|
|
| Line 1: |
Line 1: |
| {{two other uses|the resultant of polynomials|the result of adding two or more [[Vector (geometric)|vectors]]|Parallelogram rule|the musical phenomenon|Resultant tone}}
| | Of course plants include a variety of leafy greens, nuts, seeds, fruits, roots, and other wild vegetables. This diet program consists largely of meat, fruit, greens, nuts and seeds. When you store your supplements at home they should be kept refrigerated to maintain the health of the bacteria therefore making them more effective. The difference between Crossfit and other workouts is that Crossfit is designed not to specialize, but to cross train so that your body does not get accustomed to any one workout and is continually developing new skills and muscles. Mother Natures Diet Recipes macro-nutrition for Breakfast, Lunch and Dinner. <br><br>So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Ready-made paleo foods items are starting to find their way into conventional grocery stores. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. Following a low-carb diet dramatically reduced the spread of cancer, said Martin, who published his findings in the medical journal Cell. <br><br>And what's my Paleo Diet weight loss result so far. What the Paleo guides you thru is referred to as a back-to-basics mentality. In other words, when you have sufficient protein you will stop feeling hungry and your body will continually be working off of old fat cells. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. Varied cooking used in dessert, shakes, and drinks and baking. <br><br>Grass-fed items have less than one third the fat than grain-fed items of a similar cut. Foods had been valued by the energy and calories they will imparted. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. Lots of types of oil are also recommended, with the most common being the olive oil. With these advantages, who would not wish to have Paleo diet cookbooks. <br><br>Or at least the leftover carcass of a wolf-pack kill. I don't recall the exact number of recipes, but they do have several paleo diet cake recipes. Regardless, of the primitive place of shelter Caveman was constantly looking for food. The occurrence of diabetes, for example, is almost unknown to aboriginal people even now who naturally follow this sort of diet. Heart disease and stroke is the leading cause of death and disability in America and the prime reason behind this is the sedentary lifestyle and a diet high in calories and saturated fat. <br><br>There is scientific evidence that there are strong links between what you eat and health. This choice for those who want to lose weight or who just want to become healthier overall. Even though most of us are not going to be able to afford to switch to a diet that is loaded with the amount of fish we need, there are solutions available and the best on the market today is Omega Daily which provides all we need in a single capsule designed to be put to immediate use by our bodies. You know, the Mediterranean diet is considered to be one of the healthiest diets in the world today. There's a good reason for that, as most diets really do incorporate ingredients that are usually tasteless, and fill your life with constant test of willpower and boring calorie counting.<br><br>If you adored this article and also you would like to collect more info regarding paleo food list; [http://gritsandgroceries.info/ please click the following website], kindly visit our own website. |
| | |
| In [[mathematics]], the '''resultant''' of two [[polynomial]]s is a [[polynomial expression]] of their coefficients, which is equal to zero if and only if the polynomials have a common [[root of a function|root]] (possibly in a [[field extension]]), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called '''eliminant'''.{{sfn|Salmon|1885|loc=lesson VIII, p. 66}}
| |
| | |
| The '''resultant''' is widely used in [[number theory]], either directly or through the [[discriminant]], which is essentially the resultant of a polynomial and its derivative. The '''resultant''' of two polynomials with [[rational number|rational]] or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of [[computer algebra]], and is a built-in function of most [[computer algebra system]]s. It is used, among others, for [[cylindrical algebraic decomposition]], [[Symbolic integration|integration]] of [[rational function]]s and drawing of [[curve]]s defined by a [[Polynomial#Number of variables|bivariate]] [[polynomial equation]].
| |
| | |
| The '''resultant''' of ''n'' [[homogeneous polynomial]]s in ''n'' variables or '''multivariate resultant''', sometimes called '''Macaulay's resultant''', is a generalization of the usual resultant introduced by [[Francis Sowerby Macaulay|Macaulay]]. It is, with [[Gröbner basis|Gröbner bases]], one of the main tools of effective [[elimination theory]] (elimination theory on computers).
| |
| | |
| == Definition ==
| |
| | |
| For [[univariate]] [[monic polynomial]]s <var>P</var> and <var>Q</var> over a [[Field (mathematics)|field]] <var>k</var>, the resultant res(<var>P</var>,<var>Q</var>) is a [[polynomial function]] of their coefficients. It is defined as the [[Product (mathematics)|product]]
| |
| | |
| :<math>\prod_{(x,y):\,P(x)=Q(y)=0} (x-y)</math>
| |
| | |
| of the differences of their roots in an [[algebraic closure]] of <var>k</var>; in the case of multiple roots, the factors are repeated according to their multiplicities. It results that the number of factors is always the product of the degrees of ''P'' and ''Q''.
| |
| For non-monic polynomials with [[leading coefficient]]s <var>p</var> and <var>q</var>, respectively, the above product is multiplied by <var>p</var><sup>deg<var>Q</var></sup><var>q</var><sup>deg<var>P</var></sup>.
| |
| | |
| See the section on computation below, for a proof that res(<var>P</var>,<var>Q</var>) is a polynomial function of their coefficients.
| |
| | |
| == Properties ==
| |
| | |
| * The resultant is zero if and only if the two polynomials have a common root in an [[algebraically closed field]] containing the coefficients.
| |
| * Since the resultant is a [[polynomial]] with integer coefficients in terms of the coefficients of <var>P</var> and <var>Q</var>, it follows that
| |
| ** The resultant is well defined for polynomials over any [[commutative ring]].
| |
| ** If <var>h</var> is a [[homomorphism]] of the ring of the coefficients into another commutative ring, which preserve the degrees of <var>P</var> and <var>Q</var>, then the resultant of the image by <var>h</var> of <var>P</var> and <var>Q</var> is the image by <var>h</var> of the resultant of <var>P</var> and <var>Q</var>.
| |
| * If <var>P</var> and <var>Q</var> are two polynomials ''over a commutative ring <var>R</var>'', then there exist two polynomials <var>A</var> and <var>B</var> in the variable <var>X</var> over <var>R</var> such that <var>AP</var> + <var>BQ</var> = res(<var>P</var>, <var>Q</var>) (with the right hand side being interpreted as a constant polynomial). This result is a kind of substitute for [[Bézout's identity]] for polynomials over arbitrary commutative rings, where the usual version of Bézout's identity doesn't generally hold.
| |
| * The resultant of two polynomials with coefficients in an [[integral domain]] is null if and only if they have a [[Greatest common divisor of two polynomials|common divisor]] of positive degree.
| |
| * res(<var>P</var>,<var>Q</var>) = (-1)<sup>deg<var>P</var>deg<var>Q</var></sup>res(<var>Q</var>,<var>P</var>)
| |
| * res(<var>PR</var>,Q)=res(<var>P</var>,<var>Q</var>)res(<var>R</var>,<var>Q</var>)
| |
| * If <math> P'=P+RQ</math> and <math> \deg P'=\deg P</math>, then <math>\mathrm{res}(P',Q)=\mathrm{res}(P,Q)</math>.
| |
| * If <var>X</var>, <var>Y</var>, <var>P</var>, <var>Q</var> have the same degree and <var>X</var>=<var>a</var><sub>00</sub><var>P</var>+<var>a</var><sub>01</sub><var>Q</var>, <var>Y</var>=<var>a</var><sub>10</sub><var>P</var>+<var>a</var><sub>11</sub><var>Q</var>,
| |
| :then <math>\mathrm{res}(X,Y) = \det{\begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix}}^{\deg P} \cdot \mathrm{res}(P,Q)</math>
| |
| * <math> \mathrm{res}(P(-z),Q(z))=\mathrm{res}(Q(-z),P(z))</math>
| |
| | |
| ==Computation==
| |
| | |
| Since the resultant depends polynomially (with integer coefficients) on the roots of <var>P</var> and <var>Q</var>, and it is invariant with respect to permutations of each set of roots, it must be possible to calculate it using an (integer) polynomial formula on the coefficients of <var>P</var> and <var>Q</var>. See [[elementary symmetric polynomial]] for details.
| |
| | |
| More concretely, the resultant is the [[determinant]] of the [[Sylvester matrix]] (and of the [[Bézout matrix]]) associated to <var>P</var> and <var>Q</var>. This is the standard definition of the resultant over a commutative ring.
| |
| | |
| The above definition of the resultant can be rewritten as
| |
| ::<math>p^{\deg(Q)}\prod_{P(x)=0} Q(x),</math>
| |
| so it can be expressed polynomially in terms of the coefficients of <var>Q</var> for each fixed <var>P</var>. By the symmetry of the defining formula, the resultant is also a polynomial in the coefficients of <var>P</var> for each fixed <var>Q</var>. It follows that the resultant is a polynomial in the coefficients of <var>P</var> and <var>Q</var> jointly.
| |
| | |
| This expression remains unchanged if <var>Q</var> is replaced by the remainder <var>P</var> mod <var>Q</var> of the [[Polynomial greatest common divisor#Euclidean division|Euclidean division]] of <var>Q</var> by <var>P</var>.
| |
| If we set <var>P'</var> = <var>P</var> mod <var>Q</var>, then this idea can be continued by swapping the roles of <var>P'</var> and <var>Q</var>. However, <var>P'</var> has a set of roots different from that of <var>P</var>. This can be resolved by writing res(<var>P'</var>,<var>Q</var>) as a determinant again, where <var>P'</var> has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient <var>q</var> of <var>Q</var> appears: res(<var>P</var>,<var>Q</var>)=<var>q</var><sup>deg<var>P</var>-deg<var>P'</var></sup> res(<var>P'</var>,<var>Q</var>). Continuing this procedure ends up in a variant of the [[Polynomial greatest common divisor#Euclid's algorithm|Euclid's algorithm]].
| |
| | |
| This procedure needs a number of arithmetic operations on the coefficients that is of the order of product of the degrees. However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient.
| |
| | |
| The [[Polynomial greatest common divisor#Subresultant pseudo-remainder sequence|subresultant pseudo-remainder sequence]]s have been introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism of the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many [[prime number]]s, and then reconstruct the result with [[Chinese remainder theorem]].
| |
| | |
| ==Applications==
| |
| | |
| * If ''x'' and ''y'' are [[algebraic numbers]] such that <math>P(x)=Q(y)=0</math> (with degree of ''Q'' = ''n''), we see that <math>z=x+y</math> is a root of the resultant (in ''x'') of <math>P(x)</math> and <math>Q(z-x)</math> and that <math>t=xy</math> is a root of the resultant of <math>P(x)</math> and <math>x^nQ(t/x)</math> ; combined with the fact that <math>1/y</math> is a root of <math>y^nQ(1/y)</math>, this shows that the set of algebraic numbers is a field.
| |
| | |
| * The [[discriminant]] of a polynomial is the quotient by its leading coefficient of the resultant of the polynomial and its derivative.
| |
| | |
| * Resultants can be used in [[algebraic geometry]] to determine intersections. For example, let
| |
| ::<math>f(x,y)=0</math>
| |
| :and
| |
| ::<math>g(x,y)=0</math>
| |
| :define [[algebraic curve]]s in <math>\mathbb{A}^2_k</math>. If <math>f</math> and <math>g</math> are viewed as polynomials in <math>x</math> with coefficients in <math>k[y]</math>, then the resultant of <math>f</math> and <math>g</math> is a polynomial in <math>y</math> whose roots are the <math>y</math>-coordinates of the intersection of the curves and of the common asymptotes parallel to the <math>x</math> axis.
| |
| | |
| * In [[computer algebra]], the resultant is a tool that can be used to analyze modular images of the [[greatest common divisor]] of integer polynomials where the coefficients are taken modulo some prime number <math>p</math>. The resultant of two polynomials is frequently computed in the [[Daniel Lazard|Lazard]]–Rioboo–[[Barry Trager|Trager]] method of finding the [[integral]] of a ratio of polynomials.
| |
| | |
| * In [[Wavelet|wavelet theory]], the resultant is closely related to the determinant of the [[transfer matrix]] of a [[refinable function]].
| |
| | |
| == Generalizations and related concepts ==
| |
| | |
| The resultant is sometimes defined for two homogeneous polynomials in two variables, in which case it vanishes when the polynomials have a common non-zero solution, or equivalently when they have a common zero on the projective line. More generally, the '''multivariate resultant''' or '''[[Francis Sowerby Macaulay|Macaulay's]] resultant''' of ''n'' homogeneous polynomials in ''n'' variables is a polynomial in their coefficients that vanishes when they have a common non-zero solution, or equivalently when the ''n'' hypersurfaces corresponding to them have a common zero in ''n''–1 dimensional projective space. The multivariate resultant is, with [[Gröbner basis|Gröbner bases]], one of the main tools of effective [[elimination theory]] (elimination theory on computers).
| |
| | |
| == See also ==
| |
| *[[Elimination theory]]
| |
| *[[Subresultant]]
| |
| | |
| == Notes ==
| |
| | |
| {{Reflist}}
| |
| | |
| ==References==
| |
| | |
| *{{Citation | last1=Gelfand | first1=I. M. | last2=Kapranov | first2=M.M. | last3=Zelevinsky | first3=A.V. | title=Discriminants, resultants, and multidimensional determinants | publisher=Boston: Birkhäuser | isbn=978-0-8176-3660-9 | year=1994}}
| |
| *{{citation|first=F. S.|last= MacAulay|title=Some Formulæ in Elimination|journal=Proc. London Math. Soc. |year=1902|volume=35|pages= 3–27|doi=10.1112/plms/s1-35.1.3 }}
| |
| *{{Citation | last1=Salmon | first1=George | title=Lessons introductory to the modern higher algebra | origyear=1859 | url=http://archive.org/details/salmonalgebra00salmrich | publisher=Dublin, Hodges, Figgis, and Co. | edition=4th | isbn=978-0-8284-0150-0 | year=1885}}
| |
| | |
| ==External links==
| |
| * {{MathWorld |urlname=Resultant |title=Resultant}}
| |
| | |
| [[Category:Polynomials]]
| |
| [[Category:Determinants]]
| |
| [[Category:Computer algebra]]
| |
Of course plants include a variety of leafy greens, nuts, seeds, fruits, roots, and other wild vegetables. This diet program consists largely of meat, fruit, greens, nuts and seeds. When you store your supplements at home they should be kept refrigerated to maintain the health of the bacteria therefore making them more effective. The difference between Crossfit and other workouts is that Crossfit is designed not to specialize, but to cross train so that your body does not get accustomed to any one workout and is continually developing new skills and muscles. Mother Natures Diet Recipes macro-nutrition for Breakfast, Lunch and Dinner.
So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Ready-made paleo foods items are starting to find their way into conventional grocery stores. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. Following a low-carb diet dramatically reduced the spread of cancer, said Martin, who published his findings in the medical journal Cell.
And what's my Paleo Diet weight loss result so far. What the Paleo guides you thru is referred to as a back-to-basics mentality. In other words, when you have sufficient protein you will stop feeling hungry and your body will continually be working off of old fat cells. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. Varied cooking used in dessert, shakes, and drinks and baking.
Grass-fed items have less than one third the fat than grain-fed items of a similar cut. Foods had been valued by the energy and calories they will imparted. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. Lots of types of oil are also recommended, with the most common being the olive oil. With these advantages, who would not wish to have Paleo diet cookbooks.
Or at least the leftover carcass of a wolf-pack kill. I don't recall the exact number of recipes, but they do have several paleo diet cake recipes. Regardless, of the primitive place of shelter Caveman was constantly looking for food. The occurrence of diabetes, for example, is almost unknown to aboriginal people even now who naturally follow this sort of diet. Heart disease and stroke is the leading cause of death and disability in America and the prime reason behind this is the sedentary lifestyle and a diet high in calories and saturated fat.
There is scientific evidence that there are strong links between what you eat and health. This choice for those who want to lose weight or who just want to become healthier overall. Even though most of us are not going to be able to afford to switch to a diet that is loaded with the amount of fish we need, there are solutions available and the best on the market today is Omega Daily which provides all we need in a single capsule designed to be put to immediate use by our bodies. You know, the Mediterranean diet is considered to be one of the healthiest diets in the world today. There's a good reason for that, as most diets really do incorporate ingredients that are usually tasteless, and fill your life with constant test of willpower and boring calorie counting.
If you adored this article and also you would like to collect more info regarding paleo food list; please click the following website, kindly visit our own website.