articles unfinished

8/11/2019 Articles Unfinished

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ARTICLES UNFINISHEDFirst edition

Is about the gift of the fsica celestial number 3

Jose luis armenta16/11/2013


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1.-quantic webs and pauli matrix.................................................................4,5

2.-faa di bruno formula have 3 m 3 n and 2 j and 2 g.............................................6

3.-the force of young or young module............................................................7

4.-the first vector is stand up and the second is bent over......................................8

5.-the electric current density is a function of the area........................................9

6.-the quantum power.............................................................................10

7.-relative error as i-j/i.......................................................................11

7.-first came the electricity later the magnetism................................................12

8.-the nemotecnics of christoffel symbols........................................................13

9.-contracted of two index in dalembertian......................................................14

10.-metric tensor is not degenerate if is not a circle function..................................15

11.-the gibbs was helmholtz and later gibbs......................................................16

12.-achx10-bc is the mnemotecnic of boltzmann constant.........................................17

13.-potential dwell in alternating current.......................................................18

14.-direct current is a constant.................................................................19

15.-beta minus decreasing the protons............................................................20

16.-beta minus increasing protons................................................................21

17.-the magic numbers are odds...................................................................22

18.-crash of uv particles........................................................................23

19.-cofactors and the big bang theory............................................................24

20.-hyperbolic sine is tha past and the future is hyperbolic cosine..............................25

21.-pauli matrix are dilpotents..................................................................26

22.-Gamma function depends of z, t ,e , -1 ,0,infinite............................................27

23.-Beta function depends of 1,0,x,t,-1,y.......................................................28

24.-mgsintetha is for pendulums..................................................................29

25.-mgcosinetehta is for plane...................................................................30

26.-acceleration of a pendulum ..................................................................31


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27.-each summand have a constant.................................................................32

28.-luminous signals and parabolas...............................................................33

29.-dirac delta and variance simettries..........................................................34

30.-cauchy schwartz inequality as a product......................................................35

31.-triangle inequality and the sum of everything................................................36

32.-the covariant derivative goes alphabetically.................................................37

33.-the coriolis force have 3 croice products....................................................38

34.-Single-dimensional approximation and quantum energy..........................................39

35.-mean value theorem and slopes................................................................40

36.-bolzano and mean theorem.....................................................................41

36.-hessian matrix goes like covariant multiple tensor...........................................42

37.-observables hamiltonians.....................................................................43

38.-taylor polynomial and dirac delta function simetrie..........................................44

39.-relative error as inverse of the capacitance.................................................45

40.-concave is goes to down coordinates axis.....................................................46

41.-convex is goes to up coordinates axis........................................................47

42.-lipschitz inequality are inverse to triangle inequality......................................48

43.-seismograph in one floor.....................................................................49

44.-the quantum pipet............................................................................50

45.-two derivatives is two characteristic equations..............................................51

46.-the minus one plus two trick................................................................52

47.-hiperboloyd of one sheet at the beginnig of the universe.....................................53

48.-lagrangian of gravity as 2 spheres...........................................................54

49.-the components of the metric are covariants..................................................55

50.-lie bracket is only for positives bosons.....................................................56

51.-dark matter and dark energy have a lot of hot but we only perceive the color.................57

52.-abelian product is biunivoc..................................................................58


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53.-the grade from w is i+1 of the k/m in angular frequency for the armonic oscillator...........59

54.-sturm liouville and the rect line as the second and third factors in linear acelerators.....60

55.-monotonic barrer in -1 spinors...............................................................61

56.-material derivative not use croice product...................................................62

57.-Helmholtz descomposition not use dot product.................................................63

58.-lie algebra and fractals.....................................................................64

59.-three lines in jacobi identity...............................................................65

60.-biharmonique funcions as double integrals in wirtinger analysis..............................66

61,-the cauchy sequence goes to infinite.........................................................67

62.-for n=0 the degree of freedom is one fermion in euclidean group..............................68

63.-for spin-foam the multiplication goes like his own matrix....................................69

64.-current density in casimir effect............................................................70

65.-the christoffel symbol number 4 goes in the gamma ..........................................71

66.-the beta function and the fourier bessel function............................................72

67.-debye model in cubic stray...................................................................73


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Quantum webs and Pauli matrices

The Pauli matrices are:

For a spin 1/2 particle, the spin operator is given by J = /2 . It is possible to form generalization of Pauli

matrices in order to describe higher spin systems in three spatial dimensions. For arbitrarily large j , the Pauli

matrices can be calculated using the spin operator and ladder operators The spin matrices for spin 1 andspin 3/2 are given below:

:

:


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And the quantum webs are:


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Faa di Bruno have 3 ns 2 ms 2 js and 2 gs


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Young module as young force

Were rro is a pressure I mean force between area


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The first vector in a matrix is stand up and the second is bent over


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The current density is a function of the area

Electric current density J is simply the electric current I (SI unit: A) per unit area A (SI unit: m 2). Its magnitudeis given by the limit:


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Quantum potency

A potency is One watt is the rate at which work is done when an object's velocity is held constant at onemeter per second against constant opposing force of one newton.

But quantically is w= /s having the units:

1.054571726(47)10 34 j.s/s = armentas =a


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Relative error as i-j/i

where the vertical bars denote the absolute value If the relative error is


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In the big bang first came the electricity later the magnetism

electric field (see electrostatics): an especially simple type of electromagnetic field produced by anelectric charge even when it is not moving (i.e., there is no electric current). The electric field produces

a force on other charges in its vicinity. Moving charges additionally produce a magnetic field

due to Faraday law need to exist a paralell electric field first of all


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the christoffel symbols and nemotecnics

The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the secondkind and the metric ,[3]

The last index in the metric tensor and the upper chrystoffel symbol are i=I or 1=1
http://en.wikipedia.org/wiki/Christoffel_symbols#cite_note-ludvigsen-3http://en.wikipedia.org/wiki/Christoffel_symbols#cite_note-ludvigsen-3http://en.wikipedia.org/wiki/Christoffel_symbols#cite_note-ludvigsen-3http://en.wikipedia.org/wiki/Christoffel_symbols#cite_note-ludvigsen-3


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Contraction of two index in dalembertian

In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box:), also called the d'Alembertian or the wave operator , is the Laplace operator of Minkowski space. Theoperator is named for French mathematician and physicist Jean le Rond d'Alembert In Minkowski space in

standard coordinates ( t , x , y , z) it has the form:

The index contracted are miu and niu


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Metric tensor non degenerate is have a constant x_p

g p is nondegenerate . A bilinear function is nondegenerate provided that, for every tangentvector X p 0, the function

So the constant x p is a constant different of zero in that case will be a circle function


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First was helmholtz later gibbs

First was helmholtz

The Helmholtz energy is defined as:

Later was cause the adding of one term I mean pv the Gibbs free energy is defined as:

G(p,T) = U + pV TS


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Boltzman constant mnemotecnic

1.3806488(13)10 23

achx10-bc

each letter means a ordinal number I mean a=1 b=2 c=3 d=4 e=5 f=6 g=7 h=8


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potential dwell as alternating current

In alternating current (AC, also ac ), the flow of electric charge periodically reverses direction. In directcurrent ( DC, also dc ), the flow of electric charge is only in one direction.

The abbreviations AC and DC are often used to mean simply alternating and direct , as when theymodify current or voltage

W e have 2 potentials dwells one above and one under


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Direct current is a constant

Direct current (DC) is the unidirectional flow of electric charge. Direct current is produced by sources such

as batteries, thermocouples solar cells, and commutator-type electric machines of the dynamotype. Direct

current may flow in a conductor such as a wire, but can also flow through semiconductors,insulators, or

even through a vacuum as in electron or ion beams. The electric current flows in a constant direction,

distinguishing it from alternating current (AC). A term formerly used for direct current was galvanic current .


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Beta decay negative if the protons are increasing

In nuclear physics beta decay ( decay) is a type of radioactive decay in which a beta particle (an electron ora positron) is emitted from an atomic nucleus. Beta decay is a process which allows the atom to obtain the

optimal ratio of protons and neutrons

Beta decay is mediated by the weak force. There are two types: beta minus and beta plus . In the case of

beta decay that produces an electron emission, it is referred to as beta minus (

An example of decay is shown when carbon-14 decays into nitrogen-14

+


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Beta plus decay decreasing protons

In nuclear physics, beta decay ( decay) is a type of radioactive decay in which a beta particle (an electron ora positron) is emitted from an atomic nucleus. Beta decay is a process which allows the atom to obtain the

optimal ratio of protons and neutronsBeta decay is mediated by the weak force in the case of a positron emission as beta plus ( +).

An example of positron ( + decay) is shown with magnesium-23 decaying into sodium-23

+


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magic numbers are odds

In nuclear physics, a magic number is a number of nucleons (either protons or neutrons) such that they arearranged into complete shells within the atomic nucleus. The seven most widely recognised magic numbers

as of 2007 are 2, 8, 20, 28, 50, 82, and 126 (sequence A018226 in OEIS). Atomic nuclei consisting of such amagic number of nucleons have a higher average binding energy per nucleon than one would expect based

upon predictions such as the semi-empirical mass formula and are hence more stable against nuclear decay.


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Ultraviolet colitions

is having a device of condensate ultraviolet particles and made explote under a soup of ultraviolet lamp or

crash particles ultraviolet versus ultraviolets particles

this is the device

This is the device of the accelerator of uv rays

Ultraviolet UV 400 100 nm 3.10 12.4 eV


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Cofactors and big bang theory

Cause in a line of time we have minus 1 zero plus one here is the cofactor definition:

If A is a square matrix, then the minor of the entry in the i -th row and j -th column (also called the (i , j ) minor ,

or a first minor ) is the determinant of the submatrix formed by deleting the i-th row and j-th column. This

number is often denoted M i,j . The (i,j) cofactor is obtained by multiplying the minor by .


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Hyperbolic sine is the past and cosine is future in the bigbang theory

The hyperbolic functions are:

Hyperbolic sine:

Hyperbolic cosine:

As we see the past have two sign minus and the cosine have one plus in this form

-1 0 +1


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Pauli matrix are idempotent

When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds

since [ I M ][I M] = I M M + M2= I M M + M = I M .

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 2 complex matrices which are Hermitian and unitary Usually indicated by the Greek letter sigma ( ), they

are occasionally denoted with a tau ( ) when used in connection with isospin symmetries. They are:


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Gamma function depends of f(z, t ,e ,-1 ,0 )

The gamma function is defined for all complex numbers except the negative integers and zero. For complexnumbers with a positive real part, it is defined via a convergent improper integral:


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Beta function depends of 1,0,x,t,-1,y

In mathematics, the beta function , also called the Euler integral of the first kind, is a special function definedby

for


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pendulum is function of sine tetha


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Incline plane is cosine theta

Instead of the pendulum we use a incline plane for cosine is very useful to think as a pendulum or incline

plane


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Acceleration in pendulums

A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolatedsystem using the following assumptions:

The rod or cord on which the bob swings is massless, inextensible and always remains taut;

Motion occurs only in two dimensions i.e. the bob does not trace an ellipse but an arc.

The motion does not lose energy to friction or air resistance.

The differential equation which represents the motion of a simple pendulum is

where is acceleration due to gravity, is the length of the pendulum, and is the angulardisplacement.


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each summand have a constant

a + b = b + a .

i mean that is it: ka+kb=kb+ka were k is a constant directly proportional


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Variance and dirac delta

This is the variance:

If the random variable X is continuous with probability density function f ( x ), then the variance is

given by

And this is the dirac delta is:

The integral of the time-delayed Dirac delta is given by :

As we see there is a rest or a minus sign between boths


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Cauchy Schwartz inequality is a dot product

The Cauchy Schwarz inequality states that for all vectors x and y of an inner product space it istrue that

where is the inner product also known as dot product. Equivalently, by taking the squareroot of both sides, and referring to the norms of the vectors, the inequality is written as


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The sum of everything is not the sum over z in triangle inequality


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The derivative covariant goes alphabetically

.See goes ijkl i mean 1234 is the same


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The coriolis force have 3 croice products

The apparent motion of a distant star as seen from Earth is dominated by the Coriolis andcentrifugal forces. Consider such a star (with mass m) located at position r , with declination , so

r = | r | sin( ), where

is the Earth's rotation vector. The star is observed to rotate about theEarth's axis with a period of one sidereal day in the opposite direction to that of the Earth's rotation,making its velocity v = r . The fictitious force, consisting of Coriolis and centrifugal forces, is:


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Single-dimensional approximation and quantum energy

So far, one usually considers the single-dimensional case of this phenomenon, that is when the

potential has translational symmetry in two directions (say and ), such that only a singlecoordinate (say ) is important. In this case one can examine the specular reflection of a slowneutral atom from a solid state surface . Where one has an atom in a region of free space close to amaterial capable of being polarized, a combination of the pure van der Waals interaction, and therelated Casimir-Polder interaction attracts the atom to the surface of the material. The latter forcedominates when the atom is comparatively far from the surface, and the former when the atomcomes closer to the surface. The intermediate region is controversial as it is dependent upon thespecific nature and quantum state of the incident atom.

The condition for a reflection to occur as the atom experiences the attractive potential can be givenby the presence of regions of space where the WKB approximation to the atomic wave-function

breaks down. If, in accordance with this approximation we write the wavelength of the gross motionof the atom system toward the surface as a quantity local to every region along the axis,

Clearing E= /h

Since the kinetic energy is p 2/2m


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Mean value theorem and slopes

In calculus the mean value theorem states, roughly: that given a planar arc between twoendpoints, there is at least one point at which the tangent to the arc is parallel to the secant through

its endpoints.The theorem is used to prove global statements about a function on an interval starting from localhypotheses about derivatives at points of the interval.

More precisely, if a function f is continuous on the closed interval [ a , b], where a < b, anddifferentiable on the open interval ( a , b), then there exists a point c in ( a , b) such that


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Bolzano theorem and mean value theorem

The mean value theorem says:

In calculus, the mean value theorem states, roughly: that given a planar arc between two

endpoints, there is at least one point at which the tangent to the arc is parallel to the secant throughits endpoints.

The theorem is used to prove global statements about a function on an interval starting from localhypotheses about derivatives at points of the interval.

More precisely, if a function f is continuous on the closed interval [ a , b], where a < b, anddifferentiable on the open interval ( a , b), then there exists a point c in ( a , b) such that

And the Bolzano theorem says:

In mathematics, specifically in real analysis, the Bolzano Weierstrass theorem , named

after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-

dimensional Euclidean space R n. The theorem states that each bounded sequence in R n has

aconvergent subsequence. An equivalent formulation is that a subset of R n is sequentially

compact if and only if it is closed and bounded


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Hessian matrix goes that covariant multiple tensors

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partialderivatives of a function. It describes the local curvature of a function of many variables. The

Hessian matrix was developed in the 19th century by the German mathematician Ludwig OttoHesse and later named after him. Hesse originally used the term "functional determinants".

Given the real-valued function

if all second partial derivatives of f exist and are continuous over the domain of the function, thenthe Hessian matrix of f is

where x = ( x 1, x 2, ..., x n) and Di is the differentiation operator with respect to the i th argument. Thus


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Observables as unitaries or hermitian matrices

Physical observables are represented by Hermitian matrices on H.

See the following example:

The diagonal elements must be real , as they must be their own complex conjugate.
http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_number


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Taylor polynomial and dirac equation

The taylor polynomial is:

If a real-valued function f is differentiable at the point a then it has a linear approximation at thepoint a . This means that there exists a function h1 such that

and the dirac delta is:

The integral of the time-delayed Dirac delta is given by :

This is sometimes referred to as the sifting property or the sampling property . The delta functionis said to "sift out" the value at t = T .

It follows that the effect of convolving a function f (t ) with the time-delayed Dirac delta is to time-delay f (t ) by the same amount :

(using (4):

)
http://en.wikipedia.org/wiki/Dirac_delta_function#math_4http://en.wikipedia.org/wiki/Dirac_delta_function#math_4http://en.wikipedia.org/wiki/Dirac_delta_function#math_4http://en.wikipedia.org/wiki/Dirac_delta_function#math_4


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Relative error and capacitance simetries

where the vertical bars denote the absolute value. If the relative error is

Capacitance is the ability of a body to store an electrical charge. Any object that can be electricallycharged exhibits capacitance. A common form of energy storage device is a parallel-plate capacitor.In a parallel plate capacitor, capacitance is directly proportional to the surface area of the conductorplates and inversely proportional to the separation distance between the plates. If the charges onthe plates are + q and q, and V gives the voltage between the plates, then the capacitance C isgiven by

Capacitance is qi-qj/vj and in relative error is the inverse


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Concave is goes to down axis coordenates


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Convex is goes to up coordinates axis


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Seismograph in one floor

Need to be in one floor cause the waves of the steps persons will crate a noise of waves here is a picture of

one kinematics seismograph


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Quantum pipet

A wave potential have a limit to an asintot madding a quantum pipet for give inks to atoms


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Every characteristic equation is one derivative

If we have 2 characteristic equations we have 2 characteristic equations.

In mathematics, the characteristic equation (or auxiliary equation ) is an algebraic equation

of degree on which depends the solutions of a given th-order differential equation Thecharacteristic equation can only be formed when the differential equation is linear,homogeneous,and has constant coefficients Such a differential equation, with as the dependentvariable and as constants

will have a characteristic equation of the form

where are the roots from which the general solution can be formed.This

method of integrating linear ordinary differential equations with constant coefficients was discoveredby Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic'equation. The qualities of the Euler's characteristic equation were later considered in greater detailby French mathematicians Augustin-Louis Cauchy and Gaspard Monge


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the -1+2 = numerator

-1+2/1=1 is a old trick to made two divitions is useful in limits partial fractions


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Hyperboloid of one sheet at the beginning of the universe

This is because in the beginning of the universe there was a tunnel effect like says efrain rojas marcial


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Gravitational lagrangian as 2 spheres

The Lagrangian (density) is in Jm 3 . The interaction term m is replaced by a term involving acontinuous mass density in kgm 3 . This is necessary because using a point source for a field

would result in mathematical difficulties. The resulting Lagrangian for the classical gravitational fieldis:

As we have a rea of a sphere in 4r 2 so we have 8 so we have 2 spheres


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The metric components are covariants

The components of the metric in any basis of vector fields, or frame, f = ( X 1, , X n ) are given by


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Lie bracket only useful for positive bosons

A Lie algebra is a vector space over some field F together with a bynary operation

called the Lie bracket , which satisfies the following axioms:

Bilinearity

for all scalars a , b in F and all elements x , y , z in .

Alternating on :

for all x in .

The Jacobi identity:

for all x , y , z in .

Will be more clear In this graphic


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Dark energy and dark mattery are so hot in other dimension but we only perceive the color

In physical cosmology and astronomy, dark energy is a hypothetical form of energy that permeatesall of space and tends to accelerate the expansion of the universe Dark energy is the most

accepted hypothesis to explain observations since the 1990s that indicate that the universeis expanding at anaccelerating rate According to the Planck mission team, and based onthe standard model of cosmology the total mass energy of the universe contains 4.9% ordinarymatter 26.8% dark matter and 68.3% dark energy.

Dark matter is a type of matter hypothesized in astronomy and cosmology to account for a largepart of the total mass in the universe. Dark matter cannot be seen directly with telescopes; evidentlyit neithermits nor absorbs light or other electromagnetic radiation at any significant level. Instead, itsexistence and properties are inferred from its gravitational effects on visible matter, radiation, andthe large-scale structure of the universe. According to the Planck mission team, and based on

the standard model of cosmology, the total mass energy of the known universe contains4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy Thus, dark matter is estimated toconstitute 84.5% of the total matter in the universe and 26.8% of the total content of the universe.


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Abelian product as a biunivoc relation

The product of an abelian variety A of dimension m, and an abelian variety B of dimension n, overthe same field, is an abelian variety of dimension m + n. An abelian variety is simple if it is

not isogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenousto a product of simple abelian varieties.


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The i+1 in the angular frequency in the harmonic oscillator

An object attached to a spring will oscillate. Assuming that the spring is ideal and massless with nodamping then the motion will be simple and harmonic with an angular frequency given by:

where

k is the spring constant

m is the mass of the object.


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the last 2 terms on sturm liouville and the light trayectory In a line

the last 2 terms I mean the second and third element are a line in a Cartesian plane so is a

rect line in a lineal accelerator or laser beam and the fotomultiplier at the sight


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monotonic barrer in -1 spinors

Some simple examples of spinors in low dimensions arise from considering the even-gradedsubalgebras of the Clifford algebra C p , q(R). This is an algebra built up from an orthonormal basis

of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1and q of which have norm 1, with the product rule for the basis vectors

is a monotonic barrer cause in +1 is from 1..2 but for 2+1 goes only for 3..3 so is less its acts like a barrier


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material derivative not use croice product

The material derivatives of a scalar field ( x, t ) and a vector field u ( x, t ) are defined respectivelyas:


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Helmholtz descomposition not use dot product

Let F be a vector field on a bounded domain V in R 3, which is twice continuously differentiable.Then F can be decomposed into a curl-free component and a divergence-free component :[11]
http://en.wikipedia.org/wiki/Helmholtz_decomposition#cite_note-11http://en.wikipedia.org/wiki/Helmholtz_decomposition#cite_note-11http://en.wikipedia.org/wiki/Helmholtz_decomposition#cite_note-11http://en.wikipedia.org/wiki/Helmholtz_decomposition#cite_note-11


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Three lines in jacobi integral

One for each bracket

In a Lie algebra, the objects that obey the Jacobi identity are infinitesimal motions. When acting onan operator with an infinitesimal motion, the change in the operator is the commutator.

The Jacobi Identity

which can be changed into the following form by Bilinearity and Alternating

This formula can be expatiated with plain words: "the infinitesimal motion of B followed by theinfinitesimal motion of A ([A,[B, ]]), minus the infinitesimal motion of A followed by the infinitesimalmotion of B ([B,[A, ]]), is the infinitesimal motion of [A,B] ([[A,B], ]), when acting on any arbitrary

infinitesimal motion C (thus, these are equal)".


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Biharmonique derivatives as double derivatives in wirtinger analysis

Then he writes the equation defining the functions he calls biharmonique , previously writtenusing partial derivatives with respect to the real variables , with , ranging from 1 to ,

exactly in the following way


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A Cauchy sequence goes to infinite

To define Cauchy sequences in any metric space X, the absolute value is replaced by

thedistance

(whered

: X

X

R

with some specific properties, see Metric (mathematics)) between and .

Formally, given a metric space ( X , d ), a sequence

is Cauchy, if for every positive real number > 0 there is a positive integer N such that for all positiveintegers m ,n > N , the distance

Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests thatthe sequence ought to have a limit in X . Nonetheless, such a limit does not always exist within X .


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for n=0 the degree of freedom is one fermion in euclidean group

The number of degrees of freedom for E( n) is

n(n + 1)/2,

which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translationalsymmetry, and the remaining n(n 1)/2 to rotational symmetry


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spin foam how the multilplication goes behind each matrix

The partition function for a spin foam model is, in general,

So for mulplication over f the matrix is sub f and for multilplication c the matrix goes sub c , for themultlplication for v goes for matrix sub v


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Current density goes like casimir effect

Electric current density J is simply the electric current I (SI unit: A) per unit area A (SI unit: m 2). Itsmagnitude is given by the limit

When the area goes to limit equal too zero is like casimir effect cause the vacuum wavessurrounded


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The first kind christoffel symbol and the number 4 in the gamma variable

The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of thesecond kind and the metric,

Where a=1 b=2 c=3 and d=4


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Debye model in 3 d

The Debye model is a solid-state equivalent of Planck's law of black body radiation , where onetreats electromagnetic radiation as a gas of photons in a box . The Debye model treats atomicvibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical.

Consider a cube of side . From the particle in a box article, the resonating modes of the sonicdisturbances inside the box (considering for now only those aligned with one axis) havewavelengths given by

where is an integer. The energy of a phonon is

where is Planck's constant and is the frequency of the phonon. Making theapproximation that the frequency is inversely proportional to the wavelength, we have:

in which is the speed of sound inside the solid. In three dimensions we will use:

Sustituing 2L by 3L we have :

= )2( + + )
http://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiationhttp://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiationhttp://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiationhttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Gas_in_a_boxhttp://en.wikipedia.org/wiki/Gas_in_a_boxhttp://en.wikipedia.org/wiki/Gas_in_a_boxhttp://en.wikipedia.org/wiki/Phononhttp://en.wikipedia.org/wiki/Phononhttp://en.wikipedia.org/wiki/Phononhttp://en.wikipedia.org/wiki/Particle_in_a_boxhttp://en.wikipedia.org/wiki/Particle_in_a_boxhttp://en.wikipedia.org/wiki/Particle_in_a_boxhttp://en.wikipedia.org/wiki/Planck%27s_constanthttp://en.wikipedia.org/wiki/Planck%27s_constanthttp://en.wikipedia.org/wiki/Planck%27s_constanthttp://en.wikipedia.org/wiki/Planck%27s_constanthttp://en.wikipedia.org/wiki/Particle_in_a_boxhttp://en.wikipedia.org/wiki/Phononhttp://en.wikipedia.org/wiki/Gas_in_a_boxhttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiation


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The Bessel-fourier function and beta function

Because said, differently scaled Bessel Functions are orthogonal with respect to the inner product

according to

,

the coefficients can be obtained from projecting the function f(x) onto the respective Besselfunctions:

where the plus or minus sign is equally valid.

And the beta function is:

In mathematics, the beta function , also called the Euler integral of the first kind, is a specialfunction defined by

for

The beta function was studied by Euler and Legendre and was given its name by JacquesBinet; its symbol is a Greek capital rather than the similar Latin capital B.

Conclution:

If we are clever we see that the limit of the integral is going from zero to 1 like the beta function

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