​Every educator knows the drill: exam season approaches, and you find yourself digging through dozens of nested folders, trying to remember where you saved that perfect physics problem or that tricky multiple-choice set from two years ago. Recreating work, retyping equations, and accidentally reusing the same question three terms in a row takes up valuable teaching energy.

That is why we built Qubase, an AI-powered, Apple-native app designed for professional educators who want to transform scattered worksheet collections into a structured, searchable question database.

From Static PDFs to a Dynamic Question Bank
Instead of manually sorting through files, simply import your existing PDF worksheets. Powered by Google’s Gemini AI, Qubase automatically parses your documents into individual, reusable questions. The AI handles the complex heavy lifting that traditional text extractors struggle with, seamlessly processing:

• Complex Formatting: Math equations are converted into TeX, and multi-row tables are formatted into clean, structured Markdown.
• Academic Content: Multiple-choice options, chemical formulas, and diagram labels are cleanly separated into text and answer keys.
• Scanned & Handwritten Notes: Extract usable text directly from older scanned documents or handwritten prep notes.
• Data Privacy: Google acts solely as a secure data processor for Qubase and never uses your uploaded worksheet data to train its models.

Smarter Test Prep with Similarity Tracking
Once your library is in Qubase, preparing revision materials and internal exams becomes effortless. Qubase automatically flags duplicate or closely related questions across your library, complete with similarity scores and side-by-side comparisons. You can track how often a specific concept or question variant has appeared over time, ensuring your assessment sets stay fresh, balanced, and free of accidental repetition.

Built for Collaboration and Apple Workflows
Teaching is a team sport. Qubase lets user export portable question libraries as shareable files. Colleagues can open, edit, and search these libraries for free. Designed natively across iPhone, iPad, and Mac, Qubase takes full advantage of multi-window and split-screen workflows on larger displays, letting you drag, compare, and edit resources effortlessly.

Transparent, Fair Pricing
We believe in pricing that respects how teachers work. Qubase runs on a transparent pay-as-you-go credit system. Your initial download includes free credits to get started, and credits are only used when importing and parsing new worksheets. Searching, organizing, editing, and sharing your question bank are 100% free, and your purchased credits never expire.

Ready to turn folder clutter into an organized assessment library? Try Qubase today on iOS, iPadOS, and macOS, and see how much time you can save on your next exam cycle! ​

Polynomial regression is a technique to model the dependence of data collected in an experiment. The dependence is represented by a polynomial:
$$f(x)=a_0x^{n}+a_1x^{n-1}+\cdots+a_{n-1}x+a_n,$$
where \(x\) is a real number. The integer \(n\geq 0\) is called the order of the polynomial, which is defined as the highest power among the terms with non-zero coefficient. For examples, 
$$-5,\quad x+1,\quad 2x^3+x^2+1$$
are zeroth (\(n=0\)), first (\(n=1\)) and third (\(n=3\)) order polynomials, respectively. Note that \(0\cdot x^2+x+1\) is a first order polynomial since the first term has zero coefficient.

​To illustrate the idea of polynomial regression, we consider data collected during an experiment to determine the change in thrust efficiency (in percent) as the divergence angle of a rocket nozzle changes (see [1], p. 530):

Interpolation requires the polynomial passing through all given data points, i.e., \(y_i=f(x_i)\) for all \(i\). However, this often requires a high order polynomial because the order grows with the number of data points:

​In this above figure, the blue dots represent samples of the black curve. There are in total 10 points and hence interpolation requires a 9th order polynomial, as shown by the red curve. We see that the red curve is oscillatory while the original curve (black) is not.

As we have seen, it is sometimes undesirable to force a polynomial to strictly pass through all data points. A better approach is to look for a polynomial that is "good enough". That means, we find a polynomial such that the error 
$$e=\sqrt{\sum_{i=1}^N\left[y_i-f(x_i)\right]^2}$$
is minimised. Obviously, for interpolation, the error is zero. When interpolation is undesirable, we will accept a non-zero but minimised error \(e\).

To perform polynomial regression:

1. Select a suitable polynomial order \(n\). The order \(n\) should be small, such as, 1, 2 or 3.
2. Find the coefficients \(a_0,a_1,\ldots,a_n\) such that the error \(e\) defined above is minimised.
3. Evaluate the fitness of the regression. If needed, choose a different \(n\) and perform Step 2 again.

The coefficients \(a_0,a_1,\ldots,a_n\) in Step 2 can be obtained by solving a least square problem using QR decomposition. A brief description can be found in the document page of polyfit included in SIMO or Console.

The coefficients can be obtained by SIMO or Console using the function polyfit, as shown below.

​The output coefficients are 17.680815 and -47.420696. Therefore, the polynomial obtained is
$$f(x)=17.680815x-47.420696.$$
The error \(e\) is given by S.normr, which is 6.9987482. As shown below, the error can be improved to 4.971942 by using a second order polynomial
$$f(x)=1.4670472x^2+1.3837119x-4.4594937.$$
For the meanings of s.R and s.df, please see the document page of polyfit included in the apps.

That is it for the post. If you have any question, feel free to leave comments. Thanks.

[1] Montgomery Runger Hubele, Engineering Statistics, 5th edition, Wiley, 2011.
[2] Gene H. Golub and Charles F. Van Loan, Matrix Computations, 4th edition, The Johns Hopkins University Press, 2013.

As we can see from the scatter plot above, the data \((x,y)\) tend to fall along a line with positive slope. This  might suggest that \(x\) and \(y\) are somewhat linearly correlated. However, we have to be careful when making such an assumption. The validity of the assumption is always doubtful, unless we have conducted analyses to establish the adequacy of the linear model. Analyses can be performed using the function corrcoef in SIMO or Console. This is explained in the following section.

Correlation Coefficient
Correlation coefficient is used to measure the linear relationship between two variables. For the variables \(x\) and \(y\), their correlation coefficient is defined as 
$$\rho_{xy}=\frac{\sigma_{xy}}{\sigma_y\sigma_x},$$
where \(\sigma_{xy}\) is the covariance, and \(\sigma_x\) and \(\sigma_y\) are the standard deviations of \(x\) and \(y
\), respectively. 

• The variables \(x\) and \(y\) are called correlate if \(\rho_{xy}\neq 0\).
• It always holds that \(-1\leq\rho_{xy}\leq 1\).
• Complete positive correlation means \(\rho_{xy}=1\), which suggests that \((x,y)\) fall along a line with positive slope.
• Complete negative correlation means \(\rho_{xy}=-1\), which suggests that \((x,y)\) fall along a line with negative slope. 

The function corrcoef can be used to compute the sample correlation coefficient \(r_{xy}\), which is an estimation of \(\rho_{xy}\). This is shown in the example below.

​Today is Valentine's Day. So, we are going to draw some hearts, and then send them all to our lovers. Of course, we are going to do it by coding and math. This is what we are going to accomplish today:

The outline of a heart is a path of points \((x,y)\), which are given by the following parametric equations:
$$
\begin{align}
x&=x_c+16\sin(t)^3,\\
y&=y_c+13\cos(t)-5\cos(2t)\\
​&\quad\quad\quad-2\cos(3t)-\cos(4t),
\end{align}
$$
for \(-\pi\leq t\leq \pi\). Here, \((x_c,y_c)\) is the center of the heart. You can change the location of the heart by changing \((x_c,y_c)\). We implement the above parametric equations in SIMO or Console as a function like the following one: 

Now, you can run the function above with center \((0,0)\) to produce the following heart. This heart has only 30 path points, and therefore is not very smooth. You can make it smoother by increasing the number of path points.

Next, we need to duplicate this heart many times with different centers \((x_c,y_c)\). The centers are in fact path points on a bigger heart. The first part of the following code generates path points of the bigger heart. We need to scale it up 10 times bigger than the normal heart. These path points will serve as the centers of the smaller hearts, which will be drawn in the for-loop. It loops through each of the path points of the bigger hearts, and each of them is the center point of a smaller heart.

If you run the above script, you will get the following:.

Finally, you can save the plot as an image using the export feature of the app. Then, you can find the saved picture in the Photos app on your iPhone or iPad. You may want to do some editing like cropping and filters. Finally, you can send it to your lover! Happy Valentine's Day!

The Chinese community is celebrating the Chinese New Year in this week. Each year of the Chinese calendar is associated with a particular animal. This year is the Pig. ​To celebrate the new year and to brush up our programming/math skill, let us draw a few pigs by code. The pigs we are going to draw actually consist of some ellipses and parabolas. In case you forget, an ellipse is basically a flattened circle like this one.

This ellipse has the center \((u,v)\), width \(a\), and height \(b\). For a general ellipse, \(a\) is not necessarily equal to \(b\). However, when \(a=b\), we call it a circle and \(a\) the radius.

An ellipse can be represented by the equation:
$$
\frac{(x-u)^2}{a^2}+\frac{(y-v)^2}{b^2}=1.
$$
We can also use parametric equations to represent an ellipse:
$$
\begin{align}
x&=u+a\cos(t),\\
y&=v+b\cos(t),\quad 0\leq t\leq 2\pi.
\end{align}
$$
The parametric form is actually more convenient for plotting since it gives the \((x,y)\) point lying on the ellipse. Each \((x,y)\) point is associated with the parameter \(t\). Here, \(t\) goes from 0 to 180 degrees (i.e., 0 to \(2\pi\) in radians) and it means the full ellipse. If we just want to plot part of the ellipse, we can use other different intervals of \(t\). The following function draws an ellipse with given center \((u,v)\), width \(a\), height \(b\), and line color.

Then, we can plot an ellipse with center \((u,v)=(0,1)\), width \(a=5\), height \(b=6\), and red line, as shown in the following.

Our pigs consist of several ellipses, which represent the body, head, eyes, eye balls, nose, and nostrils. The mouth and ears are shifted and scaled parabolas. Finally, the tail is simply a cosine curve. The whole construction of a single pig with center \((x_c,y_c)\) is shown in the following function. 

For example, we can use the function to draw a pig centered at \((x,y)=(2,3)\). The result is shown below.

Finally, we want to plot 4 pigs with centers \((0,0)\), \((35,0)\), \((35,30)\), \((0,30)\). This can be done using the script below. And, finally, we have 4 cute pigs to celebrate the new year!