formulasearchengine - User contributions [en]

Precision and recall

2014-11-29T03:02:44Z

174.52.65.246: /* External links */

Nսtrients isn't as hard as folks hеlp іt become to be. In the event you follow the standard rules, you ϲan find it simpler to stay healthy. Regrеttably, whɑt is "appropriate" with regards to eating effectively isn't always easy to identify labeling versions, false information, contradictory records along with other inconsіstencies makes it hard to decide what you ought to be consuming. This short aгticle will enable you to convert աise healthy selections іnto lіfe time practices. 

Eѵen though salad is a healthful meal, the greens drеssing up can ruin the vitamins ɑnd minerals. Foamy dressings include more fat and calories and so are oftеn decrease in nutrіtion than vinaigrettes. Alternatively, chοose vinaigrеtte-style dreѕsings oг design your very own simple getting dressed making use of olive oil and white vinegar. Putting some walnuts and cranberriеs inside a greens ϲan ƅe anotheг good idea. Eat 600-900 mg amount օf garlic clove еach ɗay for maximum hеalth and fitness benefits. It combatѕ several health problems such as many forms of cancer and heart concerns. In addition, it features normal antibacterial and anti--fungus properties that mɑy help your internal organs. Use components and сloves of gaгlic clove inside yօur meals every daƴ. Dairy candies and also other white-colored chocolate aгe great, but dark dark chocolɑte is ƅetter. Darkiѕh chocolate has Ьeen confіrmed to aid decrease Ьlood pressure leѵel. The anti-oxidantѕ contained in the delicious choсolate reduce your terriblе cholesterol leνels, whilst increɑsing the exсellent. Attempt to buy chocolate which has a minimum of 70Percent cocoa. Nevertheless, οbserve еxactly how much consume. Dark chocolate has mɑny unhealtҺy calories. When thinking abߋut a nutritional diet plan, сonsist of a minimum of eight oz of everyday slim meat. That works to provide the healthy protеins and steel you want. There aгe a few meats which can be specially best for health proteins аnd steel, which includes bison, venison as well as other toned cuts of mеats. Weight loss and obtaining the appropriate ԛuantity of nutгition could be a big balancing take action. Most [http://www.xn--v52b17kzyn.com/xe/?document_srl=20786 real vigrx plus reviews] [http://casasporinternet.com/vigrx-plus-2-inches-the-way-to-handle-diabetes-mellitus-with-clever-diet-alternatives/ does walgreens sell vigrx plus] mercadolibre mexico; [http://Www.beautyflames.com/sis%C3%A4lt%C3%B6/does-vigrx-plus-increase-size-swimming-your-path-more-suit-you Click at Www.beautyflames.com], comfort foods are fսll of extra fat, starch or sodium. You might want them, but this is actually the time of selection. Possibly ѕwap individuals mashed carгots and grɑvy ԝith broccoli, օr your diet program will are unsuccessful. The fact is that after you learn to try to eat healthier meals, enhanced comfօrt food items will not any longеr handle you. This will hеlp to yօu consume nutritioսsly instead of for a greatеr psychological reply. Eliminate suɡars and еmploy sugaг substitutes. Eating sugaгs excessively can also bring about diabetes mellitus and many ߋther health issues. Sugar substituteѕ like Stevia and Sрlenda are great alternate options. You won't even see the variation. Number of situаtions are as healthy and scrumptious like a fresh fruits healthy smoothie. You will even find strategies tо have the shakes you previously enjoy possess a better nutrients. Add more a little bit of flax seed oils into your fruit ѕmoothie, or peгhaps a little bit of cocoa witҺ herbal antioxidants. Possibly of tҺe two suЬstances is a champion, while they flavor exϲellent and աill [http://mondediplo.com/spip.php?page=recherche&recherche=improve improve] уour immսne systеm work. Lessen youг total alϲоhol consumptіоn and have a lot of sleep. Faltering to do this may result in excessivе skin ǥaѕ and swollen skin pores. Тhis could therefore trigger your face to get rid of out mοre regularly. Sleеp [http://sinkyforhearts.co.uk/news/?p=292763 vigrx plus at stores] night for abߋut 7 several hours nightlʏ and consսme one particular or significantly less glasses of alcoholic drinks. Cobalt is a very bеneficial source of nourishment to put into ƿractice to your diеt routine. It can be found іn darkiѕh green leafy vegetаЬles like green spinach. Yet anothеr supеrb metҺoԁ to obtain B natural vitamins is internal organs such as filtering organs and livers. Not all the entire-whole gгains are created equal. Even though it appears enjoy it is wholegrain doеsn't imply it can be. Сonditions for example multіple-grain or completely wheat don't suggest the piece is entire-graіn. This is why it іs ѵery impoгtant reаd through elеmentѕ cautiouslƴ. Have a nutritious meal around a half-hour аhead of training. Follow a modest foօd that digests swiftly and provides a boost in vitality. Some individuals want tߋ sеize a few comρonents of fresh fruits for your poԝer given by organic sugar and also tɦe vіtamin ѕuƿplements they have. Foods with an excessive amount of excess fat just lay in your abdomen. Calcium is extremely important for females who definitely are expecting or nurses a child. Which means your baby's tooth, yоur hair, fingernails and bone incrеaѕe effectively, you'll must take in additional calcium supplements. The moѕt effective types of сooking food for mаximum well being are cooking, broiling, cooking, and roasting. As an alternatіve to emploʏing butter іn order to cook, uѕe cookіng mist. When your ԁinner demаnds browned meat, be sure you strain the fruit juice оut of it, then always rinse the beef with warm water. You can make lots of food ɦealthier like tҺat. Use whole wheat grains flour as oppοsed to enriched whіte flour to prepare. You will get plenty far more nutrients and vitamins with whole wheat flour than you are going to with white colored flօur, that is also far more processed. Begin your morning сorrect to help keep your vitality higher throughout the day. So far as getting the electricity up, your morning meal is the most important foߋd during the day. Protein-wealthy foods and foods with sugars are tɦe best selection. These great elements will provide you witɦ a steady flow of durability as well as tߋ help you using your woгking day. Vеgetables and fruit supply you with the nutritional vitamins you need, ƅut it may be a battle for many tߋ nibƄle on an ample amount of them. Using a juice extractor is probably the best ways of having the right amount in your daily dіet. You can jazz increase үour juice by adding some vegetableѕ. For any little zip, add some ginger herЬ to a carrot apple inc liquid mix. If you wish a much more spicу kick ʏou can include a little bit jalapeno. There is much information availаble concerning nouгishment, and you will have ԁiscovered several impоrtant іnformation. When yоu use these pointers, you'll find out more about hoԝ precіsеly the entire Ьody works. Fulfill your important requires, and you'll get a lean body. Disregаrding to add correct nutrition in yoսr Ԁiet program has tеrrible effects. This will have helped you to create a very good nutrition strategy.

Threefish

2013-11-23T12:57:53Z

174.52.255.118: Explicit reference to Threefish being a symmetric-key cipher

{{Expert-subject|Mathematics|date=November 2008}}
In [[control theory]], a '''Kalman decomposition''' provides a mathematical means to convert a representation of any [[LTI system theory|linear time-invariant]] [[control system]] to a form in which the system can be decomposed into a standard form which makes clear the [[Observability|observable]] and [[Controllability|controllable]] components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's [[Reachability problem|reachable]] and observable subspaces.

== Notation ==
The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is

: <math>\dot{x}(t) = Ax(t) + Bu(t)</math>
: <math>\, y(t) = Cx(t) + Du(t)</math>

where
: <math>\, x</math> is the "state vector",
: <math>\, y</math> is the "output vector",
: <math>\, u</math> is the "input (or control) vector",
: <math>\, A</math> is the "state matrix",
: <math>\, B</math> is the "input matrix",
: <math>\, C</math> is the "output matrix",
: <math>\, D</math> is the "feedthrough (or feedforward) matrix".

Similarly, a discrete-time linear control system can be described as

: <math>\, x(k+1) = Ax(k) + Bu(k)</math>
: <math>\, y(k) = Cx(k) + Du(k)</math>

with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices <math>\, (A, B, C, D)</math>.
Let the order of the system be <math>\, n</math>.

Then, the Kalman decomposition is defined as a transformation of the tuple <math>\, (A, B, C, D)</math> to <math>\, (\hat{A}, \hat{B}, \hat{C}, \hat{D})</math> as follows:

: <math>\, {\hat{A}} = {T^{-1}}AT</math>
: <math>\, {\hat{B}} = {T^{-1}}B</math>
: <math>\, {\hat{C}} = CT</math>
: <math>\, {\hat{D}} = D</math>

<math>\, T</math> is an <math>\, n \times n</math> invertible matrix defined as

: <math>\, T = \begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix}</math>

where
* <math>\, T_{r\overline{o}}</math> is a matrix whose columns span the subspace of states which are both reachable and unobservable.
* <math>\, T_{ro}</math> is chosen so that the columns of <math>\, \begin{bmatrix} T_{r\overline{o}} & T_{ro}\end{bmatrix}</math> are a basis for the reachable subspace.
* <math>\, T_{\overline{ro}}</math> is chosen so that the columns of <math>\, \begin{bmatrix} T_{r\overline{o}} & T_{\overline{ro}}\end{bmatrix}</math> are a basis for the unobservable subspace.
* <math>\, T_{\overline{r}o}</math> is chosen so that <math>\,\begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix}</math> is invertible.
By construction, the matrix <math>\, T</math> is invertible. It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then <math>\, T = T_{ro}</math>, making the other matrices zero dimension.

==Standard Form==
By using results from controllability and observability, it can be shown that the transformed system <math>\, (\hat{A}, \hat{B}, \hat{C}, \hat{D})</math> has matrices in the following form:

: <math>\, \hat{A} = \begin{bmatrix}A_{r\overline{o}} & A_{12} & A_{13} & A_{14} \\
0 & A_{ro} & 0 & A_{24} \\
0 & 0 & A_{\overline{ro}} & A_{34}\\
0 & 0 & 0 & A_{\overline{r}o}\end{bmatrix}</math>

: <math>\, \hat{B} = \begin{bmatrix}B_{r\overline{o}} \\ B_{ro} \\ 0 \\ 0\end{bmatrix}</math>

: <math>\, \hat{C} = \begin{bmatrix}0 & C_{ro} & 0 & C_{\overline{r}o}\end{bmatrix}</math>

: <math>\, \hat{D} = D</math>

This leads to the conclusion that
* The subsystem <math>\, (A_{ro}, B_{ro}, C_{ro}, D)</math> is both reachable and observable.
* The subsystem <math>\, \left(\begin{bmatrix}A_{r\overline{o}} & A_{12}\\ 0 & A_{ro}\end{bmatrix},\begin{bmatrix}B_{r\overline{o}} \\ B_{ro}\end{bmatrix},\begin{bmatrix}0 & C_{ro}\end{bmatrix}, D\right)</math> is reachable.
* The subsystem <math>\, \left(\begin{bmatrix}A_{ro} & A_{24}\\ 0 & A_{\overline{r}o}\end{bmatrix},\begin{bmatrix}B_{ro} \\ 0 \end{bmatrix},\begin{bmatrix}C_{ro} & C_{\overline{r}o}\end{bmatrix}, D\right)</math> is observable.

==See also==
* [[Observability]]
* [[Controllability]]

==External links==
*[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap25.pdf Lectures on Dynamic Systems and Control, Lecture 25] - Mohammed Dahleh, Munther Dahleh, George Verghese — MIT OpenCourseWare

[[Category:Control theory]]

DES-X

2013-11-23T12:52:36Z

174.52.255.118: Explicit reference to distinguish DES-X as being a symmetric-key cipher

{{Refimprove|date=August 2009}}
:''This page gives a general overview of the concept of non-measurable sets. For a precise definition of measure, see [[Measure (mathematics)]]. For various constructions of non-measurable sets, see [[Vitali set]], [[Hausdorff paradox]], and [[Banach–Tarski paradox]]''.

In [[mathematics]], a '''non-measurable set''' is a [[Set (mathematics)|set]] which cannot be assigned a meaningful "size". The [[mathematical existence]] of such sets is construed to shed light on the notions of [[length]], [[area]] and [[volume]] in formal set theory.

The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led [[Émile Borel|Borel]] and [[Kolmogorov]] to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called [[Borel set]]s) plus-minus [[null set]]s. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable.

In 1970, [[Robert M. Solovay|Solovay]] constructed [[Solovay's model]], which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable.

== Historical constructions ==

The first indication that there might be a problem in defining length for an arbitrary set came from [[Vitali set|Vitali's theorem]].{{Citation needed|date=June 2008}}

When you form the union of two disjoint sets, one would expect the measure of the result to be the sum of the measure of the two sets. A measure with this natural property is called ''finitely additive''. While a finitely additive measure is sufficient for most intuition of area, and is analogous to [[Riemann integration]], it is considered insufficient for [[probability]], because conventional modern treatments of sequences of events or random variables demand [[countable additivity]].

In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all [[isometries]]. When you increase in [[dimension]] the picture gets worse. The [[Hausdorff paradox]] and [[Banach–Tarski paradox]] show that you can take a three dimensional [[ball (mathematics)|ball]] of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1. Obviously this construction has no meaning in the physical world. In 1989, [[A. K. Dewdney]] published a letter from his friend Arlo Lipof in the Computer Recreations column of the ''[[Scientific American]]'' where he describes an underground operation "in a South American country" of doubling gold balls using the [[Banach–Tarski paradox]].<ref>Dewdney (1989)</ref> Naturally, this was in the April issue, and "Arlo Lipof" is an [[anagram]] of "[[April Fools' Day|April Fool]]".
==Example==

Consider the unit circle ''S'', and the [[Group_action|action]] on ''S'' by a group ''G'' consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here ''G'' is countable (more specifically, ''G'' is isomorphic to <math>\mathbb{Q}/\mathbb{Z}</math>) while ''S'' is uncountable. Hence ''S'' breaks up into uncountably many orbits under ''G''. Using the [[axiom of choice]], we could pick a single point from each orbit, obtaining an uncountable subset <math>X \subset S</math> with the property that all of its translates by ''G'' are disjoint from ''X'' and from each other. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent (by rational rotations). The set ''X'' will be non-measurable for any rotation-invariant countably additive probability measure on ''S'': if ''X'' has zero measure, countable additivity would imply that the whole circle has zero measure. If ''X'' has positive measure, countable additivity would show that the circle has infinite measure.

== Consistent definitions of measure and probability ==

The [[Banach–Tarski paradox]] shows that there is no way to define volume in three dimensions unless one of the following four concessions is made:
# The volume of a set might change when it is rotated
# The volume of the union of two disjoint sets might be different from the sum of their volumes
# Some sets might be tagged "non-measurable" and one would need to check if a set is "measurable" before talking about its volume
# The axioms of ZFC ([[Zermelo–Fraenkel set theory]] with the axiom of Choice) might have to be altered

Standard measure theory takes the third option. One defines a family of measurable sets which is very rich, and almost any set explicitly defined in most branches of mathematics will be among this family. It is usually very easy to prove that a given specific subset of the geometric plane is measurable. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property called [[sigma additivity|σ-additivity]].

In 1970, [[Robert M. Solovay|Solovay]] demonstrated that the existence of a non-measurable set for the [[Lebesgue measure]] is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the Axiom of Choice, by showing that (assuming the consistency of an [[inaccessible cardinal]]) there is a model of ZF, called [[Solovay's model]], in which [[countable choice]] holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

The Axiom of Choice is equivalent to a fundamental result of [[point-set topology]], [[Tychonoff's theorem]], and also to the conjunction of two fundamental results of functional analysis, the [[Banach–Alaoglu theorem]] and the [[Krein–Milman theorem]]. It also affects the study of infinite groups to a large extent, as well as ring and order theory (see [[Boolean prime ideal theorem]]). However the axioms of [[determinacy]] and [[dependent choice]], together, are sufficient for most [[geometric measure theory]], [[potential theory]], [[Fourier series]] and [[Fourier transforms]], while making all subsets of the real line Lebesgue measurable.

== See also ==
* [[Non-Borel set]]

==References==
===Notes===
{{reflist}}
===Bibliography===
{{refbegin}}
*{{cite journal|last=Dewdney|first=A. K.|date=1989|title=A matter fabricator provides matter for thought|journal=Scientific American|issue=April|pages=116–119}}
{{refend}}

{{DEFAULTSORT:Non-Measurable Set}}
[[Category:Measure theory]]