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Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set ( the solution set of a collection of polynomials in n variables is generally a geometric object ( such as a curve or a surface ) in n-space ).
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Geometrically and any
where v is any fixed vector satisfying Av = b. Geometrically, this says that the solution set to Ax = b is the translation of the null space of A by the vector v. See also Fredholm alternative.
* Geometrically reduced and having k geometric connected components ( for any fixed k ).
Geometrically, the construction goes like this: for any point ( cos φ, sin φ ) on the unit circle, draw the line passing through it and the point (− 1, 0 ).
Geometrically we notice that any point Q common to C and C ′ is also on each of the conics of the linear system.
Geometrically and infinite
Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy ( in the infinite dimensional field space ), much like the saddle point of the surface in three dimensional analytic geometry.
Geometrically and may
Geometrically two edges meeting at a corner are required to form an angle that is not straight ( 180 °); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed.
Geometrically this says that fibres of good mappings may have ' infinitesimal ' structure.
Geometrically and be
Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized ( at least in the case when r is an integer ) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.
Geometrically, the trace can be interpreted as the infinitesimal change in volume ( as the derivative of the determinant ), which is made precise in Jacobi's formula.
Geometrically, the swastika can be regarded as an irregular icosagon or 20-sided polygon.
Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry.
Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection ( from the new position of the sphere ) to the plane.
Geometrically, it can be described as an arrow from the origin of the space ( vector tail ) to that point ( vector tip ).
Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution.
Geometrically, this solution will be a vertex of the convex polytope consisting of all feasible points.
Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using convex analysis.
Geometrically, the degrees of freedom can be interpreted as the dimension of certain vector subspaces.
Geometrically and finite
** Geometrically finite groups
Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K < sub > v </ sub >- rational points for all places v of K. The Selmer group is finite.
Geometrically and with
Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles.
Geometrically, ( x < sub > 1 </ sub >, 0 ) is the intersection with the x-axis of a line tangent to f at ( x < sub > 0 </ sub >, f ( x < sub > 0 </ sub >)).
Geometrically, the eigenvalue corresponds to the point in affine k-space with coordinates with respect to the basis given by
Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together.
f ( x ) over some interval containing a. Geometrically, the graph defined by R ( x, y ) = 0 will overlap locally with the graph of some equation y = f ( x ).
Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length ; this follows from the Pythagorean theorem.
Geometrically, the locus defined by will overlap locally with the graph of a function ( an explicit function, see article on implicit functions ).
Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal in the fiber product of Spec ( S ) with itself over Spec ( S ) → Spec ( R ).
Geometrically, the operation consists in cutting each vertex of the polyhedron with a plane cutting all edges adjacent to the vertex at their midpoints ; it is sometimes named rectification.
Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a, b, and c, with bivectors, and matching the parallelogram faces of the parallelepiped.
Geometrically and same
Geometrically, the theorem states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation.
Geometrically, split-complex numbers are related to the modulus ( x < sup > 2 </ sup > − y < sup > 2 </ sup >) in the same way that complex numbers are related to the square of the Euclidean norm ( x < sup > 2 </ sup > + y < sup > 2 </ sup >).
Geometrically and is
Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero.
Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state.
Geometrically, the relative position vector R < sub > B / A </ sub > is the vector from point A to point B.
Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image.
Geometrically, four-acceleration is a curvature vector of a world line.
Geometrically, a diagonalizable matrix is an inhomogeneous dilation ( or anisotropic scaling ) – it scales the space, as does a homogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis ( diagonal entries ).
Geometrically, it is the
Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x
Geometrically, a ribbon is a piece of the envelope of the osculating planes of the curve.
Geometrically, it is possible to " roll " a plane along the ribbon without slipping or twisting so that the regulus always remains within the plane.
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