P^2+Σ2P

The smallest pair of amicable numbers is (220, 284). You expand it infinitely. M and N are amicable. σ(M) is the sum of proper divisors. σ(220)=1+2+4+5+10+11+20+22+44+55+110+220=504. Moreover, σ(N)=σ(284)=1+2+4+71+142+284=504. σ(M)-M=N and σ(N)-N=M In this case, M=220 and N=284. σ(220)-220=504-220=284 σ(284)-284=504-284=220 ∴ σ(M)=M+N=σ(N) This is crossing over. M=apq, N=ar pqr are different prime numbers, and a is the prime relatively. Leonhard Euler found it. σ(p)=p+1, p is the prime. σ(pq)=σ(p)σ(q)=1+p+q+pq=(1+p)+q(1+p)=(1+p)(1+q), p and q are prime numbers. Therefore, σ(M)=σ(N), σ(apq)=σ(ar) σ(a)σ(p)σ(q)=σ(a)σ(r) σ(p)σ(q)=σ(r) ∴ (p+1)(q+1)=r+1 This is based on prime numbers. Then, you put x=p+1, y=q+1 . xy=r+1, r=xy-1 σ(M)=M+N=apq+ar=a(pq+r) σ(M)=σ(a)σ(p)σ(q)=σ(a)(p+1)(q+1)=a(pq+r) σ(a)xy=a[(x-1)(y-1)+(xy-1)] ax=[2ax-σ(a)x-a]y y=ax/[2ax-σ(a)x-a] Then, you put a/[2a-σ(a)]=b/c . 2a-σ(a)=ac/b, σ(a)=2a-(ac/b) ∴ y=ax/[(ac/b)x-a]=bx/(cx-...

Euler's formula is based on Maclaurin's expansion . The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. x is 0, so f(x)=(sin x,cos x,-sin x,-cos x,・・・)=(0,1,0,-1,・・・)=sin x and f(x)=(cos x,-sin x,-cos x,sin x,・・・)=(1,0,-1,0,・・・)=cos x Euler's identity is well known as the beauty of mathematics.

This is the process of data streaming with less memory. x∈R^n This is high dimensional vector. You update it. xi←xi+v You have O(n) space, and the whole stream is O(m). You compress the data by streaming . A=(1±ε)B (1-ε)B≦A≦(1＋ε)B ∴ σi,j = ±1, E[σi,j] = 0

This is Inclusion-Exclusion . Then, this is the Zeta transform. Moreover, this is the inversion. You prove this. At first, you use the Zeta transform. You put the inversion. You remove the gray area. You divide S and R. This is Inclusion-Exclusion. You got f(R). Therefore, the Zeta transform is feasible.

OPT(U,s,t) is the length of s and t , and you choose the cheapest. You split the path into two halves in all possible ways. This is called Exponential Divide and Conquer. m is an intermediate point. ｜S｜=｜｜U｜/2｜+1 T∪S=U T∩S=m n is nodes . ∴ ε>0 The exponential space is huge.

In a flow network , you need to choose the cheapest path. This is the directed graph. G=(V,E) (u,v)∈E This is the edge. c(u,v)>0 (c is the capacity) f(u,v)≧0 (f is the flow) a(u,v) is the cost of the path. The cost of sending this flow is f(u,v)*a(u,v). You choose the cheapest. This is the total cost of the flow over all edges. f(u,v)≦c(u,v) Then, you think the maximum cardinality matching in G that has minimum cost. G=(A∪B,E) G'=(V'=A∪B,E'=E) The capacity of all the new edges is 1 and their costs is 0.

Blocking flows make a tree because of O(logU) . However, this is dynamic trees, so you can delete and insert nodes. There is no direction basically. This is the algorithm. initially, makeTree() n times while(true): v = findRoot(s) if (v == t) //(z,parent(z)) is a min capacity edge with weight x (z,x) = findMin(s) subtract(s,x) cut(z) delete(z,parent(z)) from the level graph continue else //try to advance if v has an outgoing edge to some w in L: link((v,w),capacity(v,w)) else if (v == s): break else for every child y of v cut(y) delete(y,v) from L The cost of splaying nodes is O(logU). The total number of preferred child changes is O(m logU).