Geometrically, the eigenvalue corresponds to the point in affine k-space with coordinates with respect to the basis given by

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, this corresponds to the common intersection of the varieties.

Geometrically, there is a rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension ( general type ) corresponds to negative curvature.

Geometrically, a nontrivial idempotent e corresponds to the function on X which is equal to 1 on some connected component ( s ) and 0 on others.

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 u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function ; the contours on which u and v are constant cross at right angles.

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, in singular cohomology of a manifold, one can interpret the product dually in terms of bounding manifolds and intersections, following Poincaré duality: dual to cocycles are cycles, often representable as closed manifolds ( without boundary ), dual to product is intersection, and dual to the subtraction of the bounding products is gluing the two bounding manifolds together along the boundary, obtaining a closed manifold which represents the homology class dual of the Massey product.

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 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.

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 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, 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 ).

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, 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, 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.

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 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, it is the

Geometrically, the theorem states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation.

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.

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, these subspaces are points, lines, planes, and so on, that pass through the point 0.

4 for squaring Curve lines Geometrically " ( emphasis added ).

Geometrically, as we move farther and farther to the right along the x-axis, the value of 1 / x < sup > 2 </ sup > approaches 0.

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, 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, this means that a horizontal tangent line to the graph at one point ( x

Observe that the characteristic function is not differentiable at the origin: this corresponds to the fact that the Cauchy distribution does not have an expected value.

In fact, every ESS corresponds to a Nash equilibrium, but some Nash equilibria are not ESSes.

This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus.

For positive J < sub > 0 </ sub > the thermodynamics of the Mattis spin glass corresponds in fact simply to a ferromagnet, just because these systems have no " frustration “ at all.

The non-abelian character of the SU ( 3 ) corresponds thereby to the non-trivial " chemical links “, which glue different loop segments together, and " asymptotic freedom " means in the polymer analogy simply the fact that in the short-wave limit, i. e. for ( where R < sub > c </ sub > is a characteristic correlation-length for the glued loops, corresponding to the above-mentioned " bag radius ", while λ < sub > w </ sub > is the wavelength of an excitation ) any non-trivial correlation vanishes totally, as if the system had crystallized.

In physical terms, this corresponds to the fact that there is no measurement protocol which can specify a basis of the spin space, even if a preferred basis of V is given.

In fact, the modern explanation of the uncertainty principle, extending the Copenhagen interpretation first put forward by Bohr and Heisenberg, depends even more centrally on the wave nature of a particle: Just as it is nonsensical to discuss the precise location of a wave on a string, particles do not have perfectly precise positions ; likewise, just as it is nonsensical to discuss the wavelength of a " pulse " wave traveling down a string, particles do not have perfectly precise momenta ( which corresponds to the inverse of wavelength ).

The fact that the sensation of Haidinger's brush corresponds with the visual field correlate of the macula means that it can be utilised in training people to look at objects with their macula.

Both authors agree that the sign was not readily understandable to denote Christ, which corresponds to the fact that there is no certain evidence of the use of the letters chi and rho as a Christian sign before Constantine.

Engel's version of the correspondence theory of truth explains that what makes a sentence true is that it corresponds to a fact.

The associated electronics use this fact to create a signal exactly at the frequency which corresponds to the external field.

This alludes to the fact that they all shine opposite the western, which corresponds to the Jewish people .”

In fact in northern Thailand the divide roughly corresponds to a long section of the political border of the country today.

While color derives from the electromagnetic scale that corresponds to the magnitudes of energy expressed by musical pitch, in fact, the neurological occidentals by which we experience color make it seem multidimensional, while musical pitch ( not timbre, volume, or duration ) is experienced as a linear relationship ... Stanczak's ' gift is for layering.

He argues that ' an environment which can be rendered ' corresponds to a set of mathematical questions whose answers can be calculated, and discusses various forms of the Turing Principle, which in its initial form refers to the fact that it is possible to build a universal computer which can be programmed to execute any computation that any other machine can do.

The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon.

A large part of the brain of whisker-specialist mammals is involved in the processing of nerve impulses from vibrissae, a fact that presumably corresponds to the important position the sense occupies for the animal.

In fact, the number and orders of these trees are uniquely determined by the number of nodes n: each binomial tree corresponds to one digit in the binary representation of number n. For example number 13 is 1101 in binary,, and thus a binomial heap with 13 nodes will consist of three binomial trees of orders 3, 2, and 0 ( see figure below ).

In clearness's second grade ( the " nominal " grade ), he defines truth as the correspondence of a sign ( in particular, a proposition ) to its object, and the real as the object ( be it a possibility or quality, or an actuality or brute fact, or a necessity or norm or law ) to which a true sign corresponds, such that truth and the real are independent of that which you or I or any actual, definite community of inquirers think.

In fact, this date in the Chinese / Japanese calendar corresponds to the date January 12, 1865, in our Western calendar.

In fact, one has the following: every closed subscheme of corresponds bijectively to a homogeneous ideal such that.

Mathematically, this corresponds to the fact that spherical coordinates do not define a coordinate chart on the sphere at zenith and nadir.

That this map is a homomorphism follows from the fact that the multiplication of unit complex numbers corresponds to addition of angles:

Indeed, the Future Value in this case grows linearly ( it's a linear function of the initial investment ): it doesn't take into account the fact that the interest earned might be compounded itself and produce further interest ( which corresponds to an exponential growth of the initial investment-see below -).