Algebraic curve: Difference between revisions

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{{Other uses|Definition (disambiguation)}}
Anyone that has been on a diet utilizes 'calorie math': in the event you burn more calories than we eat, we will lose weight. But the math will get a small complicated. There are bodily plus mental challenges to figuring out the proper number of calories to take in to lose weight plus maintain which fat loss.<br><br>Calculating your BMR with a [http://safedietplans.com/bmr-calculator bmr calculator] is an significant step. This informs you how countless calories you burn a day by simply existing. Breathing, hearts beating, kidneys functioning and everything the bodies do takes calories. Knowing how numerous calories these functions utilize is significant knowledge inside you weight reduction system.<br><br>If the basal metabolic rate is 2,300 calories a day, 1,200 calories of food consumption is not enough food for you to function properly. Doing this for too lengthy usually damage your metabolism! Having a healthy body at a healthy fat plus acceptable body fat percentage are the most important points. You need a healthy, lean body which will provide we the greatest chance to live healthy for a lengthy time.<br><br>The struggle is frequently harder than it must be because you don't always have a obvious picture of what is happening in the bodies. Fat cells act only like a piggy bank, in which they sit all about each blood vessel plus capillary inside our bodies, and when they "see" excess food molecules coming past, they just snatch them up plus deposit them inside their small round sac. As the millions of sacs of adipose tissue expand, our bodies grow to house the heighten of fat.<br><br>Body Surface Area: Your height and fat lead a lot in determining bmr. The greater is the body surface area, the high is the BMR. Thin, tall persons have a high BMR.<br><br>Calories In is easy. This is merely the number of calories you eat and drink every day, regardless where they come from. There are many methods to look these up. Fitday.com and Calorie-Count.com are two great internet resources. You do need to track your Calories In. Fitday.com has tools for this, or you are able to make an Excel spreadsheet, or write them in a notebook. However you do it, keep track of the Calories In daily. As a side benefit, recognizing we will have to write down that piece of cake assists motivate we to not eat it.<br><br>There are 3 ways to create this deficit: diet, exercise, or a combination. If you combine methods you have a better chance of sticking with it. Just think, instead of completely cutting out which afternoon snack, or risking injury by working out too hard, you might just have a lighter snack and take the stairs. Its simpler to make smaller changes that usually add up to 500 calories a day.
 
In [[mathematics]], an expression is '''well-defined''' if it is unambiguous and its objects are independent of their representative. More simply, it means that a mathematical statement is sensible and definite. In particular, a function is well-defined if it gives the same result when the form (the way in which it is presented) but not the value of an input is changed. The term well-defined is also used to indicate whether a logical statement is unambiguous, and a solution to a [[partial differential equation]] is said to be well-defined if it is [[continuity (mathematics)|continuous]] on the boundary.<ref name="MathWorld" />
 
==Well-defined functions==
 
In [[mathematics]], a function is well-defined if it gives the same result when the form but not the value of the input is changed.  For example, a function on the [[real number]]s must give the same output for 0.5 as it does for 1/2, because in the real number system 0.5 = 1/2.  An example of a "function" that is not well-defined is "''f''(''x'') = the first digit that appears in ''x''".  For this function, ''f''(0.5) = 0 but ''f''(1/2) = 1. A "function" such as this would not be considered a function at all, since a function must have exactly one output for a given input.<ref> Joseph J. Rotman, ''The Theory of Groups: an Introduction'', p.287 "...a function is "single-valued," or, as we prefer to say ... a function is ''well defined''.", Allyn and Bacon, 1965.</ref>  
 
In [[group theory]], the term well-defined is often used when dealing with [[coset]]s, where a function on a [[quotient group]] may be defined in terms of a [[coset representative]].  Then the output of the function must be independent of which coset representative is chosen.  For example, consider the group of [[modulo arithmetic|integers modulo 2]]. Since 4 and 6 are congruent modulo 2, a function defined on the integers modulo 2 must give the same output when the input is 6 that it gives when the input is 4. 
 
A function that is not well-defined is not the same as a function that is [[undefined (mathematics)|undefined]].  For example, if ''f''(''x'') = 1/''x'', then ''f''(0) is undefined, but this has nothing to do with the question of whether ''f''(''x'') = 1/''x'' is well-defined.  It is; 0 is simply not in the domain of the function.
===Operations===
In particular, the term well-defined is used with respect to (binary) [[operation (mathematics)| operation]]s on cosets. In this case one can view the operation as a function of two variables and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some ''n'' can be defined naturally in terms of integer addition.  
:<math>[a]\oplus[b] = [a+b]</math>
The fact that this is well-defined follows from the fact that we can write any representative of <math>[a]</math> as <math>a+kn</math>, where k is an integer. Therefore,  
:<math>[a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b] = [a]\oplus[b]</math>
and similarly for any representative of <math>[b]</math>.
 
==Well-defined notation==
For real numbers, the product <math>a \times b \times c</math> is unambiguous because <math>(ab)c= a(bc)</math>. <ref name="MathWorld">{{cite web | last = Weisstein | first = Eric W. | title = Well-Defined | publisher = From MathWorld--A Wolfram Web Resource | url=http://mathworld.wolfram.com/Well-Defined.html | accessdate = 2 January 2013 }}</ref> In this case this notation is said to be ''well-defined''. However, if the [[operation (mathematics)|operation]] (here <math>\times</math>) did not have this property, which is known as [[associativity]], then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined. The [[subtraction]] operation, <math>-</math>, is not associative, for instance. However, the notation <math>a-b-c</math> is well-defined under the convention that the <math>-</math> operation is understood as addition of the opposite, thus <math>a-b-c</math> is the same as <math>a + -b + -c</math>. Division is also non-associative. However, <math>a/b/c</math> does not have an unambiguous conventional interpretation, so this expression is ill-defined.
 
==See also==
* [[Definitionism]]
* [[Existence]]
* [[Uniqueness]]
* [[Uniqueness quantification]]
* [[Undefined_(mathematics)|Undefined]]
 
==References==
===Notes===
{{Reflist}}
===Books===
* ''Contemporary Abstract Algebra'', Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, ISBN 0-618-51471-6.
 
{{DEFAULTSORT:Well-Defined}}
[[Category:Definition]]
[[Category:Mathematical terminology]]

Latest revision as of 17:21, 13 December 2014

Anyone that has been on a diet utilizes 'calorie math': in the event you burn more calories than we eat, we will lose weight. But the math will get a small complicated. There are bodily plus mental challenges to figuring out the proper number of calories to take in to lose weight plus maintain which fat loss.

Calculating your BMR with a bmr calculator is an significant step. This informs you how countless calories you burn a day by simply existing. Breathing, hearts beating, kidneys functioning and everything the bodies do takes calories. Knowing how numerous calories these functions utilize is significant knowledge inside you weight reduction system.

If the basal metabolic rate is 2,300 calories a day, 1,200 calories of food consumption is not enough food for you to function properly. Doing this for too lengthy usually damage your metabolism! Having a healthy body at a healthy fat plus acceptable body fat percentage are the most important points. You need a healthy, lean body which will provide we the greatest chance to live healthy for a lengthy time.

The struggle is frequently harder than it must be because you don't always have a obvious picture of what is happening in the bodies. Fat cells act only like a piggy bank, in which they sit all about each blood vessel plus capillary inside our bodies, and when they "see" excess food molecules coming past, they just snatch them up plus deposit them inside their small round sac. As the millions of sacs of adipose tissue expand, our bodies grow to house the heighten of fat.

Body Surface Area: Your height and fat lead a lot in determining bmr. The greater is the body surface area, the high is the BMR. Thin, tall persons have a high BMR.

Calories In is easy. This is merely the number of calories you eat and drink every day, regardless where they come from. There are many methods to look these up. Fitday.com and Calorie-Count.com are two great internet resources. You do need to track your Calories In. Fitday.com has tools for this, or you are able to make an Excel spreadsheet, or write them in a notebook. However you do it, keep track of the Calories In daily. As a side benefit, recognizing we will have to write down that piece of cake assists motivate we to not eat it.

There are 3 ways to create this deficit: diet, exercise, or a combination. If you combine methods you have a better chance of sticking with it. Just think, instead of completely cutting out which afternoon snack, or risking injury by working out too hard, you might just have a lighter snack and take the stairs. Its simpler to make smaller changes that usually add up to 500 calories a day.

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