Precalculus challenge: can we just cancel out the sine? | Summary and Q&A
This video explains how to solve equations involving sine and cosine identities using the double angle identity and the reference triangle method.
Questions & Answers
Q: How do you use the double angle identity to solve equations with sine?
To solve an equation like sine(2θ) = sine(θ), apply the double angle identity, which states that sine(2θ) = 2sin(θ)cos(θ). Rearrange the equation and consider different values of θ to find all possible solutions.
Q: How do you solve equations involving cosine and πθ?
Equations like cosine(πθ) = cosine(θ) require considering cases and adding odd multiples of π to the angles. By doing so, you can find all the solutions to the equation.
Q: Can you cancel out the sine in equations like sine(πθ) = sine(θ)?
No, you cannot cancel out the sine in equations with different inputs. The video explains that canceling out the sine is not valid because sine is not injective. Instead, you need to consider different cases and add multiples of π to the angles.
Q: Where can I learn more about solving equations with trigonometric identities?
The video suggests checking out Brilliant, a problem-solving website and app that offers courses in math, science, and computer science. They have courses that can help you deepen your understanding of trigonometric identities and other topics.
Summary & Key Takeaways
The video introduces two equations involving sine and cosine: sine(2θ) = sine(θ) and cosine(πθ) = cosine(θ).
The traditional way to solve the first equation is to use the double angle identity, which results in infinitely many solutions.
The second equation requires considering different cases and adding odd multiples of π to the angles.


IXL | Reflection, rotation, and translation | 5th grade mathPresentation is determined by the report that is given in front of the class.Print out these two worksheets to learn how to use a grid to draw and change the size of just about any picture.
In nonstandard analysis, let be a set of urelements, and let be the superstructure with individuals in.The guiding principles of the Mathematics syllabus direct that Mathematics as taught in Caribbean schools should be relevant to the existing and anticipated needs of Caribbean society, related to the abilities and interests of.Note: here we call it resizing, but other people call it dilation, contraction, compression, enlargement or even expansion.
4.1 Enlargements and Reductions - PBworksAn enlargement is a type of transformation in which lengths are multiplied whilst directions and angles are preserved.
Enlargement Worksheet - Lesson PlanetGeometry Worksheets and Printables Circles, spheres, squares, cubes, triangles, rectangles, octagons, polygons, parallelograms, quadrilaterals—oh, my.
An enlargement is a transformation which changes the size of an object without changing its shape, rather as a projector enlarges the details of a photographic slide onto a screen.The list of sources on the Math Awareness web site is a great place to start.Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets.
Exam Style Questions - Corbettmaths
It can be used as a first introduction to trigonometry from similarity in the middle school to teaching trigonometry in the high school.Enlargement, sometimes called scaling or dilation, is a kind of transformation that changes the size of an object.
Mathematics and Art -- So Many ConnectionsIndeed historical figures such as Newton and Gauss are difficult to classify as purely physicists or.Find scale factor enlargement lesson plans and teaching resources.O p MMza Udyeo ewsiWt7h9 EIJn Of0i zn giPt4eW dGWeXoJm ue 9terVyR.U Worksheet by Kuta Software LLC.
The Fibonacci sequence starts with 1, 1,. and each subseque.Some of the worksheets for this concept are Enlargement work, Work on years, Mathematics linear 1ma0 enlargement, Enlargement grid work, Enlargements and reductions work, Enlargement grid work, Fully describe the single transformation from the triangle, Scale.
Enlargement and Reductions | Naomi's BlogDefine enlargement and reduction. A. A dilation is a type of dilation where the value of the scale factor is greater than 0.
042-055 Ch04 097344 ML9 - THSS MathThe lesson takes students from enlargement via similar triangles to trigonometry.Similarity and Congruence Two objects are similar if they have the same shape, so that one is an enlargement of the other.
NAEP Grade 12 Mathematics Practice Questions
Two objects are congruent if they are the same shape and size.Indeed, the many interconnections between mathematics and art provide a wealth of material from which organizers of Math Awareness Month events can select.Enlargement (in fiction) is a theme in fiction, especially in science fiction and fantasy.
Similarity transformation | Define SimilarityWelcome to the geometry worksheets page at Math-Drills.com where we believe that there is nothing wrong with being square.
Extremity | Definition of Extremity by Merriam-Webster
Mathematics: To solve this task successfully, students must relate different scale factors as fractions, decimals, and percents to.


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Being from Physics, I have been trained to look into a lot of problems from a geometrical point of view. For example the differential geometry of manifolds in dynamical systems etc. When I read the foundations of computer science, I always try to find geometric interpretations. Like a plausible geometric interpretation of recursively enumerable sets (I worked on a part where I tried to connect them with Algebraic Geometry by exploiting equivalence with Diophantine Sets but the connection seemed forced and I could not find a "natural" expression of the facts in that formulation) or a beautiful geometric result for a simple algorithm for sorting numbers. Though I am not an expert I have read surveys on Geometric Complexity Theory and it is surely an interesting program but I am more interested in having a geometric view of extremely fundamental concepts like the dynamics of a Turing Machine, Lambda Calculus or the structure of (un)computable sets (rather than specific problems). Is it a hopeless job to find geometrical structure in these objects or can one expect some intricate results? Is there any formulation of TCS which treats it geometrically?
The semantics of computer programs can be understood geometrically in three distinct (and apparently incompatible) ways.
The oldest approach is via domain theory. The intuition behind domain theory arises from the asymmetry behind termination and nontermination.
When treating programs extensionally (ie, only looking at their I/O behavior, and not their internal structure), it is always possible to confirm in finite time that a program halts -- you just wait until it stops. However, it's not possible to confirm that a program doesn't halt, because no matter how long you wait, there is always a halting program that will run for a few more steps than you waited.
As a result, halting and looping can be viewed as forming a topological space (the Sierpiński space). This lifts to richer notions of observation (via the Scott topology), and you can thereby interpret programs as elements of topological spaces. These spaces are generally quite surprising from a traditional point of view -- domains are generally not Hausdorff.
The best topological introduction I know to these ideas is Steve Vickers' short and extremely accessible Topology via Logic. It can be understood as a sort of warm-up for Peter Johnstone's significantly more formidable Stone Spaces.
If you are looking for online lecture notes, let me suggest Martin Escardo's Synthetic Topology of Data Types and Classical Spaces.
Another view arises from concurrency theory. A concurrent program can be understood as having multiple valid executions (sequences of states), depending on how races are resolved. Then, the set of executions can be viewed as a space, with each possible sequence of states understood as a path through this space. Then, methods from algebraic topology and homotopy theory can be applied to derive invariants about the program execution.
Nir Shavit and Maurice Herlihy uses this idea to prove the impossibility of certain distributed algorithms, for which they won the 2004 Gödel prize. (See The Topological Structure of Asynchronous Computation.) Eric Goubault has a survey paper explaining the relevant ideas in Some Geometric Perspectives in Concurrency Theory.
Most recently, it has been observed that the structure of the identity type in dependent type theory corresponds very closely with the notion of homotopy type in homotopy theory -- so closely, in fact, that dependent type theory can actually be seen as a sort of "synthetic homotopty theory"! (Vladimir Voevodsky has joked that he spent several years developing a new calculus for homotopy theory, only to discover that his colleagues in the CS department were already teaching it to undergraduates.)
See cody's link above to the homotopy type theory book.
Interestingly, these three views seem incompatible with each other, or at least very difficult to reconcile. Dependent type theory is a total language, so nontermination (and the Scott topology) does not arise in it. It is also confluent, so the view of computations-as-spaces doesn't arise either. Similarly, formulating concurrency in terms of domain theory has proved ferociously difficult, and a completely satisfactory account is still an open problem.
As it just so happens, there have been recent development in the theory of dependent types, in which a types, which traditionally represent a static invariant for a computer program, can be interpreted to be a topological space, or rather an equivalence class of such spaces (a homotopy type).
This has been the subject of intense research over the last few years, which culminated in a book.
Older work has attempted to give a description of models of computation systems, like the pure $\lambda$-calculus, in terms of certain topological spaces called domains. The Wikipedia article gives a good overview.
You are aware of GCT, but you might not be aware of Mulmuley's earlier work on showing a separation between a subset of PRAM-computations and P, which uses geometric ideas of how a computation can be viewed as carving up a space.
Many lower bounds for problems in the algebraic decision tree model reduce to reasoning about the topology of underlying spaces of solutions (Betti numbers show up as a relevant parameter).
In one sense, ALL of optimization is geometric: linear programs involve finding the lowest point of a polytope in high dimensions, SDPs are linear functions over the space of semidefinite matrices, and so on. Geometry is used heavily in the design of algorithms here.
On that theme, there's a long and deep connection between our ability to optimize certain functions on graphs and our ability to embed metric spaces in certain normed spaces. This is a vast literature now.
Finally, in recent years there's been a great deal of interest in so-called "lift-and-project" mechanisms for solving optimization problems, and these make heavy use of the underlying geometry and lifts to higher dimensional spaces: notions from algebraic geometry play an important role here.
Computation is about information processing. The intrinsic nature of information and information processing naturally leads to topological notions (see Neel's answer about domain theory), but these are not directly of a geometric nature, as the resulting topological spaces tend not to be Hausdorff (or even $T_1$). They are "directed" in a sense, so one would have to come up with directed geometry to account for the phenomenon. And there are tricks to be played which symmetrize the situation (essentially you stand on your head).
One way to understand the relationship between information processing (also known as "computation") and geometry is that information processing is preceeds geometry. This view should be familiar from certain parts of physics. For instance in relativity theory we study both the causal structure of spacetime (its information processing) as well as its geometric structure. Many would consider the latter to be more basic than the former.
These connections have been noticed in the past and several years ago there was an effort to connect the information-theoretic aspects of computer science with relativity theory. One of the tasks people wanted to solve was: starting from the causality structure of spacetime (which is just a partial order on spacetime), reconstruct the topology of spacetime, or possibly the geometry as well. Recovering topology from a partial order is the sort of thing that domain theory is good at, so there was some success.
Computational Structures for Modelling Space, Time and Causality, Dagstuhl Seminar 06341, August 20–25, 2006
Key Martin and Prakash Panangaden: Spacetime geometry from causal structure and a measurement
Nielsen et al. showed that quantum computing has a geometrical interpretation. Specifically, they showed that finding a short quantum circuit to perform a target unitary $U$ is equivalent to finding a short geodesic in a particular curved geometry. See the following papers for details: http://arxiv.org/abs/quant-ph/0603161 and http://arxiv.org/abs/quant-ph/0701004
creatively interpreting your question, some possibilities other than GCT as you mention come to mind. one way is to look for undecidable problems (aka Turing completeness) which are quite ubiquitous.
Cellular automata which also increasingly are shown to have deep connections to physics, many related undecidable problems, proven TM complete, and they're naturally interpreted as (and converted between) TM computational tableaus.
Undecidability in dynamical systems (Hainry), again sometimes closely connected to physics. dynamical systems generally have a multidimensional geometric interpretation.
Visual programming languages. a program can be seen as a type of (directed?) graph with different types of vertices (eg conditional, arithmetic operation) etc.


Hi, could someone show me how I would go about doing this one?
Find all solutions of and plot them in the complex plane.
How many solutions should I be getting? I know that they should lie on some sort of circle... should I include this circle in my graph?
Any help would be greatly appreciated!


Areas of Polygons, Triangles, and Trapezoids
In this area worksheet, students find the area of 10 polygons including estimating the area. Students find the area of 6 triangles and trapezoids followed by 2 area word problems.
7th - 11th Math 14 Views 68 Downloads
Areas of Geometric Shapes with the Same Perimeter
A triangle, square, hexagon, and circle all have a perimeter of one unit. Eighth graders must attempt to find the area of each and arrange them in order from least to greatest areas. A terrific task for applying four different area...
7th - 9th Math CCSS: Designed
Finding Areas of Polygons, Variation 1
An exercise in finding the area of polygons by decomposing into triangles and composing into rectangles is the focus of this resource. Make several copies of the worksheet for each participant. Ask learners to use colored pencils to...
5th - 9th Math CCSS: Designed
Find the Area of Polygons by Decomposing into Triangles, Rectangles, Parallelograms, and Trapezoids
Take a multi-sided figure, such as a pentagon, and break it into smaller shapes to calculate the area. The video demonstrates this idea visually by breaking figures into small pieces and calculating each area. Also included, is a review...
6 mins 5th - 7th Math CCSS: Designed
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100 recipes that teach you how to perform various machine learning tasks in the real world
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Understand which algorithms to use in a given context with the help of this exciting recipe-based guide
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The Complete Idiot's Guide to Statistics, 2nd Edition
Statistics, statistics everywhere, but not a single word can we understand! Actually,
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Complex Variables and the Laplace Transform for Engineers
Widely acclaimed text on essential engineering mathematics. Theory of complex variables, Cauchy-Riemann equations, conformal mapping, multivalued functions, etc. Also Fourier and Laplace Transform theory, its applications to engineering, including integrals, linear integrodifferential equations, Z Transform, much more. Many excellent...
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Student Mathematical Library
2000; 380 pp; softcover
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Problems in Mathematical Analysis II: Continuity and Differentiation - W J Kaczor and M T Nowak
Problems in Mathematical Analysis III: Integration - W J Kaczor and M T Nowak
Moscow Mathematical Olympiads, 1993-1999 - Roman Fedorov, Alexei Belov, Alexander Kovaldzhi and Ivan Yashchenko
We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study.
Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often several consecutive exercises are concerned with different aspects of one mathematical problem or theorem. This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided.
The book covers three topics: real numbers, sequences, and series, and is divided into two parts: exercises and/or problems, and solutions. Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products.
Undergraduates, graduate students, and instructors interested in analysis.
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Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.
Recommended - GCSE & iGCSE
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Unformatted text preview: range=12-1=11 mean=6.71 stdev=sqrt(121-6.71^2)=sqrt(76)=8.71 8. Find the z-score of an exam of 76 if the mean of the entire class was 82 and the std. deviation was 4. z=(82-76)/4=3/2=1.5 9. What is the probability that a student in the class of problem #8 above scored less than 76? p=0.067 10. Tom scored a 56 on a quiz where the average was 44 and the std. deviation was 12; his brother Tim scored 63 in a different class where the average was 52 and the std. deviation was 10. Who scored relatively higher? zT=(56-44)/12=1 zT=(63-52)10=1.1 Tim scored relatively higher...
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This note was uploaded on 12/30/2010 for the course DSA 12 taught by Professor Jhonecaine during the Winter '10 term at Aberystwyth University.
- Winter '10


A continuous dependence result for a nonstandard system of phase field equations
- Colli, Pierluigi
- Gilardi, Gianni
- Krejčí, Pavel
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35K61 35A05 35B30
- nonstandard phase field system, nonlinear differential equations, uniqueness, continuous dependence
The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.
- Math. Methods Appl. Sci., 37 (2014) pp. 1318--1324.


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This paper addresses a unified mathematical expression describing a class of chaotic and hyper-chaotic systems, for which the problem of synchronization between different chaotic and hyper-chaotic systems with uncertain parameters is studied. Based on the Lyapunov stability theory, a novel, simple and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Two different chaotic systems and a hyper-chaotic system are employed as the examples to show the effectiveness of the proposed adaptive synchronization scheme. Additionally, the robustness of the proposed method and the effect of noise are investigated by measuring the mean squared error of the systems.


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A group consists of four men and five women. Three people are selected to attend a conference.
a. In how many ways can three peop le be selected from this group of nine?
b. In how many ways can three women be selected from the five women?
c. Find the probability that the selected group will consist of all women.


is a type of mathematics
that deals with methods of factoring numbers, by finding their factor
s using different methods of eliminating certain possibilites. Sieve theory is used primarily in cracking encryption
and finding large prime
Sieve methods were created to attack the well-known Goldbach and twin primes problems. It turns out that there are excellent reasons why sieve methods alone cannot solve these problems, but they give partial information on these and many other problems where the `deeper' methods of analytic number theory, such as exponential sums, will not work. For example pairs of consecutive odd numbers which are either prime or very hard to factorise do keep on occurring.
Sieve methods can be purely combinatorial like the sieve of Eratosthenes, or more complex. The (Multiple Polynomial) Quadratic Seive is the most famous example of recent Sieve Theory, and works, essentially, in three steps:
- Find a factor base and solve the congruences x 2º n (mod p) for each prime p in the factor base.
- Perform the sieving operation to find sufficient f(r)’s which can be completely factored over the factor base.
- Use Gaussian elimination to find a product of the f(r)’s which is a perfect square.
More complex methods, especially the number field sieve, are faster for larger numbers (more than 100 or so digits.)
Although the specifics of the method are beyond me, the abstract for the original paper introducing the sieve reads as follows:
The number field seive is an algorithm to factor integers of the form re ± s for small positive r and s... This leads to a general purpose factoring algorithm that is asymptotically substantially faster than the fastest factoring algorithms known so far, such as the multiple polynomial quadratic seive... The algorithm has proved quite practical: it took us only a few weeks to factor numbers that would have taken several years, had we used the multiple polynomial quadratic seive.


Definition of composite function
: a function whose values are found from two given functions by applying one function to an independent variable and then applying the second function to the result and whose domain consists of those values of the independent variable for which the result yielded by the first function lies in the domain of the second
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an act or instance of editing or removing
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‘Science and Nature’
Birthday parties are fun. And when you’re as old as the Methuselah Tree, you deserve to have at least one party! Read on to find out just how old this plant is, and branch out with some tree math!
Bedtime Math fan Valeria M. in Spain just asked us, how long would it take to run all the way around the world? Believe it or not, it’s been done! But how long would it take if you didn’t have to take any breaks? Read on to run through the math of world record runs.
Bedtime Math fan Sophia Y. asked us how many ants would it take to pick you up and carry you? Read on to see how the numbers answer this great, antsy question – and get carried away with the math!
What better way to fall asleep than to count sheep — and better yet, a whole parade of them? Every year nearly 1 million sheep are lead through Madrid in search of warmer weather. Even though this may cause a traffic baa-ck up, it must be interesting to see (and count!) so many sheep in the city. Read on to parade your way through the sheep-y math.
We got an awesome question from a Bedtime Math fan: Carmel P. wants to know, could you drink a whole lake in an hour? Read on to see what the numbers say – and drink in the math!
We are loving the geometry in the Anthem Veterans Memorial sculpture in Arizona. What makes this sculpture so mathematically special? Read on to find out – and see how math can meet nature to create memorable and honoring tributes.
Have you ever wondered how cranberries get from the farm and into your cup of juice? Unlike other fruits, like apples, that you pick straight from the tree, farmers have a unique – and wet! – way of picking cranberries. Read on to get the scoop on cropping cranberries, and do the “berry” wild math.
Dogs come in all kinds of shapes and sizes, from really big to really small! But just how small is the smallest dog? Read on to find out – and do the math behind petite pups.
We have jelly, and we have fish…and then we have jellyfish. It’s a great name for this slimy sea creature. Jellyfish can be smaller than your fingernail, or...
When it’s really clear at night and there aren’t a lot of lights around, you can look up at the sky and see lots and lots of stars. But you’d be amazed by just how many star are up there that your eyes can’t see! Read on to count up the stars and be star-struck by the math in the sky.


Calculate the area bounded by the circumscribed and inscribed circle in a triangle with sides 16 cm, 20 cm, and 15 cm.
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Square with sides 83 cm is circumscribed and inscribed with circles. Determine the radiuses of both circles.
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How was the Babylonian number system created?
The Babylonians writing and number system was done using a stylus which they dug into a clay tablet. This explains why the symbol for one was not just a single line, like most systems.
Why do the Babylonians use base 60?
“Supposedly, one group based their number system on 5 and the other on 12. When the two groups traded together, they evolved a system based on 60 so both could understand it.” That’s because five multiplied by 12 equals 60. The base 5 system likely originated from ancient peoples using the digits on one hand to count.
Who developed the Babylonian number system?
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.
What is the base 12 number system?
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. As a result, duodecimal has been described as the optimal number system.
Why do we count base 10?
Nature gave us ten fingers, and so it is natural for us to count in tens. Machines count bigger numbers in the same way we do: by counting how many times they run out of digits. This system is called binary and the binary number 10 means the machine ran out of digits one time. A human would call this number two.
Why is base 60 better than 10?
To be clear, base 60 has a big advantage over base 10: 60 is divisible by 3, and 10 isn’t. It’s easy to write the fractions 1/2, 1/4, and 1/5 in base 10: they’re 0.5, 0.25, and 0.2, respectively. But 1/3 is 0.3333…. Its decimal representation doesn’t terminate.
Why did Mesopotamians use base 60?
The numerical system based on 60 originated with the Sumerian civilisation, 4500 years ago. The theory is that the Sumerian and Babylonians were keen on the number 60 because 60 has many factors, including the first six numbers, one to six, but also 12, 15, 20, 30. This meant dividing by 60 often gives easy fractions.
What was the Babylonian number system used for?
The Babylonians developed a system for writing down numbers, using symbols for singles, tens, and hundreds, showing that they probably used a decimal system for everyday life. This system allowed them to handle large numbers comfortably and perform all of the major arithmetical functions.
What number system did the Babylonian’s use?
Babylonian Mathematics and the Base 60 System Base 10 or Base 60. The system surfaced circa 3100 BCE, according to The New York Times. History. Babylonian math has roots in the numeric system started by the Sumerians, a culture that began about 4000 BCE in Mesopotamia, or southern Iraq, according to USA Today Measuring Time.
What is the rule of the Babylonian number system?
The Babylonian system uses base-60, meaning that instead of being decimal, it’s sexagesimal. Both the Babylonian number system and ours rely on position to give value. The two systems do it differently, partly because their system lacked a zero.
What is the Babylonian numeral system?
The Babylonian system is credited as being the first known positional numeral system, in which the value of a particular digit depends both on the digit itself and its position within the number.


Generate a NaN...on purpose
Sum of series V
Matlab Basics - Pick out parts of a vector
Find scalar product of two polynomials a and b, given as vector array.
Pointwise multiplication of vectors.
Select primes from the matrix.
Convert array of decimal numbers into binary numbers array.
Divide polynomial p1 by p2.
For given vectors x,y find coresponding spline coefficients.
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Before reading on, you should start here: Dan Meyer’s summative post on mathematical modeling.
What CCSS says about Standard of Mathematical Practice 4 in the elementary grades:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation.
Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
What modeling can look like in a flow chart:
Elementary 3-Act Resources:
- Graham Fletcher (K-8)
- Robert Kaplinsky (3-12)
- Dane Ehlert’s (3-12)
- Dan Meyer (3-12)
- Kyle Pearce (3-12)
- Mike Wiernicki (3-9)
- 101 Questions (hosted by Dan Meyer)
Student Recording Sheets
Modeling posts worth reading:
I hope this section will continue to grow. Please drop me a line and let me know of a modeling post that needs to be added here. All of us are smarter than one of us.
Q: I have an idea for a 3-Act but don’t have a blog?
A: Don’t need one. 101questions is an excellent website for you to begin experimenting and sharing your ideas with others trying to get their model on.


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Classifying and Sorting Shapes
In this early childhood problem solving activity, students discover how to sort shapes with corners, sides and 4 corners and then respond to 3 classification questions.
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If They Ever Revive High Rollers, You’ll Be Ready With This Strategy Guide
What’s the optimal strategy for taking on the Big Numbers?
Game show fan and math aficionado Dave Mattingly has a fantastic in-depth read on the optimal strategy for playing the bonus round of the game show High Rollers.
For those that don’t know, High Rollers’ Big Numbers game has players trying to shut off lights labeled from 1 through 9. The player rolls the dice, and whatever they roll, they have to remove from the board. So, if they roll an 11, they can remove the 9 and the 2; or the 8, the 2 and the 1. Rolling a doubles gives the player insurance against a roll they can’t remove from the board, but a bad roll and no insurance means the game is over. It’s based on a popular, traditional board game called Shut the Box.
So how do you do it? Well, we’ll let Dave explain:
Since there are nine different numbers, and each number has two states (either still on the board or removed), that means that there are 512 (2 to the 9th power) possible combinations of numbers that you could be left with at some point during the game.
Um… shit. That’s a lot of math. You better read Dave’s whole article to work out exactly how to win (and play around with his interactive High Rollers widget).


Math Skills Training System
101 Games and Activities
101 Games and Activities is a book designed for teachers, caregivers and therapists of children with Autism, Asperger’s and Sensory Processing Disorders to teach them through interactive games how to make eye contact, stay focused, strengthen their motor skills, associate words with objects and improve language and numerical skills. Successfully engages the child. The author shows the user how to teach children by moving their bodies through play. These interactive games are quick to learn but w
2+2 is a math skills training system designed for use with children with low vision or cognitive disabilities. This program focuses on early arithmetic problems and uses an errorless learning method. The program features both drill and test modes and are designed to enable the instructor to enter vertical or horizontal problems for drilling and testing. This program can be accessed using the standard keyboard, IntelliKeys (not included; see separate entry), or a single switch (not included). COM
3-D Pentomino Puzzle
The 3-D Pentomino Puzzle is a math skills game designed for use by children with cognitive disabilities. This 12-piece three-dimensional puzzle forms hundred of combination, sharpening geometry and problem-solving skills. The set includes a guide with fifty puzzles in three levels of difficulty.
Base Ten Place Value Frame
The Base Ten Place Value Frame is designed for children ages five to seven with learning disabilities to help them learn early math concepts. A good, hands-on way to demonstrate early place value concepts. Model ones, tens, and a hundred for base ten activities. For use with the standard 1-centimeter base ten blocks (sold separately).
Circus Program, The, & The Circus Kit (Models Mw845 & Mw840)
The Circus Program and The Circus Kit, models MW845 & MW840, are language and math skills training systems for children with learning disabilities. Appropriate for group or one-on-one tutorials, the Circus Progam consists of a magnetized display classboard, 200 metallic-backed visuals that adhere to the board, a progam manual, an audio cassette featuring songs and activities, make-your-own plastic for creating customized visuals and Wytebord erasable marking pens. The Circus Kit includes jus
Colored Counting Cubes And Full-Color Task Cards
The Colored Counting Cubes and Full-Color Task Cards (sold separately) are a perceptual training activity designed for use with children with cognitive, perceptual, learning, and fine motor disabilities.The set of 100 hardwood cubes can be used to create three-dimensional designs to reinforce fine motor and visual discrimination skills and introduce basic math concepts. A set of 36 Full-Color Task Cards with designs and activities is available. DIMENSIONS: The cubes are one inch. COLOR: The bloc
Colorful Clips (Model Cc-13F)
Colorful Clips, model CC-13F, is a reading tutorial program and training system for matching/sorting/assembly designed to teach basic counting, pattern-following, and word recognition skills to people with cognitive disabilities. The activities require students to follow written and illustrated directions for matching clips to pattern cards or activity mats. The set includes 162 plastic clips, 2 activity mats, 72 pattern cards, marking pencil, storage box, and instructions. COLOR: The clips come
Colorful Concepts Manipulatives
The Colorful Concepts Series, model CC-20Q, is a set of educational toys, training systems for matching/sorting/assembly, math skills training systems, and vocabulary tutorial programs designed to develop fine motor, direction-following, cooperative learning, attention, vocabulary, and discrimination skills in children with cognitive disabilities. The series is comprised of Colorful Words, model CC-10F ($29), Colorful Numbers, model CC-12F ($29), Colorful Clips, model CC-11F ($29), and Colorful
Colorful Numbers (Model Cc-12F)
Colorful Numbers, model CC-12F, is an educational toy, a training system for matching/sorting/assembly, and a math skills training system designed to teach counting, number recognition, and addition to people with cognitive disabilities. Dice in three colors can be used for matching, counting, number recognition, and simple addition activities. The set includes 100 dice, 2 activity mats, marking pencil, storage box, and instructions. COLOR: The set includes red, green, and white dice. WARRANTY:
Differentiated Math Lessons
Differentiated Math Lessons is a software program and book for teachers for lesson planning in math. Features all lesson plans and student materials needed for teaching the these units: graphing, linear measurement, volume and capacity, weight, time, and money. The Instructor's Guide gives step-by-step lessons with measurable objectives, vocabulary lists, pretests and post-tests plus grade level maps. Student Materials include unity stories, Research It!, vocabulary cards, and tiered worksheets.
Exploring Patterns is a math skills training system designed for use by children with physical and cognitive disabilities. Using simulated manipulatives, the program enables students to copy, complete, and create hundreds of patterns of increasing complexity. Accessible with a mouse, a switch, or IntelliKeys (see separate entry), this program offers 140 carefully sequenced activities, a choice of three manipulatives for most activities, continuous auditory feedback, and a comprehensive Teacher's
Exploring Patterns Activity Book
The Exploring Patterns Activity Book is a math skills training system designed for use by children with physical and cognitive disabilities. The program enables students to create their own pattern rules and discover patterns within patterns. Accessible with a mouse, a switch, or IntelliKeys (see separate entry), this program offers thirty illustrations that can be colored using IntelliPaint and provide a real-life context for learning about patterns. Categories include Everyday Objects, Nature,
Giant Textured Beads With Pattern Matching Cards
Giant Textured Beads with Pattern Matching Cards help teachers and parents working with young students in preschool, kindergarten and early elementary grades who are blind and have visual impairments. These Pattern Matching Cards reinforce the tactile continuum from real object (most basic level) to raised-line graphics (most advance level). Students in higher grades will be challenged by the increasing difficulty posed by the “Platform” and “3-D View” matching cards.
How Many Cows on the Farm Counting Game
iTouchiLearn Numbers is a math tutorial program and a math skills training system designed for use by parents working with children who have cognitive, learning, physical or speech disabilities. This application (or app) allows a parent to use their Apple iPad, iPhone or iPod Touch to help a child understand math concepts and enable them to apply what they learn to new situations through a blend of curriculum based early learning with verbal directions and animated games. These games use multipl
Leap Into Math Activity Set
The Leap into Math Activity Set is a math skills training system designed for use by children with cognitive disabilities. The set assists with early counting, sorting and matching, and addition and subtraction skills. The set includes 48 counters in the shape of frogs and 12 double-sided activity cards. Information on four species of frogs and their habitats are also included. OPTIONS: An extra set of 72 frog counters is available. COLOR: The counters come in six colors.
Learn More About Math
Learn More About Math is a math tutorial program designed for use by children with learning disabilities or children with upper extremity disabilities and fine motor disabilities who use switch input. The program has six activities that reinforce essential early math skills and concepts. In these activities, students count to nine; count animated creatures; complete a sequence of numbers; play a memory game using numerals; practice analog and digital time telling; and practice addition and subtr
Learning Resources Bake & Learn
Learning Resources Bake & Learn is a playset designed for children ages three and up with learning disabilities to enhance fine motor skills, learn letters, shapes and numbers while pretending to bake brownies, cookies or a cake.
Learning Resources Counting Cows Toy Set, 20 Pieces
Learning Resources Counting Cows Toy Set, 20 Pieces is a set of toys designed for children ages two and up with learning disabilities to learn to count and to learn number identification and more. Color coding on the cows helps children find the correct solutions.
Link 'n' Learn Count & Measure
The Link 'N' Learn Count & Measure is a math skills training system designed for use with children with cognitive disabilities. The set includes 78 links, 22 hands-on activities, and two links games. The games and activities use playful animals to help children build counting, sorting, and measuring skills. COLOR: The links come in assorted colors.
Link 'n' Learn Math Activity Card Set
The Link 'N' Learn Math Activity Card Set is a math skills training system designed for use by children with cognitive disabilities. The set assists with counting, sorting, patterning, and graphing activities. The set includes 102 links and 16 double-sided activity cards. OPTIONS: An extra set of 102 links is available. COLOR: The links come in six colors.
Making Sense With Numbers
Making Sense with Numbers is a math tutorial program designed for use by children with learning disabilities and children with upper extremity disabilities or fine motor disabilities who use switch input. The program has nine activities related to early number work: Number recognition (1-9), counting, dot-to-dot, dominoes, memory activities, and painting by numbers. Activities are accompanied by graphics and animations, speech, and sound. Teachers and parents can configure the program for level
Math Workshop is a math tutorial program designed to devleop and practive critical math skills for children with a developmental or physical disability. By combining games, music and animated characters, children have fun as they develop problem solving, strategic thinking and computation skills. Each activity has multiple levels of difficulty designed to build both confidence and motivation. There are seven activities including Bowling for Numbers, Puzzle Patterns, Sticklers and Rhythm Generato
Math...To Go! is an auditory output, electronic educational toy designed to help children develop skills in equation solving, math tables, and critical-thinking problems while practicing addition, subtraction, multiplication, and division. The toys memory has more than 50,000 math problems on three challenge levels. Correct answers are rewarded with musical tunes and graphics on the 7-character LCD display. The toy also has a calculator built-in so the device can be used to complete school lesso
Mathematics Made Easy For Children With Visual Impairment
Mathematics Made Easy for Children with Visual Impairment, by Dr. M.N.G. Mani, Aree Plernchaivanich, G.R. Ramesh, and Larry Campbell, is a manual on math training for blind students. This book helps children who are blind or have low vision to master secondary level mathematics by providing their teachers and parents with a resource for improving instructional strategies. The material presented in this publication is divided into five modules. Module 1 deals with the commonly used methods of tea
Mathosaurus is a math tutorial program designed to build early math concepts and computation skills. The four disk series includes 16 activities where animated dinosaurs entertain children as they learn. Students match sets to corresponding number symbols 1 to 10, create matching sets and compare sets to determine which is more and which is less. They learn the meaning of numbers to 20, ordinal numbers first through tenth and the concepts of greater than and less than. They practice counting to
Mini Motors Activity Card Set
The Mini Motors Math Activity Card Set is a math skills training system designed for use by children with cognitive disabilities. The set assists with early counting, sorting, and matching skills. The set includes 48 mini motor counters in the shapes of various vehicles, including cars, planes, boats, trucks, etc. and 12 double-sided activity cards. OPTIONS: An extra set of 72 mini motors counters is available. COLOR: The counters come in four colors.
Number Navigator is a math skills training program designed for use by individuals with upper extremity and fine motor disabilities. This mathematical software enables children to record their arithmetic and algebra exercises and homework via the button bar, keyboard, or switches without having to resort to the complexities of a spreadsheet. Numbers are entered into grid locations on the screen, using the keyboard or switches with SAW (Switch Access for Windows) Selection Sets. The pupil then na
NumberPack is a math skills training program designed for children with upper extremity and fine motor disabilities. This software program contains three simulated pieces of classroom equipment: a teaching clock, an abacus, and a number line. These simulations are keyboard-operated stand-alone units with their own printing functions. The clock is a re-creation of a classroom teaching clock with a number of additional features. The clock can be moved forwards or backwards in a number of adjustabl
Peg-It Facts Board
The Peg-It Facts Board is a math skills training system designed for use with individuals with learning and other cognitive disabilities. The set offers students visual and tactile reinforcement of addition and subtraction concepts using numbers up to 10. The set includes Tactilmat pegboards printed with addition and subtraction problems, Easy-Grip pegs, number and symbol tiles, and a 36-page reproducible workbook which introduces addition and subtraction concepts and provides reviews. Students
Real Life Math Collection
Real Life Math Collection books teach simple addition, subtraction and multiplication through the use of real-life situations, while making instruction fun and effective. Students refer to colorful price lists to pay for food at a restaurant, market, or department store. From pizza to sunglasses, the easy-to-read price lists generate hundreds of number and word problems, sure to improve any student’s math skills.
School Smart Pattern Block Set
The School Smart Pattern Block Set is designed for children with Autism Spectrum Disorder to teach them geometry, symmetry, fractions and other math concepts. Each set comes in six shapes and colors. 250 pieces. Good for creating pictures of everyday objects. Six shapes and colors.
Shapes And Numbers Thematic Activity Module (Models 2842-7 & 2834-5)
The Shapes and Numbers Thematic Activity Module is a math skills and language training system using a circus theme to develop early skills for students at the Kindergarten and first grade levels, beginning with whole numbers and progressing through the characteristics of four basic shapes. Circus performers are used to teach counting, identification and comparison. The system utilizes a magnetized display board and metallic-backed visuals to illustrate concepts and encourage interactive learning
Super Speak & Math
Super Speak & Math is a math skills game and training system designed to develop an understanding of important basic math concepts. This electronic, educational toy has a memory with more than 250,000 problems on three difficulty levels. This portable unit has a built-in handle, 3-line LCD display that helps children visualize the "Base 10" picture method, and calculator-like keyboard with color-coded activity keys. Auditory output includes music and video arcade sound effects and speech syn
Supermarket Program, The & The Supermarket Kit (Models Mw835A, Mw835B, Mw820A & Mw820B)
The Supermarket Program and The Supermarket Kit, models MW835A, MW835B, MW820A & MW820B, is a math, language and independent living skills tutorial for children with learning disabilities. The Supermarket Program includes a magnetized diplay classboard, 200 visuals that adhere to the board, a program manual, math lesson cards and Wytebord marking pens. The Supermarket Kit includes just the visual overlays and either a regular education or English as a Second Language (ESL) program manual. By
Talking Math is a language tutorial program designed for use with children and adults with cognitive and communication disabilities. Contained in a sturdy storage box, this set of 48 large cards is designed to teach the language and concepts necessary for math success, including sorting, sequencing, position, number, shape, and size. Teaching points and follow-up activities are included on the back of each full-color card. DIMENSIONS: 8 x 5.75 inches.
Tanagramables Book And Tanagrams Set
The Tanagramables Book and Tanagrams Set is a perceptual training activity designed for use by children with cognitive, perceptual, sensory, and fine motor disabilities. This set includes a seven-piece plastic Tanagram set for creating designs and patterns and a 96-page book with 75 reproducible activity sheets organized by level of difficulty.
Three Bear Family Count & Match
The Three Bear Family Count & Match is a training system for matching / sorting / assembly designed for use by children with cognitive disabilities. The set includes 16 plastic circus bears for counting, sorting and matching; 27 hands-on activities, and two games.
Virtual Pencil is math training software designed for use by students who have difficulty using a pencil, including students who are blind or have low vision, upper extremity disabilities, or learning disabilities. This software can be used by students to solve mathematics problems without the use of a pencil, including addition, subtraction, multiplication, and division. When the software is opened, a split screen appears, looking like a window with two panes. The left pane shows a tree directo
Wild Animal Sort & Compare
The Wild Animal Sort & Compare is a math skills training system designed for use with children with cognitive disabilities. The set includes 16 rubber animals, 17 hands-on activities, and two games. The games and activities help children build counting, sorting, and matching skills. COLOR: The animals come in assorted colors.


It is quite simple math that is at work. The first cases were seen in early March. If you assume that each victim spreads it to just two more people in about a week, then cases would triple each week (including the original victim each week) . That would work out roughly to the 2,500 now suspected cases in Mexico. Tripling each week would mean about a million and a half cases in three months, rising to 6 billion or so in another eight weeks, i.e. the world's population, roughly. This is less time that it will take to manufacture a vaccine. Even with a widely used vaccine, ordinary flu causes from a quarter to a half million deaths world wide each year. The fatalities from this flu would likely be higher just because of the lack of vaccine, even if it turns out to be no more deadly than normal flus. But the apparently excess deaths in Mexico indicate it might be even more dangerous than that.


Many sparse cuts via higher eigenvalues
Joint with Anand Louis, Santosh Vempala and Prasad Raghavendra (moving to Berkeley).
Topic: relating higher eigenvalues of the Laplacian to isopermetric constants.
1. Cheeger inequality
Laplacian is where is the adjacency matrix of the graph and the diagonal degree matrix. The first eigenvalue is , let be the second one.
Cheeger’s inequality deals with Riemannian manifolds. Here is its translation for graphs. Given a subset of vertices, let denote the set of edges between and its complement . Set
There is a weighted variant.
Theorem 1 (Cheeger’s inequality, due to Alon, Alon-Milman, Jerrum-Sinclair) Let be a -regular graph.
Proof: Use min-max characterization of eigenvalues. For reverse inequality, insert Cauchy-Schwarz.
2. Higher eigenvalues
Motivated by Small Set Expansion, a problem introduced by Raghavendra and Steurer in connection with the Unique Games Conjecture.
Unclear a priori what will replace edge expansion.
Theorem 2 Easy: For every disjoint subsets of vertices with, one of them has edge expansion .
Difficult: There exist disjoint subsets with size which all have edge expansion .
Open: Can one replace with exactly .
Tightness: term cannot be avoided (look at noised cube).
Proof is cumbersome but algorithm is nice. James Lee, S. Oveis Gharan and L. Trevisan have an alternate treatment (STOC’12).
Case . Let be an eigenvector. Sort vertices according to increasing . Stop when …
General case. Let be orthonormal eigenvectors. This gives a map to . Pick random Gaussian vectors in . For each vertex ,
– Project along Gaussian directions .
– Pick which maximizes .
– Move (project down) the vector to direction .
Run Cheeger rounding along each direction , output the cuts of expansion .
2.2. Questions from the audience
If one knows that eigenvalues have high multiplicity, can improve on ? Answer: I don’t know.


“Learning should be an unending process,” said Don Bushaw in an interview some years back. Anyone who knew him at all will know this was no idle observation. Bushaw, who first arrived at Washington State College in 1943 as a 17-year-old freshman and returned, a doctorate in mathematics from Princeton in hand, to teach and lead for a distinguished 43 years, passed away in Portland, Oregon, on January 15, 2012, surrounded by his wife and children.
Don Bushaw was born in Anacortes, Washington, on May 5, 1926, to Elmond and Ruth Bushaw. The family moved to Bremerton in 1930 when Elmond took a job at Puget Sound Naval Shipyard.
Bushaw graduated from Bremerton High School in 1943 and entered Washington State College. After serving for two years in the U.S. Navy, he returned to WSC, where he met and then married Sylvia Lybecker ’47 in 1946. Don graduated from WSC with a degree in mathematics in 1949, and went on to graduate school at Princeton.
At Princeton, he studied under Solomon Lefschetz, who during the early Cold War believed that the United States lagged behind in aspects of differential equations most relevant to missile technology. Bushaw’s resulting dissertation opened up the new mathematical discipline of control theory, which has many ramifications for national defense.
Bushaw, with his family, returned to Pullman in 1952 as a mathematics instructor, climbing the academic ranks to full professor in 1962. During his 43-year tenure at WSU, Don served as mathematics professor, mathematics department chairman, acting director of libraries (twice), and vice provost for instruction. He taught 62 different courses in mathematics. He also taught a University Honors class on Eastern civilization.
His research and writing ranged widely, encompassing control theory, differential equations, topology, and mathematical economics, as well as mathematics education, the history of mathematics, and mathematical biographies.
Among many other awards, Bushaw delivered the Faculty Invited Address (1968) and was the first recipient of the WSU Faculty Excellence Award for Instruction in 1983.
Apart from the time Don spent on family and professional activities, he had a passionate lifelong interest in traveling and learning languages. He had good friends all around the world. He also translated books and research articles in a variety of languages, including Russian and Chinese.
In a 1993 Daily News article celebrating Bushaw’s 50 years at WSU, Cal Long, who joined WSU’s math department in 1956, four years after Bushaw, said of his colleague, “There is simply nobody I’ve known who is as bright and knowledgeable across the board. I don’t know anybody on our faculty who’s more respected … But he never flaunted it … If I were funding a university, I’d want to staff it with Don Bushaws.”
When asked to introduce an article on great teaching for the University’s 1992 financial report, Bushaw submitted the following:
“Teaching itself can be described as making it easier for others to learn, and it can be done in many ways. Great teachers teach greatly. Active student involvement is often the key to good learning, and the great teacher may be precisely the teacher who can provide strong motivation, guidance, and intellectual resources for that involvement. In doing this, the great teachers—like all good teachers—draw on such qualities as knowledge, organizational skills, enthusiasm, and a sense of humor. And they care about the learners.”
In a handwritten note attached to his submission, Bushaw wrote: “Not very flamboyant, but (I think) honest.”
Don is survived by his wife of 65 years, Sylvia, and his sister, Shirley Hanson of Gig Harbor. He leaves daughters Amy Bushaw and Margaret Parker of Portland, sons Bruce ’74 of Richland, Gordon of Silverdale, and Tom of Kennewick. He also leaves three grandsons, Brice Bushaw, Neal Bushaw, and Robert Parker, and his great-granddaughter Natalya Parker.
Remembrances can be made to the Donald W. and Sylvia R. Bushaw Scholarship in the Glenn Terrell Scholarship Program at Washington State University.


Pure Mathematics 2: v. 2 Paperback – 1 Aug 1979
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this is the book 'further pure maths' by the same authors is based upon.
from the preface =
'pure mathematics 2 is a continuation of the work covered in volume 1 and is intended to complete a full two year course in pure mathematics. it caters for the pure mathematics content of such advanced level schemes as the university of london syllabus in further mathematics.'


the number of cells has increased by 7.2%.Let t be the number of hours since the start of the study. Let y be the of the bacteria cells.Write exponential function showing the relationship between y and t .
y = a(1+r)^x
initial amount before
r = growth rate
x = number of
time intervals that have passed
y = 2500(1+.072)^t
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An infinite sheet of surface current density is given by Js1 = 10y^ A/m (y^ is y hat) exists on the plane defined by x =1. A second infinite sheet of surface
current density given by Js2 = 14z^ (z hat) A/m is on the plane defined by x= -2. What is the magnetic field intensity H(vector) for any point where -2<x<1


No matter who you are or what you believe to be true based on your current knowledge, you live in a universe not of absolutes, but of possibilities. Even mathematics and geometry are not absolute, in fact, far from it. But this isn’t my idea, or even a new idea.
Using our knowledge of triangles, a basic and simple shape, here is one example that can be easily understood once you’ve heard it. What most of us believe regarding triangles is what we have been taught in school, and that is that no matter what shape a triangle is, and how long and short its sides, and what angle each of its interior corners make—the sum of its three angles will always be 180 degrees. Wow, that’s cool, but wrong. Now, imagine this, it’s mind opening when you do. Let’s say you were marking off land to be used to create a park or nature reserve, and that the area was to be in the shape of a triangle. Plot out the outline of its perimeter on the ground. To make it simple, use an Equilateral triangle which has three equal sides and three equal angles. Make each side 100 yards long. Now, measure the three angles. They will be 60 degrees each, making a total of 180 degrees that make up the triangle. If you drew an Isosceles or Scalene triangle, the result would be the same, a sum of 180 degrees for the angles of your triangle.
Okay, now let’s say you decided that your triangle would be even bigger—here’s where it gets interesting—say you plot out a perimeter of triangle sides that are 1000 miles long. Or just imagine a triangle of straight lines running from Los Angeles to New York City and then to Miami and back to Los Angeles. Now, measure the three angles. Magically, they are greater than a sum total of 180 degrees. But how can this be possible? This is not what you’ve been told; you’ve been told that math and geometry are absolutes that followed laws and can guarantee certainty.
So, do you want to know why the larger a triangle is, the greater the sum of its angles? The answer is intrinsic to the fact that nothing and no thing is absolute. The answer is that the angles of a triangle were never absolutely 180 degrees—they vary—because life, the world, the universe is not two-dimensional. As you draw out a large triangle on the ground, the ground bends past the horizon, wrapping around the bend of our globe, stretching and warping the angles of how a triangle is shaped and how it comes together…and everything that we thought we knew. It’s not noticeable in smaller triangles—unless you are able to take measurements on the quantum level—and hardly noticeable in triangles even as large as the sides of an Egyptian pyramid. Yet, in actual fact there are no absolutes, only fields of possibilities when you open your scope of awareness. What appears, in our limited understanding, to be a straight line from Los Angeles to New York City, is not a straight line at all. And, what appears to be a straight line from New York City to Miami is not a straight line. Of course, what appears to be a straight line from Miami back to Los Angeles is, also, not a straight line. All three are bent line, with a sum total of angles of varying degrees. And it doesn’t stop on Earth. While this phenomena might seem as though it is associated with the fact that the Earth is round, in actuality it has to do with a greater reality: the curvature and possibilities of everything.
Let’s take a look at gravity as an absolute. A fairly certain scientific fact, don’t you think? Not so fast. In the 4th century BC, the Greek philosopher Aristotle enlightened us to cause and effect: there is no effect or motion without a cause. In his system, heavy bodies are not attracted to the Earth by an external force of gravity, but tend toward the center of the universe because of an inner gravitas or heaviness. Smart guy, and that theory held for a long time. During the 17th century, Galileo found that, counter to Aristotle’s teachings, all objects accelerated equally when falling. But then came along Sir Isaac Newton, the mathematician and physicist whom we hail as having discovered gravity in 1687 BC. Newton’s law of universal gravitation states that every mass attracts every other mass in the universe, and the gravitational force between two bodies is proportional to the product of their masses, and inversely proportional to the square of the distance between them. But Newton only had a piece of the puzzle. His theory was not accurate, and yet to this day most of us remember that image of an apple falling on his head and so we believe that Newton cracked the mystery of gravity. (We’re almost at the point, and it’s worth the walk down history’s memory lane.) Next, the metaphorical baton of gravity was passed to scientist and mathematician Carl Friedrich Gauss in the early 19th century, who passed it to his pupil Bernhard Riemann. Together their insights led to the idea that space is bent.
Finally, Einstein came up with his theory of general relativity between 1905 and 1915. It described what we perceive as the force of gravity as in fact a geometric property arising from the curvature of space and time, or spacetime, and it’s the theory that we are currently going with. And still, most people don’t understand what it is.
To break it down: gravity works because the space around the Earth responds to the presence of the Earth. In the presence of matter and energy, this geometry can evolve, stretch and warp. Otherwise said, the added possibility called Earth actually alters space. Imagine, little old us in this big universe, altering space and time? Today, many scientists describe the universe as moving from order to disorder, however, the magic of viral energy says that it’s moving from possibilities to more possibilities.
Space is malleable, everything bends and adjusts to variables. You are one of those variables and you can cause reality to change. Remember what quantum physicist Dr. Fred Alan Wolf told me about the bar magnet and iron filings? “The mind is a process, and it’s related to what is happening at the level of what we call the quantum field of reality. At this unobservable invisible level of reality we are dealing with a field of possibilities, not a gathering of physical objects. Changing the field of possibilities changes the pattern of behavior.” You see, even what was fact evolves as we grasp the next discovery. This begs the question, what’s next?
Whatever we create is next, more questions to be answered, then answered again and further, and of course—more possibilities.
There is a question, a loose end so to speak, that we have answered. It’s the question posed in the 1800s by William Herschel and his brain pack of stargazers: Is our galaxy everything? The groundwork was set by the discovery in 1893 of ‘variable stars’ made by Henrietta Swan Leavitt, a female American astronomer at Harvard. Variable stars provided a key for how to measure distance in far space, and to determine what was inside our galaxy, and what was beyond it. This key was then used in 1923, by American astronomer Edwin Hubble when he tackled the question using the 100-inch Hooker Telescope on Mount Wilson near Los Angeles, California, and calculated that the Andromeda nebula was really the Andromeda galaxy. We now know—thanks to the open minds of many people over many hundreds of years—that yes, indeed, there are hundreds of billions of other galaxies besides our own Milky Way galaxy. At long last, we can observe that we are not the sole galaxy in existence.
Everything is out there. All possibilities are present. Everything does exist. If you can imagine anything at all, it exists somewhere in space and time. So dream big, the biggest that you can. Audrey Hepburn said, “Nothing is impossible, the word itself says—‘I’m possible’!” You can discover your idea, or create it. Say it is so, and for you, it will be.
THIS ARTICLE IS AN EXCERPT FROM THE BOOK, THE MAGIC OF VIRAL ENERGY: SPIRITUALITY MEETS QUANTUM THEORY, WRITTEN BY PENELOPE JEAN HAYES. PROPRIETARY; ALL RIGHTS RESERVED.
Categories: Viral Energy Sparks


This is in part a preparation for my 25-minutes talk in a workshop here at Princeton next week. (Never given a short talk before…I’m super nervous about this >.<) In this little survey post I wish to list some background and historical results which might appear in the talk.
Let me post the (tentative) abstract first:
Title: Volume preserving extensions and ergodicity of Anosov diffeomorphisms
Abstract: Given a self-diffeomorphism of a compact subset in , from Whitney’s extension theorem we know exactly when does it extend to . How about volume preserving extensions?
It is a classical result that any volume preserving Anosov di ffeomorphism of regularity is ergodic. The question is open for . In 1975 Rufus Bowen constructed an (non-volume-preserving) Anosov map on the 2-torus with an invariant positive measured Cantor set. Various attempts have been made to make the construction volume preserving.
By studying the above extension problem we conclude, in particular the Bowen-type mapping on positive measured Cantor sets can never be volume preservingly extended to the torus. This is joint work with Charles Pugh and Amie Wilkinson.
A diffeomorphism is said to be Anosov if there is a splitting of the tangent space that’s invariant under , vectors in are uniformly expanding and vectors in are uniformly contracting.
In his thesis, Anosov gave an argument that proves:
Theorem: (Anosov ’67) Any volume preserving Anosov diffeomorphism on compact manifolds with regularity or higher on is ergodic.
This result is later generalized to Anosov diffeo with regularity . i.e. with an -holder condition on the derivative.
It is a curious open question whether this is true for maps that’s strictly .
The methods for proving ergodicity for maps with higher regularity, which relies on the stable and unstable foliation being absolutely continuous, certainly does not carry through to the case:
In 1975, Rufus Bowen gave the first example of an Anosov map that’s only , with non-absolutely continuous stable and unstable foliations. In fact his example is a modification of the classical Smale’s horseshoe on the two-torus, non-volume-preserving but has an invariant Cantor set of positive Lebesgue measure.
A simple observation is that the Bowen map is in fact volume preserving on the Cantor set. Ever since then, it’s been of interest to extend Bowen’s example to the complement of the Cantor set in order to obtain an volume preserving Anosov diffeo that’s not ergodic.
In 1980, Robinson and Young extended the Bowen example to a Anosov diffeomorphism that preserves a measure that’s absolutely continuous with respect to the Lebesgue measure.
In a recent paper, Artur Avila showed:
Theorem: (Avila ’10) volume preserving diffeomorphisms are dense in volume preserving diffeomorphisms.
Together with other fact about Anosov diffeomorphisms, this implies the generic volume preserving diffeomorphism is ergodic. Making the question of whether such example exists even more curious.
In light of this problem, we study the much more elementary question:
Question: Given a compact set and a self-map , when can the map be extended to an area-preserving diffeomorphism ?
Of course, a necessary condition for such extension to exist is that extends to a diffeomorphism (perhaps not volume preserving) and that has determent on . Whitney’s extension theorem gives a necessary and sufficient criteria for this.
Hence the unknown part of our question is just:
Question: Given , s.t. for all . When is there a with ?
There are trivial restrictions on i.e. if separates and switches complementary components with different volume, then can never have volume preserving extension.
A positive result along the line would be the following slight modification of Moser’s theorem:
Theorem: Any diffeomorphism on can be extended to a area-preserving diffeomorphism on the unit disc .
For more details see this pervious post.
Applying methods of generating functions and Whitney’s extension theorem, as in this paper, in fact we can get rid of the loss of one derivative. i.e.
Theorem: (Bonatti, Crovisier, Wilkinson ’08) Any diffeo on the circle can be extended to a volume-preserving diffeo on the disc.
With the above theorem, shall we expect the condition of switching complementary components of same volume to be also sufficient?
No. As seen in the pervious post, restricting to the case that only permute complementary components with the same volume is not enough. In the example, does not separate the plane, can be extended, the extension preserves volume on , and yet it’s impossible to find an extension preserving the volume on the complement of .
The problem here is that there are ‘almost enclosed regions’ with different volume that are being switched. One might hope this is true at least for Cantor sets (such as in the Bowen case), however this is still not the case.
Theorem: For any positively measured product Cantor set , the Horseshoe map does not extend to a Holder continuous map preserving area on the torus.
Hence in particular we get that no volume preserving extension of the Bowen map can be possible. (not even Holder continuous)


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Reinforce or review concepts taught using the animated lessons. Construct your own mathematical understanding, verify Mathematical concepts and connectthinking and Mathematics with the help of our video lessons.
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Many spectral unmixing approaches ranging from geometry, algebra to statistics have been proposed, in which nonnegative matrix factorization (NMF)-based ones form an important family. The original NMF-based unmixing algorithm loses the spectral and spatial information between mixed pixels when stacking the spectral responses of the pixels into an observed matrix. Therefore, various constrained NMF methods are developed to impose spectral structure, spatial structure, and spectral-spatial joint structure into NMF to enforce the estimated endmembers and abundances preserve these structures. Compared with matrix format, the third-order tensor is more natural to represent a hyperspectral data cube as a whole, by which the intrinsic structure of hyperspectral imagery can be losslessly retained. Extended from NMF-based methods, a matrix-vector nonnegative tensor factorization (NTF) model is proposed in this paper for spectral unmixing. Different from widely used tensor factorization models, such as canonical polyadic decomposition CPD) and Tucker decomposition, the proposed method is derived from block term decomposition, which is a combination of CPD and Tucker decomposition. This leads to a more flexible frame to model various application-dependent problems. The matrix-vector NTF decomposes a third-order tensor into the sum of several component tensors, with each component tensor being the outer product of a vector (endmember) and a matrix (corresponding abundances). From a formal perspective, this tensor decomposition is consistent with linear spectral mixture model. From an informative perspective, the structures within spatial domain, within spectral domain, and cross spectral-spatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF-based unmixing methods. Index Terms-Hyperspectral imagery (HSI), spectral unmixing, spectral-spatial structure, tensor factorization.
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