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Introduction to Structural Equation Modeling Using IBM SPSS Statistics and Amos (2nd ed.)
Blunch 2. As an alternative to existing books on the subject, which are customarily very long, very high-level and very mathematical, not to mention expensive, Niels Blunch's introduction has been designed for advanced undergraduates and Masters students who are new to SEM and still relatively new to statistics. Besides the identification of latent factors, the task of the factor analysis is also to determine the level, to which the each observed variable can be explained by each factor.
As mentioned before, there are two basic types of factor analyses: exploratory factor analysis EFA and confirmatory factor analysis CFA. The first one is useful, if we are only interested in searching for a structure among a set of variables or when we want to employ the data reduction method. On the other hand, the second one is useful, when we already have some preconceived ideas about the actual structure of the data, based on certain theoretical support or prior research. So the confirmatory approach can asses the degree, which explains how the data meet fit the expected structure .
The conceptual block diagram for exploratory factor analysis The conceptual EFA block diagram is shown in Fig. After the research design is applied and all the necessary assumptions are investigated, the EFA procedure starts due to the given factor model structure. The appropriate estimation method must be selected and then the factors can be extracted by the means of the estimation procedure. In this process, the factor loadings are also estimated through the estimation of factor matrix.
When selecting how many factors to include in a model, we must provide that their number still provides the sufficient variability in the underlying data. If the unrotated factor solutions do not provide enough information to unambiguously interprete the factors related to the examined variables, the rotation of factors must be additionally employed by the means of orthogonal or oblique methods .
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If there is any variable a candidate for deletion, or we want to change the number of factors or the type of estimation method, respectively, then the model respicification must be done. Naturally, an appropriate evaluation of the model's validity and reliability must be also investigated in the processing of the factor analysis procedure.
The conceptual EFA block diagram B. The presentation of the factor model structure EFA is designed for the situation when the connections between the indicator variables and the latent factors are unknown or uncertain . The calculation of this matrix then also enables 2 the estimation of the communalities hi , which appear in expression 4. This way, we get the ability to identify the minimum possible number of factors that account for covariation among the observed variables . All of these methods have the specific properties, with accompanying advantages and weaknesses.
The decision about the factor extraction estimation method is related to the objectives of factor analysis and the theoretical knowledge about some basic characteristics of the relations between the variables . The main advantage of ML method is the ability to calculate the wide range of indexes for testing the model's goodness of fit. This method also enables the use of the statistical significance tests for the correlations and factor loadings, as well as the determination of confidence intervals .
On the other hand, the main limitation of the maximum likelihood method is the assumption of multivariate normality, which can cause the calculation of false results . Regarding the PAF method, it does not rely on the assumption of multivariate normality, which can definitely be treated as an advantage.
But this method also has certain weaknesses, since it can not enable such a wide range of goodness-of-fit indexes as it is provided in the case of ML method . Factor rotation When the factor matrix is extracted, the judgment must be done about the interpretation of the corresponding factors. The factor loadings indicate the degree of association between the variable and the factor, which means that they enable the interpretation of relationships between each variable with respect to each factor .
In many cases the unrotated factor solutions are not capable of providing the information that offers the most adequate interpretation of the observed variables and the corresponding factors. Then the rotation of factors must be addtionally applied, where the factors reference axes are turned about the origin until some other position is reached . By using this maneuver the redistribution of the variance from the earlier factors to the later ones is achieved, which usually also gives us a simpler and theoretically more visible factor pattern.
The examination of literature shoes that the several rotation methods have been developed, such as: the varimax method, the equimax method, the orthomax method, the quartimax method, the promax method, the procrustes method, and so on . These factor rotation methods can be either orthogonal or oblique.
All the details about the rotation methods can be found in the appropriate literature [4, 5, 6, 51]. Verification of the appropriateness of applying the factor analysis Since it must be ensured that the data matrix has sufficient correlations to justify the use of factor analysis, the latter is in principle not appropriate in the case of low or approximately equal correlations . By other words, if no substantial number of correlations are greater than 0.
ISBN 13: 9781446248997
To check the suitability of the factor analysis application, the so-called Bartlett test of sphericity can be also employed, which investigates the statistical significance of correlations among the variables. If the test confirms that the correlation matrix is not equal to the identity matrix, then the observed variables are significantly correlated to the certain level and the factor analysis can be applied.
Another test, which checks the appropriateness of factor analysis application, is so-called Kaiser-Meyer-Olkin KMO test. The latter measures the strength of the intercorrelations and a value of 0. The suitability of the data checked by the KMO test, which is conducted to help us in decision, whether to apply or not to apply the factor analysis, is ilustrated in table 4 .
Table 4. The appropriateness of the data checked by the KMO test to help us in decision whether to apply or not to apply the factor analysis KMO value Appropriateness 0. Criteria for the number of factors to extract It is important to choose the number of factors which still provides the sufficient variability in the corresponding data in the sense of maximization of the proportion of explained variance.
At the same time we also want to limit the number of these factors, so there must be a compromise found between how much variability we want to keep and how many variables we want to exclude from the further analysis. There are a number of procedures designed to determine the optimal number of factors to retain in factor analysis, like the Kaiser's "eigen value bigger than one rule" , Cattell's scree plot, percentage of variation criterion PVC , and so on . For example, in the case of PVC approach we require that at least a specified amount of variance must be explained from the derived latent factors.
A slightly modified version of this criterion involves a selection of enough factors to achieve the pre-specified communality for each of the original measured variables in order to adequately represent each of them . The convergent validity and assessing the statistical significance of factor loadings The convergent validity means that there can be found the convergences between the variables, which measure the same phenomenon. Namely, the certain measure is valid, if truly measure what it should measure .
The convergent validity is usually reflected by the size of the factor loadings, where the variables within a single factor must be highly correlated . As it turns out, the significance of the factor loadings depend on the sample size of the data. In general, the smaller is the sample size, the higher must be the required loadings. Table 5 represents the thresholds of the sample sizes needed for each factor loading value to be considered significant .
Thresholds of the sample sizes needed for factor loadings to be considered significant Sample size needed for Factor loading significance 0. Interpretation of the factor matrix In the process of interpretation of factor loadings, most factor solutions unfortunately do not result in a simple structure, where each variable would have a single high loading on only one factor. If the variable has more than one significant loading, than we are dealing with a so-called cross-loading.
If we want to make each variable loaded only on one factor, the different types of rotations are usually required, which would eliminate the cross-loadings and thus enable a simpler structure. But if the variable after applied rotation still has significant cross-loadings, then it is a candidate for deletion . Besides this, the variable is also the candidate for deletion, if it does not have at least one significant loading. Also, the communalities must be inspected for each variable to check whether the variables meet an acceptable level of explanation.
- Introduction to Structural Equation Modeling Using IBM SPSS Statistics and Amos;
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As suggested by Hair , the variables are the candidates for deletion, if their communalities are smaller than the value 0. The discriminant validity The discriminant validity means that the observed phenomenon is independent and separated from the other phenomonons. Or by other words, the variables, which measure certain phenomenon, are not highly correlated with the variables, which measure some other phenomenon .
This implies that the variables should relate more strongly to their own factor than to some other factor. The discriminant validity is usually investigated by the observation of so-called factor correlation matrix, where the correlations between factors must not exceed certain pre- specified level . The reliability The reliability investigates the level of consistency between multiple measurements of a certain variable.
The commonly used measure is so-called internal consistency, which means that the "reliable" set of variables will consistently load on the same factor. This can be deduced from the fact that the individual indicators of the scale should all measure the same factor and thus should be highly intercorrelated . The most widely used reliability measure is so-called Cronbach's alpha reliability coefficient, which is usually calculated for each factor .
According to Hair , the generally agreed minimum lower level for Cronbach's alpha coefficient is the value 0. So this method enables us to test how well the observed variables represent a smaller number of constructs . This type of analysis is similar to EFA in many aspects, but philosophically it is essentialy different. Namely, in CFA, the number of factors for a given set of variables must be defined in advance, as well as to which factor the variables will load on. It must be also specified whether these factors are correlated or not.
Evenmore, the variabled are assigned to only a single factor and cross-loadings are not present at all. Naturally, all the present factors can be also covariated among themselves. This means that the CFA enables some sort of confirmatory test of our measurement theory, which verifies the consistency of our systematic representation of a theoretical factor model with the data reflected in observed variables. After that, the measurement theory can be combined with a so-called structural theory in order to fully specify a SEM model , which will be more precisely introduced in the next section.
As it was already mentioned in the introductory section, the CFA provides the means to construct the measurement part of the SEM model. Since it is pointless to validate the structural model, if the measurement model is not validated at first, the structural modeling is usually a two- stage process. In the first stage, the measurement model is constructed and validated by the means of CFA, while in the second stage the design of the whole structural model is completed by adding the structural part of model and appropriately validating the entire model structure.
All the details about the design and construction of CFA model, the estimation methods used in CFA analysis, and the appropriate evaluation of the CFA model, respectively, can be found in the corresponding literature [, ]. There can be also found, how the expressions, related to the CFA model, can be applied, similarly, as it is illustrated for the case of EFA model c.
The convergent and discriminant validity and reliability in the CFA analysis It is always necessary to investigate the convergent and discriminant validity, as well as the reliability, when a CFA analysis is conducted . The thresholds for these measures are shown in table 6 . Table 6. If all AVE estimates are bigger than the corresponding SIC estimates, this could indicate a good evidence of discriminant validity . For this purpose, the structure of interrelationships expressed in a set of equations, similar to a system of multiple regression equations, is examined.
These equations reveal all of the relationships among the constructs the dependent and independent factors , which are involved in the analysis. Similarly as in case of factor analysis, the constructs are unobservable latent factors, which can be represented by the multiple variables .
The main origin of the SEM comes from two familiar multivariate techniques: factor analysis and multiple regression analysis. Thus, the SEM can be treated as a unique combination of both types of these techniques. SEM is a method which gains in the popularity because it combines the confirmatory factor analysis and the regression analysis simultaneous equations models in order to depict a variety of different relationships between the unmeasurable latent factors .
In SEM, variables can be either exogenous or endogenous, and the whole set of interrelations like direct, indirect, multiple and reversed, can be applied. An additional explanation about the variables involved in SEM and the characteristics of their relationships can be found in the literature . The main definitions of the variables involved in the SEM modeling, and the specifics of their relationships A.
From Fig. After that, the model must be specified, which means the definition of all causal paths between the variables due to some theory and basic theoretical knowledge. The model identification and estimation are the next steps in the procedure.
IBM SPSS - Advanced: Structural Equation Modelling using AMOS
During the model identification, the care must be taken about over-identification, which means that there are more parameters available than it is the number of parameters, we want to estimate. Concerning the model estimation, the suitable estimation method must be chosen in order to achieve the succesful model verification, which is the next step of SEM procedure.
Herein, the model is tested and its fit quality is evaluated, where the different fit indices can be applied. If these indices indicate a poor fit performance, the post-hoc modifications of the model specification must be additionaly proceeded. Otherwise, the model is ready for an appropriate interpretation and the report of achieved results.
Even more clear presentation of all significant steps in the SEM modeling and how it incorporates the CFA measurement model, is given in Fig. For this purpose, the corresponding measurement model must be converted to the structural model at first. In the next step, the structural model validity must be assesed, where the goodness of fit indices and the direction of all paths are investigated, as well as the structural parameters are estimated.
If the structural model passes the validity test, it can be used to make some conclusions.
Otherwise, the model refinement is needed. The example includes four latent factors and 16 observed indicator variables. Each factor is investigated via four indicators, and the covariances for the each pair of factors are given in the CFA model. When the conversion to the SEM model is completed, some factors and their indicators have an exogenous independent charater, while the others have an endogenous dependent character.
In Fig. The further details about the SEM model structure, all the variables involved, and their relationships and characteristics, will be depicted more precisely in the sequell. The structure of the SEM model represented by the path diagram Fig. The meaning of the corresponding variables and coefficients is shown in table 7. Obviously in this case we are dealing with 6 exogenous indicators, 4 endogenous indicators, 3 exogenous latent factors and 2 endogenous latent factors.
The exogenous latent factors are supposed to be covariated among each other, and the directions of causal paths related to the constructs can be cleary seen. Since the latter is not very convenient for the theoretical purposes, it can be further converted into the analogue system of matrix equations. All the details about the mathematical structure of SEM models, their matrix analysis and the derivation of variance- covariance matrices can be found in the appropriate literature [, ]. The problem of normality in the process of SEM estimation When estimating the parameters of SEM model, the maximum likelihood ML method, which is the default estimation method in the most of SEM softwares, is usually used.
This method demands the certain statistical assumptions about the multivariate normality and in principle provides the accurate estimates, if the data are continuous and normally distributed. According to Kline , the multivariate normality includes cited from  : 1.
Introduction to Structural Equation Modeling Using SPSS and AMOS by Niels J. Blunch
Since it is difficult to investigate all of the aspects of multivariate normality, at least the inspection of univariate distribution of each observed variable is requiered . The inspection of non- normality is usually proceeded by the observation of the skewness and kurtosis of the measurements. There has been certain disagreement detected in the literature about the criterions concerning the non-normality, which is stil admissible for an efficient usage of the ML method.
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float flownoise(vector xyz, float flow)
vector flownoise(vector xyz, float flow)
float flownoise(vector4 xyzt, float flow)
vector flownoise(vector4 xyzt, float flow)
float flownoise(float x, float y, float flow)
vector flownoise(float x, float y, float flow)
This operator generates 1D and 3D Perlin Flow noise from 3D and 4D data. There are two forms of Perlin flow noise: a non-periodic noise which changes randomly throughout the N-dimensional space, and a periodic form which repeats itself over a given range of the space. The periodic form can be used to generate patterns which tile over N-dimensional space, such as a noise-based texture map which repeats seamlessly.
The noise has a range of (0, 1) with a median value of 0.5. The distribution of the noise depends on the dimension, with higher dimensions approaching a Gaussian distribution of noise values.
Flow noise is very similar to Perlin noise, as in Periodic Noise, but with an extra flow parameter. The flow parameter can be thought of as an extra dimension, but a dimension whose period is always 1. Moving through the flow dimension rotates the noise vectors rather than adjusting slices through a noise space, which generates a more flowing appearance to the animation.


Standard MGB engine has a bore of 80.26mm and a stroke of 88.9mm. Say an overbore is plus 40 thousandsth of an inch, or 0.04 inch which converts to 1.016mm, so the bore of the engine is now 80.26 + 1.016 = 81.276 mm
The swept volume of a single cylinder is the circular area of the cylinder multiplied by the stroke of the piston. Circular area is PI x radius squared.
Radius = 81.276/ 2 = 40.638 Squared = 1651.447
Multiplied by PI (3.14159) = 5188.169 sq mm
Multiplied by the stroke (88.9) = 461228.27 sq mm or 461.2 sq cm (cc)
Multiplied by 4 (four cylinders) = 1845 cc
Probably represents an additional 2 or 3 bhp


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On the figure hereafter, subfigure a depicts 3 different cases of shape optimization with a radius of curvature constraint; subfigure b shows the polar plot of the effective in-plane Poisson ratio; subfigure c is a plot of the convergence of negative Poisson’s ratio with respect to the iteration steps of the optimization scheme.


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This Tutorial will Explain you that how data can be analyze using Discriminant analysis with the help of Python coding.
Data Science is very growing field in current technology. because it is helpful to expand business , future prediction , business analysis , Machine Learning ,etc..
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- Discriminant Analysis ( DA ) is one type of Machine Learning Algorithm to Analyzing and prediction of Data.
- This algorithm is used t Discriminate between two or multiple groups .
Types of Discriminant Algorithm
- When there is dependent variable has two group or two categories then it is known as Two-group discriminant analysis.
- When there is dependent variable has more then two group or categories then it is known as Multiple discriminant analysis.
Similarity and Difference between Anova , Regression and Discriminant Analysis
|Similarity:- No of Dependent Variable||single||single||single|
|Similarity:- No of Dependent Variable||Multiple||Multiple||Multiple|
|Difference:- Nature of Dependent Variable||Continuous||Continuous||Categorical|
|Difference:- Nature of Independent Variable||Categorical||Continuous||Continuous|
Objectives of Discriminant analysis
- Development of discriminant functions, or linear combinations of the predictor or independent variables, which will best discriminate between the categories of the criterion or dependent variable (groups).
- Examination of whether significant differences exist among the groups, in terms of the predictor variables.
- Determination of which predictor variables contribute to most of the inter group differences.
The discriminant analysis model involves linear combinations of the following form:
D = b0 + b1X1 + b2X2 + b3X3 + . . . + bkXk
D = discriminant score
b ‘s = discriminant coefficient or weight
X ‘s = predictor or independent variable
Deciding factor Wilks‘λ
Wilks‘ λ :- Sometimes also called the U statistic, Wilks’ λ for each predictor is the ratio of the within-group sum of squares to the total sum of squares. Its value varies between 0 and 1. Large values of (near 1) indicate that group means do not seem to be different. Small values of (near 0) indicate that the group means seem to be different.
Basic Python code Step for DA
import pandas as pd
import numpy as np
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
- Import Above Three packages for Dataset and discriminant analysis algorithm loading.
- Load the dataset in which you want to perform DA.
- Descide Dependent and Independent Variable
- Take Dependent variable on y and Independent on X.
Thus you will get Accuracy Score for Prediction accuracy.


Conference on Constructive Approximation of Functions 3
A conference in honor of Stefano De Marchi on his 60th birthday
The conference will be held in Krakow, Poland, 22-24 February 2023.
The conference is organized by the Department of Mathematics of the University of Applied Sciences in Tarnow and the Faculty of Mathematics and Computer Science of the Jagiellonian University.
The conference will be held in a hybrid form: although it will bepossible to participate online, personal presence will be welcome.
The event will take at the Faculty of Mathematics and Computer Science of Jagiellonian University (ul. Prof. S. Łojasiewicza 6, 30-348 Kraków).
The general theme of the meeting concerns various aspects of complex analysis, especially: approximation theory, numerical analysis, constructive theory of functions, pluricomplex analysis, (pluri)potential theory, polynomial properties. Another important aspect of the CAF meetings, previously held in Będlewo 2014, and Tarnów 2021, is to promote the creation and maintenance of contacts with scientists from diverse cultures.
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The organizers are very proud that the following eminent mathematicians have agreed to deliver the plenary lectures:
Mirosław Baran, Pedagogical University of Cracow, Poland
András Kroó, Hungarian Academy of Sciences and Budapest University of Technology and Economics, Hungary
Michael A. Slawinski, Memorial University of Newfoundland, Canada
Gianluca Vinti, Università degli Studi di Perugia, Italy
Other participants are welcome to deliver a talk (20 min) after acceptance by the Scientific Committee.
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First-year mathematics is really a sub-set of the Department of Mathematics. However, since OU offers a large number of first-year mathematics courses with close to 6000 students enrolled in these classes every fall semester, there are certain structures in place that are specific to first-year mathematics. If you would like additional information related to first-year mathematics including details related to some of these structures, please go to the tabs on the black bar above.
Deborah Moore-Russo, First-Year Mathematics Director


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Some are questioning whether requiring proficiency in intermediate algebra for community college enrollment and completion is helping or hurting students, with a number of educators in California’s.
The VirtualSC Course Catalog is a list of all potential courses offered by VirtualSC. Click on the Current Course Offerings button below to view the courses that are offered each semester.
May 9, 2018. This online course provides students a thorough study in the arithmetic of real numbers. MTH 102 Intermediate Algebra With Review 5 Cr Hrs.
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It’s that time of year, and we’re hearing… WHY ANOTHER TEST!?!
Since many of our parents have been contacting us at Mathnasium asking if we can help improve their students’ score on the MAPS (Measures of Academic Progress) test that determines if a child is selected for the Honor’s and accelerated track for Math in many school districts, we have decided to put together this resource guide. Many kids Practice MAPS Tests at home before the actual test, but keep in mind that Students must score at least 95% on these tests to get into these advanced courses. Is your child on the borderline to achieve that score? This guide may help you with navigating the getting that last few percent to get to the top. First, a little background on what MAPS is.
This “MAPS” test is aligned with the Common Core Standards that Ohio schools have adopted for the 2013-2014 school year. Unlike the OAA (Ohio Achievement Assessments), MAPS does not determine how much information has been learned at a particular point in time, but rather one individual student’s progress. Each student completes MAPS testing three times per year.
Many kids ask, why another test!? Well, the MAPS test is designed to provide feedback that is used to specifically to modify the teaching and learning activities during instruction. MAPS tests are assessments designed specifically for learning. They are customized to the individual student, similar to Mathnasium’s method. You may be asking yourself, what exactly do the MAPS tests measure?
MAPS tests are given to students in kindergarten through 10th grade. The tests are a very thorough evaluation, covering everything from Problem Solving, Number Sense, Computation, Measurement and Geometry, all the way through Algebra and Functions, and Data Analysis for the more advanced grade levels just like Mathnasium. Those happen to be all the concepts that Mathnasium is currently teaching. The Mathnasium program is all about Number Sense.
A person is said to have Number Sense when he/she can count from any number to any number, by any number, split any number into halves, fourths, thirds, tenths, and hundredths; and compare any two numbers by subtractions and division. The way that the Mathnasium curriculum is designed will positively and directly affect any child’s ability to score well on a MAPS test. A Learning Center, like the one the Mathnasium that I own, will can raise a child’s score on the MAPS test!
Some parents say, “I don’t understand how how MAPS work.” They can be a bit tricky. Some schools have put together guidelines to help parents and students figure out what to make of the test. Once great school that has put together a great guide to MAPS Activity is Sycamore Schools MAPS REVIEW. Not only does the Sycamore Site have great resources like the Practice MAPS Tests (some of them free MAPS practice tests) They’ve put together a fantastic resource for parents, including an orientation, study-guide, and other resources for the test. [Keep in mind that the study guide provided there is decent, but only a guide. Nothing is a substitute for the 1-on-1 learning interaction you get at a TRUE Mathnasium Learning center! (See Bellow)]
So, are you the kind of Parent that’s ready to Improve Your Child’s MAPS Test Score, more than just a simple online resources? Truly ready to take it to the next level? Well, don’t take our word for it that Mathnasium is a great place to Improve Your Child’s MAPS Test Score, read the couple of Testimonials below.
The following testimonials tell how Mathnasium is proven to boost test scores and give the children the confidence they need to succeed.
One parent reached out to us from Mathnasium (Blue Ash, OH) after his child, Liam, came to the program. Liam was able to accelerate into the honors Math program and even get a 98% on the MAPS test, siting Mathnasium as the reason. The parent stated, “Liam was promoted to the Honor’s Math class. Mathnasium(R) gave him a boost in his math skills and we saw improvement in his test scores.”
Another parent reached out to us after their student of Mathnasium (Mason, OH) was able to excel and enter the Honor’s Program at Sycamore and Mason.
Do you have a student that’s ready for that “boost” in their math skills? Is your student ready to get to that next level or even further? At learning centers where we pride ourselves on our student’s successes where we make math make sense, you child will be a success!
If you want to improve MAPS scores, contact a learning center near you to make it happen! Contact Mathnasium


Welcome to school year 2022 - 2023
Fort Sam Houston Elementary
Home of the Cougar Cubs
Room 31 – Kennedyz Kubz
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Meet the Teacher - August 11th 5:00 - 6:30 p.m. in Room 31
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A single app that we will be using for all communication between home and school
The Reflex Math program helps to develop fluency with your child's basic addition, subtraction, multiplication and division math facts.
This is a personalized online math & reading program that works as a tutor to cater to your child's current academic level. It is an excellent tool for helping to close the gaps in your child's learning.
Research is revealing that regular reading each day, for as little as 6 minutes, helps to lower stress levels! Reading with your child and asking questions to check their understanding can help them to appreciate balanced school-home support.
Thanks for who you are, and what you do :)
Please be sure to visit Greg Tang's webpage for many of his awesome online math teaching games - www.gregtang.com
Please continue to encourage and monitor your child as they make use of the iStation & IXL Math & Reading programs.
We have some awesome new spirit wear. Please use the following links to order spiritwear for your child or for yourselves.
All of the students will be set up with iStation, IXL, and Reflex Math accounts. iStation is a supplemental reading and math program that the state of Texas makes available to every student. It is a fun and engaging way for students to achieve success in reading and math. Letters will be sent home with your students with directions to guide you through setting up their iStation accounts at home so that they can continue to work on their English Language Arts , Reading (ELAR) and Math skills. Pearson Realize is the main math program that is available to every student. It also provides progress monitoring, intervention and enrichment opportunities for your child.
HMH is the program that we use for Reading & Writing skills. It also shares the same benefits as the Pearson program. In addition to these programs, your child will also have opportunities to read Scholastic News articles, and to participate in targeted reading activities. Storyworks also provides your child with a chance to develop strong comprehension skills, and to compare and contrast different reading genres,
The school district has generously subscribed to many learning programs that can be accessed through the apps on their Google Classlink page.


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Unit 1 – Discussion Board-Calculus
The following will examine the concepts of one-sided limits, limits, limits at infinity, and continuity.
Recall that the limit is a way to talk about a function when a value in the function cannot simply be substituted. Post your response as a reply of 100–300 words, and then comment on your fellow students’ answers in the discussion forum.
Use the following definition to establish your answer to Part I:
If f(x) is defined for all values of x near c, then the limit of f as x approaches c exists, and it is equal to L if both of the following are true:
L is finite. Both the left- and right-sided limits of f equal L. Does f need to be continuous for the limit to exist?
Evaluate the cost function C(x) for the given values in a. Address the following:
What do you notice as the values get larger? Your response should help you answer question b. Could the city remove all of the toxins in its water now that it knows the limit of C(x) as x approaches 100? Discuss what percentage of the toxin you find acceptable, and find the cost of cleanup for that percentage.


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How many jokers are there in a standard pack of cards.
Some games use special cards called jokers. If jokers are used, there are usually one or two, producing 53 or 54 cards in the pack. Some games such as Canasta use more than one normal pack combined together as a single unit, a few games (some Canasta variants, as well as many table banking games) using as many as four or six combined packs.
Textbook solution for Mathematics: A Practical Odyssey 8th Edition David B. Johnson Chapter 2.CR Problem 26CR. We have step-by-step solutions for your textbooks written by Bartleby experts!
Question 1114279: In a deck of 52 playing cards (jokers not allowed), how many five-card poker hands containing three of a kind (but not a full house or four of a kind) are possible? Answer by Edwin McCravy(17900) (Show Source): You can put this solution on YOUR website! We can choose the suits of the three of a kind in 4C3 ways We can choose the rank of the three of a kind in 13C1 ways. That.
Each deck contains 48 different cards. These cards will consist of two each of the Nine, Ten, Jack, King, Queen and Ace cards in all four suits. Canasta Decks: Canasta decks come in the regulation poker size that uses the same 52 playing cards plus four Jokers. Canasta decks come in pairs because the game requires two sets of cards to play.
The total number of possible hands can be found by adding the above numbers in third column, for a total of 2,598,960. This means that if there are 52 cards, how many combinations of 5 cards can be drawn (answer 2,598,960 combinations). What if there is a deck of 54 cards with two joker (or wild) cards, how many possible hands can be dealt now?
In a standard deck, there are usually two Jokers. The Joker's use varies greatly. Many card games omit the card entirely; as a result, Jokers are often used as informal replacements for lost or damaged cards in a deck by simply noting the lost card's rank and suit on the joker. See Full Answer. 17. How many faces are in a deck of cards? A standard deck of playing cards has 52 cards in it.
Technically there’s no limit to how many decks of cards can be used in a game of blackjack. While a couple of friends playing at home can play a decent game of 21 using as little as one or two decks, casino blackjack is most commonly played with six to eight decks of standard 52 playing cards, excluding jokers. How the number of decks affect the house edge. The higher the deck, the higher.


A core purpose of teaching and learning at Lenah Valley Primary School is to develop each student’s intellectual capacity and the quality of their thinking. In numeracy our focus is on thinking mathematically, problem solving and mental computation. We have a whole school approach to the teaching of mental computation and problem solving. Progressively, through all grades, students learn to effectively use a range of thinking strategies for mental computation. Through our class programs, students are encouraged to develop understanding and to identify the connections between ideas rather than the rote learning of facts and procedures.
Students are encouraged to use specific mathematical vocabulary so they can communicate precisely and shared meanings can be developed. Our classroom programs use open-ended challenges and investigations to develop mathematical thinking and creative problem solving. We want all students to be able to think in ways that are considered mathematical – conjecturing, proving, convincing and justifying. We give emphasis to being able to interpret and use mathematical information. Students learn the importance of presenting information visually so their findings and justifications can be understood by others.
We value students sharing their discoveries and learning collaboratively. Through small group learning arrangements, students can engage in joint inquiries that extend each other’s’ thinking and learning.


Decimal literals are decimal values written directly in the program text.
There are two notations for writing decimals.
In general, if the sign is omitted, it is assumed to be
+. Both a
+ and a
- can be used.
There can be at most 15 decimal digits. If there are more than 15 digits, the number is rounded to 15 digits.
If there is a decimal point, there needs to be at least one digit before and after. The decimal point can be placed between any of the digits. The decimal point can also be placed first with a preceeding zero.
e, after which comes the exponent
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— Jonathan A. Handler, MD, FACEP, FAMIA
— Craig F. Feied, MD, FACEP, FAMIA
My personal library contains 100 books, 50 with red bindings and 50 with blue bindings. I hide coins inside 20 of the books. 10 of the red books each contain a penny and ten of the blue books each contain a silver dollar. I’ll let you pick 10 red books and 10 blue books from my library (20 books in all). After you select your 20 books I’ll let you open each one to see whether they contain any money, and you can keep whatever money you find. If you choose perfectly, you’ll get to keep all the money in my library!
You immediately start grabbing books off the shelves.
Every book you grab that is found to have a coin in it we will label a True Positive (TP). Every book you grab that doesn’t have a coin in it we will label a False Positive (FP). Every book you fail to grab that has a coin in it we will label a False Negative (FN). Every book you don’t grab that doesn’t have a coin in it we will call a True Negative (TN).
After you’ve chosen your 20 books, it turns out that two of your red books contain a penny, eight of your blue books contain a silver dollar, and the other 10 books contain no coin at all.
How good a chooser are you? There are several possible metrics that might be used to describe your performance at choosing books with hidden coins. Which of the following metrics describes your performance in the way that matters most to you?
- Performance = 50%: the percentage of available coins in my library that you found (you found 10 coins out of 20 available).
- Performance = 79%: the percentage of available value in my library that you found (you found $8.02 out of $10.10 available).
Using metric #1, I could achieve 50% performance by finding all 10 pennies while finding none of the dollars — or vice versa. Those two seem completely different to me, but to metric #1 they are exactly the same! I’d prefer metric #2 because it always tells me how much of the available money I got to take home. The value captured is much more important to me than just the number of coins I found.
Unfortunately, our familiar Boolean (True/False) performance metrics don’t give us this option. They are based solely on the simple counts of true positives, false positives, true negatives, and false negatives. For each prediction, a “score” of 1 is assigned to one of those “buckets” (TP, TN, FP, or FN). This “count-based” approach makes some astonishing assumptions that are almost never true in real-life clinical scenarios. One of these, demonstrated in this example, is the often false assumption that every result has the same amount of value (“utility”) as every other result. In other words, the utility is assumed to be “uniform” across all results. In this case it’s obvious that utility is not uniform across all results: some results are worth 1 penny and others are worth 1 silver dollar. The uniformity assumption of count-based metrics does not hold.
So maybe now you conclude that we shouldn’t do counts. Instead, we should weight each result depending on how much utility it created. You decide that each true positive in which a penny is found gets 0.01 “points” and each true positive in which a silver dollar is found gets 1.00 “points.” Each false negative in which a penny was missed loses 0.01 points, and each false negative in which a silver dollar was missed loses 1.00 point. Great! The problem of non-uniformity among results is solved!
Now imagine I make you a different offer. I’ve hidden a silver dollar in ten of the 100 books in my library. You can choose any ten books, but again, no peeking! After you’ve chosen, we’ll look in the books to see how many silver dollars you’ve found. You can keep anything you’ve found up to a maximum of $5. However, if you find any more than 5 silver dollars then I’m going to label you “greedy” and you can’t keep any of the money. On top of that, I’m going to fine you $5 for being so greedy.
Classic count-based metrics assume that every correct result is “good” and every incorrect result is “bad.” We call this an assumption that the results are “fully dichotomized.” However, with my new offer the sixth true positive result is actually harmful to you: it makes you lose all your winnings AND pay an additional $5. Even though the sixth true positive result is correct, it’s harmful rather than beneficial. We can see that in this case, the results are not fully dichotomized. The assumption of full dichotomization required by count-based metrics does not hold, and that creates a problem: count-based metrics will continue to calculate a better and better performance the more positive results you get, when in actuality you’ll be losing money for any positive result after the first five.
So… maybe now you conclude that, in addition to assigning points for utility, we should also stop categorizing results by whether they are correct (true) or incorrect (false) and instead categorize them by whether the result was beneficial or harmful. In other words, instead of counting true positives we’d add up all the value created by positives that were beneficial (regardless of whether correct or incorrect). Instead of counting false positives we’d sum up the value from all the positives that were harmful. Instead of counting false negatives we’d sum up the value from all the negatives that were harmful, and instead of true negatives we’d sum up the value from all the negatives that were beneficial. Great! Now the problem of results that are not fully dichotomized is solved too!
Determined to win as much money as possible, you go home and train yourself to detect the presence of a coin inside a book without peeking inside. Not so fast! Given your training, I’m changing the deal. Once again I’ve hidden a silver dollar in each of 10 blue books and a penny in each of 10 red books out of the 100 books in my library. Once again you can choose 10 red and 10 blue books. You can still keep every coin you find. However, since you claim you’re such an expert, this time you must pay me $10 for every coin you fail to find. You agree to the deal because practicing in your own home has convinced you that you are really amazing at detecting books holding hidden coins. You apply your detection technique to my library and choose 20 books, finding 1 penny and 5 silver dollars. Oops, you now owe me a bunch of money for the 14 coins you didn’t find!
Now which of the following metrics describes the performance that matters most to you?
- Performance = 30%: The percentage of books with coins in them that you found (6/20).
- Performance = 50%: The percentage of potential benefit available to you in my library that you found ($5.01/$10.10)
- Performance = 3%: The percentage of money exchanged between us that actually went in the right direction (into your wallet). You got $5.01 from me but you had to pay me $140. So, of the $145.01 in total that we exchanged, 3% went from me to you ($5.01/$145.01) and 97% went from you to me. Anytime this percentage is greater than 50% you are making money from me. Anytime it’s less than 50%, you are losing money to me. Muwahahahaha.
We’ve already decided that metric #1 doesn’t measure up because the percentage of available coins actually found does not take into account the fact that some coins are worth more than others.
Metric #2 recognizes that we care more about the money (benefit) than about the number of coins. It tells us how much of the money we found compared with how much we could have found if all our false negatives (representing missed benefit) had been true positives instead. However, in this situation each coin you failed to detect not only represents missed value, but it also costs real cash money out of your pocket (harm experienced). Metric #2 doesn’t include harm at all. The missed benefit is small and the cost you’ll have to pay is large, so #2 can’t be a very good measure of what we care about.
OTOH, metric #3 takes into account both the benefit and the harm resulting from each prediction you made. That’s exactly what we need! Of these three metrics, it’s the only one that measures what we really care about: how well (or poorly) we are doing at making money rather than losing it.
By now it’s pretty obvious that our choice of performance metric has to depend on the rules of the game. In the very first scenario above (where I let you keep all the coins you found and I did not penalize you for coins you failed to find) we cared most about the percentage of available opportunity that we captured (the value we actually received compared to the value we would have gotten if all of our false negatives had instead been true positives). In that “no-lose” situation, we’d choose metric #2 because that is exactly what metric #2 measures. The scenario where I penalized you heavily for each missed coin gives you a chance to end up better off than you started, but you also risk ending up worse off if you perform poorly. In that scenario, we’d choose metric #3 because it incorporates both the benefit you receive and the harm you incurred.
Unfortunately, with classic count-based metrics we only get a single formula to describe the relationship of true positives to false negatives. That’s the formula for metric #1:
TP/(TP + FN)
where TP is the count of True Positives and FN is the count of False Negatives.
This formula might be familiar to you. In count-based metrics, it has been given two different names: Recall and Sensitivity. Since count-based metrics use the same formula for both Recall and Sensitivity, many (perhaps most) people think of them as being “the same thing”.
However, we’ve just seen that count-based metrics ignore a lot of things that we care about. With our utility-based approach, we see that instead of the single count-based formula #1, we now have two distinctly different formulas (#2 and #3) that take utility into account. The two metrics measure different aspects of the relationship between True Positives and False Negatives. Which one to use depends on what question you’re trying to answer in a particular scenario.
Let’s look first at the formula for metric #3:
(#TP * $ per TP) / ((#TP * $ per TP) + (#FN * $ per FN))
In this formula #TP is the count of true positives and $ per TP is the amount of money we receive for each true positive, while #FN is the count of false negatives and $ per FN is the amount of money we pay out for each false negative. We can simplify this formula to:
$TP/($TP + $FN)
where $TP is the sum of all the money we received from the true positives and $FN is the sum of all the money we paid out due to the false negatives. You’ve probably already noticed that this formula looks an awful lot like the traditional count-based for metric #1, except that Instead of counting items we sum up the money (or utility) associated with each result.
Wasn’t that easy? Now let’s look at the formula for metric #2:
(#TP * $ per TP) / ((#TP * $ per TP) + (#FN * $ per TP))
In this formula the last term represents the money we would have received if we had made the “complementary” choice (positive instead of negative) for each of the false negatives. It’s the money we could have gotten if every single choice had been perfect, so all the false negatives had instead been true positives. We call this the complementary benefit of the false negatives.
If we do the same simplification as in #2, this becomes:
$TP/($TP + $C(FN))
where $C(FN) is the complementary value of the false negatives: the sum of all the dollars we could have gotten if we had made the correct guess instead of the wrong one.
You may have noticed that whenever the harm from a false negative exactly equals the benefit of a true positive (the result is symmetric), these two different equations will yield the exact same result.
if $C(FN) = $FN then
$TP/($TP + $C(FN)) = $TP/($TP + $FN)
In fact, count-based metrics not only ignore the fact that different results might have different utility (treating utility as uniform) and the fact that true predictions might cause harm instead of benefit (treating utility as fully dichotomized) but they also always treat all results as symmetric. For this reason, under count-based metrics the two equations will always be the same, whether or not this matches reality.
OTOH, any time results are asymmetric, the two equations always yield different numbers. Each equation has a different meaning, and they only happen to evaluate to the same number in the rare cases when the size of the benefit of a true positive is exactly equal to the size of the harm from the alternative false negative.
So what should we call these two distinctly different metrics? In count-based metrics formula #1 is called both Sensitivity and Recall.
Many (perhaps most) people would recognize “Recall” as the fraction of available benefit that was captured (for example, here). Recall quantifies the relationship between captured value and total available value, where total value includes both captured and missed value (lost opportunity). It makes sense, then, for us to call metric #2 “Recall” because that’s exactly what it measures!
In count-based metrics the term “Sensitivity” is often used to mean the fraction of actual items (e.g., coins, or cases of a disease) that were correctly detected or predicted. However, even though there’s only one formula, the term is used to mean different things in different contexts.
For utility-based metrics we’ll use “Sensitivity” to mean the fraction of all the utility associated with actual items that is beneficial utility. This is the beneficial utility of the actual items divided by *all* the utility of those items. If the ones we got wrong generate harm, that gets included.
- Sensitivity: A measure of the relationship between the benefit actually received (usually it comes from true positives) and any harm actually received (if there is any, usually it comes from false negatives). This is calculated as:
$TP/($TP + $FN)
Going forward, since utility is not always measured in dollars, let’s substitute “u” to more generically indicate the amount of utility, so that the equation becomes:
uTP/(uTP + uFN)
Making it even more general… remember that results are not always dichotomized: not all benefit comes from true results and not all harm comes from false results. So if we set Beneficial Positives (BP) to mean the sum of utilities from all positive results that were beneficial (whether a true positive or false positive) and Adverse Negatives (AN) to mean the sum of utilities from all negative results that were adverse or harmful (or at least would have been more beneficial had it been positive), then the final equation to allow for non-dichotomy of results becomes:
BP/(BP + AN).
- Recall: A measure of the benefit actually received from positive results compared to the benefit from positive results that would have been received from an optimally performing system. This is calculated as:
$TP/($TP + $C(FN))
or more generally (allowing for any measure of utility):
uTP/(uTP + uC(FN))
or even more generally (allowing for non-dichotomy of results):
The last term in this equation would be read as “the complementary benefit of adverse negatives.”
Our shocking discovery, one that we cannot find has ever been described previously, is that when we properly weight each result based on its utility, and the utility is asymmetric (different magnitude if predict positive than if predict negative), then there’s not just one but rather two distinct basic metrics for the relationship between true positives and false negatives. These two metrics, Sensitivity and Recall, measure different things and they also return different values in nearly every case. In medicine, the only time they give the same result is the exceedingly rare case where the harm (as opposed to loss of benefit) from a negative result exactly balances the benefit from a positive result. Taking it a step further, there’s actually an entirely new set of basic performance metrics (not previously described, to our knowledge) that we have uncovered.
To properly handle cases of non-uniformity, non-dichotomy, and asymmetry of prediction results, we also discovered that a better and more complete version of the classic count-based confusion matrix is needed.
The classic confusion matrix tallies counts of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN):
|Actual = Yes||Actual = No|
|Predict = Yes||TP||FP|
|Predict = No||FN||TN|
Our new, utility-based confusion matrix sums the utilities of positive and negative predictions rather than just tallying counts. In this way it properly handles both uniform and non-uniform utility of results. It categorizes results based on the benefit or harm created rather than the correctness or incorrectness of the result. In this way it properly handles both dichotomized and non-dichotomized results. Finally, it includes columns that explicitly sum the “complementary” utilities — the utilities that would have been realized had the opposite prediction been made. In this way it properly handles both symmetric and asymmetric results.
Here is the complete, utility-based confusion matrix that makes no assumptions about the uniformity, dichotomy, or symmetry of results:
|Realized Beneficial Utility||Realized|
|Predict = Yes||AC(BP)||BP||AP||BC(AP)|
|Predict = No||AC(BN)||BN||AN||BC(AN)|
We have published an academic paper, available here, that more fully describes and assesses:
- The entirety of this discovery of a more complete and utility-based confusion matrix, along with how to implement it in real-world use.
- Two new and important sets of key performance metrics: one for metrics of realized utility and another for the capture rates of potentially available utility.
- A method for improving the performance of Boolean predictors in real world use, often improving utility-based precision substantially for any given level of utility-based recall. We won’t spend more time on this exciting development here, but in a later post we will dive right in.
OK, let’s play the game one final time. I’ll put a new penny in each of 10 red books and a vintage gold doubloon (worth at least $1,000) in each of 10 blue books. Again, you can pick 10 red books and 10 blue books out of my library. I feel badly for taking 140 bucks from you earlier, so I’ll let you keep whatever you find — no penalty! Plus, I’ll even let you use a metal detector.
You go online and search for metal detectors. You find two that are about the same price, and both get good reviews from customers. Since my library is cramped and you won’t be able to bring both inside, you’ll have to pick one. Yikes, that means it’s time for performance metrics. But which one?
Wait a minute. With no penalty, this is a no-lose situation, so why make a big deal over the metrics? These academics in their ivory towers with their fancy equations! They’re out of touch with us down here in the real world just trying to make things a little better. You resolve to follow the “KISS” principle and just rely on classic count-based metrics to help you choose. After all, everybody uses them all the time!
The manufacturers of Metal Detector #1 claim a sensitivity (TP/TP+FN) of 50%: you interpret this to mean that it will find 50% of the coins. The manufacturers of Metal Detector #2 report a sensitivity of only 40%. Obviously #1 is better, right? You can’t argue with the numbers!
However, a nagging little voice reminds you that there’s a lot of money at stake. You decide to order both units and test them yourself before choosing one. You replicate the exact scenario you expect to encounter in my library, and your test shows that the manufacturers didn’t lie: Metal Detector #1 finds 10 coins, or 50%. Metal Detector #2 finds only 8 coins, or 40%. How can they even sell such an inferior product?.
But wait, what about utility? You look at the coins that were detected.
It turns out the classifier inside Metal Detector #1 was trained on US pocket change and is very good at detecting copper, zinc, and nickel. The 10 coins found by Metal Detector #1 were all pennies! Your captured value was just $0.10 of $10,000.10 available.
The classifier inside Metal Detector #2 was trained on recovered pirate treasure, and is good at detecting precious metals. Arrrr, matey! All 8 coins found were gold doubloons, so your captured value was $8,000 of $10,000.10 available.
No matter which metal detector you choose, you will go home with more money than you originally had. If you “keep it simple” and use count-based metrics to make your decision, you will choose Metal Detector #1 because it has a count-based sensitivity of 50%, obviously better than the 40% sensitivity of Metal Detector #2. That’s a special kind of KISS, with the emphasis on the last ‘S’. No pirate treasure for you!
Pirate treasure aside, when choosing between different predictors or different implementations of predictors that will be used in the clinical arena, patient lives are affected by those decisions. If you assess predictors using metrics that don’t reflect what you care about, you’re likely to choose the wrong predictor or design a bad implementation. In many cases, this may result in clinical harm and even in excess lives lost. Choosing metrics that accurately reflect the kind of real-world clinical value you care about will give you a better chance of selecting the right predictor and the right implementation to achieve better outcomes.
In recent years, clinicians have often complained that “machine learning” is overhyped. We often see articles claiming great performance for a predictor, when the real-world performance on the front lines doesn’t seem to be anywhere near as good. At other times, predictors that are working well in practice are abandoned because “measured performance” using count-based metrics was poor.
Our new utility-based performance metrics are designed to more closely match up with the utility that real users will actually experience from a predictor. We hope this will eventually reduce the frequency with which health systems adopt a bad predictor with good c-metrics, or abandon a good predictor that happens to have poor c-metrics. Even without the help of a machine-learning algorithm we can predict that would be a good thing for healthcare.
The opinions expressed here are personal, and not necessarily those of anyone else, including our employers.


Force Voronoi Kaleidoscope
D3.js provides a layout that simulates particles interacting via charge and gravity, called a force directed layout.
When a click lands in the iframe below, a particle is placed into a force directed layout. It's position is then mirrored to create the kaleidoscope effect.
Open in a new Window
This array of points become the vertices of a Voronoi diagram , which partitions the space by drawing line segments equidistant from the nearest points.
By then gradually changing the charge acting on the vertices, a dynamic kaleidoscope effect is created.
Check out the original work by Mike Bostock that became this toy here and here.


|3||2π / 3||120°||triangle|
|4||π / 2||90°||square|
|5||2π / 5||72°||pentagon|
|n||2π / n||360 / n||n-gon|
But suppose you want to position points around the surface of a sphere so that they're evenly spaced.
What is the angle between adjacent points? Is it possible to position the points so that the angle is the same between all adjacent points on the sphere?
Let's look at some examples. For n=4, you have a tetrahedron.
According to wikipedia, the angle between adjacent points is arccos(-1/3) or about 109.4712°.
For n=6, you have an octahedron. The angle between adjacent points is π / 2 or 90°.
To summarize and expand:
|n||faces||tri. faces||edges||tri. edges||angle||angle||name|
|6||8||8||12||12||π / 2||90°||octahedron|
So what's the answer? I don't know, but one thing we have learned is that a cube's points are not all evenly spaced. The angles between them follow a regular pattern. There are twelve edges with an angle between the points of 2 arcsin(1/√3) ≈ 70.528°, and if you include all triangle edges, there are six edges with an angle between the points of 2 arcsin(√2/√3) ≈ 109.4712°.
Continue reading Part 2


Chapter 4 - Functions
Page 1 of 15
|Functions - Function Definition - Contents|
A mathematical function is an equation that assigns a single answer to each possible question. A variable represents an unknown value in the function's equation. The function's input variable is usually represented by an x. This is the independent variable to the function. The function's output is a variable usually represented by y. This is the dependent output of the function. To be a function every x value has to have one and only one answering y value. More than one x value can yield the same y value as long as there is the only one answering value. Function mapping of a mathematical equation with two variables takes the elements of the domain set and maps them onto the function's range set on a two-dimensional grid. This grid is called a Cartesian Plane named after the French Renaissance mathematician Descartes. The grid has bisecting perpendicular coordinate number lines each representing the set of real numbers intersecting at the zero point of both number lines at the center of the grid. This is the origin of the Cartesian Plane Using this coordinate system a function's variables are assigned x and y values which can be mapped out as a two-dimensional representation. A map of a function can be continuous or discontinuous. This can be shown by mapping the function on to the Cartesian Plane to see if it is a continuous line. Continuity of a function can also be mathematically proved by examining summations and limits of the function. There are a series of fundamental mathematical functions used in basic math, geometry and trigonometry that will be examined in the following chapters.
|Functions - Function Definition - Examples|
From the function y = x squared,
(x, y) = (1, 1), (2, 4), (3, 9) . . .
|Functions - Sections - Chapters|
|1 - Function Definition||2 - Function Notation||3 - Function Domain|
|4 - Function Range||5 - Composite Functions||6 - Inverse Functions|
|7 - Linear Functions||8 - Power Functions||9 - Quadratic Functions|
|10 - Logarithmic Functions||11 - Exponential Functions||12 - Factorial Functions|
|13 - Limit Functions||14 - Summation Functions||15 - Percentage Functions|
Math Book Chapters


Director of Performance Analytics,
Andre is focused on creating mathematical models whose concepts are carefully true to the meanings ascribed to them, so that their evaluation provides reliable information to users.
He began his career in finance building fixed income pricing models, and has worked, done research and published in physics, the foundations of physics, epistemology, and broadly in ex post financial analysis with particular emphasis on decision evaluation.
He has a B.S. from The Cooper Union and a Ph.D. in theoretical physics from Boston University.


You can use these skew tetrahedra to build cubes.Print out
a whole stack and take 6 and make a CUBE, three left oriented
and three right oriented. Make a PYRAMID with 8, 4 left and 4
Take 12, six of each 'political' orientation, tape them together
in a ring, Yes I wrote tape - sacrilege - alternating between
left and right and it will flex in many funny shapes.
If you make a box that fits this shape and give it
to a friend to play with, he might have difficulties in getting
it back in the box again, because it can flex into many different
forms, but the one that fits the box is not easily found.


You can see how the VT looks like in this demo videos I just recorded.
So how does it work?
To illustrate how the VT is implemented, let's imagine our friend Felix the cat is the input of the VT in a certain moment in time. Then, the state of the wave function will be obtained by computing two projections: the one in the X axis and the one in the Y axis. To compute the projections I just consider p(x) = sum(img(x, y)) for all possible y values, and viceversa. Each projection has different meaning for the computation of the wave form. The projection in Y is used to represent a spectral analysis whereas the projection in X is used to calculate a base frequency. For the spectral analysis I just discretize the projection to a reasonable limit (16 partials) and for the base frequency I compute the mean of the distribution of the projection.
Before doing the projections, I binarize the image and optionally apply gradient detection to it.
Hasn't someone done something like that?
I am aware that very similar things exist. There are very complex proprietary pieces of software that can do almost anything with live video. Speaking about simple software like mine, this and this are using a different approach: they are getting the input of the waveform inside the webcam output, whereas in the VT the whole webcam output is the input of the waveform. I think this is a big difference. It probably has been simpler to implement (otherwise you have to first recognize some kind of object to get some input) and it leads to infinite possibilities (any image can be the input of a waveform).
Finally, this thing implemented using the Wii is really cool.


Michael Lacey was born on a sunny day on September 26, 1959. From an early age, Lacey found himself interested in mathematics and was highly competent at solving complex problems. As such, he received a doctoral degree in Mathematics at the prestigious University of Illinois at Urbana-Champaign. Read more: Michael Lacey | GAtech and Michael Lacey | Wikipedia
He received guidance from his then mentor, Walter Phillipp, an accomplished mathematician in his own rights. Lacey’s thesis mentioned Banach spaces (a functional analysis relating to vector space) and solved multiple problems related to empirical functions and iterated logarithms. In his early career, Lacey’s works focused mostly on probability and harmonic relations.
After his tenure at the University of Illinois at Urbana-Champaign in 1987, Lacey started his career at the Louisiana State University, and then the University of North Carolina at Chapel Hill.
At these universities, Lacey partnered with his old friend and mentor, Walter Phillipp to provide evidence for their highly esteemed central limit theorem, which adds that when independent variables are added to certain functions, their sums tend to create a normal distribution.
Soon, Michael Lacey acquired a position at the Indiana University, where he received several accolades, including a fellowship at the National Science Foundation (a governmental organization which supports research in various non-medical fields).
With the help of the National Science Foundation, Lacey began studying bilinear Hilbert transform, which takes the function of one subset and produces another function. At the moment, this function was debated by various mathematicians around the nation, including Alberto Calderon.
Although Lacey’s studies at Indiana University lasted only three years, he was awarded the Salem Prize (awarded to young mathematicians who have accomplished outstanding works) with his fellow mathematician, Christoph Thiele.
In 1996, Lacey decided to become a Mathematics Professor at the Georgia Institute of Technology. He continually enjoys traveling to foreign countries. In fact, he has worked with various foreign mathematicians such as Xiaochun Li, receiving awards such as the Guggenheim Fellowship (grants given to highly capable mathematicians).
Recently in 2012, Lacey became a member of the American Mathematical Society, which works to aid mathematicians around the world with research, scholarship, and a network of universities.