MATH 92: Applied Beginning and Intermediate Algebra
This course emphasizes real world applications of beginning and intermediate algebra. Topics include a review of fractions, decimals and percents, and the development of linear, quadratic, rational, radical, exponential and logarithmic functions. This course is designed for those students whose major and transfer institution requires only statistics or math for liberal arts as the transfer level math course for the degree. This course does not meet the prerequisite for trigonometry, college algebra or any other transfer level math course.
Student Learning Outcomes
Upon successful completion of this course the student will be able to demonstrate knowledge related to the interrelatedness of the equation of a quadratic function with its graph, including the vertex and x and y intercepts. Additionally, specific course objectives include:
1. Perform the basic arithmetic operations with real numbers, exponents and the order of operations.
2. Apply properties of equality to solve linear equations and related application problems.
3. Determine the equation for a linear function and graph it.
4. Perform the basic arithmetic operations with polynomials.
5. Factor polynomial expressions using a variety of methods.
6. Solve quadratic equations by factoring and use of the quadratic formula.
7. Graph quadratic functions.
8. Identify functions from their equations and graphs and use appropriate functional notation.
9. Perform the basic arithmetic operations with rational expressions.
10. Solve systems of linear equations in two variables graphically and algebraically.
11. Solve exponential and logarithmic equations and applications.
12. Use correct notation to identify, simplify and use arithmetic and geometric sequence and series.
Math 92 Links


Dr Antonie Kotze (Pr.Phys)
Antonie holds a Ph.D. in Theoretical/Mathematical Physics from the University of the Witwatersrand where Quantum Chaos Theory was his field of interest. He is a registered professional physicist (http://www.saip.org.za/index.php/news-and-events/opportunities/287-register-as-professional-physicist-pr-phys-with-saip). With more than 23 years' experience as a quantitative analyst in the South African and African financial and derivatives markets, he is a renowned expert in financial mathematics, modelling and training. Antonie is a former Rand Afrikaans University (currently the University of Johannesburg) faculty member who has facilitated many successful training workshops throughout Africa. He has recently been appointed Senior Research Associate in the department of Quantitative Finance at the University of Johannesburg and and as Extraordinary Senior Lecturer in the Department of Mathematics and Applied Mathematics at the University of Pretoria.
With his one foot firmly planted in academia, he bridges the gap between academic research and the practicalities of the trading floor and risk management within trading houses.
The span of Antonie's derivative experience includes capital, interest rate, foreign exchange, commodity (hards and softs), credit and equity markets. He has hands-on knowledge of trading derivatives, statistical analysis, Monte Carlo simulations, stress testing and scenario analysis, option trading and risk management systems, basket and relative arbitrage, futures curves, equity linked and interest rate derivative structures, foreign exchange futures, derivatives clearing and collateralised finance - the gains of working in a front office quantitative position. He is also a registered Safex and YieldX derivatives dealer.
He has been an independent derivatives pricing, valuation and model validation specialist and expert since 2003. He also conducts and facilitates specialist training courses and workshops on all topics relating to the financial markets, trading, risk management, Basel regulations and derivatives pricing and modeling.
Antonie's career history includes close work with institutional clients and hedge funds to devise investment, hedging and gearing strategies utilising complex optionality. He has developed highly sophisticated mathematical models and computer systems used by traders, risk managers, fund managers and corporate financiers. As a derivative expert he assists companies and audit firms in the structuring, modelling and valuation of complex mergers, acquisitions, take-overs, lending agreements between companies, BEE structures, structured transactions and employee share incentive schemes.
Antonie held positions as a quant at Standard Bank, RMB, Mercury Structured Products and Absa Corporate and Merchant Bank (now Absa Capital and part of Barclays Capital) before he incorporated Financial Chaos Theory (FCT) in 2003. FCT is proud to be an independent valuation service provider or pricing vendor.


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Simple calculator with Hexadecimal, Octal and Decimal conversion functions. Requirements: iOS 4.0 or later. Compatible with iPhone, iPad, and iPod touch.
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This is the perfect tool for the advanced developer Provides features such as * number format conversion (hexadecimal, decimal, octal, binary) * data type conversion (BYTE, WORD, DWORD, QWORD) * calculate logical expressions The calculator is precise up to 64 bits No decimal numbers allowed...
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Application developed to make the conversion quickly and objectively between number systems such as Hexadecimal, Decimal, Octal and Binary. Aplicacao desenvolvida para fazer conversao entre sistemas numericos de forma simples e objetiva como: Hexadecimal, Decimal, Octal e Binario Requirements:...
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Hexadecimal Calculator is a handy tool for doing number conversions using either hexadecimal, decimal or binary. It can also be used for simple calculations in any of the three bases. Simple calculations may involve addition, subtraction, division and multiplication. Also the boolean operations...
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Replace Pioneer is:A text replacement and format conversion tool; A word translation and correction tool;An encode converstion tool supporting any encoding;A html to text/link converstion tool;A binary/hex/octet/decimal conversion tool;A batch command generator;A word counting/sorting/statistics...
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Compress and extract to disk / memory / Clipboard, search files or ASCII / binary / hexadecimal / decimal / octal data in archives / non-archives, encode / decode, convert between archive formats, work with multi-volume / multi-part / nested / def64X huge 4GB archives, test, SFX, UnSFX, Edit...
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Binary Converter is a tool used for various applications in digital electronics to perform binary to decimal conversion and decimal to binary conversion. In digital computation, the binary numbers are the numeric codes to represent the decimal numbers. Whereas the decimal numbers are the numbers...
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10 0A 1010 [Ten] is a number base convertor and calculator - ALL IN ONE! *** Special Features *** View Hexadecimal, Decimal and Binary equivalents at the same time! Switch between IEEE-754 and PIC Microchip Format for 32-bit Floating Point numbers. Its late at night, the caffeine is wearing...
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Puppy Pull is a multiplayer game that allows students from anywhere in the world to compete with each other while practicing decimal conversion. The amount that your puppy pulls is determined by how fast you answer the decimal problems. Requirements: iOS 4.3 or later. Compatible with iPhone,...
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Binfar2Dec:Binary vector To Decimal vector Dec2Binfar:Decimal vector To Binary vectorReadme:Helpful mfile (exemples:how to use m functions).
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Hex Editor II is a hexadecimal editor that helps an average PC user edit large files, containing non-textual data, safely and easily. Unlike other hexadecimal editors, Hex Editor II has two inbuilt plugins: hex calculator, and base converter, as well as the ability to edit files in any of the...
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degtodec.sh is a script which does conversion between decimal numbers and degree values (degrees, minutes, seconds, or hours, minutes, seconds). Possible input methods are interactively, by parameter, or directly via stdin. The project can be used, for example, for geographical conversions or for...
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Hextrapolate is a flexible number conversion utility for programmers and other people who work with numbers in a digital setting. It converts, in real time, any number in any base representation from Base 2 (Binary) through to Base 36. Numbers can be arbitrarily large, limited only by the...
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tkalc is a pocket calculator for developers and IP network operators. Arbitrary arithmetic/logical expressions can be entered very quickly. The result is then simultaneously displayed in decimal, hexadecimal, octal, binary, ASCII, IP address, and boolean.
Added: January 15, 2006 | Visits: 5.202
With BinView you can quickly examine the binary content of a file, by dropping it from explorer onto the main window. Binview allows viewing in hexadecimal, decimal and ascii notation. File index pointers indicate what position is being shown. The number of bytes per line can be configured.
Added: March 10, 2010 | Visits: 1.221
You can buy this game Feel the scream of lasers against cold steel and prepare for the clash of armor against armor -- Heavy Gear II for Linux is here! Get ready for the ultimate in mech experiences: a thrilling combat adventure pitting robot against robot in the distant future is waiting for...
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Home buyers, if real estate and mortgage math stress you, Real Estate Calculator Suite is the help you need!
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MicroAdobe Hex Editor is the first Hex Editor tool, you can edit, cut, copy, paste, insert, and delete hex, print, and export to Txt,MicroAdobe Hex Editor you can supports drag and drop,The friendly interfacial energy causes you rapidly easily to edit hex, in work area.Edit, Cut, Copy, Paste,...
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2 edition of Homotopy theory in groupoid enriched categories found in the catalog.
Homotopy theory in groupoid enriched categories
Thesis (Ph.D.)--University of Toronto, 1987.
1. Overview: Introduction to the homotopy theory of homotopy theories To understand homotopy theories, and then the homotopy theory of them, we ﬂrst need an understanding of what a \homotopy theory" is. The starting point is the classical homotopy theory of topological spaces. In this setting, we consider. homotopy theory of T(C,E). The notation Cath(T 1,T 2) or T hT 1 2 denotes the homotopy theory of functors from the ﬁrst homotopy theory to the second, but taken in the correct homotopy theoretic way. The notation ThT 1 2 is very similar to a notation for homotopy ﬁxed point sets that will come up later on (), but I’ll use it anyway.
This article is about topology. For chemistry, see Homotopic groups. The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy. The homotopy theory of -categories infty,1 Bergner, Julia Elizabeth. The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides.
What homotopy type theory provides is a more transparent language that helps to understand the equations, proofs, and other elements that are being evaluated when looking at advanced mathematics. This allows it to be understood on a core level by more individuals, even if they are not putting in the effort to solve the equations involved. This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others. Rating: (not yet rated) 0 with reviews - Be the first.
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In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms.
Basic concepts of enriched category theory 25 A ﬁrst example 26 The base for enrichment 26 Enriched categories 27 Underlying categories of enriched categories 30 Enriched functors and enriched natural transformations 34 Simplicial categories 36 Tensors and cotensors 37 Simplicial homotopy and simplicial File Size: 1MB.
Logical constructions in type theory then correspond to homotopy-invariant constructions on spaces, while theorems and even proofs in the logical system inherit a homotopical meaning. As the natural logic of homotopy, constructive type theory is also related to higher category theory as it is used e.g.
in the notion of a higher topos. Basically, Π 1 (X, A) \Pi_1(X,A) allows for the computation of homotopy 1-types; the theory was developed in Elements of Modern Topology (), now available as Topology and Groupoids ().
These accounts show the use of the algebra of groupoids in 1-dimensional homotopy theory, for example for covering space s, and, in the later edition. History Prehistory: the groupoid model. At one time the idea that types in intensional type theory with their identity types could be regarded as groupoids was mathematical was first made precise semantically in the paper of Martin Hofmann and Thomas Streicher called "The groupoid interpretation of type theory", in which they showed that intensional type theory had a model in.
The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory.
This book provides a relatively self Cited by: 3. There is an old paper: P. Fantham and E. Moore, Groupoid enriched categories and homotopy theory, Canad. Math., 35, (), –which also examines this question and of course, some of the classical book by Gabriel and Zisman is devoted to developing GE-categories in your sense.
Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model by: This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples.
Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion. In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.
Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their. Homotopy Type Theory: Univalent Foundations of Mathematics The Univalent Foundations Program Institute for Advanced Study Buy a hardcover copy for $ [ pages, 6" × 9" size, hardcover] Buy a paperback copy for $ [ pages, 6" × 9" size, paperback] Download PDF for on-screen viewing.
[+ pages, letter size, in color, with color links]. Analytic ∞ \infty-groupoid theory seems, at least a priori, to deal more easily with morphisms than with discs, and hence might not consider spheres to be “small” examples — whereas both classical homotopy theory and homotopy type theory do consider spheres to be “small”, the one by using discs and the other by using freely.
homotopy theory categories 2-categories enriched categories, multicategories, brations, ) 1-groupoids are the raw material of higher-dimensionality.
[Homotopy type theory is] a new conception of foundations of mathematics, with intrinsic homotopical content, [and] an. homotopy theory for lie ∞-groupoids & application to integration 3 L ∞ -algebras have a good notion of a homotopy theory which can be modeled in a variety of ways, e.g. via enriched category theory , or as the biﬁbrant objects.
8 Some category theory Locally presentable categories Monadic projection Miscellany about limits and colimits Diagram categories Enriched categories Internal Hom Cell complexes The small object argument Injective coﬁbrations in diagram categories 9 Model categories THE HOMOTOPY THEORY OF A ∞-CATEGORIES GEOFFROYHOREL Abstract.
We put a left model structure on the category of A∞-categories enriched in a. The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory.
In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories.
Starting with a cohesive overview of the many different approaches currently used. In fact, commencing with the work of Gabriel-Zisman , various aspects of the homotopy theory familiar to Top, have been developed in the setting of groupoid enriched categories.
Simplicial Homotopy Theory by Goerss-Jardine is great, though it won't talk in terms of higher higher groupoids. The idea that Kan complexes are equivalent to higher groupoids is an incarnation of the homotopy hypothesis, one of the fundamental problems of higher category theory.
But there's not yet a convincing algebraic development of the. arXivv3  2 Oct Homotopy theory with ∗-categories UlrichBunke∗ October3, Abstract We construct model category structures on various types of (markeCited by: 4.
Under the equivalence of categories between small 2-categories and double categories with connection given in [BM] the homotopy double groupoid corresponds to the homotopy 2-groupoid, G 2 (X.Homotopy theory, and change of base for groupoids and multiple groupoids⁄y Ronald Brown March 6, Abstract This survey article shows how the notion of \change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues.The main novelty when doing category theory in homotopy type theory is that you have more freedom in how you treat equality of objects in a category.
The obvious definition of a category A A has a type of objects, say A 0: Type A_0:Type, and a family of types of morphisms, say hom A: A 0 × A 0 → Type hom_A: A_0 \times A_0 \to Type.


A-Level maths tuition is a great way to boost your chances of getting a high grade at the end of your course. These tutors have a proven track record of preparing students for their final year exams and building their confidence.
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The local Kumon Lowton Study Centre offers individualised maths and English study programmes for children of all ages and abilities within the local community.
There is no minimum age requirement at the Lowton Study Centre, as the Kumon Method of Learning can help to develop children into confident, independent learners from a young age. Children as young as 3-years-old currently attend the study centre.
The sooner a child begins their Kumon journey, the sooner they are able to start developing effective study skills, which will remain with them for the rest of their lives. The Kumon Method is suitable for children of all ages as the programmes are tailored to suit the needs of each individual child. For this reason, no starting point is the same. Kumon students work through their study programmes at their own pace, which helps them to realise their full potential.
The Kumon Lowton Study Centre has a dedicated learning area specifically for its youngest students. The early learner area is consistently manned by a senior class assistant, who dedicates their time to helping the younger students individually as they begin their Kumon journey. Alongside the Kumon Maths and English Programmes, younger students are able to complete additional pencil skills worksheets, which have colourful pictures to help children practise pencil skills by holding and using a pencil.
Elizabeth Forrester, the Instructor at the Kumon Lowton Study Centre, has seen many younger students blossom as they have progressed on their individual Kumon programmes. Elizabeth says:
“I believe that the earlier children learn, the better. Our unique study programmes offer fun, engaging learning for children as young as two. For many children enrolled in Kumon early learning levels, the centre is their first introduction to an academic setting. Kumon helps develop their maths, reading, and pencil skills and allows them to experience feelings of success as they progress which sparks a confidence that inspires a love of learning.
By introducing learning into their daily routine in advance of starting primary school, you can give them a head start. They will begin school with boosted confidence in their skills and ability allowing them to adjust quickly and get their academic experience off to a flying start.
Kumon develops children through the development of independent learning skills, rather than being taught in the conventional sense. They develop the ability to learn for themselves, so they do not rely on the skills and knowledge of others.
A 2020 study, conducted by the Education Policy Institute in England has shown that Kumon students significantly outperform their peers in their Key Stage 2 SATs; achieving scores that place them over a year ahead of the national average, and over six months ahead of their matched peers in Maths learning, so there has never been a better time to try Kumon out for yourself!
Last year marked 30 years since Kumon were officially established in the UK. Kumon is now the largest supplementary education provider in the UK and Ireland with more than 60,000 Kumon students learning at more than 630 study centres across the UK and Ireland. We can help give your child the tools for a lifelong love of learning.
Kumon now has its very own app to support learning, too!
The latest innovation at the Kumon Centre is the introduction of a new app – Kumon Connect – which allows students to complete their work digitally, using a tablet and stylus. This allows their Kumon tutor to set them tailored digital worksheets which can be downloaded at home, making the process convenient and allowing students to become independent learners.
Because the Kumon Connect app is used with a stylus, it means the student still enhances their cognitive and writing skills, whilst also familiarising themselves with digital learning tools.
How much is it to study with Kumon?
Each student that signs up to Kumon pays a small registration fee of £40. You’ll then choose which subjects your child needs help with, and the fee per subject is £70 per month.
Contact the Kumon Study Centre in Lowton for more information
The Kumon Lowton Study Centre is located at St Mary’s Church Community Hall, Newton Road, Lowton, WA3 1DQ, with classes running on Mondays 2.30pm – 5.30pm, Tuesdays 2.30pm – 7pm and Wednesday (online only) 5pm – 7pm.


The Applicability of Currriculum Based-Measurement In Math Computation in Jordan
The proper assessment of math computational skills is essential for monitoring progress, predicting achievement, and identifying students with disabilities. The current study extends previous research on assessment of curriculum-based measurement in mathematics(M-CBM) . The purpose of this research was to examine the effects of the M-CBM computation assessment on improving third-grade math achievement. This paper presents a comparison study of two classrooms; one used a M-CBM computation in addition to the summative assessment and one used summative assessment only. Each class consisted of 35 students; three who had a Specific Learning Disability in math. The results of a 15-week CBM process demonstrated the effectiveness of using the M-CBM with third- grade students. Furthermore, when compared to the traditional way of assessment, the use of the MCBM produced significant gains in students’ achievement, specifically, for the students who were struggling with math.


I am writing this post, after having provided my next bit of support, as a tutor, on a course. Before that I was doing some marking (where I fell asleep!) So, life is proving rather busy at the moment. (Have I taken too much on? – maybe!)
With my reference to Part 4 of motivation, I had a pleasant surprise from one of the 9/10 year old’s today. They said that they were now beginning to enjoy maths, when it comes to telling the time. (They have even been practising at home, it seems!) Another student, however, said they hate maths!
There seems to be such a disparity between different students’ perceptions about maths when they are young. Some like it and some hate it! I wonder if other subjects are the same?
Just so that you are aware, my Mathematics Question – Number 17 might be a little delayed due to how busy I am. Please keep an eye out here and on my website.
Bye for now.
For those of you following the Mathematics Questions, Number 16 is now available to download from https://nilsbirdtraining.com/mathematics-resources.
As I said in a previous post, from Mathematics Question – Number 17 onwards, the questions and any additional information that would ordinarily appear in the Mathematics Questions – Newsletters, will become a feature of the blog. The newsletters will then become a summary of the blog posts. You can sign up to receive the Mathematics Questions – Newsletters here.
A new feature, coming soon, will be a dedicated domain name for the Mathematics Questions.
Mathematics Question – Number 15 can now be found at https://nilsbirdtraining.com/mathematics-resources. This time the combination of topics is – geometric sequences, ratio and surds.
From Mathematics Question – Number 17, onwards, I plan to post each question on the blog, as well, with extra comments that will then appear in the Mathematics Questions – Newsletters. So, basically, each question, plus comments, will become part of the blog and then the newsletter will appear periodically, bringing together, each time, three questions plus comments (as it does now). Hope that makes sense!
See you soon,


Posts tagged equation
The Cooke Triplet is a system of three lenses designed in the 19th century to reduce distortion. In 1950 a specific triplet was invented and patented by Eastman Kodak (EF=100mm, f/1.9) and we will look at how to recreate it in Geomagic Design using scripting. We will create a script that can generate any type of thick lens, including how to solve the lensmaker’s equation. Hopefully the techniques demonstrated here will be useful to others solving and creating mathematical shapes in Geomagic Design.
Download, install and run WizoScript 1.70 or later. Start Geomagic Design. This script was tested with version 22.214.171.12490.
First we must enable a couple of useful libraries and make sure our script is using millimeters to match the values in the patent.
import sympy from sympy import * import math Units.Current = UnitTypes.Millimeters
We are going to generate three lens parts and save them so we need to specify a location to save to. Change this to make your PC.
OutputFolder = "P:\\temp"
The patent completely describes the three lenses but we are going to make it a bit more interesting by calculating the radius of the back of the lens from the focal length. To do that we need to use the lensmaker’s equation.
In the formula the radius of the back of the lens is R2. This appears in two locations in the formula. We could rearrange it to give us R2 = … but that is a bit tedious. Instead we can leave WizoScript to worry about it instead. Here is how we enter the equation into the script using ‘rb’ to represent the radius of the back of the lens.
# calculates the back radius of a lens based on the other parameters # uses the lensmaker's equation def GetBackRadius(FrontRadius, Thickness, RefractiveIndex, FocalLength): rb = Symbol('rb') Equation = Eq((RefractiveIndex-1)*((1/FrontRadius)-(1/rb)+(((RefractiveIndex-1)*Thickness)/(RefractiveIndex*FrontRadius*rb))),(1/FocalLength)) # we don't bother rearranging the equation, instead we let the sympy module do that for us BackRadius = solve(Equation) return BackRadius
WizoScript rearranges the equation and calculates the value we need using the single line:
BackRadius = solve(Equation)
The equation is placed into a function called ‘GetBackRadius’ which calculates the value when given a set of parameters, such as the thickness of the lens, refractive index of the material, etc. We can call the function repeatedly for different parameters.
Now we need to start creating a part for a single lens. First we tell Geomagic Design to create a part. We are going to do that in a new function.
# creates a part representing a lens based on a set of parameters def GenerateLens(Name, Folder, FrontRadius, Thickness, RefractiveIndex, FocalLength, Diameter): # get missing parameter BackRadius = GetBackRadius(FrontRadius, Thickness, RefractiveIndex, FocalLength) # check diameter is small enough if Diameter > abs(FrontRadius): print "%s diameter is larger than the front radius %f" % (Name, FrontRadius) return if Diameter > abs(BackRadius): print "%s diameter is larger than the back radius %f" % (Name, BackRadius) return # create new part Lens = Part(Name)
The general approach we will take is to draw a 2D profile of half of the lens on the XY plane and then create a revolve boss around the X axis. The X axis is the optical axis (i.e. the center of rotation for the lens).
# start creating the 2d sketch representing the shape of the lens Profile = Lens.AddSketch("Profile", Lens.GetPlane("XY-Plane")) LensRadius = Diameter / 2.0
First we will draw the front of the lens. There are three possibilities, the front is flat (infinite radius), bulging out aka convex (positive radius) or indented aka concave (negative radius). We treat each case individually. Simple trigonometry is used to calculate the end points of the front. Note that infinity is represented in the script by ‘oo’.
# draw shape of the front of the lens, it can be flat, convex or concave if FrontRadius == oo: FEndX = 0 Profile.AddLine(0, 0, FEndX, LensRadius, False) elif FrontRadius > 0: Angle = asin(LensRadius / FrontRadius) FEndX = FrontRadius - (cos(Angle) * FrontRadius) Profile.AddArcCenterStartEnd(FrontRadius, 0, FEndX, LensRadius, 0, 0, False) else: FrontRadius = -FrontRadius Angle = asin(LensRadius / FrontRadius) FEndX = (cos(Angle) * FrontRadius) - FrontRadius Profile.AddArcCenterStartEnd(-FrontRadius, 0, 0, 0, FEndX, LensRadius, False)
The back of the lens is treated in the same way except the sign of the radius is reversed. This means that a convex shape is a negative radius and a concave shape is a positive radius.
# draw shape of the back of the lens, it can be flat, convex or concave if BackRadius == oo: BEndX = Thickness Profile.AddLine(Thickness, 0, BEndX, LensRadius, False) elif BackRadius < 0: BackRadius = -BackRadius Angle = asin(LensRadius / BackRadius) BEndX = (cos(Angle) * BackRadius) - BackRadius + Thickness Profile.AddArcCenterStartEnd(Thickness - BackRadius, 0, Thickness, 0, BEndX, LensRadius, False) else: Angle = asin(LensRadius / BackRadius) BEndX = -(cos(Angle) * BackRadius) + BackRadius + Thickness Profile.AddArcCenterStartEnd(Thickness + BackRadius, 0, BEndX, LensRadius, Thickness, 0, False)
Before we can add the revolve boss we must make sure the sketch is closed by drawing the line for the outside of the lens and the line along the optical axis
# check diameter of lens again if FEndX > BEndX: print "%s diameter is too large" % Name return # if this is not a "thin" lens then draw top of profile if FEndX != BEndX: Profile.AddLine(FEndX, LensRadius, BEndX, LensRadius, False) # draw line of profile along x axis (optical axis) Profile.AddLine(0, 0, Thickness, 0, False)
Now for the revolve boss.
# create lens by revolving sketch around x axis Lens.AddRevolveBoss("Lens", Profile, Lens.GetAxis("X-Axis"), 360.0)
Creating two reference planes for the front and back of the lens will help with aligning them to create the triplet. Finally we save and close the part.
# add reference planes to aid in aligning this lens with others Lens.AddPlane("Front", Lens.GetPlane("YZ-Plane"), 0); Lens.AddPlane("Back", Lens.GetPlane("YZ-Plane"), Thickness); # save and close Lens.Save(Folder) Lens.Close()
The last step is to generate the three lenses needed for the triplet using the values from the patent along with the focal length.
# lens 1 GenerateLens("Lens1", OutputFolder, 62.63, 25.56, 1.745, 73.2, 40.0) # lens 2 GenerateLens("Lens2", OutputFolder, -66.47, 4.51, 1.647, -37.74, 36.0) # lens 3 GenerateLens("Lens3", OutputFolder, 119.5, 19.92, 1.697, 54.88, 30.0)
The complete script can be obtained from the script library. Running it will generate three part files, one for each lens.
Create a new assembly and add the three lenses in the order lens1, lens2 and lens3. Align the X axis of all three lenses. Align the front plane of lens2 7.89mm from the back plane of lens1. Align the front plane of lens3 14.14mm from the back plane of lens2.


Trigonometry is the branch of mathematics, which mainly deals with the studying of triangles their identities and the relationships between distances and the angles. The term Trigonometry was derived from Greek word meaning measurements of triangles. It is an applied mathematics, mainly used to calculate the angles and sides of a given triangle. The Hipparchus, Greek astronomer and mathematician is known as a father of Trigonometry and he was the first person to create the Trigonometric table.
The application of trigonometry is widely used in numerous fields including engineering, science, and others. As per the students, many of them face difficulties in learning and remembering the formulae but it cannot be ignored as it is the most important fields of mathematics.
Image Source: Pixabay
To help students in learning the Trigonometry Formula, here are some important tips to memorize Trigonometry formulas.
- The best way to learn any topic is by having a basic understanding of all the concepts. Therefore, be thorough with all the basics. Learn all the six Trigonometric formulae-sin, cos cot, cosec, sec, tan.
- Be regular to the class and stay focused on the teacher’s explanations. Make a note of all the important points discussed in the class session.
- For clear and better understandings of the Trigonometric formulae and identities, students can refer NCERT Solutions for Class 9 Maths.
- Solve more problem on a daily basis, as by solving more problems and different questions on the same formulas, students can easy memorize formulas.
- The best and fastest way to learn and remember the formulae is by memorizing a trigonometric ratio table and by making a note of all the formulas, which you need to be memorized in a sheet of paper and practice it on a daily basis.
These were some easy trick to learn trigonometry formula. Trigonometric formulas are quite difficult but not impossible. There are many other techniques, which can make the formula easier. The best trick is by practicing more problems with different formulas and identities. Once we are perfect with this formulas, we can remember them throughout our life.
For more numerical problems on trigonometry, students can refer to many books and to other online YouTube videos. Students can also download free NCERT textbooks, NCERT solutions, and exemplar which provides a number of solved numerical problems and help students to make the concepts crystal clear. Students can learn more about NCERT solutions by visiting the official websites.


PROBLEM 9. The light-year is the distance traveled by light during one Earth year. To three significant digits, the speed of light is 3.00 x 10^5 km/s. Find the length of the light-year in km and in AU.
Solution:1 Earth year = 365.25 days = 365.25 x 24 x 60 x 60 seconds. In one year, light travels 3.00 x 10^5 x 365.25 x 24 x 60 x 60 km = 9.47 x 10^12 km.
To express this distance in AU,
1 light-year = 9.47 x 1O^I2 km x 1 AU/(1.50 x lO^8km)
= 6.31 x 10^4 AU.
The "parsec" is the astronomical unit of distance that relates to observational measurements. In order to define this unit, we must consider the fact that when we observe the heavens, we have no direct perception of depth or distance. A useful model developed to portray the heavens is the celestial sphere. In this model, Earth is surrounded by an imaginary sphere with infinite radius. A coordinate system, similar to latitude and longitude, is imposed on the celestial sphere by projecting Earth's rotation axis on the sphere to identify the celestial north pole (CNP) and celestial south pole (CSP) as shown in Fig. 2.1. Since the radius of the celestial sphere is infinite, all parallel lines point to the same spot on the sphere, and so every line parallel to Earth's rotation axis also points to the celestial north and south poles.
Every star or celestial object can now have its position identified by the ordered pair of angles in the previous example. Because Earth rotates with respect to the celestial sphere, the time of observation must also be known in order to use the coordinate system. Differences in the positions of two objects on the celestial sphere are expressed in terms of the angle subtended at Earth by the arc joining these points. As Earth revolves around the Sun, very distant stars show no discernible changes in position, but closer stars will show apparent motion with respect to the celestial sphere when viewed from different points in Earth's orbit, as shown in Fig. 2.3. This apparent motion is called "parallactic motion", and the change in position is called the "parallax angle". In this context, 1 parsec is defined as the distance at which the radius of Earth's orbit subtends an angle measuring 1 arc-second (see Fig. 2.4).


What is geometry?
Geometry is one of the major fields of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the most important and common fields of mathematics because it concerns the real world application of mathematics, and it’s critical in fields like architecture and construction.
Who uses geometry?
It’s used commonly by architects, construction workers, carpenters, and other fields related to building because of the importance of geometric balance in a building.
When will I need to use geometry?
Well if you’re reading this you probably have to use it now! Geometry is a common math class students have to take so there’s an excellent chance you’ll encounter it sometime in school. In professional life geometry can be used in the fields mentioned above. In reality geometry we use basic geometry every day when we’re moving or placing things.
Is geometry difficult?
Though it depends on the specific geometry subject and level of the geometry, but generally geometry is going to be difficult. The basic geometry you learned in elementary school may not have been much a challenge, but the complex ideas of advanced geometry can lose even the best math student, and that’s the kind of geometry you’ll most likely encounter in college.
Can I get help with geometry?
Yes! That’s what geometryhelp.biz is here for, to get you the best assistance and expertise on the web!
Should I enlist the help of an online service?
It depends on your circumstances, if you’re on a tight schedule and lack the time, motivation, or background knowledge to complete your geometry assignments then enlisting the help of an online service may be the right move.
How do I choose which service is right for me?
Know what it is you need and find a service which can help you with all these needs, it’s that simple!
How can your geometry help writing service help me?
We’re the premier geometry help service on the web because we can do almost anything related to geometry for you, we built this service making sure it could help anyone with any geometry needs!


Mee Genius is an app for the iPad that allows students (4-6) practice reading. Mee Genius will let you read a book or it can read it to you. This app is good because students can learn at their own pace. Mee Genius is a great way for students to read at home or own at school.
Math Ninja has skills, strategy and math. This is a great game for you to play. If your child is having a hard time in math, Math Ninja is there for you! It is free in the app store. Math Ninja is putting First things First for your kids and making learning fun for your child and for you!


quadratic parabola function graph transformations notes charts bacterial transformation workflow–4 main steps the inverse trigonometric functions – she loves math graphs of rational functions horizontal asymptote video function table worksheets answers parent functions and all transformations worksheet answers elegant fun transformations semi log plot transformations of the sine and cosine graphs graphs of logarithmic functions video function transformations chart inspirational l & od as a function the inverse trigonometric functions – she loves math acceleration of of generalized fourier series by the d quadratic parabola function graph transformations notes charts
This graphic (Beautiful Neb Turbo Petent E Coli High Efficiency) over can be labelled with: graph transformation in backus bnf, put up through Dianna at 2018-12-30 19:44:35. To determine all photos within Best Of Graph Transformations Chart
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Rules for Miller Indices:
For example, if the x-, y-, and z- intercepts are 2, 1, and 3, the Miller
indices are calculated as:
Determine the intercepts of the face along the crystallographic axes, in
terms of unit cell dimensions.
Take the reciprocals
Reduce to lowest terms
Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis,
its intercept is at infinity and its Miller index is zero. A generic Miller
index is denoted by (hkl).
Take reciprocals: 1/2, 1/1, 1/3
Clear fractions (multiply by 6): 3, 6, 2
Reduce to lowest terms (already there)
If a plane has negative intercept, the negative number is denoted by
a bar above the number. Never alter negative numbers. For example,
do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that
the crystal may not have!
Some General Principles
If a Miller index is zero, the plane is parallel to that axis.
The smaller a Miller index, the more nearly parallel the plane is to the
The larger a Miller index, the more nearly perpendicular a plane is to
Multiplying or dividing a Miller index by a constant has no effect on the
orientation of the plane
Miller indices are almost always small.
Why Miller Indices?
of aminated gif above. Click the pause button, then step through the planes
Using reciprocals spares us the complication of infinite intercepts.
Formulas involving Miller indices are very similar to related formulas
from analytical geometry.
Specifying dimensions in unit cell terms means that the same label can
be applied to any face with a similar stacking pattern, regardless of the
crystal class of the crystal. Face 111 always steps the same way regardless
of crystal system.


I was writing a C# class library to manage finite groups. A group is a set G of elements with a defined binary operation *, where:
a * b is in G (closure)
a * (b * c) = (a * b) * c (associative)
a * a’ = e (inverse and identity)
a * e = e * a (identity)
The solution code is in my AjCodeKata Google Project, under trunk/AjGroups:
The key interfaces are:
IElement has a method Multiply, that receives another IElement and returns the result IElement. IElement.Order is the number of elements to be multiplied to get the identity:
a * a * a ….. * a = e
IGroup.Elements is the collection of IElements that are the elements of the group. IGroup.Order is the count of elements.
There are two implementations of that interfaces: one based on permutations, another based on named elements (such “a”, “e”) and a table describing their multiplications.
There are methods to:
– Get the subgroups of a group
– Get the normal subgroups of a group
– Answer if two groups are isomorphic
I followed TDD ideas during development. All tests in green:
Good code coverage:
I should make some refactoring, but the project is taking form. I will write posts describing the permutation implementation, and the abstract elements implementation.


The Math For Honor video game has begun the development of the web version of their game due to private donations and crowdfunding.
New York, NY – October 30 2015 – The Founder of Bunnycave Studios, Dante Hin-Gasco, is excited to announce that after their first media push and crowdfunding efforts, Bunnycave Studios is able to start the development of the web version of the Math For Honor video game.
The current version of the game is currently available for download on www.mathforhonor.com, but will soon be available online due to the funding from private donations and their GoFundMe campaign. Due to funding efforts, Bunnycave Studios was able to hire developer Michael Blout to further their vison of making the Math For Honor game available online as well as using Agile House for their coding incubator.
Bunnycave Studios is a VERY EARLY stage company with a vision to release more educational games in the future, as well as more games in general. However, they need more skilled professionals like artists, web and video game developers, and professionals to accomplish their goals. They need more support and donations to continue with the development of Math for Honor. Please help in the developing for the Math for Honor game by donating to their GoFundMe page: http://www.gofundme.com/mathforhonor
About Bunnycave Studios
Bunnycave Studios is a educational video game design company founded by Dante Hin-Gasco.
The Math for Honor video game has won the top prize in the high school division of MassDigi 2015, as well as a Regional Gold and National Silver keys in the Scholastic Arts and Writing Awards 2015.
Dante Hin-Gasco, a 11th grader that has been a video game developer since 2012, understands that many of his colleagues have trouble in classes just because they don’t have enough hands on practice in the subjects. Dante made Math for Honor to make it fun to practice and enhance your math skills. Bunnycave Studios needs skilled professionals like artists, web and video game developers, and professionals to continue with converting Math for Honor into a web-based educational video game. Please support Bunnycave Studios in developing Math for Honor by donating to their GoFundMe campaign or learning more about them on their website http://www.mathforhonor.com/
To help fund their GoFundMe campaign please visit:
New York, NY


Isobel is a little girl who loves solving problems. She enjoys figuring out the answers to addition, subtraction, multiplication, and division problems. Although, there is one problem is having trouble solving – one that concerns noisy neighbors.
Every time Isobel sits down to do her math homework, she can’t concentrate with the racket next door. She wonders if it’s a marching band, a basketball team, or even a family of elephants! When she does figure out who the boisterous people (or animals) living next door are, they become best friends!
Isobel Adds It Up by Kristy Everington is a sweet story emphasizing imagination. It’s a story that will keep young kids in suspense, excited to learn who the neighbors turn out to be. This book comes out June 29 with Random House. I got this advanced digital copy from the author.
I loved the fantastical aspect of this book, I think that side will interest kids who like envisioning things like subtractions getting soaked in the rain, and elephants being able to cook and do math. The illustrations by AG Ford also made the story more magical, you could really see into Isobel’s creative mind.
This book is great for elementary-age classrooms, and it reminds me of The Word Collector by Jason Reynolds. Both of them had a strong passion for a subject – for Jerome, words; for Isobel, math. Overall, please pick up this book!!
Four out of four roses!


Trend 53109 Fraction Fun Flash Card
Fraction Fun Flash Cards offer a thorough way to learn and understand fractions. One side has a numerical representation, while the other is divided to illustrate the fraction. Learn wholes through twelfths with symbols and graphic cues, plus 11 cut-apart cards for manipulatives. Durable cards are self-checking, and feature rounded corners for quick sorting. Ideal for first-grade through fourth-grade (ages 6 to 10).
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Most people consider the lottery a game of luck. All you have to do to play is to select random numbers. However, it should be remembered that just luck is not enough to guarantee big wins in the unpredictable lottery. Therefore, you still need to arm yourself with some important tips on how you can easily win the lottery.
Here is a list of notable tips you can take into account when playing and winning the lottery:
- Avoid choosing numbers that are significant in your life, such as your birthday, anniversary, or a favorable number of horoscopes. Yes, they can be very important, but they cannot even increase your chances of winning the lottery.
- Choose lottery numbers that appear frequently. There are currently several online sites that list the numbers that most often appear in lottery games. You must consider and control these numbers to increase your chances of winning the lottery without much effort.
- Use probability theory. In fact, you can use probability theory by playing the lottery, even if you are not a certified math genius. All you have to do is understand the basics of probability and get a good set of odd and even numbers in your lottery ticket.
- Never try to create sequences or patterns when choosing numbers for betting in the lottery. The winning lottery becomes 1, 5, 10, 15, 20 or 3, 6, 9, 12. In addition, it is completely impossible for you to win with a lotto ticket that has oblique lines and comic shapes.
- Do not try to use computer programs just to get a good set of lottery numbers. In fact, there are many programs on the Internet that promise to give you a winning streak of lottery numbers.


Associate Professor of Mathematical Sciences
Ph.D in Applied Mathematics, Brown University, 2004
B.S and M.S. in Computational Mathematics, Peking University, 1997 and 2000
NSF CAREER Award, 2009
Alfred P. Sloan Research Fellow, 2008
Stella Dafermos Award, Brown University, 2004
Simon Ostrach Fellowship, Brown University, 2003-2004
- Local-structure-preserving discontinuous Galerkin methods; divergence cleaning techniques.
- Computational methods for Maxwell equations, magnetohydrodynamics (MHD) equations and Hamilton-Jacobi equations.
- Fast sweeping methods for static Hamilton-Jacobi equations.
- Other high order numerical methods (weighted ENO methods, spectral methods) for hyperbolic conservation laws; computational fluid dynamics.
- F. Li and C.-W. Shu, A local-structure-preserving local discontinuous Galerkin method for the Laplace equation, Methods and Applications of Analysis, to appear.
- S. C. Brenner, F. Li and L.-Y. Sung, A locally divergence-free nonconforming finite element method for the reduced time-harmonic Maxwell equations, Mathematics of Computation , to appear.
- F. Li and C.-W. Shu, Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Applied Mathematics Letters , v18 (2005), pp.1204-1209.
- F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for MHD equations, Journal of Scientific Computing, v22-23 (2005), pp.413-442.
- B. Cockburn, F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, Journal of Computational Physics, v194 (2004), pp.588-610.
- L.-A. Ying and F. Li, Exterior Problem of the Darwin Model and its Numerical Computation, Mathematical Modelling and Numerical Analysis, v37 (2003), pp.515-532.


This week is the last chance to bet on Super Bowl 53 futures before the AFC and NFC title games are played on January 20th.
And to help you with your betting models, we decided to find out at the chances of the four remaining teams in the 2019 NFL Playoffs to win the big game, according to oddsmakers.
We’ll do this by examining the odds of the Saints, Chiefs, Patriots, and Rams and do a little math to find out their implied probability a.k.a. the Super Bowl chances of each team winning it all by converting it to a percentage.
BOVADA: SUPER BOWL 53 – ODDS TO WIN
New Orleans Saints +175
Kansas City Chiefs +275
New England Patriots +333
Los Angeles Rams +350
The first step to discovering the Super Bowl chances for each team is to convert the odds to a percentage. We can do this by using the implied probability formula.
Implied probability = 100 / (Odds + 100) x 100
After putting all the odds through the formula, you’ll get these percentages which will tell you how likely each team is to win the Super Bowl, according to Bovada.
- New Orleans Saints 36.36%
- Kansas City Chiefs 26.67%
- New England Patriots 23.09%
- Los Angeles Rams 22.22%
So that’s it, right? Wrong.
If you add up all the percentages we just converted, you’ll see that it adds up to 108.34%, which isn’t what we want since it’s mathematically impossible to have a chance of something happening over 100% for anything – sports or not.
The 8.34% extra is called vigorish or “juice,” which is the extra percentage baked into the odds and is essentially a fee all sportsbooks have in the odds for letting you bet on any given line.
So, if the odds are slightly skewed, then how can we find the “real” chance a sportsbook is giving a team?
To do this, simply divide each team’s implied probability into the total implied probability, and then multiply by 100.
Real Odds = team implied probability/total implied probability x 100
- New Orleans Saints 33.56%
- Kansas City Chiefs 24.62%
- New England Patriots 21.31%
- Los Angeles Rams 20.51%
And that’s all there is to it. These odds add up to 100%, which is exactly what we’re looking for when determining the four remaining teams’ Super Bowl chances, and this formula is good for any positive odds (anything with a plus sign in front of the number) you’ll find at a sportsbook.
Whether it’s for the Super Bowl or for odds on any other sport or event, the implied probability formula is a good tool to have and has a 100% chance of assisting you when trying to beat the odds.


You will become perfect in addition , subtraction, multiplication , division and comparison of Integers , fractions and decimals .
Finding H.C.F , LCM and prime factorization will become faster and accurate by applying different methods
Solve real life word problems based on fractions, H.C.F and LCM with full understanding.
Solve problems easily by applying Estimation and writing numbers in Scientific Notation
Percentage will become very easy for you and you will be able to solve all problems related to percentage easily.
Able to solve difficult problems of direct and inverse variations
Able to apply BOD-MAS rules on simplification problems without making mistakes.
By applying divisibility tests , you will be easily reduce fraction to lowest terms .
Locate Rational numbers on number line and write in decimal form using shortcut methods
Will be able to write rounded off answers accurately to 1/2/3 d.p. or to given significant digits
Able to solve problems on Ratio and proportion topic
You will be able to solve all kind of word problems on Profit , Loss, Discount and Tax.
In this Math Fundamentals course, you will be able to simply solve problems using various techniques. This course will equip you with key mathematical knowledge.
Learn how to solve real life math problems easily, compare integers and identify fractions by completing this course.
Upon completion, you will be able to confidently carry out math problems. You will find solutions to direct and inverse variations, ratios and much more. This course will teach you how to work out profit, loss and tax amounts with ease. Boost your confidence in solving key math problems today by completing this course in Math Fundamentals.
Math Fundamentals is a best selling course developed by industry experts and already helped tons of students like you. It is suitable for anyone who wants to improve their knowledge and skills in the or relevant sector.
This course is accredited by CPD,so you will get a career boost upon completing this course. Our Math Fundamentals is packed with 67 modules and 8 hours, 19 minutes of study. You will be awarded with a certificate of completion, the proof of your expertise in this field. If you want to get a job or looking for professional skills to excel in this field, a certificate from this course will help you appear as a strong candidate. You can also validate your certification from our website.
It doesn’t matter if you are willing to study full-time or part-time. This course is designed for any type of student and you can even complete it at your own pace. The materials are accessible from anyplace, any device and anytime.
Besides that, our experienced tutors will help you throughout the comprehensive syllabus of this course and answer all your queries through email.
After successfully completing the course, you will be able to get the UK and internationally accepted certificate to share your achievement with potential employers or include it in your CV. The PDF Certificate + Transcript is available at £6.99 (Special Offer - 50% OFF). In addition, you can get a hard copy of your certificate for £12 (Shipping cost inside the UK is free, and outside the UK is £9.99).
Integers ( Directed Numbers)
Integers and ordering of integers
Addition and Subtractions of Integers
Operations on Integers
Multiplication and Division of Integers
Powers of Integers (Exponents)
Factors and Multiples
Complete knowledge about factors and multiples
Divisibility tests for 2,3,4,5,6
Divisibility rules 7,8,9,10,11
Highest common factor HCF
Word problems based on HCF
Least common multiple LCM
Relation between lcm and HCF
Classification of Fractions
Convert mixed to improper and improper to mixed fractions
Comparing Fractions after converting to like fractions
Reducing a fraction to lowest term
Addition and subtraction OF fractions
Multiplication and division of fractions
Word problems on fractions
Simplification Rule : BODMAS
Bodmas and rules to remove brackets
simplify expressions using BODMAS rules
simplify expressions involving fractions using BODMAS
Decimal numbers in relation to fractions
Like and unlike Decimals
Ordering of Decimal numbers
Addition and subtraction of Decimal
Multiplication of Decimal
Division of Decimal
Representation of Rational Numbers on Number line
Determining which fraction is terminating or non-terminating
shortcut method of writing decimal expansion of fraction to decimal
Rounding to whole number
Rounding to required number of decimals (1 d.p. )
Rounding to significant digits
Practice question on three methods of approximation
Estimation and Scientific Notation or Standard form
Operation on numbers when they are in scientific notation
Percentage to fraction and fraction to percentage
Percentage of a quantity and conversation to decimal
Expressing one quantity as percentage of another quantity
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