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From Wikipedia

Cube (algebra)

In arithmetic and algebra, the cube of a number n is its third power— the result of the number multiplying by itself three times:

n3 = n× n× n.

This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

A perfect cube (also called a cube number, or sometimes just a cube) is a number which is the cube of an integer.

The sequence of non-negative perfect cubes starts :

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97736, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112, 205379, 216000, 226981, 238328...

Geometrically speaking, a positive number m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The pattern between every perfect cube from negative infinity to positive infinity is as follows,

n3 = (n− 1)3 + (3n− 3)n + 1.

Cubes in number theory

There is no smallest perfect cube, since negative integers are included. For example, (−4) × (−4) × (−4) = −64. For any n, (−n)3 = −(n3).

Base ten

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers, for example 64 is a square number (8 × 8) and a cube number (4 × 4 × 4); this happens if and only if the number is a perfect sixth power.

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root1, 8 or 9. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

  • If the number is divisible by 3, its cube has digital root 9;
  • If it has a remainder of 1 when divided by 3, its cube has digital root 1;
  • If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Waring's problem for cubes

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

Fermat's last theorem for cubes

The equation x3 + y3 = z3 has no non-trivial (i.e. xyz≠ 0) solutions in integers. In fact, it has none in Eisenstein integers.

Both of these statements are also true for the equation x3 + y3 = 3z3.

Sums of rational cubes

Every positive rational number is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.

Sum of first ''n'' cubes

The sum of the first n cubes is the nthtriangle number squared:

1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^2.

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

1^3+2^3+3^3+4^3+5^3 = 15^2 \,

A similar result can be given for the sum of the first yodd cubes,

1^3+3^3+\dots+(2y-1)^3 = (xy)^2

but {x,y} must satisfy the negative Pell equation x^2-2y^2 = -1. For example, for y = 5 and 29, then,

1^3+3^3+\dots+9^3 = (7*5)^2 \,
1^3+3^3+\dots+57^3 = (41*29)^2

and so on. Also, every evenperfect number, except the first one, is the sum of the first 2(p−1)/2odd cubes,

28 = 2^2(2^3-1) = 1^3+3^3
496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3
8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3

Sum of cubes in arithmetic progression

There are examples of cubes in arithmetic progression whose sum is a cube,

3^3+4^3+5^3 = 6^3
11^3+12^3+13^3+14^3 = 20^3
31^3+33^3+35^3+37^3+39^3+41^3 = 66^3

with the first one also known as Plato's number. The formula F for finding the sum of an n number of cubes in arithmetic progression with common difference d and initial cube a3,

F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+...+(a+dn-d)^3

is given by,

F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)

A parametric solution to,

F(d,a,n) = y^3

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d> 1, such as d = {2,3,5,7,11,13,37,39}, etc.

History

Determination of the Cube of large numbers was very common in many ancient civilizations. statistical signal processing and physics, the spectral density, power spectral density (PSD), or energy spectral density (ESD), is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per Hz, or energy per Hz. It is often called simply the spectrumof the signal. Intuitively, the spectral density captures the frequency content of astochastic process and helps identify periodicities.

Explanation

In physics, the signal is usually a wave, such as an electromagnetic wave, random vibration, or an acoustic wave. The spectral density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, per unit frequency, known as the power spectral density (PSD) of the signal. Power spectral density is commonly expressed in watts per hertz (W/Hz) or dBm/Hz.

For voltage signals, it is customary to use units of V2Hz−1 for PSD, and V2sHz−1 for ESD or dBμV/Hz.

For random vibration analysis, units of g2Hz−1 are sometimes used for acceleration spectral density.

Although it is not necessary to assign physical dimensions to the signal or its argument, in the following discussion the terms used will assume that the signal varies in time.

Definition

Energy spectral density {{Anchor|energy spectral density}}

The energy spectral density describes how the energy (or variance) of a signal or a time series is distributed with frequency. If f(t) is a finite-energy (square integrable) signal, the spectral density \Phi(\omega) of the signal is the square of the magnitude of the continuous Fourier transform of the signal (here energy is taken as the integral of the square of a signal, which is the same as physical energy if the signal is a voltage applied to a 1-ohm load, or the current).

\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt\right|^2 = \frac{F(\omega)F^*(\omega)}{2\pi}

where \omega is the angular frequency (2\pi times the ordinary frequency) and F(\omega) is the continuous Fourier transform of f(t), and F^*(\omega) is its complex conjugate.

If the signal is discrete with values f_n, over an infinite number of elements, we still have an energy spectral density:

\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^\infty f_n e^{-i\omega n}\right|^2=\frac{F(\omega)F^*(\omega)}{2\pi}

where F(\omega) is the discrete-time Fourier transform of f_n.

If the number of defined values is finite, the sequence does not have an energy spectral density per se, but the sequence can be treated as periodic, using a discrete Fourier transform to make a discrete spectrum, or it can be extended with zeros and a spectral density can be computed as in the infinite-sequence case.

The continuous and discrete spectral densities are often denoted with the same symbols, as above, though their dimensions and units differ; the continuous case has a time-squared factor that the discrete case does not have. They can be made to have equal dimensions and units by measuring time in units of sample intervals or by scaling the discrete case to the desired time units.

As is always the case, the multiplicative factor of 1/2\pi is not absolute, but rather depends on the particular normalizing constants used in the definition of the various Fourier transforms.

Power spectral density

The above definitions of energy spectral density require that the Fourier transforms of the signals exist, that is, that the signals are integrable/summable or square-integrable/square-summable. (Note: The integral definition of the Fourier transform is only well-defined when the function is integrable. It is not sufficient for a function to be simply square-integrable. In this case one would need to use the Plancherel theorem.) An often more useful alternative is the power spectral density (PSD), which describes how the power of a signal or time series is distributed with frequency. Here power can be the actual physical power, or more often, for convenience with abstract signals, can be defined as the squared value of the signal, that is, as the actual power dissipated in a load if the signal were a voltage applied to it. This instantaneous power (the mean or expected value of which is the average power) is then given by

P(t) = s(t)^2

for a signal s(t).

Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener–Khinchin theorem provides a simple alternative. The PSD is the Fourier transform of the autocorrelation function, R(\tau), of the signal if the signal can be treated as a wide-sense stationary random process.

This results in the formula,

S(f)=\int_{-\infty}^\infty \,R(\tau)\,e^{-2\,\pi\,i\,f\,\tau}\,d \tau=\mathcal{F}(R(\tau)).

The ensemble average of the average periodogram when the averaging time interval T→∞ can be proved (Brown & Hwang) to a


From Yahoo Answers

Question:I would like to know how to graph a cube into a 3D graph.

Answers:http://en.wikipedia.org/wiki/Cube

Question:

Answers:density is mass per unit volume density = mass / volume volume is different for different shapes

Question:

Answers:If we assume that a (111) plane is the plane that cuts through the body center cube diagonally, then we see that along this plane, there are 5 atoms. This means that we ((4/8)+1) or 1.5 atoms contributing to the planar density. PS - the reason I take only (1/8) contribution from 4 atoms is because these 4 are shared in the BCC crystal lattice, whereas the center atom is not shared.

Question:You ve decided to spend your savings on a 1000 cm3 cube of Unobtainium, a rare and valuable imaginary metal whose properties I have made up for the sake of this problem. Unobtainium has a density of 15 g/cm3 at 0 C and an average coefficient of linear expansion of 10-3 C^(-1). (Side note that really isn t important to the solution of this problem: relatively speaking, this is a very high coefficient of linear expansion, ~100 times higher than that of steel.) What is the mass of a 1000 cm3 cube of Unobtanium at 0 C? What is the density of Unobtainium at 30 C? (hint think about the volume expansion) What is the mass of a 1000 cm3 cube of Unobtanium at 30 C? Based on this answer, should you purchase your 1000 cm3 cube of Unobtanium on a cold day or a hot day, and why?

Answers:Let m0, d0, V0 be mass, density and volume at 0 C, and m0, d, V the same at 30 C (of course, the mass m0 doesn't change during thermal expanson). We know that d0 = 15 g/cm , and that the linear expansion coefficient is 10-3 C-1, which implies a volumetric espansion coefficient k = 3x10-3 C-1. a) V0 = 1000 cm , so m0 = d0 V0 = 15 x 1000 = 15000 g = 15 kg b) According to thermal volumetric expansion formula, V = V0 (1 + k T) = 1000 + 1000 x 3 x 10-3 x 30 = 1090 cm Then d = m0/V = 15000 / 1090 = 13.761 g/cm c) For a volume V' = 1000 cm and a density d = 13.761 g/cm , the mass is m = d V' = 13.761 x 1000 = 13761 g The answer to the last question depends in the first place on what you need the cube for. If you need a V cm cube of unobtanium to be used at a temperature of T degrees, you MUST buy a V cm cube when the temperature is T degrees, whether the day is hot or cold. If instead you need a certain amount (i.e. mass) of unobtanium, than the sale conditions come into consideration: - if unobtanium is sold by weight (which should be the case for so a rare and valuable metal), you pay the same if you buy it on a cold or a hot day - in the unlikely case that unobtanium is sold by volume, it's advisable to buy it on a cold day, when the same volume holds a greater mass.

From Youtube

Factoring Sums and Differences of Cubes :Factoring Sums and Differences of Cubes. In this video, I use the formulas for factoring sums and differences of cubes. I factor two expressions using the formulas.

Cross Cube Formulas [PowerOLAP Formulas #07] :A cross-cube formula can be used to calculate values in a Cube using data from another Cube or even two or more Cubes. For example, Cubes A and B contain sales data for Year 1 and Year 2 (Year 2 being current year). Cube C might contain forecasts for Year 3. A cross-cube formula, created in Cube C, could use values in Cubes A and B to average their sales data and dynamically calculate a percentage-based increase in sales, showing results in Cube C. The fact that the calculations are dynamic (true for all PowerOLAP formula calculations) means that changing a variable will automatically recalculate the formula driving the forecast in Year 3 (Cube C), therefore, forecasts would be updated as new "Actuals" are recorded for Year 2 (Cube B). Another example worth mentioning particularly beneficial for multinational companies would be the use of a Cube that stores monthly foreign exchange calculations. One could multiply the data in this Cube by the figures in a Financials cube, where company units report in local currency, to determine all figures, by month, in US dollars, back in the Financials cube.