Presently, the main results of econophysics comprise the explanation of the " fat tails " in the distribution of many kinds of financial data as a universal self-similar scaling property ( i. e. scale invariant over many orders of magnitude in the data ), arising from the tendency of individual market competitors, or of aggregates of them, to exploit systematically and optimally the prevailing " microtrends " ( e. g., rising or falling prices ).

Certain neurophenomenological research groups, such as the Biogenetic Structuralism group, suggest that invariant patterns and structures discovered in first-person explorations of consciousness may find their explanation in the physiology and functioning of the brain.

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.

: Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.

The current state of psychological study about the nature of religiousness suggests that it is better to refer to religion as a largely invariant phenomenon that should be distinguished from cultural norms ( i. e. " religions ").

Keeping our aim at linear, time invariant systems, we can also characterize the multipath phenomenon by the channel transfer function, which is defined as the continuous time Fourier transform of the impulse response

( B ) each Af is invariant under T ; ;

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.

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.

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.

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.

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.

The property of unit-treatment additivity is not invariant under a " change of scale ", so statisticians often use transformations to achieve unit-treatment additivity.

Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance ( As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant ).

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

A characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G. That is,

A subgroup of G that is invariant under all inner automorphisms is called normal.

Since a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal.

Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry.

Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity.

All standards developed by the Free Standards Group ( FSG ) were released under open terms ( the GNU Free Documentation License with no cover texts or invariant sections ) and test suites, sample implementations and other software were released as free software.

A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rota and his school.

In mathematics, the symbolic method in invariant theory is an algorithm developed by,,, and in the 19th century for computing invariants of algebraic forms.

In 1930, Paul Dirac developed a new version of the Wave Equation which was relativistically invariant ( unlike Schrödinger's one ), and predicted the magnetic moment correctly, and at the same time treated the electron as a point particle.

There are many well developed techniques for dealing with the response of linear time invariant systems, such as Laplace and Fourier transforms.

It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory.

Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometry questions.

In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.

The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i. e., which is diffeomorphism invariant, is infinite.

This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory.

Therefore, the genus is an important topological invariant of a Riemann surface.

Topological ideas in the realm of gravity date back to Riemann, Clifford and Weyl and found a more concrete realization in the wormholes of Wheeler characterized by the Euler-Poincare invariant.

For instance, by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula.

For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multiply valued harmonic function as a single-valued function on a branched cover of R < sup > n </ sup > or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or orbifold.

On Riemann surfaces, isometries are identical to changes of coordinate: that is, both the Laplace – Beltrami operator and the curvature are invariant under isometries.

The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors.

Therefore, the Riemann surface, or more simply its genus is a birational invariant.

These homologies are closely related to the Donaldson and Seiberg invariants of 4-manifolds, as well as to Taubes's Gromov invariant of symplectic 4-manifolds ; the differentials of the corresponding three-manifold homologies to these theories are studied by considering solutions to the relevant differential equations ( Yang – Mills, Seiberg – Witten, and Cauchy – Riemann, respectively ) on the 3-manifold cross R. The 3-manifold Floer homologies should also be the targets of relative invariants for four-manifolds with boundary, related by gluing constructions to the invariants of a closed 4-manifold obtained by gluing together bounded 3-manifolds along their boundaries.