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In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion. Linear algebra is the branch of Mathematics concerned with In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional

For a fixed positive real number r, the mapping

(x,y) → (r x, y / r )

is the squeeze mapping with parameter r. Since

\{(u,v)\ : \ u v = \mathrm{constant}\}

is a hyperbola, if u = r x and v = y / r, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions For this reason it is natural to think of the squeeze mapping as a "hyperbolic rotation", as did Émile Borel in 1913. Félix Édouard Justin Émile Borel ( January 7, 1871 in Saint-Affrique, France &ndash February 3, 1956 in Paris

Contents

Group theory

If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. In Mathematics, a composite function represents the application of one function to the results of another Therefore the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak An additive view of this group arises from consideration of hyperbolic sectors and hyperbolic angles. A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and ( A hyperbolic angle in standard position is the Angle at (0 0 between the ray to (1 1 and the ray to ( x, 1/ x) where x > 1

Literature

The myth of Procrustes is linked with this mapping in an educational (SMSG) publication:

Among the linear transformations, we have considered similarities, which preserve ratios of distances, but have not touched upon the more bizarre varieties, such as the Procrustean stretch (which changes a circle into an ellipse of the same area). "Damastes" redirects here For the huntman spider see Damastes or Sparassidae. The School Mathematics Study Group (SMSG was an academic Think tank focused on the subject of reform in Mathematics education.
Coxeter & Greitzer, pp. 100, 101.

In his 1999 monograph Classical Invariant Theory, Peter Olver discusses GL(2,R) and calls the group of squeeze mappings by the name the isobaric subgroup. An isobaric process is a Thermodynamic process in which the pressure stays constant \Delta p = 0 The term derives from the Greek isos "equal"

Applications

In studying linear algebra there are the purely abstract applications such as illustration of the singular-value decomposition or in the important role of the squeeze mapping in the structure of real matrices (2 x 2). In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix The 2 x 2 real matrices are the Linear mappings of the Cartesian coordinate system into itself by the rule (xy \mapsto (xy\begin{pmatrix}a & c These applications are somewhat bland compared to two physical and a philosophical application:

Fluid flow

In fluid dynamics one of the fundamental motions of an incompressible flow involves bifurcation of a flow running up against an immovable wall. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion In Fluid mechanics or more generally Continuum mechanics, an incompressible flow is Solid or Fluid flow in which the Divergence of Representing the wall by the axis y = 0 and taking the parameter r = exp(t) where t is time, then the squeeze mapping with parameter r applied to an initial fluid state produces a flow with bifurcation left and right of the axis x = 0. The same model gives fluid convergence when time is run backward. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Indeed, the area of any hyperbolic sector is invariant under squeezing. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and ( In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance.

For another approach to this flow with hyperbolic streamlines, see the article potential flow, section "Power law with n = 2". Fluid flow is described in general by a Vector field in three (for steady flows or four (for non-steady flows including time dimensions In Fluid dynamics, a potential flow is a Velocity field which is described as the Gradient of a scalar function the velocity potential

Relativistic spacetime

Select (0,0) for a "here and now" in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity track closer to the original timeline (0,t). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a Lorentz boost. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to This insight follows from a study of split-complex number multiplications and the "diagonal basis" which corresponds to the pair of light lines. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real Formally, a squeeze preserves the hyperbolic metric expressed in the form \ \ xy\ \ \!; in a different coordinate system. This application in the Theory of relativity was noted in 1912 by Wilson and Lewis (see footnote p. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. 401 of reference).

Bridge to transcendentals

The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function:

Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b). The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and (

Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d).

Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d).

Theorem (Gregoire de Saint-Vincent 1647) If bc = ad , then the quadrature of the hyperbola xy = 1 against the asymptote has equal areas between a and b compared to between c and d. Grégoire de Saint-Vincent ( March 22 1584 Bruges – June 5 1667 Ghent) a Jesuit, was a Mathematician

Proof: An argument adding and subtracting triangles of area ½, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma.

Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asyptote increase in geometric sequence. Alphonse Antonio de Sarasa was a Jesuit Mathematician who contributed to the understanding of Logarithms particularly as Areas under a Hyperbola Thus the areas form logarithms of the asymptote index.

For instance, on may ask “When is the hyperbolic angle in standard position equal to one?” The standard position angle runs from (1,1) to (x, 1/x). The answer is “When x = E (mathematical constant)" which is a transcendental number. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of area one. The geometric sequence

1,e, e2, e3, … en, …

corresponds to the asymptotic index achieved with each sum of areas

1,2,3, …, n,. . .

which is a proto-typical arithmetic progression A + nd where A = 0 and d = 1 . In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members

See also

References

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