While “Software to discover equations and mathematical relationships in data” isn’t at the top of our christmas wish list, we have to admit that Eureqa is pretty cool. Developed at Cornell University, Eureqa uses machine learning algorithms to determine the underlying math behind data sets. It derived Newton’s second law of motion in a few hours on a standard desktop computer, which isn’t bad at all for a cold unfeeling robot mind. There probably aren’t many applications for this in most hacks, but what hacker wouldn’t want Sir Issac Newton’s brain in their toolbox? The software can be downloaded for free from their website.

[Via Wired]

Assignment due at the beginning of Night 4

1. Page 383  #  1- 12,  25-30, 33-48
2. Page 142  #  19-43 odd
3. Page  148  # 5 – 39 odd
4. Page 158  # 5 – 7,  21 – 27,  37 – 47, 59 – 87 odd
5. Page 167  # 11 – 21 odd and 33
6. Help one another in understanding the difficult concepts from this week’s homework.

Introduction

We can represent functions in three ways*: graphs, tables (or list) and equations.  It is often better to view simultaneously the representations of a function to be able to understand its behavior, but it is quite hard – or expensive – to find a software with such functionality.  However, with some knowledge on formulas in MS Excel, it can actually be done.

In this tutorial, we are going to represent functions of the form y = mx + b by letting the user enter and change the values of m and b, then generate the equation, the table, and the graph at the same time. Although we are going to use Microsoft Excel 2007 in this tutorial, but this can be done using lower versions. The expected output of this tutorial can be downloaded here.

The prerequisite of this tutorial is basic knowledge of Microsoft Excel, but I designed this tutorial to be stand-alone, so inexperienced Excel users could follow it step by step.

Preliminaries

The first thing that we are going to do is to place the letters m in A2 and b in A3.  A2 and A3 are examples of cell addresses.

As you can observe, a worksheet is composed of columns labeled with letters and rows labeled with numbers.  The intersection of a column and a row is called a cell and its location is determined by its cell address which is its corresponding column letter and row number.

Setting up for user input boxes

1. To place the instruction, click cell A1, then type Type m and b in the colored boxes: Note that when you click a cell, the cell pointer appears. The cell pointer is the rectangular box where your text appears when you type in the worksheet.
2. Type m in cell A2 and type b in cell A3.
3. Place the cell pointer in B2, and choose the color you want from the theme color box(see Figure below). Using the Theme color box, change also the color of B3. You may want to change the font of the text you have written should you want.

Figure 1 - Theme Colors dialog box

Generating the Equation

We want the equation y = mx + b to appear when the user put the value of m in B2 and the value of b in B3. To do this, we will use the concatenate command. The concatenate command joins series of strings (characters, words or numbers). In our case, we will join the string  “y =”, the content of B2 (which is our m), the string “x +” and the content of B3.  We will put our equation in B7 so that we can put our graph on the right side of our worksheet. The logic is that we want the equation to appear like y = content of B2 x + content of B3, where the content of B2 is the number typed by the user in cell B2 and the content of B3 is the number typed by the user in B3. Notice that the different colors indicate the string groups.

1. Type 3 in B2 and 5 in B3. This will enable us to see later if our concatenate command will work.
2. Type Equation in A6.
3. Place the cell pointer in A7. Type =CONCATENATE(“y =”, B2,”x +  “, B3). Then press the ENTER key on your keyboard.

Notes on the concatenate command:

1. Words concatenated are separated by commas.
2. Texts are enclosed by quotations(such as “y =”)
3. Values of cells are distinguished by their address. For example, B2 means that it will display the content of cell B2 which is 3.
4. If you want to edit the equation, you can double click the cell or press the F2 function key on your keyboard.

Generating the Table

In this section, we will display the values of the x- from -5 to 5 from B10 all the way to L10 and compute for its corresponding y-values. from B11 to L11.

1. Type Table in cell A9 and type x in cell A10 and y in cell A11.
2. Next, we adjust the width of the cells containing the table. Highight A10 – B10. In the Home tab of the Excel toolbar, click Format from the Cells block, then click Column width.
   

   Figure 2 - The Format Column Width command

3. In the column width dialog box, type 3, then press the OK button.
4. Type the -5 in B10, -4 in C10, and so on all the way to 5 in L10. Your table should look like the table below.

Figure 3 - Values of x

Finding the values of y

Next, we want to get the corresponding values of y in the equation y = mx + b. To do this, we  multiply the value of m (located in B2) to the value of x (located in B10) then add it to  the value in b (located in B3).  Our expression should be, B2*B10 + B3. In Microsoft Excel, formulas always start with an equal sign. So we will type type =B2*B10 + B3.

1. To compute for the value of y in B11, click cell B11, then type =B2*B10 + B3 then press the ENTER key.
2. We can type one-by-one the other corresponding values of y but we can do better than that. We will just copy the values of cell B11 to C11 all the way through cell L11 to get the other y-values. But before that, we must modify the formula written in B11. We should edit that formula in B11 and change it to =$B$2*B10+$B$3. Notice that we add dollar signs in B2 and B3. For the explanation of this see ** below.
3. To Edit the formula in B10, double click cell B11 (or click cell B10, then press the F2 function key on your keyboard), then change the formula to =$B$2*B10+$B$3
4. Next we copy the formula in B11 and paste it from C11 all the way to L11. To do this, place the cell pointer in B11, then click the Copy icon from the toolbar (or press CTRL  C).
5. Highlight C11 to L11, then click the Paste icon from the toolbar (or press CTRL V)
6. If you have done the procedures correctly, and have not changed the values of x and y you inputted earlier, your table should look like the diagram below.

Figure 4 - Table of values of the equation y = 3x + 5

Q1: Change the values of m and b in cells B2 and B3. What do you observe?

Constructing the Graph

1. To create the graph, highlight the table from B10 to L11 as shown below



2. In the Charts section of the Insert tab of the Excel, click Scatter and choose the graph on the second graph in the first row. A graph should appear on your worksheet.



3. Try to change the value of m and b several times. What do you observe about the graph?

Formatting the Graph

If you can observe, the axes of the graph automatically adjust to the value of m and b. Entering the bigger values of m also increases the range of the y-axes. To remedy this problem, we must set the axes to constant range regardless of the values of m and b. We will set the range of the y-axis from – 20 to 20, and the range of the x-axis from -5 to 5.

Setting the Axis Range

1. To set the domain and range of the graph of the y-axes, click the y-axis and be sure that a rectangular box appears as shown below.


Figure 7 - The y-axes selected

2. Right-click the rectangular box surrounding the y-axis, then click Format Axis from the pop-up menu.
3. In the Axis Options, select all the Fixed option button, then copy the numbers shown below.



Figure 8 - The Format Axis dialog box

4. Click the Close button.
5. Now, adjust the Axis options of the x-axis by right-clicking the x-axis and choosing Format Axis. Copy the numbers as shown below.


Figure 9 - Axis options for the x-axis

6. Click the Close button.
7. Change the values of m and b. What do you observe about the equation, the table and the graph?

Q2: How does the value of m and b affect the graph?

_____________________________________________________________

*Actually, functions can also be represented verbally, so that makes the number of representations 4.

** When you copy formulas in Excel, the locations of cells also adjusts.  For example, if I copy the formula in B11, which is =B2*B10 + B3, to C11 – that is one cell to the right – all the cell addresses in the formula will also adjust one cell to the right. If you do that, the formula in C11 will be =C2*C10+C3 where C2 is now our m and C3 our bwhich is different from their original locations. The dollar sign tells Microsoft Excel to points to a constant location – in our case, B2 and B3.

My jealousy is driving me mad. I wish it would just go away and let my mind be at peace. I made mathematical equations for jealousy.

Anger + Pride = Jealousy

She is not attractive. Why does he like her? I am better than her. I dread every minute of her presence.

Fear + Confusion = Jealousy

How could he do this to me? I gave everything I had to him. Will he leave me for her?

417242     สมการเชิงอนุพันธ์เบื้องต้น     Introductory Differential Equations

สมการเชิงอนุพันธ์แบบเชิงเส้น สมการเชิงเส้นไม่เอกพันธุ์ ผลการแปลงลาปลาซและการผกผัน ผลเฉลยของสมการเชิงเส้นที่เป็นอนุกรมกำลัง ระบบสมการเชิงเส้น การแปรผันของตัวแปรเสริม

(Linear differential equations, non-homogeneous linear equations, Laplace transforms and inverses, power series, solutions of linear differential equations, system of linear equations, variation of parameters.)

(417242 มหาวิทยาลัยเกษตรศาสตร์)

417342     สมการเชิงอนุพันธ์ระดับกลาง     Intermediate Differential Equations

สมการเชิงอนุพันธ์ย่อย ปัญหาประเภทแพร่กระจาย เชิงไฮเพอร์โบลา และเชิงวงรี

(Partial differential equations, diffusion-type problems, hyperbolic-type problems, elliptic-type problems.)

(417342 มหาวิทยาลัยเกษตรศาสตร์)

417371     ผลต่างสืบเนื่องและสมการผลต่างสืบเนื่อง     Finite Differences and Difference Equations

ผลต่างอันตะ ปริพันธ์อันตะ การประมาณค่าในช่วง สมการผลต่าง การประยุกต์

(Finite differences, finite integration, interpolation, difference equations, applications.)

(417371 มหาวิทยาลัยเกษตรศาสตร์)

I have a simple question about a Mathematics related topic that you may have first heard of in Elementary or Middle School called CROSS MULTIPLICATION. 

When do you use cross multiplication?

Can you explain and/or define cross multiplication? 

Do you know of any published resources or online links that explain cross muliplication?

Let us know!

Avery (Edu.  Consultant & Trainer)

Inquiry into the surface tectonics began with studying the piecemeal seed sourced by George Legendre.  A quick set of manipulations and I chose to halt at the current stage (parametrics indicated in the image below).

A careful study into the equations and its subsequent outcomes indicate an increase in the frequency of the period along the “j” threads results in the thread knotting around itself. When carefully stitched along the subsequent three sides throws up interesting phenomena. There is suddenly a close resemblance to the tectonics of a mobius strip.

Using a custom script developed by George in Lisp, for autocad, these threads could be then imported into autocad for further study.

It was not long before the realization dawned in that the implementation of thickness into these threads would indicate an inability for the threads in the corners to “knot” around. Further, each of these threads are inclined with the “horizontals” that are not perpendicular to it. Hence, I had to resort to scripting on grasshopper to ease the process of extruding the rectangular cross section along these threads. This enabled me to test the threads in the corner for the failure of knots and I eliminated them for the purposes of fabrication. This further enhanced the peculiarities of the “knot” which increases from a near zero in the corners to a certain maximum on the other end.

Scripting in grasshopper further eased the process of implementing boolean transformations on the threads. This was critical since otherwise, it is impossible to guess how the threads would lock up against each other. The “notches” installed into the threads are all inclined at different angles and dimensions.

The poly-surfaces so derived were then exploded. I was gambling with using the laser cutter to install these notches and this meant the impossibility of creating a swarf cut in the material (for the purposes of this study, I was working with an eighth of an inch – chip board). I tried making all the notches larger so as to accommodate the “incoming” threads. These were then laid out in autocad to be laser cut.

And finally, there was the catastrophe!! Everything seemed to work to a certain extent. But, pretty soon, it ended up being impossible to notch the threads one against another. This problem could also be attributed in part to the depth of horizontals not being too large. There was an intrinsic problem towards making the depth fairly large, since the knotting of the thread imposed a certain maximum on it. In all probability, my best guess would be an unavoidable swarf cutting or perhaps an extremely thin material.

So, after some excruciatingly large amounts of time and energy exhausted with this process, net result seems be tending towards zero……. LOL!!

And yes, I would be henceforth willing to share all code written be me, if requested (base Mathcad seeds do not belong to me. Apologies. Anything else. Yes)

If you are an IT professional, you are probably used to reading about computers, software, and other things of that nature. I am used to reading about science myself, and I always dread reading poetry because I can never understand what is going on. My attitude towards poetry was like that of one of my favorite physicists, Paul Dirac, who said, “The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way.” When you can understand it, poetry can be interesting, but I can seldom understand it without help. The Fairie Queene is an example of  incomprehensible poetry, that completely baffled me when I first read it. However, I gradually began to understand. Here are some tips that someone with a background of science can use to read The Fairie Queene:

1. Read each passage slowly, and read it at least twice

When read quickly, the poem seems like a haphazard jumble of strange words, without any meaning or discernable story. After multiple readings, sometimes you can pick up meanings or themes that you may have missed the first time.

2. Look up words that you do not understand

You cannot ignore words you do not know, just as when working an equation, you must look up constants that are necessary for calculations. Although you may think that skipping a word every now and then won’t affect your understanding of the poem, they can be crucial to it.

3. Understand the symbols

The poem is full of words that are not meant to be taken literally, but instead are symbols for something else. In science, equations are also full of letters or symbols that stand for a number or quantity. It is important to understand from the context what these symbols mean if you do not know what they are.

4. Get help from others

Sometimes, no matter how much you try, you cannot understand a passage. At this point, it is best to consult with a peer, someone who does understand the poem, sort of like having a science paper peer-reviewed by other scientists.

Like an extremely complicated equation, The Faerie Queene takes a lot of effort to read or understand, and you may not even want to read it. If you do, however, these tips can help you enjoy and comprehend the poem a little bit better.

-Kashyap Saxena

Strange book. It is somewhere between a popular science book and a textbook. Maybe the best use is for the  not number-theorist mathematicians to understand what is going on there.

It is an easy read, at least for the first 2 parts. I understood the new concept quickly and the structures that I did know beforehand are also explained very nicely and quickly. Sometimes their strategy is that say something firm, easy to understand/remember then to admit that it is a lie, the situation is a bit more complicated. It seems to be a useful technique.

So what is achieved in the first 2 parts? Group theory with permutation and matrix representations, elliptic functions, Galois groups, characters. Then it all comes together in reciprocity laws, using Galois representations for understanding the set of solutions of equations… but at this point, in Part 3 it becomes quite steep. New concepts are coming in a pace that full understanding is not possible any more.  Understanding Wiles’ proof after reading a thin book is not feasible under normal circumstances. Most probably  that is not the aim here, it is rather wetting the appetite for learning more, and that is done well.

An index of symbols would be very useful. The authors keep referring back to previous definitions, reflect on notations, so the omission of this is quite surprising.

The text contains some philosophical reflection on math at some random points, and these are quite good, though I was quite surprised to hear that some people think that mathematics is finished. I never met those people.

The preface for the paperback edition is great fun. It is a pity that that sense of humour is not pursued in the main text.

publisher’s page

This evening, while eating a pizza with tartufo at the pizzeria, I got this astonishing enlightenment. Let me quickly explain the following equations and the truth about the real essence of “work”!

Phi(W)=frustration at work; t=time; a=duration of your job contract in years; W(t)=work; sigma(t)=satisfaction

alpha(n)=achievement; n=number of achievements

F=force; d=displacement

From these 3 easy equations, two expected conclusions and one amazing and revolutionary postulate are obtained:

If (n) of achievements equals to zero, or if time t tends to infinite, frustration Phi(w) goes infinite. More unexpected, if the force F applied to work W is zero, frustration is zero! The postulate is: do not work and live happily!

THE LAMP POST

the formulae you leave on
scrawled on
unused disused paper around my house
its written in an ancient, understandable
language to you
but its not for me to understand
unless i am tutored or even schooled
the root of the problem is
the formulae you leave on
scrawled on
unused disused paper in my house
its written beautifully, and as it remains
un-understandable to me it becomes
more beautiful
as you write it
and later
i’ll find it

Picture by: n Submitted by: drinkmorecoffee via Fail Uploader

This is to show you that, except for the last one, they come easily from the straight-line v-t graph.  I will not ask you to perform these first-principles developments on a test, but it is comforting to know that everything here is mathematically consistent.

Kinematics_Equations_and_Derivations

These variants have been developed from the original seed (top most image) sourced by George L. Legendre. This is work in progress, part of an ongoing investigation into the methods of “writing surfaces”. All the variants have been developed from a single piecemeal seed that builds on 5 separate functions describing each of the 4 “wall surfaces” and the single “top surface”. Reference reading – ijp : The book of surfaces.

Apparently, this hasn’t been a successful attempt of hard coding math yet. There seems to be an extremely close resemblance to free form control point deformation in any of the other animation packages and unfortunately, does not reflect the rigors of coding different math for each of the 5 surfaces and yet managing to stitch them . I was vaguely aware of this issue ever since I began working on the seed, and quite naturally, attracted a critique on the same lines.

More updates to follow later.

I need to give this to someone someday. Epic win.