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An equation ( z = e < sup > sT </ sup >) maps continuous s-plane poles ( not zeros ) into the z-domain, where T is the sampling period.
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equation and z
Illustration of a simple equation, x, y, z are real numbers, analogous to weights.
In terms of Cartesian coordinates p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the ellipse is
Electrons behave as beams of energy, and in the presence of a potential U ( z ), assuming 1-dimensional case, the energy levels ψ < sub > n </ sub >( z ) of the electrons are given by solutions to Schrödinger ’ s equation,
The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity I as a function of position z in the gain medium:
To solve, we first rearrange the equation in order to separate the variables, intensity I and position z:
If the right-hand side is specified as a given function, f ( x, y, z ), i. e., if the whole equation is written as
This set is a cycle in the dual number plane ; since the equation setting the difference in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola.
An example is the associative axiom for a binary operation, which is given by the equation x * ( y * z ) = ( x * y ) * z.
( The equation in example I was z = 0, and the equation in example II was x = y.
The equation (*) is a result of the identity, for z
The Nernst equation has a physiological application when used to calculate the potential of an ion of charge z across a membrane.
In other words, the defining equation for W ( z ) is
Suppose z is defined as a function of w by an equation of the form
In this case, the column space is precisely the set of vectors ( x, y, z ) ∈ R < sup > 3 </ sup > satisfying the equation z = 2x ( using Cartesian coordinates, this set is a plane through the origin in three-dimensional space ).
T ( x ', z, t ) and use the heat equation:
It may be noted that the above equation implies that gravitational acceleration is a constant over z since it is placed outside of the integral.
* Letting x: y: z be a variable point in trilinear coordinates, an equation for the nine-point circle is
To start with a simple example the variables x, y and z in an equation such as can refer to any number.
For a function u ( x, y, z, t ) of three spatial variables ( x, y, z ) ( see cartesian coordinates ) and the time variable t, the heat equation is
equation and =
As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.
The end result of antimatter meeting matter is a release of energy proportional to the mass as the mass-energy equivalence equation, E = mc < sup > 2 </ sup > shows.
For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.
These points form a line, and y = x is said to be the equation for this line.
This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).
The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.
The basic accounting equation is assets = liabilities + equity.
; Associativity: For all a, b and c in A, the equation ( a • b ) • c = a • ( b • c ) holds.
Consider the graph of the equation y = 1 / x shown to the right.
The oblique asymptote, for the function f ( x ), will be given by the equation y = mx + n.
Assuming that all the particles start from the origin at the initial time t = 0, the diffusion equation has the solution
Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.
Bessel himself originally proved that for non-negative integers n, the equation J < sub > n </ sub >( x ) = 0 has an infinite number of solutions in x.
The equation of a circle is ( x − a )< sup > 2 </ sup > + ( y − b )< sup > 2 </ sup > = r < sup > 2 </ sup > where a and b are the coordinates of the center ( a, b ) and r is the radius.
For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 4.
Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.
Trivially, if y = 0 then y '= 0, so y = 0 is actually a solution of the original equation.
We note that y = 0 is not allowed in the transformed equation.
But we have independently checked that y = 0 is also a solution of the original equation, thus
From the equation for density ( ρ = m / V ), mass density must have units of a unit of mass per unit of volume.
equation and e
Atomic orbitals can be the hydrogen-like " orbitals " which are exact solutions to the Schrödinger equation for a hydrogen-like " atom " ( i. e., an atom with one electron ).
The mathematical equation for an ideal gas undergoing a reversible ( i. e., no entropy generation ) adiabatic process is
; Identity element: There exists an element e in A, such that for all elements a in A, the equation holds.
Subsequently, the accelerative force on any given ion is controlled by the electrostatic equation, where n is the ionisation state of the ion, and e is the fundamental electric charge.
Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).
In 1930, Stoner derived the internal energy-density equation of state for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately ( for μ < sub > e </ sub >= 2. 5 ) 2. 19 · 10 < sup > 30 </ sup > kg.
Given that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age ( i. e. BA ), leads to the equation 19B-8A
A linear equation with two variables has many ( i. e. an infinite number of ) solutions.
With the reduced state variables, i. e. V < sub > r </ sub >= V < sub > m </ sub >/ V < sub > c </ sub >, P < sub > r </ sub >= P / P < sub > c </ sub > and T < sub > r </ sub >= T / T < sub > c </ sub >, the reduced form of the Van der Waals equation can be formulated:
An important example is the case where is at a steady state, i. e. the concentration does not change by time, so that the left part of the above equation is identically zero.
The Born-Landé equation gives a reasonable fit to the lattice energy of e. g. sodium chloride where the calculated value is − 756 kJ / mol which compares to − 787 kJ / mol using the Born-Haber cycle.
; Identity element: There exists an element e in S, such that for all elements a in S, the equation holds.
Since E < sub > 2 </ sub >-E < sub > 1 </ sub > ≫ kT, it follows that the argument of the exponential in the equation above is a large negative number, and as such N < sub > 2 </ sub >/ N < sub > 1 </ sub > is vanishingly small ; i. e., there are almost no atoms in the excited state.
He independently derived Tsiolkovsky's rocket equation, did basic calculations about the energy required to make round trips to the Moon and planets, and he proposed the use of atomic power ( i. e. Radium ) to power a jet drive.
For pure rocket approaches Tsiolkovsky's rocket equation shows that dead weight will prevent reaching orbit unless the ratio of propellant to structural mass ( called mass ratio ) is very high — between about 10 and 25 ( i. e. 24 parts propellant weight to 1 part structural weight ; depending on propellant choice ).
The converse is not true: not all irrational numbers are transcendental, e. g. the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x < sup > 2 </ sup > − 2
Using a uniqueness theorem and showing that a potential satisfies Laplace's equation ( second derivative of V should be zero i. e. in free space ) and the potential has the correct values at the boundaries, the potential is then uniquely defined.
Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein – Gordon equation, and describes a spinless particle field ( e. g. pi meson ).
With time reversed we have the situation of two objects pushed away from each other, e. g. shooting a projectile, or a rocket applying thrust ( compare the derivation of the Tsiolkovsky rocket equation ).
The wavelength dependence in the grating equation shows that the grating separates an incident polychromatic beam into its constituent wavelength components, i. e., it is dispersive.
Since the frequency range of the typical noise experiment ( e. g. 1 Hz – 1 kHz ) is low compared with typical microscopic " attempt frequencies " ( e. g. 10 < sup > 14 </ sup > Hz ), the exponential factors in the Arrhenius equation for the rates are large.
0.253 seconds.