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.
The image of a circle with radius r and centered at C(−2f,0) through a thin lens at x=0 with focal length f and centered at O(0,0) is a conic section with the focus being (2f,0), the directrix being line x=f, and the eccentricity being fr.
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.
There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for E<0 is u:=p2+p02p2−p02n^+p2+p022p0p, where u is the position of the particle on 3-sphere (a 4-dimensional vector),
p is the momentum of the original particle in Kepler problem, n^ is a vector perpendicular to the 3-dimensional hyperplane where p lies, and p0:=−2mE.
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…
To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit T+e−C0C(T0−T), where T is the initial temperature of the object, T0 is the temperature of the hot water, C is the heat capacity of the object, and C0 is the heat capacity of the hot water.
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.
The image of a circle with radius and centered at through a thin lens at with focal length and centered at is a conic section with the focus being , the directrix being line , and the eccentricity being .
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.
After some comparison among solutions, I use KaTeX to typeset math in my emails.
There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for is where is the position of the particle on 3-sphere (a 4-dimensional vector),
is the momentum of the original particle in Kepler problem, is a vector perpendicular to the 3-dimensional hyperplane where lies, and .
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…
To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit , where is the initial temperature of the object, is the temperature of the hot water, is the heat capacity of the object, and is the heat capacity of the hot water.
I have created a community rhythm game called Dododo. It is a rhythm game with musical rhythm notations.