end of metadata Go to start of metadata
The DNA/RNA Graphs Package draws contextual graphs for sequences. The DNA/RNA Graphs Package is available for the Standard DNA and Standard RNA alphabets.
Open a sequence in the Sequence View and click the Graphs icon on the toolbar. The popup menu appears:
To see a graph select the corresponding graph item in the popup menu. A new area with the graph appears right above the Sequence zoom view:
Each point on a graph is calculated for a window of a specified size. The window is moved along the sequence by a step. See Graph Settings for instructions on how to modify these parameters.
It is possible to get information about each point of a graph. When a mouse is moved in the Graphs area, a small circle shows on the graph. A coordinates hint shows above it. When you hold Shift and
click on a graph, the circle and the hint locks:
To remove it click on the hint. Also you can delete all labels by Graph->Delete all labels context menu. To select all extremum points use the Graph->Select all extremum points context menu item.
All graphs are always aligned to the range shown in the Sequence zoom view. It means that if you change the visible range in the overview (either by zooming or scrolling) the graph will also be
updated. The minimum and maximum values of the visible range are shown at the right lower and upper corners of the graph.
To close a graph, uncheck its item in the popup menu. | {"url":"https://doc.ugene.net/wiki/pages/viewpage.action?pageId=10289585","timestamp":"2024-11-14T21:47:33Z","content_type":"text/html","content_length":"51253","record_id":"<urn:uuid:a2fa4cca-01cb-4fbd-b7b0-884dc987b49d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00512.warc.gz"} |
DAY 21: How many good friends do you have?
To ask the quantity, there are 2 question words in Chinese which express "how many". One is 多少(duō shǎo) which is introduced in Lesson 16, and the other is 几(jǐ). The main difference lies in that
几(jǐ) is often to ask the number which is supposed to be less than 10, and 多少(duō shǎo) is used to ask the number larger than 10.
Structure:Subject + 有 (yǒu) + 几 (jǐ) + Measure word + Noun
How many sth does sb have? | {"url":"https://www.learn-chinese.com/ev-lesson-21-how-many-good-friends-do-you-have/","timestamp":"2024-11-05T00:57:59Z","content_type":"text/html","content_length":"40801","record_id":"<urn:uuid:a1c4a251-7f20-48c3-b590-fc46707aa2e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00408.warc.gz"} |
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Data Update
May you notice any obsolete information about one or more SISSA alumni, please write at webmaster.math@sissa.it.
Name Origin Year Phd Denomination Position Country
Nirav Vasant Shah India 2022 Mathematical Analysis, Modelling and Applications Postdoc United Kingdom
Laura Meneghetti Italy 2022 Mathematical Analysis, Modelling and Applications Postdoc Italy
Stefano Baranzini Italy 2022 Geometry and Mathematical Physics Postdoc Italy
Michele Graffeo Italy 2022 Geometry and Mathematical Physics Postdoc Italy
Alice Fiaschi Italy 2008 Applied Mathematics Postdoc Italy
Massimo Fonte Italy 2005 Functional Analysis and Applications Postdoc Austria
Chengbo Li China 2009 Applied Mathematics Postdoc China
Cristina Manolache Romania 2009 Geometry Postdoc United Kingdom
Davide Masoero Italy 2010 Mathematical Physics Postdoc Portugal
Fabio Musso Italy 2003 Mathematical Physics Postdoc Italy
Cheikh Birahim Ndiaye Senegal 2007 Mathematical Analysis Postdoc Switzerland
Chiara Pagani Italy 2005 Mathematical Physics Postdoc Italy
Alessandro Selvitella Italy 2010 Mathematical Analysis Postdoc Canada
Monica Ugaglia Italy 1998 Mathematical Physics Postdoc Italy
Paolo Ventura Italy 2023 Mathematical Analysis, Modelling and Applications Postdoc Italy
Davide Manini Italy 2023 Mathematical Analysis, Modelling and Applications Postdoc Israel
Ivan Prusak Ukraine 2023 Mathematical Analysis, Modelling and Applications Postdoc Italy
Martina Zizza Italy 2023 Mathematical Analysis, Modelling and Applications Postdoc Germany
Francesco Romor Italy 2023 Mathematical Analysis, Modelling and Applications Postdoc Germany
Simone Carano Italy 2023 Mathematical Analysis, Modelling and Applications Postdoc United Kingdom
Luca Gennaioli Italy 2020 Postdoc
Nicholas Rungi Italy 2020 Geometry and Mathematical Physics Postdoc France
Dario Andrini Italy 2022 Mathematical Analysis, Modelling and Applications Postdoc Italy
Alessandro Nobile Italy 2022 Geometry and Mathematical Physics Postdoc Luxembourg
Alessandro Scagliotti Italy 2022 Mathematical Analysis, Modelling and Applications Postdoc Germany
Carolina Biolo Italy 2017 Mathematical Analysis, Modelling and Applications Teacher Italy
Benjamin Kipkirui Kikwai Kenya 2016 Geometry Teacher Kenya
Jacopo Vittorio Scalise Italy 2016 Geometry Teacher United Kingdom
Paolo Dall'Aglio Italy 1998 Functional Analysis and Applications Teacher Italy
Monique Annie Gradolato France 1993 Functional Analysis and Applications Teacher Italy | {"url":"https://www.math.sissa.it/alumni?page=10&order=field_alumni_position&sort=asc","timestamp":"2024-11-06T21:09:33Z","content_type":"application/xhtml+xml","content_length":"64557","record_id":"<urn:uuid:0f7390f5-f579-4c49-a057-b6e8044f169b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00030.warc.gz"} |
We introduce a new masked spectral bound for the maximum-entropy sampling problem. This bound is a continuous generalization of the very effective spectral partition bound. Optimization of the masked
spectral bound requires the minimization of a nonconvex, nondifferentiable function over a semidefiniteness constraint. We describe a nonlinear affine scaling algorithm to approximately minimize the
bound. … Read more
Distance Weighted Discrimination
High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine
suffers from “data piling” at the margin, which can diminish generalizability. This leads naturally to the development of Distance Weighted Discrimination, which is based on … Read more
Bounds on measures satisfying moment conditions
Given a semi algebraic set S of R^n we provide a numerical approximation procedure that yields upper and lower bounds on mu(S), for measures mu that satisfy some given moment conditions. The bounds
are obtained as solutions of positive semidefinite programs that can be solved via standard software packages like the LMI MATLAB toolbox. Citation … Read more | {"url":"https://optimization-online.org/category/applications-science-engineering/statistics/page/19/","timestamp":"2024-11-04T01:12:18Z","content_type":"text/html","content_length":"88648","record_id":"<urn:uuid:ce7d43ae-8eff-4daa-8b85-1ea9894e1d5d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00011.warc.gz"} |
3.7 Two-dimensional Shapes and Perimeter
Unit Goals
• Students reason about shapes and their attributes, with a focus on quadrilaterals. They solve problems involving the perimeter and area of shapes.
Section A Goals
• Reason about shapes and their attributes.
Section B Goals
• Find the perimeter of two-dimensional shapes, including when all or some side lengths are given.
Section C Goals
• Solve problems involving perimeter and area, in and out of context.
Section D Goals
• Apply geometric understanding to solve problems.
Read More | {"url":"https://im-beta.kendallhunt.com/k5/teachers/grade-3/unit-7/lessons.html","timestamp":"2024-11-04T12:26:00Z","content_type":"text/html","content_length":"84609","record_id":"<urn:uuid:2d8d3259-3812-4ee7-b3bf-e99ddc33a512>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00215.warc.gz"} |
Items where Author is "
Number of items: 14.
Adilah Abdul Ghapor, and Yong Zulina Zubairi, and Al Mamun, Sayed Md. and Siti Fatimah Hassan, and Elayaraja Aruchunan, and Nurkhairany Amyra Mokhtar, (2023) Identifying multiple outliers in linear
functional relationship model using a robust clustering method. Sains Malaysiana, 52 (5). pp. 1595-1606. ISSN 0126-6039
Nur Ain Al-Hameefatul Jamaliyatul, and Basri Badyalina, and Nurkhairany Amyra Mokhtar, and Adzhar Rambli, and Yong Zulina Zubairi, and Adilah Abdul Ghapor, (2023) Modelling wind speed data in Pulau
Langkawi with functional relationship. Sains Malaysiana, 52 (8). pp. 2419-2430. ISSN 0126-6039
Azuraini Mohd Arif, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, (2022) Outlier detection in balanced replicated linear functional relationship model. Sains Malaysiana, 51 (2). pp. 599-607. ISSN
Norli Anida Abdullah, and Yong Zulina Zubairi, and Afera Mohamad Apandi, and Mohd Iqbal Shamsudheen, (2021) COVRATIO statistic as a discrimination method for multivariate normal distribution. Sains
Malaysiana, 50 (7). pp. 2079-2084. ISSN 0126-6039
Nor Hafizah Moslim, and Nurkhairany Amyra Mokhtar, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, (2021) Understanding the behaviour of wind direction in Malaysia during monsoon seasons using
replicated functional relationship in von Mises distribution. Sains Malaysiana, 50 (7). pp. 2035-2045. ISSN 0126-6039
Siti Zanariah Satari, and Nur Faraidah Muhammad Di, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, (2021) Comparative study of clustering-based outliers detection methods in circular-circular
regression model. Sains Malaysiana, 50 (6). pp. 1787-1798. ISSN 0126-6039
Nur Arina Bazilah Kamisan, and Muhammad Hisyam Lee, and Abdul Ghapor Hussin, and Yong Zulina Zubairi, (2020) Imputation techniques for incomplete load data based on seasonality and orientation of the
missing values. Sains Malaysiana, 49 (5). pp. 1165-1174. ISSN 0126-6039
Nor Hafizah Moslim, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, and Siti Fatimah Hassan, and Nurkhairany Amyra Mokhtar, (2019) A comparison of asymptotic and bootstrapping approach in
constructing confidence interval of the concentration parameter in von mises distribution. Sains Malaysiana, 48 (5). pp. 1151-1156. ISSN 0126-6039
Azuraini Mohd Arif, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, (2019) On robust estimation for slope in linear functional relationship model. Sains Malaysiana, 48 (1). pp. 237-242. ISSN
Nur Arina Bazilah Kamisan, and Muhammad Hisyam Lee, and Suhartono, Suhartono and Abdul Ghapor Hussin, and Yong Zulina Zubairi, (2018) Load forecasting using combination model of multiple linear
regression with neural network for Malaysian city. Sains Malaysiana, 47 (2). pp. 419-426. ISSN 0126-6039
Nurkhairany Amyra Mokhtar, and Yong Zulina Zubairi, and Abdul Ghapor Hussin, (2017) Estimation of concentration parameter for simultaneous circular functional relationship model assuming unequal
error variance. Sains Malaysiana, 46 (8). pp. 1347-1353. ISSN 0126-6039
Adilah Abdul Ghapor, and Yong Zulina Zubairi, and A.H.M. Rahmatullah Imon, (2017) Missing value estimation methods for data in linear functional relationship model. Sains Malaysiana, 46 (2). pp.
317-326. ISSN 0126-6039
Nuradhiathy Abd Razak, and Yong Zulina Zubairi, and Rossita M. Yunus, (2014) Imputing missing values in modelling the PM10 concentrations. Sains Malaysiana, 43 (10). pp. 1599-1607. ISSN 0126-6039
Nur Arina Basilah Kamisan, and Abdul Ghapor Hussin, and Yong Zulina Zubairi, (2010) Finding the best circular distribution for southwesterly monsoon wind direction in Malaysia. Sains Malaysiana, 39
(3). pp. 387-393. ISSN 0126-6039 | {"url":"http://journalarticle.ukm.my/view/creators/Yong_Zulina_Zubairi=3A=3A=3A.html","timestamp":"2024-11-07T22:06:20Z","content_type":"application/xhtml+xml","content_length":"16887","record_id":"<urn:uuid:a3b259f6-5ea6-4203-8f23-06594e0002e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00160.warc.gz"} |
Two-Sample t Test (Independent Samples with a Common Variance) - Tests of Hypotheses | Biostatistics
Power function for a test that a normal population has mean zero versus a two-sided alternative
Recall from Section 8.5 the use of the appropriate t statistic for a confidence interval under the following circumstances: the parent populations have normal distributions and common variance that
is unknown. In this situation, we used the pooled variance estimate, s2p, calculated by the formula Sp2 = {St2(nt – 1) + Sc2(nc – 1)}/[nt + nc – 2]
Figure 9.2. Power function for a test that a normal population has mean zero versus a two-sided alternative when the sample size n = 25, n = 100, and the significance level α = 0.05.
Suppose we want to evaluate whether the means of two independent samples selected from two parent populations are significantly different. We will use a t test with s[p]^2 as the pooled variance
estimate. The corresponding t statistic is t = . The formula for t is obtained by replacing the common in the formula for the two sample Z test with the pooled estimate S[p]. The resulting statistic
has Student’s t distribution with n[t] + n[c] – 2 degrees of freedom. This sample t statistic is used for hypothesis testing. For a two-sided test the steps are as follows:
1. State the null hypothesis H[0]: μ[t] = μ[c] versus the alternative hypothesis H[1]: μ[t]
≠ μ[c].
2. Choose a significance level α= α[0] (often we take α[0] = 0.05 or 0.01).
3. Determine the critical region, that is, the region of values of t in the upper and lower α/2 tails of the sampling distribution for Student’s t distribution with n[t] + n[c] – 2 degrees of freedom
when μ[t] = μ[c] (i.e., the sampling distribution when the null hypothesis is true).
4. Compute the t statistic: n[t] and n[c], where X[t] is the sample mean for the treatment group, [c] is the sample mean for the control group, and S[p] is the pooled sample standard deviation.
5. Reject the null hypothesis if the test statistic t (computed in step 4) falls in the rejection region for this test; otherwise, do not reject the null hypothesis.
We will apply these steps to the pig blood loss data from Section 8.7, Table 8.1. Recall that S[p]^2 = {S^2[t](n[t] – 1) + S[c]^2(n[c] – 1)}/[n[t] + n[c] – 2] = {(717.12)^2 9 + (1824.27)^2 9}/18,
since n[t] = n[c] = 10, S[t] = 717.12, and S[c] = 1824.27. So S[p]^2 = 2178241.61 and taking the square root we find S[p] = 1475.89. As the degrees of freedom are n[t] + n[c] – 2 = 18, we find that
the constant C from the table of the Student’s t distribution is 2.101. Applying steps 1–5 to the pig blood loss data for a two-tailed (two-sided) test, we have:
1. State the null hypothesis H[0]: μ[t] = μ[c] versus the alternative hypothesis H[1]: μ[t] ≠ μ[c].
2. Choose a significance level α = α[0] = 0.05.
3. Determine the critical region, that is, the region of values of t in the upper and lower 0.025 tails of the sampling distribution for Student’s t distribution with 18 degrees of freedom when μ[t]/
μ[c] (i.e., the sampling distribution when the null hypothesis is true).
4. Compute the t statistic: n[t] = 10 and n[c] = 10, respectively. Under the null hypothesis, μ[t] – μ[c] = 0 and [t] – X[c] = 1085.9–2187.4 = –1101.5 and s[p], the pooled sample standard deviation,
is 1475.89. Since √{(1/n[t]) + (1/n[c])]} = √(2/20) = √0.1 = 0.316, t = –1101.5/(1475.89)0.316 = –2.362.
5. Now, since –2.362 < –C = –2.101, we reject H[0]. | {"url":"https://www.pharmacy180.com/article/two-sample-t-test--independent-samples-with-a-common-variance--2946/","timestamp":"2024-11-07T13:21:15Z","content_type":"application/xhtml+xml","content_length":"49417","record_id":"<urn:uuid:843c8fca-3378-4c27-b823-39b11016062b>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00856.warc.gz"} |
Each lesson includes a cool-down (also known as an exit slip or exit ticket) to be given to students at the end of the lesson. This activity serves as a brief checkpoint to determine whether students
understood the main concepts of that lesson. Teachers can use this as a formative assessment to plan further instruction.
What if the feedback from a cool-down suggests students haven’t understood a key concept? Choose one or more of these strategies:
• Look at the next few lessons to see if students have more opportunities to engage with the same topic. If so, plan to focus on the topic in the context of the new activities.
• During the next lesson, display the work of a few students on that cool-down. Anonymize their names, but show some correct and incorrect work. Ask the class to observe some things each student
did well and could have done better.
• Give each student brief, written feedback on their cool-down that asks a question that nudges them to re-examine their work. Ask students to revise and resubmit.
• Look for practice problems that are similar to, or involve the same topic as the cool-down, then assign those problems over the next few lessons.
Here is an example. For a lesson in grade 6, unit 2, the learning goals are
• Understand that doubling, tripling, or halving a recipe yields something that tastes the same.
• Understand that “doubling, tripling, or halving the recipe” means “doubling, tripling, or halving each ingredient.”
The cool-down reads:
Usually when Elena makes bird food, she mixes 9 cups of seeds with 6 tablespoons of maple syrup. However, today she is short on ingredients. Think of a recipe that would yield a smaller batch of
bird food but still taste the same. Explain or show your reasoning.
A number of students responded with 8 cups of seeds and 5 tablespoons of maple syrup, and did not provide an explanation or show their reasoning. Here are some possible strategies:
• Notice that this lesson is the first of several that familiarize students with contexts where equivalent ratios carry physical meaning, for example, the taste of a recipe or the result of
mixing paint colors. Over the next several lessons, there are more opportunities to reason about multiple batches of a recipe. When launching these activities, pause to assist students to
interpret this correctly. Highlight the strategies of any students who use a discrete diagram or other representation to correctly represent multiple batches.
• Select the work of one student who answered correctly and one student whose work had the common error. In the next class, display these together for all to see (hide the students’ names). Ask
each student to decide which interpretation is correct, and defend their choice to their partner. Select students to share their reasoning with the class who have different ways of representing
that \(9:6\) is equivalent to \(3:2\), \(6:4\), or \(4\frac12:3\).
• Write feedback for each student along the lines of "If this recipe is 3 batches, how could you make 1 batch?" Allow students to revise and resubmit their work.
• Look for practice problems in upcoming lessons that require students to generate examples of different numbers of batches equivalent to a given ratio, and be sure to assign those problems. | {"url":"https://im.kendallhunt.com/MS/teachers/cool_downs.html","timestamp":"2024-11-01T23:58:28Z","content_type":"text/html","content_length":"77197","record_id":"<urn:uuid:9eb1fcd7-c610-4c17-9860-bcd4385bc908>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00827.warc.gz"} |
Need help with formula to alert when Actual Progress is behind Projected Progress
I have been trying to nail this down for the past couple of weeks but falling short. I need a column that shows Red, Green, or Yellow based on Actual % Complete vs Projected % Complete. I am calling
it Progress Alert. I need it to show the following:
If Actual % Complete is 25%+ behind Projected % Complete show Red
If Actual % Complete is 15%+ behind Projected % Complete show Yellow
If Actual % Complete is 10% or less behind Projected % show Green
If Actual % Complete is ahead of Projected % at all show Green
Background- I currently have the Progress Alert working with the Progress column Harvey balls but the last criteria listed above doesn't work. If the Progress is ahead of Projected % Complete the
Progress Alert is showing Red which is our main issue. I added an Actual % Complete which converts the Harvey balls to % hoping that may make it easier to do the formula. The Projected Complete is
calculated based on the days on the Planned Dates.
Any help would be much appreciated!
Best Answer
Try this...
=IF([Projected % Complete]@row - .25>= [Actual % Complete]@row, "Red", IF([Projected % Complete]@row - .15>= [Actual % Complete]@row, "Yellow", "Green"))
Please note: There is a gap in your logic between Yellow and Green. Yellow is more than 15% but Green is less than 10%. Writing a formula to match this exactly would leave a blank for any row
between those two differences. The formula above shows Yellow for more than 15% and Green for less than 15% (instead of less than 10%).
• Exactly how are your percentages entered? If through formulas, can you provide those formulas? If through manual entry, are you manually keying the % or are you just entering a number into a text
/number column that has been formatted for percentages?
• Actual % Complete- =IF(Progress@row = "Empty", 0, IF(Progress@row = "Quarter", 0.25, IF(Progress@row = "Half", 0.5, IF(Progress@row = "Three Quarter", 0.75, 1))))
Projected % Complete- =IF(TODAY() < [Planned START]@row, 0, IF(TODAY() > [Planned END]@row, 1, (TODAY() - [Planned START]@row) / ([Planned END]@row - [Planned START]@row)))
Try this...
=IF([Projected % Complete]@row - .25>= [Actual % Complete]@row, "Red", IF([Projected % Complete]@row - .15>= [Actual % Complete]@row, "Yellow", "Green"))
Please note: There is a gap in your logic between Yellow and Green. Yellow is more than 15% but Green is less than 10%. Writing a formula to match this exactly would leave a blank for any row
between those two differences. The formula above shows Yellow for more than 15% and Green for less than 15% (instead of less than 10%).
• Thank you, Paul! This is what I have needed for weeks!
• Happy to help. 👍️
The Community is always a great place to come if you ever hit a roadblock! There are a lot of knowledgeable people wandering around here.
• Hi, this is really useful, Id like to know how to add the Blue as showing that the task is complete: can you advise?
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/77088/need-help-with-formula-to-alert-when-actual-progress-is-behind-projected-progress","timestamp":"2024-11-09T01:02:23Z","content_type":"text/html","content_length":"420379","record_id":"<urn:uuid:9fd66f6a-0345-4fc8-805c-4e9101bd8c6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00799.warc.gz"} |
Gradient Operator/Riemannian Manifold
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.
Let $f \in \map {\CC^\infty} M$ be a smooth mapping on $M$.
The gradient of $f$ is defined as:
\(\ds \grad f\) \(:=\) \(\ds \nabla f\)
\(\ds \) \(=\) \(\ds g^{-1} \d_{\d R} f\)
where $\d_{\d R}$ is de Rham differential.
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.
Let $f \in \map {C^\infty} M : M \to \R$ be a smooth mapping on $M$.
The gradient of $f$ is the vector field obtained from the differential $\rd f$ obtained by raising an index:
$\grad f := \paren {\rd f}^\sharp$
Further research is required in order to fill out the details.
In particular: equivalence
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Research}} from the code.
The gradient of a scalar field $U$ is usually vocalised grad $U$.
Also see
• Results about the gradient operator can be found here.
During the course of development of vector analysis, various notations for the gradient operator were introduced, as follows: | {"url":"https://proofwiki.org/wiki/Definition:Gradient_Operator_on_Riemannian_Manifold","timestamp":"2024-11-08T06:10:09Z","content_type":"text/html","content_length":"49031","record_id":"<urn:uuid:6b86afbe-38f2-449f-89dc-e9b0d12469eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00332.warc.gz"} |
Abstract Algebra
Hw10 Problem 8
Return to Homework 10, Homework Problems, Glossary, Theorems
Problem 8
Let $G$ be a group, $h \in G$ and $n \in \mathbb{Z}^{+}$. Let $\phi: \mathbb{Z}_n \rightarrow G$ be defined by $\phi(i) = h^{i}$ for $0 <= i <= n$. Give a necessary and sufficient condition (in terms
of $h$ and $n$) for $\phi$ to be a homomorphism. Prove your assertion.
The map $\phi$ is a homomorphism if and only if $h ^ n = e$ where $e$ is the identity of $G$.
(=>) If $\phi$ is a homomorphism then $e = \phi(0) = \phi(1)^n = h ^ n$.
(<=) Suppose that $h^n = e$, so that $\langle h \rangle \cong Z_{m}$ where $m$ is a divisor of $n$, and let $i, j \in \mathbb{Z}_n$. By the division algorithm, $\phi(i+j) = \phi(i)\phi(j)$, and then
$\phi$ is a homomorphism. | {"url":"http://algebra2014.wikidot.com/hw10-problem-8","timestamp":"2024-11-09T06:33:01Z","content_type":"application/xhtml+xml","content_length":"28510","record_id":"<urn:uuid:caf9ed56-bb28-4541-9f7d-116c9132948e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00140.warc.gz"} |
Coronavirus (COVID-19) related deaths by ethnic group, England and Wales methodology
This technical appendix provides the detail around the data and methods used in the article Updating ethnic contrasts in deaths involving the coronavirus (COVID-19), England and Wales: deaths
occurring 2 March to 28 July 2020.
Nôl i'r tabl cynnwys
These analyses are based on a unique linked dataset that encompasses Census 2011 records, death registrations in England and Wales, and hospital episode statistics (HES) with England coverage only.
It was created by:
• linking the 2011 Census to NHS Patient Register (PR) records between 2011 and 2013, where NHS number was added to those Census records identified in the Patient Register
• using NHS number and a deterministic match key linkage method where NHS number was unavailable – death registrations were linked to 2011 Census records up to 24 August 2020
• joining HES records from April 2017 onto the census-deaths linked data using a combination of date of birth and NHS number
The linked population has a very similar distribution across a range of characteristics as the full census population, and so can be considered representative of the general population of England and
Wales in 2011. Examination of linkage rates for ethnic groups showed distributions at 2011 Census and the linked population were relatively consistent across all categories, although there was more
significant variation in unlinked records. For all ethnic groups, linkage rates of NHS number exceeded 80% in all cases.
The study population included all usual residents coded to an ethnic group in 2011 and not known to have died before 2 March 2020 (number surveyed (N) equals 48,468,645). Those enumerated in 2011
answering the “Intention to Stay” question, because they had entered the UK in the year before the 2011 Census took place, were excluded from the analyses because of their high propensity to have
left the UK before the analysis period under investigation. However, this leaves uncertainty in the extent of emigration of usual residents between 27 March 2011 and 2 March 2020, which is dealt with
later in this section. Analyses using HES data were limited to usual residents thought to be alive on 2 March 2020 in England only (number surveyed (N) equals 45,842,599).
We use data from the Office for National Statistics (ONS) Longitudinal Study and the International Passenger Survey (IPS) to estimate emigration between March 2011 and March 2020 by broad age group
and ethnicity. As we only have IPS data up to year-ending March 2019, we assume emigration rates observed between March 2019 and March 2020 are the same as those observed in the previous year.
These emigrations and deaths are used to ensure that the analysis refers to people still in the population of England and Wales and at risk of the coronavirus (COVID-19) from 2 March 2020, by
applying out-migration adjustment factors to deplete the population sizes resulting from expected emigration since the 2011 Census.
The number of deaths occurring between 2 March 2020 and 28 July 2020 that were registered by 24 August 2020 amounted to 253,194. Of these, 229,983 were successfully linked to a 2011 Census record
(90.8%). However, only 229,929 were usable because 48 were linked to non-usual residents and six to individuals over 110 years of age, which we excluded from our study population. Of these, 216,406
were resident in England and 13,523 were resident in Wales.
Causes of death were defined using the International Classification of Diseases, 10th Revision (ICD-10). Deaths involving COVID-19 include those with an underlying cause, or any mention, of ICD-10
codes U07.1 (COVID-19, virus identified) or U07.2 (COVID-19, virus not identified).
The study population is not currently refreshed with new births or immigrations. Some COVID-19 deaths will therefore have occurred to immigrants entering the country since 2011; deaths involving
COVID-19 to those born since the 2011 Census and resident in England and Wales will be very small as they will be nine years old or younger.
Nôl i'r tabl cynnwys
3. Hospital episode statistics
For this analysis, we used hospital episode statistics (HES) data from April 2017 sourced from three datasets: Accident and Emergency (AE), Outpatients (OP) and Admitted Patient Care (APC). The
information within these three datasets is at episode level (each finished period of care under a consultant). We created a person-level dataset from the record-level HES data to preserve all
information when linking to 2011 Census and deaths data.
The analytical variables derived from HES were:
• flags for ICD10 diagnoses codes of interest in the OP and APC datasets
• the total number of episodes per NHS number and date of birth (our method to identify an individual) for all datasets.
• the number of first admission episode flags in the APC dataset to derive the number of admissions per person.
• the number of days spent in admitted patient care from the APC dataset
These were then aggregated up to the person level by stacking and deduplicating all datasets on NHS number and date of birth, to create one row per individual. Records with blank or invalid NHS
numbers and/or dates of birth were dropped, as these could not be linked to the Census. The total number of individuals in our HES data was 43,562,505. The HES data was then linked to the Census and
deaths data through a simple deterministic link on NHS number and date of birth. 31,903,383 of the HES records linked to the 2011 Census (73.2%). The remaining unlinked 26.8% are likely to have not
been registered on the 2011 Census, because they were born after 27 March 2011, migrated to England after that date or were not enumerated at the 2011 Census despite being a resident. In addition,
some individuals in the unlinked group may not have been able to have an NHS number assigned to their Census record. This could be due to conflicting addresses, name changes or other reasons, and
thus the deterministic and probabilistic linkage methods would have failed, though this is only in a small number of cases.
Nôl i'r tabl cynnwys
4. Age-standardisation method
This Microsoft Excel template demonstrates how age-standardised rates and 95% confidence intervals are calculated.
Age-standardised rates are calculated as follows:
• i is the age group
• w[i] is the number, or proportion, of individuals in the standard population in age group i
• r[i] is the observed age-specific rate in the subject population in age group i, given by:
• d[i]is the observed number of deaths in the subject population in age group i
• n[i] is the population at risk in age-group i
The age-standardised rate is a weighted sum of age-specific death rates where the age-specific weights represent the relative age distribution of the standard population (in this case the 2013
European Standard Population (ESP)). The variance is the sum of the age-specific variances and its standard error is the square root of the variance:
• r[i] is the crude age-specific rate in the local population in age group i
• d[i] is the number of deaths in the local population in age group i
Confidence intervals
The mortality data in this release are not subject to sampling variation as they were not drawn from a sample. Nevertheless, they may be affected by random variation, particularly where the number of
deaths or probability of dying is small. To help assess the variability in the rates, they have been presented alongside 95% confidence intervals.
The choice of the method used in calculating confidence intervals for rates will, in part, depend on the assumptions made about the distribution of the deaths data these rates are based on.
Traditionally, a normal approximation method has been used to calculate confidence intervals on the assumption that deaths are normally distributed. However, if the number of deaths is relatively
small (fewer than 100), it may be assumed to follow a Poisson probability distribution. In such cases, it is more appropriate to use the confidence limit factors from a Poisson distribution table to
calculate the confidence intervals instead of a normal approximation method.
The method used in calculating confidence intervals for rates based on fewer than 100 deaths was proposed by Dobson and others (1991) as described in APHO (2008). In this method, confidence intervals
are obtained by scaling and shifting (weighting) the exact interval for the Poisson distributed counts (number of deaths in each year). The weight used is the ratio of the standard error of the
age-standardised rate to the standard error of the number of deaths.
The lower and upper 95% confidence intervals are denoted as ASR lower and ASR upper, respectively, and calculated as:
• D[l] and D[u] are the exact lower and upper confidence limits for the number of deaths, calculated using confidence limit factors from a Poisson probability distribution table
• D is the number of deaths in each year
• v(ASR) is the variance of the age-standardised rate
• v(D) is the variance of the number of deaths
Where there are 100 or more deaths in a year, the 95% confidence intervals for age-standardised rates are calculated using the normal approximation method:
ASR[LL/UL] = ASR± 1.96*SE
ASR[LL/U] represents the upper and lower 95% confidence limits, respectively, for the age-standardised rate and SE is the standard error.
Nôl i'r tabl cynnwys
We use Cox proportional hazard models to assess how the risk of dying with coronavirus (COVID-19) varies among ethnic groups once we adjust for a range of geographical, demographic, socio-economic,
household, occupational exposure and health-related factors. Most individual characteristics are retrieved from the 2011 Census, except for pre-existing health conditions which are derived from
hospital episode statistics (HES) records from April 2017 onwards.
We model the hazard of dying with COVID-19 between 2 March 2020 and 28 July 2020. In our analytical dataset, we include all those who died of any cause during this period and a weighted random sample
of those who did not (the sampling fractions are 5% for the White population and 20% among other ethnic groups combined). The regression estimates are further weighted using the probability not to
have migrated between 2011 and 2020.
We estimate separate models for males and females, as the risk of death involving COVID-19 differs markedly by sex. We also estimate separate models for people in private households and in care homes
according to place of residence in 2011 and the Patient Register in 2019. We present results from several models, adding different control variables step by step. This allows us to see how the
differences across ethnic groups vary as we include further explanatory variables.
All our models are adjusted for age. We include age as a second-order polynomial to account for the non-linear relationship between age and the hazard of death involving COVID-19.
We then adjust for geographical factors. The probability to be infected by COVID-19 is likely to vary by region of residence. We therefore allow the baseline mortality hazard to vary by local
authority district. We also adjust for population density for the Lower Super Output Area (LSOA) of residence at the time of the 2011 Census. To account for the non-linear relationship between
population density and the hazard of death involving COVID-19 we include population density as a second-order polynomial, allowing for different slopes for the top 1% of the population density
distribution to account for outliers.
We then account for deprivation and wider measures of socio-economic status. We adjust for neighbourhood deprivation by adding decile of the Index of Multiple Deprivation (IMD) 2015 at the time of
the 2011 Census in our model. The IMD is an overall measure of deprivation based on factors such as income, employment and health.
We also adjust for the level of household deprivation, a summary measure of disadvantage based on four selected household characteristics (employment, education, health and housing). We include in
our model the highest level of qualification (degree, A-level or equivalent, GCSE or equivalent, no qualification) of the individual, and the National Statistics Socio-Economic Classification
(NS-SEC) of the household head (higher managerial, administrative and professional occupations, intermediate occupations, routine and manual occupations, never worked or long-term unemployed, not
We further adjust for household composition and circumstances. We include in our models the number of people in the household, the family type (not a family, couple with children, lone parent), and
binary variables for living in a multigenerational household (defined as three generations living together) or with any children (aged 18 years or under). We also adjust for the tenure of the
household (owned outright, owned with mortgage, social rented, private rented, other).
In addition, we adjust for a set of measures of occupational exposure. We include binary variables indicating if the individual is a key worker, and if so, what type. This data is taken from
occupation as recorded on the 2011 Census. We also include a binary variable indicating if anyone in the household is a key worker. We account for exposure to diseases and contact with others using
scores ranging from 0 (no exposure) to 100 (maximum exposure). Exposure to disease and physical proximity scores were originally obtained using O*NET data based on US Standard Occupational
Classification (SOC) codes and were mapped to UK SOC codes. The derivation of the scores is in line with the methodology previously used by the Office for National Statistics (ONS). We include these
scores for all individuals with a valid occupation and derive the maximum value amongst all household members.
Finally, we adjust for several measures of health. We include in the model self-reported health status (very good, good, fair, bad, very bad) and whether the individual has activity limitation
(disability) (not limited, daily activity limited a lot, daily activity limited a little) as recorded in the 2011 Census. We also adjust for pre-existing conditions derived from HES records from
April 2017 onwards, as discussed in Section 4 of the main article:
• history of cancer
• history of cardiovascular disease
• history of digestive system conditions
• history of mental health conditions
• history of metabolic conditions
• history of musculoskeletal conditions
• history of neurological conditions
• history of renal conditions
• history of respiratory conditions
• number of admitted patient care (APC) admissions (0, 1, 2 to 3, 4 to 5, 6 to 9, 10 plus)
• number of days spent in APC (0, 1, 2 to 4, 5 to 9, 10 to 19, 20 to 39, 40 to 69, 70 plus)
To allow for the effect of all these health-related factors to vary depending on the age of the individuals, we interact each of them with a binary variable indicating if the individual is aged 70
years or over.
In the article we report the hazard ratios for each ethnic minority group relative to the White population for people in private households in England, after adjusting for age, geographical factors,
socio-economic factors and health-related variables. The corresponding model goodness-of-fit statistics can be found in the dataset.
We find that much of the difference in COVID-19 mortality risk across ethnic groups that is attributable to health-related factors can be explained by self-reported health and disability status in
2011. For most ethnic groups, further adjusting for hospital-based comorbidities has little impact on the hazard ratios, though there is notable attenuation for Bangladeshi and Pakistani males. For
females, including hospital-based comorbidities increases the hazard ratios for several ethnic groups, most notably for those of Black African or Chinese ethnic background.
Figure 1: Rate of death involving COVID-19 by ethnic group and sex relative to the White population for people in private households, England, 2 March to 28 July 2020
Download the data
1. Cox proportional hazards models adjusting for age, geography (local authority and population density), socio-economic factors (area deprivation, household composition, socio-economic position,
highest qualification held, household tenure, multigenerational household flags and occupation indicators (including key workers and exposure to others), and health (self-reported health and
disability status in March 2011, and hospital-based comorbidities since April 2017).
2. Figures based on death registrations up to 24 August 2020 that occurred between 2 March 2020 and 28 July 2020 and could be linked to the 2011 Census.
3. Deaths were defined using the International Classification of Diseases, 10th Revision (ICD-10). Deaths involving COVID-19 include those with an underlying cause, or any mention, of ICD-10 codes
U07.1 (COVID-19, virus identified) or U07.2 (COVID-19, virus not identified).
4. Other ethnic group encompasses Asian other, Black other, Arab, and other ethnic group categories in the classification.
5. Error bars not crossing the x axis at value 1.0 denote a statistically significant difference in relative rates of death.
Nôl i'r tabl cynnwys
Manylion cyswllt ar gyfer y Methodoleg
Chris White and Daniel Ayoubkhani
Ffôn: +44 (0)1633 455865 | {"url":"https://cy.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/methodologies/coronaviruscovid19relateddeathsbyethnicgroupenglandandwalesmethodology","timestamp":"2024-11-13T14:54:51Z","content_type":"text/html","content_length":"110546","record_id":"<urn:uuid:60ef8801-25ac-487e-82d9-a7e217a803a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00666.warc.gz"} |
Introduction to NACA Airfoil Aerodynamics Using Python
Written on
Chapter 1: Understanding NACA Airfoils
This article aims to elucidate the essential features of NACA airfoils, particularly for students new to aerodynamics. Initially, we will explore the fundamental principles underlying airfoil
geometry. Subsequently, we will look into how these equations can be implemented in Python to compute numerical attributes for visualizing a NACA 4-Series 2D wing profile using Matplotlib.
Airfoils represent the cross-sectional shapes of wings. The National Advisory Committee for Aeronautics (NACA) created and assessed a range of airfoils known as NACA airfoils. The following figure
illustrates various samples of these wing sections.
The four-digit and five-digit series are commonly studied in introductory aerodynamics courses, although six-digit models also exist. This article concentrates on the four-digit series, specifically
the NACA 4415 airfoil.
Airfoil Geometry
Below is a diagram of a symmetrical airfoil, highlighting its key geometric parameters:
• Leading and Trailing Edges: The foremost and rearmost points of an airfoil.
• Chord: The straight line connecting the leading and trailing edges.
• x: The horizontal distance along the chord, starting from zero at the leading edge.
• y: The vertical height relative to the horizontal x-axis.
A cambered airfoil is depicted in the next figure, where camber refers to the curvature of the airfoil.
• Mean Camber Line: This line lies midway between the upper and lower surfaces and serves as the geometric centerline.
• Thickness (t): This refers to the distribution of height along the length of the airfoil.
From these illustrations, it is clear that the two main variables defining the geometric profile of the airfoil surface are camber and thickness.
A significant aspect of the design is that the 4-Series airfoil shapes are derived from analytical equations that describe the mean camber line and the thickness distribution of the section. In
contrast, later families, such as the 6-Series, are established using more complex theoretical methods.
4-Series Equations
The NACA 4415 is an example of the 4-Series family. The digits 4415 represent the two-dimensional profile.
• Equation 1: The first digit indicates the maximum camber (m) as a percentage of the chord. For the 4415 airfoil, this maximum camber is 4% of the chord length.
• Equation 2: The second digit, 4, specifies the distance (p) of maximum camber from the leading edge in tenths of the chord. For the 4415 airfoil, the maximum camber is located at 40% of the chord
• Equation 3: The last two digits (15) denote the maximum thickness (t) as a percentage of the chord. Thus, the thickness of the 4415 airfoil is 15% of the chord length.
The following Python code (Gist 1) defines three functions to extract numerical characteristics based on the four-digit NACA code.
For symmetric airfoils, the first two digits are zero, for example, 0015 implies m = 0 and p = 0. Two equations specify the mean camber line, depending on whether the x-coordinate is less than or
greater than the maximum camber position (p), as illustrated in Equation 4.
It is crucial to note that the equations presented here are analytical, developed by NACA through extensive research and experimentation.
Gist 2 provides the Python code for implementing Equation 4.
The variable yₜ represents the thickness distribution. The thickness values above (+) and below (-) the mean camber line are determined by Equation 4.
The x⁴ coefficient changes based on whether the trailing edge is open or closed. For a closed surface, the coefficient is -0.1036, while for a finite thickness trailing edge, it changes to -0.1015.
The thickness values are computed in Python using Gist 3.
To compute the upper and lower coordinates of the airfoil surfaces, use Equations 8-11, where θ represents the angle obtained from the inverse tangent of the derivative of the mean camber line.
Gist 6 contains the code for determining the final (x, y) coordinates for the upper and lower surfaces of the airfoil.
Plotting Results
The final wing profile can be visualized using the resulting (xᵤ, yᵤ) and (xₗ, yₗ) values. Below is a plot of the NACA 4415 generated using Matplotlib.
Another example of a cambered 4-Series airfoil, the NACA 2412, is illustrated below. By visually comparing the 4415 and 2412, one can note the differences in their geometric properties, particularly
in relation to the y-axis scale.
As previously mentioned, these analytical equations apply to symmetrical airfoils. Both the mean camber line and thickness distribution are perfectly aligned with the chord, as evident in the plot of
a 0015 shown below.
Each equation is versatile and can be parameterized with any four integers to visualize any member of the 4-Series NACA family. This article has outlined basic properties of airfoils and demonstrated
how to implement the geometric expressions to visualize the 2D surface profile of a wing.
Thank you for reading! If you're interested in more articles related to aerodynamics, please let me know.
[1] Fundamentals of Aerodynamics. Sixth Edition. John D. Anderson, Jr. Curator of Aerodynamics. National Air and Space Museum. Smithsonian Institution.
[2] NACA Airfoils — NASA. Last Updated: Aug 7, 2017, Editor: Bob Allen
[3] The NACA airfoil series (AA200_Course_Material) — Stanford
[4] Explained: NACA 4-Digit Airfoil [Airplanes] — Josh The Engineer
The first video illustrates a tutorial on Python scripting in FreeCAD, focusing on modeling NACA airfoils, which can help deepen your understanding of airfoil geometry.
The second video demonstrates how to plot a NACA 4-digit aerofoil, offering visual insights into the plotting process discussed in this article. | {"url":"https://dogmadogmassage.com/naca-airfoil-aerodynamics-python.html","timestamp":"2024-11-08T01:12:51Z","content_type":"text/html","content_length":"16481","record_id":"<urn:uuid:e13d2e39-33e6-41be-9f25-4d1bc7724dde>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00299.warc.gz"} |
A Gentle Introduction to Maximum Likelihood Estimation for Machine Learning - MachineLearningMastery.comA Gentle Introduction to Maximum Likelihood Estimation for Machine Learning - MachineLearningMastery.com
Density estimation is the problem of estimating the probability distribution for a sample of observations from a problem domain.
There are many techniques for solving density estimation, although a common framework used throughout the field of machine learning is maximum likelihood estimation. Maximum likelihood estimation
involves defining a likelihood function for calculating the conditional probability of observing the data sample given a probability distribution and distribution parameters. This approach can be
used to search a space of possible distributions and parameters.
This flexible probabilistic framework also provides the foundation for many machine learning algorithms, including important methods such as linear regression and logistic regression for predicting
numeric values and class labels respectively, but also more generally for deep learning artificial neural networks.
In this post, you will discover a gentle introduction to maximum likelihood estimation.
After reading this post, you will know:
• Maximum Likelihood Estimation is a probabilistic framework for solving the problem of density estimation.
• It involves maximizing a likelihood function in order to find the probability distribution and parameters that best explain the observed data.
• It provides a framework for predictive modeling in machine learning where finding model parameters can be framed as an optimization problem.
Kick-start your project with my new book Probability for Machine Learning, including step-by-step tutorials and the Python source code files for all examples.
Let’s get started.
This tutorial is divided into three parts; they are:
1. Problem of Probability Density Estimation
2. Maximum Likelihood Estimation
3. Relationship to Machine Learning
Problem of Probability Density Estimation
A common modeling problem involves how to estimate a joint probability distribution for a dataset.
For example, given a sample of observation (X) from a domain (x1, x2, x3, …, xn), where each observation is drawn independently from the domain with the same probability distribution (so-called
independent and identically distributed, i.i.d., or close to it).
Density estimation involves selecting a probability distribution function and the parameters of that distribution that best explain the joint probability distribution of the observed data (X).
• How do you choose the probability distribution function?
• How do you choose the parameters for the probability distribution function?
This problem is made more challenging as sample (X) drawn from the population is small and has noise, meaning that any evaluation of an estimated probability density function and its parameters will
have some error.
There are many techniques for solving this problem, although two common approaches are:
• Maximum a Posteriori (MAP), a Bayesian method.
• Maximum Likelihood Estimation (MLE), frequentist method.
The main difference is that MLE assumes that all solutions are equally likely beforehand, whereas MAP allows prior information about the form of the solution to be harnessed.
In this post, we will take a closer look at the MLE method and its relationship to applied machine learning.
Want to Learn Probability for Machine Learning
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Maximum Likelihood Estimation
One solution to probability density estimation is referred to as Maximum Likelihood Estimation, or MLE for short.
Maximum Likelihood Estimation involves treating the problem as an optimization or search problem, where we seek a set of parameters that results in the best fit for the joint probability of the data
sample (X).
First, it involves defining a parameter called theta that defines both the choice of the probability density function and the parameters of that distribution. It may be a vector of numerical values
whose values change smoothly and map to different probability distributions and their parameters.
In Maximum Likelihood Estimation, we wish to maximize the probability of observing the data from the joint probability distribution given a specific probability distribution and its parameters,
stated formally as:
This conditional probability is often stated using the semicolon (;) notation instead of the bar notation (|) because theta is not a random variable, but instead an unknown parameter. For example:
• P(x1, x2, x3, …, xn ; theta)
This resulting conditional probability is referred to as the likelihood of observing the data given the model parameters and written using the notation L() to denote the likelihood function. For
The objective of Maximum Likelihood Estimation is to find the set of parameters (theta) that maximize the likelihood function, e.g. result in the largest likelihood value.
We can unpack the conditional probability calculated by the likelihood function.
Given that the sample is comprised of n examples, we can frame this as the joint probability of the observed data samples x1, x2, x3, …, xn in X given the probability distribution parameters (theta).
• L(x1, x2, x3, …, xn ; theta)
The joint probability distribution can be restated as the multiplication of the conditional probability for observing each example given the distribution parameters.
• product i to n P(xi ; theta)
Multiplying many small probabilities together can be numerically unstable in practice, therefore, it is common to restate this problem as the sum of the log conditional probabilities of observing
each example given the model parameters.
• sum i to n log(P(xi ; theta))
Where log with base-e called the natural logarithm is commonly used.
This product over many probabilities can be inconvenient […] it is prone to numerical underflow. To obtain a more convenient but equivalent optimization problem, we observe that taking the
logarithm of the likelihood does not change its arg max but does conveniently transform a product into a sum
— Page 132, Deep Learning, 2016.
Given the frequent use of log in the likelihood function, it is commonly referred to as a log-likelihood function.
It is common in optimization problems to prefer to minimize the cost function, rather than to maximize it. Therefore, the negative of the log-likelihood function is used, referred to generally as a
Negative Log-Likelihood (NLL) function.
• minimize -sum i to n log(P(xi ; theta))
In software, we often phrase both as minimizing a cost function. Maximum likelihood thus becomes minimization of the negative log-likelihood (NLL) …
— Page 133, Deep Learning, 2016.
Relationship to Machine Learning
This problem of density estimation is directly related to applied machine learning.
We can frame the problem of fitting a machine learning model as the problem of probability density estimation. Specifically, the choice of model and model parameters is referred to as a modeling
hypothesis h, and the problem involves finding h that best explains the data X.
We can, therefore, find the modeling hypothesis that maximizes the likelihood function.
Or, more fully:
• maximize sum i to n log(P(xi ; h))
This provides the basis for estimating the probability density of a dataset, typically used in unsupervised machine learning algorithms; for example:
Using the expected log joint probability as a key quantity for learning in a probability model with hidden variables is better known in the context of the celebrated “expectation maximization” or
EM algorithm.
— Page 365, Data Mining: Practical Machine Learning Tools and Techniques, 4th edition, 2016.
The Maximum Likelihood Estimation framework is also a useful tool for supervised machine learning.
This applies to data where we have input and output variables, where the output variate may be a numerical value or a class label in the case of regression and classification predictive modeling
We can state this as the conditional probability of the output (y) given the input (X) given the modeling hypothesis (h).
Or, more fully:
• maximize sum i to n log(P(yi|xi ; h))
The maximum likelihood estimator can readily be generalized to the case where our goal is to estimate a conditional probability P(y | x ; theta) in order to predict y given x. This is actually
the most common situation because it forms the basis for most supervised learning.
— Page 133, Deep Learning, 2016.
This means that the same Maximum Likelihood Estimation framework that is generally used for density estimation can be used to find a supervised learning model and parameters.
This provides the basis for foundational linear modeling techniques, such as:
• Linear Regression, for predicting a numerical value.
• Logistic Regression, for binary classification.
In the case of linear regression, the model is constrained to a line and involves finding a set of coefficients for the line that best fits the observed data. Fortunately, this problem can be solved
analytically (e.g. directly using linear algebra).
In the case of logistic regression, the model defines a line and involves finding a set of coefficients for the line that best separates the classes. This cannot be solved analytically and is often
solved by searching the space of possible coefficient values using an efficient optimization algorithm such as the BFGS algorithm or variants.
Both methods can also be solved less efficiently using a more general optimization algorithm such as stochastic gradient descent.
In fact, most machine learning models can be framed under the maximum likelihood estimation framework, providing a useful and consistent way to approach predictive modeling as an optimization
An important benefit of the maximize likelihood estimator in machine learning is that as the size of the dataset increases, the quality of the estimator continues to improve.
Further Reading
This section provides more resources on the topic if you are looking to go deeper.
In this post, you discovered a gentle introduction to maximum likelihood estimation.
Specifically, you learned:
• Maximum Likelihood Estimation is a probabilistic framework for solving the problem of density estimation.
• It involves maximizing a likelihood function in order to find the probability distribution and parameters that best explain the observed data.
• It provides a framework for predictive modeling in machine learning where finding model parameters can be framed as an optimization problem.
Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.
14 Responses to A Gentle Introduction to Maximum Likelihood Estimation for Machine Learning
1. Seun Animasahun October 25, 2019 at 7:19 am #
Thanks for your explanation. Highky insightful.
I want to ask that in your practical experience with MLE, does using MLE as an unsupervised learning to first predict a better estimate of an observed data before using the estimated data as
input for a supervised learning helpful in improving generalisation capability of a model ?
□ Jason Brownlee October 25, 2019 at 1:46 pm #
It is not a technique, more of a probabilistic framework for framing the optimization problem to solve when fitting a model.
Such as linear regression:
2. George October 25, 2019 at 3:07 pm #
This product over many probabilities can be inconvenient […] it is prone to numerical underflow. To obtain a more convenient but equivalent optimization problem, we observe that taking the
logarithm of the likelihood does not change its arg max but does conveniently transform a product into a sum
— Page 132, Deep Learning, 2016.
This quote is from Page 128 – based on the edition of the book in the link
□ Jason Brownlee October 26, 2019 at 4:34 am #
Thanks George.
3. Jose CyC November 5, 2019 at 12:04 pm #
“We can state this as the conditional probability of the output X given the input (y) given the modeling hypothesis (h).”
Shouldn’t this be “the output (y) given the input (X) given the modeling hypothesis (h)”?
Given that we are trying to maximize the probability that given the input and parameters would give us the output.
It would be consistent with maximize L(y|X ; h)
□ Jason Brownlee November 5, 2019 at 1:41 pm #
Yes, that’s a typo.
Fixed. Thanks for pointing it out!
4. BRT February 27, 2020 at 8:30 pm #
How can we know the likelihood function from the data given?
□ Jason Brownlee February 28, 2020 at 6:05 am #
It is for an algorithm, not for data.
5. Manjit July 22, 2021 at 10:50 pm #
good explanation. I like how you link same technique in different fields like deep learning and unsupervised learning etc. ultimately if you understand you will know the underlying mechanism the
same. thanks for the article
□ Jason Brownlee July 23, 2021 at 5:59 am #
You’re welcome.
6. Yunhao(Ron) June 17, 2022 at 1:56 am #
Dear Jason,
I have a question about ‘MLE applied to solve problem of density function’ and hope to get some help from you.
For the definition of MLE,
– it is used to estimate ‘Parameters’ that can maximize the likelihood of an event happened.
– For example, Likelihood (Height > 170 |mean = 10, standard devi. = 1.5). The MLE is trying to change two parameters ( which are mean and standard deviation), and find the value of two
parameters that can result in the maximum likelihood for Height > 170 happened.
When we use MLE to solve the problem of density function, basically we just
(1) change the ‘mean = 10, standard devi. = 1.5’ into –> ‘theta (θ) that defines both the choice of the probability density function and the parameters of that distribution.’
(2) change the ‘Height > 170’ into –> sample of observation (X) from a domain (x1, x2, x3, · · · , xn),
Simply, we just use the logic/idea/framework of MLE to solve the problem of density function (Just change some elements of MLE framework).
After modifying the framework of MLE, the parameters (associated with the maximum likelihood or peak value) represents the parameters of probability density function (PDF) that can best fit for
probability distribution of the observed data.
Is that correct?
7. Yunhao June 17, 2022 at 5:42 am #
Hi Jason,
In the book, you write ‘MLE is a probabilistic framework for estimating the parameters of a model. We wish to maximize the conditional probability of observing the data (X) given a specific
probability distribution and its parameters’
May I ask why ‘ parameters that maximize the conditional probability of observing the data’ are parameters that result in/belong to the best-fit Probability Density (PDF)?
I cannot understand how to figure out the relationship between maximum likelihood and best-fit.
8. Suraj March 30, 2024 at 11:56 pm #
Hi Jason,
Firstly, thank you for the detailed explanation; it really helped clarify the topic. However, I find myself a bit confused about the notation used for likelihood. In your post, you denoted
likelihood as L(X | theta), implying that we aim to find theta parameters that maximize the likelihood of observing the given data. Therefore, wouldn’t it be more appropriate to use L(theta|X)
instead? I also find this sort of notation more commonly used across different articles and blog posts.
□ James Carmichael March 31, 2024 at 1:40 am #
Hi Suraj… Your question brings up a common point of confusion in statistics regarding the notation and interpretation of likelihood and probability. Let’s clarify this.
### Probability vs. Likelihood
– **Probability** of observing data \(X\) given parameters \(\theta\), denoted as \(P(X | \theta)\), quantifies the probability of seeing the data \(X\) if the parameters \(\theta\) are
known. This is what you’d use in a probabilistic model to predict data outcomes based on known parameters.
– **Likelihood** of parameters \(\theta\) given observed data \(X\), denoted as \(L(\theta | X)\) or sometimes simply \(L(\theta)\), is a function of \(\theta\) for a fixed \(X\). It
represents how “likely” different parameter values are given the data you have observed. Unlike a probability, likelihood is not constrained to sum up to 1 over \(\theta\).
### Why \(L(\theta | X)\) and not \(L(X | \theta)\)?
The notation \(L(\theta | X)\) is indeed used to represent the likelihood of parameters given the data, but it’s crucial to understand that this is not the same as conditional probability
notation even though it looks similar. The likelihood function \(L(\theta | X)\) is a function of \(\theta\) with \(X\) held fixed, essentially flipping the conditioning found in probability
### The Essence of the Confusion
The confusion often arises because the likelihood function is mathematically equivalent to the probability of observing the data given the parameters, i.e., \(L(\theta | X) = P(X | \theta)\),
but with a different interpretation. When we talk about likelihood, we’re focusing on how the parameters \(\theta\) fit the observed data \(X\), not on the probability of the data per se.
– **In probability:** We know \(\theta\) and ask, “What’s the probability of seeing \(X\)?”
– **In likelihood:** We know \(X\) and ask, “How likely are different values of \(\theta\)?”
### Maximizing Likelihood
When we say we aim to find the parameters \(\theta\) that maximize the likelihood of observing the given data \(X\), we’re seeking the parameter values under which the observed data would be
most probable. This process is central to many statistical estimation techniques, including maximum likelihood estimation (MLE).
In summary, while the notation might suggest a probability statement, it’s essential to recognize the distinct interpretation and purpose of likelihood in statistical analysis. The use of \(L
(\theta | X)\) is correct in the context of estimating parameters that make the observed data most likely.
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Digital Electronics
Module 1.3
Binary Arithmetic
• After studying this section, you should be able to:
• Understand the rules used in binary calculations.
• • Addition.
• • Subtraction.
• • Use of carry, borrow & pay back.
• Understand limitations in binary arithmetic.
• • Word length.
• • Overflow.
Binary Addition Rules
Arithmetic rules for binary numbers are quite straightforward, and similar to those used in decimal arithmetic. The rules for addition of binary numbers are:
Fig. 1.3.1 Rules for Binary Addition
Notice that in Fig. 1.3.1, 1+1 = (1)0 requires a ‘carry’ of 1 to the next column. Remember that binary 10[2] = 2[10] decimal
Fig. 1.3.2 Simple Binary Addition
Binary addition is carried out just like decimal, by adding up the columns, starting at the right and working column by column towards the left.
Fig. 1.3.3 Binary Addition with Carry
Just as in decimal addition, it is sometimes necessary to use a ‘carry’, and the carry is added to the next column. For example, in Fig. 1.3.3 when two ones in the right-most column are added, the
result is 2[10] or 10[2], the least significant bit of the answer is therefore 0 and the 1 becomes the carry bit to be added to the 1 in the next column.
Binary Subtraction
The rules for subtraction of binary numbers are again similar to decimal. When a large digit is to be subtracted from a smaller one, a ‘borrow’ is taken from the next column to the left. In decimal
subtractions the digit ‘borrowed in’ is worth ten, but in binary subtractions the ‘borrowed in’ digit must be worth 2[10] or binary 10[2].
After borrowing from the next column to the left, a ‘pay back’ must occur. The subtraction rules for binary are quite simple even if the borrow and pay back system create some difficulty. Depending
where and when you learned subtraction at school, you may have learned a different subtraction method, other than ‘borrow and payback’, this is caused by changing fashions in education. However any
method of basic subtraction will work with binary subtraction but if you do not want to use ‘borrow and payback’ you will need to apply your own subtraction method to the problem.
Fig. 1.3.4 Rules for Binary Addition
Binary Subtraction Rules
The rules for binary subtraction are quite straightforward except that when 1 is subtracted from 0, a borrow must be created from the next most significant column. This borrow is then worth 2[10] or
10[2] because a 1 bit in the next column to the left is always worth twice the value of the column on its right.
Fig. 1.3.5 shows how binary subtraction works by subtracting 5[10] from 11[10] in both decimal and binary. Notice that in the third column from the right (2^2) a borrow from the (2^3) column is made
and then paid back in the MSB (2^3) column.
Note: In Fig 1.3.5 a borrow is shown as ^10,and a pay back is shown as 0^1. Borrowing 1 from the next highest value column to the left converts the 0 in the 2^2 column into 10[2] and paying back 1
from the 2^2 column to the 2^3 adds 1 to that column converting the 0 to 01[2].
Fig. 1.3.5 Binary Subtraction
Once these basic ideas are understood, binary subtraction is not difficult, but does require some care. As the main concern in this module is with electronic methods of performing arithmetic however,
it will not be necessary to carry out manual subtraction of binary numbers using this method very often. This is because electronic methods of subtraction do not use borrow and pay back, as it leads
to over complex circuits and slower operation. Computers therefore, use methods that do not involve borrow. These methods will be fully explained in Number Systems Modules 1.5 to 1.7.
Subtraction Exercise
Just to make sure you understand basic binary subtractions try the examples below on paper. Don’t use your calculator, click the image to download and print the exercise sheet. Be sure to show your
working, including borrows and paybacks where appropriate. Using the squared paper helps prevent errors by keeping your binary columns in line. This way you will learn about the number systems, not
just the numbers.
Fig. 1.3.6 Limits of 4 Bit Arithmetic
Limitations of Binary Arithmetic
Now back to ADDITION to illustrate a problem with binary arithmetic. In Fig. 1.3.6 notice how the carry goes right up to the most significant bit.
This is not a problem with this example as the answer 1010[2] (10[10]) still fits within 4 bits, but what would happen if the total was greater than 15[10]?
Fig. 1.3.7 The Overflow Problem
As shown in Fig 1.3.7 there are cases where a carry bit is created that will not fit into the 4-bit binary word. When arithmetic is carried out by electronic circuits, storage locations called
registers are used that can hold only a definite number of bits. If the register can only hold four bits, then this example would raise a problem. The final carry bit is lost because it cannot be
accommodated in the 4-bit register, therefore the answer will be wrong.
To handle larger numbers more bits must be used, but no matter how many bits are used, sooner or later there must be a limit. How numbers are held in a computer system depends largely on the size of
the registers available and the method of storing data in them, however any electronic system will have a way of overcoming this ‘overflow’ problem, but will also have some limit to the accuracy of
its arithmetic. | {"url":"https://learnabout-electronics.org/Digital/dig13.php","timestamp":"2024-11-08T05:06:29Z","content_type":"text/html","content_length":"19780","record_id":"<urn:uuid:e384d2d2-9086-42ba-a3ef-fc8ec86dd206>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00173.warc.gz"} |
4036 Form: Fill
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PDF Editor FAQ
What are all the ordered integer pairs (a,b) which satisfy [math]\frac{1}{a}+\frac{1}{b}=\frac{3}{2018}[/math]?
Writing the left side as a single fraction, we get [math]\frac{a+b}{ab}=\frac{3}{2018}[/math]. Since [math]\frac{3}{2018}[/math] is a fraction in its simplest form, any equivalent fraction must be of
the form [math]\frac{3k}{2018k}[/math] where [math]k[/math] is an integer, but not necessarily positive. Therefore we have the simultaneous equations [math]a+b=3k[/math] and [math]ab=2018k[/math]. We
can rewrite this as [math]a+\frac{2018k}{a}=3k[/math], which boils down to the quadratic equation [math]a^2-3ak+2018k=0[/math]. Using the quadratic formula, we find[math]\displaystyle a = \frac{1}{2}
\left(3k \pm \sqrt{9k^2-8072k} \right) \tag*{(1)}[/math]I know you say [math](a,b)[/math] is an ordered pair, but to save time I’ll assume [math]a[/math] is the larger integer and [math]b[/math] is
the smaller — at the end, we’ll just remember the pairs can be either way round. This means we pick the [math][/math][math]+[/math] sign for [math]a[/math] and [math]-[/math] sign for [math]b[/math].
Since [math]a \in \mathbb{Z}[/math], the right side of [math](1)[/math] must be an integer. Therefore [math]9k^2-8072k=x^2[/math] must be a perfect square. Multiplying through by [math]9[/math] and
rearranging a little shows[math]\displaystyle (9k-4036)^2 - (3x)^2 = 16289296 \tag*{}[/math]Writing [math]16289296[/math] in its prime factorisation form and making the substitutions [math]\alpha =
9k-4036[/math] and [math]\beta = 3x[/math] shows[math]\displaystyle \alpha^2 - \beta^2 = (\alpha+\beta)(\alpha-\beta) = 2^4 \times 1009^2 \tag*{}[/math]Since each bracket is an integer, we can split
into a finite number of simultaneous equations, where [math]\alpha+\beta[/math] and [math]\alpha-\beta[/math] are a factor pair of [math]16289296[/math]. To save time, we only need consider the cases
when [math]\alpha+\beta[/math] is the bigger factor, and since each bracket has the same parity we can eliminate the cases when one factor is odd. I’ll walk you through one of the equations so you
get the idea, but I won’t do them all here. Consider the case when [math]\alpha+\beta=8144648[/math] and [math]\alpha-\beta=2[/math]. Then [math]\alpha=4072325[/math]. Therefore [math]k=\frac{\
alpha+4036}{9}=452929[/math]. Plugging this back into [math](1)[/math] shows [math]a=1358114[/math] and [math]b=673[/math]. You can use exactly the same process to solve the other equations (only
some of which admit integer solutions), and make sure you remember the cases when [math]\alpha+\beta[/math] and [math]\alpha-\beta[/math] are negative.The only integer pairs are[math]\displaystyle
(a,b) = (1358114,673), [/math][math][/math][math]\, (340033,674), [/math][math][/math][math]\, (2018,1009), [/math][math][/math][math]\, (672, -678048) \tag*{}[/math]Finally, don’t forget each pair
can be either way round!
What is the remainder when 2017^2018 is divided by 11?
Since [math]2017 \equiv 2^2\pmod{11}[/math] and [math]2^5 \equiv -1\pmod{11}[/math], we have[math]2017^{2018} \equiv 2^{4036} = 2 \cdot \big(2^5\big)^{807} \equiv 2 \cdot (-1) \pmod{11}[/math].The
remainder is [math]9[/math]. [math]\blacksquare[/math]
Evaluate the integral[math]\quad \displaystyle\int \dfrac{2017 x^{2016} [/math][math][/math][math]+ 2018 x^{2017}}{1 [/math][math][/math][math]+ x^{4034} [/math][math][/math][math]+ 2 x^{4035} [/
math][math][/math][math]+ x^{4036}}\,dx[/math]?
We want to evaluate[math]I = \displaystyle \int \dfrac{2017 x^{2016} [/math][math][/math][math]+ 2018 x^{2017}}{1 [/math][math][/math][math]+ x^{4034} [/math][math][/math][math]+ 2 x^{4035} [/math]
[math][/math][math]+ x^{4036}}\,dx. \tag*{}[/math]To this end, we rewrite it as[math]I = \displaystyle \int \dfrac{2017 x^{2016} [/math][math][/math][math]+ 2018 x^{2017}}{1 [/math][math][/math]
[math]+ (x^{2017} [/math][math][/math][math]+ x^{2018})^2} \,dx. \tag*{}[/math]Letting [math]w = x^{2017} [/math][math][/math][math]+ x^{2018}[/math], we obtain[math]\begin{align*} I &= \displaystyle
\int \dfrac{1}{1 [/math][math][/math][math]+ w^2} \,dw\\ &= \arctan{w} [/math][math][/math][math]+ C\\ &= \arctan(x^{2017} [/math][math][/math][math]+ x^{2018}) [/math][math][/math][math]+ C. \end
{align*} \tag*{}[/math] | {"url":"https://cocodoc.com/form/form-4036","timestamp":"2024-11-11T16:53:53Z","content_type":"text/html","content_length":"62774","record_id":"<urn:uuid:9d616bc3-e26c-4d91-ba1b-47d8b3cad3f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00529.warc.gz"} |
Nonlinear system - (Thinking Like a Mathematician) - Vocab, Definition, Explanations | Fiveable
Nonlinear system
from class:
Thinking Like a Mathematician
A nonlinear system is a type of mathematical system in which the relationship between variables is not a linear combination, meaning that changes in the input do not produce proportional changes in
the output. In these systems, small variations can lead to significant and unpredictable effects, making them complex and often difficult to analyze. Nonlinear systems are prevalent in real-world
phenomena, where interactions are more intricate than simple linear relationships.
congrats on reading the definition of nonlinear system. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Nonlinear systems can exhibit behavior such as bifurcations, where a small change in a parameter can lead to a sudden qualitative change in the system's behavior.
2. In nonlinear differential equations, solutions may not exist or may not be unique, complicating the analysis and prediction of system behavior.
3. Stability analysis of nonlinear systems often requires techniques such as Lyapunov's methods, as traditional linear stability methods do not apply.
4. Many real-world systems, including ecological models and economic systems, are inherently nonlinear, reflecting the complexity of interactions within those systems.
5. Nonlinear systems can demonstrate phenomena like limit cycles and chaos, where the system's behavior becomes unpredictable over time.
Review Questions
• How do nonlinear systems differ from linear systems in terms of their behavior and analysis?
□ Nonlinear systems differ from linear systems primarily in how their outputs respond to inputs. In linear systems, changes in input result in proportional changes in output, making them
predictable and easier to analyze. In contrast, nonlinear systems can produce disproportionate responses, leading to complex behaviors such as bifurcations or chaos. This complexity often
requires specialized analytical techniques for understanding and predicting their dynamics.
• Discuss the implications of chaos theory within the context of nonlinear systems and how it affects predictability.
□ Chaos theory plays a crucial role in understanding nonlinear systems as it reveals how small changes in initial conditions can lead to vastly different outcomes. This sensitivity makes
long-term predictions challenging since even slight variations can result in unpredictable behavior. For instance, weather patterns are often modeled as nonlinear systems, where chaotic
dynamics prevent accurate long-range forecasting. This highlights the importance of understanding chaos when dealing with real-world nonlinear phenomena.
• Evaluate how the concepts of equilibrium points and stability analysis apply to nonlinear systems, including their relevance in practical applications.
□ Equilibrium points are essential in nonlinear systems as they indicate states where the system can potentially remain stable. Analyzing stability at these points helps determine whether small
perturbations will return the system to equilibrium or lead it away into chaotic behavior. Techniques like Lyapunov's methods are vital for assessing stability since traditional linear
approaches fail for nonlinear cases. This analysis is crucial in fields such as engineering and economics, where ensuring stability can be critical for maintaining desired outcomes.
"Nonlinear system" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/thinking-like-a-mathematician/nonlinear-system","timestamp":"2024-11-10T15:12:06Z","content_type":"text/html","content_length":"156096","record_id":"<urn:uuid:02572c0d-a5ec-4847-b49e-2e4cd529aad5>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00347.warc.gz"} |
Show Me
Understanding cardinal numbers
Matching numerals and amounts
Children often enjoy finding things which are the same.
Adults could provide lots of different images and resources to show the value of numbers and numerals.
The Activity
Hold up five fingers, a giant dotty dice or a large numeral and ask the children to show you that number in different ways. Put out lots of different things which children can use to show the
numbers, including countable items like conkers, small world toys, large blocks, multilink, dot images like dice and dominoes, structured apparatus like Numicon, Cuisenaire or an abacus, things in
packs like egg boxes and crayon cartons, and number symbols including washing lines, number lines and 100 squares.
Encouraging mathematical thinking and reasoning:
How does this five look different from that five?
What does this pattern of five look like?
What can you see?
How did you make five with two hands?
How do you know these are the same number?
What is the same and what is different about these fives?
Opening Out
Can you show me five fingers a different way? Is there another way? What different patterns can you make with five counters?
Can you see any numbers hidden inside this pattern of five?
Can you show me 15? How do you know it is 15?
Can you draw or record your patterns?
Can you put something on the paper to show how many there are? Can you put some numbers to show what hidden numbers you see?
The Mathematical Journey
Counting and cardinality:
• using counting to check
• subitising - recognising the number of items without counting
• conservation - rearranging items and explaining that the number is the same because the arrangement can be returned to the original and none have been added or taken away
• matching two groups one-to-one to show that they have the same number
Matching numerals and amounts:
• selecting number symbols to match the total or numbers inside numbers
Composition of numbers:
• talking about numbers being made up of other numbers: "It's six because I see three and three"
• knowing number facts e.g. "Five and one more makes six"
Development and Variation
Ask children to show different numbers.
Have a display table for the number of the day or week, or where children can choose a number to make a display for. Show number symbols in different forms and scripts e.g. on calculator.
Number hunt: hide numerals and bags with numbers of things in (e.g. conkers).
Use laminated cards with dots or pictures and ask children to find a numeral and then items or pictures with the same number.
Make different patterns for the same number with objects on trays; take photos.
Use overlapping digit cards for teen numbers (see picture) and same colour sticks of 10.
Countable items: conkers, small world toys, large blocks, multilink and pennies.
Dot pattern images: dice, dominoes and Hungarian number pictures.
Structured resources: Numicon, Cuisenaire, unifix with same colour sticks of 10, 10p coins.
Things in pairs or packs: pairs of baby socks, egg boxes, packets of crayons or multipacks.
Numerals in different styles, on tiles, washing lines, giant number tracks, 100 square mats, overlapping place value cards for teen numbers, calculators.
Numerals on everyday objects like birthday cards, football shirts, calendars, clocks and measuring equipment e.g. height charts.
Displays of numbers in different arrangements with numerals e.g. staircases of rods, conkers on strings, number lines or tracks with numerals and dot patterns.
Download a
of this resource.
Acknowledgements: Jenni Back and Janine Davenall | {"url":"https://nrich.maths.org/eyfs-activities/show-me","timestamp":"2024-11-09T02:32:07Z","content_type":"text/html","content_length":"43964","record_id":"<urn:uuid:86e13f14-6437-4556-b2f4-65b5f8d2aae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00416.warc.gz"} |
Lattice Multiplication
for iPad, iPhone, Mac, and Windows 10 PCs
This app can be used to teach and study the ancient lattice multiplication method. Solving lattice multiplication problems is also excellent times tables practice. The app is very easy to use and it
has an intuitive interface with customizable colors and other settings. The user can solve random and custom multiplication problems with small and large numbers.
Lattice multiplication is algorithmically equivalent to long multiplication. A lattice (a grid) guides the calculation. All the multiplications are done first and then the additions. The method was
introduced to Europe in 1202 in Fibonacci's Liber Abaci.
In this groundbreaking book Fibonacci presented many algorithms for working with Arabic numerals. Ancient Indians and Chinese originally invented some of the algorithms. Fibonacci presented both the
current standard long multiplication and also an originally Indian method called lattice multiplication, which is faster and more compact for working with larger numbers.
The Lattice Multiplication app
The Lattice Multiplication app allows the user to solve a lattice multiplication problem step by step and animates all the steps. In the steps the user will multiply or add. The correct answer will
fly to the right place. If the user presses the wrong button the answer will appear above the keyboard but it will not move.
• The multiplicand can have up to 5 digits
• The multiplier can have up to 3 digits
• The current operation for each step can be hidden
• The operands for the current operation can be highlighted
• There are 3 different themes: black, gray and gold
• The speed of the animations can be set.
Lattice Multiplication Videos
Lattice multiplication with two digit numbers
54 x 73
Lattice multiplication with three and five digit numbers
567 x 66439
Lattice Multiplication in the Apple VPP Store for Education
The Volume Purchase Program allows participating educational institutions to purchase iDevBooks math apps in volume and distribute them to students. All iDevBooks math apps offer special 50% discount
for purchases of 20 apps or more for participating educational institutions. | {"url":"http://idevbooks.com/apps/lattice.php","timestamp":"2024-11-06T21:44:51Z","content_type":"text/html","content_length":"6739","record_id":"<urn:uuid:c8f9f02f-77f4-4bfb-9b7c-ec81758104f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00730.warc.gz"} |
NFL Betting Angles #5: Injuries
Every year there are a few teams that manage an impressive injury list by late-season. This year, two examples would be the Redskins and Broncos, which followers of the blog picks would know we are
usually on, although these teams managed to play badly enough, and lose players at key enough positions, that we don't have them this week (a shame in the case of the Redskins). In general, we value
defensive injuries highly, but if a team is hurt on offense ex-QB, we will be on them. And if you look at the popular Massey-Peabody picks, they are fading injuries every single week.
The fine book Sharper introduced the idea of using Madden ratings to calculate injuries. Oddly enough, I had never used the ratings for injuries before reading the book, but more as a way to measure
early-season talent levels before stats really start to work in the NFL. It turns out they are effective in injury calculation as well.
A metric I have developed to measure how injured a team is involves taking the best 10 players by Madden rating on that team on offense (excluding the FB), the best 11 on defense, and the best QB who
played in the game, and using these as overall unit Madden ratings in each game. Then, one can take the maximum Madden rating that any such unit has had up to that point in time of the season, and
compare the rating in the current game to that maximum. Those teams with a rating in the current game far below their season maximum are likely suffering from a large number of injuries or
suspensions/trades, which are for the most part the same thing.
Ideally, we'd then include these injury impacts as part of a fundamental handicapping model that takes into account team statistics and other situational factors. For one example of how to do this,
buy my book. But for now, we can look at whether injury impacts have been correctly rated by the market, through an ATS analysis of games between 2012 and 2017.
Each of the tables below are shown in terms of each team's Madden rating for that particular unit in that game, subtracted from the maximum Madden rating that that team's unit has had up to that
point in the season (we exclude Week 1). We then take the difference in the two numbers, to determine how relatively injured each team is. For example, suppose in a game the home team is starting an
offense with an average Madden rating of 78, whose highest Madden rating thus far was 81, and they are facing a team whose average Madden rating in that game is 82, whose highest Madden rating thus
far was 83. The home team has a differential of 78 - 81 or -3, while the away team has a differential of 81 - 82 or -1. Subtracting the two we get -2, meaning the home team's offense is two points
more injured, relative to their highest Madden rating, compared to the away team.
We start with offensive injuries:
We can see from the ATS table above that on offense, fading offensive injuries has been a huge winning angle. If the home team is two points or more injured on offense than the away team, they have
covered the spread by 1.2 points on average and have won 9 units over 219 games, while if the away team is two points or more injured, they have covered by 0.9 points and have won 14.5 units over 208
games. Overall the angle would have won 23.5 units over 427 games, a very solid return.
Looking at the average point spread, the market appears to value offensive injuries highly, as there is a difference of 3.2 points between the two categories of games. This would indicate that our
Madden approach has done a decent job of figuring out when players are hurt. But at least over this sample of games, this hasn't translated to nearly as large of a difference in performance - the
market prices in 3.2 points, but the difference between the "away injured" and "home injured" groups is only about 1.1 points. Perhaps this is why systems like the Massey-Peabody that ignore injuries
have done pretty well. Although offensive injuries do matter, you would have been much better off completely ignoring them than over-accounting for them has the market has done.
Moving on to defense:
On defense we don't see any obvious winning angle. If anything, defensive injuries seem to be slightly under-valued by the market, as betting against the team that is more hurt on defense would have
shown a slight profit of 2.2 units in 337 games, and covered by about 0.7 points per game. But this is well within statistical variance, and it appears these injuries are being priced in correctly.
Finally, quarterbacks, perhaps the most interesting category:
Here we have what may be another winning angle. While there is over a 6 point difference between home QB-injured teams and away QB-injured teams, suggesting the average QB injury is given an impact
of 3 points, it turns out this wasn't enough. Betting against the team with the QB out would have won 23.3 units in 417 games, with a very healthy average cover margin of 1.6 points per game.
The teams with the injured QB in these situations play only very slightly worse than the market expects, with the main driver of the difference being that they average about 0.01 more turnovers per
difference in 1 Madden injury point after adjusting for the market point spread. One turnover is worth about 4.2 points, so this ends up being about 0.04 points per game, per Madden injury point.
There is also a very slight difference in yards per play differential as well. Overall with such small differences it is tough to say whether backups have simply had a tough run of luck in these
games, but the angle seems promising. | {"url":"https://www.quantumsportssolutions.com/blogs/handicappers-library/nfl-betting-angles-5-heavily-injured-teams","timestamp":"2024-11-14T21:32:04Z","content_type":"text/html","content_length":"75895","record_id":"<urn:uuid:5a1e079c-b816-4a64-ad13-52f9a62eec5c>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00187.warc.gz"} |
Tolstoy Loves Me
Brendan McKay
I demonstrate a remarkable matching between my name and my date of birth in the Hebrew translation of War and Peace. A rigorous method of analysis produces a significance level of less than 1 in
5000. As supporting evidence, my place of residence is also shown to be remarkably well matched to my name.
The first 78064 letters of War and Peace have been used as a standard control text since they were so-used in the paper of Witztum et al [WRR]. The length is taken to be the same as the length of the
Koren edition of the Book of Genesis.
Given two words w,w', a complicated procedure in [WRR] defines a "distance" c(w,w') in the range 0-1 (though sometimes it is "undefined"). To good approximation, random words w,w' will produce a
value of c(w,w') uniformly distributed between 0 and 1. Also in [WRR] are several procedures by which a set of numbers in the interval [0,1] are converted into a single number in the same interval.
These procedures are called P1 and P2. P1 and P2 cannot be correctly interpretted as probabilities, but in [WRR] they are used as measures by which several different sets of numbers can be compared.
Small values of P1 and P2 indicate "close matching", and large values indicate lack of matching.
I will use the Michigan-Clairmont scheme for transliterating Hebrew letters. Here is the definition:
) = Alef B = Beit G = Gimmel D = Dalet H = Hey W = Waw
Z = Zain X = Chet + = Tet Y = Yud K = Kaf L = Lamed
M = Mem N = Nun S = Samech ( = Ain P = Pey C = Tzadi
Q = Kuf R = Reish $ = Shin T = Tav
My name is Brendan McKay, which in Hebrew is most correctly written BRNDN MQYY. With regard to spellings, there is no question about BRNDN, as the final "a" of Brendan is not pronounced at all, and
certainly is not "o" or "oo". The name McKay can conceivably be spelt in other ways, but the use of only one K or Q correctly reflects the pronunciation, which is not Mc'Kay but M'Kay. In any case,
the spelling MQYY was the one independently chosen by the Israeli newspaper Maariv on the single occasion that I have been named in the Israeli press (May 29, 1997).
My place of residence for the past 14 years has been Canberra, Australia. The Hebrew spellings of these names, according to the Encyclopedia Hebraica are QNBRH and )WS+RLYH.
My date of birth was October 26, 1951. In the Jewish calendar, that was 26 Tishri 5712.
To perform the experiment, we must convert the data into words, limiting ourselves to the range 5-8 letters as in [WRR].
Firstly, following [WRR], we make a list of appellations. There are only two obvious ones: BRNDN and DR MQYY. Others are too long (BRNDN MQYY, DR BRNDN MQYY) or too short (MQYY). (The method of [WRR]
requires 5-8 letters.) The use of DR for representing my title is common in Israel; for example it appears on office doors in university departments. I do not hold the rank of Professor according to
the rules of my University, and so am not entitled to use appellations with that title. [Note added: I was promoted to Professor in September 1998, but since PRWP MQYY does not have an ELS in War and
Peace the results of this article do not change.]
To make the date 26 Tishri 5712 into words, we use precedents from earlier experiments. Firstly, we follow [WRR] in writing 26 Tishri as KWT$RY, BKWT$RY and KWBT$RY. Secondly, we follow [HNGM] in
writing 5712 as BT$YB, HT$YB, BHT$YB, $NTT$YB, B$NTT$YB, and $NTHT$YB. (The other two, T$YB and B$NTHT$YB are outside the 5-8 length range.)
When the procedure c(w,w') from [WRR] is applied, a considerable number of small distances are found.
w = BRNDN DR MQYY
KWT$RY 0.1120 0.0480
BKWT$RY 0.2542 0.1441
KWBT$RY 0.1557 0.1393
BT$YB 0.3200 0.4160
HT$YB 0.0720 0.0160
BHT$YB 0.0240 0.0720
$NTT$YB 0.1368 0.0211
B$NTT$YB undef undef
$NTHT$YB 0.7500 0.8750
QNBRH 0.0880 0.0081
)WS+RLYH undef undef
Values marked "undef" are undefined according to the rules of [WRR].
Significance Levels
The procedure from [WRR] cannot be used because there is only one personality (myself) in the experiment. Therefore we must devise another way to determine how unlikely it is to obtain so many small
values. In all cases we will use the measures P1 and P2. Because the place of residence is somewhat different in nature from the date, we will perform all tests twice: once with only the date, and
once with date and place combined.
The values of P1 and P2 are:
Date only Date and Place
P1 : 0.0000326 0.00000252
P2 : 0.0002415 0.00001971
The first method of obtaining a significance level will be to try alternates to the two appellations, namely rearrangements of the letters. For example, instead of BRNDN we try BNNDR, NDBRN etc..
This approach was used in [M] to analyse the famous Aaron cluster.
Since the calculation of c(w,w') uses words running both forwards and backwards, and each appellation contains one letter twice, there are 30 distinct permutations of BRNDN and 180 of DRMQYY. In
combination, there are 30*180 = 5400 distinct pairs of appellations. We tried every one of these appellation pairs and calculated the P1 and P2 scores. Here are the results:
1. Using the date only
P1 P2
Best: BRNDN, DRMQYY 0.0000326 0.0002415
Next best: DBRNN, DRMQYY 0.0002476 0.0008575
Best derangement: DBRNN, RDQYMY 0.0070036 0.0030297
The correct spelling wins by a factor of 7.6 for P1, and by a factor of 3.6 for P2. The third line shows the best combination of two misspellings. It is worse by a factor of 215 for P1, and a
factor of 12.5 for P2.
2. Using both the date and the place
P1 P2
Best: BRNDN, DRMQYY 0.0000025 0.0000197
Next best: BRNDN, RYQDMY 0.0000224 0.0000767
Best derangement: BDRNN, RYQDMY 0.0009109 0.0006115
The correct spelling wins by a factor of 9.0 for P1, and by a factor of 3.9 for P2. The third line shows the best combination of two misspellings. It is worse by a factor of 364 for P1, and a
factor of 31 for P2.
In either case, we find that the correct spelling scores a clear and convincing win. The result remains the same either with or without the place names, and for both P1 and P2. This event clearly has
a probability of at most 1 in 5400.
A Confirmatory Computation
In order to demonstrate that the remarkably low significance level computed in the previous section is not too much an artifact of the computational method, we also tried another approach. Keeping
the spellings constant, we randomly permuted the letters of War and Peace. We did that 10,000 times, and for each text we computed the P1 and P2 scores. Here are the numbers of texts, out of 10,000,
performing better than the real text of War and Peace:
Using the date only Using both date and place
P1 P2 P1 P2
Here again we see consistent significance levels below 1 in 1000, completely confirming our expectations.
What am I Claiming?
The lesson to be drawn from this paper is clear enough. Anyone with the skill and the perseverance can make ELS experiments that seem to show remarkable results. In this paper we found a significance
level well below 1/1000 from a single name and a single date.
Did it happen by chance? Yes!
[WRR] D. Witztum, E. Rips and Y. Rosenberg, Equidistant Letter Sequences in the Book of Genesis, Statistical Science, Vol 9 (1994) 429-438.
[M] D. Michaelson, Reading the Torah at Equal Intervals, B'Or Torah, 1987.
[HNGM] D. Bar-Natan, A. Gindis, A. Levitan and B. McKay, Report on new ELS tests of Torah, 1997, here.
The image at the top of the page shows "Canberra" and "Dr McKay".
Back to the Mathematical Miracles page
Copyright Brendan McKay (1997), bdm@cs.anu.edu.au. | {"url":"https://users.cecs.anu.edu.au/~bdm/codes/wpmckay.html","timestamp":"2024-11-10T15:45:02Z","content_type":"text/html","content_length":"9284","record_id":"<urn:uuid:1bf0a6af-89d1-4eb6-875f-ba0a0e1a3c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00133.warc.gz"} |
How can I tick a box based on two criteria?
I'd like to tick a checkbox based on another checkbox being ticked and for a specific task name. I have the following formulas that work individually. How can I combine them?
=IF([Checkbox]@row = 1, 1,0)
=IF([Task name]@row = "XYZ", 1,0)
Best Answers
• I think I found my own answer - FYI for anyone who has the same query:
=IF(Checkbox@row = 1, IF([Task Name]@row = "XYZ", 1, 0))
• Hi @Stephen Everiss,
Your formula would look like this:
=IF(AND(Checkbox@row = 1, [Task Name]@row = "XYZ"), 1, 0)
Hope this helps, but if you've any problems/questions then let us know!
• I think I found my own answer - FYI for anyone who has the same query:
=IF(Checkbox@row = 1, IF([Task Name]@row = "XYZ", 1, 0))
• Hi @Stephen Everiss,
Your formula would look like this:
=IF(AND(Checkbox@row = 1, [Task Name]@row = "XYZ"), 1, 0)
Hope this helps, but if you've any problems/questions then let us know!
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Microsoft Excel is a fundamental tool for data analysis,used by organizations across the globe. Whether you're applying for a dataanalyst position or a related role, Excel knowledge is essential.
Below are theTop 50 Microsoft Excel Interview Questions along with comprehensiveanswers to help you ace your next interview.
1. What is Microsoft Excel and its use in data analysis?
Microsoft Excel is a spreadsheet program that allows users to store, organize,and manipulate data. In data analysis, Excel is used to perform calculations,analyze large datasets, and create charts
and visualizations for reporting.
2. What are the most common data analysis functions inExcel?
Some of the most common data analysis functions include:
● SUM(): Adds numbers in a range.
● AVERAGE(): Calculates the mean of a range.
● COUNT(): Counts the number of cells with numbers.
● IF(): Performs conditional operations.
● VLOOKUP(): Searches for a value in the first column of a table.
● INDEX/MATCH: Advanced lookup and reference functions.
3. What is the difference between absolute and relativecell references in Excel?
● Absolute Cell Reference: Refers to a fixed cell, marked with a $ symbol (e.g., $A$1), so it does not change when copied to other cells.
● Relative Cell Reference: Refers to cells relative to the position of the formula, and it adjusts when copied to another cell (e.g., A1).
4. How would you use PivotTables in data analysis?
PivotTables allow users to summarize and analyze large datasets by groupingdata and performing operations like summing, averaging, or counting. They areideal for analyzing trends, comparing
categories, and generating quickinsights.
5. What is the difference between a PivotTable and aPivotChart?
● PivotTable: Summarizes data in tabular form.
● PivotChart: A visual representation of a PivotTable that allows interactive data visualization.
6. What is VLOOKUP, and how do you use it?
VLOOKUP (Vertical Lookup) is used to search for a value in the firstcolumn of a table and return a value in the same row from a specified column.Syntax:
=VLOOKUP(lookup_value, table_array, col_index_num, [range_lookup])
7. Can you explain what INDEX and MATCH functions do inExcel?
● INDEX: Returns the value of a cell in a given range based on row and column numbers.
● MATCH: Searches for a value in a range and returns its relative position. When combined, INDEX and MATCH can perform advanced lookups, overcoming the limitations of VLOOKUP.
8. How do you remove duplicates in a dataset in Excel?
To remove duplicates, select the dataset, then go to the Data tab andclick on Remove Duplicates. You can choose which columns to check forduplicate values.
9. How do you use conditional formatting in Excel?
Conditional formatting allows you to format cells based on their values. Youcan apply color scales, data bars, or specific rules (e.g., highlight cellsgreater than a certain value). Navigate to Home
> Conditional Formattingto set rules.
10. What is the use of the IF function?
The IF() function performs a logical test and returns one value if thetest is true and another if it is false. Syntax:
=IF(logical_test, value_if_true, value_if_false)
11. How can you handle missing data in Excel?
To handle missing data, you can:
● Remove rows or columns with missing values.
● Use the IF function to replace missing data with a placeholder or calculated value.
● Use Go To Special to locate and fill blank cells.
12. What is a Data Validation in Excel?
Data Validation restricts the type of data that can be entered into a cell. Forexample, you can set a cell to only allow numbers, dates, or values from apredefined list. This ensures data consistency
and accuracy.
13. How do you create a drop-down list in Excel?
To create a drop-down list:
1. Select the cells where the list will appear.
2. Go to Data > Data Validation.
3. In the Allow box, choose List and enter the list values or select a range.
14. How would you use the CONCATENATE function in Excel?
The CONCATENATE() function joins two or more text strings into one. Forexample:
=CONCATENATE(A1, " ", B1) combines the contents of cells A1and B1 with a space between them. In newer Excel versions, use the TEXTJOIN()function.
15. How do you protect a worksheet in Excel?
To protect a worksheet:
1. Go to Review > Protect Sheet.
2. Set a password and select the elements users can edit.
16. What is the use of the Goal Seek function in Excel?
Goal Seek is a feature that allows you to find the input value needed toachieve a specific goal. For example, if you know the desired output of aformula but need to determine the input, Goal Seek
helps find the solution.
17. How can you filter data in Excel?
You can filter data by selecting the dataset, going to the Data tab, andclicking Filter. Drop-down arrows appear, allowing you to filter bytext, numbers, or dates.
18. How would you create a histogram in Excel?
To create a histogram:
1. Select your data.
2. Go to Insert > Insert Statistic Chart > Histogram.
19. What is the difference between COUNT, COUNTA, andCOUNTBLANK?
● COUNT(): Counts cells with numbers.
● COUNTA(): Counts non-empty cells.
● COUNTBLANK(): Counts empty cells in a range.
20. How do you calculate percentage in Excel?
To calculate a percentage, use the formula:
= (Part/Total) * 100
For example, = (A2/B2) * 100 calculates the percentage of value in A2relative to B2.
21. What is the purpose of using charts in Excel?
Charts in Excel provide a graphical representation of data, making it easier tovisualize patterns, trends, and comparisons. Common chart types include barcharts, line charts, and pie charts.
22. How do you use the PivotTable slicer feature inExcel?
Slicers are visual filters for PivotTables. To add a slicer:
1. Select your PivotTable.
2. Go to PivotTable Tools > Insert Slicer.
3. Choose the fields to create slicers for filtering the data.
23. What is the purpose of the TEXT function?
The TEXT() function converts numbers to text in a specified format. Forexample, =TEXT(A1, "0.00") will format the value in A1 withtwo decimal places.
24. How do you split data into multiple columns in Excel?
You can split data using the Text to Columns feature:
1. Select the data.
2. Go to Data > Text to Columns.
3. Choose the delimiter or fixed width to split the data into different columns.
25. What is the difference between Sort and Filter inExcel?
● Sort: Organizes data in a particular order (ascending or descending).
● Filter: Displays only the data that meets certain criteria while hiding the rest.
26. What are array formulas in Excel, and how are theyused?
Array formulas allow you to perform multiple calculations on one or more itemsin an array. These formulas return either a single result or multiple results.They are entered by pressing Ctrl Shift
Enter instead of just Enter.For example, =SUM(A1:A5 * B1:B5) would multiply each pair of correspondingvalues from columns A and B, then sum the results.
27. How do you use the IFERROR function in Excel?
The IFERROR() function is used to trap and handle errors in formulas. Itreturns a specified value if an error is found, and the result of the formulaotherwise. Syntax:
=IFERROR(value, value_if_error)
For example, =IFERROR(A1/B1, "Error") will return "Error"if B1 is 0, preventing division by zero errors.
28. Explain how the LOOKUP function works in Excel.
The LOOKUP() function searches for a value in a column or row andreturns a corresponding value from another column or row. It works for bothvertical and horizontal searches. Syntax:
=LOOKUP(lookup_value, lookup_vector, result_vector)
Unlike VLOOKUP, LOOKUP assumes the data is sorted and returns anapproximate match if an exact match isn't found.
29. What is the purpose of the CHOOSE function in Excel?
The CHOOSE() function returns a value from a list based on an indexnumber. For example, =CHOOSE(2, "Red", "Green","Blue") would return "Green" because it's the second itemin the list. It’s useful for
selecting from multiple outcomes.
30. How do you create a dynamic named range in Excel?
A dynamic named range automatically adjusts as data is added or removed. Tocreate it:
1. Go to Formulas > Name Manager.
2. Define a name and use a formula like =OFFSET(Sheet1!$A$1,0,0,COUNTA(Sheet1!$A:$A),1) to make the range dynamic based on the number of filled cells.
31. What is Power Query, and how does it help in dataanalysis?
Power Query is an Excel tool that allows for the importation, transformation,and cleaning of large datasets from various sources (e.g., databases, webpages, files). It enables users to automate data
preparation tasks and managelarge volumes of data efficiently.
32. How does the INDIRECT function work in Excel?
The INDIRECT() function returns the reference specified by a textstring. For example, =INDIRECT("A1") would refer to the value in cellA1. This function is useful when you want to dynamically
reference cellsor ranges.
33. What is data modeling in Excel?
Data modeling in Excel refers to the creation of relationships betweendifferent tables of data, allowing users to analyze and visualize combineddatasets. Power Pivot is used for this purpose,
allowing for the creation of adata model with calculated fields and measures.
34. How do you use the SUMIF and SUMIFS functions inExcel?
● SUMIF() adds cells based on a single condition. Syntax:
=SUMIF(range, criteria, [sum_range])
● SUMIFS() adds cells based on multiple conditions. Syntax:
=SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2, criteria2], ...)
35. What is the purpose of the SUBTOTAL function inExcel?
The SUBTOTAL() function performs various calculations (sum, average,count, etc.) on a filtered dataset. It can exclude hidden rows, making it idealfor summarizing data in filtered lists. Syntax:
=SUBTOTAL(function_num, ref1, [ref2], ...)
For example, =SUBTOTAL(9, A1:A10) will sum the visible rows.
36. What are the benefits of using Excel Tables for dataanalysis?
Excel Tables offer structured references, automatic range expansion, andbuilt-in filtering and sorting options. Tables allow for dynamic references informulas and make it easier to manage and analyze
data, especially as rows areadded or deleted.
37. How do you use the OFFSET function in Excel?
The OFFSET() function returns a range of cells that is a specifiednumber of rows and columns from a reference cell. It's often used with dynamicranges and is structured as:
=OFFSET(reference, rows, cols, [height], [width])
For example, =OFFSET(A1, 2, 3, 1, 1) refers to the cell 2 rows down and 3columns over from A1.
38. What is a Waterfall Chart, and how is it used inExcel?
A Waterfall Chart is used to visualize how sequential positive ornegative values contribute to a final result, making it easier to see thecumulative effect of values. It’s commonly used in financial
analysis. You caninsert a Waterfall Chart from Insert > Waterfall Chart.
39. How do you use the PMT function in Excel?
The PMT() function calculates the periodic payment for a loan based onconstant payments and a constant interest rate. Syntax:
=PMT(rate, nper, pv, [fv], [type])
For example, =PMT(5%/12, 60, 10000) calculates the monthly payment for a loanof 10,000 with an annual interest rate of 5% over 5 years.
40. How would you troubleshoot a formula in Excel?
To troubleshoot a formula:
1. Use Error Checking from the Formulas tab.
2. Utilize Evaluate Formula to step through the formula's calculations.
3. Check for common errors like incorrect cell references, missing parentheses, or circular references.
4. Use F9 to evaluate parts of the formula manually.
41. How do you create a macro in Excel?
A macro is a set of instructions to automate repetitive tasks in Excel. Tocreate one:
1. Go to View > Macros > Record Macro.
2. Perform the tasks you want to automate.
3. Stop recording. The macro can be replayed from the Macro menu.
42. What is Power Pivot, and how is it used in Excel?
Power Pivot is an advanced Excel add-in that allows users to create datamodels, establish relationships between datasets, and perform complexcalculations with large volumes of data. It enables users
to create moresophisticated reports and dashboards.
43. How do you use the NETWORKDAYS function in Excel?
The NETWORKDAYS() function calculates the number of working days(excluding weekends and specified holidays) between two dates. Syntax:
=NETWORKDAYS(start_date, end_date, [holidays])
For example, =NETWORKDAYS(A1, B1) calculates the working days between the datesin A1 and B1.
44. Explain the difference between RANK, RANK.EQ, andRANK.AVG functions in Excel.
● RANK(): Returns the rank of a number in a dataset.
● RANK.EQ(): Returns the rank of a number, giving the same rank to identical values.
● RANK.AVG(): Returns the rank of a number, averaging ranks for identical values.
45. What are Excel Add-ins, and how are they useful?
Add-ins are tools that extend Excel’s functionality. Some popular add-insinclude Analysis ToolPak for statistical analysis, Solver foroptimization problems, and third-party add-ins that enhance data
analysis,visualization, and automation.
46. How do you create a Gantt chart in Excel?
A Gantt chart can be created by using stacked bar charts in Excel. You createtasks with start dates and durations, then use a stacked bar chart to visuallyrepresent the timeline of tasks.
47. What is a Nested IF statement, and how is it used?
A Nested IF statement allows you to place multiple IF functions insideone another to evaluate several conditions. For example,
=IF(A1>90, "A", IF(A1>80, "B", "C"))
checks if A1 is greater than 90 (returns "A"), greater than 80(returns "B"), or otherwise "C".
48. How can you use Solver in Excel for optimization?
Solver is an Excel tool used for optimization problems. It helps youfind the optimal value for a formula in one cell (called the objective) subjectto constraints. Solver can be found in Data > Solver
49. What is the purpose of using Sparklines in Excel?
Sparklines are mini-charts embedded within a cell that provide a visualrepresentation of data trends. They are useful for showing trends at a glance.You can insert Sparklines from the Insert tab.
50. How do you troubleshoot circular references in Excel?
A circular reference occurs when a formula refers to itself, either directly orindirectly. To resolve this, you can:
1. Look for circular reference warnings in the Formulas tab.
2. Trace the dependency arrows to find the cell causing the issue.
3. Adjust the formula so it no longer depends on itself. | {"url":"https://www.csdt.co.in/CsdtNewBlog/Community/Answer.aspx?Uid=144","timestamp":"2024-11-12T16:39:37Z","content_type":"application/xhtml+xml","content_length":"119932","record_id":"<urn:uuid:fba713e1-833d-40de-a8eb-391a27146290>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00749.warc.gz"} |
Braess’s Paradox as a Routing Problem
Written by Luca M. Tittel
Braess’s Paradox is a phenomenon rarely witnessed in street-networks or even similar networks like power-networks or some mechanical Systems. It describes the occurrence of the overall flow of a
network increasing while the ”capacity” decreases. Or conversely, it describes the decrease in flow for an increase in ”capacity”.
Figure 1: Picture of 42nd Street, New York. Source: Wikimedia Commons
In 1990 Lucius J. Riccio, the transportation commissioner of New York City, decided to close 42nd street for the festivities of Earth Day that year. As 42nd street seemed to always be congested, the
common conception was, that this was going to be a horrible experience for the motorists in New York City. However, the traffic on Earth Day seemed to flow better than before [5]. How is this
possible? Can we recreate this in other networks? Can we adapt routing-strategies to abuse this phenomenon?
Analysing the Paradox
User vs. Socially Optimized States
To understand how this is possible, we need to understand the difference between a user optimized (UO) state and a socially optimized (SO) state. The first one being a state of a street-network in
which a change in strategy of a user, i.e., that user taking a different route, would result in them taking longer to get to the destination. An SO state is when a change in user strategy would
result in the average travel time of all users increasing. These two states not always being compatible is what allows for Braess’s Paradox to occur. There is a simple example for the UO state not
being the SO state within a 4-intersection-network.
A network is made up of intersections (vertices) and streets (edges) between them. Each edge has a latency or travel time function associated with it. This function tells us the time it takes a
driver to travel the respective street, given the number of drivers choosing to also travel on this street. These functions are of the form \(\text{latency}_{e}(f)=a\cdot f + b \), where \(e\) is the
street’s name and \(f\) is the amount of drivers taking this street. The constants \(a\) and \(b\) can be understood as indicators of capacity and length of the street, respectively. A long street
with very high capacity may have a low \(a\)-value but a high \(b\)-value, as it is not that sensitive to the amount of drivers on it, but in any case takes a lot of time to travel. Note that we can
easily define the travel time for a path \(e_0 \dots e_n\) by defining:
\(\text{latency}_{e_0\dots e_n} = \text{latency}_{e_0} +\dots + \text{latency}_{e_n}\)
The argument \(f=(f_\alpha)_{\alpha\in A}\) is the family that saves how many drivers are taking which path in the system. Here A is the set of all possible paths between all allowed start-
destination combinations. If we know this, we also know how many drivers are going to take which street in the system. So we especially know how many drivers are taking which street on the path \(e_0
\dots e_n\). That is, we know \((f_0,\dots ,f_n)\). Here it is important to take away that for computing \(\text{latency}_{e_0\dots e_n}(f)\) we need to know how many drivers are taking which path in
the system, and not just how many drivers are taking \(e_0 \dots e_n\).
Figure 2: Different Networks (Mountain Cities)
In our street network with 4 intersections, let’s say 4000 drivers are trying to get from intersection a to intersection b. Every state is defined by a strategy profile, which for each driver
determines which exact path they are going to take to reach their destination. Figure 2-left shows a version of the graph, where the SO state is the UO state. In both states all drivers divide
equally between paths a1b and a2b. The proof is straightforward:
Proof: In the first step, we demonstrate that we have a UO state. We call the number of drivers taking path a1b or a2b, \(f_1\) or \(f_2\) respectively.
Let’s say the network is in the state \(f_1=2000=f_2\). If one user changes their strategy for example from taking path a1b to taking path a2b, their travel time would increase. Their former travel
time would have been:
\(\text{latency}_{a1b}(f_1=2000,f_2=2000)=\frac{f_1}{100}+45=\frac{2000}{100} + 45=65\)
Their new travel time would be
\(\text{latency}_{a2b}(f_1=1999,f_2=2001)=\frac{f_2}{100}+45=\frac{2001}{100} + 45=65.01\)
So we have a UO state because any strategy change from the strategy profile \(f_1=f_2\) would increase that users travel time. Now we check whether we have an SO state, which we do by analysing the
average travel time. The average travel time is the total travel time of all drivers added up and divided by the number of drivers. Using \(f_1+f_2=4000\) we get:
\(\text{latency}_{\text{total}}(f_1,f_2)= \frac{\text{latency}_{a1b}(f_1)\,\cdot f_1 + \text{latency}_{a2b}(f_2)\,\cdot f_2}{4000}\)
\(\dots = \frac{f_1^2}{400000}+ \frac{f_2^2}{400000}+45\)
\(\dots = \frac{f_1^2}{2000}-2\cdot f_1+4045\)
Basic optimisation tells us, that this travel time function has a minimum for \(f_1=2000\) and \(f_2=2000\). And again we have an SO state for the strategy profile \(f_1=2000\) and \(f_2=2000\).
Notice that in this optimised state, every driver takes 65 minutes to travel from a to b.
Let’s take a look at Figure 2-right. Here we just added a very quick and large one- directional street between intersections 1 and 2 resulting in \(\text{latency}_{12}(f)=0\cdot f + 0\). Due to us
just adding capacity to the network the new SO state can only have the same or a smaller average travel time. Our SO average travel time must be smaller than 65 minutes. The UO state of this network
is that all 4000 drivers take the path a12b to their destination. This results in everybody’s travel time being:
\(\text{latency}_{a12b}(f_1=0,f_2=0,f_{12}=4000)=\frac{4000}{100} + 0 + \frac{4000}{100}=80\)
Is this true? I will outline the proof concept again for a better understanding of what a UO state is:
Proof (Outlined): What would happen if a single adventurous driver chose a different path, for example a1b. We encode this in \(f_{12}=3999\) and \(f_{1}=1\). The adventurer would have a travel time
This is to say, it is not worth it for an individual to change from taking the route a12b. A change in strategy from everybody taking a12b would be an increase in travel time for that specific
driver, hence we have a UO state. Notice our UO average travel time is 80 minutes and our SO tavel time is smaller than 65 minutes.
Simple Model for Analysing Street Networks
The difference between the UO and SO state can be explained via game theory. We can understand the process of every driver choosing the quickest path to their destination as an egotistical, rational,
uncooperative and time absent strategy game with perfect information. Such a game consists of a set of players (users or drivers), a set of strategies for each player (paths to take from their start
to their destination) and for each player a preference function. In our case this function is given by all the travel time functions that tell us the time it takes to travel a path in the current
network state. So the preference function tells us which state of the game, i.e., strategy profile, i.e., which driver taking which path, the specified player prefers to other strategy profiles [8,
p.11]. Within this model a UO state is called a Nash-Equilibrium.
The description ”egotistical, rational, uncooperative, time absent strategy game with perfect information” means that our players are looking for the quickest route for themselves and do not care
about the travel times of others. They always choose the quickest route disregarding of qualitative factors, like the view along the route. They do not cooperate before choosing their paths to, for
example, split up traffic evenly. They all choose at once and do not change their route while traveling. And they always know how long every route takes to the destination and how many players are
choosing that route.
All of these assumptions might seem random, but as is written by Osborne: ”As always, the proof of the pudding is in the eating: if a model enhances our understanding of the world, then it serves its
purpose.” [8, p.7]. And modern routing algorithms, while avoiding traffic jams for you, do look for YOUR ”best” route [6]. Google does take into account a variety of factors while planning your
quickest route, like traffic state, predicted traffic state, road conditions and route efficiency, such as authoritative data on speed limits, etc. [6]. So one might be misled in thinking their
algorithms are cooperative. It is true that they are interacting, in sending positional data or broadcasting traffic jams and so on, but they do not follow a common plan to decrease everybody’s
travel times and instead choose routes egotistically. Their method is UO and fits the description: ”egotistical, rational, uncooperative and with perfect information”. The method for the greater good
(and often also individual good), i.e., socially optimized, is a problem of multi-variable optimization [7, p.5-6].
Finding the UO State Algorithmically
Now that we have a fitting model, we can look for an algorithmic approach to exploring this phenomenon. To better understand when Braess’s Paradox occurs, I wanted an algorithm that for a given small
street network determines the UO state, or the Nash-Equilibrium. Then, I could experiment with some capacity changes and see if this would have the paradoxical effect on the average travel time I was
hoping for. Let’s take a look at the pseudo code: (implementation available in github)
Algorithm 1 Find Equilibrium
procedure EQUILIBRIUM(streetNetwork: Graph, demands: Demand[])
while network not in equilibrium do
for demand ∈ demands do
quickestRoute ← dijkstraSearch(streetNetwork, demand.start,
for route ∈ demand do
demand.redistribute(streetNetwork, route, quickestRoute)
The Graph saves the information about the street network, i.e., intersections and streets along with their travel time functions and drivers using them. A demand keeps track on how many drivers are
traveling from a specified demand.start to a specified demand.destination. It also knows how many drivers are taking which route to the destination. The “dijkstraSearch”-function finds the quickest
path from demand.start to a demand.destination given the current distribution of drivers in the street network. And the procedure “demand.redistribute” allows drivers taking “route” to their
destination to change their strategy to taking ”quickestRoute” instead. Drivers will start changing onto “quickestRoute” till “quickestRoute” becomes so slow, that change is not viable anymore. The
“quickestRoute” slows down, when drivers change onto it, because of the travel time functions having components proportionate to the amount of drivers taking a street within the route.
The idea is, that for every start-destination pair, we allow drivers not taking the quickest route, to change onto the quickest route, till it is slowed down.
Believable Real World Scenarios
We are now able to compute a UO state for a given street network with certain demands. We will look at two examples and try to understand when Braess’s Paradox is likely to occur in the real world.
Mountain Cities
For the change to the street network seen in Figure 2 the algorithm finds our exact solutions. We can see the UO states determined by the algorithm in Figure 3:
Figure 3: different UO states for different networks (Mountain Cities)
We can see that the SO state seen on the left-hand side of the figure is actually faster for everybody than the UO state on the right-hand side, which is counter intuitive, because for it to exist,
nobody is allowed to actively choose the quickest available path for themselves. If, for example, one driver would choose a12b, they would decrease their travel time to
but the average travel time would increase. But what scenario could lead to this exact situation? This graph fits the scenario of two cities on opposite sides of a Mountain depicted in Figure 3.
Imagine you want to get from point a to point b. You can either first go through city A to point 1 and then take a long road with a lot of capacity to your destination b at the edge of city B. Or you
could first take a long road with high capacity to city B and then move through the inner city to get to your destination on the other side. Looking at the algorithms results, we have learned that if
we have a large demand between a and b and we have the option to connect the cities A and B via a big tunnel between 1 and 2 while a tunnel between a and b is not feasible for some reason, we should
opt not to do it. Simply because such a tunnel would congest the cities so badly, traffic would flow worse overall.
To further understand our modeling approach we will now discuss in more detail, how we can compare our mountain cities to the 4-vertex-graph. Our first option (first city, then long road) corresponds
to taking path a1b and the second is equal to taking path a2b. The travel time function of moving through the city can be interpreted as proportionate to the amount of drivers going through the city.
This is because the inner city will congest quickly. Taking a big, long road on the other hand can be analysed as a constant travel time function, because it has high capacity, meaning it does not
congest easily but still takes a certain amount of time to travel. This notion of adding linear components for signifying congestion and adding constant components is generally accepted in simple
models. Look for example at this approach of analysing uncongested streets with congested intersections [9, p.419]. For understanding why congestion is analysed by linear components of the travel
time function, it is helpful to visualise a traffic light with congestion.
Figure 4: congested flow through traffic light. Source: github
Let’s say for each lane 2 cars can pass during every green phase and we have one lane. For the first two cars, the time to pass the green light will be the length of one red-green phase (\(l_r\)
minutes). For the next car it will already take \(2\cdot l_r\) minutes. The fourth car will also take \(2\cdot l_r\) minutes and the fifth \(3\cdot l_r\) minutes. We can roughly say that for \(f\)
cars approaching the light, it will take \(\frac{f}{2}\cdot l_r\) minutes for all cars to pass the light. The number of cars per lane being able to pass during a green phase is \(n_g=2\). And we can
denote the number of lanes at the intersection with \(k=1\). This results in the rough total travel time for this intersection being proportionate to the amount of drivers on the intersection \(\text
{latency}(f)=\frac{l_r}{k\cdot n_g}\cdot f\). An increase in capacity could be an increase of \(k\cdot n_g\) the number of lanes \(k\). Look at Figure 5 to see how the same amount of drivers move
through intersections with different lanes at different times.
Figure 5: decongesting of traffic lights
Congestion is a major part of Braess’s Paradox. It is possible to demonstrate that in uncongested networks Braess’s Paradox does not occur [7, p.7].
Bypass Outer City
When I found the following graph in [2, p.245] I was unsure of how the Paradox could occur. Starting with some first guesses, I received Figure 6-left in which 200 drivers want to go from a to c, 50
people want from a to b and 1000 drivers want to go from b to c:
The idea was to encounter Braess’s Paradox by, for example, decreasing the speed limit on the street ab. The intuition from the source was that ac would be some kind of bypass road. This means, it
would be quiet long to get to the destination. And that bc would need to have high demand. Adjusting to these factors and changing demands to ac → 200, ab → 50, bc → 3650 led to the network in Figure
Figure 6: first guess for the bypass-example
Figure 7: different networks (Bypass City)
The corresponding real world scenario would be bypassing a set of small congested streets (ab and bc) via a longer uncongested bypass (ac). Let’s say bc is some kind of main street, still fairly
small but very important inside the city. The link ab could be a street for accessing the city, also prone to congestion but only used by drivers entering from your direction. To get to the point c
on the edge of town, you can either take the combination of the access street ab and then the heavily congested main street bc, or you can take a long bypass bringing you directly to your
destination. The values of the algorithm seen in Figure 8 tell us that if we artificially decrease capacity on our access street ab, by for example decreasing the speed limit, we can actually
decrease everybody’s travel time. Everybody wanting to go from a to b now egotistically chooses the bypass. By making entering the city less appealing and slower, we avoid the new drivers further
congesting our main street bc. Average travel time will then decrease.
Figure 8: different UO states for different networks (Bypass City)
Notice that for the first guess mentioned in Figure 6, Braess’s Paradox does not occur when changing the capacity of ab. In that case every driver who wants to go from a to c already takes the
bypass. Additionally it would not be as bad for the average travel time if they further congested the main street bc, because it is not as busy. The algorithm underlines that in Figure 9:
Figure 9: same UO states for different networks (Bypass City)
Bonus: Bypass Inner City
The Pigou Phenomenon [3] illustrates a similar problem. Imagine there to be two points in a city. We can either take a very short bottleneck to our destination. This might be a negligible distance,
if there is nobody else on this road, but might turn into an hour long drive, if there are 1000 drivers on this stretch. An alternative could be taking a long bypass road. This might never congest
for a reasonable amount of drivers, but takes 60 minutes in any case because of its length. This network is shown in Figure 10-left.
Let’s say 1000 driver want to get from a to b within the street network given in Figure 10-left. We can see, that the average travel time in Figure 11 only increases if the travel time on our
bottleneck increases as seen in Figure 10-right. This is because the number of drivers on the bottleneck decreases with the decrease in capacity of it. This shows that we cannot increase the average
travel time by decreasing capacity in this scenario.
Figure 10: different networks (Bypass Bonus)
However, there is a more socially optimal distribution of drivers, than seen in the user optimized state in Figure 11-left.
Figure 11: different UO states for different networks (Bypass Bonus)
Let’s say, in the network seen in Figure 10-left, instead of just slowing down the bottleneck as in Figure 10-right, we only allow 500 drivers to pass through it. This means \(f_0=500\) drivers will
take the bypass and \(f_1=500\) the bottleneck, because of their own egotistical need to take the quickest available route for themselves as in Figure 12. The resulting average travel time will be:
\(\frac{f_0\cdot \text{latency}_{\text{bypass}}(f_0,f_1)+ f_1\cdot \text{latency}_{\text{bottle}}(f_0,f_1)}{f_0+f_1} = 45 \)
Here we can see, that by changing the capacity of our network, we may try to change the egotistical routing strategies of drivers into a more social distribution, but by truly forcing egotistical
drivers to route socially we can increase efficiency significantly. Can you define the multi-variable optimization problem of which the solution is the SO state of the network? The solution should be
the pair \((f_0,f_1)\) of how many drivers are taking the bypass or the bottleneck.
Figure 12: UO state for socially forced drivers
Looking at the naive examples explored here, but also seeing the clearly visible effect of Braess’s Paradox, one could ask if this phenomenon is the key to effectively navigating our large and
complex street networks. Some research suggests that the paradox occurs more likely in more complex graphs, and efforts are being made to better understand Braess’s Paradox in more natural settings.
In [4], a proof is outlined that in a random network model, as the number of vertices goes to infinity, there is a set of edges whose removal decreases latencies in the system. In general [1, p.1]
lists some sources in favor of Braess’s Paradoxes likelihood in real world scenarios.
One has to understand that Braess’s Paradox is simply tricking egotistical drivers into a more social behaviour. The reason we found good results in the first 4-vertex example by removing a street,
was that the user optimized state changed to the socially optimized state. During our examples, we have to keep in mind that while the user optimized state might move closer to the socially optimized
state after removing capacity from our network, we are still removing capacity. And removing capacity from our network can only have a negative or zero impact on its socially optimized state. Hence,
we are actually worsening the social optimum, to trick egotistical drivers instead of changing them. This leads to the more important question. Instead of asking: ”How can we abuse Braess’s Paradox
to trick egotistical drivers into a more social behaviour?”, we should be asking: ”How much better is socially optimised routing to user optimized routing? And how can we achieve socially optimised
routing in a large scale?”
A next step would be to look at routing as a multi-variable optimization problem as discussed in [7] and compare this to the user optimized approach of modern routing algorithms. We have the
infrastructure to allow our devices to communicate and follow common plans, so a change to social optimisation may be a very possible one in advancing quality of life, especially in cities and for
1. Viktor Avrutin Arianna Dal Forno Ugo Merlone. “Dynamics in Braess Paradox with Nonimpulsive Commuters”. In: Discrete Dynamics in Nature and Society (2014).
2. Stefano Pallottino Caroline Fisk. “Empirical Evidence for Equilibrium Paradoxes with Implications for optimal Planning Strategies”. In: Centre de Recherche sur les Transports, Universite de
Montreal (1979).
3. Étienne Ghys. “Les Applications GPS, plus lib ́erales que sociales”. In: L’Humanité Magazine (2023).
4. Tim Roughgarden Greg Valiant. “Braess’s Paradox in Large Random Graphs”. In: Random Structures Algorithms (2006).
5. Gina Kolata. “What if They Closed 42d Street and Nobody Noticed?” In: The New York Times, Dec. 25th, 1990 (1990).
6. Johann Lau. “Google Maps 101: How AI helps predict traffic and determine routes”. In: https://blog.google/ (2020).
7. Kenneth M Monks. “Braess’ Paradox in City Planning: An Application of Multi- variable Optimization”. In: MAA Convergence (2020).
8. Martin J. Osborne. An Introduction to Game Theory. Self-Published Draft, 2000.
9. M. J. Smith. “In a Road Network, Increasing Delay Locally can Reduce Delay Globally”. In: Transportation Research (1977). | {"url":"https://hegl.mathi.uni-heidelberg.de/braesss-paradox-as-a-routing-problem/","timestamp":"2024-11-05T12:09:59Z","content_type":"text/html","content_length":"122438","record_id":"<urn:uuid:f55e2386-0103-40bf-ac4a-be08f2dd372f>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00439.warc.gz"} |
Estimating and Solving for Volume of Prisms
In previous grades, you learned that the volume of an object is the number of cubic units in the interior of a three-dimensional figure.
In this lesson, you will investigate other ways to calculate the volume of rectangular prisms and triangular prisms. You will use volume formulas to solve problems involving volume of prisms.
To begin this lesson, watch the video below that compares the area of a two-dimensional figure with the volume of a related three-dimensional figure. The video also compares the volume of triangular
prisms to the volume of rectangular prisms.
Based on what you saw in the video, answer the following questions:
Solving Problems Involving the Volume of Rectangular Prisms
In the introduction, you reviewed some area formulas that you can use to calculate the area of rectangles or triangles. You also reviewed the volume formula for rectangular prisms.
In this section, you will investigate the volume formula for a rectangular prism more fully, and use the volume formula to solve problems.
Use the interactive below to explore the volume formula for rectangular prisms. Use what you see to answer the questions that follow.
Pause and Reflect
1. The number of cubes in the bottom layer is the area of the base of the prism, B. The number of layers is the height of the prism, h. Write a formula that relates the volume of the prism, V, to the
area of the base of the prism, B, and the height of the prism, h.
2. A rectangular prism has a base with dimensions 6 centimeters by 8 centimeters and a height of 4 centimeters. What is the volume of the prism?
Solving Problems Involving the Volume of Triangular Prisms
In the last section, you developed the volume formula for a prism, V = Bh, where B represents the area of the base of the prism, and h represents the height of the prism. You also used the volume
formula to solve problems involving rectangular prisms.
However, not all prisms are rectangular prisms. Consider the skyscrapers from different cities that are shown below.
┃ The Flatiron Building in New York City is a triangular prism since │ The JPMorgan Chase Tower in Houston is a pentagonal prism since │ The Tower in Fort Worth is an octagonal prism since the ┃
┃ the roof and street outline are congruent right triangles. │ the roof and street outline are congruent pentagons. │ roof and street outline are congruent octagons. ┃
In this section, you will focus on triangular prisms, which are prisms with triangular bases.
Use the interactive to create several triangular prisms. Use the dimensions in the interactive to make the calculations to complete the table below. Record the volume of the prism from the
interactive. Use the table to answer the questions that follow.
Copy the table below into your notes or a word processing document to enter the data into the table.
┃ Prism Number │ Area of Base │ Height of Prism │ Volume of Prism ┃
┃ 1 │ 6.18 │ 5 │ 30.9 ┃
┃ 2 │ │ │ ┃
┃ 3 │ │ │ ┃
┃ 4 │ │ │ ┃
1. How did you calculate the area of the base of the triangular prism?
2. When calculating the area of the triangle, how did you know which two dimensions to use?
3. In the table, how does the product of the area of the base and the height of the prism compare to the volume of the prism from the interactive?
4. Write a volume formula that can be used to calculate the volume of a triangular prism. Use B for the area of the base and h for the height of the prism.
┃Pause and Reflect ┃
┃ ┃
┃1. How is calculating the volume of a triangular prism like calculating the volume of a rectangular prism? ┃
┃ ┃
┃2. How is calculating the volume of a triangular prism different from calculating the volume of a rectangular prism?┃
For questions 1–3, each composite figure is broken into different component regions. Identify the area formula required to calculate the area of each component region.
In this lesson, you learned how to apply volume formulas for prisms in order to solve problems involving rectangular prisms and triangular prisms. As you noticed, there are also many other types of
prisms. However, you will learn more about determining the volumes of those prisms in later lessons. | {"url":"https://texasgateway.org/resource/estimating-and-solving-volume-prisms","timestamp":"2024-11-05T22:13:06Z","content_type":"text/html","content_length":"125273","record_id":"<urn:uuid:a6218a4f-e472-480c-a7f0-a7ed23c85cfd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00489.warc.gz"} |
A.6A Activities
This post explores some ideas for reviewing concepts for TEKS A.6A.
determine the domain and range of quadratic functions and represent the domain and range using inequalities
Staar Performance
On recent STAAR tests, here is how students across the state of Texas have performed.
Active, Playful Learning
• Active
• Engaging
• Meaningful
• Social
• Iterative
• Joyful
These six principles, together with a clear learning goal, help students learn.
Students learn through active, engaged, meaningful, socially interactive, iterative and joyful experiences in the classroom and out. When we add a learning goal or engage in guided play we
achieve Active Playful Learning.
In other words, math review doesn't have to be boring STAAR prep or mindless worksheets. Instead, students' learning is enhanced when playing with numeracy and algebraic concepts in a guided context.
Who says math can't be fun? You can watch a video to learn more about Active, Playful learning here.
Here's a walkthrough of all the activities on this blog post.
Talk a Mile a Minute
Learning objective: Students will name terms associated with key ideas related to the domain and range of quadratic functions.
1. Pair students together.
2. Display a term related to TEKS A.6A on the board (e.g., domain, range, x-axis, y-axis, x-value, y-value, axis of symmetry).
3. Partner A has one minute to name as many phrases related to the shown term while partner B writes them down.
4. Show another term for one minute while partner B names related terms and partner A writes them down.
5. Give partners 3 minutes to circle, from each list, one term they think will be on someone else's list and one term they think will not be on anyone else's list.
6. As a class, go over the circled terms from each list. Pairs earn one point for each term they correctly guess will or will not be on another list (4 points max).
7. Have students provide a rationale for obscure terms and let the class decide whether or not they relate to the given term.
• Provide a list of closely related terms to ban for each target word, forcing students to cast a wider net for points.
• Create a list of bonus words that earn an extra point if they appear on students' lists, regardless of whether or not they are selected as a circled term.
• Have pairs work on different words simultaneously, rotating several turns until every pair as gone through every term before the class debrief. This will forestall pairs from picking up words
from their neighbors.
• Allow pairs to select two words that will be on other's lists to reinforce common terms related to target vocabulary.
Quadratic Sliders
Learning objective: Students will use an online graphing tool to manipulate the vertex of a quadratic equation.
1. On their laptops/Chromebooks, have students open Desmos.
2. Ask students to enter the quadratic vertex form (y = a(x - h)^2 + k) and add sliders for the values a, h, and k.
3. Give students a series of clues so that their graph can match the clues by further adjusting the values h or k. For example, move the sliders so that the vertex is in quadrant III.
4. Have students change the a value as well to match the clues. For example, make the function open downward and be very wide [or narrow].
5. For each manipulation, have students share their changes and discuss commonalities as a class.
• Have students restrict the domain and/or range using curly braces (e.g., {x >= l}{y <= r}) and add l and r as sliders for further manipulation.
• Allow students to lead the activity, giving directions and/or restrictions to the class and then checking for correctness.
• Have students transfer the values in the sliders to an equation (written in vertex form) that represents the function they've graphed on Desmos.
• Show a graphed function on Desmos and hide the sliders. Allow students to work together to approximate the values of a, h, k, l, and r based on the graph before revealing them.
Axis of Symmetry Match
Learning objective: Students will find the axis of symmetry of a quadratic function given the standard form.
Note: Students can find the vertex of a quadratic function when they identify the axis of symmetry given standard form and substitute the x-value into the equation to find the y-value.
1. Pair students up and give them the axis of symmetry match cards.
2. Without calculator, pencil, or paper, have students match the standard forms with their corresponding axes of symmetry.
3. Allow pairs to check their work with another pair.
• Allow students to use pencil/paper or calculator to check their work.
• After matching all the cards, ask students to take two pairs and find the vertex for each of them by substituting the x-value into the equation to find the y-value. When this is complete,
students should note the sign for a and write the range.
• Allow students to use the cards individually or with pairs as a memory match game.
• Ask students to graph the functions on a graphing calculator to check to see if they correctly identified the axis of symmetry.
Race to the Inequality
Learning objective: Students work collaboratively to describe the range of a quadratic equation with inequalities.
1. Ask for (or designate) 13 volunteers. The first 9 volunteers will each represent the digits 1 - 9 (either printed up or written on white boards). The remaining four volunteers will represent y, a
negative sign (i.e., - ), and the inequality symbols ≥ and ≤.
2. Split the rest of the class into two teams.
3. Display the graph of a quadratic equation on the board or interactive white board. The 13 volunteers should each be facing the class with their backs to the board.
4. The first team's task is to represent the range. As quickly as they can, the team directs the 13 volunteers (not all will be used) to arrange themselves to represent the range.
5. Reset the volunteers. The second team works on a different quadratic equation, trying to beat the time of the first team.
• Instead of displaying the graph of a quadratic equation, display the equation in vertex form [y = a(x - h)^2 + k].
• For alternate scoring, give teams points based on whether or not they solve within stepped time frames. For example, 5 points for solving in 20 seconds or less, 4 points for solving in 30 seconds
or less, etc.
• For an extra challenge, have a team arrange the volunteers to represent both the domain and range in succession under a predetermined time limit. Be sure to add volunteers that represent positive
and negative infinity.
• Use range limitations that require two digits (e.g., 21). Add an extra volunteer to represent zero and ensure that a digit is not used more than once.
Table Range
Learning objective: Students will identify the range of a quadratic function given a table of values.
Materials: Give each team two half-sheets of paper, one printed with y ≥ and the other printed with y ≤.
1. Split class into teams of four.
2. Display a table of values for a quadratic function (e.g., 2023 STAAR #32)
3. Give teams a small time limit (e.g., 20 seconds) to converse with each other and identify the range.
4. When time is up, each team sends a representative up to the board/screen and places either y ≥ or y ≤ on the dependent variable value that represents the limit of the range.
5. Ask students how they knew that the value they chose was the limit of the range.
• Replace the half-sheets with y ≥ and y ≤ with half-sheets that read all real numbers greater than or equal to and all real numbers less than or equal to.
• Have a coordinate grid on the board. After identifying the range, ask another team to come up, plot the points, and sketch a graph.
• Use a mixture of decimals, mixed numbers, and improper fractions for the dependent variable values in the tables.
• Change up the directionality of the tables. Have some read horizontally and some read vertically. | {"url":"https://www.fiveminutemath.net/post/a-6a-activities","timestamp":"2024-11-01T20:38:03Z","content_type":"text/html","content_length":"1050049","record_id":"<urn:uuid:13e48fac-cb83-4730-8e0d-b388d999e7bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00315.warc.gz"} |
Craig Hane, Ph.D., aka Dr. Del
Dr. Del has taught mathematics at virtually all levels from the very basic to the most advanced math required by engineers, scientists and theoretical mathematicians.
While in high school, Dr. Del studied at DePauw University and tutored fellow students in math. He also taught trig identities to his own trig class.
Dr. Del earned a B.A. from Oberlin College in Math and English. During that time, he was a math problem session instructor.
After college, Dr. Del taught high school math at Western Reserve High School in Ohio and taught math at DePauw University while Math Department Chairman, Dr. Clint Gass, was on sabbatical.
Dr. Del returned to school and earned a Ph.D. in Algebraic Number Theory from Indiana University. While attending, he served as a Teaching Associate teaching Calculus and Logic classes.
After earning his Ph.D., Dr. Del taught all of their Advanced Theoretical Math at Indiana State University. He then went on to teach Theory plus Calculus and Differential Equations at Rose-Hulman
Institute of Technology.
Dr. Del then founded Hane Training and taught technical subjects to adults in many large companies such as Ford, Caterpillar, General Motors, and State Farm, just to name a few.
Most recently, Dr. Del is the subject matter expert and founder of Triad Math and the STEM Math Academy.
You may contact Dr. Del at craig@hane.com | {"url":"https://stemmathmadeeasy.com/team_member-member-name/craig-hane-ph-d-aka-dr-del-founder-and-subject-matter-expert/","timestamp":"2024-11-12T21:53:01Z","content_type":"text/html","content_length":"53528","record_id":"<urn:uuid:d3ec7769-7218-43b3-8bbe-6b85a3029ab4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00857.warc.gz"} |
TDCOSMO IV: Hierarchical time-delay cosmography - joint inference of the Hubble constant and galaxy density profiles
The H0LiCOW collaboration inferred via strong gravitational lensing time delays a Hubble constant value of H[0] = 73.3[−1.8]^+1.7 km s^−1 Mpc^−1, describing deflector mass density profiles by either
a power-law or stars (constant mass-to-light ratio) plus standard dark matter halos. The mass-sheet transform (MST) that leaves the lensing observables unchanged is considered the dominant source of
residual uncertainty in H[0]. We quantify any potential effect of the MST with a flexible family of mass models, which directly encodes it, and they are hence maximally degenerate with H[0]. Our
calculation is based on a new hierarchical Bayesian approach in which the MST is only constrained by stellar kinematics. The approach is validated on mock lenses, which are generated from
hydrodynamic simulations. We first applied the inference to the TDCOSMO sample of seven lenses, six of which are from H0LiCOW, and measured H[0] = 74.5[−6.1]^+5.6 km s^−1 Mpc^−1. Secondly, in order
to further constrain the deflector mass density profiles, we added imaging and spectroscopy for a set of 33 strong gravitational lenses from the Sloan Lens ACS (SLACS) sample. For nine of the 33 SLAC
lenses, we used resolved kinematics to constrain the stellar anisotropy. From the joint hierarchical analysis of the TDCOSMO+SLACS sample, we measured H[0] = 67.4[−3.2]^+4.1 km s^−1 Mpc^−1. This
measurement assumes that the TDCOSMO and SLACS galaxies are drawn from the same parent population. The blind H0LiCOW, TDCOSMO-only and TDCOSMO+SLACS analyses are in mutual statistical agreement. The
TDCOSMO+SLACS analysis prefers marginally shallower mass profiles than H0LiCOW or TDCOSMO-only. Without relying on the form of the mass density profile used by H0LiCOW, we achieve a ∼5% measurement
of H[0]. While our new hierarchical analysis does not statistically invalidate the mass profile assumptions by H0LiCOW – and thus the H[0] measurement relying on them – it demonstrates the importance
of understanding the mass density profile of elliptical galaxies. The uncertainties on H[0] derived in this paper can be reduced by physical or observational priors on the form of the mass profile,
or by additional data.
Dive into the research topics of 'TDCOSMO IV: Hierarchical time-delay cosmography - joint inference of the Hubble constant and galaxy density profiles'. Together they form a unique fingerprint. | {"url":"https://researchportal.port.ac.uk/en/publications/tdcosmo-iv-hierarchical-time-delay-cosmography-joint-inference-of","timestamp":"2024-11-06T04:36:56Z","content_type":"text/html","content_length":"68742","record_id":"<urn:uuid:39dbe50b-128a-4c05-a9bc-64343c3d0fff>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00152.warc.gz"} |
Paper Columns
Renee Allen Louis Wirth Experimental
4959 S Blackstone
Chicago IL 60615
(Eighth Grade Level)
1. Students will learn that the principles of mathematics and physics
are used in building.
2. Students will learn how to change:
(A) ounces to pounds
(B) ounces to grams
3. Students will learn the importance of columns when used in con-
books - 40 or more (paperbacks) Same dimensions
graph paper
pencils (#2)
plain paper (21cm X 28cm)
1. Divide students in groups of three or four.
2. Place two sheets of plain paper (no holes), a stack of books, and a
meter stick or a ruler with metric units with each group.
3. Each group should construct two columns not shorter than ll cm. These
columns should be constructed to support the paperback books.
4. When students are ready to test their columns, teacher or assistant
will watch as students place one book at a time on columns. (Example..
..If columns fall with the 5th book, then credit is given for columns
supporting four books.)
5. Each group should know the MASS of one book in grams. Then each student
should compute the MASS of books supported by the columns. It also might
be interesting to convert the total MASS to ounces.(28.35g.=1 oz.)
6 Put each group's support MASS on the board, and have students make
a line graph of all the support MASS.
Note: Each paper column should be made with one sheet of plain paper.
and no tape, pins, or etc. should be used.
Performance Assessment:
Students should be able to write out a conclusion that the shorter or
reinforced (folded paper..double thickness) paper columns will support
more MASS.
Art- drawing columns.................
LA- Writing papers on columns..............
Social Studies/Multi Cultural - Studying history of columns in different
Return to SMILE Plus Index | {"url":"https://smileprogram.info/pl9601.html","timestamp":"2024-11-11T17:20:35Z","content_type":"text/html","content_length":"2666","record_id":"<urn:uuid:52d3485d-59e3-49a3-bc1d-14140e2f660a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00084.warc.gz"} |
SSC General Math Question Solution Mymensingh Board 2023
SSC General Math Question Solution Mymensingh Board 2024 All Set | Mymensingh Board SSC General Math Solution 2024
SSC General Math Question Solution Mymensingh Board has already been published on our website. This year, 1.1 lakh candidates participated in the SSC examination. Today, 9th May, they have completed
SSC general math exam under 11 education boards of Bangladesh.
The SSC general math question paper is not the same for all 11 education boards. The mark distribution of all education boards is the same.
The variation of the question pattern demands solving the question. In this post, we have focused on the SSC general math question solution for the candidates of the Mymensingh Board.
Stay with us and read the post to solve SSC general math question 2024.
SSC General Math Question Solution Mymensingh Board 2024
Are you looking for the SSC General math question solution of Mymensingh Board 2024? Mymensingh Board has made the question of SSC general math for the candidates. The SSC general math question paper
is not so hard to solve all the questions.
If you are a candidate for the Mymensingh board, look at the below section. We have solved all the questions there.
SSC General Math Question Answer Mymensingh Board 2024
In this section, we will focus on the SSC general Math question answer of the Mymensingh board. We have got the question paper and have tried to solve the SSC general math question. We already have
solved the full question paper correctly. Look at the below section to find out it.
SSC General Math Exam Details Mymensingh Board 2024
SSC general math exam detail is an essential issue for all candidates to increase their knowledge. So we have included it in this post.
Name of the Exam SSC (Secondary School Certificate)
Name of the Subject SSC General Math
The Creative Question Answer 70 marks
The MCQ Test 30 marks
Total Mark 100
Pass Mark 33
SSC General Math Question Solution Mymensingh Board
SSC General Math Exam Mark Distribution 2024
SSC general math mark distribution 2024 is given below. The mark distribution is the same for all education boards.
• বীজগণিত-20 marks
• ত্রিকোণমিতি-10 marks
• পরিমিতি-10 marks
• পরিসংখ্যান-10 marks
• জ্যামিতি-20 marks
MCQ Test-30 marks
• Every multiple questions bear 1 mark.
SSC General Math Question Solution Mymensingh Board 2024 Creative Question
In this section, we will discuss SSC general math creative question section of the Mymensingh board. This section consists of 4 divisions like Kha division, kh division, ga division, and so on. It is
mandatory to answer from all divisions. The creative section bears 70 marks.
Look at the below image to check the answer to the creative question section. If you want to know the SSC general math MCQ question solution, go to the next section.
SSC General Math Question Answer Mymensingh Board 2024 MCQ Test
SSC general math question answer of the MCQ section of the Mymensingh board is given below section. If you want to solve it, go there.
Also Read,
SSC General Math MCQ Test Question Solution Mymensingh Board 2024
The SSC general math question solution Mymensingh board MCQ section will discuss in this part. If you want to know the solution to the MCQ question of the Mymensingh board, look at the below image.
In the image, we have solved all the questions for the candidates for the Mymensingh board.
We are at the end of our post. In this post, we have discussed the SSC General Math Question Solution Mymensingh Board 2024. If you want to see other question solutions, drop a line in the comment
box below.
Q. How to check SSC General Math Question Solution Mymensingh Board 2024?
Ans. If you are a candidate for the Mymensingh Board you can visit our website because on our website we have made a post on SSC General Math Question Solution Mymensingh Board 2024.
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Mastering Matrix Calculations – A Comprehensive Guide to Using a Matrix Cofactor Calculator
When it comes to matrix calculations, understanding the various aspects and techniques involved is essential. One important concept within matrix calculations is the computation of matrix cofactors.
In this blog post, we will delve into the world of matrix cofactor calculations, discussing their significance and applications. We will also explore how a matrix cofactor calculator can streamline
the process, making it easier and more efficient.
Understanding Matrix Cofactors
Definition and Role
Matrix cofactors are integral components in matrix calculations. They are derived from subsets of a given matrix and play a crucial role in determining properties such as the inverse and determinant
of the matrix. Essentially, matrix cofactors offer a simplified and systematic approach to solving complex matrix problems.
Properties and Characteristics
Matrix cofactors possess several noteworthy properties and characteristics. They are generally derived from minors, which are determinants of smaller matrices obtained by excluding the row and column
of a selected element in the original matrix. This leads to an interesting pattern of alternating signs in the cofactor matrix.
Calculating the Inverse
One of the primary applications of matrix cofactors is in determining the inverse of a matrix. The cofactor matrix, along with the determinant of the original matrix, can be used to find the inverse.
This is particularly useful in solving systems of linear equations and performing other matrix operations.
Using a Matrix Cofactor Calculator
Introduction to Matrix Cofactor Calculators
A matrix cofactor calculator simplifies the process of computing matrix cofactors, making it accessible to individuals who may not have a strong mathematical background. These digital tools are
designed to provide accurate and efficient results.
Step-by-Step Guide
To utilize a matrix cofactor calculator effectively, follow these steps:
1. Inputting Matrix Values: Enter the values of the matrix into the calculator, ensuring they are entered correctly to obtain accurate results.
2. Performing the Cofactor Calculation: Initiate the cofactor calculation function on the calculator, which will process the entered values and generate the cofactor matrix.
3. Obtaining the Result: Once the calculation is complete, the matrix cofactors will be displayed as a matrix on the calculator screen, which can be used for further computations or analysis.
Tips and Tricks
Maximize your efficiency when using a matrix cofactor calculator with the following tips:
• Double-check Inputs: Avoid errors by verifying the accuracy of the inputted matrix values before initiating the calculation.
• Understand Calculator Functions: Familiarize yourself with the various features and functions of the matrix cofactor calculator. Explore additional capabilities it may have, such as determinant
and inverse calculations.
• Practice Regularly: Consistent practice will enhance your proficiency in using the matrix cofactor calculator and increase your overall understanding of matrix operations.
Applications of Matrix Cofactor Calculations
Solving Systems of Linear Equations
Matrix cofactors can be utilized to solve systems of linear equations. By representing the system of equations as a matrix and using matrix cofactor calculations, the coefficients and variables can
be efficiently solved for, offering a simpler and organized solution method.
Finding the Determinant
The determinant of a matrix can be determined using matrix cofactor calculations. The determinant is a scalar value that describes important properties of the matrix, such as whether it is invertible
or singular.
Calculating Area and Volume
Matrix cofactors can also be applied to finding the area and volume of various geometric shapes. By creating a matrix with specific elements, the cofactor calculations can yield accurate measurements
of these properties, providing a mathematical approach to solving real-world problems.
Common Mistakes and Troubleshooting
Identifying Common Mistakes
While using a matrix cofactor calculator, it is crucial to be aware of common mistakes that can occur:
• Incorrect Matrix Inputs: Typing errors or incorrect arrangement of values can lead to inaccurate results.
• Misinterpreting the Output: Misunderstanding the meaning or interpretation of the obtained matrix cofactors can contribute to further errors in subsequent calculations.
• Confusion in Calculator Functions: Lack of familiarity with the functionalities of the matrix cofactor calculator can result in incorrect usage and calculations.
Solutions and Tips
To troubleshoot errors and optimize your use of a matrix cofactor calculator:
• Verify Input Accuracy: Double-check all inputs to ensure they are precise and correspond to the matrix being analyzed.
• Refer to User Manuals or Guides: Consult the user manual or online guides provided with the calculator to gain a better understanding of its features and functionalities.
• Seek Online Resources: Access additional resources such as video tutorials or forums where you can find solutions to specific calculator-related problems.
Recapping Importance and Encouragement
Mastering matrix calculations and utilizing a matrix cofactor calculator can greatly enhance your understanding of mathematics and its applications. The concepts explored in this blog post provide a
solid foundation for performing matrix cofactor calculations effectively to solve complex problems.
Continuing the Journey
As you delve deeper into the world of matrix calculations, further resources such as textbooks, online courses, and practice problems can facilitate your learning journey. Continually practice your
skills to strengthen your proficiency and explore new areas of application. | {"url":"https://skillapp.co/blog/mastering-matrix-calculations-a-comprehensive-guide-to-using-a-matrix-cofactor-calculator/","timestamp":"2024-11-09T19:45:12Z","content_type":"text/html","content_length":"111286","record_id":"<urn:uuid:7084ae45-a589-48ca-9cf7-d38922a13649>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00822.warc.gz"} |
Proximal Distance Algorithms: Theory and Practice
Kevin L. Keys, Hua Zhou, Kenneth Lange.
Year: 2019, Volume: 20, Issue: 66, Pages: 1−38
Proximal distance algorithms combine the classical penalty method of constrained minimization with distance majorization. If $f(x)$ is the loss function, and $C$ is the constraint set in a
constrained minimization problem, then the proximal distance principle mandates minimizing the penalized loss $f(x)+\frac{\rho}{2}dist(x,C)^2$ and following the solution $x_{\rho}$ to its limit as $\
rho$ tends to $\infty$. At each iteration the squared Euclidean distance $dist(x,C)^2$ is majorized by the spherical quadratic $\|x-P_C(x_k)\|^2$, where $P_C(x_k)$ denotes the projection of the
current iterate $x_k$ onto $C$. The minimum of the surrogate function $f(x)+\frac{\rho}{2}\|x-P_C(x_k)\|^2$ is given by the proximal map $prox_{\rho^{-1}f}[P_C(x_k)]$. The next iterate $x_{k+1}$
automatically decreases the original penalized loss for fixed $\rho$. Since many explicit projections and proximal maps are known, it is straightforward to derive and implement novel optimization
algorithms in this setting. These algorithms can take hundreds if not thousands of iterations to converge, but the simple nature of each iteration makes proximal distance algorithms competitive with
traditional algorithms. For convex problems, proximal distance algorithms reduce to proximal gradient algorithms and therefore enjoy well understood convergence properties. For nonconvex problems,
one can attack convergence by invoking Zangwill's theorem. Our numerical examples demonstrate the utility of proximal distance algorithms in various high-dimensional settings, including a) linear
programming, b) constrained least squares, c) projection to the closest kinship matrix, d) projection onto a second-order cone constraint, e) calculation of Horn's copositive matrix index, f) linear
complementarity programming, and g) sparse principal components analysis. The proximal distance algorithm in each case is competitive or superior in speed to traditional methods such as the interior
point method and the alternating direction method of multipliers (ADMM). Source code for the numerical examples can be found at https://github.com/klkeys/proxdist.
PDF BibTeX | {"url":"https://jmlr.org/beta/papers/v20/17-687.html","timestamp":"2024-11-06T12:06:12Z","content_type":"text/html","content_length":"8562","record_id":"<urn:uuid:9ec62494-e93c-4d5b-82f2-dd735ddf3d29>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00831.warc.gz"} |
Identification: What is it?
Iavor Bojinov
Identification and identifiable are two extremely commonly used words in Statistics and Economics. But, have you ever wondered, what does it really mean to say that a quantity is identifiable from
the data? Statisticians seem to agree on a definition in the context of parametric models --- saying a parameter is identifiable if each parameter corresponds to a distinct parametric model --- but
beyond that things get a little tricky. Unfortunately, this is a place where looking at the existing academic literature leaves us with more questions than answers: Are parameters the only quantities
that can be identified? Is the concept of identification meaningful outside of parametric statistics? Does it even require the notion of a statistical model? Many authors have tried to answer these
questions, but have tended to provide either partial or idiosyncratic answers in specific contexts.
In a recent paper, Guillaume Basse and I propose a unifying theory of identification that incorporates existing definitions for parametric and nonparametric models and formalizes the process of
identification analysis. In this blog post, I am going to provide a brief and (hopefully) easy to understand description of our general theory. I will leave the identification analysis for another
blog post, but if you are interested then just read the paper!
We make two main contributions:
1. We provide a single general and mathematically rigorous definition of identifiability. Abstracting away the specifics of each domain allows us to recognize the commonalities and make the concepts
more transparent as well as easier to extend to new settings.
2. We use our definition to develop a set of results and a systematic approach for determining whether a quantity is identifiable and, if not, what is its identification region (i.e., the set of
values of the quantity that are coherent with the data and assumptions).
Statistical inference teaches us ``how'' to learn from data, whereas identification analysis explains ``what'' we can learn from it. Although ``what'' logically precedes ``how,'' the concept of
identification has received relatively less attention in the statistics community.
A general theory of identification
Set up
Our framework consists of three elements:
• A statistical universe that contains all of the objects relevant to a given problem (S).
• An estimand mapping that describes what aspect of the statistical universe we are trying to learn about (𝛉: S↦𝚹)
• An observation mapping that tells us what parts of our statistical universe we observe (𝝀: S↦𝚲)
We then define identification by studying the inherent relationship between the estimand mapping and the observation mapping using the induced binary relation R. Intuitively, the induced binary
relation connects ``what we know'' to ``what we are trying to learn'' through the ``statistical universe'' in which we operate.
Example: In parametric models where P(ϑ) is a distribution indexed by ϑ∈𝚹,
• The statistical universe is S = {(P(ϑ),ϑ) ϑ∈𝚹}
• The estimand mapping is 𝛉(S) = ϑ
• The observation mapping is 𝝀(S) = P(ϑ).
The induced binary relation then maps R:ϑ ↦P(ϑ).
Definition of Identification
We say that 𝛉 is identifiable from 𝝀 if the induced binary relation R is injective. That is, if there is a 1-1 relationship between what we are trying to estimate and what we observe, then the
estimand mapping is identifiable from the observation mapping. Of course, there is a bunch of maths to make this definition exact, but this is the essence of it.
Our main results show that our general formulation allows us to work directly with both parametric and nonparametric models, without having to introduce separate definitions. What's more, our setup
allows us to cleanly define identification in finite population settings without having to revert to using a nonparametric sampling argument. | {"url":"https://www.ibojinov.com/post/identification-what-is-it","timestamp":"2024-11-03T01:08:15Z","content_type":"text/html","content_length":"1009996","record_id":"<urn:uuid:187b0790-4fbd-4ad4-b603-063741b90c08>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00510.warc.gz"} |
How do you set a title in Pyplot?
How do you set a title in Pyplot?
Matplotlib Labels and Title
1. Add labels to the x- and y-axis: import numpy as np. import matplotlib.pyplot as plt.
2. Add a plot title and labels for the x- and y-axis: import numpy as np.
3. Set font properties for the title and labels: import numpy as np.
4. Position the title to the left: import numpy as np.
How do you change the title of a subplot?
Add Title to Subplots in Matplotlib
1. Set_title() Method to Add Title to Subplot in Matplotlib.
2. title.set_text() Method to Set Title of Subplots in Matplotlib.
3. plt.gca().set_title() / plt.gca.title.set_text() to Set Title to Subplots in Matplotlib.
How do I add a title to my Seaborn plot?
How to Add a Title to Seaborn Plots?
1. set method takes 1 argument “title” as a parameter which stores Title of a plot.
2. suptitle method takes a string which is title of plot as parameter.
3. set_title method takes a string as parameter which is title of plot.
What is the use of Pyplot?
pyplot is a collection of functions that make matplotlib work like MATLAB. Each pyplot function makes some change to a figure: e.g., creates a figure, creates a plotting area in a figure, plots some
lines in a plotting area, decorates the plot with labels, etc.
What is a Suptitle?
suptitle() method. The suptitle() method figure module of matplotlib library is used to Add a centered title to the figure.
What does the function subplot return?
Matplotlib – Subplots() Function The function returns a figure object and a tuple containing axes objects equal to nrows*ncols. Each axes object is accessible by its index.
Is it possible to create a Coloured scatterplot using matplotlib?
Combining two scatter plots with different colors To change the color of a scatter point in matplotlib, there is the option “c” in the function scatter.
What is SNS set?
Seaborn is a Python data visualization library based on matplotlib (it is the go to library for plotting in Python). Seaborn provides a high-level interface for drawing attractive and informative
statistical graphics.
What should I title graph?
Titling the Graph The proper form for a graph title is “y-axis variable vs. x-axis variable.” For example, if you were comparing the the amount of fertilizer to how much a plant grew, the amount of
fertilizer would be the independent, or x-axis variable and the growth would be the dependent, or y-axis variable.
Which is used for main title in graph?
Explanation: Main is used for the main title in the graphics. | {"url":"https://www.curvesandchaos.com/how-do-you-set-a-title-in-pyplot/","timestamp":"2024-11-01T22:32:04Z","content_type":"text/html","content_length":"47908","record_id":"<urn:uuid:0ea9ccf6-ca9e-449b-a57c-ef33640e4733>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00880.warc.gz"} |
parametric Tests
Nonparametric Tests
Several nonparametric tests are available. As usual, this section mentions only a few possibilities. Currently, these refer to an outcome variable that indicates ranks (or that can, and should, be
ranked, such as a non-normal metric variable), and a grouping variable. Note that if your data do not represent ranks, Stata will do the ranking for you.
Two groups
For two groups, you might use Wilcoxon's rank sum test (which is equivalent to Mann and Whitney's U test), or the median test. The commands for these tests go like this:
ranksum income, by(sex)
median income, by(sex)
For the median test, option exact is available that computes an exact test according to Fisher.
More than two groups
In this case, you might use the Kruskal-Wallis test. If the groups represent some kind or order, a test for trend across groups is available.
kwallis income, by(firmsize)
nptrend income, by(firmsize)
© W. Ludwig-Mayerhofer, Stata Guide | Last update: 21 Dec 2010 | {"url":"https://wlm.userweb.mwn.de/Stata/wstatnpa.htm","timestamp":"2024-11-15T00:19:35Z","content_type":"text/html","content_length":"7805","record_id":"<urn:uuid:223859a6-b303-4146-9b7a-829a59fa1f2c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00434.warc.gz"} |
Weekly Average Graphed by Month
I am looking to graph sales data of the weekly average of items sold per month, not sure of the best way to accomplish this. The current dataset that I am working with is essential a transaction
detail dataset coming out of NetSuite Analytics Connect, my query references several different tables but I have joined them all together.
I have attached an image of the graph that was generated from an excel sheet that we are looking to recreate.
Best Answers
• Keep in mind this is not a fully fleshed out solution, but does work if you are just looking at general trends. Depends how exact you want/need to be in your weekly average. But one possible
solution is to graph by month, and then have a beast mode like this.
This is likely better than just dividing by 4.
You could also adjust this as needed to dial in how you want to approach weekly average.
Then just set it up in a stacked bar chart with your group as the series and weekly_average as the y-axis. It's important to note that this approach isn't super flexible. You'll need to always
graph by month, or adjust your beast mode. If you wanted a more robust option, that is possible, but would use a date dimension table and ETL.
David Cunningham
** Was this post helpful? Click Agree π , Like π οΈ , or Awesome β €οΈ below **
** Did this solve your problem? Accept it as a solution! β οΈ **
• Here's what I came up with on how to solve this:
If I solved your problem, please select "yes" above
• Keep in mind this is not a fully fleshed out solution, but does work if you are just looking at general trends. Depends how exact you want/need to be in your weekly average. But one possible
solution is to graph by month, and then have a beast mode like this.
This is likely better than just dividing by 4.
You could also adjust this as needed to dial in how you want to approach weekly average.
Then just set it up in a stacked bar chart with your group as the series and weekly_average as the y-axis. It's important to note that this approach isn't super flexible. You'll need to always
graph by month, or adjust your beast mode. If you wanted a more robust option, that is possible, but would use a date dimension table and ETL.
David Cunningham
** Was this post helpful? Click Agree π , Like π οΈ , or Awesome β €οΈ below **
** Did this solve your problem? Accept it as a solution! β οΈ **
• I don't have a solution at the moment, but I think the solution will depend on how your data is structured, if there are records for every date, and how you want to handle when a week straddles
two different months.
• Here's what I came up with on how to solve this:
If I solved your problem, please select "yes" above
• @ColemenWilson Thanks for your help on this! This is exactly what I needed.
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hwloc_topology_get_topology_cpuset: CPU and node sets of entire topologies - Linux Manuals (3)
hwloc_topology_get_topology_cpuset (3) - Linux Manuals
hwloc_topology_get_topology_cpuset: CPU and node sets of entire topologies
hwlocality_helper_topology_sets - CPU and node sets of entire topologies
static hwloc_const_cpuset_t hwloc_topology_get_complete_cpuset (hwloc_topology_t topology)
static hwloc_const_cpuset_t hwloc_topology_get_topology_cpuset (hwloc_topology_t topology)
static hwloc_const_cpuset_t hwloc_topology_get_online_cpuset (hwloc_topology_t topology)
static hwloc_const_cpuset_t hwloc_topology_get_allowed_cpuset (hwloc_topology_t topology)
static hwloc_const_nodeset_t hwloc_topology_get_complete_nodeset (hwloc_topology_t topology)
static hwloc_const_nodeset_t hwloc_topology_get_topology_nodeset (hwloc_topology_t topology)
static hwloc_const_nodeset_t hwloc_topology_get_allowed_nodeset (hwloc_topology_t topology)
Detailed Description
Function Documentation
static hwloc_const_cpuset_t hwloc_topology_get_allowed_cpuset (hwloc_topology_t topology) [inline], [static]
Get allowed CPU set.
the CPU set of allowed logical processors of the system. If the topology is the result of a combination of several systems, NULL is returned.
The returned cpuset is not newly allocated and should thus not be changed or freed, hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_nodeset_t hwloc_topology_get_allowed_nodeset (hwloc_topology_t topology) [inline], [static]
Get allowed node set.
the node set of allowed memory of the system. If the topology is the result of a combination of several systems, NULL is returned.
The returned nodeset is not newly allocated and should thus not be changed or freed, hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_cpuset_t hwloc_topology_get_complete_cpuset (hwloc_topology_t topology) [inline], [static]
Get complete CPU set.
the complete CPU set of logical processors of the system. If the topology is the result of a combination of several systems, NULL is returned.
The returned cpuset is not newly allocated and should thus not be changed or freed; hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_nodeset_t hwloc_topology_get_complete_nodeset (hwloc_topology_t topology) [inline], [static]
Get complete node set.
the complete node set of memory of the system. If the topology is the result of a combination of several systems, NULL is returned.
The returned nodeset is not newly allocated and should thus not be changed or freed; hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_cpuset_t hwloc_topology_get_online_cpuset (hwloc_topology_t topology) [inline], [static]
Get online CPU set.
the CPU set of online logical processors of the system. If the topology is the result of a combination of several systems, NULL is returned.
The returned cpuset is not newly allocated and should thus not be changed or freed; hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_cpuset_t hwloc_topology_get_topology_cpuset (hwloc_topology_t topology) [inline], [static]
Get topology CPU set.
the CPU set of logical processors of the system for which hwloc provides topology information. This is equivalent to the cpuset of the system object. If the topology is the result of a
combination of several systems, NULL is returned.
The returned cpuset is not newly allocated and should thus not be changed or freed; hwloc_bitmap_dup() must be used to obtain a local copy.
static hwloc_const_nodeset_t hwloc_topology_get_topology_nodeset (hwloc_topology_t topology) [inline], [static]
Get topology node set.
the node set of memory of the system for which hwloc provides topology information. This is equivalent to the nodeset of the system object. If the topology is the result of a combination of
several systems, NULL is returned.
The returned nodeset is not newly allocated and should thus not be changed or freed; hwloc_bitmap_dup() must be used to obtain a local copy.
Generated automatically by Doxygen for Hardware Locality (hwloc) from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-hwloc_topology_get_topology_cpuset/","timestamp":"2024-11-04T14:22:21Z","content_type":"text/html","content_length":"14221","record_id":"<urn:uuid:c2eeffe8-dcd1-48da-8ad3-52c341f737f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00763.warc.gz"} |
Honors Calculus by Frank Jones
Honors Calculus
by Frank Jones
Publisher: Rice University 2004
The goal is to achieve a thorough understanding of vector calculus, including both problem solving and theoretical aspects. The orientation of the course is toward the problem aspects, though we go
into great depth concerning the theory behind the computational skills that are developed.
Download or read it online for free here:
Download link
(multiple PDF files)
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d02gbc (bvp_fd_lin_gen)
NAG CL Interface
d02gbc (bvp_fd_lin_gen)
1 Purpose
d02gbc solves a general linear two-point boundary value problem for a system of ordinary differential equations using a deferred correction technique.
2 Specification
d02gbc (Integer neq,
void (*fcnf)(Integer neq, double x, double f[], Nag_User *comm),
void (*fcng)(Integer neq, double x, double g[], Nag_User *comm),
double a, double b, double c[], double d[], double gam[], Integer mnp, Integer *np, double x[], double y[], double tol, Nag_User *comm, NagError *fail)
The function may be called by the names: d02gbc or nag_ode_bvp_fd_lin_gen.
3 Description
solves the linear two-point boundary value problem for a system of
ordinary differential equations in the interval
. The system is written in the form
$y ′ = F (x) y + g (x)$ (1)
and the boundary conditions are written in the form
$Cy (a) + Dy (b) = γ$ (2)
$F (x)$
matrices, and
$g (x)$
component vectors. The approximate solution to
is found using a finite difference method with deferred correction. The algorithm is a specialisation of that used in the function
which solves a nonlinear version of
. The nonlinear version of the algorithm is described fully in
Pereyra (1979)
You need to supply an absolute error tolerance and may also supply an initial mesh for the construction of the finite difference equations (alternatively a default mesh is used). The algorithm
constructs a solution on a mesh defined by adding points to the initial mesh. This solution is chosen so that the error is everywhere less than your tolerance and so that the error is approximately
equidistributed on the final mesh. The solution is returned on this final mesh.
If the solution is required at a few specific points then these should be included in the initial mesh. If, on the other hand, the solution is required at several specific points, then you should use
the interpolation functions provided in
Chapter E01
if these points do not themselves form a convenient mesh.
4 References
Pereyra V (1979) PASVA3: An adaptive finite-difference Fortran program for first order nonlinear, ordinary boundary problems Codes for Boundary Value Problems in Ordinary Differential Equations.
Lecture Notes in Computer Science (eds B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76 Springer–Verlag
5 Arguments
1: $neq$ – Integer Input
On entry
: the number of equations; that is
is the order of system
Constraint: $neq≥2$.
2: $fcnf$ – function, supplied by the user External Function
must evaluate the matrix
$F (x)$
at a general point
The specification of
void fcnf (Integer neq, double x, double f[], Nag_User *comm)
1: $neq$ – Integer Input
On entry: the number of differential equations.
2: $x$ – double Input
On entry: the value of the independent variable $x$.
3: $f[neq×neq]$ – double Output
On exit
: the
th element of the matrix
$F (x)$
, for
$i , j = 1 , 2 , … , neq$
$F ij$
is set by
$f[ (i-1) × neq + (j-1) ]$
. (See
Section 10
for an example.)
4: $comm$ – Nag_User *
Pointer to a structure of type Nag_User with the following member:
p – Pointer
On entry/exit
: the pointer
should be cast to the required type, e.g.,
struct user *s = (struct user *)comm → p
, to obtain the original object's address with appropriate type. (See the argument
Note: fcnf
should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
. If your code inadvertently
return any NaNs or infinities,
is likely to produce unexpected results.
3: $fcng$ – function, supplied by the user External Function
must evaluate the vector
$g (x)$
at a general point
The specification of
void fcng (Integer neq, double x, double g[], Nag_User *comm)
1: $neq$ – Integer Input
On entry: the number of differential equations.
2: $x$ – double Input
On entry: the value of the independent variable $x$.
3: $g[neq]$ – double Output
On exit
: the
th element of the vector
$g (x)$
, for
. (See
Section 10
for an example.)
4: $comm$ – Nag_User *
Pointer to a structure of type Nag_User with the following member:
p – Pointer
On entry/exit
: the pointer
should be cast to the required type, e.g.,
struct user *s = (struct user *)comm → p
, to obtain the original object's address with appropriate type. (See the argument
Note: fcng
should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
. If your code inadvertently
return any NaNs or infinities,
is likely to produce unexpected results.
If you do not wish to supply
the actual argument
must be the NAG defined null function pointer
4: $a$ – double Input
On entry: the left-hand boundary point, $a$.
5: $b$ – double Input
On entry: the right-hand boundary point, $b$.
Constraint: $b>a$.
6: $c[neq×neq]$ – double Input/Output
7: $d[neq×neq]$ – double Input/Output
8: $gam[neq]$ – double Input/Output
On entry
: the arrays
must be set to the matrices
must be set to the vector
On exit
: the rows of
and the components of
are re-ordered so that the boundary conditions are in the order:
1. (i)conditions on $y (a)$ only;
2. (ii)condition involving $y (a)$ and $y (b)$; and
3. (iii)conditions on $y (b)$ only.
The function will be slightly more efficient if the arrays
are ordered in this way before entry, and in this event they will be unchanged on exit.
Note that the boundary conditions must be of boundary value type, that is neither
may be identically zero. Note also that the rank of the matrix
must be
for the problem to be properly posed. Any violation of these conditions will lead to an error exit.
9: $mnp$ – Integer Input
On entry: the maximum permitted number of mesh points.
Constraint: $mnp≥32$.
10: $np$ – Integer * Input/Output
On entry
: determines whether a default or user-supplied initial mesh is used.
np is set to a default value of 4 and a corresponding equispaced mesh $x[0] , x[1] , … , x[np-1]$ is used.
You must define an initial mesh using the array x as described.
Constraint: $np=0$ or $4≤np≤mnp$.
On exit: the number of points in the final (returned) mesh.
11: $x[mnp]$ – double Input/Output
On entry
: if
above), the first
elements must define an initial mesh. Otherwise the elements of
need not be set.
$a = x[0] < x[1] < ⋯ < x[np-1] = b ,$ (3)
On exit
$x[0] , x[1] , … , x[np-1]$
define the final mesh (with the returned value of
) satisfying the relation
12: $y[neq×mnp]$ – double Output
On exit
: the approximate solution
$z j ( x i )$
, on the final mesh, that is
$y[(j-1)×mnp+i-1] = z j ( x i ) , i = 1 , 2 , … , np ; j = 1 , 2 , … , neq ,$
is the number of points in the final mesh.
The remaining columns of
are not used.
13: $tol$ – double Input
On entry
: a positive absolute error tolerance.
$a = x 1 < x 2 < ⋯ < x np = b$ (4)
is the final mesh,
$z j ( x i )$
is the
th component of the approximate solution at
$x i$
, and
$y j ( x i )$
is the
th component of the true solution of equation
Section 3
) and the boundary conditions, then, except in extreme cases, it is expected that
$| z j ( x i )- y j ( x i )| ≤ tol , i = 1 , 2 , … , np ; j = 1 , 2 , … , neq$ (5)
Constraint: $tol>0.0$.
14: $comm$ – Nag_User *
Pointer to a structure of type Nag_User with the following member:
p – Pointer
On entry/exit
: the pointer
, of type Pointer, allows you to communicate information to and from
. An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer
by means of a cast to Pointer in the calling program, e.g.,
comm.p = (Pointer)&s
. The type pointer will be
void *
with a C compiler that defines
void *
char *
15: $fail$ – NagError * Input/Output
The NAG error argument (see
Section 7
in the Introduction to the NAG Library CL Interface).
6 Error Indicators and Warnings
On entry, $b=⟨value⟩$ while $a=⟨value⟩$. These arguments must satisfy $b>a$.
Dynamic memory allocation failed.
More than
columns of the
are identically zero. i.e., the boundary conditions are rank deficient. The number of non-identically zero columns is
At least one row of the
is a linear combination of the other rows, i.e., the boundary conditions are rank deficient. The index of the first such row is
One of the matrices $C$ or $D$ is identically zero, i.e., the problem is of initial value and not of the boundary type.
At least one row of the
is a linear combination of the other rows determined up to a numerical tolerance, i.e., the boundary conditions are rank deficient. The index of first such row is
. There is some doubt as to the rank deficiency of the boundary conditions. However even if the boundary conditions are not rank deficient they are not posed in a suitable form for use with this
function. For example, if
$C = ( 1 0 1 ε ) , D = ( 1 0 1 0 ) , γ = ( γ 1 γ 2 )$
is small enough, this error exit is likely to be taken. A better form for the boundary conditions in this case would be
$C = ( 1 0 0 1 ) , D = ( 1 0 0 0 ) , γ = ( γ 1 ε −1 ( γ 2 - γ 1 ) )$
of the array
and the corresponding row of array
are identically zero, i.e., the boundary conditions are rank deficient.
A finer mesh is required for the accuracy requested; that is
is not large enough.
The Newton iteration failed to converge on the initial mesh. This may be due to the initial mesh having too few points or the initial approximate solution being too inaccurate. Try using
Solution cannot be improved due to roundoff error. Too much accuracy might have been requested.
On entry, $mnp=⟨value⟩$.
Constraint: $mnp≥32$.
On entry, $neq=⟨value⟩$.
Constraint: $neq≥2$.
On entry,
. The argument
must satisfy either
$4 ≤ np ≤ mnp$
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
for assistance.
On entry, the left boundary value
, has not been set to
$x[0] = ⟨value⟩$
The sequence
is not strictly increasing:
$x[⟨value⟩] = ⟨value⟩$
$x[⟨value⟩] = ⟨value⟩$
On entry,
must not be less than or equal to 0.0:
On entry, the right boundary value
, has not been set to
$x[np-1] = ⟨value⟩$
7 Accuracy
The solution returned by the function will be accurate to your tolerance as defined by the relation
except in extreme circumstances. If too many points are specified in the initial mesh, the solution may be more accurate than requested and the error may not be approximately equidistributed.
8 Parallelism and Performance
Background information to multithreading can be found in the
d02gbc is not threaded in any implementation.
9 Further Comments
The time taken by the function depends on the difficulty of the problem, the number of mesh points (and meshes) used and the number of deferred corrections.
In the case where you wish to solve a sequence of similar problems, the use of the final mesh from one case is strongly recommended as the initial mesh for the next.
10 Example
We solve the problem (written as a first order system)
with boundary conditions
for the cases
$ε = 10 −1$
$ε = 10 −2$
using the default initial mesh in the first case, and the final mesh of the first case as initial mesh for the second (more difficult) case. We give the solution and the error at each mesh point to
illustrate the accuracy of the method given the accuracy request
$tol = 1.0e−3$
10.1 Program Text
10.2 Program Data
10.3 Program Results | {"url":"https://support.nag.com/numeric/nl/nagdoc_latest/clhtml/d02/d02gbc.html","timestamp":"2024-11-10T17:36:16Z","content_type":"text/html","content_length":"61409","record_id":"<urn:uuid:84f7fe7a-261d-4707-8083-1c84748b9150>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00656.warc.gz"} |
12 Useful Formula for Industrial Engineers in Garment Manufacturing (Poster)
I love formulas for mathematical calculations. Because a formula helps me get the result quicker and it specifically mentions all data I need to collect to find the result of the desired thing. In
this post, I have shared 12 important garment production formulas for engineers.
In garment manufacturing, industrial engineers, production managers, and planners measure factory performance on a daily basis. Performances are measured using standard formulas. I guess you are
already using such formulas for calculating performance.
Don't you think, it would be a good idea to have a poster of commonly used formulas in your workspace? A poster is a very useful tool for many reasons. I have made one poster for you and sharing it
with you.
The following are the common formulas those are used by garment industry professionals.
Formula#1: Calculating the daily production target of a line
Daily Line Target = (Total shift hours X 60 X No. of operators in a line X Line Efficiency%)/Garment SAM
Here is an example to understand this calculation.
Daily shift Hours of a factory = 8 Hours
Number of operators working in that line= 30 operators
Average Efficiency of that line: 60%
Garment SAM: 20 Minutes
Production target of this line (Daily) = (8 X 60 X 30 X60%)/20
=(480 x 30 x60) / (100 X20)
= 432 Pieces
Formula#2: Operator-wise production target calculation
Individual operator target = (Total working minutes in a day X Line Efficiency %)/Operation SAM
In Formula #1, you have the formula for calculating the daily production target for a line.
But operators working in that line will be working on different operations and the SAM of those operations will be different.
So, you need to calculate individual operator targets as well based on Line efficiency and operation SAM.
Refer to this example for the calculation operator target for the specific line.
Factory shift: 8 Hours (480 Minutes)
Line efficiency = 60% (considering it is the same line in formula#1)
Operation SAM: 0.5 Minutes
Production target of that operator: (480 Minutes x 60%)/0.50
= (480 X 0.60)/0.5
= 576 Pieces per 8 hours per day.
In case you use operator performance level (efficiency%) and operation SAM and calculate the production target of that operator, the result is called the operator's production capacity.
Normally, an operator-wise target is not given based on the individual operators' performance, instead, it is calculated based on the line efficiency. See the below example of the operator's
Factory shift: 8 Hours (480 Minutes)
Operator efficiency = 80%
Operation SAM: 0.5 Minutes
Production capacity of that operator: (480 Minutes x 80%)/0.50
= (480 X 0.80)/0.5
= 768 Pieces per 8 hours per day.
Read more
about the operator's target calculation.
Formula#3: Calculating operator efficiency
Individual operator Efficiency% = (Units Produced X Operation SAM X 100)/Total minutes worked
When you develop the skill matrix for sewing operators you need to measure individual performance.
Secondly, if you plan to start a performance-based incentive scheme for the individual operator, measuring individual operator efficiency is essential.
Let's say you need to calculate the efficiency of an operator based on the last production day's data.
Operator Produced total unit: 500 pieces
SAM of the operation: 0.60 Minutes
The operator worked: 480 Minutes (full shift hours)
Operator Efficiency = ((500 x 0.60)/480 )X100
Formula#4: Line efficiency
Line Efficiency% = (Line output X Garment SAM X 100)/(Number of operators X Minute worked in a day)
Note: Include helpers and workers doing manual operations in case you have included SAM of those operations.
Line A produced 600 units (Style Z)
SAM of style Z is 20 Minutes
Attendance in Line A = 30 Operators
Shift Time: 8 Hours
Line Efficiency (Overall Efficiency%)= (600 x 20 x 100)/(30X8 x60)
= (1200000/14400)
= 83.33%
Here is another example of line efficiency calculation.
Note: include helpers and workers doing manual operations in case you have included SAM of those operations. See
an example of line efficiency calculation.
Formula#5: Machine productivity
Machine Productivity: (Line output / No. of machine used in producing those garments)
Machine productivity is measured in production per machine per shift day. In
this article
, I have discussed more about machine productivity calculation with examples.
Considering Line A has production of 600 units.
Numer of sewing machines used: 27 Machines
Machine productivity:
= (600/27)
= 22.22 units per machine
Formula#6: Labour productivity
Labor Productivity = Line output / No. of total manpower (operators +helpers)
Considering Line-A has production of 600 units
Numer of sewing operators used: 30 and helpers 5. Therefore labor productivity
= (600/35)
= 17.14 units per labor
Line WIP (work in process) = Total pieces lying on the line for a particular order line
WIP of the line of an order is equal to the total pieces loaded till date minus Total pieces out till date. The WIP calculation method and Excel report template are shown in
this post
Standard Time = (Observed time X observed rating) + Allowances
Allowances – Relaxation allowance, contingency allowance
Machine utilization% = (Actual Machine running Time X 100) / Time available
Cost per minute = Total cost incurred in labor / Total available working minute in a day X no. of labors
Production Cost per unit = Total cost incurred in production in a day/ no. of garment produced in a day
Man to Machine ratio = Total manpower of the factory / Total no. of sewing machines (utilized)
Learn more about
Man to Machine Ratio
Poster Download:
12 useful performance measuring formulas (Click to enlarge the image and save the image file).
To download the poster save the image after clicking on it.
Note: Our purpose is to provide you with correct information. Still, if you find any formula that is not correct you may comment in the below comment box. | {"url":"https://www.onlineclothingstudy.com/2013/10/12-useful-formulas-for-industrial.html","timestamp":"2024-11-14T13:24:33Z","content_type":"application/xhtml+xml","content_length":"177408","record_id":"<urn:uuid:0a9f8280-050a-45d2-84ad-d0da1b27a5c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00785.warc.gz"} |
Probability-Generating Functions
I have long struggled with understanding what probability-generating functions are and how to intuit them. There were two pieces of the puzzle missing for me, and we’ll go through both in this
There’s no real reason for anyone other than me to care about this, but if you’ve ever heard the term pgf or characteristic function and you’re curious what it’s about, hop on for the ride!
Imagine you are holding five regular playing cards in your hand. Maybe your hand is QQA97, i.e. a pair of queens, an ace, a nine, and a seven. We’re playing some sort of weird poker variant where I
get to blindly draw one of your cards. We’re curious about the probability distribution of the outcome of that draw.
In words, most cards (e.g. 2, 4, 8, J and others) have a probability of zero of being drawn from your hand (because they are not in your hand.) Some cards (ace, seven, nine) have a 20 % probability
of being drawn, and then there’s a 40 % probability that a queen is drawn, since you have two of them. | {"url":"https://folu.me/post/ragebcvpgubhtugf-d-dpbz/probability-generating-functions","timestamp":"2024-11-13T16:15:26Z","content_type":"text/html","content_length":"235595","record_id":"<urn:uuid:399c8a45-c70e-40d9-953c-e64ca38012fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00049.warc.gz"} |
Meta-analysis is widely used to summarize estimated effects sizes across multiple statistical tests. Standard fixed and random effect meta-analysis methods assume that the estimated of the effect
sizes are statistically independent. Here we relax this assumption and enable meta-analysis when the correlation matrix between effect size estimates is known. Fixed effect meta-analysis uses the
method of Lin and Sullivan (2009), and random effects meta-analysis uses the method of Han, et al. 2016.
Install from GitHub | {"url":"https://cran.case.edu/web/packages/remaCor/readme/README.html","timestamp":"2024-11-06T17:56:15Z","content_type":"application/xhtml+xml","content_length":"5699","record_id":"<urn:uuid:34cff446-a1ae-4654-a1d3-ad56e44a931b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00348.warc.gz"} |
Prediction of Carcass Composition Using Carcass Grading Traits in Hanwoo Steers
Article information
Asian-Australas J Anim Sci. 2016;29(9):1215-1221
Received 2015 September 09; Revised 2015 October 29; Accepted 2015 November 26.
The prediction of carcass composition in Hanwoo steers is very important for value-based marketing, and the improvement of prediction accuracy and precision can be achieved through the analyses of
independent variables using a prediction equation with a sufficient dataset. The present study was conducted to develop a prediction equation for Hanwoo carcass composition for which data was
collected from 7,907 Hanwoo steers raised at a private farm in Gangwon Province, South Korea, and slaughtered in the period between January 2009 and September 2014. Carcass traits such as carcass
weight (CWT), back fat thickness (BFT), eye-muscle area (EMA), and marbling score (MAR) were used as independent variables for the development of a prediction equation for carcass composition, such
as retail cut weight and percentage (RC, and %RC, respectively), trimmed fat weight and percentage (FAT, and %FAT, respectively), and separated bone weight and percentage (BONE, and %BONE), and its
feasibility for practical use was evaluated using the estimated retail yield percentage (ELP) currently used in Korea. The equations were functions of all the variables, and the significance was
estimated via stepwise regression analyses. Further, the model equations were verified by means of the residual standard deviation and the coefficient of determination (R^2) between the predicted and
observed values. As the results of stepwise analyses, CWT was the most important single variable in the equation for RC and FAT, and BFT was the most important variable for the equation of %RC and
%FAT. The precision and accuracy of three variable equation consisting CWT, BFT, and EMA were very similar to those of four variable equation that included all for independent variables (CWT, BFT,
EMA, and MAR) in RC and FAT, while the three variable equations provided a more accurate prediction for %RC. Consequently, the three-variable equation might be more appropriate for practical use than
the four-variable equation based on its easy and cost-effective measurement. However, a relatively high average difference for the ELP in absolute value implies a revision of the official equation
may be required, although the current official equation for predicting RC with three variables is still valid.
Carcass composition is important for value-based marketing of a carcass and for the establishment of an optimal feeding scheme (Drennan et al., 2008; Minchin, et al., 2009). However, the actual
values of composition obtained from carcass dissection may not facilitate the determination of carcass pricing because the pricing of carcasses is typically completed prior to the carcass dissection.
Thus, many studies have focused on the development of equations that predict carcass composition based on carcass traits, such as carcass weight (CWT), back-fat thickness (BFT), eye-muscle area
(EMA), marbling score (MAR), and the percentage of kidney, pelvic and heart fat (%KPH) (Greiner et al., 2003b). Various efforts have been made to improve the prediction equation using a large dataset
and focusing on the reduction of differences between the predicted and observed values with a low standard error for the estimate (Kauffman et al., 1975) and further ultrasonic measurement of fat has
been widely used for the development of a prediction equation (Anderson et al., 1983; Herring et al., 1994). However, ultrasound technology may not be available at the farm level and the official
equation includes three independent carcass variables (CWT, BFT, and EMA) and is widely used to estimate the percentage of retail cut for the Hanwoo carcasses (MAFRA, 2009a); several studies have
been conducted to develop an equation with improved predictive ability for Hanwoo carcasses (Lee et al., 2005; 2008; Choy et al., 2010). However, the development and proof of prediction equations
must be continuously reconfirmed due to the environmental changes in the beef production system and the endless efforts to genetically improve the cattle to ensure high productivity, which could lead
to biological changes in Hanwoo cattle that could affect feed efficiency, growth performance, and body composition (Kim and Lee, 2000).
Thus, this study aimed to develop equations to predict carcass composition, such as retail cut, trimmed fat and separable bone weights and percentages, of Hanwoo steer using carcass traits (CWT, BFT,
EMA, and MAR) on the steer field data collected from the private farm of Korea and to verify the predictive ability of the model equation developed.
Animals and traits
Data were obtained from 7,907 Hanwoo steers raised at a private farm in Gangwon Province, South Korea, and slaughtered in the period between January 2009 and September 2014. The number of animals per
slaughter year were 1,162, 1,227, 1,857, 1,450, 1,168, and 1,043 for year 2009, 2010, 2011, 2012, 2013, and 2014, respectively, and mean age in month was 31.97±3.22. The feeding and management of the
steers has been explained by Koh et al. (2014).
Following the standard normal industrial procedures recommended by the Korean government (MAFRA, 2009a), the slaughtering of the steers was conducted at an abattoir that was located at a transport
distance less than an hour from the steer farm by truck. After slaughtering, each carcass was immediately halved and held in a chilling room at 4°C for one night, and then both sides of the carcasses
were taken from the chilling room and transferred to the evaluation venue for the weight measurement. The left sides were dissected at the position between the last rib and the first lumbar vertebra,
and these were used for the measurements of BFT, EMA, and MAR. The BFT was measured at the three-fourths position of the longissimus muscle from the spinal column. The EMA was measured using a
transparent grid. The MAR was evaluated from 1 (poor) to 9 (best) according to the Korea Beef Marbling Standard, and the estimated retail yield percentage (ELP) was calculated according to the Korea
Beef Grading Standards as follows.
Carcass fabrication procedures were performed at the adjacent commercial packing plant based on the standards recommended by the Korean government (MAFRA, 2009b). The entire carcass was dissected
into several sub-primal cuts, and all bones were removed with the exception of the ribs. Rib bones were left in the rib. After the de-boning, the excessive fat and intermuscular fat were closely
All sub-primal cuts were summed as retail cut weight (RC), while the bones removed from the sub-primal cuts and tail were summed as bone weight (BONE). The trimmed fat weight (FAT) was calculated by
subtracting the RC and BONE from the CWT, since the measurement of FAT was not possible due to the conditions of the packing plant. The percentages for the retail cuts (%RC), separated bones (%BONE),
and trimmed fats (%FAT) were obtained based on the cold CWT.
After data collection, the unrealistic values that were regarded as recording errors were deleted, and the furthest values from the normal, as determined by 3 standard deviations from the mean values
in the ratios of RC to FAT, RC to BONE, and FAT to BONE, were excluded (Hickey et al., 2007; McPhee et al., 2008). A total of 7,152 carcass records were used in the subsequent statistical analyses.
Statistical analyses
A randomly determined half of the 7,152 records (n = 3,576) was assigned to the development of prediction equations, while the other half of the dataset (n = 3,576) was assigned to verifying the
equations developed. The simple statistics for the development and test datasets are presented in Table 1.
The prediction equations for the six carcass compositional traits—RC, %RC, FAT, %FAT, BONE, and %BONE—were developed using a stepwise regression procedure, using CWT, BFT, EMA, and MAR as independent
variables that had to retain statistical significance (p<0.05) within the model equation. The predictive ability of the equations developed was determined by the coefficient of determination (R^2)
and the residual standard deviation (RSD), and the equations with the highest R^2 and the lowest RSD values were considered to have the best predictive ability.
The difference and correlation coefficients between the predicted and observed values were calculated using the dataset and the equations developed in this study and compared to the predictive
ability of the equations. In addition, since ELP represents the predicted percentage of retail yield calculated by the current official equation for Hanwoo carcasses, the difference and correlation
coefficients between the ELP and %RC observed were compared to those values calculated from the %RC equations developed in this study, whereby the feasibility of the %RC equations were evaluated for
practical use.
All the statistical analyses, including the elementary statistics, correlation coefficients, and stepwise regressions, were performed using SAS software (Version 9.2, SAS Inst., Inc., Cary, NC, USA).
Stepwise regressions for RC and %RC
Stepwise regression equations for predicting RC and %RC from the four carcass grading traits of CWT, BFT, EMA, and MAR are shown in Table 2. For presentation purposes, the equations for each step of
the regression analyses were labeled as Eq. 1, Eq. 2, Eq. 3, and Eq. 4, respectively.
The CWT was the most important for the prediction of RC, since CWT alone governed RC values with a variation of 83.4%. For %RC, the BFT covered 10.7% of the variation of %RC (Eq. 1). The EMA was the
second most influential variable on both RC and %RC, and it increased the power of predictability an additional 2.5% and 9.2% of variations in RC and %RC, respectively (Eq. 2). The MAR was the last
variable in the equation for RC and %RC in the stepwise process, but its influence on the increase in the R^2 value was not significant.
The RSD and R^2 of Eq. 3 with three independent variables were estimated as 10.350 kg and 87.1%, respectively, and the values for Eq. 4 in which MAR was added were 10.342 kg and 87.1%, respectively.
The RSD and R^2 between Eq. 3 and Eq. 4 showed little difference for the dependent variables of RC and %RC.
Stepwise regressions for FAT and %FAT
The results of the stepwise regression analyses for predicting FAT and %FAT are presented in Table 3. The CWT again governed the equation, accounting for 59.7% of the variations for FAT. The second
variable, BFT, was added to the equation for FAT, which resulted in an increase of the R^2 value by 5.3%, while the further addition of EMA increased the R^2 value to 68.8%. The MAR was the last
variable in the equation, but its contribution to the variation in FAT was insignificant and induced an increase of only 0.2% in the equation evaluation with three variables (Eq. 3).
In the stepwise analyses for %FAT, the most influential factors in the equation were BFT, EMA, CWT, and MAR, in this order, and the addition of EMA and CWT in the equation with BFT resulted in an
increase of the R^2 by 9.9% in the equation (from 17.7% in Eq. 1 to 27.6% in Eq. 3). However, the addition of MAR showed an increase in the R^2 of only 0.6% for the equation fit (from 27.6% in Eq. 3
to 28.2% in Eq. 4).
Stepwise regressions for BONE and %BONE
The results of the stepwise analyses for predicting BONE and %BONE are delineated in Table 4. The CWT and EMA were the first and last variables, respectively, entered into the equation for predicting
BONE and %BONE. As the third variable, MAR with CWT and BFT improved BONE and %BONE by 1.3% and 2.0%, respectively.
The R^2 values in Eq. 3 and Eq. 4 for BONE were 53.3% and 53.4%, respectively, which corresponded with 27.5% and 27.6% for %BONE, respectively. The differences in R^2 and RSD between the final
four-variable equation (Eq. 4) and the three-variable equation in the third step (Eq. 3) were not very significant in BONE and %BONE.
Evaluation of the equation
The three- and four-variable equations (Eq. 3 and 4, respectively) were selected as the best probable equations because of their high R^2 and small RSD, and they were applied to the test dataset for
the evaluation of the equations. Each dependent variable predicted from Eq. 3 and Eq. 4 were compared with an extra retail cut percentage from the official equation (ELP).
Table 5 indicated the means predicted from Eq. 3 and Eq. 4 and observed from the test dataset. Eq. 3 and Eq. 4 were evaluated using the average differences and the correlation coefficients. The
differences between the predicted and observed values were calculated by subtracting the observed value from the predicted value.
The correlation coefficients between the predicted and observed values in Eq. 3 were almost the same as those in Eq. 4 for every dependent variable, and the correlation coefficients of RC, FAT, and
BONE were higher than those of %RC, %FAT, and %BONE in both Eq. 3 and Eq. 4.
The differences between the predicted and observed values for %RC and %BONE were statistically determined using a T-test in Eq. 3 and for %RC, %FAT, and %BONE in Eq. 4. Eq. 3, with three variables,
overestimated %RC and %BONE, while Eq. 4 underestimated %RC, BONE, and %BONE. Using the ELP (official equation) for prediction, the average differences of %RC, with an absolute value of 1.01, were
higher than the differences of 0.15 and 0.28 from Eq. 3 and Eq. 4, respectively, and the average differences from Eq. 3 tended to be smaller than those from Eq. 4 for all dependent variables, except
RC, for which the average difference in Eq. 3 was slightly higher.
The model equation in the final step was functions of all four carcass-grading traits (CWT, BFT, EMA, and MAR) in order to obtain each dependent variables. The MAR was the last variable in the
equation for RC, %RC, FAT, and %FAT, while the EMA was the last for BONE and %BONE. However, compared to the RSD and R^2 values, there were no practical differences between Eq. 3 and Eq. 4, which
indicates that the last variables in the equation rarely contributed to the predictive ability of the equation. The inclusion of MAR, as the last variable in Eq. 3, with CWT, BFT, and EMA, increased
the R^2 value of the equation for %RC and %FAT by only 0.02% and 0.06%, respectively (Tables 2 and 3). The inclusion of EMA, as the last variable in the Eq. 3, with CWT, BFT, and MAR increased the R^
2 value of the equation for both BONE and %BONE by only 0.01% (Table 4).
The low contribution of MAR to the predictive ability of the equation for %RC and %FAT differs from the results of Griffin et al. (1999) and Greiner et al. (2003a), where the addition of MAR to the
equation for %RC, which consisted of four independent variables (CWT, BFT, the percentage of kidney, pelvic and heart fat [%KPH], and EMA), increased the model R^2 values by sizable amounts (2% to
4%). The low contribution of MAR in predicting %RC and %FAT in this study could be due to a low correlation between MAR and %RC and %FAT in Hanwoo steer carcasses. Koh et al. (2014) reported a small
and positive phenotypic correlation between MAR and %RC and %FAT in Hanwoo steer data (r = +0.04 and +0.07, respectively), of which similar and positive correlation coefficients of +0.02 and +0.03
were estimated in the preliminary analyses of this study (data not shown).
On the contrary, many previous studies in the US have shown a negative and moderate correlation between MAR and %RC and a moderate positive correlation between MAR and %FAT (Herring et al., 1994;
Shackelford et al., 1995; Johnson and Rogers, 1997; Griffin et al., 1999; May et al., 2000; Greiner et al., 2003b).
When compared to the R^2 values of the equations for %RC, %FAT, and %BONE measured in percentages units, the R^2 values corresponding to the RC, FAT, and BONE in kg units, respectively, were higher,
of which trends had generally been shown in previous studies (Herring et al., 1994; Shackelford et al., 1995; Williams et al., 1997; Dikeman et al., 1998; Realini et al., 2001; Greiner et al., 2003b;
Lee et al., 2005; Maeno et al., 2014). These results imply that the equations for RC or FAT constructed using carcass traits might be more accurate than the equations for %RC and %FAT. Further, CWT
among the independent variables showed the strongest correlation in the equation with the highest R^2 value as the best single predictor. The correlation coefficients for RC, FAT, and BONE with CWT,
which were obtained using the square root of the R^2 values of Eq. 1 for each weight variable, were 0.91, 0.77, and 0.69, respectively, in this study.
The official Korean equation for predicting %RC includes the three independent variables of CWT, BFT, and EMA, which were the same independent variables used to construct Eq. 3 for RC and %RC in this
study. Lee et al. (2005 and 2008) and Choy et al. (2010) also developed equations using the same three independent variables to predict the RC and %RC for Hanwoo steer carcasses. The R^2 values in
Eq. 3 for RC and %RC were 87.1% and 23.5%, respectively, in the present study, and these seem to concur with the values reported by Lee et al. (2005). However, the R^2 value of 23.5% for %RC is lower
than the 54% reported by Choy et al. (2010). On the other hand, the R^2 values reported for equations for %RC with exotic beef carcasses ranged from 32.2% (Williams et al., 1997) to 75% (Cannell et
al., 1999), which are generally higher than the R^2 values from Eq. 3 and Eq. 4 in the present study. The different R^2 values from the various studies might be due to a number of factors, including
different cattle types, feeding management, fat trimming level, carcass fatness, variables for equation development, and cutting procedures. Another plausible reason, which might be exclusive to the
study on commercial data, was the inconsistent fat trimming level due to the purchaser’s demand. In this study, the fat trimming of the retail cut was conducted within the 6 mm fat cover, but the fat
trimming level could differ based on the purchaser’s demand.
When the overall predictability of an equation is evaluated in terms of accuracy and precision, the RSD or the difference between the predicted and observed values were used to determine the level of
accuracy, and the precision was determined through the comparison of R^2 values or the correlation coefficients of the predicted and observed values (Johnson and Rogers, 1977; Tedeschi, 2006). In
this study, almost identical correlation coefficients were found for Eq. 3 and Eq. 4 regarding the predicted and observed values for all the dependent variables, which indicates that equal precision
was achieved by Eq. 3 and Eq. 4. The small average difference in the absolute value for Eq. 3 implies that Eq. 3 is more accurate than Eq. 4 in predicting %RC, BONE, and %BONE; furthermore, the three
variables in the current official equation for predicting RC were found to still be valid. However, compared to Eq. 3 and Eq. 4 for predicting %RC, a relatively high absolute value in the average
difference for the ELP suggests that further study may be necessary to revise the official equation.
In stepwise process, the CWT was the most important trait for predicting RC, FAT, BONE, and %BONE, and the BFT was the most important trait for predicting %RC, %FAT.
There are no differences in the model precision between the three- and four-variable equations for predicting all six dependent variables, and the model accuracy is similar for the prediction of RC,
FAT, and %FAT; however, three-variable equation (Eq. 3) is more accurate than the four-variable equation for predicting %RC, BONE, and %BONE.
We concluded that Eq. 3, which has three variables, might be the optimal choice for practical use in predicting the six dependent variables, and this should facilitate the investigation of new
variables with easy and cost-effective measurements in order to increase the precision of the equations for predicting percentage variables, such as %RC, %FAT, and %BONE. The three variable equations
included CWT, BFT, and EMA may be used for predicting lean and fat composition of Hanwoo carcasses and might be target variables for improving lean yield productivity of Hanwoo.
This study was supported by 2014 Research Grant from Kangwon National University (No. 120141398).
We certify that there is no conflict of interest with any financial organization regarding the material discussed in the manuscript.
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How to Code Independent Cascade Model of Information Diffusion | Suman Kundu
How to Code Independent Cascade Model of Information Diffusion
Jul 25, 2013· · 10 min read
Social Network is increasing its popularity, and one of the important research areas within this field is Information Diffusion. Information diffusion is the process by which a new idea or innovation
spread over the networks by the means of communication among the social entities [6]. There are two widely used information diffusion mode (i) Threshold Model of Diffusion [3] and (ii) Cascade Model
of Diffusion [1, 2]. Simplest and popular form of Cascade Model is Independent Cascade Model (ICM) [1].
Figure 1: Graph Representing Social Network
The article briefly describes what Independent Cascade Model is and provides an overview on how to code a simulator for it. Java is used for all coding work for the research; however I will try to
write it as general as possible. To understand the article you do not require to know Java (mostly), however basic knowledge of any one programming language would be necessary. Few concepts of
Object-Oriented Programming would be used. Readers may refer Object Oriented Programming Fundamentals for basic information regarding OOP.
Note that my PhD work is on Directed Social Networks, so the later part will mostly be focused on it. I believe it could be easily modified to accept other types of Social Networks too.
Social Network
A Social Network is made up of individuals (person, organizations, etc.) and their ties. A social tie can be of different types like friendship, common interests, financial exchange, dependencies,
travel routes, etc. A social network normally represented by a graph $G(V,E)$. Where $V$ represents the set of nodes and $E$ represents set of relationships. Figure 1↑ shows a directed social
network. The set $V=\\{1,2,3,4,5,6,7,8,9,10\\}$ is the set of nodes, and the set $E=\\{(7,1),(6,7),(6,5),(7,5),$ $(7,2),(7,8),(4,7), (3,7),(3,8),(3,9),(3,1),(3,10),$
$(8,9),(2,3)\\}$ is the set of edges. An undirected social network is the social network where edges are undirected all others remain same.
Independent Cascade Model of Information Diffusion
Independent Cascade Model (ICM) is a stochastic information diffusion model where the information flows over the network through Cascade. Nodes can have two states, (i) Active: It means the node
already influenced by the information in diffusion. (ii) Inactive: node unaware of the information or not influenced.
The process runs in discrete steps. At the beginning of ICM process, few nodes are given the information known as seed nodes. Upon receiving the information these nodes become active. In each
discrete step, an active node tries to influence one of its inactive neighbors. In spite of its success, the same node will never get another chance to activate the same inactive neighbor. The
success depends on the propagation probability of their tie. Propagation Probability of a tie is the probability by which one can influence the other node. In reality, Propagation Probability is
relation dependent, i.e., each edge will have different value. However, for the experimental purpose, it is often considered to be same for all ties.
The process terminates when no further nodes became activated from inactive state.
Figure 2: Class Diagram of Vertex and DirectedGraph
Data structures
A social graph is required to run ICM over it. In this section, a data structure DirectedGraph will be defined. A directed graph consists nodes; so a type Vertex will also be defined in this section.
In addition to that some standard data structures available to the Java users will be discussed. This will help the readers to find similar data structures in their favorite programming language.
Figure 2↑ shows a class diagram of DirectedGraph and Vertex.
A Vertex has two Sets of Vertex, (i) In Bound Neighbors and (ii) Out Bound Neighbors. Propagation Probabilities are store into a one-to-one map for each in links. This is to note here is that there
is no propagation probability for out links. It is so because one can influence its followers but not whom (S)he is following. In the Figure 3↑ the yellow nodes $1,2,3,4$ are the followers of the
node $U$ and it is following nodes in green $a,b,c$. So, each green node will save the propagation probability at which they can influence $U$ and $U$ will store propagation probability of each in
links, i.e. $(1,U),(2,U),(3,U),(4,U)$.
Each incoming node, is of type Vertex, and each outgoing node is of type Vertex. Those who are interested to implement it with procedural language you may use Sparse Matrix with three columns (Col 1:
From Node ID, Col 2: To Node ID, Col 3: Propagation Probability at which To Node influences From Node) to store links with corresponding propagation probabilities.
DirectedGraph stores the set of nodes. As we only require to traverse from one nodes to its followers, or followings we kept DirectedGraph clean, i.e., no edge information is stored here. This way we
can reduce the use of memory.
It is an Abstract Data Type (ADT) resembles the concepts of mathematical finite set. Elements of set are not in order, i.e., elements can be stored in any order. A set does not contain any repetitive
elements. Read more about this ADT in Wiki on Set.
List ADT resembles the concepts of finite sequence. It is an ordered collection of values. There might be repetitive elements in a list. Read more on Wiki on List.
Map :
Map ADT is a collection of (key, value) pairs where keys are unique. That is, any key appears exactly once in the collection. Other similar ADTs are Associative Array, Symbol Table and Dictionary.
Read more about this on Wiki on Map.
Stack is a basic computer science data structure. It is a List in which an element can only be added or removed from one end, i.e., it is Last In First Out (LIFO) List.
ICM Simulator
ICM is a stochastic process. So, it requires to run the simulation for a sufficiently large number of times for accurately determine the information diffusion spread. For my work, I run 20000 times
and took an average of all the values.
Simulation with a large scale social network requires high CPU. Further, you need results in steps. For example, if your algorithm selects $50$ seeds, you might have to calculate results for
$10,20,30,40$ seeds as well. If your algorithm is like mine (provide sequence of nodes instead of set of nodes based on their ranks), you can use the following trick to execute everything at one
go. Evaluate each nodes starting from the top ranked node separately and saved partial data after each node’s evaluation finishes.
Code listed in Listing 4↓ shows a java code for a single diffusion. Following paragraphs describes the code in a general approach.
Single Diffusion
public Map<Int, Int> singleDiffusion(DirectedGraph graph, List seeds) {
//will store the active nodes
Set<Vertex> active = new HashSet();
//will store unprocessed nodes during intermediate time
Stack<Vertex> target = new ArrayStack();
//will store the results
Map<Int, Int> result = new HashMap<Int, Int>();
foreach(Vertex s in seeds) {
while (target.size() > 0) {
Vertex node = target.pop();
foreach(Vertex follower in node.getInBoundNeighbor()) {
float randnum = ran.nextFloat();
if (randnum <= node.getPropagationProbability(follower)) {
if (!active.contains(follower)) {
result.put(result.size() + 1, active.size());
return result;
As the name signifies the function “singleDiffusion” runs a single simulation of information diffusion. It takes two parameters (i) The Graph (In Code: graph) and (ii) list of seeds (In Code: seeds).
A point to note here is that seeds are passed as a list (generally higher ranked nodes are stored in lower offset) and iterated accordingly, i.e., top ranked nodes are evaluated first.
Active nodes are stored temporarily in a Set (In Code: active). One can use an array for this purpose. Set has been used here to ensure no duplicate entries. Those interested to use array, please
check the duplicate entry explicitly. A stack (In Code: target) is required for storing intermediate nodes during evaluation. Results are stored in a Map (In Code: result) whose keys and values are
in integer.
Each node is evaluated as per the following steps,
1. Push it to the stack
2. while stack is not empty
1. pop the node ($v$) from stack
2. consider it as active and put it into active sec
3. Repeat following for each inbound neighbor ($u$) of node
1. generate a random floating point number ($rand$ between 0 to$1$).
2. if (propagation_probability($u,v$) > $rand$)
1. if ($u$ is not in active set) push $u$ to stack
When stack become empty evolution of the node is done. Size of the active set is the total nodes influenced due to the node in evaluation and all previously evaluated nodes. The cardinality of the
active set (value) is stored into the map by the number of nodes (key) evaluated so far.
This singleDiffusion with the same parameters needs to be run for several times. In my work, I use 10000/20000 times depending upon the time taken to the execution. Then calculate the average of all
as Listing 4↓. Higher number of simulation provides higher accurate results.
Code for calculating avarage
Map<Int, Int>[] results = new Map<Int, Int>[20000];
float[] avg = new float[seeds.size()];
for(int i = 0; i < 20000; i++){
results[i] = singleDiffusion(graph, seeds);
for(int j = 1; j <= seeds.size(); j++){
avg[j-1] += results[i].get(j);
for(int i = 0; i < seeds.size(); i++){
avg[i] = avg[i] / 20000;
Discussion & Conclusion
The article briefly described what Independent Cascade Model of Information Diffusion is and provided an implementation, in details, which has been used for the work [4, 5]. Coding is done in Java.
Descriptions of the data structures (slandered and specific) are reported in details so that interested readers can easily port it to their favorite programming language.
Method described here can directly be used for any directed social networks. With simple modification, it can also be generalized to work with undirected social networks as well. Method works in a
single Threaded environment. This help us to record intermediate results in one go. Whoever interested to use multi-threaded environment for singleDiffusion must take care of this separately. One may
run each singleDiffusion in a different thread and calculate the average upon completion of all. Proper multi-threading implementation would run faster than the above listed code.
The article did not provide any information regarding memory footprint or CPU usage. With the modern equipment, it will run within a reasonable time span even for a graph with million nodes and
If this article helps in your research work then cite any of the following work
The paper addresses the problem of finding top k influential nodes in large scale directed social networks. We propose two new …
The paper addresses the problem of finding top k influential nodes in large scale directed social networks. We propose a centrality …
[1] : “Talk of the network: A complex systems look at the underlying process of word-of-mouth”, Marketing Letters, pp. 211—223, 2001.
[2] : “Using complex systems analysis to advance marketing theory development: Modeling heterogeneity effects on new product growth through stochastic cellular automata”, Academy of Marketing Science
Review, pp. 1—18, 2001.
[3] : “Threshold models of collective behavior”, The American Journal of Sociology, pp. 1420—1443, 1978.
[4] : “A New Centrality Measure for Influence Maximization in Social Networks”, 4th international conference on Pattern recognition and machine intelligence (PReMI'11), pp. 242—247, 2011.
[5] : “Centrality Measures, Upper Bound, and Influence Maximization in Large Scale Directed Social Networks”, Fundamenta Informaticae, .
Assistant Professor of Computer Science and Engineering
My research interests include social network analysis, network data science, streaming algorithms, big data, granular computing, soft computing, fuzzy and rough sets. | {"url":"https://sumankundu.info/articles/how-to-code-independent-cascade-model-of-information-diffusion/","timestamp":"2024-11-11T14:27:12Z","content_type":"text/html","content_length":"122613","record_id":"<urn:uuid:6ac9109a-1311-4763-86ec-9da91954cbf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00036.warc.gz"} |
Reynolds Number Calculation
Reynolds Numbers
Pipes that have a smooth wall such as glass, copper, brass and polyethylene cause less fritional resistance and hence they produce a smaller frictional loss than those pipes with a greater internal
roughness, such as concrete, cast iron and steel.
The velocity profile of fluid flow in a pipe shows that the fluid at the centre of the stream moves more quickly than the fluid flow towards the edge of the stream. Therefore friction occurs between
layers within the fluid.
Fluids with a high viscosity flow more slowly and generally not produce eddy currents, thus the internal roughness of the pipe has little or no effect on the frictional resistance to flow in the
pipe. This condition is known as laminar flow.
Reynolds Number Calculation
The Reynolds number (Re) of a flowing fluid is calculated by multiplying the fluid velocity by the internal pipe diameter (to obtain the inertia force of the fluid) and then dividing the result by
the kinematic viscosity (viscous force per unit length).
Kinematic viscosity = dynamic viscosity/fluid density
Reynolds number = (Fluid velocity x Internal pipe diameter) / Kinematic viscosity
Laminar Flow in a Pipe
Laminar flow occurs when the calculated Reynolds number is less than 2300 and in this case the resistance to flow is independent of the pipe wall roughness.
Turbulent Flow in a Pipe
Turbulent flow occurs when the Reynolds number calculation exceeds 4000.
When Eddy currents occur within the flow, the ratio of the pipe's internal roughness to the internal diameter of the pipe needs to be considered to calculate the friction factor, which in turn is
used to calculate the friction loss that occurs.
For pipes with a small diameter, the internal roughness can have a major influence on the friction factor. For pipes with a large diameter the overall effect of the eddy currents are less
You can use this link to view information on the internal roughness for various pipe materials.
The relative roughness of the pipe and the Reynolds number can be used to plot the friction factor chart.
When flow occurs between the Laminar and Turbulent flow conditions (Re 2300 to Re 4000) the flow condition is known as critical and is difficult to predict. Here the flow is neither wholly laminar
nor wholly turbulent. It is a combination of the two flow conditions.
The Colebrook-White equation is used to calculate the friction factor for turbulent flow.
The friction factor is then used in the Darcy-Weisbach formula to calculate the fluid frictional loss in a pipe. | {"url":"https://www.pipeflow.com/pipe-pressure-drop-calculations/reynolds-numbers","timestamp":"2024-11-03T16:41:10Z","content_type":"text/html","content_length":"20559","record_id":"<urn:uuid:feee3b2a-1471-420e-a687-552d456c3c2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00598.warc.gz"} |
The Logic of Modern Physics
Percy Bridgman (1927)
The Logic of Modern Physics
Source: The Logic of Modern Physics (1927), publ. MacMillan (New York) Edition, 1927.
One of the most noteworthy movements in recent physics is a change of attitude toward what may be called the interpretative aspect of physics. It is being increasingly recognised, both in the
writings and the conversation of physicists, that the world of experiment is not understandable without some examination of the purpose of physics and of the nature of its fundamental concepts. It is
no new thing to attempt a more critical understanding of the nature of physics, but until recently all such attempts have been regarded with a certain suspicion or even sometimes contempt. The
average physicist is likely to deprecate his own concern with such questions, and is inclined to dismiss the speculations of fellow physicists with the epithet "metaphysical." This attitude has no
doubt had a certain justification in the utter unintelligibility to the physicist of many metaphysical speculations and the sterility of such speculations in yielding physical results. However, the
growing reaction favouring a better understanding of the interpretative fundamentals of physics is not a pendulum swing of the fashion of thought toward metaphysics, originating in the upheaval of
moral values produced by the great war, or anything of the sort, but is a reaction absolutely forced upon us by a rapidly increasing array of cold experimental facts.
This reaction, or rather new movement, was without doubt initiated by the restricted theory of relativity of Einstein. Before Einstein, an ever increasing number of experimental facts concerning
bodies in rapid motion required increasingly complicated modifications in our naive notions in order to preserve self-consistency, until Einstein showed that everything could be restored again to a
wonderful simplicity by a slight change in some of our fundamental concepts. The concepts which were most obviously touched by Einstein were those of space and time, and much of the writing
consciously inspired by Einstein has been concerned with these concepts. But that experiment compels a critique of much more than the concepts of space and time is made increasingly evident by all
the new facts being discovered in the quantum realm.
The situation presented to us by these new quantum facts is two-fold. In the first place, all these experiments are concerned with things so small as to be forever beyond the possibility of direct
experience, so that we have the problem of translating the evidence of experiment into other language. Thus we observe an emission line in a spectroscope and may infer an electron jumping from one
energy level to another in an atom. In the second place, we have the problem of understanding the translated experimental evidence. Now of course every one knows that this problem is making us the
greatest difficulty.
The experimental facts are so utterly different from those of our ordinary experience that not only do we apparently have to give up generalisations from past experience as broad as the field
equations of electrodynamics, for instance, but it is even being questioned whether our ordinary forms of thought are applicable in the new domain; it is often suggested, for example, that the
concepts of space and time break down.
The situation is rapidly becoming acute. Since I began writing this essay, there has been a striking increase in critical activity inspired by the new quantum mechanics of 1925-26, and it is common
to hear expositions of the new ideas prefaced by analysis of what experiment really gives to us or what our fundamental concepts really mean. The change in ideas is now so rapid that a number of the
statements of this essay are already antiquated as expressions of the best current opinion; however I have allowed these statements to stand, since the fundamental arguments are in nowise affected
and we have no reason to think that present best opinions are in any way final. We have the impression of being in an important formative period; if we are, the complexion of physics for a long time
in the future will be determined by our present attitude toward fundamental questions of interpretation. To meet this situation it seems to me that something more is needed than the hand-to-mouth
philosophy that is now growing up to meet special emergencies, something approaching more closely to a systematic philosophy of all physics which shall cover the experimental domains already
consolidated as well as those which are now making us so much trouble. It is the attempt of this essay to give a more or less inclusive critique of all physics. Our problem is the double one of
understanding what we are trying to do and what our ideals should be in physics, and of understanding the nature of the structure of physics as it now exists. These two ends are together furthered by
an analysis of the fundamental concepts of physics; an understanding of the concepts we now have discloses the present structure of physics and a realisation of what the concepts should be involves
the ideals of physics. This essay will be largely concerned with the fundamental concepts; it will appear that almost all the concepts can profit from re-examination.
The material of this essay is largely obtained by observation of the actual currents of opinion in physics; much of what I have to say is more or less common property and doubtless every reader will
find passages that he will feel have been taken out of his own mouth. On certain broad tendencies in present day physics, however, I have put my own interpretation, and it is more than likely that
this interpretation will be unacceptable to many. But even if not acceptable, I hope that the stimulus of combating the ideas offered here may be of value.
Certain limitations will have to be set to our inquiry in order to keep it within manageable compass. It is of course the merest truism that all our experimental knowledge and our understanding of
nature is impossible and non-existent apart from our own mental processes, so that strictly speaking no aspect of psychology or epistemology is without pertinence. Fortunately we shall be able to get
along with a more or less naive attitude toward many of these matters. We shall accept as significant our common sense judgment that there is a world external to us, and shall limit as far as
possible our inquiry to the behaviour and interpretation of this "external" world. We shall rule out inquiries into our states of consciousness as such. In spite, however, of the best intentions, we
shall not be able to eliminate completely considerations savouring of the metaphysical, because it is evident that the nature of our thinking mechanism essentially colours any picture that we can
form of nature, and we shall have to recognise that unavoidable characteristics of any outlook of ours are imposed in this way.
Chapter I
Broad Points of View
WHATEVER may be one's opinion as to our permanent acceptance of the analytical details of Einstein's restricted and general theories of relativity, there can be no doubt that through these theories
physics is permanently changed. It was a great shock to discover that classical concepts, accepted unquestioningly, were inadequate to meet the actual situation, and the shock of this discovery has
resulted in a critical attitude toward our whole conceptual structure which must at least in part be permanent. Reflection on the situation after the event shows that it should not have needed the
new experimental facts which led to relativity to convince us of the inadequacy of our previous concepts, but that a sufficiently shrewd analysis should have prepared us for at least the possibility
of what Einstein did.
Looking now to the future, our ideas of what external nature is will always be subject to change as we gain new experimental knowledge, but there is a part of our attitude to nature which should not
be subject to future change, namely that part which rests on the permanent basis of the character of our minds. It is precisely here, in an improved understanding of our mental relations to nature,
that the permanent contribution of relativity is to be found. We should now make it our business to understand so thoroughly the character of our permanent mental relations to nature that another
change in our attitude, such as that due to Einstein, shall be forever impossible. It was perhaps excusable that a revolution in mental attitude should occur once, because after all physics is a
young science, and physicists have been very busy, but it would certainly be a reproach if such a revolution should ever prove necessary again.
The first lesson of our recent experience with relativity is merely an intensification and emphasis of the lesson which all past experience has also taught, namely, that when experiment is pushed
into new domains, we must be prepared for new facts, of an entirely different character from those of our former experience. This is taught not only by the discovery of those unsuspected properties
of matter moving with high velocities, which inspired the theory of relativity, but also even more emphatically by the new facts in the quantum domain. To a certain extent, of course, the recognition
of all this does not involve a change of former attitude; the fast has always been for the physicist the one ultimate thing from which there is no appeal, and in the face of which the only possible
attitude is a humility almost religious. The new feature in the present situation is an intensified conviction that in reality new orders of experience do exist, and that we may expect to meet them
continually. We have already encountered new phenomena in going to high velocities, and in going to small scales of magnitude: we may similarly expect to find them, for example, in dealing with
relations of cosmic magnitudes, or in dealing with the properties of matter of enormous densities, such as is supposed to exist in the stars.
Implied in this recognition of the possibility of new experience beyond our present range, is the recognition that no element of a physical situation, no matter how apparently irrelevant or trivial,
may be dismissed as without effect on the final result until proved to be without effect by actual experiment.
The attitude of the physicist must therefore be one of pure empiricism. He recognises no a priori principles which determine or limit the possibilities of new experience. Experience is determined
only by experience. This practically means that we must give up the demand that all nature be embraced in any formula, either simple or complicated. It may perhaps turn out eventually that as a
matter of f act nature can be embraced in a formula, but we must so organise our thinking as not to demand it as a necessity.
Einstein's Contribution in Changing Our Attitude Toward Concepts
Recognising the essential unpredictability of experiment beyond our present range, the physicist, if he is to escape continually revising his attitude, must use in describing and correlating nature
concepts of such a character that our present experience does not exact hostages of the future. Now here it seems to me is the greatest contribution of Einstein. Although he himself does not
explicitly state or emphasise it, I believe that a study of what he has done will show that he has essentially modified our view of what the concepts useful in physics are and should be. Hitherto
many of the concepts of physics have been defined in terms of their properties. An excellent example is afforded by Newton's concept of absolute time. The following quotation from the Scholium in
Book I of the Principia is illuminating:
I do not define Time, Space, Place or Motion, as being well known to all. Only I must observe that the vulgar conceive those quantities under no other notions but from the relation they bear to
sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into Absolute and Relative, True and Apparent, Mathematical and Common.
(1) Absolute, True, and Mathematical Time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called Duration.
Now there is no assurance whatever that there exists in nature anything with properties like those assumed in the definition, and physics, when reduced to concepts of this character, becomes as
purely an abstract science and as far removed from reality as the abstract geometry of the mathematicians, built on postulates. It is a task for experiment to discover whether concepts so defined
correspond to anything in nature, and we must always be prepared to find that the concepts correspond to nothing or only partially correspond. In particular, if we examine the definition of absolute
time in the light of experiment, we find nothing in nature with such properties.
The new attitude toward a concept is entirely different. We may illustrate by considering the concept of length: what do we mean by the length of an object? We evidently know what we mean by length
if we can tell what the length of any and every object is, and for the physicist nothing more is required. To find the length of an object, we have to perform certain physical operations. The concept
of length is therefore fixed when the operations by which length is measured are fixed: that is, the concept of length involves as much as and nothing more than the set of operations by which length
is determined. In general, we mean by any concept nothing more than a set of operations; the concept is synonymous with a corresponding set of operations. If the concept is physical, as of length,
the operations are actual physical operations, namely, those by which length is measured; or if the concept is mental, as of mathematical continuity, the operations are mental operations, namely
those by which we determine whether a given aggregate of magnitudes is continuous. It is not intended to imply that there is a hard and fast division between physical and mental concepts, or that one
kind of concept does not always contain an element of the other; this classification of concept is not important for our future considerations.
We must demand that the set of operations equivalent to any concept be a unique set, for otherwise there are possibilities of ambiguity in practical applications which we cannot admit.
Applying this idea of "concept" to absolute time, we do not understand the meaning of absolute time unless we can tell how to determine the absolute time of any concrete event, i.e., unless we can
measure absolute time. Now we merely have to examine any of the possible operations by which we measure time to see that all such operations are relative operations. Therefore the previous statement
that absolute time does not exist is replaced by the statement that absolute time is meaningless. And in making this statement we are not saying something new about nature, but are merely bringing to
light implications already contained in the physical operations used in measuring time.
It is evident that if we adopt this point of view toward concepts, namely that the proper definition of a concept is not in terms of its properties but in terms of actual operations, we need run no
danger of having to revise our attitude toward nature. For if experience is always described in terms of experience, there must always be correspondence between experience and our description of it,
and we need never be embarrassed, as we were in attempting to find in nature the prototype of Newton's absolute time. Furthermore, if we remember that the operations to which a physical concept are
equivalent are actual physical operations, the concepts can be defined only in the range of actual experiment, and are undefined and meaningless in regions as yet untouched by experiment. It follows
that strictly speaking we cannot make statements at all about regions as yet untouched, and that when we do make such statements, as we inevitably shall, we are making a conventionalised
extrapolation, of the looseness of which we must be fully conscious, and the justification of which is in the experiment of the future.
There probably is no statement either in Einstein or other writers that the change described above in the use of "concept" has been self-consciously made, but that such is the case is proved. I
believe, by an examination of the way concepts are now handled by Einstein and others. For of course the true meaning of a term is to be found by observing what a man does with it, not by what he
says about it. We may show that this is the actual sense in which concept is coming to be used by examining in particular Einstein's treatment of simultaneity.
Before Einstein, the concept of simultaneity was defined in terms of properties. It was a property of two events, when described with respect to their relation in time, that one event was either
before the other, or after it, or simultaneous with it. Simultaneity was a property of the two events alone and nothing else; either two events were simultaneous or they were not. The justification
for using this term in this way was that it seemed to describe the behaviour of actual things. But of course experience then was restricted to a narrow range. When the range of experience was
broadened, as by going to high velocities, it was found that the concepts no longer applied, because there was no counterpart in experience for this absolute relation between two events. Einstein now
subjected the concept of simultaneity to a critique, which consisted essentially in showing that the operations which enable two events to be described as simultaneous involve measurements on the two
events made by an observer, so that "simultaneity" is, therefore, not an absolute property of the two events and nothing else, but must also involve the relation of the events to the observer. Until
therefore we have experimental proof to the contrary, we must be prepared to find that the simultaneity of two events depends on their relation to the observer, and in particular on their velocity.
Einstein, in thus analysing what is involved in making a judgment of simultaneity, and in seizing on the act of the observer as the essence of the situation, is actually adopting a new point of view
as to what the concepts of physics should be, namely, the operational view.
Of course Einstein actually went much further than this, and found precisely how the operations for judging simultaneity change when the observer moves, and obtained quantitative expressions for the
effect of the motion of the observer on the relative time of two events. We may notice, parenthetically, that there is much freedom of choice in selecting the exact operations; those which Einstein
chose were determined by convenience and simplicity with relation to light beams. Entirely apart from the precise quantitative relations of Einstein's theory, however, the important point for us is
that if we had adopted the operational point of view, we would, before the discovery of the actual physical facts, have seen that simultaneity is essentially a relative concept, and would have left
room in our thinking for the discovery of such effects as were later found.
Detailed Discussion of the Concept of Length
We may now gain further familiarity with the operational attitude toward a concept and some of its implications by examining from this point of view the concept of length. Our task is to find the
operations by which we measure the length of any concrete physical object. We begin with objects of our commonest experience, such as a house or a house lot. What we do is sufficiently indicated by
the following rough description. We start with a measuring rod, lay it on the object so that one of its ends coincides with one end of the object, mark on the object the position of the other end of
the rod, then move the rod along in a straight line extension of its previous position until the first end coincides with the previous position of the second end, repeat this process as often as we
can, and call the length the total number of times the rod was applied. This procedure, apparently so simple, is in practice exceedingly complicated, and doubtless a full description of all the
precautions that must be taken would fill a large treatise. We must, for example, be sure that the temperature of the rod is the standard temperature at which its length is defined, or else we must
make a correction for it; or we must correct for the gravitational distortion of the rod if we measure a vertical length; or we must be sure that the rod is not a magnet or is not subject to
electrical forces. All these precautions would occur to every physicist. But we must also go further and specify all the details by which the rod is moved from one position to the next on the object
its precise path through space and its velocity and acceleration in getting from one position to another. Practically of course, precautions such as these are not mentioned, but the justification is
in our experience that variations of procedure of this kind are without effect on the final result. But we always have to recognise that all our experience is subject to error, and that at some time
in the future we may have to specify more carefully the acceleration, for example, of the rod in moving from one position to another, if experimental accuracy should be so increased as to show a
measurable effect. In principle the operations by which length is measured should be uniquely specified. If we have more than one set of operations, we have more than one concept, and strictly there
should be a separate name to correspond to each different set of operations.
So much for the length of a stationary object, which is complicated enough. Now suppose we have to measure a moving street car. The simplest, and what we may call the "naive" procedure, is to board
the car with our meter stick and repeat the operations we would apply to a stationary body. Notice that this procedure reduces to that already adopted in the limiting case when the velocity of the
street car vanishes. But here there may be new questions of detail. How shall we jump on to the car with our stick in hand? Shall we run and jump on from behind, or shall we let it pick us up from in
front? Or perhaps does now the material of which the stick is composed make a difference, although previously it did not? All these questions must be answered by experiment. We believe from present
evidence that it makes no difference how we jump on to the car, or of what material the rod is made, and that the length of the car found in this way will be the same as if it were at rest. But the
experiments are more difficult, and we are not so sure of our conclusions as before. Now there are very obvious limitations to the procedure just given. If the street car is going too fast, we can
not board it directly, but must use devices, such as getting on from a moving automobile; and, more important still, there are limitations to the velocity that can be given to street cars or to meter
sticks by any practical means in our control, so that the moving bodies which can be measured in this way are restricted to a low range of velocity. If we want to be able to measure the length of
bodies moving with higher velocities such as we find existing in nature (stars or cathode particles), we must adopt another definition and other operations for measuring length, which also reduce to
the operations already adopted in the static case. This is precisely what Einstein did. Since Einstein's operations were different from our operations above, his "length" does not mean the same as
our "length." We must accordingly be prepared to find that the length of a moving body measured by the procedure of Einstein is not the same as that above; this of course is the fact, and the
transformation formulas of relativity give the precise connection between the two lengths.
Einstein's procedure for measuring the length of bodies in motion was dictated not only by the consideration that it must be applicable to bodies with high velocities, but also by mathematical
convenience, in that Einstein describes the world mathematically by a system of coördinate geometry, and the "length" of an object is connected simply with quantities in the analytic equations.
It is of interest to describe briefly Einstein's actual operations for measuring the length of a body in motion; it will show how operations which may be simple from a mathematical point of view may
appear complicated from a physical viewpoint. The observer who is to measure the length of a moving object must first extend over his entire plane of reference (for simplicity the problem is
considered two-dimensional) a system of time coördinates, i.e., at each point of his plane of reference there must be a clock, and all these clocks must be synchronised. At each clock an observer
must be situated. Now to find the length of the moving object at a specified instant of time (it is a subject for later investigation to find whether its length is a function of time), the two
observers who happen to coincide in position with the two ends of the object at the specified time on their clocks are required to find the distance between their two positions by the procedure for
measuring the length of a stationary object, and this distance is by definition the length of the moving object in the given reference system. This procedure for measuring the length of a body in
motion hence involves the idea of simultaneity, through the simultaneous position of the two ends of the rod, and we have seen that the operations by which simultaneity are determined are relative,
changing when the motion of the system changes. We hence are prepared to find a change in the length of a body when the velocity of the measuring system changes, and this in fact is what happens. The
precise numerical dependence is worked out by Einstein, and involves other considerations, in which we are not interested at present.
The two sorts of length, the naive one and that of Einstein, have certain features in common. In either case in the limit, as the velocity of the measuring system approaches zero, the operations
approach those for measuring the length of a stationary object. This, of course, is a requirement in any good definition, imposed by considerations of convenience, and it is too obvious a matter to
need elaboration. Another feature is that the operations equivalent to either concept both involve the motion of the system, so that we must recognise the possibility that the length of a moving
object may be a function of its velocity. It is a matter of experiment, unpredictable until tried, that within the limits of present experimental error the naive length is not affected by motion, and
Einstein's length is.
So far, we have extended the concept of length in only one way beyond the range of ordinary experience, namely to high velocities. The extension may obviously be made in other directions. Let us
inquire what are the operations by which we measure the length of a very large object. In practice we probably first meet the desirability of a change of procedure in measuring large pieces of land.
Here our procedure depends on measurements with a surveyor's theodolite. This involves extending over the surface of the land a system of coördinates, starting from a base line measured with a tape
in the conventional way, sighting on distant points from the extremities of the line, and measuring the angles. Now in this extension we have made one very essential change: the angles between the
lines connecting distant points are now angles between beams of light. We assume that a beam of light travels in a straight line. Furthermore, we assume in extending our system of triangulation over
the surface of the earth that the geometry of light beams is Euclidean. We do the best we can to check the assumptions, but at most can never get more than a partial check. Thus Gauss checked whether
the angles of a large terrestrial triangle add to two right angles and found agreement within experimental error. We now know from the experiments of Michelson that if his measurements had been
accurate enough he would not have got a check, but would have had an excess or defect according to the direction in which the beam of light travelled around the triangle with respect to the rotation
of the earth. But if the geometry of light beams is Euclidean, then not only must the angles of a triangle add to two right angles, but there are definite relations between the lengths of the sides
and the angles, and to check these relations the sides should be measured by the old procedure with a meter stick. Such a check on a large scale has never been attempted, and is not feasible. It
seems, then, that our checks on the Euclidean character of optical space are all of restricted character. We have apparently proved that up to a certain scale of magnitude optical space is Euclidean
with respect to measures of angle, but this may not necessarily involve that space is also Euclidean with respect to measures of length, so that space need not be completely Euclidean. There is a
further most important restriction in that our studies of non-Euclidean geometry have shown that the percentage excess of the angles of a non-Euclidean triangle over 180° may depend on the magnitude
of the triangle, so that it may well be that we have not detected the non-Euclidean character of space simply because our measurements have not been on a large enough scale.
We thus see that the concept of length has undergone a very essential change of character even within the range of terrestrial measurements, in that we have substituted for what I may call the
tactual concept an optical concept, complicated by an assumption about the nature of our geometry. From a very direct concept we have come to a very indirect concept with a most complicated set of
operations. Strictly speaking, length when measured in this way by light beams should be called by another name, since the operations are different. The practical justification for retaining the same
name is that within our present experimental limits a numerical difference between the results of the two sorts of operations has not been detected.
We are still worse off when we make the extension to solar and stellar distances. Here space is entirely optical in character, and we never have an opportunity of even partially comparing tactual
with optical space. No direct measures of length have ever been made, nor can we even measure the three angles of a triangle and so check our assumption that the use of Euclidean geometry in
extending the concept of space is justified. We never have under observation more than two angles of a triangle, as when we measure the distance of the moon by observation from the two ends of the
earth's diameter. To extend to still greater distance our measures of length, we have to make still further assumptions, such as that inferences from the Newtonian laws of mechanics are valid. The
accuracy of our inferences about lengths from such measurements is not high. Astronomy is usually regarded as a science of extraordinarily high accuracy, but its accuracy is very restricted in
character, namely to the measurement of angles. It is probably safe to say that no astronomical distance, except perhaps that of the moon, is known with an accuracy greater than 0.19. When we push
our estimates to distances beyond the confines of the solar system in which we are assisted by the laws of mechanics, we are reduced in the first place to measurements of parallax, which at best have
a quite inferior accuracy, and which furthermore fail entirely outside a rather restricted range. For greater stellar distances we are driven to other and much rougher estimates, resting for instance
on the extension to great distances of connections found within the range of parallax between brightness and spectral type of a star, or on such assumptions as that, because a group of stars looks as
if it were all together in space and had a common origin, it actually is so. Thus at greater and greater distances not only does experimental accuracy become less, but the very nature of the
operations by which length is to be determined becomes indefinite, so that the distances of the most remote stellar objects as estimated by different observers or by different methods may be very
divergent. A particular consequence of the inaccuracy of the astronomical measures of great distances is that the question of whether large scale space is Euclidean or not is merely academic.
We thus see that in the extension from terrestrial to great stellar distances the concept of length has changed completely in character. To say that a certain star is 105 light years distant is
actually and conceptually an entire different kind of thing from saying that a certain goal post is 100 meters distant. Because of our conviction that the character of our experience may change when
the range of phenomena changes, we feel the importance of such a question as whether the space of distances of 10 5 light years is Euclidean or not, and are correspondingly dissatisfied that at
present there seems no way of giving meaning to it.
We encounter difficulties similar to those above, and are also compelled to modify our procedures, when we go to small distances. Down to the scale of microscopic dimensions a fairly straightforward
extension of the ordinary measuring procedure is sufficient, as when we measure a length in a micrometer eyepiece of a microscope. This is of course a combination of tactual and optical measurements,
and certain assumptions, justified as far as possible by experience, have to be made about the behaviour of light beams. These assumptions are of a quite different character from those which give us
concern on the astronomical scale, because here we meet difficulty from interference effects due to the finite scale of the structure of light, and are not concerned with a possible curvature of
light beams in the long reaches of space. Apart from the matter of convenience, we might also measure small distances by the tactual method.
As the dimensions become smaller, certain difficulties become increasingly important that were negligible on a larger scale. In carrying out physically the operations equivalent to our concepts,
there are a host of practical precautions to be taken which could be explicitly enumerated with difficulty, but of which nevertheless any practical physicist is conscious. Suppose, for example, we
measure length tactually by a combination of Johanssen gauges. In piling these together, we must be sure that they are clean, and are thus in actual contact. Particles of mechanical dirt first engage
our attention. Then as we go to smaller dimensions we perhaps have to pay attention to adsorbed films of moisture, then at still smaller dimensions to adsorbed films of gas, until finally we have to
work in a vacuum, which must be the more nearly complete the smaller the dimensions. About the time that we discover the necessity for a complete vacuum, we discover that the gauges themselves are
atomic in structure, that they have no definite boundaries, and therefore no definite length, but that the length is a hazy thing, varying rapidly in time between certain limits. We treat this
situation as best we can by taking a time average of the apparent positions of the boundaries, assuming that along with the decrease of dimensions we have acquired a corresponding extravagant
increase in nimbleness. But as the dimensions get smaller continually, the difficulties due to this haziness increase indefinitely in percentage effect, and we are eventually driven to give up
altogether. We have made the discovery that there are essential physical limitations to the operations which defined the concept of length. [We perhaps do not regard the substitution of optical for
tactual space on the astronomical scale as compelled by the same sort of physical necessity, because I suppose the possible eventual landing of men in the moon will always be one of the dreams of
humanity.] At the same time that we have come to the end of our rope with our Johanssen gauge procedure, our companion with the microscope has been encountering difficulties due to the finite wave
length of light; this difficulty he has been able to minimise by using light of progressively shorter wave lengths, but he has eventually had to stop on reaching X-rays. Of course this optical
procedure with the microscope is more convenient, and is therefore adopted in practice.
Let us now see what is implied in our concept of length extended to ultramicroscopic dimensions. What, for instance, is the meaning of the statement that the distance between the planes of atoms in a
certain crystal is 3 x 10^-1 cm.? What we would like to mean is that 1/3 x 10^8 of these planes piled on top of each other give a thickness of 1 cm.; but of course such a meaning is not the actual
one. The actual meaning may be found by examining the operations by which we arrived at the number 3 x 10^-8. As a matter of fact, 3 x 10^-8 was the number obtained by solving a general equation
derived from the wave theory of light, into which certain numerical data obtained by experiments with X-rays had been substituted. Thus not only has the character of the concept of length changed
from tactual to optical, but we have gone much further in committing ourselves to a definite optical theory. If this were the whole story, we would be most uncomfortable with respect to this branch
of physics, because we are so uncertain of the correctness of our optical theories, but actually a number of checks can be applied which greatly restore our confidence. For instance, from the density
of the crystal and the grating space, the weight of the individual atoms may be computed, and these weights may then be combined with measurements of the dimensions of other sorts of crystal into
which the same atoms enter to give values of the densities of these crystals, which may be checked against experiment. All such checks have succeeded within limits of accuracy which are fairly high.
It is important to notice that, in spite of the checks, the character of the concept is changing, and begins to involve such things as the equations of optics and the assumption of the conservation
of mass.
We are not content, however, to stop with dimensions of atomic order, but have to push on to the electron with a diameter of the order of 10^-12 cm. What is the possible meaning of the statement that
the diameter of an electron is 10^-13 cm.? Again the only answer is found by examining the operations by which the number 10^-13 was obtained. This number came by solving certain equations derived
from the field equations of electrodynamics, into which certain numerical data obtained by experiment had been substituted. The concept of length has therefore now been so modified as to include that
theory of electricity embodied in the field equations, and, most important, assumes the correctness of extending these equations from the dimensions in which they may be verified experimentally into
a region in which their correctness is one of the most important and problematical of present-day questions in physics. To find whether the field equations are correct on a small scale, we must
verify the relations demanded by the equations between the electric and magnetic forces and the space coördinates, to determine which involves measurement of lengths. But if these space coördinates
cannot be given an independent meaning apart from the equations, not only is the attempted verification of the equations impossible, but the question itself is meaningless. If we stick to the concept
of length by itself, we are landed in a vicious circle. As a matter of fact, the concept of length disappears as an independent thing, and fuses in a complicated way with other concepts, all of which
are themselves altered thereby, with the result that the total number of concepts used in describing nature at this level is reduced in number. A precise analysis of the situation is difficult, and I
suppose has never been attempted, but the general character of the situation is evident. Until at least a partial analysis is attempted, I do not see how any meaning can be attached to such questions
as whether space is Euclidean in the small scale.
It is interesting to observe that any increased accuracy in knowledge of large scale phenomena must, as far as we now can see, arise from an increase in the accuracy of measurement of small things,
that is, in the measurement of small angles or the analysis of minute differences of wave lengths in the spectra. To know the very large takes us into the same field of experiment as to know the very
small, so that operationally the large and the small have features in common.
This somewhat detailed analysis of the concept of length brings out features common to all our concepts. If we deal with phenomena outside the domain in which we originally defined our concepts, we
may find physical hindrances to performing the operations of the original definition, so that the original operations have to be replaced by others. These new operations are, of course, to be so
chosen that they give, within experimental error, the same numerical results in the domain in which the two sets of operations may be both applied; but we must recognise in principle that in changing
the operations we have really changed the concept, and that to use the same name for these different concepts over the entire range is dictated only by considerations of convenience, which may
sometimes prove to have been purchased at too high a price in terms of unambiguity. We must always be prepared some day to find that an increase in experimental accuracy may show that the two
different sets of operations which give the same results in the more ordinary part of the domain of experience, lead to measurably different results in the more unfamiliar parts of the domain. We
must remain aware of these joints in our conceptual structure if we hope to render unnecessary the services of the unborn Einsteins.
The second feature common to all concepts brought out by the detailed discussion of length is that, as we approach the experimentally attainable limit, concepts lose their individuality, fuse
together, and become fewer in number, as we have seen that at dimensions of the order of the diameter of an electron the concepts of length and the electric field vectors fuse into an amorphous
whole. Not only does nature as experienced by us become different in character on its horizons, but it becomes simpler, and therefore our concepts, which are the building stones of our descriptions,
become fewer in number. This seems to be an entirely natural state of affairs. How the number of concepts is often kept formally the same as we approach the horizon will be discussed later in special
A precise analysis of our conceptual structure has never been attempted, except perhaps in very restricted domains, and it seems to me that there is room here for much important future work. Such an
analysis is not to be attempted in this essay, but only some of the more important qualitative aspects are to be pointed out. It will never be possible to give a clean-cut logical analysis of the
conceptual situation, for the nature of our concepts, according to our operational point of view, is the same as the nature of experimental knowledge, which is often hazy. Thus in the transition
regions where nature is getting simpler and the number of operationally independent concepts changes, a certain haziness is inevitable, for the actual change in our conceptual structure in these
transition regions is continuous, corresponding to the continuity of our experimental knowledge, whereas formally the number of concepts should be an integer.
The Relative Character of Knowledge
Two other consequences of the operational point of view must now be examined. First is the consequence that all our knowledge is relative. This may be understood in a general or a more particular
sense. The general sense is illustrated in Haldane's book on the Reign of Relativity. Relativity in the general sense is the merest truism if the operational definition of concept is accepted, for
experience is described in terms of concepts, and since our concepts. are constructed of operations, all our knowledge must unescapably be relative to the operations selected. But knowledge is also
relative in a narrower sense, as when we say there is no such thing as absolute rest (or motion) or absolute size, but rest and size are relative terms. Conclusions of this kind are involved in the
specific character of the operations in terms of which rest or size are defined. An examination of the operations by which we determine whether a body is at rest or in motion shows that the
operations are relative operations: rest or motion is determined with respect to some other body selected as the standard. In saying that there is no such thing as absolute rest or motion we are not
making a statement about nature in the sense that might be supposed, but we are merely making a statement about the character of our descriptive processes. Similarly with regard to size: examination
of the operations of the measuring process shows that size is measured relative to the fundamental measuring rod.
The "absolute" therefore disappears in the original meaning of the word. But the "absolute" may usefully return with an altered meaning, and we may say that a thing has absolute properties if the
numerical magnitude is the same when measured with the same formal procedure by all observers. Whether a given property is absolute or not can be determined only by experiment, landing us in the
paradoxical position that the absolute is absolute only relative to experiment. In some cases, the most superficial observation shows that a property is not absolute, as, for example, it is at once
obvious that measured velocity changes with the motion of the observer. But in other cases the decision is more difficult. Thus Michelson thought he had an absolute procedure for measuring length, by
referring to the wave length of the red cadmium line as standard, it required difficult and accurate experiment to show that this length varies with the motion of the observer. Even then, by changing
the definition of the length of a moving object, we believe that length might be made to reassume its desired absolute character.
To stop the discussion at this point might leave the impression that this observation of the relative character of knowledge is of only a very tenuous and academic interest, since it appears to be
concerned mostly with the character of our descriptive processes, and to say little about external nature. [What this means we leave to the metaphysician to decide.] But I believe there is a deeper
significance to all this. It must be remembered that all our argument starts with the concepts as given. Now these concepts involve physical operations; in the discovery of what operations may be
usefully employed in describing nature is buried almost all physical experience. In erecting our structure of physical science, we are building on the work of all the ages. There is then this purely
physical significance in the statement that all motion is relative, namely that no operations of measuring motion have been found to be useful in describing simply the behaviour of nature which are
not operations relative to a single observer; in making this statement we are stating something about nature. It takes an enormous amount of real physical experience to discover relations of this
sort. The discovery that the number obtained by counting the number of times a stick may be applied to an object can be simply used in describing natural phenomena was one of the most important and
fundamental discoveries ever made by man.
Meaningless Questions
Another consequence of the operational character of our concepts, almost a corollary of that considered above, is that it is quite possible, nay even disquietingly easy, to invent expressions or to
ask questions that are meaningless. It constitutes a great advance in our critical attitude toward nature to realize that a great many of the questions that we uncritically ask are without meaning.
If a specific question has meaning, it must be possible to find operations by which an answer may be given to it. It will be found in many cases that the operations cannot exist, and the question
therefore has no meaning. For instance, it means nothing to ask whether a star is at rest or not. Another example is a question proposed by Clifford, namely, whether it is not possible that as the
solar system moves from one part of space to another the absolute scale of magnitude may be changing, but in such a way as to affect all things equally, so that the change of scale can never be
detected. An examination of the operations by which length is measured in terms of measuring rods shows that the operations do not exist (because of the nature of our definition of length) for
answering the question. The question can be given meaning only from the point of view of some imaginary superior being watching from an external point of vantage. But the operations by which such a
being measures length are different from the operations of our definition of length, so that the question acquires meaning only by changing the significance of our terms in the original sense the
question means nothing.
To state that a certain question about nature is meaningless is to make a significant statement about nature itself, because the fundamental operations are determined by nature, and to state that
nature cannot be described in terms of certain operations is a significant statement.
It must be recognised, however, that there is a sense in which no serious question is entirely without meaning, because doubtless the questioner had in mind some intention in asking the question. But
to give meaning in this sense to a question, one must inquire into the meaning of the concepts as used by the questioner, and it will often be found that these concepts can be defined only in terms
of fictitious properties, as Newton's absolute time was defined by its properties, so that the meaning to be ascribed to the question in this way has no connection with reality. I believe that it
will enable us to make more significant and interesting statements, and therefore will be more useful, to adopt exclusively the operational view, and so admit the possibility of questions entirely
without meaning.
This matter of meaningless questions is a very subtle thing which may poison much more of our thought than that dealing with purely physical phenomena. I believe that many of the questions asked
about social and philosophical subjects will be found to be meaningless when examined from the point of view of operations. It would doubtless conduce greatly to clarity of thought if the operational
mode of thinking were adopted in all fields of inquiry as well as in the physical. Just as in the physical domain, so in other domains, one is making a significant statement about his subject in
stating that a certain question is meaningless.
In order to emphasise this matter of meaningless questions, I give here a list of questions, with which the reader may amuse himself by finding whether they have meaning or not.
(1) Was there ever a time when matter did not exist ?
(2) May time have a beginning or an end?
(3) Why does time flow?
(4) May space be bounded?
(5) May space or time be discontinuous?
(6) May space have a fourth dimension, not directly detectible, but given indirectly by inference?
(7) Are there parts of nature forever beyond our detection?
(8) Is the sensation which I call blue really the same as that which my neighbour calls blue? Is it possible that a blue object may arouse in him the same sensation that a red object does in me and
vice versa?
(9) May there be missing integers in the series of natural numbers as we know them?
(10) Is a universe possible in which 2+2 = 4?
(11) Why does negative electricity attract positive?
(12) Why does nature obey laws?
(13) Is a universe possible in which the laws are different?
(14) If one part of our universe could be completely isolated from the rest, would it continue to obey the same laws?
(15) Can we be sure that our logical processes are valid?
To adopt the operational point of view involves much more than a mere restriction of the sense in which we understand "concept," but means a far-reaching change in all our habits of thought, in that
we shall no longer permit ourselves to use as tools in our thinking concepts of which we cannot give an adequate account in terms of operations. In some respects thinking becomes simpler, because
certain old generalisations and idealisations become incapable of use; for instance, many of the speculations of the early natural philosophers become simply unreadable. In other respects, however,
thinking becomes much more difficult, because the operational implications of a concept are often very involved. For example, it is most difficult to grasp adequately all that is contained in the
apparently simple concept of "time," and requires the continual correction of mental tendencies which we have long unquestioningly accepted.
Operational thinking will at first prove to be an unsocial virtue; one will find oneself perpetually unable to understand the simplest conversation of one's friends, and will make oneself universally
unpopular by demanding the meaning of apparently the simplest terms of every argument. Possibly after every one has schooled himself to this better way, there will remain a permanent unsocial
tendency, because doubtless much of our present conversation will then become unnecessary. The socially optimistic may venture to hope, however, that the ultimate effect will be to release one's
energies for more stimulating and interesting interchange of ideas.
Not only will operational thinking reform the social art of conversation, but all our social relations will be liable to reform. Let any one examine in operational terms any popular present-day
discussion of religious or moral questions to realize the magnitude of the reformation awaiting us. Wherever we temporise or compromise in applying our theories of conduct to practical life we may
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Quantitative SAR (QSAR) Analysis | Quizgecko
Quantitative SAR (QSAR) Analysis
Document Details
Uploaded by SaintlyChaparral
QSAR drug design physicochemical properties biological activity
This document describes Quantitative Structure-Activity Relationship (QSAR) methods to determine how physicochemical properties of a drug impact its biological activity. It explores various
parameters like hydrophobicity (log P), steric effects, and electronic effects and how they can be predicted to improve drug design strategies.
Full Transcript
Quantitative SAR (QSAR) QSAR studies attempt to identify/quantify physicochemical properties of a drug to establish whether these properties have an effect on the drug’s biological activity If
there are relationships between biological activity and a physicochemical property, it might be possibl...
Quantitative SAR (QSAR) QSAR studies attempt to identify/quantify physicochemical properties of a drug to establish whether these properties have an effect on the drug’s biological activity If
there are relationships between biological activity and a physicochemical property, it might be possible to describe this relationship with an equation Such equation can then be used to predict
whether a novel molecule has biological activity Two advantages: Saves effort/time in synthesizing molecules that are predicted to have poor activities If a bioactive molecule is found that
does not fit the equation, it implies that some other feature or property is important (starting point for further drug development) In practice it is best to focus on two (or three)
physicochemical properties in QSAR, because relationships (equations) should be established with molecules that vary in one property at a time while the other ones should remain roughly constant
(note: that is easier said than done). 1 In the simplest case, molecules are produced that vary in one parameter (e.g., log P) The biological activity is typically expressed as log (1/C) where C
= concentration of the molecule required to achieve a defined level of biological activity In the example below, the relationship appears to be linear The equation describing this relationship is
therefore: k1 and k2 are constants Notes: The data should be fit using various equations (linear, parabolic, exponential, …) using non‐linear least‐squares regression The trendline denotes the
best fit to the linear relationship The error or standard deviations are important (R value) since large errors would make reasonable predictions for new/unsynthesized molecules difficult 2
Physicochemical parameters used in QSAR Hydrophobicity The hydrophobic character of a drug‐like molecule is important when it comes to crossing membranes and to interactions with a target One of
the QSAR parameters is the partition coefficient, P P can be determined for a variety of similar compounds with various substituents If the log P range into which these compounds fall is small
(e.g., from 1 to 3), a linear relationship is obtained Example: Binding of 42 drugs to human serum albumin follows the following linear equation In this study, the log P values ranged from 0.8 to
3.8. 3 If the hydrophobicity is increased (beyond log P = 4), a drug might not be soluble or might remain in the lipid bilayer This would imply that the biological activity must decrease beyond a
certain log P value (which implies that the function cannot be linear until infinity) In reality, a parabolic function is often obtained (with the maximum denoting the highest biological activity)
dominated by dominated by Example: Equation for the general anaesthetic properties of ethers: ca. 2.3 for general anaesthetics 4 General anaesthetics need to pass the blood brain barrier to get
into the CNS It was found that a log P of 2.3 affords the highest biological activities For example, the log P values for ether (least effective), chloroform and halothane (most effective) are
0.98, 1.97, and 2.3 Example: Changing log P can remove CNS side effects: Compound (I) is a cardiotonic* agent that has the side effect of producing bright visions → related to the drug entering the
CNS. The log P value is 2.6. Replacing the methoxy group with the similarly‐sized, but more polar SOMe group decreases log P to 1.2 Compound (II) does not display CNS side effects (too polar to
enter the CNS) *cardiotonic agents improve heart muscle contraction (improved blood flow) 5 Log P requires experimental determination (i.e., compounds need to be synthesized and then measured) It
is possible to estimate log P values using substituent hydrophobicity constants () is a measure of how hydrophobic a substituent is relative to hydrogen (H) values are experimentally
determined for a standard compound such as benzene, with and without a variety of substituents (X) using > 0: substituent X is more hydrophobic than H < 0: substituent X is less hydrophobic than
H 6 For a lead compound, log P needs to be determined experimentally, but when the lead compound is modified with different substituents, their log P values can be calculated (ClogP) based on
values Example: = 2.13 + 0.71 – 1.49 = 1.35 (exp: 1.5) (CONH2) = 0.64 – 2.13 = – 1.49 7 Substituent electronic effects Electronic effects can have profound effects on a drug’s polarity or
ionization For aromatic substituents, electronic effects are expressed through the Hammett substituent constant () is a measure of the electron‐donating or withdrawing ability of the
substituent Example: benzoic acid >0 0) Final note on aliphatic substituents: Hammett parameters are determined by measuring the effect of substituents on the rate of ester hydrolysis
Resonance does not play a role with aliphatic substituents! 11 Steric effects Size and shape can influence how a drug binds to its target Steric properties are more difficult to quantify because
a bulky substituent may increase or decrease target affinity There are a variety of approaches to treat steric effects Taft’s steric factor (ES): Determined by comparing rates of hydrolysis of
substituted (X) aliphatic esters against a methylester (reference; k0) 12 Another measure of steric effects is the molar refractivity (MR) n = refractive index; MW = molecular weight; d = density
MR correlates to the volume occupied by an atom or group Correction factor defining how easily a substituent can be polarized Defines the volume 13 The Hansch Equation The biological activity of
most drugs is related to a combination of physicochemical properties The Hansch equation incorporates multiple properties (previously discussed) including log P, , and a steric factor
Examples: (For log P values covering a larger range; parabolic behaviour) Example: QSAR equation for no dependence on steric effects! Inhibitors of adrenergic activity 14 The Craig Plot The Craig
Plot is a good visualization tool for and values (instead of looking at extensive tables) The plot on the right shows the values for para‐ aromatic substituents Advantages of using the plot:
The plot shows that there is no relationship between and values (they are spread over all quadrants) It is easy to see which substituents have similar or values (see red and blue lines) 15
The Craig Plot The plot is useful for planning QSAR studies (in general analogues with and values from all four quadrants should be synthesized to get to the most accurate equations) After
the derivation of the Hansch equation, it is easy to see whether and values should be positive or negative to improve biological activity (helps in lead optimization) Final note: Craig plots
can be made to compare other parameters (e.g., hydrophobicity and ES or MR) 16 The Craig Plot for meta aromatic substituents 17 The Topliss Scheme Topliss scheme for aromatic substituents Sometimes
it is not feasible to generate a large enough number of compounds to obtain a Hansch equation (e.g., difficult syntheses) In this case, it is possible to follow a flow diagram (Topliss scheme) to
synthesize a compound, then analyze its biological activity, and then plan the next synthesis, and so on. Topliss scheme for aliphatic substituents There are two general Topliss schemes (one for
aromatic substituents, one for aliphatic ones) 18 Topliss scheme for aromatic substituents M = more activity E = equally active L = lower activity The scheme assumes that a lead compound has
biological activity and contains a monosubstituted aromatic ring The starting point is to synthesize the p‐Cl derivative (Cl is more hydrophobic and electron‐withdrawing than H; i.e., and
values are positive) When the p‐Cl derivative is tested for activity, there are 3 possible outcomes: L, E or M 19 Topliss scheme for aromatic substituents M = more activity E = equally active L =
lower activity The possible outcome of the testing of the p‐Cl derivative (L, E or M) determines which path to follow for the next synthesis e.g., for M: The next step would be to add another
chlorine atom in the 3‐position to see whether even more positive and values enhance the biological effect If that is not the case (L or E branch), it is possible that steric effects or
excessive hydrophobicity play a role This can now be tested by making 4‐CF3 or 4‐Br (same but different values – see Craig Plot) 20 Example: Topliss Scheme for a series of sulfonamides Step
3: Addition of a 3‐Cl substituent decreased activity Indicates that the decrease could be due to steric effects or that the hydrophobicity is too high Step 4: 4‐Br derivative (equal activity
compared to 3,4‐Cl2) This means that the 4‐Br derivative is less potent than the 4‐Cl derivative! Br has a larger hydrophobicity (), but the same electronic effect () as Cl (see Craig Plot)
Interpretation: The lower activity of 4‐Br could be due to the larger hydrophobicity Step 5: 4‐Nitro derivative (highest activity) The NO2 group has a much smaller hydrophobicity (than Br) and a
large electron‐withdrawing effect (large ) Overall, it seems that high activities are obtained with a large value of , and a smaller value of 21 Bioiosteres in QSAR Tables of substituent
constants are useful to decide which bioisosteres to use in drug design Example: Scenario 1 (p is most important for activity) COCH3 (0.5) is a good bioisostere of SOCH3 (0.49) – and vice versa
Example: Scenario 2 ( is most important for activity) COCH3 (‐0.55) is not a good bioisostere of SOCH3 (‐1.58) SO2CH3 (‐1.63) is a good bioisostere of SOCH3 (‐1.58) 22 Planning of QSAR studies
At the beginning it needs to be decided which parameters to study Most often QSAR studies start with and (and potentially ES) Based on the parameters, a number of compounds need to be
synthesized such that there is a considerable variation in the parameters (note: Craig Plots are very useful here) Rule‐of‐thumb: At least 5 molecules should be synthesized per parameter studied
It is best not to use substituents in initial QSAR studies that could ionize (CO2H, NH2) or be metabolized (esters, nitro group) After the first QSAR equation is generated, more and more analogues
are prepared to refine the equation (e.g., introduction of new parameters) The refinement is an iterative process (synthesis → activity → refinement) 23 Final example of a QSAR study on
antiallergic pyranenamines The initial study included 19 compounds and gave the following equation The negative constant (‐0.14, for ) and the dependence on 2 are quite unusual (for 2: both
electron‐donating and withdrawing substituents decrease activity) A refined QSAR expression was generated with 61 compounds, and was then further refined with 98 compounds F‐5 345‐HBD M‐V 4‐OCO
HB‐intra F‐test (7 variables) inductive effect at 5 H‐bonding (at 3,4,5) volume of m‐substituents para‐acyloxy group ortho‐standing HB groups 24 | {"url":"https://quizgecko.com/uploads/11-qsarpdf-ro0jh6","timestamp":"2024-11-04T11:14:17Z","content_type":"text/html","content_length":"168884","record_id":"<urn:uuid:2739cdf6-a2a9-44c4-89fc-b9410afde09a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00079.warc.gz"} |
Math Mama Writes...
126 = 6*21 =
If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.
Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?
126 = 2
- 2
(difference of powers of 2)
126 = 4
+ 5
+ 6
+ 7
(sum of consecutive squares)
126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers)
This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to
. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)
I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7.
But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's
perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then
find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges.
If you want to get better at geometric construction (straightedge and compass style), play with it at
You can improve your geometric reasoning skills with the puzzles in
Geometry Snacks
More Geometry Snacks
), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.
Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously
offered me his time over breakfast. He advised me to download his
Geometry Labs
book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.
I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I
tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!) I then found another proof that followed a very different
path. Can you prove it?
Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?
One of my favorites for seeing the geometry in math topics you didn't know were geometric is
Magic Pi - math animations
. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of
their animations in
. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.
Making Your Own Math
At the beginning, I mentioned having students make up their own story problems.
Here's a lovely post from Arithmophobia No More
about just that.
Here's another angle on teaching story problems
, from Jen at Math State of mind. Leaving out the numbers helps students to slow down.
This blog post, by Amy at When Life Gave Us Lemons
, is about her son making up his own math games. And
John Golden has a whole class make up variations
on a game he shared with them.
Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in
this post
. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher
asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to
a discussion of his own with a student
. And that made me think about
Bob Kaplan's guide to 'becoming invisible'
(or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!) | {"url":"https://mathmamawrites.blogspot.com/2019/04/","timestamp":"2024-11-10T19:23:39Z","content_type":"application/xhtml+xml","content_length":"106247","record_id":"<urn:uuid:d12008ce-20a1-4b7b-83fe-0de2669be66c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00744.warc.gz"} |
Factorio - Fluid System Questions
I just went over this Factorio Fluid Mechanics study (
https://drive.google.com/file/d/0B7aC5k ... Iwdmc/view
), and, while it is an incredible work, it is still not 100% reliable (or so it seems to me at least. I may be wrong. Therefore I appologize in advance just in case), thus not really worth the effort
required to put into to actually understand and implement it into your gameplay (plus I don't know if it still applies and I really have no time nor mood to test it out). Since we want to actually be
able to play the game instead of studying it at an academic level, a simple table with information would suffice and the most reliable information would be directly from the source.
Question 1: "Is the table found on
under Pipelines accurate? (for Vanilla Factorio)"
Question 2: "What about this formula for pipe throughput
https://www.reddit.com/r/factorio/comme ... ation_015/
Question 3: "If the formula from question 2 is accurate is there a way to use it to calculate throughput for a line of pipes composed of not only 1 line, but of n pipes next to each other (e.g. a
3x10 pipeline being 3 pipes wide and 10 long)?"
Question 4: "Is there any practical and efficient way to calculate pipe throughput for a pipeline n * m (n long, m wide)? Is there any clear and concise formula/formulas to help us do so?" (I mean,
there should be, given the fact that Factorio is coded, and there MUST exist a specific formula/formulas which the game uses to compute the fluid flow through pipes)
Please share that formula/formulas/way/ways so I can start focusing on playing the game again instead of using my free time trying to understand the damn fluid mechanics, which I can't really wrap my
head around, and since I am a chronic perfectionist I can't go on playing until I am 100% sure I have a precise, accurate and efficient way to control the fluid flow in my factory.
If you will come with the answer "Check
out" then please also care to explain the Flow to Distance formula, since it makes no sense to me whatsoever.
Thanks in advance for your time and shared wisdom!
Re: Factorio - Fluid System Questions
1. The table on the wiki is correct. It is the same as viewtopic.php?f=18&t=19851, but swaps the axis and converts to seconds.
4. Side-by-side pipes make the fluid simulation worse in every way. Don't do it.
5. Precise control is impossible in 0.16, because the fluid system uses fractions of a unit that can't be measured accurately.
Re: Factorio - Fluid System Questions
If you have troubles with long distance fluid transport, remember that you can always use barrelling. Barrels on underground belts resemble underground pipes and in addition don't block movement. For
really long distances you can use trains.
Re: Factorio - Fluid System Questions
Rather than worry about exact formulats, I just maintain full speed pipe transport with a pump every three spans of ug pipe (pump-ug ... ug-ug ... ug-ug ... ug-pump-ug ...) and build close to water
Re: Factorio - Fluid System Questions
Thank you for your fast replies. That table is enough for now. I hope that in the future they will come up with a solution to make fluids easier to control even through pipes. | {"url":"https://forums.factorio.com/viewtopic.php?f=18&t=62658&p=380131","timestamp":"2024-11-04T04:55:02Z","content_type":"text/html","content_length":"54708","record_id":"<urn:uuid:725c3b86-de45-4d4c-8d95-4a3a73df18a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00361.warc.gz"} |
The Underbelly of the Graph – THINK Magazine
How does Facebook suggest new long lost friends? And, how does Google get your searches right so often. The answer is Graph Theory, an area of mathematics being investigated by Prof. Josef Lauri. Dr
Claude Bajada finds out more.
A mathematician sits alone at his desk. Hunched over a stack of papers he plays with numbers as a child would work at a jigsaw puzzle. The scene dematerialises in front of your eyes. A new reality
builds up. People are connected to one another by virtual links. They have access to every convenience at the click of a button. Computers know their likes and dislikes. Algorithms find cures for
diseases. Security agencies know your every step. The world is safe. The world is good. There is nowhere to hide.
As I interview mathematician Prof. Josef Lauri (University of Malta), these thoughts race through my mind. Lauri works on Graph Theory, which involves solving complex puzzles. He starts the interview
by drawing an odd shape on a blank piece of paper; dots with lines connecting them. While drawing and explaining a particularly difficult problem he is tackling, Lauri tells me that his fondness for
Graph Theory is akin to his son’s passion for the football players Lionel Messi and Cristiano Ronaldo. ‘Neither of them have solved any medical problem, but they have made a lot of people happy.’
Abstract mathematical problems make pure mathematicians like Lauri happy. But the difference between a goal by Messi and a puzzle of the pure mathematician is that the joys of the mathematician can
change the world.
Leonhard Euler established Graph Theory in 1737. Like Lauri, Euler set out to solve a puzzle. Königsberg (now Kaliningrad, part of Russia) was a city with two islands and seven bridges. Was it
possible to walk through the city using each bridge only once? Euler created a mathematical description of the way objects (land) related to one another (bridges). We use the term ‘node’ to describe
the land areas in Königsberg and the term ‘edge’ to describe the bridges. By abstracting the problem into a mathematical framework, Euler was able to prove that there was no way of taking the
suggested path. More importantly, this solution allowed other problems to be solved.
Computers revolutionised Graph Theory. Until then, it was ‘a specialised subject with no applications. The applications of mathematics were oriented towards the physical sciences,’ says Lauri. But in
1996 a paradigm shift occurred. Larry Page and Sergey Brin used Graph Theory to organise the world wide web. They developed PageRank and Google was born. In the old days, search engines ranked pages
by the amount of keywords present on that page. If someone looked up the phrase ‘Think magazine’ on the Internet, search engines would crawl the web looking for websites that contained those keyword.
The search engine assumed that the page that used that phrase the most was the most important result. A problem with this method was that a site that contains a keyword many times was not necessarily
the most important site. Spammers could pad their site with keywords of their choice and appear high on the ranking. Brin and Page’s new algorithm ranked websites (the nodes of their graph) according
to the number of other sites that linked to them (the edges). Lauri tells me that to understand the way this is done one needs to learn about fancy things like eigenvectors, but PageRank, like much
in Graph Theory, is simple to understand: a page is important if other important pages link to it.
Like Google, other Internet companies have seen value in using Graph Theory. Facebook and LinkedIn create social networks, in other words, social graphs. Each person is a node and a ‘friendship’ is
an edge. Many people have experienced looking up an old school friend on Facebook. The site then starts recommending other people from your class. An algorithm tailors its suggestions to your
specific situation by using your social graph. ‘It looks simplistic but it is amazing how well it works, you can see it yourself!’ exclaims Lauri. In this way advertising is becoming personalised.
Digital assistants such as Siri, Cortana, and Google Now suggest new movies for you to watch using the same principles that underlie Lauri’s puzzles. Sociologists caught the bug before the modern day
Internet giants. They have been analysing sociograms since the 1930s. They analyse social problems by looking for abnormal network structure. The reach of Graph Theory has been immense, scientists
now use Graph Theory to analyse medical data and find new ways of curing diseases.
Governments have jumped on the bandwagon. They use Graph Theory for public security. In 2001, the National Security Agency (NSA) started collecting metadata from US telephone calls. This allowed them
to build a communications network that is like a Facebook friendship network. Imagine having access to everyone’s Facebook network. NSA have access to this level of information. The NSA analyses this
data using Graph Theory coupled with raw computing power. The agency can then root out suspected criminals and prevent terrorist activity. But, there is always collateral damage. Using the
mathematics of Graph Theory, governments know a lot more about their citizens than ever before.
As the interview ends, the dystopic visions disassemble. Once again, I see the pure mathematician sitting in front of me. The interview has opened a door to perceive both the amazing and scary
applications of Lauri’s work. When I leave the room I cannot help but think: all the giants of our modern world, Facebook, Google, and NSA, are standing high on the shoulders of the work of pure
What is Graph Theory?
Graph Theory is a branch of Mathematics that describes how networks work. Networks are basically the relationship of a group of separate objects to one another. For example, five objects labeled as
1, 2, 3, 4, 5 can interact as follows:
Each of the objects are called vertices or nodes. The lines that connect them are called edges or links. This graph can then be used to describe many different types of networks. For example, the
nodes could represent people and the edges may be friendships. Alternatively they could be websites and weblinks. The way the graph is drawn does not matter as long as the connections are the same.
The graph can either be described as it is, or more often, it can be transformed into a matrix in order to do more complicated mathematics on it. This can identify which node is more important (a
hub) or which group of nodes belong to the same group (or cluster). For example, in Think magazine’s facebook network. The University of Malta page is a hub in the same cluster as Think’s page.
Further reading
• Barabasi, A. (2015). Network Science. [online] Barabasi.com. Available here.
• Brin, S. and Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30(1-7), pp.107-117.
• Seven Bridges of Königsberg – Woodside High School. (2013). Video available here.
Comments are closed for this article! | {"url":"https://thinkmagazine.mt/the-underbelly-of-the-graph/","timestamp":"2024-11-04T20:13:14Z","content_type":"text/html","content_length":"129922","record_id":"<urn:uuid:2e9edbc5-5000-4e24-8888-9769353a8cfe>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00472.warc.gz"} |
Description of the Hooke stress potential
The Hooke stress potential describes the linear elastic part of the behaviour of an isotropic or orthotropic material.
Evolution of the elastic strain
This stress potential relies on the fact that the behaviour is based on the strain split hypothesis.
The elastic strain must be defined as the first integration variable. The associated variable must be called eel and its glossary name must be ElasticStrain. This is automatically the case with the
@Implicit dsl.
The total strain increment \(\Delta\,\underline{\epsilon}^{\mathrm{to}}\) is automatically substracted to the equation associated with the elastic (\(f_{\underline{\epsilon}^{\mathrm{el}}}\)), which
is equivalent to the following statement:
Computation of the stress
If the elastic behaviour is orthotropic, the stiffness tensor must be available available (using the keyword @RequireStiffnessTensor) or computed by the behaviour (using the keyword
@ComputeStiffnessTensor). If those keywords are not explicitly used, the stress potential will automatically sets the attribute requireStiffnessTensor to true which has the same effect than the
@RequireStiffnessTensor keyword.
Thus, two cases arise:
• the stiffness tensor is available (using keyword @RequireStiffnessTensor) or computed by the behaviour (@ComputeStiffnessTensor).
• the behaviour has is an isotropic elastic behaviour and the stiffness tensor is not available.
First case: the stiffness tensor is available
Computation of the stress at \(t+\theta\,dt\)
At \(t+\theta\,dt\), the stress are computed using:
\[ {\left.\sigma\right|_{t+\theta\,\Delta\,t}}=\underline{\mathbf{D}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\theta\,\Delta\,t}} \]
Computation of the final stress at \(t+dt\)
If the stiffness tensor is avaible using the @RequireStiffnessTensor, the final stress \({\left.\sigma\right|_{t+\Delta\,t}}\) is computed using the following formula :
\[ {\left.\sigma\right|_{t+\Delta\,t}}=\underline{\mathbf{D}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}} \]
If the stiffness tensor is computed using @ComputeStiffnessTensor, the final \({\left.\sigma\right|_{t+\Delta\,t}}\) stress is computed using:
\[ {\left.\sigma\right|_{t+\Delta\,t}}={\left.\underline{\mathbf{D}}\right|_{t+\Delta\,t}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}} \]
Second case: the stiffness tensor is not available
In this case, the elastic behaviour of the material is isotropic. The computation of the stress requires the definition of the first Lamé coefficient and the shear modulus (second Lamé coefficient).
The Lamé coefficients are derived from the Young modulus and Poisson ratio. They can be defined using:
• the @ElasticMaterialProperties keyword. In this case, the Implicit dsl already automatically computes the following variables (See the documentation of the @ElasticMaterialProperties keyword):
□ young: the Young modulus at \(t+\theta\,dt\)
□ nu: the Poisson ratio modulus at \(t+\theta\,dt\)
□ lambda: the first Lamé coefficient at \(t+\theta\,dt\)
□ mu: the second Lamé coefficient at \(t+\theta\,dt\)
□ young_tdt: the Young modulus at \(t+dt\)
□ nu_tdt: the Poisson ratio modulus at \(t+dt\)
□ lambda_tdt: the first Lamé coefficient at \(t+dt\)
□ mu_tdt: the second Lamé coefficient at \(t+dt\)
• the Young modulus and Poisson ratio has been defined as material properties or parameters. In this case, the names of those variables must be young and the nu and the glossary names associated
with those variables must be respectively YoungModulus and PoissonRatio. The Lamé coefficients will be computed and stored in a data structure used internally by the stress potential.
If the material properties are not defined using one of those two ways, the appropriate material properties will be automatically defined by the stress potential.
Computation of the stress at \(t+\theta\,dt\)
At \(t+\theta\,dt\), the stress are computed using the following formula: \[ {\left.\sigma\right|_{t+\theta\,\Delta\,t}}=\lambda\,{\mathrm{tr}{\left({\left.\underline{\epsilon}^{\mathrm{el}}\right|_
{t+\theta\,\Delta\,t}}\right)}}+2\,\mu\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\theta\,\Delta\,t}} \] where \(\lambda\) and \(\mu\) are respectively the values of the first and second
Lamé coefficients at \(t+\theta\,dt\)
Computation of the final stress at \(t+dt\)
The final stress \({\left.\sigma\right|_{t+\Delta\,t}}\) is computed using the following formula :
\[ {\left.\sigma\right|_{t+\Delta\,t}}={\left.\lambda\right|_{t+\Delta\,t}}\,{\mathrm{tr}{\left({\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}\right)}}+2\,{\left.\mu\right|_{t+\Delta
\,t}}\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}} \]
Enforcement of the plane stress conditions: computation of the axial strain
If the user has explicitly specified that the axisymmetric generalised plane stress modelling hypothesis must be supported by the behaviour using the @ModellingHypothesis keyword or the
@ModellingHypotheses keyword, this support is performed by automatically introducing an additional state variable: the axial strain. The associated variable is etozz, although this variable shall not
be used by the end user. The glossary name of this variable is AxialStrain.
The introduction of the variable modify the strain split equation like this: \[ feel(2) += detozz; \] where \(detozz\) is the increment of the axial strain. The associated jacobian term is added if
The plane stress condition is enforced by adding an additional equation to the implicit system ensuring that: \[ {\left.\sigma_{zz}\right|_{t+\Delta\,t}}=0 \]
This equation is appropriately normalised using one of the elastic properties. The associated jacobian term are added if necessary.
Enforcement of the generalised plane stress conditions: computation of the axial strain
If the user has explicitly specified that the axisymmetric generalised plane stress modelling hypothesis must be supported by the behaviour using the @ModellingHypothesis keyword or the
@ModellingHypotheses keyword, this support is performed by automatically introducing an additional state variable, the axial strain and an additional external state variable, the axial stress.
The variable associated to the axial strain is etozz, although this variable shall not be used by the end user. The glossary name of this variable is AxialStrain.
The variable associated to the axial stress is sigzz, although this variable shall not be used by the end user. The glossary name of this variable is AxialStress.
The introduction of the variable modify the strain split equation as follows:
where \(\epsilon^{\mathrm{to}}_{zz}\) is the increment of the axial strain. The associated jacobian term is added if necessary.
The plane stress condition is enforced by adding an additional equation to the implicit system ensuring that: \[ {\left.\sigma_{zz}\right|_{t+\Delta\,t}}-\sigma^{zz}-d\sigma^{zz}=0 \]
where \(\sigma^{zz}\) is the value of the axial stress at the beginning of the time step and \(d\sigma^{zz}\) is the value of the increment of the axial stress.
This equation is appropriately normalised using one of the elastic properties. The associated jacobian terms are added if necessary.
Generic computation of the tangent operator
The elastic and secant operator are equal to the elastic stiffness matrix at the end of the time step. How this elastic stiffness matrix is obtained depends on the many cases described before.
The consistent tangent operator is computed by multiplying the elastic stiffness matrix at the end of the time step by a partial invert of the jacobian matrix. This procedure is discussed in depth in
the MFront manuals.
Computation of the elastic prediction of the stress
The Hooke stress potential automatically defines the computeElasticPrediction method which computes a prediction of the stress under the assumption that all states variables are equal to their values
at the beginning of the time step except the elastic strain which are assumed to be equal to \({\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t}}+\Delta\,\underline{\epsilon}^{\mathrm{to}}\).
Options of the stress potential
The Hooke stress potential supports the following options:
• young_modulus
• young_modulus1
• young_modulus2
• young_modulus3
• poisson_ratio
• poisson_ratio12
• poisson_ratio23
• poisson_ratio13
• shear_modulus12
• shear_modulus23
• shear_modulus13
• thermal_expansion
• thermal_expansion1
• thermal_expansion2
• thermal_expansion3
• thermal_expansion_reference_temperature
• plane_stress_support
• generic_tangent_operator
• generic_prediction_operator | {"url":"https://thelfer.github.io/tfel/web/HookeStressPotential.html","timestamp":"2024-11-02T11:41:08Z","content_type":"text/html","content_length":"31962","record_id":"<urn:uuid:563ef7fe-63ff-4e2f-8af8-10b163d2a52d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00556.warc.gz"} |
Eco-evolutionary dynamics of clonal multicellular life cycles
The evolution of multicellular life cycles is a central process in the course of the emergence of multicellularity. The simplest multicellular life cycle is comprised of the growth of the propagule
into a colony and its fragmentation to give rise to new propagules. The majority of theoretical models assume selection among life cycles to be driven by internal properties of multicellular groups,
resulting in growth competition. At the same time, the influence of interactions between groups on the evolution of life cycles is rarely even considered. Here, we present a model of colonial life
cycle evolution taking into account group interactions. Our work shows that the outcome of evolution could be coexistence between multiple life cycles or that the outcome may depend on the initial
state of the population – scenarios impossible without group interactions. At the same time, we found that some results of these simpler models remain relevant: evolutionary stable strategies in our
model are restricted to binary fragmentation – the same class of life cycles that contains all evolutionarily optimal life cycles in the model without interactions. Our results demonstrate that while
models neglecting interactions can capture short-term dynamics, they fall short in predicting the population-scale picture of evolution.
This article models the evolution of simple multicellular life cycles using evolutionary game theory. The authors discuss natural selection between different life cycles by modeling growth,
fragmentation, and interactions between propagules, discovering conditions for selection of a single life cycle or coexistence of multiple ones. Overall, the model is biologically intuitive, the
results are rigorous, and the implications for the evolution of multicellularity are interesting.
Multicellular organisms are found everywhere. In all major branches of complex multicellularity (animals, plants, fungi, red and brown algae), organisms are formed by cells staying together after
cell division – unlike unicellular species, in which cells part their ways before the next division occurs (Márquez-Zacarías et al., 2021; Herron et al., 2022). However, organisms have to reproduce
as otherwise their species will eventually go extinct. For a multicellular organism, this means that some cells must depart in order to develop into an offspring individual. The combination of
organism growth and reproduction constitutes a clonal life cycle. Emergence of clonal multicellular life cycles was the central innovation in the earlier stages of the evolution of multicellularity.
There, traits, which do not even exist for unicellular species, become crucial for long-term success of even the most primitive colony of cells (Maynard Smith and Szathmáry, 1995, Michod, 2007).
These include the number of cells in the colony, how often cells depart to give rise to new colonies, how large the released propagules are, and how many of them are produced. As the reproduction
and, consequently, fitness of simple cell colonies are dependent on these traits, they immediately become subjected to natural selection, favoring some life cycles over others. Since complex
multicellular life descends from those loose cell colonies, the understanding of the prior evolution of primitive life cycles is essential to our understanding of the later evolution of complex
traits (Ratcliff et al., 2012; De Monte and Rainey, 2014; Hammerschmidt et al., 2014; Doulcier et al., 2020).
There are several theoretical approaches to the modeling of the evolution of multicellular life cycles. The mechanistically simplest class of models assumes that natural selection operates by means
of growth competition. Colonies are born small, but due to cell divisions they increase in size and, eventually, fragment, so the number of colonies in the population increases. The life cycle
maximizing the population growth rate has a selective advantage as it outgrows all competitors (Roze and Michod, 2001; Libby et al., 2014; Pichugin et al., 2017; Pichugin et al., 2019; Staps et al.,
2019; Gao et al., 2019; Pichugin and Traulsen, 2020; Gao et al., 2021; Pichugin, 2022; Pichugin and Traulsen, 2022). For groups made of identical cells, growth competition models of evolution predict
that some life cycles cannot be the winners of this growth competition under any conditions. For instance, if the fragmentation event is instantaneous and its execution does not cost anything to the
group, only fragmentation into two pieces can evolve (Pichugin et al., 2017; Pichugin and Traulsen, 2020). And indeed, the division into two pieces is, by a large margin, the most common reproductive
strategy among microscopic life forms.
However, these models, due to their conceptual simplicity, assume unconstrained (exponential) growth of the population, which cannot be sustained for a prolonged period of time, because resources and
space are limited. Other models consider density-dependent growth (Rossetti et al., 2011; Tarnita et al., 2013; van Gestel and Nowak, 2016; Doulcier et al., 2020, Henriques et al., 2021), where the
population growth decreases with the number of groups. A similar approach is the Moran birth–death process on the group level, where whenever a new group emerges, one other group dies (Traulsen and
Nowak, 2006; Matias Rodrigues et al., 2012; Simon et al., 2013; Luo, 2014; Kaveh et al., 2016; Olejarz et al., 2018; Cooney, 2019; Cooney, 2020). While the population dynamics of density-dependent
population growth is vastly different from the exponential explosion found in models of unconstrained growth, these two approaches lead to identical results for life cycle evolution: as shown in
Pichugin and Traulsen, 2020, the dynamics of the fraction of a given life cycle in a population are identical in models with unconstrained and density-dependent growth. Therefore, even in models with
density-dependent growth, the evolutionary success of the life cycle is still fully determined by the population growth rates.
Nevertheless, density-dependent growth is also a simplification as different groups may differ in their competitiveness. For instance, large-cell colonies are able to block single cells from access
to vital resources (Rainey and Travisano, 1998; Rainey and Rainey, 2003; Hammerschmidt et al., 2014), which may even lead to a complete extinction of solitary cells. Thus, the population dynamics of
multicellular life cycles is not necessarily density dependent, but could be frequency dependent – the impact of resource limitation on the population growth depends on both the size and the
composition of the population.
Hence, the evolution of multicellular life cycles cannot always be reduced to growth competition, but may arise from eco-evolutionary dynamics.
From a broader empirical perspective, frequency-dependent dynamics is found to be common among microbial populations (Levin, 1988; Ribeck and Lenski, 2015; Healey et al., 2016; Friedman et al., 2017
). From the perspective of the theoretical ecology, frequency-dependent evolutionary dynamics arising from interactions between diverse population members has also been considered in detail (May,
1972; Wangersky, 1978; Bomze, 1983; Huang et al., 2015; Huang et al., 2017; Bunin, 2017; Barbier et al., 2018; Kotil and Vetsigian, 2018; Tarnita, 2018; Farahpour et al., 2018; Park et al., 2019;
Park et al., 2020). The impact of interactions between individuals is recognized in the context of the emergence of aggregative multicellularity, where cells come together to form collectives (Garcia
and De Monte, 2013; De Monte and Rainey, 2014; Garcia et al., 2015; Miele and De Monte, 2021). However, both empirical and theoretical ecology approaches tend to overlook frequency dependence in the
context of clonal life cycles, where the organism’s growth in the course of the life cycle may cause a change in its role in interactions (but see an example in Tverskoi and Gavrilets, 2022 modeling
evolution of germ–soma differentiation).
In this article, we developed a model of the evolution of clonal life cycles under frequency-dependent dynamics, implemented in the form of frequency-dependent colony death rate. We focus on three
• What is the population dynamics of a single life cycle?
• What kind of evolutionary outcomes does frequency-dependent selection bring?
• Are there any patterns or constraints among possible evolutionary outcomes that are universal for multiple forms of frequency dependence?
We first address these questions in the context of simpler models with unconstrained growth. In these models, a population performing any life cycle grows exponentially, the competition between life
cycles always results in a single one outcompeting all others (which life cycle will be the winner depends only on the growth/death rates but not on the initial composition of population), and
finally, the winner always comes from a limited subset of life cycles (in the simplest version of the model with costless fragmentation – it must be a fragmentation mode producing exactly two
offspring). We found that including frequency-dependent interactions between organisms performing different life cycles and thus constraining the total population size changes the answers to these
questions. First, the population dynamics leads to a stationary state with a finite population size. Second, we found that interactions between groups allow for situations with bistability or
coexistence of multiple life cycles – scenarios impossible in the unconstrained growth model. Third, evolutionary stable strategies in our present model always belong to a limited subset of life
cycles – the same one containing possible winners of growth competition in models without group interactions. Thus, we found that despite the fundamental differences between our present model and
simpler models with unconstrained growth, some of their results have a direct analogy in a much more general eco-evolutionary context considered here.
Population dynamics of a single life cycle
We consider a population consisting of cell groups that grow in size and fragment, giving rise to new groups. Cells within a group of size $i$ divide at rate $bi$, thus a group of size $i$ grows at
rate $ibi$. Groups also die due to both external environmental factors and within-population competition for resources or space. The death rate of groups of size $i$ due to external factors is $di$.
Frequency-dependent competition is modeled as the death of groups of size $i$ upon encounter with groups of size $j$ at rate $Ki,j$ (see Figure 1A).
Model of clonal life cycles.
Whenever a group of maturity size $m$ grows to $m+1$ cells, it immediately fragments. The fragmentation always occurs by the same pattern and determines the life cycle of a population. We represent a
fragmentation pattern by $κ$ – a partition of a number $m+1$. For example, the fragmentation pattern of the unicellular life cycle, in which two daughter cells always go apart, is $κ=1+1$ (see Figure
1B). Other fragmentation patterns correspond to multicellular life cycles. The simplest of them are the life cycles in which groups grow up to two cells, but fragment upon reaching size three. Such a
fragmentation can be performed in two ways: either detachment of a single cell, leading to the fragmentation pattern $κ=2+1$, or fission into three solitary cells, $κ=1+1+1$ (see Figure 1B). For
simplicity, we assume that the cell number does not change during fragmentation (no cell loss), the sum of a fragmentation pattern $κ$ is equal to $m+1$.
If we denote the abundance of cell groups containing $i$ cells as $xi$, then the dynamics of population is described by a system of differential equations
(1) $\begin{array}{ll}\frac{d{x}_{1}}{dt}& =\phantom{\rule{63pt}{0ex}}-{b}_{1}{x}_{1}-{d}_{1}{x}_{1}+m{b}_{m}{\pi }_{1}\left(\kappa \right){x}_{m}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule
{thinmathspace}{0ex}}-{x}_{1}\sum _{j=1}^{m}{k}_{1,j}{x}_{j},\\ {\frac{d{x}_{i}}{dt}|}_{i>1}& =\left(i-1\right){b}_{i-1}{x}_{i-1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}
-i{b}_{i}{x}_{i}-{d}_{i}{x}_{i}+m{b}_{m}{\pi }_{i}\left(\kappa \right){x}_{m}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-{x}_{i}\sum _{j=
where the first two terms $(i−1)bi−1xi−1−ibixi$ describe the growth of groups – the positive term represents growth from size $i-1$ to $i$ and the negative term represents growth from $i$ to $i+1$.
The third term $-dixi$ is the environmentally caused death. The term $mbmπi(κ)xm$ describes the birth of new groups of size $i$ via fragmentation of larger groups, where $πi(κ)$ is the number
of groups of size $i$ produced in the result of that fragmentation. Finally, $-xi∑j=1mKi,jxj$ is the death of groups due to the competition between groups.
Summarizing the dynamics into matrix notation, the system (1) can be written as
(2) $\frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}-\mathrm{diag}\left(\mathbf{K}\mathbf{x}\right)\mathbf{x}.$
Here, $x$ is the column vector of group abundances
(3) $\begin{array}{cc}\hfill x={\left({x}_{1},{x}_{2},{x}_{3},\mathrm{\dots },{x}_{m}\right)}^{T}.& \end{array}$
The linear term $Ax$ represents the processes of group growth, fragmentation, and frequency-independent death. The matrix $A$ of size $m×m$ is called a population projection matrix in the field of
formal demography – in the sense of the projection of the current state into the future state. For an arbitrary life cycle, matrix $A$ is given by
(4) $\begin{array}{cc}\hfill A=\left(\begin{array}{cccccc}\hfill -{b}_{1}-{d}_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill & \hfill m{b}_{m}{\pi }_{1}\
left(\kappa \right)\hfill \\ \hfill {b}_{1}\hfill & \hfill -2{b}_{2}-{d}_{2}\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill & \hfill m{b}_{m}{\pi }_{2}\left(\kappa \right)\
hfill \\ \hfill 0\hfill & \hfill 2{b}_{2}\hfill & \hfill -3{b}_{3}-{d}_{3}\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill & \hfill m{b}_{m}{\pi }_{3}\left(\kappa \right)\hfill \\ \hfill 0\
hfill & \hfill 0\hfill & \hfill 3{b}_{3}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill 0\hfill & \hfill m{b}_{m}{\pi }_{4}\left(\kappa \right)\hfill \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\
hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\
hfill & \hfill \left(m-1\right){b}_{m-1}\hfill & \hfill m{b}_{m}{\pi }_{m}\left(\kappa \right)-m{b}_{m}-{d}_{m}\hfill \end{array}\right).& \end{array}$
The elements of the population projection matrix $Ai,j$ represent changes to number of groups of size $i$ due to processes occurring with groups of size $j$ (but not due to interactions). Hence, the
population projection matrix has nonzero elements only on the main diagonal (group death and growth of groups to larger sizes), lower subdiagonal (growth of smaller groups), and rightmost column
(fragmentation at the end of the life cycle). The elements of the competition matrix $K$ are given by $Ki,j$ for $i,j=1,…,m$. The operation $diag(⋅)$ takes an input vector of length $m$ and
transforms it into a diagonal matrix of size $m×m$ with the entries of the input vector on the main diagonal.
Population dynamics of multiple life cycles
To investigate the eco-evolutionary dynamics of clonal life cycles, we consider a composition of subpopulations executing various life cycles: $κ(1),κ(2),…κ(r)$. In this composite population, the
cell growth ($bi$), environmentally caused (constant) group death ($di$), and group fragmentation ($πi(κ)$) occur independently in each subpopulation. However, frequency-dependent death due to
competition entangles the dynamics of the subpopulations as groups with different life cycles growing together have to compete with each other. If we denote with $xi(j)$ the number of groups
containing i cells in a subpopulation executing the life cycle $κ(j)$, the dynamics of the composite population is described by the differential equations
(5) $\begin{array}{ll}\frac{d{x}_{1}^{\left(j\right)}}{dt}& =\phantom{\rule{63pt}{0ex}}-{b}_{1}{x}_{1}^{\left(j\right)}-{d}_{1}{x}_{1}^{\left(j\right)}+{m}^{\left(j\right)}{b}_{{m}^{\left(j\right)}}
{\pi }_{1}\left({\kappa }^{\left(j\right)}\right){x}_{{m}^{\left(j\right)}}^{\left(j\right)}-{x}_{1}^{\left(j\right)}\sum _{p=1}^{r}\sum _{k=1}^{n}{K}_{1,k}{x}_{k}^{\left(p\right)},\\ {\frac{d{x}_{i}
^{\left(j\right)}}{dt}|}_{i>1}& =\left(i-1\right){b}_{i-1}{x}_{i-1}^{\left(j\right)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-i{b}_{i}{x}_{i}^{\left(j\right)}-{d}_{i}{x}_
{i}^{\left(j\right)}+{m}^{\left(j\right)}{b}_{{m}^{\left(j\right)}}{\pi }_{i}\left({\kappa }^{\left(j\right)}\right){x}_{{m}^{\left(j\right)}}^{\left(j\right)}-{x}_{i}^{\left(j\right)}\sum _{p=1}^{r}
\sum _{k=1}^{n}{K}_{i,k}{x}_{k}^{\left(p\right)},\end{array}$
where $m(j)$ is the maturity size of the life cycle $κ(j)$, and $n=max(m(1),m(2),…,m(r))$ is the maximal maturity size in the composite population. The difference between the one life cycle system
(1) and the system of multiple life cycles (5) is in the last term, where groups from every competing subpopulation contribute to frequency-dependent death.
In vector form, the state of the composite population is described by a concatenation of vector states of each subpopulation:
(6) $\begin{array}{rl}\stackrel{~}{\mathbf{x}}=& \left({\mathbf{x}}^{\left(1{\right)}^{T}},{\mathbf{x}}^{\left(2{\right)}^{T}},\dots ,{\mathbf{x}}^{\left(r{\right)}^{T}}{\right)}^{T}\\ =& \left({x}_
{1}^{\left(1\right)},\dots ,{x}_{n}^{\left(1\right)},{x}_{1}^{\left(2\right)},\dots ,{x}_{n}^{\left(2\right)},\dots ,{x}_{1}^{\left(r\right)},\dots ,{x}_{n}^{\left(r\right)}{\right)}^{T},\end{array}$
where $x~$ is the column vector describing the state of the composite population, $x(j)$ is the column vector describing $j$th subpopulation in a form (3). Note that the last entries of any $x(j)T$
will be zero if $m(j)<n$. The dynamics of the composite population in Equation (5) can be represented in the vectorized form, similar to Equation (2):
(7) $\frac{d\stackrel{~}{\mathbf{x}}}{dt}=\stackrel{~}{\mathbf{A}}\stackrel{~}{\mathbf{x}}-\mathrm{diag}\left(\stackrel{~}{\mathbf{K}}\stackrel{~}{\mathbf{x}}\right)\stackrel{~}{\mathbf{x}}.$
Here, the composite population projection matrix representing the cell growth, environmentally caused (constant) group death, and group fragmentation is a diagonal block matrix
(8) $\begin{array}{cc}\hfill \stackrel{~}{A}=\left(\begin{array}{ccccc}\hfill {A}^{\left(1\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill \\ \hfill 0\
hfill & \hfill {A}^{\left(2\right)}\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {A}^{\left(3\right)}\hfill & \hfill \mathrm{\
dots }\hfill & \hfill 0\hfill \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill 0\hfill & \hfill 0\
hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill {A}^{\left(r\right)}\hfill \end{array}\right),& \end{array}$
where $A(i)$ is the population projection matrix of the life cycle $κ(i)$ extended to size $n×n$ ($n$ is the maximal maturity size across all competing life cycles). If the maturity size of the life
cycle $i$ is $m(i)=n$, this matrix has a form exactly as in Equation (4). If the maturity size is smaller, $m(i)<n$, then the top-left $m(i)×m(i)$ has the form (4), while the remaining elements are
nonzero at the main diagonal and the lower subdiagonal, as dictated by Equation (5).
The composite competition matrix $K~$ is constructed as
(9) $\begin{array}{cc}\hfill \stackrel{~}{K}=\left(\begin{array}{ccccc}\hfill K\hfill & \hfill K\hfill & \hfill K\hfill & \hfill \mathrm{\dots }\hfill & \hfill K\hfill \\ \hfill K\hfill & \hfill K\
hfill & \hfill K\hfill & \hfill \mathrm{\dots }\hfill & \hfill K\hfill \\ \hfill K\hfill & \hfill K\hfill & \hfill K\hfill & \hfill \mathrm{\dots }\hfill & \hfill K\hfill \\ \hfill \mathrm{⋮}\hfill &
\hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill K\hfill & \hfill K\hfill & \hfill K\hfill & \hfill \mathrm{\dots }\hfill & \hfill
K\hfill \end{array}\right),& \end{array}$
where each block $K$ is a competition matrix.
In the general case, the investigation of the composite population dynamics given by Equation (7) is a very complex problem without a general solution. Hence, in our study we consider a specific
class of initial conditions – invasion from rare, where the composite population contains only two subpopulations: the abundant ‘resident’ executing life cycle $κ(R)$ and rare ‘invader’ executing
different life cycle $κ(I)$. This scenario represents an emergence of a mutant in previously stable population of the resident. The population changes if this mutant is capable of invading the
resident – otherwise, the mutant goes extinct and the resident population remains the same.
In this scenario, the composite dynamics in Equation (7) can be disentangled into resident and invader components. Since the invader population is small, its contribution to frequency-dependent
competition is negligible. The members of the resident population compete predominantly between themselves, so the resident dynamics is effectively a single-species scenario,
(10) $\frac{d{\mathbf{x}}^{\left(R\right)}}{dt}\approx \left[{\mathbf{A}}^{\left(R\right)}-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\left(R\right)}\right)\right]{\mathbf{x}}^{\left(R\right)}=\
mathbf{0}={\mathbf{A}}^{\left(R,R\right)}{\mathbf{x}}^{\left(R\right)\ast },$
where the vector $x(R)$ represents the composition of the resident population, $A(R)$ is the population projection matrix of the resident, $x(R)*$ is the equilibrium composition, and we introduced
the self-invasion population projection matrix $A(R,R)=A(R)-diag(Kx(R)*)$. Since the resident is assumed to be at a stable equilibrium in the absence of invaders, the self-invasion matrix $A(R,R)$
has an eigenvalue that is zero, and the equilibrium population composition $x(R)$ is given by the corresponding eigenvector. The resident population dynamics can be obtained by solving the nonlinear
Equation (10), which in the general case can be performed only numerically.
The rare invader population also competes primarily with the resident and self-competition is negligible. Thus, its dynamics is given by
(11) $\frac{d{\mathbf{x}}^{\left(I\right)}}{dt}\approx \left[{\mathbf{A}}^{\left(I\right)}-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\left(R\right)\ast }\right)\right]{\mathbf{x}}^{\left(I\right)}=
where vector $x(I)$ represents the composition of the invader population, and we introduced the invasion matrix
(12) ${\mathbf{A}}^{\left(I,R\right)}={\mathbf{A}}^{\left(I\right)}-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\left(R\right)\ast }\right).$
Unlike the resident dynamics, the dynamics of the invader population is linear – the invasion population projection matrix $A(I,R)$ is independent from the invader population state $x(I)$. The linear
dynamics of clonal life cycles has been extensively studied in previous work (Pichugin et al., 2017). If the largest eigenvalue of the invasion matrix $A(I,R)$ is positive, then the invader
population will increase in numbers, independently of its initial demography. Otherwise, the invader population goes extinct.
The assumption of a negligible impact of the invader population on competition limits the analysis to the early stages of invasion, when the invader population is small. Nevertheless, this makes it
possible to investigate the stability of resident life cycles with respect to invasions.
We first briefly recap the population dynamics and evolution under a more basic model with unconstrained growth ($Kij=0$) (Pichugin et al., 2017; Pichugin and Traulsen, 2020). This model has three
main features. First, a population executing a single life cycle grows exponentially in the long run. The population growth rate is given by the leading eigenvalue of the population projection matrix
($A$) and the demographic composition is given by the associated eigenvector. Second, selection always finds a single winner. In a composite population, where different subpopulations execute
different life cycles, only one life cycle survives in the long run – the one that has the largest growth rate. This outcome is independent of the initial state of the population. Third, some life
cycles cannot be optimal under any combination of cell division rates ($bi$) and group death rates ($di$). In the simplest case of instant and costless group fragmentation, life cycles with more than
two offspring cannot win the growth competition.
Next, we consider how these features transfer to a system taking into account competition between groups. We begin with the dynamics of a single life cycle (section ‘Dynamics of a single life cycle’)
in a system with a population size constraint, which is very different from exponential growth. Then, we consider how the competition proceeds in our model: for two (section ‘Competition between
pairs of life cycles may result in coexistence or bistability’) and multiple life cycles (section ‘Competition between multiple life cycles’), where a rich spectrum of possible stationary states is
found. After that, we outline the restrictions imposed on evolutionary stable strategies (section ‘The set of possible evolutionary stable strategies is restricted’). We conclude with presenting
scenarios of a special interest: interactions with killer and victim kernels, where the evolutionary dynamics is reduced to the competition for growth rate and carrying capacity respectively (section
‘Killer and victim kernels guarantee a dominance of a single life cycle’), and investigate the competition between unicellular and multicellular life cycles (section ‘Conditions promoting the
evolution of multicellular life cycles’).
Dynamics of a single life cycle
For the simplest unicellular life cycle ($κ=1+1$), where population is composed only of solitary cells, the dynamics of our model given by Equation (2) reduces to logistic growth,
(13) $\frac{d{x}_{1}}{dt}=\left({b}_{1}-{d}_{1}\right){x}_{1}-{K}_{1,1}{x}_{1}^{2},$
where the net growth rate is equal to $b1-d1$ and the carrying capacity is $(b1-d1)/K1,1$. The characteristic feature of logistic growth is that the population approaches the carrying capacity with
time, starting from either high or low populations size (see Figure 2A). The population dynamics of more complex life cycles also bears similarity to the logistic growth of the unicellular life
cycle. If a population is small, the competition term is small, so the population grows exponentially. While population size increases, so does the competition term – group mortality rises and the
overall population growth slows down. The growth stops when the population reaches a stationary state $x*$, where
(14) $\frac{d{\mathbf{x}}^{\ast }}{dt}=\mathbf{A}{\mathbf{x}}^{\ast }-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\ast }\right){\mathbf{x}}^{\ast }=\mathbf{0}.$
A single life cycle comes to a stationary state.
Numerical simulations show that a population executing a single life cycle always comes to the same stationary state $x*$ from any initial distribution of group sizes (see Figure 2B and C).
Competition between pairs of life cycles may result in coexistence or bistability
A composite population containing several subpopulations executing different life cycles also reaches a stationary state. In the simplest case, only one life cycle survives, while others go extinct.
However, we found that the stationary state may contain more than one life cycle (coexistence). Also, the stationary state may depend on the initial state of the population (multistability). Neither
of these can occur in the linear model without competition.
To illustrate these effects, we focus on a pair of life cycles ($κ(1),κ(2)$) with the special initial conditions, where one of these life cycles is abundant, while the other one is rare. The life
cycle $κ(1)$ can invade from rare into the abundant $κ(2)$ if the largest eigenvalue of the invasion matrix $A(1,2)$ is positive; see Equation (12). Otherwise, the rare life cycle $κ(1)$ goes
extinct. For a pair of life cycles, there are four possible scenarios of invasion from rare (see the outline in Table 1 and corresponding dynamics in Figure 3).
Competitive interactions can lead to the bistability between life cycles (A), a dominance of either of them (B,C), or their coexistence (D).
Life cycle $κ(1)$ dominates life cycle $κ(2)$ if $κ(1)$ spreads from rare, but $κ(2)$ does not. This is equivalent to the leading eigenvalue of the invasion matrix $A(1,2)$ being positive, while the
leading eigenvalue of $A(2,1)$ is negative. Then, independently of the initial conditions, only the life cycle $κ(1)$ survives in the long run (see Figure 3B). The opposite signs result in the
dominance of $κ(2)$ over $κ(1)$ (see Figure 3C).
Beyond a dominance of one life cycle over another, it is possible that each life cycle is able to spread in the abundance of another. This happens when the leading eigenvalues of both invasion
matrices $A(1,2)$ and $A(2,1)$ are positive. There, the result of interactions between life cycles is a coexistence of both – an outcome impossible in the model without competition (see Figure 3D).
Finally, the leading eigenvalues of both invasion matrices could be negative – then neither of the life cycles can invade into another. Then, the result is a bistability between life cycles, where
the outcome of interactions depends on the initial conditions – another result impossible in the model without competition (see Figure 3A).
The competition between groups plays a major role in determining which of the four invasion patterns occurs. For instance, it is possible that a life cycle having an advantage in the raw growth rate
(i.e., dominating in the unconstrained growth model) is dominated by the result of the competition (see Figure 3C), where the life cycle 1 + 1 has a faster growth but 2 + 1 still dominates due to the
advantage of multicellular groups in competition.
Competition between multiple life cycles
Extending our analysis beyond just a pair of life cycles, we considered the evolutionary dynamics in a population with multiple of them. We numerically investigated the evolutionary dynamics in a
population containing all life cycles in which groups do not exceed a size of three cells. There are seven such life cycles: unicellular (1 + 1), two with bicellular groups (2 + 1 and 1 + 1 + 1), and
four with groups of three cells (3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1) (see Figure 4A). We generated a set of 13,000 randomly drawn combinations of growth and competition rates from an
exponential distribution with unit rate parameter. For each set, we simulated 100 independent replicates of population dynamics that differ in the initial conditions (abundances and demographic
composition of each of the seven life cycles).
Evolution in a population with multiple competing life cycles.
These runs were classified by the state of population at the end of simulation (see Figure 4B and Table 2). The majority of simulations (about 75%) resulted in the survival of a single life cycle
across all 100 replicates. The next most common outcome is a coexistence between two or more life cycles – found in about 17% of simulations. Also, a multistability between two or three life cycles
was observed in about 6.5% of simulations. Here, coexistence describes a situation where the stationary state of the population is composed of the same set of at least two life cycles in every
replicate. Multistability describes a situation, where in every replicate the stationary state contained only a single life cycle, but there were different stationary states among the replicates.
More complex outcomes were also observed – these were the compositions of multistability and coexistences, which contributed to only about 1.5% of simulations. The most common of these composite
situations is a minimal combination of bistability and coexistence: there are two possible stationary states, one is a single life cycle and another is a coexistence of two other life cycles (0.7% of
The numerical investigation of evolutionary dynamics of multiple life cycles revealed that a diverse range of outcomes are possible, including multistability and coexistence of life cycles, as well
as their combinations. At the same time, the most common result is still the dominance of a single life cycle.
The set of possible evolutionary stable strategies is restricted
Between simple dominance and multistability, about 80% of evolutionary simulations ended with the survival of a single life cycle. This happens if a life cycle is an evolutionary stable strategy
(ESS), for example, if it is abundant, it cannot be invaded by any other life cycle. In the basic model without competition, the life cycle with the maximal growth rate also satisfies the definition
of an ESS. Here, we found that the set of evolutionary stable strategies is similarly restricted – it also contains only fragmentations into exactly two pieces.
To show this restriction, we consider a triplet of life cycles $κ(1)$, $κ(2)$, $κ(3)$. If the life cycle $κ(1)$ is a resident, there are four variants of its stability against invasions: (i) either
it is stable against invasions from both $κ(2)$ and $κ(3)$ (then $κ(1)$ is an ESS), or (ii) stable against only $κ(2)$, or (iii) stable against only $κ(3)$, or (iv) both $κ(2)$ and $κ(3)$ can invade.
Similar four variants exist for the two other life cycles. As a result, for the whole triplet, there are $43=64$ possible pairwise invasion patterns, which could feature 0, 1, 2, or 3 evolutionary
stable strategies.
Numerical simulations show that all 64 patterns can be expressed for the same triplet of life cycles for some combination of cell birth, group death, and competition rates (see Figure 5A). We
generated a set of 40,000 randomly drawn combinations of these rates from an exponential distribution with unit rate parameter and analyzed the pairwise invasion patterns for each. The six most
frequent patterns, comprising 77% of the generated dataset, feature a hierarchical dominance, where the life cycles can be ordered in a way that higher-order life cycle dominates (always invade)
lower-order life cycles. These six patterns are all possible hierarchical dominance triplets as there are exactly six ways how three items can be placed in order. If we use the same analysis for the
basic, linear model with unrestricted growth, we will only observe hierarchical dominance as larger growth rate results in domination there. On the opposite side of the frequency spectrum, the two
most rare patterns feature cyclic dominance, together comprising only 0.015% of the dataset. There, in any pair of life cycles one dominates another but the whole triplet follows a
‘rock–paper–scissors’ dynamics with no evolutionary stable strategies present (cf. Park et al., 2020).
Constrained triplets demonstrate fewer patterns of pairwise invasion.
While an arbitrary triplet of life cycles may demonstrate up to 64 invasion patterns, some triplets, which we will call ‘constrained,’ feature much smaller diversity of patterns. A triplet is
constrained if the fragmentation rule of one (constrained) life cycle $κ(M)$ can be represented as a combination of fragmentation rules of two other (constraining) life cycles $κ(C1)$ and $κ(C2)$.
The simplest example is the triplet $κ(C1)=2+1$, $κ(C2)=1+1$, $κ(M)=1+1+1$, where the fragmentation of a three-celled group into three solitary cells ($3→1+1+1$) can be presented as a combination
of the detachment of a single cell ($3→2+1$) immediately followed by the dissolution of the two-cell group ($2→1+1$). A lot of constrained triplets can be constructed, for example, in our
illustrations we use the triplet $κ(C1)=2+2,κ(C2)=4+4,κ(M)=4+2+2$. Originally, constrained triplets emerged in the model with unconstrained growth (Pichugin et al., 2017), where the growth rate of
the constrained life cycle $κ(M)$ was found to be always between growth rates of the constraining life cycles $κ(C1)$ and $κ(C2)$. It follows that in the present model with competition ($Kij≠0$),
each of two constraining life cycles ($κ(C1)$ and $κ(C2)$) must be either stable against two others or unstable against both. The constrained life cycle $κ(M)$ in turn is always invaded by one of
constraining life cycles and is stable against the other (see Appendix 1). Hence, the number of possible pairwise invasion patterns for such a triplet is limited to $2⋅2⋅2=8$ (see Figure 5B). Among
these eight patterns, two feature hierarchical dominance, where the life cycles can be ordered in a way that a higher-order life cycle dominates the lower-order life cycles (with the constrained life
cycle $κ(M)$ always being in the middle of hierarchy) (see Figure 6A). In a larger dataset (66,000 entries) with random birth, death, and competition rates, hierarchical dominance was observed in
about 78% of entries. Two patterns feature bistability between constraining life cycles $κ(C1)$ and $κ(C2)$ (see Figure 6B). These patterns appear in about 11% of the dataset. Two more patterns
feature a coexistence between all three life cycles (see Figure 6C). Note that unlike a pair of life cycles with unique coexistence equilibrium considered in section ‘Competition between pairs of
life cycles may result in coexistence or bistability,’ the triplet features a range of stable coexistence states. The coexistence pattern is similarly frequent, observed in 11% of the dataset.
Finally, the least frequent two patterns are non-hierarchical dominance, where one constraining life cycle dominates another, but in the abundance of the constrained life cycle, the invasion pattern
is inverse (see Figure 6D). They appear with three orders of magnitude lower frequency, smaller than 0.01% of cases.
There are only eight patterns of pairwise invasion in a constrained triplet of life cycles.
The fundamental feature of constrained triplets is that any constrained life cycle ($κ(M)$) can always be invaded by exactly one constraining life cycles ($κ(C1)$ and $κ(C2)$). Hence, any life
cycle, which can be constrained by two others, cannot be an ESS. We found that any life cycle with more than two offspring can be constrained (see Appendix 1 for the proof). As a result, only binary
fragmentation life cycles can be evolutionary stable strategies.
Killer and victim kernels guarantee a dominance of a single life cycle
As shown above, the evolutionary dynamics of interacting life cycles can be quite complex. In this context, a special interest attracts these cases, where the complexity of dynamics is limited. In
this section, we present two forms of interaction matrices ($K$), which guarantee that the evolutionary outcome is a straightforward domination of a single life cycle.
The first special case is the killer kernel, defined as
(15) ${K}_{ij}={k}_{j}.$
There, the probability of a group to die in an encounter depends only on the size of the opponent group ($j$), hence the name.
For an arbitrary killer kernel, a single life cycle has the same demographic composition in the stationary state as it has in the no-interactions model ($Kij=0$) (see Appendix 2 for the proof).
This feature of the demography leads to the result that the outcome of evolution under a killer kernel is also the same as in the no-interaction model. To show that, consider a composite population
containing multiple life cycles; its dynamics is described by Equation (7) and depends on the composite projection ($A~$) and competition ($K~$) matrices. If the competition matrix $K$ is a killer
kernel, the composite competition matrix $K~$ defined in Equation (9) is a killer kernel as well. Since the population dynamics of both single and multiple life cycles are governed by equations with
the same structure (Equation (2) and (7), respectively), the results of Appendix 2 carry over to a composite population. Specifically, the stationary state of the composite population is proportional
to the eigenvector of the composite population projection matrix $A~$ corresponding to its leading eigenvalue. The composite population projection matrix $A~$ defined in Equation (8) is a block
diagonal matrix, composed of population projection matrices of the life cycles constituting the composite population. Thus, the leading eigenvalue of $A~$ is the largest among the eigenvalues of all
population projection matrices comprising $A~$. Additionally, the corresponding eigenvector has nonzero components only at the positions in $x~$ associated with the block having the maximal leading
eigenvalue – that is, only that life cycle constitutes the stationary state. This rule is equivalent to the choice of the fastest growing life cycle in the linear model. Thus, the evolution of life
cycles competing by a killer kernel ($Kij=kj$) can be reduced to growth competition, which results in the survival of only one life cycle.
Another special case is the victim kernel defined as
(16) ${K}_{ij}={k}_{i}.$
There, the chance of a group to die depends only on the size of that group ($i$). For an arbitrary victim kernel, the carrying capacity of a single life cycle can be explicitly found. The total
number of groups at the stationary state $N*=∑ixi*$ is equal to the growth rate of this life cycle in the no-interaction model ($Kij=0$) with modified cell division rates $bi′=bi/ki$ and group death
rates $di′=di/ki$ (see Appendix 3 for the proof).
We found that in the composite population with many life cycles interacting by a victim kernel, only one survives in the long run (see the proof in Appendix 3). In such a scenario, each life cycle
grows in numbers if the total population size is smaller than the carrying capacity of that life cycle (derived in Appendix 3). If the whole population size exceeds this value, the life cycle
gradually dies out. Hence, selection favors the life cycle with the largest carrying capacity because it can grow in a dense population, when all other life cycles die from intense competition.
In both killer and victim kernels, the evolutionary dynamics is reduced to optimization of a single trait: growth rate for the killer kernel and carrying capacity for the victim kernel. Thus, the
result of selection in these scenarios is a dominance of a single life cycle over all suboptimal competitors.
Conditions promoting the evolution of multicellular life cycles
We conclude our findings with one of the most biologically interesting cases – the evolution of multicellular life cycles in a population dominated by unicellular organisms. This is also one of the
simplest scenarios as it deals with the simplest, unicellular life cycle 1 + 1.
If a unicellular population is abundant in the system, its stationary state can be explicitly found (see Appendix 4). Hence, for an arbitrary invading multicellular life cycle $κ(M)$, its invasion
matrix ($A(M,1)$) can be found explicitly. As a result, a multicellular invader can spread from rare in a population of unicellular residents if this life cycle has a positive growth rate in a model
with unconstrained growth with modified group death rates,
(17) ${d}_{i}^{\mathrm{\prime }}={d}_{i}+\frac{{b}_{1}-{d}_{1}}{{K}_{1,1}}{K}_{i,1}.$
The successful multicellular invader drives the unicellular life cycle to extinction if the unicellular life cycle cannot invade from rare. If the unicellular life cycle is an invader, its invasion
matrix ($A(1,M)$) has size $1×1$ and the invasion rate is just equal to its only element. The unicellular life cycle cannot invade into a multicellular resident life cycle if
(18) ${b}_{1}-{d}_{1}-\sum _{i=1}^{{m}^{\left(M\right)}}{K}_{1,i}{x}_{i}^{\left(M\right)\ast }<0,$
where $m(M)$ is the maximal group size in the life cycle $κ(M)$, and $xi(M)*$ is the composition of the population when $κ(M)$ is abundant. Derivations of both conditions are presented in Appendix
In this article, we have added an ecological dimension to the problem of life cycle evolution. Specifically, we are interested in three aspects of the eco-evolutionary dynamics: (i) What is a
population dynamics of a single life cycle? (ii) What is the evolutionary dynamics of multiple life cycles? (iii) What are the possible constraints on the outcomes of that evolution? All three
questions has been extensively studied for the model with unconstrained growth (Pichugin et al., 2017), and it is natural to contrast our current findings with these results.
First, the introduction of group competition completely changes the population dynamics of the life cycle growth. Instead of the exponential explosion of the population size occurring in the model
with unconstrained population size, the growth in our current model slows down with the population size. Eventually, the population approaches a stationary state with the limited population size and
constant composition (see Figure 2B).
Second, frequency-dependent selection arising from competition allows diverse outcomes of evolution: the stationary state can feature a coexistence of multiple life cycles or multistability between
several possible end states, as shown in Figure 3. By contrast, evolution in the model with unconstrained growth always results in a survival of a single life cycle, independently of the initial
Third, despite all the differences in the population and evolutionary dynamics, the restrictions on the possible evolutionary stable strategies are exactly the same in the models with and without
population size constraints (see section ‘The set of possible evolutionary stable strategies is restricted’). In both models, an ESS may only feature a fragmentation into exactly two pieces. Any life
cycle producing more than two offspring can always be invaded by another life cycle with a smaller number of offspring.
Beyond the scope of our model, life cycles with fragmentation into multiple pieces may become evolutionary stable strategies if fragmentation is costly (e.g., imposes a risk of cell loss). There,
binary fragmentation is no longer special as a fragmentation into multiple pieces can win growth rate competition. Nevertheless, constrained triplets of life cycles still exist under a costly
fragmentation scenario but the constraining condition is different from the one presented here. There, a life cycle containing two different subsets of offspring with the same combined size is
constrained between two life cycles, which use either of these subsets twice (Pichugin and Traulsen, 2020). For instance, $κ(M)=2+1+1$ is constrained as it has different offspring subsets 2 and $1+1$
with the same combined size. This life cycle is constrained between life cycles $κ(C1)=2+2$ and $κ(C2)=1+1+1+1$, which use one of these subsets twice. Since the scenario of costly fragmentation
still contains constrained triplets, life cycles satisfying the rule above, such as 2 + 1 + 1, cannot be evolutionary stable strategies there. Note that this rule works in the present model too (so
there are actually two ways to construct constraining triplets), but it is a weaker condition than the rule presented in section ‘The set of possible evolutionary stable strategies is restricted’ and
does not allow to construct any additional constraining triplets.
In the broad context of the eco-evolutionary dynamics, our dynamical Equations (2) and (7) bear a similarity with the generalized Lotka–Volterra (GLV) equations widely used in ecology: both contain a
linear growth term and a nonlinear competition term (typically of the second order) balancing out the linear growth. However, our equations are not equivalent to the GLV. In the GLV, individuals
corresponding to different elements of the population vector reproduce independently, that is, in our terms, the population projection matrix $A$ is diagonal. In our model, however, an individual
group changes its state in the course of the life cycle and the population projection matrix $A$ is not diagonal. Of course, if the population projection matrix $A$ can be diagonalized, we can
perform a linear transformation $x→Cy$ ($C$ is a matrix) to make the linear term in our model diagonal as in GLV. However, in this case, the interaction term will lose the GLV form of the
modification of the growth rate ($-xi∑jKijxj$, see Equation (1)) and will become a general second-order term instead ($-∑jkKijxjxk$). Given that our system is not a GLV in disguise, it is
surprising how much of the analysis presented here has been performed using the approaches designed to analyze GLV systems.
In this article, we found that the competition plays a major role in the evolution of multicellular life cycles. Our choice of the competition matrix values was driven by theoretical aspects of this
article: we either use randomly drawn values for numerical simulations or choose specific forms leading to analytical results. Yet, competition in natural populations is neither random nor fine-tuned
to mathematically beautiful outcomes. What might empirical competition matrices in a stage-structured population look like? Unfortunately, the demographics of simple multicellular species is not
sufficiently studied experimentally. Still, we can consider an example of emergence of Pseudomonas fluorescens colonies, where a competition plays a major role in the population dynamics and
evolution (Rainey and Travisano, 1998; Rainey and Rainey, 2003; Hammerschmidt et al., 2014). In a still liquid media, these initially unicellular bacteria evolve a ‘glue’ production, which causes
formation of multicellular aggregates. These aggregates float atop the media, gaining an exclusive access to oxygen. Once the entire surface is covered by continuous bacterial biofilm, the
unicellular phenotype living in the body of the media is completely denied the oxygen access and dies out. In the framework of our study, the competition matrix $K$ is determined by the capability to
block oxygen and surface area access. Naturally, the more cells there are in the group, the more stress they apply to others and, at the same time, the more resistant to competition they are. In the
limit of small population size, single cells have almost no impact on others ($Ki,1≪1$) and are the most susceptible to oxygen denial ($K1,i≫1$). In opposite limit, an established mat of millions of
cells just drives everything else to extinction ($Ki,j≫1≫1$). For arbitrary competitors size, the terms $Kij$ should increase with the size of an opponent group ($Kij<Ki,j+1$) but decrease with the
size of the focal group ($Kij>Ki+1,j$). Similar competition matrices should arise in scenarios where being bigger is better.
Invasion of life cycles and restrictions on ESS
In this section, we show that any resident life cycle with fragmentation into multiple parts can always be invaded by at least one life cycle with a smaller number of offspring.
Consider a resident life cycle $κ(R)$, in which more than two offspring groups are produced as a result of fragmentation. The initial dynamics of any life cycle $κ(I)$ invading from rare can be
described by a linear model with death rates modified as
(19) ${d}_{i}\to {d}_{i}+\sum _{j}{K}_{i,j}{x}_{j}^{\left(R\right)}.$
The invasion is successful if the leading eigenvalue of the corresponding population projection matrix $A(I,R)$, defined as in Equation (11), is positive.
In the analysis of the linear model (Pichugin et al., 2017), it was shown that if the fragmentation preserves the number of cells (no cell loss), then for any multiple fragmentation life cycle, there
exist two constraining life cycles with a smaller number of offspring. For any combination of cell division rates (b[i]) and group death rates (d[i]), one of the constraining life cycles has a larger
growth rate than the focal multiple fragmentation life cycle, while another has a smaller growth rate.
Now consider the invasion from rare in our present model with competition. The initial invasion rate computed to the leading eigenvalue of the matrix $A(I,R)$ is equal to the growth rate of the
invader life cycle in an environment modified according to Equation (19). Therefore, for any resident population, the invasion rate of the constrained life cycle is always in between the invasion
rates of the constraining life cycles. If the resident population is formed by the constrained life cycle itself, its self-invasion rate is zero. Hence, one of the constraining life cycles has a
larger invasion rate (i.e., it is positive), while another has a smaller invasion rate (negative). As a result, the constrained life cycle can always be invaded by exactly one of its constraining
life cycles. No constrained life cycle can be an ESS. Since any life cycle with more than two offspring is constrained, only binary fragmentation can be an ESS.
To conclude Appendix 1, we consider the resident population formed by a constraining life cycle. If the constrained life cycle has a positive invasion rate, then another constraining life cycle must
have a positive invasion rate as well. Alternatively, the constrained life cycle has a negative invasion rate, then another constraining life cycle also has a negative invasion rate. Thus, a
constraining resident is either resistant to invasion from both other life cycles in a triplet or can be successfully invaded by both of them.
Population dynamics under the killer kernel
In this section, we show that the demographic distribution of populations in a stationary regime is identical in the linear model ($Ki,j=0$) and under the killer kernel ($Ki,j=kj$).
First, we consider a linear model, where the population dynamics is governed by the population projection matrix $A$:
(20) $\frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}.$
A number of useful properties of this dynamics comes from the Perron–Frobenius theorem for irreducible non-negative matrices. Here, non-negative matrix means a matrix with all elements greater or
equal to zero. Matrix $M$ is irreducible if its representation by a directed graph, where node $i$ has an edge to node $j$ only if $Mij>0$, is a strongly connected graph, that is, there is a path
from any node to any other node. Since the states $i$ and $j$ represent groups of different sizes, and the population projection matrix $A$ describes the dynamics of the system, the irreducibility of
$A$ means that group of any given size $i$ can reach any other size $j$ through growth and fragmentation.
However, the population projection matrix $A$ itself is not non-negative as its main diagonal is negative ($Aii=-ibi-di$). It is not always irreducible as well: if the fragmentation mode produces
only multicellular offspring (e.g., $κ=2+2$), no group can ever reach unicellular state $j=1$. Both these limitations can be resolved. First, since the negative elements are located only on the main
diagonal, the matrix $A′=A+I⋅max(ibi+di)$ is non-negative. The matrix $A′$ has the same eigenvectors as $A$, and its eigenvalues ($λ′$) relate to eigenvalues of the population projection matrix (
$λ$) as $λ′=λ+max(ibi+di)$. Second, the irreducibility arises only when there are groups sizes smaller than the smallest produced offspring. The number of groups of these sizes will continuously
decrease in time: they will grow into larger sizes and spontaneously die. Since fragmentation is not able to resupply these small groups, the population will eventually get rid of them, so these
small sizes can be discarded from consideration of the long-term dynamics. Thus, without loss of generality, we can truncate the population projection matrix to take into account only the sizes
actually emerging in a life cycle. The resulting modified matrix is non-negative and irreducible, hence, the Perron–Frobenius theorem applies. From this theorem, it immediately follows that for
arbitrary birth rates (b[i]), death rates (d[i]), and the life cycle executed ($κ$), the population projection matrix has a simple leading eigenvalue $λ$, its eigenspace is one-dimensional, all
components of the eigenvector are positive, and no other eigenvectors of this matrix have all their components positive.
Therefore, in the linear model, the stationary regime is an exponentially growing population (Pichugin et al., 2017)
(21) $\mathbf{x}\left(t\right)={\mathbf{w}}^{\ast }{e}^{\lambda t},$
where $λ$ is the leading eigenvalue of the matrix $A$, and $W*$ is the corresponding eigenvector,
(22) ${\mathbf{A}\mathbf{w}}^{\ast }=\lambda {\mathbf{w}}^{\ast }.$
Under the killer kernel, the death rates due to competition are the same for groups of all sizes. Hence, following Equation (2), the population dynamics of a life cycle under a killer kernel is
(23) $\begin{array}{rl}\frac{d\mathbf{x}}{dt}& =\mathbf{A}\mathbf{x}-\mathrm{diag}\left(\mathbf{K}\mathbf{x}\right)\mathbf{x}\\ & =\mathbf{A}\mathbf{x}-\mathrm{diag}\left(\sum _{j}{k}_{j}{x}_{j}\
mathbf{1}\right)\mathbf{x}\\ & =\left(\mathbf{A}-\sum _{j}{k}_{j}{x}_{j}\mathbf{I}\right)\mathbf{x},\end{array}$
where 1 is the vector of ones, and $I$ is the identity matrix ($diag(1)=I$). It can be shown by a direct calculation that the vector
(24) ${\mathbf{x}}^{\ast }=\frac{\lambda }{\sum _{i}{k}_{i}{w}_{i}}{\mathbf{w}}^{\ast }$
is the stationary state of the dynamics (23):
(25) $\begin{array}{rl}\frac{d{\mathbf{x}}^{\ast }}{dt}& =\mathbf{A}{\mathbf{x}}^{\ast }-\sum _{j}{k}_{j}{x}_{j}^{\ast }{\mathbf{x}}^{\ast }\\ & =\frac{\lambda }{\sum _{i}{k}_{i}{w}_{i}}\mathbf{A}{\
mathbf{w}}^{\ast }-{\left(\frac{\lambda }{\sum _{i}{k}_{i}{w}_{i}}\right)}^{2}\sum _{j}{k}_{j}{w}_{j}{\mathbf{w}}^{\ast }\\ & =\frac{{\lambda }^{2}}{\sum _{i}{k}_{i}{w}_{i}}{\mathbf{w}}^{\ast }-\frac
{{\lambda }^{2}}{\sum _{i}{k}_{i}{w}_{i}}{\mathbf{w}}^{\ast }\\ & =\mathbf{0}.\end{array}$
Note that while any other eigenvector $W′$ of the population projection matrix $A$ can be used in Equation (25), only the leading eigenvector $W*$ can represent a biologically meaningful population,
as all other eigenvectors have negative elements, which would mean negative number of groups of some size at the stationary state.
Since the stationary state $x*$ under the killer kernel is proportional to the vector describing the stationary distribution $W*$ in the linear model, the population compositions in both scenarios
are the same.
Population dynamics under the victim kernel
Dynamics of a single life cycle
In this section, we show that the task of finding the equilibrium population size ($N*=∑ixi*$) under the victim kernel ($Ki,j=ki$) is mathematically equivalent to the task of finding the population
growth rate in the linear model ($Ki,j=0$) with modified cell birth rates ($bi→bi/ki$) and group death rates ($di→di/ki$).
In the linear model, the population growth rate is found as the leading eigenvalue of the population projection matrix determined by Pichugin et al., 2017
(26) $\begin{array}{cc}\hfill 0& =det\left(A-\lambda I\right)\hfill \\ & =|\begin{array}{ccccc}\hfill -{b}_{1}-{d}_{1}-\lambda \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill
& \hfill m{b}_{m}{\pi }_{1}\left(\kappa \right)\hfill \\ \hfill {b}_{1}\hfill & \hfill -2{b}_{2}-{d}_{2}-\lambda \hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill m{b}_{m}{\pi }_{2}\
left(\kappa \right)\hfill \\ \hfill 0\hfill & \hfill 2{b}_{2}\hfill & \hfill -3{b}_{3}-{d}_{3}-\lambda \hfill & \hfill \mathrm{\dots }\hfill & \hfill m{b}_{m}{\pi }_{3}\left(\kappa \right)\hfill \\ \
hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \
mathrm{\dots }\hfill & \hfill m{b}_{m}{\pi }_{m}\left(\kappa \right)-m{b}_{m}-{d}_{m}-\lambda \hfill \end{array}|.\hfill \end{array}$
Under the victim kernel, the death rate due to the competition depends only on the size of the outcompeted group. Hence, following Equation (2), the stationary state of a life cycle under a victim
kernel satisfies
(27) $\begin{array}{rl}\frac{d{\mathbf{x}}^{\ast }}{dt}& ={\mathbf{A}\mathbf{x}}^{\ast }-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\ast }\right){\mathbf{x}}^{\ast }\\ & ={\mathbf{A}\mathbf{x}}^{\ast
}-\mathrm{diag}\left(\mathbf{k}\sum _{j}{x}_{j}^{\ast }\right){\mathbf{x}}^{\ast }\\ & =\left(\mathbf{A}-{N}^{\ast }\mathrm{diag}\left(\mathbf{k}\right)\right){\mathbf{x}}^{\ast }\\ & =0,\end{array}$
where $k$ is a vector constructed from any column of the competition matrix $K$ (they are all identical), and $N*=∑ixi*$ is the equilibrium population size.
The last equality in Equation (27) implies that by Fredholm alternative one out of two conditions is satisfied (Hoffman, 1971):
(28) $\begin{array}{c}{\mathbf{x}}^{\ast }=0,\\ \text{or}\\ det\left(\mathbf{A}-{N}^{\ast }\mathrm{diag}\left(\mathbf{k}\right)\right)=0\end{array}$
Limiting ourselves to scenarios where the stationary state $x*$ is not an empty population, we can conclude that the population at the equilibrium satisfies
(29) $\begin{array}{cc}\hfill 0& =det\left(A-{N}^{*}\mathrm{diag}\left(k\right)\right)\hfill \\ & =|\begin{array}{ccccc}\hfill -{b}_{1}-{d}_{1}-{k}_{1}{N}^{*}\hfill & \hfill 0\hfill & \hfill 0\hfill
& \hfill \mathrm{\dots }\hfill & \hfill m{b}_{m}{\pi }_{1}\left(\kappa \right)\hfill \\ \hfill {b}_{1}\hfill & \hfill -2{b}_{2}-{d}_{2}-{k}_{2}{N}^{*}\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\
hfill & \hfill m{b}_{m}{\pi }_{2}\left(\kappa \right)\hfill \\ \hfill 0\hfill & \hfill 2{b}_{2}\hfill & \hfill -3{b}_{3}-{d}_{3}-{k}_{3}{N}^{*}\hfill & \hfill \mathrm{\dots }\hfill & \hfill m{b}_{m}
{\pi }_{3}\left(\kappa \right)\hfill \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill 0\hfill & \
hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill m{b}_{m}{\pi }_{m}\left(\kappa \right)-m{b}_{m}-{d}_{m}-{k}_{m}{N}^{*}\hfill \end{array}|\hfill \\ & ={k}_{1}\cdot {k}_{2}\cdot
{k}_{3}\cdot \mathrm{\dots }\cdot {k}_{m}\hfill \\ & \cdot |\begin{array}{ccccc}\hfill -\frac{{b}_{1}}{{k}_{1}}-\frac{{d}_{1}}{{k}_{1}}-{N}^{*}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \
mathrm{\dots }\hfill & \hfill m\frac{{b}_{m}}{{k}_{m}}{\pi }_{1}\left(\kappa \right)\hfill \\ \hfill \frac{{b}_{1}}{{k}_{1}}\hfill & \hfill -2\frac{{b}_{2}}{{k}_{2}}-\frac{{d}_{2}}{{k}_{2}}-{N}^{*}\
hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill m\frac{{b}_{m}}{{k}_{m}}{\pi }_{2}\left(\kappa \right)\hfill \\ \hfill 0\hfill & \hfill 2\frac{{b}_{2}}{{k}_{2}}\hfill & \hfill -3\frac
{{b}_{3}}{{k}_{3}}-\frac{{d}_{3}}{{k}_{3}}-{N}^{*}\hfill & \hfill \mathrm{\dots }\hfill & \hfill m\frac{{b}_{m}}{{k}_{m}}{\pi }_{3}\left(\kappa \right)\hfill \\ \hfill \mathrm{⋮}\hfill & \hfill \
mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\ddots }\hfill & \hfill \mathrm{⋮}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill m\frac
{{b}_{m}}{{k}_{m}}{\pi }_{m}\left(\kappa \right)-m\frac{{b}_{m}}{{k}_{m}}-\frac{{d}_{m}}{{k}_{m}}-{N}^{*}\hfill \end{array}|,\hfill \end{array}$
where in the last step, we divided the $i$th column by k[i] for all $i$. Comparing the determinants in Equations (26) and (29), we find that they are identical after the substitution
(30) $\begin{array}{rl}\lambda & \to {N}^{\ast },\\ {b}_{i}& \to \frac{{b}_{i}}{{k}_{i}},\\ {d}_{i}& \to \frac{{d}_{i}}{{k}_{i}}.\end{array}$
Thus, the equilibrium population size ($N*$) under a victim kernel can be found as a population growth rate in the linear model with modified cell birth and group death rates.
Competition of multiple life cycles
In this section, we show that in a composite population, in which multiple ($r$) life cycles compete through a victim kernel ($Ki,j=ki$), only one life cycle survives to the stationary state, and
this is the same life cycle that has the maximal carrying capacity if grown in isolation, and the same life cycle that would be evolutionarily optimal in the linear model ($Ki,j=0$) with modified
cell division ($bi→bi/ki$) and group death ($di→di/ki$) rates.
If the competition matrix $K$ constitutes a victim kernel, then the composite competition matrix $K~$ defined as in Equation (9) is a victim kernel as well. Hence, our results from section ‘Dynamics
of a single life cycle’ hold also for a composite population. In particular, the population size at the stationary state ($N~*=∑p=1r∑i=1nxi(p)$) is determined by
(31) $\begin{array}{rl}0& =det\left(\stackrel{~}{\mathbf{A}}-{\stackrel{~}{N}}^{\ast }\mathrm{diag}\left(\stackrel{~}{\mathbf{k}}\right)\right)\\ & =\prod _{p=1}^{r}det\left({\mathbf{A}}^{\left(p\
right)}-{\stackrel{~}{N}}^{\ast }\mathrm{diag}\left(\mathbf{k}\right)\right),\end{array}$
where $k~$ is the vector made from a single column of the composite competition matrix $K~$ (it is a concatenation of $r$ vectors $k$). In the second step of Equation (31), we used the property of
block-diagonal matrices: the determinant of a block-diagonal matrix is the product of determinants of its blocks. Equation (31) is satisfied if one of the multipliers is equal to zero. The $p$th
multiplier becomes zero when the composite population reaches a size equal to the carrying capacity of the $p$th life cycle; cf. Equation (28). At that moment, the $p$th life cycle can neither grow
or decay, the life cycles with lower carrying capacity decay as they cannot keep up with competition caused by overcrowding, and only the life cycles with carrying capacity larger than the population
size can grow in numbers. Equation (31) has $r$ possible solutions with respect to $N~*$ – one for each competing life cycle. However, only one solution can represent a stationary population, where
no life cycle can grow in numbers – the one with the maximal population size. There, one life cycle is stationary, while all others decrease in numbers due to overcrowding. Thus, the outcome of life
cycle competition under a victim kernel is the survival of a single life cycle, which has the maximal equilibrium population size among all competitors. According to section ‘Dynamics of a single
life cycle,’ these population sizes are equal to the growth rates of the corresponding life cycles in a linear model with the modified cell division and group death rates. Consequently, the maximal
population size corresponds to the fastest growing life cycle in that modified linear model. The population corresponding to the highest eigenvalue takes over and will dominate the system. This means
that the life cycle with the largest population size in isolation dominates over all other life cycles in the competition through a victim kernel.
Invasion into unicellular resident and invasion of the unicellular invader
If the resident is unicellular ($κ(R)=1+1$), its steady state is given by the solution of
(32) $\left({b}_{1}-{d}_{1}\right){x}_{1}^{\ast }-{K}_{1,1}{x}_{1}^{\ast 2}=0,$
equal to
(33) ${x}_{1}^{\ast }=\frac{{b}_{1}-{d}_{1}}{{K}_{1,1}}.$
Then, for an arbitrary invader multicellular life cycle $κ(M)$, the invasion matrix is given by
(34) $\begin{array}{rl}{\mathbf{A}}^{\left(M,R\right)}& ={\mathbf{A}}^{\left(M\right)}-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\left(R\right)}\right)\\ & ={\mathbf{A}}^{\left(M\right)}-\mathrm
{diag}\left({K}_{i,1}{x}_{1}^{\ast }\right)\\ & ={\mathbf{A}}^{\left(M\right)}-\frac{{b}_{1}-{d}_{1}}{{K}_{1,1}}\mathrm{diag}\left({K}_{i,1}\right).\end{array}$
This is equivalent to the linear growth of the invader life cycle in an environment with modified death rates
(35) ${d}_{i}^{\mathrm{\prime }}={d}_{i}+\frac{{b}_{1}-{d}_{1}}{{K}_{1,1}}{K}_{i,1}.$
If the invader is unicellular ($κ(I)=1+1$), then the invasion matrix is effectively a $1×1$ matrix since the invader contains only isolated cells. Even if $A(I,M)$ formally has a larger size, it is a
block matrix, with the element $A1,1(I,M)$ being a block, with a value that is equal to the growth rate of the unicellular life cycle. This value is
(36) $\begin{array}{rl}{\mathbf{A}}_{1,1}^{\left(I,M\right)}& ={\mathbf{A}}_{1,1}^{\left(I\right)}-\mathrm{diag}\left({\mathbf{K}\mathbf{x}}^{\left(M\right)\ast }{\right)}_{1,1}\\ & ={b}_{1}-{d}_{1}-
\sum _{i=1}^{{m}^{\left(M\right)}}{K}_{1,i}{x}_{i}^{\left(M\right)\ast },\end{array}$
where $xi(M)$ is the number of groups of size $i$ in the resident population, and $m(M)$ is the maximal group size of the resident life cycle. The unicellular invader cannot spread in a resident
population, when $A1,1(I,M)<0$.
Parameters of calculation used in figures
In Figure 2A, we used $b1=1,d1=0,K11=1$. The initial population size was drawn from the uniform distribution $U(0.1,2)$.
In Figure 2B, we used $b=(1,1)$ and $d=(0,0)$. The competition matrix was
$K=\left(\begin{array}{cc}1& 0.2\\ 0.2& 0.5\end{array}\right).$
The initial number of groups of each size was randomly drawn from the uniform distribution $U(0.1,2)$.
In Figure 3, we considered the life cycles 1 + 1 and 2 + 1. We used $b=(1,0.5)$ and $d=(0,0)$. For each plotted trajectory, the populations were initialized with $x1+1=(s1)$, $x2+1=(s2,0)$, where
$s1,s2∈{0.1,0.2,0.3,…,1.0}$. The dynamics shown in the four panels differ by the competition matrix used:
• Panel A
$K=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$
• Panel B
$K=\left(\begin{array}{cc}\hfill 3\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$
• Panel C
$K=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 1\hfill \end{array}\right)$
• Panel D
$K=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0.6\hfill & \hfill 0.4\hfill \end{array}\right)$
In Figure 6, we considered the life cycles $κ(C1)=2+2$, $κ(C2)=4+4$, $κ(M)=4+2+2$. Panels differ in birth, death, and competition rates. Trajectories in each panel have different initial states.
For each initial state, the composite population contains all three life cycles with different fractions $s(C1),s(C2),s(M)$, such that $s(C1)+s(C2)+s(M)=1$. The initial group sizes distribution
is proportional to the equilibrium population of that life cycle alone:
(37) $\begin{array}{rl}{\stackrel{~}{\mathbf{x}}}_{t=0}=& \left({s}^{\left(C1\right)}{{\mathbf{x}}^{\mathbf{\ast }\mathbf{\left(}\mathbf{C}\mathbf{1}\mathbf{\right)}}}^{T},{s}^{\left(C2\right)}{{\
mathbf{x}}^{\mathbf{\ast }\mathbf{\left(}\mathbf{C}\mathbf{2}\mathbf{\right)}}}^{T},{s}^{\left(M\right)}{{\mathbf{x}}^{\mathbf{\ast }\mathbf{\left(}\mathbf{M}\mathbf{\right)}}}^{T}{\right)}^{T},\end
where the vectors $x∗(C1)T$, $x∗(C2)T$, $x∗(M)T$ are equilibrium population states of corresponding life cycles grown in isolation, computed according to Equation (14).
• In panel A (hierarchic dominance), we used $bi=1.0$, $di=0$ and the competition matrix $Kij=0.1$:
(38) ${K}_{\text{Hier. dom.}}=\left(\begin{array}{ccccccc}\hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill
0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\
hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\
\hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill
0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 1.1\hfill & \hfill 1.1\hfill & \hfill 1.1\hfill & \hfill 1.1\hfill & \hfill 1.1\hfill & \hfill 1.1\hfill & \hfill 1.1\
hfill \end{array}\right),$
or equivalently
(39) ${K}_{ij}=\left\{\begin{array}{cc}1.1,\hfill & i=7\hfill \\ 0.1,\hfill & \text{otherwise}\hfill \end{array}$
• In panel B (bistability), we used $bi=1$ and $di=0$, while the competition matrix was
(40) ${K}_{\text{Bi-stability}}=\left(\begin{array}{ccccccc}\hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \
hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 10\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1
\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \
\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill
0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 10\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill
or equivalently
(41) ${K}_{ij}=\left\{\begin{array}{cc}10,\hfill & \left(i,j\right)=\left(2,7\right)\text{or}\left(7,2\right)\hfill \\ 0.1,\hfill & \text{otherwise}\hfill \end{array}$
• In panel C (coexistence), we used $bi=1$ and $di=0$, while the competition matrix was
(42) ${K}_{\text{Coex.}}=\left(\begin{array}{ccccccc}\hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0.1\
hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \
hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\
hfill & \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0.1\hfill & \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \
hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \\ \hfill 0\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill & \hfill 0.1\hfill \end{array}\
or equivalently
(43) ${K}_{ij}=\left\{\begin{array}{cc}0,\hfill & i+j=8\hfill \\ 0.1,\hfill & \text{otherwise}\hfill \end{array}$
• In panel D (non-hierarchical dominance), we used $di=0$,
(44) ${b}_{\text{Nonh. dom.}}=\left(6.48076,2.79693,2.3088,5.34057,1.0,1.62478,1.32615\right)$
(45) ${K}_{\text{Nonh. dom.}}=\left(\begin{array}{ccccccc}\hfill 0.12491\hfill & \hfill 0.16453\hfill & \hfill 0.40972\hfill & \hfill 0.13981\hfill & \hfill 0.03496\hfill & \hfill 0.79364\hfill & \
hfill 0.06097\hfill \\ \hfill 0.36477\hfill & \hfill 0.10859\hfill & \hfill 0.09099\hfill & \hfill 0.70391\hfill & \hfill 0.01714\hfill & \hfill 0.53354\hfill & \hfill 0.49778\hfill \\ \hfill 0.16432
\hfill & \hfill 0.58285\hfill & \hfill 1\hfill & \hfill 0.01918\hfill & \hfill 0.01268\hfill & \hfill 0.08071\hfill & \hfill 0.11208\hfill \\ \hfill 0.13519\hfill & \hfill 0.56771\hfill & \hfill
0.11879\hfill & \hfill 0.02601\hfill & \hfill 0.08905\hfill & \hfill 0.1172\hfill & \hfill 0.14661\hfill \\ \hfill 0.01251\hfill & \hfill 0.31353\hfill & \hfill 0.02639\hfill & \hfill 0.07433\hfill &
\hfill 0.05312\hfill & \hfill 0.22877\hfill & \hfill 0.14841\hfill \\ \hfill 0.06258\hfill & \hfill 0.43833\hfill & \hfill 0.30679\hfill & \hfill 0.3323\hfill & \hfill 0.01014\hfill & \hfill 0.09637\
hfill & \hfill 0.24751\hfill \\ \hfill 0.19945\hfill & \hfill 0.022\hfill & \hfill 0.00087\hfill & \hfill 0.2469\hfill & \hfill 0.09733\hfill & \hfill 0.08247\hfill & \hfill 0.37168\hfill \end{array}
18. Book
Linear algebra
Englewood Cliffs, NJ: Prentice-Hall.
29. Book
The Major Transitions in Evolution
Oxford: W. H. Freeman.
Article and author information
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1. Vanessa Ress
2. Arne Traulsen
3. Yuriy Pichugin
Eco-evolutionary dynamics of clonal multicellular life cycles
eLife 11:e78822.
Further reading
1. Computational and Systems Biology
2. Physics of Living Systems
Explaining biodiversity is a fundamental issue in ecology. A long-standing puzzle lies in the paradox of the plankton: many species of plankton feeding on a limited variety of resources coexist,
apparently flouting the competitive exclusion principle (CEP), which holds that the number of predator (consumer) species cannot exceed that of the resources at a steady state. Here, we present a
mechanistic model and demonstrate that intraspecific interference among the consumers enables a plethora of consumer species to coexist at constant population densities with only one or a handful
of resource species. This facilitated biodiversity is resistant to stochasticity, either with the stochastic simulation algorithm or individual-based modeling. Our model naturally explains the
classical experiments that invalidate the CEP, quantitatively illustrates the universal S-shaped pattern of the rank-abundance curves across a wide range of ecological communities, and can be
broadly used to resolve the mystery of biodiversity in many natural ecosystems.
1. Chromosomes and Gene Expression
2. Computational and Systems Biology
Genes are often regulated by multiple enhancers. It is poorly understood how the individual enhancer activities are combined to control promoter activity. Anecdotal evidence has shown that
enhancers can combine sub-additively, additively, synergistically, or redundantly. However, it is not clear which of these modes are more frequent in mammalian genomes. Here, we systematically
tested how pairs of enhancers activate promoters using a three-way combinatorial reporter assay in mouse embryonic stem cells. By assaying about 69,000 enhancer-enhancer-promoter combinations we
found that enhancer pairs generally combine near-additively. This behaviour was conserved across seven developmental promoters tested. Surprisingly, these promoters scale the enhancer signals in
a non-linear manner that depends on promoter strength. A housekeeping promoter showed an overall different response to enhancer pairs, and a smaller dynamic range. Thus, our data indicate that
enhancers mostly act additively, but promoters transform their collective effect non-linearly.
1. Computational and Systems Biology
2. Physics of Living Systems
Planar cell polarity (PCP) – tissue-scale alignment of the direction of asymmetric localization of proteins at the cell-cell interface – is essential for embryonic development and physiological
functions. Abnormalities in PCP can result in developmental imperfections, including neural tube closure defects and misaligned hair follicles. Decoding the mechanisms responsible for PCP
establishment and maintenance remains a fundamental open question. While the roles of various molecules – broadly classified into “global” and “local” modules – have been well-studied, their
necessity and sufficiency in explaining PCP and connecting their perturbations to experimentally observed patterns have not been examined. Here, we develop a minimal model that captures the
proposed features of PCP establishment – a global tissue-level gradient and local asymmetric distribution of protein complexes. The proposed model suggests that while polarity can emerge without
a gradient, the gradient not only acts as a global cue but also increases the robustness of PCP against stochastic perturbations. We also recapitulated and quantified the experimentally observed
features of swirling patterns and domineering non-autonomy, using only three free model parameters - the rate of protein binding to membrane, the concentration of PCP proteins, and the gradient
steepness. We explain how self-stabilizing asymmetric protein localizations in the presence of tissue-level gradient can lead to robust PCP patterns and reveal minimal design principles for a
polarized system. | {"url":"https://elifesciences.org/articles/78822","timestamp":"2024-11-03T12:01:20Z","content_type":"text/html","content_length":"506843","record_id":"<urn:uuid:dd1ce0a8-2428-4c8d-872c-f8052eb40488>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00186.warc.gz"} |
GRE Sample Questions
Quantitative Section : Quantitative Ability
Directions:In this section you will be given two quantities, one in column A and one in column B. You are to determine a relationship between the two quantities and mark.
A: If the quantity in column A is greater than the quantity in column B.
B: If the quantity in column B is greater than the quantity in column A.
C: If the quantities are equal.
D: If the comparison cannot be determined from the information that is given.
1. Quantity A :
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
RSTU is a parallelogram.
2. Column A Column B
.25% of .25
A: Column A’s quantity is greater.
B: Column B’s quantity is greater.
C: The quantities are the same.
D: The relationship cannot be determined from the information given.
3. Tom, Dick and Harry went for lunch to a restaurant. Tom had $100 with him, Dick had $60 and Harry had $409. They got a bill for $104 and decided to give a tip of $16. They further decided to share
the total expenses in the ratio of the amounts of money each carried. The amount of money which Tom paid more than what Harry paid is
A: 120
B: 200
C: 60
D: 24
E: 36
Ans: E
4. Arnold has enough gas to last him for thirty days. If he starts using 50% more gas, how many days will the same supply last him?
A: 10
B: 12
C: 15
D: 20
E: 25
5. If 2x > 24 and 3x < 48, which of the following is a possible value for x?
A: 12
B: 14
C: 16
D: 18
E: 20
6. Dan drives to Cheryl's house at an average speed of 40 mph. If he can drive 2/3 of the way there in an hour, how far away is Cheryl's house?
A: 60 miles
B: 20 miles
C: 80 miles
D: 50 miles
E: 55 miles
7. |-5| + |3| - |-5| =
A: -2
B: -13
C: 8
D: 3
E: 2
8. If a cube has a volume of 343 cubic inches, what is the length of one side?
A: 7 square inches
B: 30 square inches
C: 7 inches
D: 49 inches
E: 49 square inches
Looking at the figure above, if triangle ABC is an equilateral triangle and line BC is parallel to line DE, what is the measure of angle 5?
A: 60 degrees
B: 90 degrees
C: 120 degrees
D: 180 degrees
E: Not enough information is given to answer this question
10. If the radius of a circle is increased by 20% then the area is increased by :
A: 44%
B: 120%
C: 144%
D: 40%
E: None of the above
Ans : A
+ comments + 27 comments | {"url":"https://www.gre.examsavvy.com/2011/09/gre-sample-questions.html","timestamp":"2024-11-11T18:18:25Z","content_type":"application/xhtml+xml","content_length":"125061","record_id":"<urn:uuid:a57d587e-50b6-4fde-a2fb-c12b54983f02>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00252.warc.gz"} |
04559cam a2200481 i 4500 22682024 20220706131709.0 m |o d | cr ||||||||||| 111205s2015 flua ob 001 0 eng 7 cbc origcop u ncip 20 y-gencatlg acquire doabebook policy default 2020718861 9780750302630
hardback : acid-free paper DLC eng pn rda DLC QC787.P3 539.7/3 23 SCI051000 SCI040000 bisacsh Berz, M. An introduction to beam physics / M. Berz, K. Makino, Weishi Wan. Boca Raton : CRC Press/Taylor
and Francis Group, [2015] 1 electronic resource (xiv, 310 pages ) text txt rdacontent computer c rdamedia online resource cr rdacarrier Series in high energy physics, cosmology, and gravitation
Includes bibliographical references and index. Unrestricted online access "An Introduction to Beam Physics covers the principles and applications of differential algebra, a powerful new mathematical
tool. The authors discuss the uses for the computation of transfer maps for all kinds of particle accelerators or any weakly nonlinear dynamical system, such as planetary orbits. The book is of
interest to graduate students and researchers working in a broad range of disciplines, including applied mathematics, beam physics (accelerator physics, particle optics, geometric light optics),
astronomy, and electrical engineering. Topics covered include transfer matrices, mechanics and electrodynamics, nonlinear motion, differential algebra, the structure of the classes, computer
implementations, nonlinear maps, one pass systems, and repetitive systems"-- Provided by publisher. "Preface It has been 8 years since we started this book project, which originated from the lecture
note of a graduate level course taught by my coauthors at Michigan State University. Compared to the lecture note, the present book is more than twice as long, which is the result of a few
contributing factors. The obvious reason is the requirement that a book has to be more self contained than a lecture note. The more important reason is that, over the past decade, the field saw
significant development in a few areas and new materials have been added to reflect the change. A couple of examples are an overview of the development of aberration-corrected electron microscopes
and the treatment of the chicane bunch compressor. The last reason is more pesonal in nature. Over the past decades, the field of beam physics have become so diverse that each area has developed it's
own way of treating the problem and communications among different areas have been problematic. It's been our belief that modern map method is a good tool to reunite this divese field and that this
book offers the best platform to realize this goal. On one hand, we cover as widely as possible the topics in different areas of the field of beam physics, ranging from electron telescopes,
spectrometers to particle accelerators. On the other hand, we attempt to present traditionally more advanced topics, such as the resonances in circular accelerators, in an introductroy book using
modern map method, hence avoiding the elegant but more involved Hamiltonian formalism. The result is a book that requires no prior knowledge of beam physics and only basic understanding of college
level classical machenics, calculus and ordinary differential equations"-- Provided by publisher. Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International CC BY-NC-ND 4.0 https://
creativecommons.org/licenses/by-nc-nd/4.0/legalcode Description based on print version record; resource not viewed. Particle accelerators. Particle beams. Particles (Nuclear physics) Mathematics.
SCIENCE / Mathematical Physics. bisacsh SCIENCE / Nuclear Physics. bisacsh Makino, Kyoko. Wan, Weishi. Print version: An introduction to beam physics Boca Raton : CRC Press/Taylor & Francis Group,
[2015] 9780750302630 (hardback : acid-free paper) (DLC) 2011045943 gdcebookspublic 2020718861 https://hdl.loc.gov/loc.gdc/gdcebookspublic.2020718861 gdc/doab s-Online Electronic Resource | {"url":"https://lccn.loc.gov/2020718861/marcxml","timestamp":"2024-11-04T01:59:07Z","content_type":"application/xml","content_length":"9532","record_id":"<urn:uuid:1cb6c5a5-06d1-497a-853b-69b101ce8649>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00283.warc.gz"} |
Equations are being turned into graphics
This problem can occur when authors insert equations with the Equation Builder feature in Word 2007 or 2010, but you’re using Word 2003 for editorial. The Word 2007/2010 equations are not backward
compatible with Word 2003. Whenever you get a DOCX file:
• Open the file with Word 2007/2010.
• Run eXtyles Font Audit. Font Audit will add this message, “This document has equations created with the Microsoft Office 2007 Equation Builder” to the alert if there are any problematic
• If this message doesn’t appear, then you can just process the file as you normally do. However, if it does appear, use MathType 6.8 to convert all of the Word 2010 equations to MathType.
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What Is The Youngest Age Cats Can Have Kittens?
1 Answers
What Is The Youngest Age Cats Can Have Kittens?
Cats can be pregnant after getting their first heat cycle. First heat cycle in cats can occur at the age of 5-12 months with an average of 6 months. SO, cats can become pregnant as early as 5 months
old. Following are some information about cats heat cycle.
1. Cats can have 3-4 heat periods in one year
2. Every heat period in cats can be of 2-3 weeks
3. In case of breeding, heat period can be of 4 days
4. In case of unsuccessful mating heat period can be of 7-10 days
5. Heat period in cats can occur after 1-6 weeks of delivery
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Risk Management
Risk management is perhaps the most important part of trading. Traders can spend a lot of time researching or investing in the perfect trading system that will set them on the road to riches. No
system exists, however, that doesn’t produce losses and if it did, who would publicise such a holy grail. Regardless of your skill level or the effectiveness of your system, it will all come to
nought unless it is accompanied by appropriate rules of risk management. This section will explore concepts behind effective money and risk management. The table below is worth imprinting in memory.
Risk Management in Trading
The table shows the percentage profit required to bring your trading balance back to break even following losses. If you lose 10% of your capital you have to make 11% profit on your remaining
capital just to get back to the initial balance. If you lose 20% you have to make 25% and so on. It’s just not going to happen. Berkshire Hathaway, founded by legendary investor Warren Buffet,
averaged an annual growth in book value of 20.3% to its shareholders for over 40 years. Soros Fund Management, founded by George Soros, known as the man who ‘broke the Bank of England’ returned an
average of 20% over four decades.
The HFRI Fund Weighted Composite Index is a global, equal-weighted index of over 2,000 single-manager funds. In 2011, the worst performing decile declined by 30.7%. The top decile gained 19.5 per
cent. The composite result was a decline of 5.6%, (Source: Hedge Fund Research, Inc.). Granted, 2011 was a very challenging year for traders but if these are the results that professionals produced
it should give us pause for thought. Novice traders often drop 10%, 20% or 30% of their account and think they can make up for the losses as quickly as they lost it. Typically they engage in
overtrading to try and “catch up” quickly and end up in an even worse financial position.
Risk/Reward Ratio:
Risk/Reward ratio is used by traders and investors to calculate the return on an investment relative to the risk. For example, a trader opens a long position and places a stop loss order below the
entry price. The trader has calculated that if the market moves against them and the stop loss is triggered they will lose $1000. On the other hand, if the trade is profitable and reaches the exit
target, the potential profit is $3000. In this scenario the trade has a risk:reward ratio of 3:1. They are risking $1000 for a potential gain of $3000. It is a convention amongst traders to refer to
the risk/reward ratio even though the potential reward is quoted first.
Win/Loss Ratio:
Win/Loss ratio is a ratio of the total number of winning trades to the total number of losing trades. Let’s say a trading system on average produces 25 trades per annum. 10 are winners and 15 are
losers which is a win/loss ratio of 2:3. This helps a trader plan their money management. Of course, it’s not guaranteed but they have a reasonable expectation that approximately 40% of their
trades will be profitable. However, you can never be sure which 40% of trades will be the winners. An attractive win/loss ratio does not guarantee profits because results can depend on the percentage
capital risked per trade and the order of successful trades.
Consider a trading system that on average generates:
• One trade per week
• Win/loss ratio of 2:3 (40% of trades are profitable)
• Risk/Reward ratio of 4:1
• 5% of capital risked on every trade
The following table shows the results after 10 weeks trading a $10,000 account.
Risk Management in Trading
A 28.57% profit (2856.80/10000 x 100) after ten trades would keep most traders content. Now examine the results of a trading system that on average generates:
• One trade per week
• Win/loss ratio of 2:3 (40% of trades are profitable)
• Risk/Reward ratio of 4:1
• 10% of capital risked on every trade
Risk Management in Trading
This approach resulted in a loss of 5581.94 or 55.82% of the account and probably a long time on the side-lines. This exercise is not an attempt to fiddle the figures to prove a point. Both trading
approaches have a win/loss ratio of 2:3 and a considerable amount of research may have gone into developing these systems. However, we cannot be sure when the winning trades will come and the
percentage of capital risked has a huge effect on the outcome.
How much should I risk per trade?
As you can see the answer to this question is not straight forward. It depends on a number of factors but important considerations are:
• Number of trades the system generates
• Win/loss ratio
• Risk/reward ratio
• Percentage of capital risked per trade
In general, the more frequently you trade the smaller the percentage capital should be risked on each trade. Frequent trading means that there is a greater chance of a string of losses in a shorter
period of time. The win/loss ratio should give a good indication of how many losses in a row the system may produce. The risk/reward ratio will give a good indication of the potential profit from
the winning trades. This has to be greater than the total losses. A reasonable risk reward ratio would be in the 3:1 or 4:1 region. Consider that a 1:1 risk/reward ratio requires that, at a minimum,
over 50% of trades are winners.
In conclusion, if you’re just starting out trading it would be wise to only risk a small amount of your capital until your skills and confidence develop. Many professional traders won’t risk more
than to 2% or 3% of their capital on a trade. As a general guideline, your account should be able to handle the potential number of losing trades in a row inherent in your trading strategy. The
drawdown should not be so significant that you cannot trade due either to the resulting damage to your financial capital or psychological capital i.e. loss of confidence in the system or your
abilities. For active trading it is wise is to start small and as your trading account increases so too can the percentage capital risked on each trade. Knowledge of Fundamental Analysis, Technical
Analysis Indicators and/or Chart Patterns can help to implement effective money management strategies.
Return to Top of Risk Management
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Summary of the Operations
Summary of the Operations#
• transform applies a function to each element of the sequence, equivalent to the functional operation map
• select takes the first N` elements of the sequence satisfying a condition (via a selection mask or a predicate function)
• unique returns unique elements within a sequence
• histogram generates a summary of the statistical distribution of the sequence
• reduce traverses the sequence while accumulating some data, equivalent to the functional operation fold_left
• scan is the cumulative version of reduce which returns the sequence of the intermediate values taken by the accumulator
• adjacent_difference computes the difference between the current element and the previous or next one in the sequence
• discontinuity detects value change between the current element and the previous or next one in the sequence
• sort rearranges the sequence by sorting it. It could be according to a comparison operator or a value using a radix approach
• exchange rearranges the elements according to a different stride configuration which is equivalent to a tensor axis transposition
• shuffle rotates the elements
• partition divides the sequence into two or more sequences according to a predicate while preserving some ordering properties
• merge merges two ordered sequences into one while preserving the order
Data Movement#
• store stores the sequence to a continuous memory zone. There are variations to use an optimized path or to specify how to store the sequence to better fit the access patterns of the CUs.
• load the complementary operations of the above ones.
• memcpy copies bytes between device sources and destinations
Other operations#
• run_length_encode generates a compact representation of a sequence
• binary_search finds for each element the index of an element with the same value in another sequence (which has to be sorted)
• config selects a kernel’s grid/block dimensions to tune the operation to a GPU | {"url":"https://rocm.docs.amd.com/projects/rocPRIM/en/docs-6.2.1/reference/ops_summary.html","timestamp":"2024-11-05T11:09:14Z","content_type":"text/html","content_length":"34656","record_id":"<urn:uuid:b22345c3-38b7-4644-9a93-d8b58a94172b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00090.warc.gz"} |
Reference the value of an active cell?
Is there a way to reference the value of an 'active cell value' in a formula or have a moving reference?
The current example code and screenshot specifically call out the desired outcome by referencing each cell in a range.
Example: Project1 will start in 2018, which is reflected on sheet 2.
formula in column 2 row 2 on sheet 2:
=IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 3, false) > 0, {SHEET 1 Range 2}, IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 4, false) > 0, {SHEET 1 Range 4}, IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 5, false)
> 0, {SHEET 1 Range 5}, "Not in 3 Year Plan")))
Is there a way I could get the formula to display the $ amount of the start year without specifically referencing each cell? I was looking to see if there was an 'active cell' type reference.
Example of thought:
=IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 3, false) > 0, ActiveCell.value, IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 4, false) > 0, ActiveCell.value, IF(VLOOKUP(Project2, {SHEET 1 Range 3}, 5, false) >
0, ActiveCell.value, "Not in 3 Year Plan")))
Any thoughts on this would be a great help. Thank you.
• I feel like an INDEX/MATCH may have the flexibility you're looking for. Let me play around with it some, and I'll get back to you...
Are you trying to pull the EARLIEST year populated?
Update: If you are trying to pull the earliest year populated, you could use the following formula updating any cross-sheet references as needed of course.
=IFERROR(INDEX({Community Test Range 1}, MATCH(Project@row, {Community Test Range 2}, 0), COUNTIFS(INDEX({Community Test Range 1}, MATCH(Project@row, {Community Test Range 2}, 0)), ISBLANK(
@cell)) + 1), "Not in 3 Year Plan")
See image below for testing of this particular solution.
• Yes, I am trying to pull the value from the EARLIEST year populated. Thanks!
• Not sure if you saw it or not, but I have updated my original post with a solution. Hope it helps.
• Thanks! This looks helpful.
Just a question, what are the cell ranges for the references "Community Test Range 1" and Community Test Range 2"?
• Oops. My apologies...
Range 1 is the year columns. Range 2 is the Project column.
Basically the way it all works is this...
The first portion of the INDEX function establishes where to pull the data from that you are wanting to display which would be your year columns.
The second portion of an INDEX function determines the row. For that we use a MATCH function to look for the Project Name.
The third portion is the column number which is where it got a little tricky. First we re-established the row with the original INDEX/MATCH as we did in parts one and two. Then we counted the
number of blank spaces and added 1. This gave us the number of columns to move from left to right to pull the data from.
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Evil Mad Scientist Laboratories
We were recently contacted by a mathematics instructor, who suggested that it might be interesting to have a program like Snowflake, but with the option of picking and choosing different symmetry
Natural snowflakes have (approximate) sixfold rotation symmetry plus reflection symmetry. However, a lot of things that you can draw by hand have absolutely no resemblance to snowflakes at all– and
it is somewhat fun to explicitly play with the rules.
Our new program, SymmetriSketch, sticks to the same basic design principles as Snowflake: it’s cross platform, open source, and able to export a true vector drawing with a closed path. However,
SymmetriSketch is a much more flexible program that allows you to play with different symmetries, and create all kinds of different things that would never be mistaken for frozen water.
Here’s what it looks like when the program first opens:
The initial shape is an overall pentagon– an object with five-fold rotation symmetry and reflection symmetry. The figure is generated by taking the editable slice– highlighted here and when you start
the program– and reflecting and rotating it to complete the full shape that you see.
Within the editable slice, you can also see three highlighted control points that can be dragged around. There is control point at every vertex and at the midpoint of every line segment between two
vertices. If you drag a control point that is the midpoint of a line segment, it turns that control point into a new vertex. That new vertex also gets new control points at the midpoints to its
Every vertex point can be moved to any location on the screen with the exception of the vertex that is initially at the top point of the pentagon– that vertex is constrained to move along the
vertical axis– the axis of reflection symmetry.
The controls are purposefully kept simple. There are two symmetry controls– for the order of rotational symmetry and to toggle reflection– which you can change in the lower left hand corner of the
The number, with its +/- controls, refers to the order of discrete rotational symmetry. If the number shown is n, then n-fold rotational symmetry is applied, which means that the displayed object is
unchanged when rotated by 360 degrees/n. In the screenshot above, 9-fold rotational symmetry is applied.
Orders from 1 to 99 are allowed– note that 1-fold rotational symmetry is “no symmetry at all” since it requires 360/1 = 360 degrees of rotation to return to the original shape.
The second control is for reflection symmetry, and toggles between “reflect” or “rot. only,” where it either does, or does not apply a mirror reflection across the vertical axis.
With reflection symmetry turned off, the figure is drawn with pure rotational symmetry. (This screenshot was taken while editing the shape, and you can see control points, indicated by little
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