To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation.

If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations.

In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension.

The affine transformation is given by:

An example of an affine transformation which is not a Euclidean motion is given by scaling.

The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M < sup >− 1 </ sup >( Ω ) does not touch or cross the ellipse.

A final wrinkle is that Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

* The Euclidean group E < sub > n </ sub >( R ) is the Lie group of all Euclidean motions, i. e., isometric affine maps, of n-dimensional Euclidean space R < sup > n </ sup >.

* The Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space.

A finite affine plane of order q, with the lines as blocks, is an S ( 2, q, q < sup > 2 </ sup >).

The projective plane of order 2 ( the Fano plane ) is an STS ( 7 ) and the affine plane of order 3 is an STS ( 9 ).

* Affine subspace, linear manifold or linear variety, is a geometric structure that generalizes the affine properties of Euclidean space.

It is a subset of a vector space closed under affine combinations of vectors in the space.

This more general type of spatial vector is the subject of vector spaces ( for bound vectors ) and affine spaces ( for free vectors ).

In geometry, an affine transformation or affine map or an affinity ( from the Latin, affinis, " connected with ") is a transformation which preserves straight lines ( i. e., all points lying on a line initially still lie on a line after transformation ) and ratios of distances between points lying on a straight line ( e. g., the midpoint of a line segment remains the midpoint after transformation ).

An affine transformation is equivalent to a linear transformation followed by a translation.

An affine map between two affine spaces is a map that induces a linear transformation on vectors, defined by pairs of points.

Since a ruled surface of order N with N concurrent generators is necessarily a cone, it follows finally that every line through a point, P, of **zg meets its image at P, as asserted.

Moreover, in this involution there is a cone of invariant lines of order Af, namely the cone of secants of **zg which pass through P.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area.

A shuttlecock ( often abbreviated to shuttle ; also called a birdie ) is a high-drag projectile, with an open conical shape: the cone is formed from sixteen overlapping feathers embedded into a rounded cork base.

Morphologically, the breast is a cone with the base at the chest wall, and the apex at the nipple, the center of the NAC ( nipple-areola complex ).

When a symbol is not marked by any of the three grammar symbols ( square, cone, inverted cone ), they may be a non material thing, a grammatical particle, etc.

A cone ( from the Greek κῶνος, Latin conus ) is a basic geometrical shape.

* Convex cone, a subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C

* Parasitic cone ( or satellite cone ) is a geographical feature found around a volcano

* Ice cream cone, an edible container in which ice cream is served, shaped like an inverted cone open at its top

If a source emits a known luminous intensity I < sub > v </ sub > ( in candelas ) in a well-defined cone, the total luminous flux Φ < sub > v </ sub > in lumens is given by

where A is the radiation angle of the lamp — the full vertex angle of the emission cone.

The memorial is a fountain in the form of a round stone inverted cone.

As ejecta escapes from the growing crater, it forms an expanding curtain in the shape of an inverted cone ; the trajectory of individual particles within the curtain is thought to be largely ballistic.

It is instead often topped with an inverted frustum of a shallow cone or a cylindrical band of carvings.

The thickest, strongest, and closed part of the cone is located near the explosive charge.

In mathematics, an ellipse ( from Greek ἔλλειψις elleipsis, a " falling short ") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve.

A typical result is the 1: 3 ratio between the volume of a cone and a cylinder with the same height and base.

Kolo is a roasted barley snack food often served in a paper cone.

`` The main bunch is outside, but there are some over there inside the wall ''.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

`` That crap is softer over here ''.

Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??

To him, law is the command of the sovereign ( the English monarch ) who personifies the power of the nation, while sovereignty is the power to make law -- i.e., to prevail over internal groups and to be free from the commands of other sovereigns in other nations.

It is the gait of the human who must run to live: arms dangling, legs barely swinging over the ground, head hung down and only occasionally swinging up to see the target, a loose motion that is just short of stumbling and yet is wonderfully graceful.

Nostalgic Yankee readers of Erskine Caldwell are today informed by proud Georgians that Tobacco Road is buried beneath a four-lane super highway, over which travel each day suburbanite businessmen more concerned with the Dow-Jones average than with the cotton crop.

Unruly hair goes straight up from his forehead, standing so high that the top falls gently over, as if to show that it really is hair and not bristle.

The information is furnished by each of the guests, is sent by oral broadcasting over the air waves, and is received by the ears.

The dweller at p is last to hear about a new cure, the slowest to announce to his neighbors his urgent distresses, the one who goes the farthest to trade, and the one with the greatest difficulty of all in putting over an idea or getting people to join him in a cooperative effort.

So in these pages the term `` technology '' is used to include any and all means which could amplify, project, or augment man's control over himself and over other men.

And the anxiety it generates is misinterpreted as anxiety over private interest and threatened social status.

The basic truth in the reactionary response is to be found in its realistic assumption of the primacy of the real over the ideational.

Going back over this ground and analyzing the composition of forces which have created the present scene is one of the tasks undertaken by the Center for the Study of Democratic Institutions, in Santa Barbara.

When I take over Taliesin, the first thing I'll do is fire you ''.

Carl thought the question over slowly and answered: `` I know a starving man who is fed never remembers all the pangs of his starvation, I know that ''.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

The `` conventional '' image of a particular time and place is not necessarily congruent with the image of the facts as established over the years by scholarly and scientific research.