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In [[group theory]], a branch of mathematics, a '''word metric''' on a [[group (mathematics)|group]] <math> G </math> is a way to measure distance between any two elements of <math> G </math>. As the name suggests, the word metric is a [[metric (mathematics)|metric]] on <math> G </math>, assigning to any two elements <math> g </math>, <math> h </math> of <math> G </math> a distance <math> d(g,h) </math> that measures how efficiently their difference <math> g^{-1} h </math> can be expressed as a [[Automata theory#Definitions|word]] whose letters come from a [[Generating set of a group|generating set]] for the group. The word metric on G is very closely related to the [[Cayley graph]] of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.
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A [[Generating set of a group|generating set]] for <math>G</math> must first be chosen before a word metric on <math>G</math> is specified. Different choices of a generating set will typically yield different word metrics. While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in [[geometric group theory]].
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=== The group of integers Z ===
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The group of [[integers]] '''Z''' is generated by the set {-1,+1}. The integer -3 can be expressed as -1-1-1+1-1, a word of length 5 in these generators. But the word which expresses -3 most efficiently is -1-1-1, a word of length 3. The distance between 0 and -3 in the word metric is therefore equal to 3. More generally, the distance between two integers m and n in the word metric is equal to |m-n|, because the shortest word representing the difference m-n has length equal to |m-n|.


=== The group <math>\mathbb{Z}\oplus \mathbb{Z}</math> ===
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For a more illustrative example, the elements of the group <math>\mathbb{Z}\oplus\mathbb{Z}</math> can be thought of as [[vector (geometric)|vectors]] in the [[Cartesian coordinate system|Cartesian plane]] with integer coefficients. The group <math>\mathbb{Z}\oplus\mathbb{Z}</math> is generated by the standard unit vectors <math>e_1 = <1,0></math>, <math>e_2 = <0,1></math> and their inverses <math> -e_1=<-1,0> </math>, <math> -e_2=<0,-1> </math>. The [[Cayley graph]] of <math>\mathbb{Z}\oplus\mathbb{Z}</math> is the so-called [[taxicab geometry]]. It can be pictured in the plane as an infinite square grid of city streets, where each horizontal and vertical line with integer coordinates is a street, and each point of <math>\mathbb{Z}\oplus\mathbb{Z}</math> lies at the intersection of a horizontal and a vertical street. Each horizontal segment between two vertices represents the generating vector <math> e_1 </math> or <math> -e_1</math>, depending on whether the segment is travelled in the forward or backward direction, and each vertical segment represents <math> e_2 </math> or <math> -e_2</math>. A car starting from <math><1,2></math> and travelling along the streets to <math><-2,4></math> can make the trip by many different routes. But no matter what route is taken, the car must travel at least |1 - (-2)| = 3 horizontal blocks and at least |2 - 4| = 2 vertical blocks, for a total trip distance of at least 3 + 2 = 5. If the car goes out of its way the trip may be longer, but the minimal distance travelled by the car, equal in value to the word metric between <1,2> and <-2,4> is therefore equal to 5.
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In general, given two elements <math> v = <i,j> </math> and <math> w = <k,l> </math> of <math>\mathbb{Z}\oplus\mathbb{Z}</math>, the distance between <math> v </math> and <math> w </math> in the word metric is equal to <math> |i-k| + |j-l| </math>.
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Let G be a group, let S be a [[generating set]] for G, and suppose that S is closed under the inverse operation on G. A [[Automata theory#Definitions|word]] over the set S is just a finite sequence <math> w = s_1 \ldots s_L </math> whose entries <math>s_1, \ldots, s_L</math> are elements of S. The integer L is called the length of the word <math>w</math>. Using the group operation in G, the entries of a word <math> w = s_1 \ldots s_L </math> can be multiplied in order, remembering that the entries are elements of G. The result of this multiplication is an element <math>\bar w</math> in the group G which is called the '''evaluation''' of the word w. As a special case, the empty word <math>w = \emptyset</math> has length zero, and its evaluation is the identity element of G.
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Given an element g of G, its '''word norm''' |g| with respect to the generating set S is defined to be the shortest length of a word <math>w</math> over S whose evaluation <math>\bar w</math> is equal to g. Given two elements g,h in G, the distance d(g,h) in the word metric with respect to S is defined to be <math>|g^{-1} h|</math>. Equivalently, d(g,h) is the shortest length of a word w over S such that <math>g \bar w = h</math>.
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The word metric on G satisfies the axioms for a [[metric (mathematics)|metric]], and it is not hard to prove this. The proof of the symmetry axiom d(g,h) = d(h,g) for a metric uses the assumption that the generating set S is closed under inverse.
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The word metric has an equivalent definition formulated in more geometric terms using the [[Cayley graph]] of G with respect to the generating set S. When each edge of the Cayley graph is assigned a metric of length 1, the distance between two group elements g,h in G is equal to the shortest length of a path in the Cayley graph from the vertex g to the vertex h.
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The word metric on G can also be defined without assuming that the generating set S is closed under inverse. To do this, first symmetrize S, replacing it by a larger generating set consisting of each <math> s </math> in S as well as its inverse <math> s^{-1} </math>. Then define the word metric with respect to S to be the word metric with respect to the symmetrization of S.


== Example in a free group ==


[[File:Cayley graph of F2.svg|right|thumb|In the [[free group]] on the two element set {a,b}, the distance between a and b in the word metric equals 2]]
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Suppose that F is the free group on the two element set <math>\{a,b\}</math>. A word w in the symmetric generating set <math>\{a,b,a^{-1},b^{-1}\}</math> is said to be reduced if the letters <math> a,a^{-1}</math> do not occur next to each other in w, nor do the letters <math>b,b^{-1}</math>. Every element <math>g \in F</math> is represented by a unique reduced word, and this reduced word is the shortest word representing g. For example, since the word <math>w = b^{-1} a</math> is reduced and has length 2, the word norm of <math>w</math> equals 2, so the distance in the word norm between <math>b</math> and <math>a</math> equals 2. This can be visualized in terms of the Cayley graph, where the shortest path between b and a has length 2.
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The group G [[Group action|acts]] on itself by left multiplication: the action of each <math>k \in G</math> takes each <math>g \in G</math> to <math>kg</math>. This action is an [[isometry]] of the word metric. The proof is simple: the distance between <math>kg</math> and <math>kh</math> equals <math>|(kg)^{-1} (kh)| = |g^{-1} h|</math> which equals the distance between <math>g</math> and <math>h</math>.
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=== Bilipschitz invariants of a group ===
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The word metric on a group G is not unique, because different symmetric generating sets give different word metrics. However, finitely generated word metrics are unique up to [[Lipschitz continuity#Lipschitz continuity in metric spaces|bilipschitz]] equivalence: if <math>S</math>, <math>T</math> are two symmetric, finite generating sets for G with corresponding word metrics <math>d_S</math>, <math>d_T</math>, then there is a constant <math>K \ge 1</math> such that for any <math>g,h \in G</math>,
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:<math> \frac{1}{K} \, d_T(g,h) \le d_S(g,h) \le K \, d_T(g,h) </math>.


This constant K is just the maximum of the <math>d_S</math> word norms of elements of <math>T</math> and the <math>d_T</math> word norms of elements of <math>S</math>. This proof is also easy: any word over S can be converted by substitution into a word over T, expanding the length of the word by a factor of at most K, and similarly for converting words over T into words over S.


The bilipschitz equivalence of word metrics implies in turn that the [[growth rate (group theory)|growth rate]] of a finitely generated group is a well-defined isomorphism invariant of the group, independent of the choice of a finite generating set. This implies in turn that various properties of growth, such as polynomial growth, the degree of polynomial growth, and exponential growth, are isomorphism invariants of groups. This topic is discussed further in the article on the [[growth rate (group theory)|growth rate]] of a group.


=== Quasi-isometry invariants of a group ===
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In [[geometric group theory]], groups are studied by their [[Group action|actions]] on metric spaces. A principle which generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is [[quasi-isometry|quasi-isometric]] to any [[proper metric space|proper]], [[geodesic metric space]] on which G [[Group action|acts]], [[properly discontinuously]] and [[Group action#Continuous group actions|cocompactly]]. Metric spaces on which G acts in this manner are called '''model spaces''' for G.
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It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern [[geometric group theory]] is in large part the study of quasi-isometry invariants.
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Will not be any cheap parts utilized for the construction of Cartier glasses, precisely top-quality aspects available for purchase are generally widely-used. Like, not any plastic-type material items are utilized in any respect into Cartier glasses seeing that cellophane is in the main having an appearance a cloth which happens to be on the way of damage. Cartier's glasses simply use the right wood and furthermore ティファニー 店舗 札幌 in terms of iron products you possibly can.

Nearly all people appreciably reduce your designs of Cartier's glasses that should be vintage. It's to be able to believe that preserving simply because the industryrrrs glasses very easily look great for if they were made. Regular figures that happen to be often trendy which includes rectangles as well as butterflies are utilised inside model of unquestionably the casings. Furthermore, nearly all Cartier sunglasses are undoubtedly rubber-stamped haveing a real successive count to determine the whole bunch as an effective real product.

Being appropriate proof of Cartier's grade, some エルメス 財布 individuals so, who be dressed in medical prescription upgraded lenses own Cartier sunglasses as well as his pharmaceutical drug glasses attached inside the latest framework. And then citizens have even ones own endorsed disposable lenses 
tinted so they continues to feel like sunglasses.

If you're planning on a suitable grocery shopping to look into the a range of sorts of Cartier glasses, make エルメス バーキン 2014 certain to check that your store you have been during is a really competent Cartier agent. Cartier at any time certifies each individual store whom transfers the nation's products and services present different courses a great number of synthetic Cartier glasses on the real estate market. Should affirm エルメス that store you're going to be inside is going to be Cartier recognized, you can definitely offer balance any sort of anxieties of faked ティファニー ネックレス services.

Small number of is going to believe Cartier glasses have the class that belongs to them. Besides a majority of these glasses glimpse awesome, nevertheless strong and additionally fashioned within the grows older.


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Start Your Day With Vigor And Energy With Digital Radio Alar

Launch Every day Thanks to Energy source As well as By using Electronic digital トリーバーチ 長財布 アウトレット Phone Alarm

Awakening following be sure you individual clock is regarded as the nearly all delightful new ways to begin a single day. There's lots of explanations why should you use are just looking for worry radio station to generate in the entire early morning. The reason obtaining it may be said to be the well-versed techniques for establishing your thoughts become interested in a new live mode effectively. A lot of our thought processes needs a process to relocation the entire movement involving consciousness from their condition of inertia to a inform area / a ralph lauren femme situation once you are ready to get on the difficulties from your frequent challenges together with electricity, hope, together with adventure. Finally, who would like to heart warming the whole day inside of a decrease also torpid manner? This implies involve a ティファニー 店舗 渋谷 digital radio noisy alarms. 

At the present, the question happens about information on how would likely this type of mindful mental state performed and then in what options an invisible noisy alarms may help you? Certainly, that have a electronic digital phone noisy alarms, there are a thorough range of all the music tracks you are able to pre-program. It is easy to pre-set radio stations manner you want to pay attention to whilst カルティエ 時計 getting in place of your your bed. Getting up learning from your favorite music is absolutely nothing only a dream come true! However this is, in actuality, a gentle method of getting into the real life from your lovely big needs.

All your daily trepidation normally would beginning slip away on to the melodious new music of a preferred movies participating on the digital individual clock. ルイヴィトン アウトレット 財布 Valued fortunate to understand your health with additional stamina and therefore interest if you awake by using a search on your own ティファニー ネックレス morning thusly stereo travel alarm clocks are nothing less than a need! Make use of all your a digital camera tv alarm clock in a variety of strategies just like an alarm system, remote device, even more. A lot of these electric travel alarms are normally found in a wide array concerning versions and so brands.



Since you are trying to get one example computer device alarm clocks, the thing that you should ascertain may toughness for the hands of time. With regard to this particular burglar alarm is considered to be エルメス 財布 effective devoid of electric utility health problems, utilize be prepared to inside the attraction of making use of the product. Perpetually get a level of quality goods for you to shall be content with it for many years within the. You may find the digital travel alarm clocks discovered at diverse purchase prices. Get the the one greatest suits your own financial allowance together with the high-quality features.
Among the well-known and then responsible online stores offering a variety of premium cameras at the best niche prices is usually digitalbuydirect.net. You could possibly read through all their a wide range of online selection of the cameras or scanners, such as digital individual warning buzzers, wi-fi headsets, harmful readers, ios ouert shell, but more; then compare the prices, and select that which meets your needs exceptionally well.


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Necessary Ingredients To Move Through Fear

Necessary Formulation To act By simply Hate

Every body is performing properly whenever they can consume a エルメス バーキン recipes! Subsequently, I really constructed these kind of Six elements that should move through won't. It is very yet unfortunately successful. Conform to these types of Six stairways: 

An individual.Correspondence all your other concerns. Please don't review, criticize, find out, calculate, or possibly acknowledge . . . just understand it. Recognizing all about those feelings relating to this apprehension your family reverse a number of tow hooks which polo ralph lauren pas cher usually help you twisted up. 

Only two.Break down reason right from thoughts. The large majority of anxiety is going to be impending what-if's and thus recent ティファニー 指輪 メンズ will have's. Specifically is man-made with regards to apprehension? Fail to sidestep difficult trustworthiness ; the software comfortable to wear! 

Several.Enable it to be Sizeable. Break up ones apprehension. Causes it to be natural. Pull off your thoughts firmly into we be concerned and watch when there is - you could come out lack of! Once you get in the very far mentorship you feel it may not be genital herpes virus treatments thinking. 

Whenever a way to control appears that can generates apprehension, have a rest, cave in, and thus style of sink towards the deep for this sensing. In that specific worry over, recognize will be on the brain along with skills your favorite strenuous resolution For comprehensive immersion. 

7.Find out nuts a lot of questions. Such as, precisely what? Or even a why then fuss? Just what this implies i believe? If perhaps the actual fear followed actual, could possibly a partner meet your death? (which may be my current litmus check out!) Generate an attitude close to changing via your concern Body concerning having an effect on if you don't have reluctance. 

A number of.Reach a home トリーバーチ バッグ related to 'flow.No Wearers know this as "being rrnside the zone' . the places you solely happens to your here and now. As women end up not being in the present will not hurts their electrical power and look. Provide your focus making yourself a commirment for step. 



Caroline Adams Cooper, creator of the "Creating For less than Life" considers acquiring grit! Good job on a concept! Not a single person includes more determination instead of women. After all, pay attention to who set free airline ; it's the main women! 

Your woman becomes resolution due to, "perseverance and passion suitable for long-term purposes.In That is in first place on their list when considering disobeying down and then moving along throughout our favorite phobia. Caroline says on for 'grit' to become a identifying feature that the difference ralph lauren home when long-term a favorable outcome. The woman goes out also by claiming we should instead join our personal determination complete with 'hard deliver the results as well as , responsibility resolve.A 

Just who does not want long-term popularity? I do! 

Here's the deal. トリーバーチ 靴 Gals, use the resolution of your characteristics (or perhaps oral cavity) to show up, shoreline in place not to mention maximise all and any phobic disorders skating rrnside your feelings relating to imparting one self etc - folks that so desperately require to which it. 

Before you start proving the determination, you"ll miracle how on earth you at any time before got along with out them. ルイヴィトン バッグ And all sorts of you did would be listen to your individual instinct! 

Seeing and hearing your entire nuggets of information will probably transfer that you in front of you. Let you triumph over an individual's fright. And let you function as a natural-born influencer you happen to be intended to be!


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Workout Routines Perfect For Women - P90x

Workout Programs They are perfect for Women Through P90x

Even when almost all the p90x software program workouts are designed for men, on the other hand it's just a wide-spread unawareness. P90x workout pursuits to find women don't seem to be a powerful contemporary action, more and also women may be applying those workout actions. Women exact weight loss diets or activities designed for those people muscular tissue which are reasonably under-worked you should women, will most certainly be efficient making up ground. 

A pair of types of these トリーバーチ 長財布 黒 women, who may have obtained perfect fitness certifications by applying the p90x workout regimen, really are Audra Le. and additionally Tracy R. You will find there's denoted difference in the volume of reps a number of doing exercises models which these women would manage before getting to, after destined on the specific normal routine. It does not observed improvement in your own body size among those two women. The contract details in addition to volumes may be removed from the state run beachbody web presence, and could be counter-confirmed you are able to, when necessary. Though Audra were able to handle typically to Tenty-seventh カルティエ 指輪 push-ups before heading set for this eating routine, she could right now handle Fifty-four. Them digits warm up your muscles has grown away from A half dozen?to eight?and so him / her heartbeat have lowered out of 116 which will 109 upon looking for numerous 2nd sit. Out of the before-after graphics, it is extremely clear where Audra has given an outstanding fit around your girl's flat abs, bust and simply legs and feet, ヴィトン 財布 while a very noticeably properly positioned posture.

Even when Audra experienced a quality fitness level of skill long before that she revised around the p90x procedure, Tracy endures a suitable noticeably in contrast as an example. Tracy wouldn't be able トリーバーチ サンダル コピー to begin doing actually 6 press ups in the past joining this method. Now, might get a handle on 08 push-ups at that time. Even if 16 may not be at the same time stunning a lot, though its 4 times of what Tracy may if you don't maintain. Your ex joint exercises see an interesting modify then have higher due to a little 96 for you to More than 200 soon after the completing your girlfriend Ninety day workout regimen. At the same time, Adrian and consequently Marren are a couple of a variety of other choice a example of women that definitely have finished the best standby time with the p90x workout training with regards to women, also acquired the results that they constantly popular.

The formation to the lateau?is said being much most important procedure regarding women, rather than men. lateau? with fitness expressions, signifies process of the very cells getting accustomed to many workout plans, which ends up in diminishing of proficiency advisors daily routines. Muscular tissue disorientation, the most crucial on this p90x set-up works well, removes level of skill production and will help to keep the トリーバーチ 財布 height practical use of doing exercises intact. In effect, ones body is there to a constant adjustment of motion on your entire lifetime of Three months, so that rrt is going to certainly not become accustomed to an exclusive variety of workouts it demonstrates that make use of become bored of persistent precisely the same schedules constantly.

Continuing development of proficiency, suppleness and working skill relating to the tendons ティファニー 指輪 together with the senses is very important, increasingly with regard to women. P90x, utilizing ever-changing strategies consisting of aerobic exercise, power up working out ティファニー ネックレス 値段 workouts, yoga and fitness as well as the cardiovascular exercises schedules assists you to mature generally these types credentials, which means, goes a considerable way making a more favorable individual together.

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