Cauchy product

In geometry and combinatorics, an arrangement of hyperplanes is a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The intersection semilattice of A, written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These subspaces are called the flats of A. L(A) is partially ordered by reverse inclusion.

If the whole space S is 2-dimensional, the hyperplanes are lines; such an arrangement is often called an arrangement of lines. Historically, real arrangements of lines were the first arrangements investigated. If S is 3-dimensional one has an arrangement of planes.

General theory

The intersection semilattice and the matroid

The intersection semilattice L(A) is a meet semilattice and more specifically is a geometric semilattice. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice. (This is why the semilattice must be ordered by reverse inclusion—rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)

When L(A) is a lattice, the matroid of A, written M(A), has A for its ground set and has rank function r(S) := codim(I), where S is any subset of A and I is the intersection of the hyperplanes in S. In general, when L(A) is a semilattice, there is an analogous matroid-like structure that might be called a semimatroid, which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if L(A) is not a lattice.

Polynomials

For a subset B of A, let us define f(B) := the intersection of the hyperplanes in B; this is S if B is empty. The characteristic polynomial of A, written p_A(y), can be defined by

$p_A(y):=\sum _B(-1{)}^{\left|B\right|}{y}^{\dim f(B)},$

summed over all subsets B of A except, in the affine case, subsets whose intersection is empty. (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with A is the Whitney-number polynomial w_A(x, y), defined by

$w_A(x,y):=\sum _B{x}^{n-\dim f(B)}\sum _C(-1{)}^{|C-B|}{y}^{\dim f(C)},$

summed over B ⊆ C ⊆ A such that f(B) is nonempty.

Being a geometric lattice or semilattice, L(A) has a characteristic polynomial, p_L(A)(y), which has an extensive theory (see matroid). Thus it is good to know that p_A(y) = yⁱ p_L(A)(y), where i is the smallest dimension of any flat, except that in the projective case it equals y^{i + 1}p_L(A)(y). The Whitney-number polynomial of A is similarly related to that of L(A). (The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)

The Orlik–Solomon algebra

The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik–Solomon algebra. To define it, fix a commutative subring K of the base field, and form the exterior algebra E of the vector space

$\underset{H\in A}{⨁}K{e}_H$

generated by the hyperplanes. A chain complex structure is defined on E with the usual boundary operator $\partial$. The Orlik-Solomon algebra is then the quotient of E by the ideal generated by elements of the form $e_{H_1}\wedge \cdots \wedge e_{H_p}$ (where $H_1,\dots ,H_p$ have an empty intersection) and by boundaries of elements of the same form for which $H_1\cap \cdots \cap H_p$ has codimension less than p.

Real arrangements

In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity. Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of A. The regions are faces because the whole space is a flat. The faces of codimension 1 may be called the facets of A. The face semilattice of an arrangement is the set of all faces, ordered by inclusion. Adding an extra top element to the face semilattice gives the face lattice.

In two dimensions (i.e., in the real affine plane) each region is a convex polygon (if it is bounded) or a convex polygonal region which goes off to infinity.

• As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set. There are four regions, none of them bounded.
• If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them bounded.
• If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded parallelograms.

A typical problem about an arrangement in n-dimensional real space is to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems are that the number of regions of an affine arrangement equals (−1)ⁿp_A(−1) and the number of bounded regions equals (−1)^{np_A(1). Similarly, the number of k-dimensional faces or bounded faces can be read off as the coefficient of xn−k in (−1)n w_A (−x, −1) or (−1)nw_A(−x, 1).}

Template:Harvtxt designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.

Another question about an arrangement in real space is to decide how many regions are simplices (the n-dimensional generalization of triangles and tetrahedra). This cannot be answered based solely on the intersection semilattice. The McMullen problem asks for the smallest arrangement of a given dimension in general position in real projective space for which there does not exist a cell touched by all hyperplanes.

A real linear arrangement has, besides its face semilattice, a poset of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, B₀, and associating with each region R the set S(R) consisting of the hyperplanes that separate R from B. The regions are partially ordered so that R₁ ≥ R₂ if S(R₁, R) contains S(R₂, R). In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak Bruhat order. In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed Template:Harv.

Complex arrangements

In complex affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) with holes where the hyperplanes were removed.

A typical problem about an arrangement in complex space is to describe the holes.

The basic theorem about complex arrangements is that the cohomology of the complement M(A) is completely determined by the intersection semilattice. To be precise, the cohomology ring of M(A) (with integer coefficients) is isomorphic to the Orlik-Solomon algebra on Z.

The isomorphism can be described rather explicitly, and gives a presentation of the cohomology in terms of generators and relations, where generators are represented (in the de Rham cohomology) as logarithmic differential forms

$\frac{1}{2\pi i}\frac{d\alpha }{\alpha }.$

with $\alpha$ any linear form defining the generic hyperplane of the arrangement.

Technicalities

Sometimes it is convenient to allow the degenerate hyperplane, which is the whole space S, to belong to an arrangement. If A contains the degenerate hyperplane, then it has no regions because the complement is empty. However, it still has flats, an intersection semilattice, and faces. The preceding discussion assumes the degenerate hyperplane is not in the arrangement.

Sometimes one wants to allow repeated hyperplanes in the arrangement. We did not consider this possibility in the preceding discussion, but it makes no material difference.

See also

• Supersoluble arrangement

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.