Algebraic invariant

Algebraic Invariants of Knots and Links, and Non-Abelian Field Models... 

Reidemeister Moves, State Model for Construction of Algebraic Invariants and Yang-Baxter Relations... 

A chemical compound might be expected to be quite as much like a proposition as like an algebraic invariant. 

The shape types Tj are usually specified by various algebraic methods, for example, by a shape group or a shape matrix, or by some other algebraic or numerical means. The algebraic invariants or the elements of the matrices are numbers, and these numbers form a shape code. The (P,W)-shape similarity technique provides a nonvisual, algebraic, algorithmic shape description in terms of numerical shape codes, suitable for automatic, computer characterization and comparison of shapes and for the numerical evaluation of 3D shape similarity. 

Hint to solution 2. Apply the theory of algebraic invariants of differential equations as outlined, for example, in the book by Sibirskii (1982, p. 91 et seq.). 

Sibirskii, K. S. (1976). Algebraic invariants of differential equations and matrices. Stiintsa, Kishinev (in Russian). 

Introduction to the theory of algebraic invariants of differential equations... 

An important property of the homotopic maps is that they become identical once we pass to the algebraic invariants that we have defined. 

In this section we define yet another family of algebraic invariants of topological spaces. We give just a very brief overview without proofs. 

For the identification of the topological state of the knot we use the Kauffman algebraic invariant K(A), which is the Laurent polynomial in A variable. We have shown the Kauffman invariant to be equal to the partition function of some special disordered Potts model [3, 4]. The number of equivalent states, and the nearest neighbor interaction constant, J /, are defined as follows ... 

A.Grosberg, S.Nechaev Algebraic invariants of knots and disordered Potts model , J.Phys.A. Math. Gen., 25 4659 (1992). 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

For each Lie algebra, one can construct a set of operators, called invariant (or Casimir, 1931) operators after the name of the physicist who first introduced them in connection with the rotation group. These operators play a very important role since they are associated with constants of the motion. They are defined as those operators that commute with all the elements of the algebra... 

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic,. Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator... 

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

The last problem of general interest in algebraic theory is the evaluation of the eigenvalues of the invariant operators in the basis discussed in Section 2.4. As mentioned before, the invariant operators commute with all the Xs. As a result, they are diagonal in the basis [A,], A, ..., A.v],... 

Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term E(). The algebra U(l) has a linear invariant... 

Since nx is an invariant, so is n. One can thus write down the most general bilinear algebraic Hamiltonian with dynamic symmetry U(l) as... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that... 

J. Li, Algebraic geometric interpretation of Donaldson s polynomial invariants, J. Differential Geom. 37 (1993) 417-466. 

In practical applications, we invariably invoke the algebraic approximation by parametrizing the orbitals in a finite basis set. This approximation may be written... 

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