Nash embedding theorem: Difference between revisions

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In [[mathematics]], a '''coefficient''' is a multiplicative factor in some [[term (mathematics)|term]] of an [[expression (mathematics)|expression]] (or of a [[series (mathematics)|series]]); it is usually a number, but in any case does not involve any [[variable (mathematics)|variable]]s of the expression. For instance in
:<math>7x^2-3xy+1.5+y</math>
the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as ''a'', ''b'', and ''c'' in
:<math>ax^2+bx+c</math>
when it is understood that these are not considered as variables.
Thus a [[polynomial]] in one variable ''x'' can be written as
:<math>a_k x^k + \dotsb + a_1 x^1 + a_0</math>
for some integer <math>k</math>, where <math>a_k, \dotsc, a_1, a_0</math> are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient.
For the largest <math>i</math> with <math>a_i \ne 0</math> (if any), <math>a_i</math> is called the '''leading coefficient''' of the polynomial. So for example the leading coefficient of the polynomial
 
:<math>\, 4x^5 + x^3 + 2x^2</math>
 
is 4.
 
Specific coefficients arise in mathematical identities, such as the [[binomial theorem]] which involves [[binomial coefficient]]s; these particular coefficients are tabulated in [[Pascal's triangle]].
 
==Linear algebra==
In [[linear algebra]], the '''leading coefficient''' of a row in a matrix is the first nonzero entry in that row. So, for example, given
 
:<math>
M = \begin{pmatrix}
1 & 2 & 0 & 6 \\
0 & 2 & 9 & 4 \\
0 & 0 & 0 & 4 \\
0 & 0 & 0 & 0
\end{pmatrix}.
</math>
 
The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.
 
Though coefficients are frequently viewed as [[constant (mathematics)|constants]] in elementary algebra, they can be variables more generally. For example, the [[coordinates]] <math>(x_1, x_2, \dotsc, x_n)</math> of a [[vector (geometric)|vector]] <math>v</math> in a [[vector space]] with [[basis (linear algebra)|basis]] <math>\lbrace e_1, e_2, \dotsc, e_n \rbrace </math>, are the coefficients of the basis vectors in the expression
:<math> v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .</math>
 
==Examples of physical coefficients==
# ''[[coefficient of thermal expansion|Coefficient of Thermal Expansion]]'' ([[thermodynamics]]) (dimensionless) - Relates the change in temperature to the change in a material's dimensions.
# ''[[Partition Coefficient]]'' (''K<sub>D</sub>'') ([[chemistry]]) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium. H2O is a coefficient
# ''[[Hall Effect|Hall coefficient]]'' (electrical physics) - Relates a magnetic field applied to an element to the voltage created, the amount of current and the element thickness. It is a characteristic of the material from which the conductor is made.
# ''[[Lift coefficient]]'' (''C<sub>L</sub>'' or ''C<sub>Z</sub>'') ([[Aerodynamics]]) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the plan-form area of the airfoil.
# ''[[Ballistic coefficient]]'' (BC) ([[Aerodynamics]]) (units of kg/m<sup>2</sup>) - A measure of a body's ability to overcome air resistance in flight. BC is a function of mass, diameter, and drag coefficient.
# ''[[Transmission coefficient (physics)|Transmission Coefficient]]'' ([[quantum mechanics]]) (dimensionless) - Represents the probability flux of a transmitted wave relative to that of an incident wave. It is often used to describe the probability of a particle [[Quantum tunnelling|tunnelling]] through a barrier.
# ''[[Damping|Damping Factor]]'' a.k.a. ''viscous damping coefficient'' (Physical Engineering) (units of newton-seconds per meter) - relates a damping force with the velocity of the object whose motion is being damped.
 
 
A coefficient is a number placed in front of a term in a [[chemical equation]] to indicate how many [[molecule]]s (or atoms)  take part in the reaction. For example, in the formula
:<math>2H_2 + O_2 \rarr 2H_2O</math>,
 
the number 2's in front of <math>H_2</math> and <math>H_2O</math> are [[Stoichiometry|stoichiometric coefficients]].
 
==See also==
*[[Degree of a polynomial]]
*[[Monic polynomial]]
 
==References==
*Sabah Al-hadad and C.H. Scott (1979) ''College Algebra with Applications'', page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0-87626-140-3 .
*Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) ''College Algebra'', 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0-534-01138-1 .
* Steven Schwartzman (1994) ''The Words of Mathematics: an etymological dictionary of mathematical terms used in English'', page 48, [[Mathematics Association of America]], ISBN 0-88385-511-9.
 
[[Category:Polynomials]]
[[Category:Mathematical terminology]]
[[Category:Algebra]]

Latest revision as of 14:53, 5 May 2014

It's going to suddenly make your ex boyfriend want you back. He will be in the best position as he doesn't have to be in a full blown committed relationship with you, but he still can see you and talk to you whenever he wants. This will also allow both of you to rethink and reevaluate what really happened why the break up occur. Please know that, of all the ways to get your ex back, this is the most important step. By being aware of his situation you can inspire trust right off the bat by not throwing yourself at your ex-boyfriend and rubbing it in his new girlfriend's face.

These improvements must be real instead of some superficial cover ups for you to trick your ex-lover into reconciling with you. She needs the justification of knowing that breaking things off made you want her more than ever before, as this verifies that she made the right decision and gives her the satisfaction of seeing you finally pay her some positive attention. Discover how you can learn powerful tricks that border on mind control and get your ex back quickly and easily. Accept that the relationship in its prior form is over. Keep In Touch with your ex boyfriend or ex girlfriend:.

He will be expecting a reaction from you, and that reaction is something he is counting on to boost his ego. The one thing that you need to know is that you can't allow yourself to lose all hope. " indicates that you're ready to extend forgiveness to that person. As soon as you call her try to keep control of the conversation by sounding vague and mysterious. It is therefore essential that you maintain a composed mind-set whenever trying to get your ex-girlfriend or boyfriend back.

s easier to accept; easier than if you ended on a bad note or worse still, there were lies or cheating involved. As contact between you and your ex boyfriend disappears entirely, he's suddenly struck with the idea of losing you forever. Relax into how good it feels not to be blindsided by this crap that was never true in the. First off, you need to know about some of the common mistakes girls make after a break up. The good news however, is that most rebound relationship fail within the first two months.

You can do some fun things together just like old days, or remember the good things happened between you and him. A thing funny, nevertheless romantic will work greatest in order to break the ice. While you shouldn't seem desperate, you have to let him know or perhaps the girl that you are attempting to try and fix the errors of the past. Word just may get back to your partner that you're out having fun without her. One of the most obvious methods of showing that you want to win back your boy or girl is by constantly telling him or her that you are still single.

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