Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

Finally, the image of a general bundle of lines is a congruence whose order is the order of the congruence of invariant lines, namely Af and whose class is the order of the image congruence of a general plane field of lines, namely Af.

The preceding observations make it clear that there exist line involutions of all orders greater than 1 with no complex of invariant lines and with a complex of singular lines consisting exclusively of the lines which meet a twisted curve Aj.

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 example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

The integral here is taken over the set of all lines incident with the point x ∈ R < sup > n </ sup >, and the measure dμ is the unique probability measure on the set invariant under rotations about the point x.

As Grünbaum writes, simplicial arrangements “ appear as examples or counterexamples in many contexts of combinatorial geometry and its applications .” For instance, use simplicial arrangements to construct counterexamples to a conjecture on the relation between the degree of a set of differential equations and the number of invariant lines the equations may have.

For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines.

It is certainly clear that the subspaces Af are invariant under T.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

Thus it follows that the secants of **zg are all invariant.

* The diffraction angles are invariant under scaling ; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.

Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angle s. The latter sort of properties are called invariant ( mathematics ) | invariant s and studying them is the essence of geometry.

In order to better understand the world of appearances and objects, phenomenology attempts to identify the invariant features of how objects are perceived and pushes attributions of reality into their role as an attribution about the things we perceive ( or an assumption underlying how we perceive objects ).

Most filters implemented in analog electronics, in digital signal processing, or in mechanical systems are classified as causal, time invariant, and linear.

The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, while others are destroyed.

Those KAM tori that are not destroyed by perturbation become invariant Cantor sets, named Cantori by Ian C. Percival in 1979.

Under a Galilean transformation the equations of Newtonian dynamics are invariant, whereas those of electromagnetism are not.

Special relativity shows that rest mass ( or invariant mass ) and rest energy are essentially equivalent, via the well-known relationship E = mc < sup > 2 </ sup >.

Since energy is dependent on reference frame ( upon the observer ) it is convenient to formulate the equations of physics in a way such that mass values are invariant ( do not change ) between observers, and so the equations are independent of the observer.

Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

Physical properties such as length, mass or time, by themselves, are not physically invariant.

However, the laws of physics which include these properties are invariant.

This law allows that stable long-run frequencies are a manifestation of invariant single-case probabilities.

For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

In classical Euclidean geometry, one is interested in studying the properties of figures in the plane which are invariant under congruence, so that if two figures are congruent then they must have the same properties.

The fixed curve is kept invariant ; the rolling curve is subjected to a continuous congruence transformation such that at all times the curves are tangent at a point of contact that moves with the same speed when taken along either curve ( another way to express this constraint is that the point of contact of the two curves is the instant centre of rotation of the congruence transformation ).

This curve is of symbol Af since it meets **yl, and hence every line of Af in the Af invariant points on **yl and since it obviously meets every line of Af in a single point.

Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of R < sup > n </ sup >.

However, since s is an unphysical parameter, physical states must be left invariant by " s-evolution ", and so the physical state space is the kernel of H − E ( this requires the use of a rigged Hilbert space and a renormalization of the norm ).

σ < sub > 3 </ sub > this reproduces the σ < sub > 1 </ sub > σ < sub > 2 </ sub > rotation considered in the previous section ; and that such rotation leaves the coefficients of vectors in the σ < sub > 3 </ sub > direction invariant, since

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field.

The mass-energy equivalence formula requires isolated systems, since if energy is allowed to escape a system, both relativistic mass and invariant mass will escape also.

However, since the vector v itself is invariant under the choice of basis,

Note that the invariant mass of a system of particles may be more than the sum of the particles ' rest masses, since kinetic energy in the system center-of-mass frame and potential energy from forces between the particles contribute to the invariant mass.

This is an important measure since it is unit invariant ( can figuratively compare apples to oranges )

The vacuum charge density should be zero, since the vacuum is Lorentz invariant, but this is artificial to arrange in Dirac's picture.

The intrinsic parity's phase is conserved for non-weak interactions ( the product of the intrinsic parities is the same before and after the reaction ), since = 0, i. e.: the Hamiltonian is invariant under a parity transformation.

In this case, the mass of the container is given by its total energy ( including the kinetic energy of the gas molecules ), since the system total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed.

If the photons are formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle ( their rest energy plus the kinetic energy ), in the center of mass frame, where they will automatically be moving in equal and opposite directions ( since they have equal momentum in this frame ).

If the principal bundle P is the frame bundle, or ( more generally ) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant R < sup > n </ sup >- valued 1-form on P, should be taken into account.

However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame.

That is because each simplex of S should be covered by exactly N in S ′ — at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler characteristic is a topological invariant.

Berry and Sierra have conjectured that since this operator is invariant under dilations perhaps the boundary condition f ( nx ) = f ( x ) for integer ' n ' may help to get the correct asymptotic results valid for big ' n '

The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism.

The characteristic and minimal polynomials of M are the same as those of A, which we would expect, since M can be obtained via a similarity transformation P < sup >− 1 </ sup > AP = M, and determinants are similarity invariant.