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12.1: Conformal Disc Model


In this section, we give new names for some objects in the Euclidean plane which will represent lines, angle measures, and distances in the hyperbolic plane. Let us fix a circle on the Euclidean plane and call it absolute. The set of points inside the absolute will be called the hyperbolic plane (or (h)-plane). Note that the points on the absolute do not belong to the h-plane. The points in the h-plane will be also called h-points.

Often we will assume that the absolute is a unit circle.

Definition: Hyperbolic lines

The intersections of the h-plane with circlines perpendicular to the absolute are called hyperbolic lines or h-lines.

By Corollary 10.5.3, there is a unique h-line that passes thru the given two distinct h-points (P) and (Q). This h-line will be denoted by ((PQ)_h).

The arcs of hyperbolic lines will be called hyperbolic segments or h-segments. An h-segment with endpoints (P) and (Q) will be denoted by ([PQ]_h).

The subset of an h-line on one side from a point will be called a hyperbolic half-line (or h-half-line). More precisely, an h-half-line is an intersection of the h-plane with arc perpendicular to the absolute that has exactly one of its endpoints in the h-plane. An h-half-line starting at (P) and passing thru (Q) will be denoted by ([PQ)_h).

If (Gamma) is the circline containing the h-line ((PQ)_h), then the points of intersection of (Gamma) with the absolute are called ideal points of ((PQ)_h). (Note that the ideal points of an h-line do not belong to the h-line.)

An ordered triple of h-points, say ((P,Q,R)) will be called h-triangle (PQR) and denoted by ( riangle_h P Q R).

Let us point out, that so far an h-line ((PQ)_h) is just a subset of the h-plane; below we will introduce h-distance and later we will show that ((PQ)_h) is a line for the h-distance in the sense of the Definition 1.5.1.

Exercise (PageIndex{1})

Show that an h-line is uniquely determined by its ideal points.

Hint

Let (A) and (B) be the ideal points of the h-line (ell). Note that the center of the Euclidean circle containing (ell) lies at the intersection of the lines tangent to the absolute at the ideal points of (ell).

Exercise (PageIndex{2})

Show that an h-line is uniquely determined by one of its ideal points and one h-point on it.

Hint

Assume (A) is an ideal point of the h-line (ell) and (P in ell). Suppose that (P') denotes the inverse of (P) in the absolute. By Corollary 10.5.1, (ell) lies in the intersection of the h-plane and the (necessarily unique) circline passing thru (P, A), and (P')

Exercise (PageIndex{3})

Show that the h-segment ([PQ]_h) coincides with the Euclidean segment ([PQ]) if and only if the line ((PQ)) pass thru the center of the absolute.

Hint

Let (Omega) and (O) denote the absolute and its center.

Let (Gamma) be the circline containing ([PQ]_h). Note that ([PQ]_h = [PQ]) if and only if (Gamma) is a line.

Suppose that (P') denotes the inverse of (P) in (Omega). Note that (O, P), and (P') lie on one line.

By the definition of h-line, (Omega perp Gamma). By Corollary 10.5.1, (Gamma) passes thru (P) and (P'). Therefore, (Gamma) is a line if and only if it pass thru (O).

Hyperbolic distance

Let (P) and (Q) be distinct h-points; let (A) and (B) denote the ideal points of ((PQ)_h). Without loss of generality, we may assume that on the Euclidean circline containing the h-line ((PQ)_h), the points (A,P,Q,B) appear in the same order.

Consider the function

(delta(P,Q):= dfrac{AQ cdot PB}{AP cdot QB}.)

Note that the right hand side is a cross-ratio; by Theorem 10.2.1 it is invariant under inversion. Set (delta(P,P)=1) for any h-point (P). Let us define h-distance as the logarithm of (delta); that is,

(PQ_h := ln[delta(P,Q)].)

The proof that (PQ_h) is a metric on the h-plane will be given later. For now it is just a function that returns a real value (PQ_h) for any pair of h-points (P) and (Q).

Exercise (PageIndex{4})

Let (O) be the center of the absolute and the h-points (O), (X), and (Y) lie on one h-line in the same order. Assume (OX=XY). Prove that (OX_h

Hint

Assume that the absolute is a unit circle.

Set (a = OX = OY). Note that (0 < a < dfrac{1}{2}), (OX_h = ln dfrac{1+ a}{1 -a}), and (XY_h = ln dfrac{(1 + 2 cdot a) cdot (1 - a)}{(1 - 2 cdot a)cdot (1 + a)}). It remains to check that the inequalities

(1 < dfrac{1 + a}{1 - a} < dfrac{(1 + 2 cdot a) cdot (1 - a)}{(1 - 2 cdot a)cdot (1 + a)})

hold if (0 < a < dfrac{1}{2}).

Hyperbolic angles

Consider three h-points (P), (Q), and (R) such that (P e Q) and (R e Q). The hyperbolic angle (PQR) (briefly (angle_h PQR)) is an ordered pair of h-half-lines ([QP)_h) and ([QR)_h).

Let ([QX)) and ([QY)) be (Euclidean) half-lines that are tangent to ([QP]_h) and ([QR]_h) at (Q). Then the hyperbolic angle measure (or h-angle measure) of (angle_h PQR) denoted by (measuredangle_h PQR) and defined as (measuredangle XQY).

Exercise (PageIndex{5})

Let (ell) be an h-line and (P) be an h-point that does not lie on (ell). Show that there is a unique h-line passing thru (P) and perpendicular to (ell).

Hint

Spell the meaning of terms "perpendicular" and "h-line" and then apply Exercise 10.5.4.


Poincaré disc model

The Poincaré disc model for ℍ 2 is the disc < ( x , y ) ∈ ℝ 2 : x 2 + y 2 < 1 >in which a point is similar to the Euclidean point and a line must be one of the following:

a diameter (excluding its endpoints ) of the unit circle

an arc (excluding its endpoints) of a circle such that it intersects the unit circle at two distinct points and the two circles are perpendicular at both intersection points.

The Poincaré disc model has the drawback that lines in the model do not resemble Euclidean lines however, it has the advantage that it is angle preserving. That is, the Euclidean of an angle within the model is the angle measure in hyperbolic geometry. For this reason, this model is also referred to as the conformal disc model. (See the entry conformal for more details.)

Some points outside of the Poincaré disc model are important for constructions within the model. The following is an example of such:

Let ℓ be a line in the Poincaré disc model that is not a diameter of the circle. The pole of ℓ is the intersection of the Euclidean lines that are tangent ( http://planetmath.org/TangentLine ) to the circle at the endpoints of ℓ .

Note that this matches the definition of pole for the Beltrami-Klein model. Also, poles are important for the same reason that they are important in the Beltrami-Klein model: Given a line ℓ that is not a diameter of the Poincaré disc model, one constructs a line perpendicular to ℓ by considering Euclidean lines passing through P ⁢ ( ℓ ) . Thus, two disjointly parallel lines ℓ and m that are not diameters of the Poincaré disc model, one constructs their common perpendicular by connecting their poles. It is actually much easier to do this construction by finding the poles of the two lines, finding the common perpendicular with respect to the Beltrami-Klein model, then converting the common perpendicular to the Poincaré disc model. See the entry on converting between the Beltrami-Klein model and the Poincaré disc model for more details.

In all pictures in this entry from this point on, blue segments are lines in the Beltrami-Klein model, and red arcs are lines in the Poincaré disc model.

Below is a picture of two disjointly parallel lines ℓ and m in the Poincaré disc model, neither of which is a diamter of the unit circle:


Contents

is an example of a real analytic and bijective function from the open unit disk to the plane its inverse function is also analytic. Considered as a real 2-dimensional analytic manifold, the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is homeomorphic to the whole plane.

There is however no conformal bijective map between the open unit disk and the plane. Considered as a Riemann surface, the open unit disk is therefore different from the complex plane.

There are conformal bijective maps between the open unit disk and the open upper half-plane. So considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") to the upper half-plane, and the two are often used interchangeably.

Much more generally, the Riemann mapping theorem states that every simply connected open subset of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.

One bijective conformal map from the open unit disk to the open upper half-plane is the Möbius transformation

Geometrically, one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk's interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the composition of two stereographic projections: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways onto a vertical half-plane touching the sphere, taking the point on the half-sphere opposite to the touching point as projection center.

The unit disk and the upper half-plane are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not.

The open unit disk forms the set of points for the Poincaré disk model of the hyperbolic plane. Circular arcs perpendicular to the unit circle form the "lines" in this model. The unit circle is the Cayley absolute that determines a metric on the disk through use of cross-ratio in the style of the Cayley–Klein metric. In the language of differential geometry, the circular arcs perpendicular to the unit circle are geodesics that show the shortest distance between points in the model. The model includes motions which are expressed by the special unitary group SU(1,1). The disk model can be transformed to the Poincaré half-plane model by the mapping g given above.

Both the Poincaré disk and the Poincaré half-plane are conformal models of the hyperbolic plane, which is to say that angles between intersecting curves are preserved by motions of their isometry groups.

Another model of hyperbolic space is also built on the open unit disk: the Beltrami-Klein model. It is not conformal, but has the property that the geodesics are straight lines.

One also considers unit disks with respect to other metrics. For instance, with the taxicab metric and the Chebyshev metric disks look like squares (even though the underlying topologies are the same as the Euclidean one).

The area of the Euclidean unit disk is π and its perimeter is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disk in the taxicab geometry is 8. In 1932, Stanisław Gołąb proved that in metrics arising from a norm, the perimeter of the unit disk can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular hexagon or a parallelogram, respectively.


10.2  Hyperbolic Cayley Transform

A hyperbolic version of the Cayley transform was used in [136]. The above formula (2) in ℝ h becomes

with some subtle differences in comparison to (3). The corresponding A , N and K orbits are given in Fig.ꀐ.3( H ). However, there is an important distinction between the elliptic and hyperbolic cases similar to the one discussed in Sectionਈ.2.

Exerciseਂ    Check, in the hyperbolic case, that the real axis is transformed to the cycle
ℝ=<( u , v )  ∣   v  =਀> → T h = < ( u , v )  ∣   l c h 2 ( u , v )=  u 2 − v 2 =𢄡>,      (8)
where the length from the centre l c h 2 is given by (6) for σ=σ c =1 and coincides with the distance d h , h  (2). On the hyperbolic unit circle, SL 2(ℝ) acts transitively and it is generated, for example, by point (0,1).

SL 2(ℝ) acts also transitively on the whole complement

to the unit circle, i.e. on its “inner” and “outer” parts together.

Recall from Sectionਈ.2 that we defined ℂ′ to be the two-fold cover of the hyperbolic point space ℝ h consisting of two isomorphic copies ℝ h + and ℝ h − glued together, cf. Fig.ਈ.3. The conformal version of the hyperbolic unit disk in ℂ′ is, cf. the upper half-plane from (1),

ℂ′ = <( u , v )∈ ℝh +   ∣   u 2 − v 2 >𢄡> ⋃  <( u , v )∈ ℝh −   ∣   u 2 − v 2 <𢄡 >.     (9)

  1. ℂ′ is conformally-invariant and has a boundary ℂ′ —two copies of the unit hyperbolas inh + andh.
  2. The hyperbolic Cayley transform is a one-to-one map between the hyperbolic upper half-plane ℂ′ + and hyperbolic unit disk ℂ′ .

We call ℂ′ the hyperbolic unit cycle in ℝ h . Figureਈ.3(b) illustrates the geometry of the hyperbolic unit disk in ℂ′ compared to the upper half-plane. We can also say, rather informally, that the hyperbolic Cayley transform maps the “upper” half-plane onto the “inner” part of the unit disk.

One may wish that the hyperbolic Cayley transform diagonalises the action of subgroup A , or some conjugate of it, in a fashion similar to the elliptic case (4) for K . Geometrically, it corresponds to hyperbolic rotations of the hyperbolic unit disk around the origin. Since the origin is the image of the point ι in the upper half-plane under the Cayley transform, we will use the isotropy subgroup . Under the Cayley map (7), an element of the subgroup becomes

where e є t = cosh t +є sinh t . The corresponding Mཫius transformation is a multiplication by e 2є t , which obviously corresponds to isometric hyperbolic rotations of ℝ h for distance d h , h and length l c h. This is illustrated in Fig.ꀐ.1(H:A’).


This project is no longer being actively maintained. If you are looking for a Windows version of Redis, you may want to check out Memurai. Please note that Microsoft is not officially endorsing this product in any way.

  • This is a port for Windows based on Redis.
  • We officially support the 64-bit version only. Although you can build the 32-bit version from source if desired.
  • You can download the latest unsigned binaries and the unsigned MSI installer from the release page.
  • For releases prior to 2.8.17.1, the binaries can found in a zip file inside the source archive, under the bin/release folder.
  • Signed binaries are available through NuGet and Chocolatey.
  • Redis can be installed as a Windows Service.
  • There is a replacement for the UNIX fork() API that simulates the copy-on-write behavior using a memory mapped file on 2.8. Version 3.0 is using a similar behavior but dropped the memory mapped file in favor of the system paging file.
  • In 3.0 we switch the default memory allocator from dlmalloc to jemalloc that is supposed to do a better job at managing the heap fragmentation.
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There are two current active branches: 2.8 and 3.0.

How to configure and deploy Redis on Windows

How to build Redis using Visual Studio

You can use the free Visual Studio 2013 Community Edition. Regardless which Visual Studio edition you use, make sure you have updated to Update 5, otherwise you will get a "illegal use of this type as an expression" error.

Open the solution file msvs edisserver.sln in Visual Studio, select a build configuration (Debug or Release) and target (x64) then build.

This should create the following executables in the msvs$(Target)$(Configuration) folder:

  • redis-server.exe
  • redis-benchmark.exe
  • redis-cli.exe
  • redis-check-dump.exe
  • redis-check-aof.exe

To run the Redis test suite some manual work is required:

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  • By default the test suite launches 16 parallel tests, but 2 is the suggested number.

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The Bill of Material (BOM) number is used to identify Dana differentials. The BOM will identify the model number, the gear ratio, the type of differential, and all component parts. Traditional BOM’s are 6 digits followed by 1 or 2 digits and start with the numbers 60 or 61. On some tags, the first 2 digits don’t appear on the tag, but they must be used to identify the axle. For instance, you might see 5561-1 for the BOM, but the 60 has been dropped, and one would need to use 605561-1. Later BOM’s may start with the first 3 digits of 200, but these are typically not dropped from the tag.

*** The B.O.M. Numbers stamped into the long tube of the axle housing are often faint and hard to read. Using a metal scraper will typically reveal the numbers most easily. Cleaning with a wire wheel is not recommended as it usually makes the numbers harder to see ***

The “Dana Expert” website is no longer available. However, the Dana Aftermarket Media Library can be used to identify all aspects of a particular axle using the Bill Of Materials number.

  1. Navigate to the Dana Aftermarket Media Library
  2. Enter the BOM number in the Keyword Search field (top left corner of the page).
  3. Click on the resulting literature link
  4. Search within the document by clicking on the Magnifying Glass at the top, or by simply using “Control-F” and re-entering the BOM number.

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12.1: Conformal Disc Model

CONTENTS Introduction Chapter I. Local projective and conformal field algebras as structure rings of quantum deformations of non-commutative coverings of the complex disc § 1. Models of Verma modules over mathrm(2,mathbb) and the Virasoro Lie algebra, and deformations of the complex disc § 2. The L-algebras L(mathrm(2,mathbb)) and L(mathbbmathrm) as structure rings of quantum deformations of the complex disc § 3. Local projective field algebras (LPFA) and local conformal field algebras (LCFA) § 4. Pseudotensorial modules over the L-algebras L(mathrm(2,mathbb)) and L(mathbbmathrm) § 5. Vertex operators in the models of Verma modules over mathrm(2,mathbb) and the Virasoro algebra § 6. The LPFA and LCFA as structure rings of quantum deformations of non-commutative coverings of the complex disc Chapter II. QPFT-operator algebras and QCFT-operator algebras as "systems of local rings" on non-commutative coverings of the complex disc § 1. QPFT- and QCFT-operator algebras § 2. Vertex operator fields in models of the Verma modules over mathrm(2,mathbb) and the Virasoro algebra mathbbmathrm § 3. The algebras mathrm(mathrm(2,mathbb)) and mathrm (mathbbmathrm c) of vertex operators for mathrm(2,mathbb) and mathbbmathrm Chapter III. The reduction of quantum conformal field theory to quantum projective theory § 1. The algebra mathrm(mathrm(2,mathbb)) of vertex operators for mathrm(2,mathbb) and the representation of QPFT-operator algebras § 2. The universal infinite-dimensional R-matrix R_>(u) of quantum projective field theory § 3. The LPFA LOEL of linear operators in the model of the Verma modules over mathrm(2,mathbb) and the representation of LPFA by matrices with coefficients in LOEL References


If we change the definition of 'distance' slightly we create a space with different properties.

In Euclidian space the distance from the origin is given by &radic(x²+y²+z²) but in Minkowski space we change the definition of distance to &radic(x²+y²+z²-t²). This gives lots of interesting properties which we look at on this page.


12.1: Conformal Disc Model

CONTENTS Introduction Chapter I. Local projective and conformal field algebras as structure rings of quantum deformations of non-commutative coverings of the complex disc § 1. Models of Verma modules over (2,) and the Virasoro Lie algebra, and deformations of the complex disc § 2. The L-algebras L((2,)) and L() as structure rings of quantum deformations of the complex disc § 3. Local projective field algebras (LPFA) and local conformal field algebras (LCFA) § 4. Pseudotensorial modules over the L-algebras L((2,)) and L() § 5. Vertex operators in the models of Verma modules over (2,) and the Virasoro algebra § 6. The LPFA and LCFA as structure rings of quantum deformations of non-commutative coverings of the complex disc Chapter II. QPFT-operator algebras and QCFT-operator algebras as "systems of local rings" on non-commutative coverings of the complex disc § 1. QPFT- and QCFT-operator algebras § 2. Vertex operator fields in models of the Verma modules over (2,) and the Virasoro algebra § 3. The algebras ((2,)) and ( c) of vertex operators for (2,) and Chapter III. The reduction of quantum conformal field theory to quantum projective theory § 1. The algebra ((2,)) of vertex operators for (2,) and the representation of QPFT-operator algebras § 2. The universal infinite-dimensional R-matrix R proj >(u) of quantum projective field theory § 3. The LPFA LOEL of linear operators in the model of the Verma modules over (2,) and the representation of LPFA by matrices with coefficients in LOEL References


There are several terms to describe disc pathologies

  1. Disc bulge ie the circumference of disc extends beyond the vertebral bodies. involves the NP. Disc herniation is significant in that it may compress an adjacent spinal nerve. A herniated disc impinges upon the nerve associated with the inferior vertebrae (e.g., L4/L5 herniation affects the L5 nerve root). The most common site of disc herniation is at L5-S1, which may be due to the thinning of the posterior longitudinal ligament towards its caudal end. There are three subtypes of herniations:
    • Disc protrusion is characterized by the width of the base of the protrusion is wider than the diameter of the disc material that is herniated.
    • In disc extrusion, the AF is damaged, allowing the NP to herniate beyond the normal bounds of the disc. In this case, the herniated material produces a mushroom-like dome that is wider than the neck connecting it to the body of the NP. The herniation may extend superior or inferiorly relative to the disc level.
    • In disc sequestration, the herniated material breaks off from the body of the NP.
  2. Disc desiccation is common in aging. It is brought about by the death of the cells that produce and maintain the ECM, including proteoglycans, such as aggrecan. The NP shrinks as the gelatinous form is replaced with fibrotic tissue, reducing its functionality, and leaves the AF supporting additional weight. This increased stress leads the AF to compensate by increasing in size. The resulting flattened disc reduces mobility and may impinge on spinal nerves leading to pain and weakness. It is thought to be due to proteoglycan breakdown, which reduces the water-retaining properties of the NP.

NB: Significant research has been put into means of replacing/re-growing the intervertebral discs. The various methods include the replacement of discs with synthetic materials, stem cell therapy, and gene therapy. [1]

Additional Points [ edit | edit source ]

There is no intervertebral disc between C1 and C2, which is unique in the spine.


Watch the video: Introduction to Poincare Disk (December 2021).