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From Wikipedia

Homogeneous (chemistry)

A substance that is uniform in composition is a definition of homogeneous (IPA: /həmɔ�dʒɪnʌs, ho�modʒi�niʌs/) in Chemistry. This is in contrast to a substance that is heterogeneous. The definition of homogeneous strongly depends on the context used. In Chemistry, a homogeneous suspension of material means that when dividing the volume in half, the same amount of material is suspended in both halves of the substance. However, it might be possible to see the particles under a microscope. In Chemistry, another homogeneous substance is air. It is equally suspended, and the particles and gases and liquids cannot be analyzed separately or pulled apart.

Homogeneity of mixtures

In Chemistry, some mixtures are homogeneous. In other words, mixtures have the same proportions throughout a given sample or multiple samples of different proportion to create a consistent mixture. However, two homogeneous mixtures of the same pair of substances may differ widely from each other and can be homogenized to make a constant. Mixtures can be characterized by being separable by mechanical means e.g. heat, filtration, gravitational sorting, etc.

Solutions

A solution is a special type of homogeneous mixture. Solutions are homogeneous because, the ratio of solute to solvent remains the same throughout the solution even if homogenized with multiple sources, and stable because, the solute will not settle out, no matter how long the solution sits, and it cannot be removed by a filter or a centrifuge. This type of mixture is very stable, i.e., its particles do not settle, or separate. As homogeneous mixture, a solution has one phase (liquid) although the solute and solvent can vary: for example, salt water. In chemistry, a mixture is a substance containing two or more elements or compounds that are not chemically bound to each other but retain their own chemical and physical identities; - a substance which has two or more constituent chemical substances. Mixtures, in the broader sense, are two or more substances physically in the same place, but these are not chemically combined, and therefore ratios are not necessarily considered.


Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of a point, even those at infinity, can be represented using finite coordinates. Often formulas involving homogeneous coordinates are simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general, one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point on the projective plane.

Introduction

The projective plane can be thought of as the Euclidean plane with additional points, so called points at infinity, added. There is a point at infinity for each direction, informally defined as the limit of a point that moves in that direction away from a fixed point. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. A given point on the Euclidean plane is identified with two ratios , so the point corresponds to the triple where . Such a triple is a set of homogeneous coordinates for the point . Note that, since ratios are used, multiplying the three homogeneous coordinates by a common, non-zero factor does not change the point represented – unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.

The equation of a line through the point may be written where l and m are not both 0. In parametric form this can be written . Let Z=1/t, so the coordinates of a point on the line may be written . In homogeneous coordinates this becomes . In the limit as t approaches infinity, in other words as the point moves away from , Z becomes 0 and the homogeneous coordinates of the point become . So are defined as homogeneous coordinates of the point at infinity corresponding to the direction of the line .

To summarize:

  • Any point in the projective plane is represented by a triple , called the homogeneous coordinates of the point, where X, Y and Z are not all 0.
  • The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
  • Conversely, two sets of homogeneous coordinates represent the same point only if one is obtained from the other by multiplying by a common factor.
  • When Z is not 0 the point represented is the point in the Euclidean plane.
  • When Z is 0 the point represented is a point at infinity.

Note that the triple is omitted and does not represent any point. The origin is represented by .

Notation

Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. The use of colons instead of commas, for example (x:y:z) instead of , emphasizes that the coordinates are to be considered ratios. Brackets, as in emphasize that multiple sets of coordinates are associated with a single point. Some authors use a combination of colons and brackets, as in [x:y:z].

Homogeneity

Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say , does not determine a function defined on points as with Cartesian coordinates. But a condition defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is k so that

f(\lambda x, \lambda y, \lambda z) = \lambda^k f(x,y,z).\,

If a set of coordinates represent the same point as then it can be written for some non-zero value of λ. Then

f(x,y,z)=0 \iff f(\lambda x, \lambda y, \lambda z) = \lambda^k f(x,y,z)=0.\,

A polynomial of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by zk, in other words by defining

f(x, y, z)=z^k g(x/z, y/z).\,

The resulting function f is a polynomial so it makes sense to extend its domain to triples where . The process can be reversed by setting , or

g(x, y)=f(x, y, 1).\,

The equation can then be thought of as the homogeneous form of and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line is

Other dimensions

The discussions in the preceding sections apply analogously to projective spaces other than the plane. So the points on the projective line may be represented by pairs of coordinates , not both zero. In this case point at infinity in this case is . Similarly the points in projective n-space are represented by (n + 1)-tuples.

Alternate definitions

Another definition of project space can be given in terms of equivalence classes. For non-zero element of R3, define to mean there is a non-zero λ so that . Then ~ is an equivalence relation and the projective plane can be defined as the equivalence classes of R3 − {0}. If is one of elements of the equivalence class p then these are taken to be homogeneous coordinates of p.

Lines in this space are defined to be sets of solutions of equations of the form where not all of a, b and c are zero. The condition depends only on the equivalence class of so the equation defines a set of points in the projective line. The mapping defines and inclusion from the Euclidean plane to the projective plane and the complement of the image is the set of points with z=0. This


From Yahoo Answers

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Answers:By definition a solution is a homogeneous mixture.

Question:can u please describe these 4 defeinitions in a way i can understand because i read a lot of definitions, but i still get confused when im doing word problems related to them. So please tell me. Thank You

Answers:HOMOGENEOUS (SAME) Homogeneous mixtures are mixtures that are the same all the way throughout, such as salt water or lemonade. No matter what size a sample is, it will be exactly the same as all other samples of the same ingredients. HETERGENEOUS (DIFFERENT) Heterogeneous mixtures are mixtures with differences in different areas of itself. Chex Mix or salad demonstrate this well. Although a salad may have tomatoes, lettuce, onions, dressing, cucumbers, carrots, etc., samples will not be of the exact composure each time the original mixture is split.

Question:explain and give examples please and name whether or not these are a heterogeneous mixture or a solution maple syrup-- [i got solution] seawater-- [i got solution] melted rocky road ice cream-- [i got heterogeneous mixture] but i'm not sure. confused, but i am. i did my homework, but i said i wasn't sure if i got it right. i kind of know the definition of these words, but it's hard to identify it. but thanks

Answers:Yes, you are correct. Homogeneous appears the same throughout. Hetro: you can see bits and pieces in it.

Question:Is air a solution or solvent because of the water vapours. Or a mixture of compounds. What ever it is please explain why.

Answers:You know that a solution is a homogeneous mixture and that in a homogeneous mixture, the solute is the substance that gets dissolved while the solvent is the substance that does the dissolving (of course you knew that). Solutions may involve any of the states of matter - we can have solutions of solids in liquids (what we usually think of), solids in gases (cigarette smoke), solids in solids (alloys), etc. Sometimes the solute can be more than the solvent, but usually the solute is the one that is present in smaller amount. Mixtures involving solids and liquids are interesting in that some of one substance can be soluble in another, but as the amount increases, we can reach a limit of solubility. Gases are not like that. Since by definition there is very little interaction between the particles of a gas, all gases are soluble in all other gases (they mix in all proportions or are miscible with each other). OK, enough generalities - air is a homogeneous mixture (solution) of nitrogen (N2) containing about 80% nitrogen, 19% oxygen (O2), 1% argon (Ar) and traces of other gases (including water vapor). Since it is homogeneous, the percentage of the usual gases is the same at the bottom of Death Valley as it is on the top of Mount Everest. Because of mixing problems, here around Los Angeles, we have some more (cough, cough) compounds dissolved in the mixture than they have in Des Moines, Iowa, but it will all be spread or precipitated eventually.

From Youtube

A Solution :The definition of the word solution is a homogeneous mixture of two or more substances. I believe that to be the answer to the problems we face today. Finding a solution becomes the solution to today's problems.