Prove the Hermitian identity, the principle of Hermites

Proof: Let v==x-[x] be the decimal part of x, then 0 v<1. k n exists such that k n v<(k+1) n. (i.e. k=[nv]).

The original formula is equivalent to: n*[x]+[v]+[v+1 n]+[v+2 n]+....v+(n-1) n]=n[x]+[nv], i.e., [v+1 n]+[v+2 n]+....v+(n-1) n]=[nv], since v+q n<(k+1) n+(n-k-1) n=1 in q<(n-k-1), so [v+q n]=0;

Therefore, only the last k term on the left side of the formula is not 0, that is, [v+m n](n-k m n-1) is not 0, a total of k terms, and this k term is 1, that is, the left side of the formula is k, and the right side of the formula is obviously k.

Therefore, the original equation is established, and the original equation is proven.

1) If x is an integer.

Rule. x]+[x+1/n]+[x+2/n]+…x+(n-1)/n]=x+x+..x=nx=[nx]

2) If x is not an integer.

then x=[x]+

And for any n, there is k such that +(k n)<1 +(k+1) n thus. x]+[x+1/n]+[x+2/n]+…x+(n-1)/n][[x]+]x]++1/n]+.x]++k/n]+[x]++k+1)/n]+.

x]++n-1)/n]

k+1)[x]+(n-k-1)([x]+1)n[x]+n-k-1

It is then known by +(k n)<1 +(k+1) n.

n-k-1 n so. nx]

n([x]+)

n[x]+n]

n[x]+[n]

n[x]+n-k-1

Therefore. x]+[x+1/n]+[x+2/n]+…x+(n-1)/n]=[nx]

Hermit was a well-rounded mathematician, and in addition to the above, he also achieved the following results in various fields of mathematics: he studied matrix theory in depth, proving that if the matrix m=m* (the conjugate transposed matrix of m), its eigenvalues are all real; He proposed a Hermitian principle belonging to the theory of algebraic functions, which was one of the special cases of the later famous Riemann-Roch theorem; There are many achievements in the invariant Zheng quantity, so much so that J J Sylvester has pointed out that "a Cayley, Hermitt, and I form a trinity of invariant quantities", for example, he proposed a "reciprocal shouting law", that is, a one-to-one correspondence between a covariant of a fixed order of the p-order of an m-order binary type and a covariant of the fixed number of m-order of a p-order binary type; Hermit popularized Gauss's method of studying quadratic coefficients and proved that the number of classes for any variable is still limited. This result is also applied to algebraic numbers, proving that if the discriminant of a number field is given, the number of paradigms is limited; He also applied this "class finiteness" to the indefinite quadratic type, and achieved some important results; His work on the Lame equation (a differential equation) was also of great significance at the time

(1) Any square matrix can be written as the sum of an Ermitian matrix and an oblique Hermitian matrix [1].

2) The eigenvalues of oblique Hermitian matrices are imaginary numbers.

3) Oblique Hermitian matrices are all regular matrices, so they are diagonal, and their different eigenvectors must be orthogonal.

4) All elements on the main diagonal of the oblique Hermitian matrix must be pure imaginary numbers or 0s

5) If a is an oblique Hermitian matrix, then ia is a Hermitian matrix.

6) If a, b are oblique Hermitian matrices, then for all real numbers a, b, aa + bb must also be oblique Hermitian matrices.

7) If a is an oblique Hermitian matrix, then a 2k is a Hermitian matrix for all positive integers k.

8) If a is an oblique Hermitian matrix, then the odd power of a is also an oblique Hermitian matrix.

9) If a is an oblique Hermitian matrix, then e a is a unitary matrix and e is the base of the natural logarithm.

10) The difference between a matrix and its conjugate transpose is an oblique Hermitian matrix.

Hermitt matrices: Self-conjugate matrices.

(1) The nth-order Hermitian matrix a is a positive definite (semi-positive) matrix, and the sufficient and necessary conditions are that all eigenvalues of a are greater than or equal to 0.

2) If a is a nth-order Hermitian matrix with an eigenvalue diagonal matrix of v, then there is a unitary matrix u, such that au=uv.

3) If a is a nth-order Hermitian matrix, the square of its Frobernian norm is equal to the sum of the squares of all its eigenvalues.

4) The conjugate transpose of the oblique Hermitian matrix is -a

The eigenvalues of oblique Hermitian matrices are all real numbers. Further, oblique Hermitian matrices are all regular matrices. Hence they are diagonal, and their different eigenvectors must be orthogonal.

<[x]≤x<[x]+1

3.[n+x]=n+[x], where n is an integer.

is a non-subtractive function.

f(x)= is a periodic function.

Its period is any positive integer.

The minimum positive period is 1

5.[x]+[y]≤[x+y]≤[x]+[y]+16.If n is positive, then [nx]n[x].

7.If n is positive, then [x n]=[x] n] 8.Hermitt identity: For any x greater than 0, there is constant [x]+[x+1 n]+[x+2 n]+....x+(n-1)/n]=[nx]。

I discovered the law of conservation in mathematics, which states that numbers and number transformations are always equal.

And a, the Gaussian function plays an important role in the definition of the Hermitt polynomial.

The Gaussian function is used as a pre-smoothing kernel in image processing (see scale space representation) and is a scalar multiple of the function that performs the Fourier transform, but it does not have an elementary indefinite integral.

Molecular orbitals used in computational chemistry are linear combinations of Gaussian functions called Gaussian orbitals (see Basis Groups in Quantum Chemistry), mathematics, and engineering. in the natural sciences.

There are applications of Gaussian beams in optical and microwave systems.

The Gaussian function is related to the vacuum state in quantum field theory.

In the field of mathematics:

in statistics and probability theory;

The Gaussian function of c2 is a characteristic function of the Fourier transform.

The Gaussian function is the wave function of the ground state of the quantum harmonic oscillator, a social science. Thereinto.

a, according to the central limit theorem it is a finite probability distribution of complex sums, and the Gaussian function is a density function of a normal distribution.

Gaussian functions are elementary functions, and examples of this include , b, and . c

is a real constant.

The indefinite integral of a Gaussian function is an error function. However, it is still possible to calculate its generalized integral on the entire real number axis (see Gaussian integral). This means that the Fourier transform of the Gaussian function is not just another form of the Gaussian function.

functions.

If you just find the eigenvalue or spectral decomposition, there is no essential difference between the real symmetric matrix and the hermite matrix, just change the orthogonal transformation to the unitary transformation, all the tools are universal, it should be said that the hermite matrix is simpler than the real symmetric matrix, the key is that you don't understand it yourself, it's not that the ready-made introduction is too little, you have to deduce it yourself, and if you don't understand the principle, you can't talk about writing a program.

Search: C program for finding eigenvalues and eigenvectors of Hermitian matrices.

h = h*(t-x(j))^2/((x(i)-x(j))^2);This grinding is to do a factorial for the fiber to crack a = a + 1 (x(i)-x(j)); This is to sum f = f + h*((x(i)-t)*(2*a*y(i)-y 1(i))+y(i))h

But it's more complicated.

This is the c implementation of the compound Simpson formula, it takes two integrals, register it, and I'm looking for the rest.

The foreign ones are Newton, Gauss, Archimedes, Fermat, Diric, Euler.

China has Zu Chongzhi, Hua Luogeng, Su Buqing, Chen Jingrun, and Yang Le.

In short, a lot.

Newton, Euler, Archimedes, Xierpaite, Chen Jingrun, Hua Luogeng.

Archimedes, Euclid, Newton, Euler, Gauss, Riemann, Poincaré, Hilbert, ......

The upstairs were all stolen from the internet!! Mine: The Five Great Mathematicians: von Neumann, Galois, Archimedes, Zu Chongzhi, Seles.

Chinese mathematicians: Chen Jingrun, Zhu Shijie, Hua Luogeng, Chen Shiingshen, Su Buqing, Yau Chengtong, Wu Wenjun, Liao Shantao, Yang Le, Chen Jiangong, Li Shanlan, Hua Hengfang, Li Xinbiao, etc.

Foreign mathematicians: Pythagoras, Morgan, Fermat, Euler, Hilbert, etc.