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This tutorial will focus on the concepts of span, basis, and dimension. Graph theory analogues will also be discussed in the context of basis.
Attached is a LaTeX typeset version for those who would prefer to view this tutorial in that format
Linear_Algebra_Part3_Tutorial.pdf (111.13K)
Number of downloads: 682
In my last tutorial, the concept of span was briefly introduced. Abstractly, given a set of vectors S, span(S) is the set of all vectors formed from linear combinations of the vectors in S. There are two predominant questions related to the span of a vector set S:
- What is the span of S?
- Does S span the vector space V?
The span of S relies on the number of linearly independent vectors in S. So if S = {(1, 0), (0, 1)}, then span(S) = R2. Clearly, the vectors are linearly independent, and they are both in the vector space R2.
Similarly, if S = {(1, 0, 0}, (0, 0, 1)}, then span(S) is the x-z plane in R3. The vectors in S are the x and z axes.
What about span(P2®), which is the span of the quadratics? Since P2® is the vector space, span(P2®) = P2®. Since linear combinations of vectors in the Vector Space belong to the Vector Space, span(V) = V for any Vector Space V.
Span is determined in a similar way as Linear Independence. However, one of the conditions for the Linear Independence of a set of vectors requires that there exist at most n vectors in the set if the dimension of the Vector Space is n. In order for a set of vectors to span the Vector Space, the set must contain at least n linearly independent vectors.
Let's unpack this intuition a little bit. Think of linearly independent vectors as coordinate axes. In R2, there are two coordinate axes- the x and y-axes. Thus, two independent vectors span the space. Any other vectors in addition to those coordinate axes won't change the fact that the axes span the space. Similarly, if just given a single vector in R2, a dependent vector won't change the fact that the set doesn't span. So if only the x-axis is given, there is no way to independently obtain the y-coordinates. Therefore, the x-axis cannot span the space. A more geometric argument is that R2 is a plane, and each axis is a line. Thus, a line is a subspace of the plane, but does not span the plane.
A basis is the minimal viable set of vectors required to span a Vector Space. It is defined as maximally linearly independent, but still spanning the space. From the last tutorial, if the dimension of the Vector Space is n, then a vector set can have no more than n vectors and still be linearly independent. Similarly, to span the Vector Space, that set of vectors must have at least n vectors. Thus, it makes sense that a basis has exactly n vectors. Looking at this from a Matroid perspective, it makes sense that all bases have the same cardinality. The third axiom of Matroids states that all maximally independent subsets of the ground set have the same cardinality. Thus, bases are all maximally independent subsets of a Vector Space.
Bases can analogously be thought of as spanning trees as well. A spanning tree on a graph is an acyclic (independent) subgraph that contains all the vertices in the graph and is connected. Not all spanning trees are isomorphic. In fact, there are usually multiple spanning trees for a given graph. Let's look at some examples. Consider the graphs below, with G as the original graph. The forest F, which is a set of trees, is clearly independent, as there are no cycles. However, since the trees in F are not connected, none of them can be thought of as a basis or a spanning tree on the graph. Both T1 and T2 are spanning trees. However, they are different. Notice how there is an edge used in T1 that is not present in T2; and similarly, there is an edge used T2 that is not used in T1. To get from T1 to T2, these edges are swapped. This is the premise for the basis substitution property.
Before unpacking the substitution property, let's determine the conditions that need to hold to determine if a vector set is a basis for the Vector Space V.
- Cardinality-Dimension Test: If the dimension of V is n, then the set must have exactly n vectors. If it does not have n vectors, then it is not a basis.
- Linear Independence: The multiples test is a quick way to determine if the vector set is linearly independent or dependent. If it passes the multiples test, then the determinant test should be used. Remember from the last tutorial that it is necessary to construct the matrix where vectors in S are the column vectors. From there, the vector set is linearly independent if and only if the determinant of the matrix is non-zero.
Now let's discuss the substitution principle for bases a little more. Essentially, as long as the vector set retains the same number of linearly independent elements, vectors can be substituted, and the vector set will remain a basis. Consider the standard basis for R2: S = {(1, 0), (0, 1)}. It's easy to see that this is linearly independent, and the vectors represent the standard x and y-axes. Now let's substitute (0, 1) for (3, 7). Thus, S is still linearly independent and has two vectors; thus, it is still a basis for R2.
Consider again S = {(1, 0), (0, 1)}. Swapping (1, 0) for (0, 5) is not an acceptable choice such that S remains a basis, as S is clearly linearly dependent by the multiples test.
So far, dimension has been referenced a lot, but not clearly defined. The dimension of a Vector Space, denoted dim(V) describes the number of basis vectors for V. For the most part, dimension is pretty easy to see. The dim(R2) = 2, and dim(Pn(F)) = n + 1, as polynomials of degree n have n+1 terms. Similarly, an m x n Matrix has dimension mn. Think about it this way- the standard basis has a single 1 and the rest of the elements in each vector are 0's. So there needs to be an element where the 1 is in each position. Thus, it is clear to see why {(1, 0), (0, 1)} is the standard basis for R2, and it is now easier to see why Matrices have dimension mn.


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Reason Why I Love Mitch Williams Number 27
On July 23, 2012, during the 7th inning of the Yankees/Mariners game, new Bomber, Ichiro Suzuki, wearing number 31 in place of his iconic 51, hit a soft grounder to second and was thrown out easily.
Mitch said, "now, see, that there… that should've been a basehit."
"The number's lighter."
Now that, my friends, is hilarity.
Don't hate me 'cuz I'm right.


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You are currently browsing the tag archive for the 'Symbolism' tag.
Introduction: Hello! Project's Central Mystery
For ages, it has defied explanation, piggybacking on one group name after another. It waits, announcing its presence in a subtle yet forceful fashion, catching the eye with its unexpected existence yet never explaining its purpose. This elusive typographical flourish is none other than the ideographic full stop (。) at the end of several Hello! Project group names: モーニング娘。 (Morning Musume。), ココナッツ娘。 (Coconuts Musume。), カントリー娘。 (Country Musume。), アイスクリー娘。 (Ice Creamusume。), ミニモニ。 (Mini Moni。), エコモニ。 (Eco Moni。)—but interestingly, not プッチモニ (Pucchi Moni), a puzzle that we may be able to solve shortly. Unlike the names that precede it, this hardy symbol has proven immune to transformations like romanization: while "モーニング娘" can be written "MORNING MUSUME" in official documents (often, but not always, in all caps), the stop itself is never replaced by its Latin counterpart (.). There is something intrinsic to the ideographic full stop, then, that would be lost in translation. But what is it? As a distinguishing feature of the flagship group's name since the very beginning, this little circular symbol lies at the heart of the Hello! Project enterprise, and understanding its true meaning may very well be the key to unlocking H!P's greatest mysteries.
Perhaps it is a simple matter of topology. The ideographic and Latin full stops can be viewed as essentially a zero-dimensional point and a one-dimensional circle, respectively. Since they are represented in two dimensions on the page, however, we can also consider them to be projections of two-dimensional objects in a three-dimensional space (the page itself is a two-dimensional Euclidean plane, so we must go to higher dimensions to have other types of two-dimensional objects). From this perspective, the ideographic full stop is homeomorphic to a 1-torus, while its Latin counterpart is homeomorphic to a sphere. These are distinct homeomorphism classes since there is no continuous deformation from one to the other, just as there is no way to continuously deform a donut into a ball.
But of course this does not explain the relevance of the full stop to Hello! Project. One important characteristic of the full stop is that it appears in the names of groups, so it is inextricably linked to H!P's concept of a group. We may therefore be able to come to a fuller understanding of H!P by studying the structure of its constituent groups.
The term group also has a mathematical meaning, of course, and while on the surface the two meanings may seem disparate and unrelated, an in-depth analysis from a group-theoretic standpoint does yield a surprising number of coincidences, suggesting that H!P groups do have inherent mathematical structure worth taking a closer look at, whether or not this is the intention of producer Tsunku♂ et al. We may be able to gain insight into the nature of Hello! Project by viewing its groups in mathematical terms, and using group theory as a tool to aid our understanding.
The Basics of Groups
(See also Wikipedia's article on elementary group theory, which may be a more useful intro to group theory.)
The mathematical discipline of group theory, a part of abstract algebra, examines the algebraic structures called groups. Groups are a central concept in algebra and other areas of mathematics, and have also found applications to physics, chemistry, computer science, and linguistics, among other fields.
Definition: A group consists of a set of objects along with a binary operator that can be applied to any two objects in the set. We can denote the set as G and the operator as ×, which for explanatory purposes we'll call "multiplication" in a general sense, though numerical multiplication is only one example of a binary operator. Furthermore, the four group axioms must hold:
- Closure: For all a, b in G, a × b is also in G. (The result of multiplying any two elements in the set produces an element in the set.)
- Associativity: For all a, b, c in G, (a × b) × c = a × (b × c). (When there is more than one multiplication, it doesn't matter which pair gets multiplied first—the result is the same.)
- Identity: There exists an element e in G such that for all a in G, a × e = e × a = a. (There is an element that can be multiplied with any element, in either order, without changing it.)
- Inverse: For each a in G, there exists an element b in G such that a × b = b × a = e. (Every element can be multiplied with some other element, in either order, to produce the identity.)
If a set and a binary operator defined on the set meet these conditions, then they form a group. We can specify further restrictions to define particular types of groups. For example, we can add commutativity to define the notion of an abelian group:
- Commutativity: For all a, b in G, a × b = b × a. (The order in which you multiply elements doesn't matter.)
One example of a group (which also happens to be abelian) is the set of real numbers, R, under addition. (Note that here the group operator is addition, not multiplication). We can easily verify that the four group axioms, as well as commutativity, hold (this is not a proof, just an example):
- Closure: The sum of two real numbers is a real number.
- Associativity: For any real numbers a, b, c, (a + b) + c = a + (b + c).
- Identity: 0 is the additive identity. You can add it to any real number without changing it.
- Inverse: For any real number a, -a is its inverse. The sum of the two is 0.
- Commutativity: For any real numbers a, b, a + b = b + a.
The set of real numbers under multiplication is not a group, however, since 0 does not have a multiplicative inverse (there's no real number you can multiply with 0 to make 1, the multiplicative identity). But you can make this a group by excluding 0.
As we will see shortly, H!P groups consist of both abelian and non-abelian groups.
Hello! Project Groups as Mathematical Groups
Set Composition
There are a number of ways in which to regard H!P groups as groups in the group-theoretic sense. A naïve approach would be to define a set consisting of the members of the group and define some appropriate binary operator over the members. Though this makes some intuitive sense and is relatively simple to start out with, we can easily show that it fails to capture the nature of H!P groups.
A counterexample would be Coconuts Musume。 after Ayaka's graduation on 2008-04-30. According to the official graduation announcement on Hello! Project's site, Ayaka, the last remaining member of the group, graduated from Hello! Project as well as Coconuts Musume。 ("2008年4月30日(水)をもちまして『ココナッツ娘。』及び『ハロー!プロジェクト』から卒業することになりました。") This implies that Coconuts Musume。 as a group did not graduate from Hello! Project. After losing its last member, it must remain as a zero-member group.
Unfortunately, there is no such thing as a zero-element group. That would violate the identity property, which states that there exists an identity element. The smallest group is the trivial group, which contains just the identity and nothing else.
And that is the key to understanding how H!P groups work in a group-theoretic sense. All groups must have an identity, and H!P groups are no exception. What Coconuts Musume。 retained after Ayaka's graduation was its identity: the name "Coconuts Musume。". The name did not graduate along with Ayaka but remained as the single persisting feature of the group. Here, the term identity is applicable in both its mathematical sense and its sense of being a name or label, an identifier.
So all true H!P groups have an identity, a name that is distinct from the members who constitute the group but is itself an element of the group. Due to the potential for confusion in terminology, I'll clarify that I'm using member to refer to a human constituent of a group and element to refer to a constituent of the group in the theoretical sense. In this study, I assume that a group's elements are solely its identity and its members, but that assumption need not hold in possible alternative group-theoretic conceptions of H!P groups, which I will not consider here. Thus the current lineup of Morning Musume。 has ten elements (that is, it's a group of order 10): nine members and an identity.
One could argue that the identity is just an insignificant auxiliary feature of a group, but I think treating it as an element in its own right does better to characterize H!P's conception of its groups. The only element of Morning Musume。 that has remained throughout its changing generations is the identity, and it is solely due to the inclusion of the identity that the current lineup can still claim older songs like "LOVE Machine" or "Renai Revolution 21" as its own, even though none of the current members were around at the time of those releases. The Hello! Project web site acknowledges the importance of the group identity: the official profile of the group has a section devoted to each of its ten elements, and each of the ten elements has its name highlighted in blue. The identity has its own section at the top, where it belongs, since it is the longest lasting and arguably most important element of the group.
Interestingly, H!P uses the term unit (ユニット) to refer to its groups. This term appears, for example, in reference to H!P's special and shuffle units, and Tsunku♂ uses it frequently. In a mathematical sense, unit can refer to some form of the number 1 (for example, the unit circle, the set of points with a distance of 1 from the origin), which is often the identity element in groups (especially when the group operator is regarded as multiplication). In a way, H!P's use of the term associates a group with its identity.
Group Operator
Having considered the question of set composition, we now turn to what kinds of binary operators exist in H!P groups. This is a crucial aspect of a group, as the group operator determines the structure of a group, and without it, a group is simply a set, an unordered, unstructured collection of distinct elements. It can also be particularly tricky to discern the existence of a group operator, since it is not always clear what the inherent structure of an H!P group is. That said, we can find examples of group operators in some H!P groups.
Let us turn to the tracklist for Morning Musume。's album Platinum 9 DISC. Several of the songs are credited to a single member (Mitsui, Michishige, Kamei) or to a subset of the members ({Takahashi, Niigaki, Tanaka}, {Kusumi, Junjun, Linlin}). Especially noteworthy is the use of an interpunct (・) to separate the names. The interpunct is commonly used in mathematics to denote multiplication (a · b), alongside other representations such as a × b, a * b, or simply ab. (My physical copy of the album uses a space to separate the names, which is another common way to denote multiplication.) We can assume that this indicates the application of the group operator to the members in question. As the group operator is associative, we need not specify an order in which pairs should be operated on, but since groups are not in general commutative, the order in which the members are listed may be relevant. (In the next section, I argue that Morning Musume。 is an abelian group, in which case the order of the members is actually not relevant.)
Given the prominence of the "Produced by Tsunku♂" credit on the cover of most of Tsunku♂'s productions, it seems sensible to call the group operator the "Tsunku♂ product" (or maybe "Tsunku♂ produce") for those groups produced by Tsunku♂.
Groups vs. Non-Groups
We have been discussing H!P groups without really defining what counts as an H!P group and what doesn't. Here is a good place to distinguish between the two. A common taxonomy of H!P divides it into two categories: "groups" and "soloists". This classification is largely consistent with our group-theoretic formulation of H!P groups, but there are a few exceptions, which I will now examine more closely. In particular, there are acts commonly classified as "soloists" that are actually groups, and there are acts consisting of more than one member that are not actually groups.
One example of the former is Tsukishima Kirari starring Kusumi Koharu (Morning Musume。). Commonly regarded as a soloist, this is actually a two-element group consisting of the identity "Tsukishima Kirari starring Kusumi Koharu (Morning Musume。)" and the single member Kusumi Koharu. The two elements are distinct from each other: Kusumi is also a member of Morning Musume。, but Tsukishima Kirari isn't.
Most soloists are not groups. The key criterion is whether or not the soloist's performing identity is distinct from their own. If so, then the soloist and the identity together form a group; if not, then the soloist is not a group.
On the other hand, we have collections of members that do not constitute groups. One example is the aforementioned "Takahashi Ai · Niigaki Risa · Tanaka Reina", who are credited for two songs on Platinum 9 DISC. In this case, there is no distinct identity associated with this trio. They are simply individual members appearing together, credited as such. If one applies the group operator to the trio, the result is an element of Morning Musume。, but which one it is depends on the definition of the group operator, a topic we will address in the next section.
In practice, there is not always a clear distinction between groups and non-groups in H!P. Some acts, like Abe Natsumi & Yajima Maimi (°C-ute), have names that are composed of individual members' names, so it may be equally valid to regard them as individual members appearing together, or as a group with a distinct identity that just happens to resemble the individual members' names. H!P acts vary a bit in this regard, and some composite names seem more like distinct group identities than others.
One important feature of an identity, however, is its immutability. If you change the name of an identity, it becomes a different identity. So this could be used to determine what counts as a distinct identity. In the case of "Takahashi Ai · Niigaki Risa · Tanaka Reina", different official representations have used interpuncts and spaces to separate the members' names, so there is no single correct representation of the name, and hence no distinct identity. On the other hand, Shigepink, Kohappink is an actual group, even if it looks as though it could be a composite of smaller groups Shigepink and Kohappink. The half-stop or comma between the names is invariable, as can be seen in the track listing for 7.5 Fuyu Fuyu Morning Musume。 Mini!. While an interpunct separates Yoshizawa, Niigaki, and Kamei in the track credit above, Shigepink, Kohappink's credit retains a half-stop.
Group Isomorphisms
Once we can characterize the structure of a group, it is useful to relate it to a more abstract notion of a group. We can say that an H!P group is isomorphic to some abstract group—that the two exhibit the same group structure. In particular, this means that there is a one-to-one correspondence between H!P group elements and elements of the abstract group such that the relationships expressed by the group operator are preserved: if Takahashi, Michishige, and Kusumi correspond to the elements a, b, and c, respectively, in another group, and multiplying Takahashi and Michishige produces Kusumi, then applying the group operator to a and b in the other group should yield c.
One can then apply what is already known about such an abstract group to the H!P group in question, and also relate different H!P groups that are isomorphic to the same abstract group (and hence to each other).
Up to this point, we have not yet conclusively defined the group operator for any group, but for groups with three or fewer elements, we do not need any additional information to determine an isomorphism, as long as we know which element is the identity.
Field Guide???
If this were, say, a field guide to H!P groups in their natural habitats (or should I say group guide, since a field is another kind of algebraic structure), each group could have its own entry with relevant info, statistics, and taxonomic classification. In this post, I will attempt to provide a brief entry for several of the groups, and if any of you are interested, you can try adding entries for the other groups.
The first three entries will include one representative group each of orders 1, 2, and 3; the groups have already been mentioned above.
(Note that the concepts of subgroup and center don't exactly correspond to their conventional H!P meanings. I'll discuss these in Part 2.)
|Coconuts Musume。 (post-Ayaka era, May 2008 — ???)
|Elements and inverses
|Cayley table
|Nontrivial proper subgroups and cosets
|Minimal generating sets
|Cycle graph
|Isomorphic to
|Trivial group
|Tsukishima Kirari starring Kusumi Koharu (Morning Musume。) (2006 — 2009)
|月島きらり starring 久住小春(モーニング娘。)
|Elements and inverses
|Cayley table
|Nontrivial proper subgroups and cosets
|Minimal generating sets
|Cycle graph
|Isomorphic to
|Cyclic group C2
|Shigepink, Kohappink (2006 — ???)
|Elements and inverses
|Cayley table
|Nontrivial proper subgroups and cosets
|Minimal generating sets
|Cycle graph
|Isomorphic to
|Cyclic group C3
The Cyclic Groups of H!P
It is interesting that Morning Musume。 was founded on a notion of "cycling" through members to establish successive generations, by inducting new members and graduating existing members. Both words in the name also reflect a notion of periodic cycles: mornings are part of the daily cycle, and musume (daughters) are part of a cycle of familial generations. The full stop also adds a notion of periodicity—both in its more common North American name, the period, and its circular shape. All of this suggests that a notion of cyclicity is integral to the concept of Morning Musume。 as a group, and perhaps to others as well, particularly those with the full stop in their names.
Indeed, groups in the mathematical sense can also be cyclic. A cyclic group is one whose elements can be generated by a single element of the group: every element can be produced by successively applying the group operator to the generator. In a multiplicative sense, every element is a power of the generator.
We can then conclude that every H!P group with a circular full stop in its name is a cyclic group. If a group has n members, then it is isomorphic to the cyclic group of order n + 1, which is isomorphic to both Zn+1, the group of integers under addition modulo n + 1, and the complex (n + 1)st roots of unity, . For example, here are the 6th complex roots of unity, a cyclic group of order 6:
(It is worth noting that the circle group, being an uncountably infinite group, is not a cyclic group, so the pictorial representation of the full stop may not be accurate if perceived as a circle. However, it is possible to generate a dense cyclic subgroup of the circle group by using any irrational number as a generator, so there is a valid cyclic group interpretation.)
The three groups detailed above are all cyclic groups, but as we go higher in order, it is possible for groups not to be cyclic, so it is useful to be able to recognize cyclic structure where it exists.
The presence of the full stop establishes Morning Musume。 as a cyclic group of order 10. However, from just this information alone, we cannot conclusively determine Morning Musume。's group structure. We know that it is isomorphic to the cyclic group of order 10, but we don't know which isomorphism is correct. Depending on our choice of isomorphism, we could identify different subsets of the members as generators. (In the previous three groups, all isomorphisms are equivalent: any permutation of the members yields the same group structure; but in Morning Musume。, different permutations of the members can yield different group structures. There are in fact 90720 possible distinct group structures you can produce depending on which of the possible permutations of the nine members you pick—I leave the proof of this fact as an exercise for the reader.)
We therefore look to the standard order of H!P's groups. Hello! Project generally orders its groups by seniority, and the standard order features prominently in musical releases, concerts, and public appearances. It is not an arbitrary order but an integral feature of each group. Accordingly, we use the standard order to enumerate the elements of an H!P cyclic group: the kth member in a group with n members is mapped to k in Zn+1 and to in the group of complex (n + 1)st roots of unity.
This yields the following entry for the current lineup of Morning Musume。:
|Morning Musume。 (Takahashi era, June 2007 — present)
|Elements and inverses
|Cayley table
|Nontrivial proper subgroups and cosets
|Minimal generating sets
|Cycle graph
|Isomorphic to
|Cyclic group C10
With this group structure, we can evaluate the combinations of group members mentioned earlier (and since all cyclic groups are abelian, the order in which we apply the group operator does not matter):
- Takahashi · Niigaki · Tanaka = 1 + 2 + 5 = 8 = Junjun
- Kusumi · Junjun · Linlin = 6 + 8 + 9 = 23 ≡ 3 (mod 10) = Kamei
Or, equivalently, treating the operation as a product (the "Tsunku♂ product") of complex numbers:
- Takahashi · Niigaki · Tanaka = = Junjun
- Kusumi · Junjun · Linlin = = Kamei
Of course, this raises newer questions that we have yet to answer, but it seems Kamei is somehow responsible for "Guruguru JUMP".
If we consider °C-ute a cyclic group as well, interpreting the degree symbol (°) to be analogous to the full stop, we have the following entry (assuming standard order to resolve among the 120 possible distinct group structures):
|°C-ute (post-Arihara era, July 2009 — present)
|Elements and inverses
|Cayley table
|Nontrivial proper subgroups and cosets
|Minimal generating sets
|Cycle graph
|Isomorphic to
|Cyclic group C7
Being a cyclic group of prime order, °C-ute is a simple group, as opposed to Morning Musume。, which is not currently a simple group but has been so several times in the past. Prior to losing Arihara, however, °C-ute was also not a simple group, and in addition, had the curious distinction of being a group whose leader (Yajima) was not a generator, since Yajima's index, 2, is not relatively prime to °C-ute's group order at the time, 8. In most groups, the leader has the highest seniority and accordingly has an index of 1, which is relatively prime to everything, so it is usually the case that the leader is also a generator.
With that, we end our discussion of cyclic groups in H!P. Not all groups are cyclic, however, and while one valid approach would be to categorize all H!P groups as cyclic (since there is a cyclic group for every positive integer order), the characteristics of some H!P groups suggest that they exhibit different group structures not isomorphic to cyclic groups. And this is where H!P groups start to get very interesting, but as I've already written way too much for this post, I'll end it here and promise to resume next time with a survey of a variety of non-cyclic Hello! Project groups, as well as a possible explanation for the lack of a full stop in Pucchi Moni's name and the meaning behind the V in Pucchi Moni V.
to be continued…
Countdown! The Top 100 Hello! Project PVs
After a three-month hiatus, the Countdown continues! And you thought I'd abandoned it!
Countdown! The Top 100 Hello! Project PVs
Like I totally don't have an exam to study for tomorrow.
Actually I don't. But I should have an exam tomorrow.
In any case, there's one Friday.


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Oh wait, that's supposed to be "Winter is Coming," isn't it… errmmm… Anyway…
Autumn IS coming. The air really is cooling off. The trees really are starting to change. September is upon us and the Summer Harvest Hunt is in full swing. The Ego Co.'s latest hunt is going strong and it's a bountiful harvest. Between the new goodies, the seasonal view off her back porch and Elymode's gorgeous WL setting, Chloe got a little lost in the moment.
(Items in the Summer Harvest Hunt will be noted in italics.)
Hair | Decoy – Aella (Fudge/Grape Fades)
Tank/Dress | Gawk! – Brown Long Tank Top (Exclusive for SHH)
Feet | SLink – Jolie Pied Barefeet (Medium)
Skin | Glam Affair – Roza (BaroQ 02, C88 August)
Tattoo | Flaunt – Fall Harvest Sleeves Tattoo (Color)
Nails | Synthetique – GoldLeaf Mini Collection (Medium Fierce Nails)
Pose | BENT – Be.Hind 10


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Feel free to explore the site and email Mary here or call her at 1-888-498-2227 (toll-free) for more information.


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Roy Hodgson has handed Liverpool FC midfielder Jonjo Shelvey his first England call-up.
The 20-year-old has been drafted into the Three Lions squad for the World Cup qualifiers with San Marino and Poland.
Chelseas Ryan Bertrand has also been added to the squad as Kieran Gibbs has been forced to pull out with a thigh injury.
VIEW pictures of Jonjo Shelvey's career so far
Bertrand's inclusion for Gibbs was an obvious one given the Chelsea defender had been involved in England's last qualifying match, the draw with Ukraine at Wembley last month.
Shelveys presence at St Georges Park is slightly more complicated.
The 20-year-old Londoner has impressed in Liverpools sluggish early season displays and was initially called up for the European Under-21 Championship play-off with Serbia by Stuart Pearce.
However, Shelvey is suspended for the first leg of that game at Norwich, and Hodgson has taken the opportunity to bring him into the senior fold.
With Frank Lampard, Michael Carrick, James Milner and Tom Cleverley all capable of playing in central midfield, Hodgson does have options, even though skipper Steven Gerrard is himself suspended for Fridays game with San Marino.
It has therefore been suggested Shelveys presence at St Georges Park is purely to further his experience.
There is also a chance he could go back to the Under-21s for the second leg of the Serbia game next Tuesday, although a decision on that will not be taken until the weekend.


text:
- Home
- City Hall
- Departments & Divisions
- Finance
- Purchasing Division
Purchasing Division
The Purchasing Division is located on the first floor in City Hall.
Department Overview
The Purchasing Division directs the efficient, legal and cost-effective procurement of goods and services for all city and Board of Lights and Water departments. The division is responsible for purchasing, with the objective of ensuring that all materials, supplies, services and equipment required are acquired in a timely manner at the lowest possible cost, consistent with the quality required and in compliance with all applicable procurement legislation.
The division regularly and routinely shops all requirements to obtain the best value for the city and is responsible for controlling and disposing of excess, obsolete and salvageable materials and equipment.
Proposals & Bids
A list of the items the Purchasing Division is accepting proposals for as well as addition information about the bid process.Link to page
Purchasing Division FAQs
This page is designed to help answer the most frequently asked questions on the city purchasing procedures and deadlines.Link to page
Vendor Registration
The City of Marietta uses Click2Gov for Purchasing and inventory.Link to page
Surplus PropertyLink to page


text:
Mad Men may take place primarily in the '60s, but when we asked Alison Brie to step into the era of big hair and heavy contour (read: decade of disco) for our latest beauty editorial, she happily obliged.
"I love the style of the '70s, it's such an eclectic decade, and so different from the '60s," Brie told us. "These looks are all looser, sexier, and more free." How does it stack up to the period she's used to working in? "The costumes are very binding because we wear all the correct undergarments, which informs your character, but this feels like you can put everything right out there. Whereas, in Mad Men, everything seems a bit hidden."
We did just that, with the help of makeup artist Jo Strettell and hairstylist Christian Marc. From big curls and Guy Bourdin-inspired colors to a full-lashed flower child, Brie makes the '70s look damn good.
Strettell's inspiration for the first look came from French photographer Bourdin. "It's a wet eye and mouth—and no cheek color," Strettell says. She used Nars' Soft Touch Pencil ($19) in Trash under a coat of Elizabeth Arden's 8 Hour Cream ($24), then reached for Hourglass' Extreme Sheen Lip Color ($23) in Icon. Manicurist Michelle Saunders created the "severe nude nail" with Essie's Nail Polish ($10) in Cocktails and Coconuts.
Meanwhile, Marc crafted soft curls with a 1-inch iron and a boar-bristle brush. "Brush through the curls after they cool with a paddle brush to create a nice wave," he says. "That's how you get that softer look."
Sophie Theallet Coral and Anemone Chantilly Lace and Chiffon Dress with Embroidery and Leather Details ($3,495)
Marc traded in his smaller iron for a 3-inch barrel for this look, again, softening the style with a brush. To complement Brie's locks, Strettell used Flower Beauty's Lip Suede Velvet Lip Chubby in Floral Coral on both her lips and cheeks. "It's a fresh face—and lots of mascara," she says. She used the same formula through our entire shoot: L'Oreal's Voluminous Mascara ($9).
Stevie Howell Ritz's Ranunculus Robe de Chambre ($395)
Brie had a hard time deciding which look was her favorite, but the faux bob and heavy contour took the prize. "It surprises me that it was my favorite, but it turned out really nice," Brie says. "It's fashion-forward and such a cool, esoteric look."
How did Marc create said faux bob? "I don't know if I want to tell you that kind of stuff! It's a secret," he joked. Needless to say, it called for a deep side part, a medium-barrel iron, and a tucked-in braid.
Derek Lam Georgette Cloque Twist Halter Gown with Wide Belt ($2,690)
Brie's reaction to the most time-consuming look of the day? "Amazing!" she says. "I don't think I've ever worn this much makeup or had this type of hairstyle. It's so glamorous."
Strettell was inspired by Bianca Jagger, so she called on colorful shadow (Nars' Soft Touch Pencil ($19) in Trash), heavy blush (Marc Jacobs' Shameless Bold Blush in Tantalizing) and a glossy red lip (Hourglass' Extreme Sheen Lip Color ($23) in Icon).
Marc applied a hefty dose of mousse to Brie's damp locks before wrapping them around thin foam rollers (try the Spiral Rollers Set ($12) by Conair) and setting with an old-fashioned blow-dry bonnet. Meanwhile, Sauders took the actress dark and dramatic with Essie's Nail Polish ($9) in Carry On.
Eres Rainbow in Now Maillot ($580).
As the creative juices flowed, Marc and Strettell stumbled across their own version of disco. "This shimmery bronzed look is sort of like…Bardot gone wild," Strettell says. She used Flower Beauty's creamy Eyes of the Prize Shadow Stick in Daffodil With It under a cocktail of powder shadows and Lancome's Juicy Tube ($39) in Simmer. Marc brushed out the curls already in Brie's hair and smoothed Rene Furterer's Molding Paste ($26) from root to tip.
Zero Maria Cornejo Lea Feather Stripe Jumpsuit ($1,025). Visit zeromariacornejo.com or call (212) 925-3849.
- Photographer: Justin Coit
- Creative Director: Liz Higgins
- Makeup Artist: Jo Strettell
- Hair Stylist: Christian Marc
- Manicurist: Michelle Saunders
- Stylist: Laurie Trott


text: Model, dancer, and business woman, Zoe Britton is one of the most multi-talented porn stars in the game today. As the co-head of New Star Productions alongside porn legend Ron Jeremy, this naturally busty beauty knows the ins and outs of porn from both sides of the camera. She filmed her first scene back in 2003, and has since earned herself two AVN Award nominations for her scintillating girl on girl scenes, not to mention appearing on the pages of over 100 men's magazines. When she's not busy fucking the hottest women in the business, Zoe loves to drink fine wines, enjoy classic art, and chase some thrills on roller coasters. This bisexual beauty is known for her sexy voice, and hearing the way she moans while getting her pretty pink pussy licked will get you off in no time flat!


text:
Uncyclopedia:Top10 07
Welcome to Uncyclopedia's Top 10 voting. The rules are simple. You go through all of the featured articles of the last year (that were by themselves hand picked ,by voters like you, to be in the top three of every month. So in theory there are like 36 articles below), and you pick your favorite 10 from those 36. You only get 10 votes, so choose wisely. Also, you may only vote once for each article you decide to vote for. After awhile, or whenever we feel like it, we'll end the voting and the 10 articles with the most votes will become Uncyclopedia's Top 10 of 2007. If there are any ties I don't know what will happen, so try to avoid ties.
All entries are in alphabetical order for easier finding. Sign your votes using #{{u|your name}}. IPs, that is, users who are not logged in and/or do not have an account, should use 3 tildes(~) to vote. IP votes are counted as half a vote. Please update the score when you add your name.
|Awards, Decorations and Honours
|Order of Uncyclopedia · Awards, Decorations and Honours · Admin Rank · Ninjastars
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Happy Monkey Competition · Other competitions


text:
- 3 Beds
- 2 Full Baths
Property TypeResidential – Townhome
SubdivisionBooth Creek Th
Listing Date06/13/18
Year Built1973
AreaBooth Creek
Lot Size0.10 Acres (4,356 SqFt)
Parking1 Car Assigned
Property Description
A preferred location with assigned parking spot just steps down the heated pathway to your front door. Upgrades, tastefully appointed, excellent condition. Kitchen offers ample granite counter space, stainless appliances (new microwave and dishwasher), heated bathroom flooring on lower level, auxiliary hot water system insures you plenty of showers and bubble baths! Enjoy the Vail fireworks from the largest deck in the complex with 180' views. Walk to fabulous Booth Creek Park! A BEST BUY!
Presented By
- Bus Route
- Meadow
- Mountains
- Valley
- Golf-Public
- Hiking/Pedestrian Trail
- On Bus Line
- Pets
- Tennis Courts
- Cable Available
- Deck
- Fireplace - Gas
- Vaulted Ceilings
- Pets
- Cable TV
- City Sewer
- City Water
- Electricity Avail.
- Natural Gas Avail.
- Phone Available
- Snow Removal
- Trash Pickup
- Baseboard
- Natural Gas
- Radiant
- Dryer
- Washer
Financial Information
About East Vail
Only four miles away from the Village and tucked under the Gore Range at the base of Vail Pass, East Vail is a favorite of locals and Front Range visitors. As a strictly residential area, it offers peace and tranquility as well as sweeping views of its high alpine surroundings. East Vail residents enjoy the many conveniently located hiking trails, a large dog park on the Eagle River, and several waterfalls along the north facing slopes. East Vail is convenient to the heart of town by the free bus system yet maintains moderate property prices in comparison.
Area Market Data
4 Beds 5 Baths
3 Beds 3 Baths
3 Beds 3 Baths
We get asked this question a lot. Below is a tool that gives you the ability to see what the market is like in your community. Find out what homes like your own are selling for right in your back yard. Simply enter in your address, choose the proximity of the search and find out market data for other homes that are selling near you.
While we'll provide you with plenty of information, this is a snapshot of your market data and there are real people behind the high-tech and automated tools. If you would like more detailed information or help in understanding the results we are here to help.
Until then, enjoy the tools!
The data relating to the real estate for sale on this website comes in part from the Internet Data Exchange program of the Vail Multi-List Service, Inc.
Copyright © 2020 Vail Multi List, Inc. (VMLS). The information displayed herein was derived from sources believed to be accurate, but has not been verified by VMLS. Buyers are cautioned to verify all information to their own satisfaction. This information is exclusively for viewers' personal, non- commercial use. Any republication or reproduction of the information herein without the express permission of the VMLS is strictly prohibited.
Listing courtesy of Slifer Smith & Frampton/Solaris


text:
Lester Kasai Skater Profile
Sponsors: Tracker, Hurley, Bones, The House
Hometown: Anaheim CA, Stance: Regular, Age: , Status: Am
Last Update: Tuesday, February 13, 2007 Submit an Update
In Stock Now With Immediate Shipping
Couldn't find any Lester Kasai products, but check out some random products from our Shop. Yeah, Lester Kasai probably shops here...
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- Contest Results - over a decade's worth of contest history.


text:
Rick's Roeder's thread http://bridgewinners.com/article/view/a-gnt-debacle-too-many-winners/is closing in on 800 comments. Thought it best to start a new thread with the official ruling shown below.
From: Ron Lien Sent: Friday, June 7, 2019 3:29 PM (6:29 PM EDT)
GNT Open Finalists,
The District has carefully reviewed the facts related to the Open GNT final held on May 26th. We take all of the issues raised very seriously. Our conclusion is that the Gupta team won the event.
There are four separate issues.
1. DIC behavior and performance
2. Conditions of Contest (COC)
3. Suggested improper player behavior
4. Actual IMP differentials in the three matches and the resulting VPs
The first three issues will be addressed, but the District and the ACBL have determined that they do not affect the outcome. The DIC behavior doesn't constitute a reason to overturn results but will be reviewed. The DIC's application of the wrong VP scale was corrected that evening.
The COC did not state a correction period. In fairness to the players, the District reviewed score cards and emails through June 6 submitted by all three teams, which resulted in an IMP score change that translated to Gupta winning by 1 VP instead of 2 VP.
Although there were statements that the COC was not available, John Kissinger, D-22 GNT Coordinator, insists there were two copies on DIC Scott Campbell's desk. Kissinger announced the format of play and type of scoring. Gupta claims their team asked and were informed about the form of scoring and played accordingly.
Several Player Memos have been filed with and initially reviewed by the ACBL. They will be handled in accordance with ACBL procedures.
To the best of our knowledge, we now know of one scoring change that has been confirmed and agreed to by all parties. The result of the Gupta match with Hiestand has been changed from +25 to +17. Although there were other suggestions of scoring errors, after 12 days none of these have been agreed upon. In an email Schafer said that Gupta agreed to a 1 IMP scoring error, but the issue was resolved with no change.
Each team played 30 boards against each of the other two teams. The attached 20 VP scale was used.
The final IMP differentials are as follows:
Schafer vs Hiestand (no changes) +10
Schafer vs Gupta (no changes) +3
Gupta vs Hiestand (originally reported as +25) +17
VP: Gupta (vs Schafer +10, vs Hiestand +13) +23
Schafer (vs Gupta +10, vs Hiestand +12) +22
The District has done all it can over the last 12 days to sort through the facts. We have also conferred with the ACBL in coming to this determination.
District 22 Board of Directors
The event ended on Sunday, May 26. The official ruling was issued this afternoon, 12 daysafter the event ended. Unprecedented!
The team that "lost"bv1 VP won two of the three matches in the 3 team final. They were told they had won. They went home thinking they had won. By total imps, they had won. But the CoCconverted imps to VPs using a scale that scored a 3 imp win as a 10-10 tie.
In my 43 years of bridge play, I have never seen such a VP scale used by the ACBL. I have never heard of a 3 way final being scored by imps converted to VP. I have never seen a team win 2 out of 3 matches in the final and be declared the loser. It boggles my mind.
But there was much more. There was no official VP scale posted in the playing area. It appears one VP scale was used, then changed. Two of the three teams in the final thought they were playing Wm/Loss. It was reported that the same CoC had existed for several years but by were ignored and imps convertedto VP were never used! So there was a clear established precedent to use Win/Loss.
There were allegations of unethical behaviour, a inept, confused Tournament Director... a complete fiasco. The "winning" team reported an incorrect score of winning by 25, later changed to winning by 17. The irregularities mounted up.
The "losing" team is considering legal action.
In my opinion, this is a disgraceful decision, one more nail in the ACBL coffin.
Perhaps Phil Clayton summed it up best:
Now I'm stranded in Honduras
I'm a desperate man
Send lawyers, guns and money
The shit has hit the fan
How does this violate our Community Guidelines?
Benefits include:
Plus... it's free!


text:
My first challenge, which is Be Inspired #18, was to make a shaped card. My mind went immediately to my "Hello Kitty Greetings" cartridge, because it has TONS of cute shapes, and actually has a card function on it. So, my first step was to cut the card shape out from black cardstock, and then to cut its layers from the two patterned papers. I chose a strawberry shaped card for my project, so after cutting the layers...I ran the green strawberry top through my Cuttlebug using the "Swiss Dots" embossing folder. I then sanded to top lightly to bring the dots out a little bit.
Once I was done with that, I layered the card together, and added some embellishments and the sentiment. And voila...here is the finished product!
I think it is pretty sweet!"You Are Berry Loved" Card
Cardstock: Bazzill Basics
Patterned Paper: American Crafts & Doodlebug Designs
Letter Stickers: American Crafts & Making Memories
Jewels: Prima
Twine: Darice
Craft Supply: button, ribbon, sandpaper
I hope you all enjoyed my first "Cuttlebug Be Inspired" challenge as a design team member! I can't wait to see what you do with the challenge!
Have a wonderful Wednesday,
41 comments
- AnonymousFebruary 11, 2009 at 7:56 AM
Oh Lea This is `berry` gorgeous...mmm`sweet`...ReplyDelete
have a lovely Day:)~x~
Oh, Lea, this is fabulous!!!!ReplyDelete
Love all the elements to this!
Congrats on being on the SDT!
Blessings, Maria
yes i agree it is pretty sweet, love the bowReplyDelete
Wow...I'm speechless...This card is gorgeous!ReplyDelete
- AnonymousFebruary 11, 2009 at 8:42 AM
I LOVE, LOVE, LOVE this card! And the extra little added bling really just makes it all pop.ReplyDelete
My mouth's watering now! This is so gorgeous Lea! I love all the detail you'e added to it, especially the bow and the bling. So happy that you are part of the DT!ReplyDelete
Hugs Mrs Jx
Fantastic!! Love this card!ReplyDelete
WOW, WOW, WOW!!!! I am speechless. This is so beautifully crafted!! Love it!ReplyDelete
Lookin' berry delicious I must say Lea! This is an adorable shape and sweet, sweet card! Fabulous!!ReplyDelete
OMG....what a cute card, Lea!! You make me want to have a Cricut too. ;-)ReplyDelete
Lea, this is gorgeous! I want that cartridge! It's been on my list of wants but after seeing your card it has definitely moved onto my list of needs!ReplyDelete
Big congrats on your new position on the team.
This is so SWEET nad summery!! Makes me want to eat a strawberry...YUM!! Love the bling on it too!ReplyDelete
This is totally cute & adorable. I LOVE it & now I think I need this cartridge:)ReplyDelete
Congratulations! So cute!ReplyDelete
This is so, so adorable! I'm loving the bows and bling!ReplyDelete
(-: Heidi
Lea I do love my Hello Kitty cartridge! And this is so funny, I had made a strawberry with mine weeks ago to use for this challenge and changed my mind because I did a strawberry card a few weeks ago and wanted to do something different! Your card is fabulous! I love it.ReplyDelete
This is really fabulous!!!!ReplyDelete
So sweet and beautiful. A nice touch with the bling.ReplyDelete
This really is a delicious card...love those swiss dots!ReplyDelete
I love this! It is 'berry' sweet!ReplyDelete
This is berry awesome. ;)ReplyDelete
Congrats Lea!!!!!!ReplyDelete
How BERRY awesome!
Adorable card Lea! Congrats on your new design gig :)ReplyDelete
Just FABALICIOUS!!! So sweet looking and I like the embellies and DP... You style of cards is very fun and fresh! Thanks for the lovely comment on my blog... I'll be visiting again soon!ReplyDelete
Con'grats on your new adventure!! :)
Such an amazing card Lea! Love love love how you've created the shaped card... XOXOReplyDelete
Soooo cute!! I am going to have to check out that cart!! Have I told you lately how much you inspire me!! I need to get some more cards done!! Hopefully tomorrow!!ReplyDelete
ABSOLUTELY amazing Lea, could NOT love it more!!!ReplyDelete
This is very cute :) I love the addition of the bling, the ribbon, button....just perfect!!ReplyDelete
Your strawberry is just wonderful! Love the bling and all of your details.ReplyDelete
that is adorable! so cute!ReplyDelete
Jealous of your Cuttlebug opportunity. LOVE that stuff soooo much! I guess I need to check them out - Your card is amazing! Totally want to take a bite.ReplyDelete
- AnonymousFebruary 12, 2009 at 9:37 PM
Lea, it's perfect! Talk about a sweet card! Welcome to the team! So glad to have you!!ReplyDelete
I agree...that card IS sweet! I just love how it turned out (my favorite part is the crystal swirls...but you probably knew I'd say that, right? ;-D). Like all your projects, it's beautiful! Anyone would be blessed to receive it.ReplyDelete
Thanks so much for the detailed instructions on creating it, Lea! I really do need to break out my Cricut more often an use it the way you do! LOL!
Have a great Valentine's Day and a wonderful weekend, Lea! :)
Wow this soooo sweet ! Well done on being a senior DT member, They made a great choice, Sarah-Jane xxReplyDelete
This is adorable! I love it, Lea!!ReplyDelete
How cute! I love your word play.ReplyDelete
This is gorgeous! Love all the bling.ReplyDelete
Lea this is the most beautiful strawberry I have ever seen I just love it and so clever. Thanks for leaving a comment on my blogReplyDelete
Lea your card is gorgeous, love all the details, scrummy!!! So cute & clever, HuGs Kim :DReplyDelete
- AnonymousFebruary 17, 2009 at 6:22 PM
Congrat's Lea!! You rock!! I have this Hello Kitty cartridge and haven't used it yet.. your strawberry is sooo sweet!ReplyDelete