A. J. Bracken

Hepatic elimination of flowing substrates: The distributed model

An earlier model of hepatic elimination with functionally identical sinusoids is extended by introducing statistical distributions of enzyme contents per sinusoid and of blood flow per sinusoid, these being either uncorrelated or closely correlated. The steady-state theory of the resulting distributed model is developed, including methods of determining experimentally the coefficients of variation of the distributions. Such determinations are made on an illustrative experimental example. Quantitative predictions of expected effects of changes in blood flow are given, including one for which the undistributed model predicts a null effect. Shapes of the postulated distributions are discussed only in relation to observable effects. Effects of the distributions are compared with maximum possible effects of incomplete equilibration of substrate within each sinusoidal cross-section, and methods for distinguishing these effects from each other are outlined.

New Supersymmetric and Exactly Solvable Model of Correlated Electrons.

A new lattice model is presented for correlated electrons on the unrestricted 4L-dimensional electronic Hilbert space n=1LC4 (where L is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin, and Schoutens. The supersymmetry algebra of the new model is superalgebra gl(2 |1). The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter U, and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreducible representations of gl(2 |1). On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.

VECTOR OPERATORS AND A POLYNOMIAL IDENTITY FOR SO(n).

It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n‐vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.

From Representations of the Braid Group to Solutions of the Yang-Baxter Equation

Abstract A systematic method is developed for constructing solutions of the Yang-Baxter equation from given braid group representations, arising from such finite dimensional irreps of quantum groups that any irrep can be affinized and the tensor product of the irrep with itself is multiplicity-free. The main tool used in the construction is a tensor product graph, whose circuits give rise to consistency conditions. A maximal tree of this graph leads to an explicit formula for the quantum R-matrix when the consistency conditions are satisfied. As examples, new solutions of the Yang-Baxter equation are found, corresponding to braid group generators associated with the symmetric and antisymmetric tensor irreps of Uq[gl(m)], a spinor irrep of Uq[so(2n)]. and the minimal irreps of Uq[E6] and Uq[E7].

Quantum group invariants and link polynomials

A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsUq(E8),Uq(so(2m+1) andUq(gl(m)) are considered as examples, and corresponding link polynomials are obtained.

Quantum Supergroups and Solutions of the Yang-Baxter Equation

A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp(1/2) and gl(m/n).

A q-analogue of Bargmann space and its scalar product

A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmanns scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.

Solutions of the Quantum Yang-Baxter Equation with Extra Nonadditive Parameters

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as Uq(su(1,1)) and type-I quantum superalgebras such as Uq(gl(1 mod 1)) and Uq(gl(2 mod 1)) are known to admit non-trivial one-parameter families of infinite-dimensional and finite-dimensional irreps, respectively, even for generic q. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples, we work out the the R-matrices for the three quantum algebras mentioned above in certain representations.

Generalized Gel’fand invariants and characteristic identities for quantum groups

The generalized Gel’fand invariants for an arbitrary quantum group are explicitly constructed, and their eigenvalues in any irreducible representation are computed. These invariants enable one to develop characteristic identities for the quantum group, and as a natural application, these identities are used to construct projection operators for tensor products of representations. To illustrate the general theory, the quantum group Uq(gl(m)) is studied in detail.

One-dimensional quantum walk with unitary noise

The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin (n=0) at time t=0, the position distribution P-t(n) at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, sigma(t) scales as sigma(t)similar tot, unlike the classical random walk for which sigma(t)similar toroott. It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be Tsimilar toalpha(-2), where alpha is the standard deviation of the noise.