I was briefly introduced to Grassmann numbers when I studied quantum field theory. I then had the natural question of how we can formally define them. In this article, I went with my intuition and tried to answer this question.
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an m-fold symmetry is compatible with translational symmetry is the same as whether φ(m)≤2.
Suppose there are n distinguishable boxes and k indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains m balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit n,k→∞ with k/n fixed, the distribution tends to be a geometric distribution.
Denote the length distribution of one’s hair to be f(l,t), where l is hair length, and t is time. Considering that each hair may be lost naturally from time to time (there is a probability of λdt for each hair to be lost within time range from t to t+dt) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.
Given your probability of breaking the combo at each note, what is the probability distribution of your max combo in the rhythm game chart? I considered the problem seriously!
By recursively integrating according to xn+1(t):=∫t0tf(xn(s),s)ds+C from x0(t0):=C, we can get the solution of the ODE
x′(t)=f(x(t),t) with initial conditions x(t0)=C as the limit of the sequence of functions.
The Hölder mean of x with weights w and a parameter p is defined as Mp,w(x):=(w⋅xp)p1, and the value at p=−∞,0,+∞ are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as p increases. This property can be used to prove HM-GM-AM-QM inequalities.
The function (1−z)n+1∑k=1∞knzk is a polynomial of degree n w.r.t. z, and the sum of its coefficients is n!. This turns out to be properties of Eulerian numbers.
Suppose P is a point on the circle ⊙C. When is the sum of distances from P to two edges of ∠O extremal? It turns out to be related to angle bisectors (the intersections of ⊙C and the bisector of ∠O or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of ⊙C and edges of ∠O) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting ⊙C at P lies, translate the region to make it center at C, and see whether P is inside the translated region.
I was briefly introduced to Grassmann numbers when I studied quantum field theory. I then had the natural question of how we can formally define them. In this article, I went with my intuition and tried to answer this question.
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an -fold symmetry is compatible with translational symmetry is the same as whether .
Suppose there are distinguishable boxes and indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit with fixed, the distribution tends to be a geometric distribution.
Bra–ket notation is a good-looking notation! I am sad that it is not generally taught in math courses. Let me introduce it to you.
Denote the length distribution of one’s hair to be , where is hair length, and is time. Considering that each hair may be lost naturally from time to time (there is a probability of for each hair to be lost within time range from to ) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.
Given your probability of breaking the combo at each note, what is the probability distribution of your max combo in the rhythm game chart? I considered the problem seriously!
By recursively integrating according to from , we can get the solution of the ODE
with initial conditions as the limit of the sequence of functions.
The Hölder mean of with weights and a parameter is defined as , and the value at are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as increases. This property can be used to prove HM-GM-AM-QM inequalities.
The function is a polynomial of degree w.r.t. , and the sum of its coefficients is . This turns out to be properties of Eulerian numbers.
Suppose is a point on the circle . When is the sum of distances from to two edges of extremal? It turns out to be related to angle bisectors (the intersections of and the bisector of or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of and edges of ) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting at lies, translate the region to make it center at , and see whether is inside the translated region.