The 4-colour theorem states that the maximum number of colours required to paint a map is 4.
The proof requires exhaustive computation with a help of a computer.
But I thought that one can visually prove the theorem in the following way;
If one replaces the map with a graph where each region...
I made a little experiment with magnets.
I got two small bars of magnets. They obey the usual attraction-repulsion rules by approaching their faces together in various permutation.
Accidentally, one of them is broken into two unequal pieces. When I managed to put the two broken pieces side by...
I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given.
It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z.
One possible...
I just thought about whether it is possible, or already exists, a method to construct a 3D optical scene using the method of backprojection.
First, I will capture the scene (say a castle) from different views using HD camera. Then using one of the app to stitch the 2D images into a cloud of...
Homework Statement
Suppose there is solute ##s## in a bowel containing fluid. There is a tiny hole near the bottom which leaks a small fixed volume of solute ##\lambda## per unit time ##dt##. In addition, there is a small added solute to the fluid in a constant rate ##\alpha## so as the volume...
This is a pic of sun rays shining on the metallic leg of the chair. The reflection comes on the floor of my room as rings. The source is just a square window. No other light sources in the room aside from reflection from the walls. The leg of the chair is cylinder in shape with smooth surface...
The Compton equation determines the shift of the wave-length of the scattered photon as a function of the ##cos\theta## of the scattered photon. It does not depend on the energy of the incident photon. With some manipulation using the equations of conservation of the momentum and energy, one...
Any pure state of spin-1/2 particle can be represented by a superposition of spin up and spin-down relative to an arbitrarily direction.
lw>=a l+> + b l-> where a^2+b^2=1
If there is no magnetic field, the measurement of the spin is random and we get 50-50 chance to be spin up and spin down...
If there is matrix that is formed by blocks of 2 x 2 matrices, what will be the relation between the eigen values and vectors of that matrix and the eigen values and vectors of the sub-matrices?
I am looking for a method that displays a see-through images of an object through an arbitrarily non-transparent surface. Suppose I have already the 3D tomographic information of an object inside a closed box and I am interested to have a holographic 3D view of that object just by looking at the...
My initial objective is to make a regression of ##y## dependent variable on a given set of ##x_1##, ##x_2##... and ##x_m## independent variables. Suppose, I am dealing with a data set of ##n## samples, I found that the variables are correlated so I decided to do factor analysis to best represent...
Homework Statement
Calculating the area of equilateral triangle using calculus.
Homework Equations
The Attempt at a Solution
The area of the triangle is the area of the circle minus 3 times the area of the sector shown in (light blue). So, the target is to calculate the pink area first...
Homework Statement
How to obtain the famous formula of velocity transformation using a chain rule.
I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule.
Homework...
I am trying to understand why the special relativity is not suitable for describing the gravity.
Consider a counterexample assuming it is the suitable and the space-time containing a gravitational mass is flat. Then one could describe the acceleration of a test particle from his inertial frame...
I can not claim that I fully understand my own question, but I feel it is worth sharing.
Suppose there is a physical system composed of MxN degree of freedom where M is the number of the states and N the number of particles. When a measurement is undertaken, the physical system collapses to...
I wish to find a closed form relation between ##\theta## and ##d## in the attached figure as a function of ##r##, the radius of the circle and ##s## the distance from the point ##p## to the center.
Thank you.
I am not sure that I fully understand even the basic aspects of the Quantum measurement and entanglement but I just came across this thought experiment and I wish to resolve it.
In a setting of two entangled spin-1/2 particles, suppose that Alice applies a uniform magnetic field ##B_0## along...
Homework Statement
Given that matrix, A can be decomposed using SVD (Singular Value Decomposition) into ##A=USV^T##`, why does always the sum of the square of cosines between v` vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1?
Homework...
In Excel, I wish to automatically delete multiple columns that have (0) in particular row location. How to do this? After the deletion, the data array should shrink to only columns that do not have (0) at that row.
Suppose we have a product formed by a multiplication of a unitary matrix U and a diagonal matrix A, can we retrieve the inverse of A without knowing either U or A?
Suppose I need to fill a row composed of 32 cells with a sequence of 4 cells with value 1 in a row followed by 4 cells with value 0 in a row and then again 4 cells of 1 and so on. How can I do this without copying those cells and repeating pasting them in their location?
Say, I have a matrix which I obtained from a website for matrix calculation, how to insert it into an excel so as for each cell in the matrix, there is a corresponding cell in the excel sheet?
Suppose I apply a particular function and I get a value of the cell-A1. How can I copy this value to another cell-C1 and the entire column under C1 to use it in a new operation provided that C1 and the entire column are always updated when A1 changes?
In Latent semantic analysis, the truncated singular value decomposition (SVD) of a term-document matrix ##A_{mn}## is
$$A=U_rS_rV^T_r$$
In many references including wikipedia, the new reduced document column vector in r-space is scaled by the singular value ##S## before comparing it with other...
I have a document-word matrix and I need to do latent semantic analysis to match a query document with a set of key-words to the best possible match among other documents in the matrix. The matrix is in the form of documents-words occurrence array of 1 and 0 input. What is the best software I...
I am looking for a software that enable me to do the basic data analysis like ANOVA, regression, factor analysis and to do nice graphing. I need something similar to SPSS with data entry in the form of variables no coding input like SAS.
I am interested to know whether there is an application to make a duty rota for employees. The application should allow all employee to select their preferred days, even shifts within the day, and the result will be the one that makes the best match to their initial choices. Of course, there...
I question whether there would be a conceptual catch in EPR proposal to get both non commutative variables define with accuracy at one location simultaneously. The idea of separability that EPR introduced to questioning the completeness of QM is that whenever one measures a physical variable q...
Cauchy integral theorem states that the contour integration of a complex harmonic function along a closed simply connected path=0. What if this simply connected path is drawn over a Riemann surface of function like ##f(z)=\sqrt z##. Will that be possible in the first place? and will the...