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It is well known that relativization cannot be used to prove or disprove NP not equal to P. This paper question this belief by giving a very simple prove of NP not equal to P using relativization.
This paper is inspired by this post. It asked how to count the number of solutions of the equation with non-negative integral solution:
x_1 + ... + x_k = n
without using the standard method of line and dot counting. I transform the problem into a partial difference equation, ...
I tried to find the n-th derivative of the function sec x + tan x in a specific form, and a partial difference equation arises from it. An attemp to solve the equation is stated in another article: Reducing a partial difference equation into a partial differential equation and solving for the generating function using method of characteristics
In this paper, I will derive an explicit formula for the Euler zigzag numbers (Up/down numbers). Euler zigzag number is the number of alternating permutation in a set. Therefore the explicit formula of Euler numbers(Secant numbers) and Bernoulli numbers are found as well. The formula involves two finite sum.
Given [A, B] = lambda, we would like to find out the expansion of (A+B)^n. This result is useful for quantum mechanics since [x,p] = i hbar. And it can be useful when solving mixed differential recurrence relation equation as well. I will give an example of it.
I am going to solve a partial difference equation in this paper, which arises from binomial expansion of non-commutative matrix/operator.
In this paper, I am going to prove the identity:
e^(A+B) = e^(-lambda/2) e^A e^B
given [A, B] = lambda.
This formula is quite useful in quantum mechanics, and I will prove it using power series expansion ...
In general there are no guaranteed methods to solve non-homogeneous ordinary differential equation with variable coefficients(NVODE) analytically. In this paper, I am going to use operator method to solve NVODE of order n.
The method is about factorizing the differential operator. In order to do so, we first need to know one of the fundamental solutions of the order n ...
Permanent of a matrix is quite similar to that of determinant, though determinant can be found in polynomial time, while permanent is #P-complete (meaning that if you solve it, you can count the number of solution of all NP problem.)
In this paper, I am going to describe a new method to calculate permanent based on a very simple principle: ...
