Well-behaved Representation Research Articles

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups G with a well-behaved representation theory it is shown that the invariant skew fields are free on |G|(d−1)+1 generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.

A Psychophysical Experiment Regarding Components of the Plausibility Illusion

We report on the design and results of an experiment investigating factors influencing Slater's Plausibility Illusion (Psi) in virtual environments (VEs). Slater proposed Psi and Place Illusion (PI) as orthogonal components of virtual experience which contribute to realistic response in a VE. PI corresponds to the traditional conception of presence as "being there," so there exists a substantial body of previous research relating to PI, but very little relating to Psi. We developed this experiment to investigate the components of plausibility illusion using subjective matching techniques similar to those used in color science. Twenty-one participants each experienced a scenario with the highest level of coherence (the extent to which a scenario matches user expectations and is internally consistent), then in eight different trials chose transitions from lower-coherence to higher-coherence scenarios with the goal of matching the level of Psi they felt in the highest-coherence scenario. At each transition, participants could change one of the following coherence characteristics: the behavior of the other virtual humans in the environment, the behavior of their own body, the physical behavior of objects, or the appearance of the environment. Participants tended to choose improvements to the virtual body before any other improvements. This indicates that having an accurate and well-behaved representation of oneself in the virtual environment is the most important contributing factor to Psi. This study is the first to our knowledge to focus specifically on coherence factors in virtual environments.

Intensity-Curvature Measurement Approaches for the Diagnosis of Magnetic Resonance Imaging Brain Tumors

This research presents signal-image post-processing techniques called Intensity-Curvature Measurement Approaches with application to the diagnosis of human brain tumors detected through Magnetic Resonance Imaging (MRI). Post-processing of the MRI of the human brain encompasses the following model functions: (i) bivariate cubic polynomial, (ii) bivariate cubic Lagrange polynomial, (iii) monovariate sinc, and (iv) bivariate linear. The following Intensity-Curvature Measurement Approaches were used: (i) classic-curvature, (ii) signal resilient to interpolation, (iii) intensity-curvature measure and (iv) intensity-curvature functional. The results revealed that the classic-curvature, the signal resilient to interpolation and the intensity-curvature functional are able to add additional information useful to the diagnosis carried out with MRI. The contribution to the MRI diagnosis of our study are: (i) the enhanced gray level scale of the tumor mass and the well-behaved representation of the tumor provided through the signal resilient to interpolation, and (ii) the visually perceptible third dimension perpendicular to the image plane provided through the classic-curvature and the intensity-curvature functional.

Function algebras on a 2-dimensional quantum complex plane

The well-behaved representations of the coordinate algebra of a 2-dimensional quantum complex plane are classified and a C*-algebra is defined which can be viewed as the algebra of continuous functions on the 2-dimensional quantum complex plane vanishing at infinity.

A Resolvent Approach to the Real Quantum Plane

Let q ≠ ± 1 be a complex number of modulus one. This paper deals with the operator relation AB = qBA for self-adjoint operators A and B on a Hilbert space. Two classes of well-behaved representations of this relation are studied in detail and characterized by resolvent equations.

Unbounded Induced Representations of ∗-Algebras

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra \({{\mathcal{A}}}\) onto a unital *-subalgebra \({{\mathcal{B}}}\) are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras \({{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g\) for which the *-subalgebra \({{\mathcal{B}}}:={{\mathcal{A}}}_e\) is commutative. Then the canonical projection \(p:{{\mathcal{A}}}\to{{\mathcal{B}}}\) is a conditional expectation and there is a partial action of the group G on the set \({{\mathcal{B}}}p\) of all characters of \({{\mathcal{B}}}\) which are nonnegative on the cone \(\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.\) The complete Mackey theory is developed for *-representations of \({{\mathcal{A}}}\) which are induced from characters of \({{\widehat{{{\mathcal{B}}}}^+}}.\) Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras \({{\mathcal{A}}}\) is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.

Effective representations of the space of linear bounded operators

<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.</p>

Down–Up Algebras

The algebra generated by the down and up operators on a differential or uniform partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down–up algebras. We show that down–up algebras exhibit many of the important features of the universal enveloping algebraU(sl2) of the Lie algebra sl2including a Poincaré–Birkhoff—Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down–up algebras and focus especially on Verma modules, highest weight representations, and category O modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.

Calculation of the cumulative reaction probability via a discrete variable representation with absorbing boundary conditions

A new method is suggested for the calculation of the microcanonical cumulative reaction probability via flux autocorrelation relations. The Hamiltonian and the flux operators are computed in a discrete variable representation (DVR) and a well-behaved representation for the Green’s operator, G(E+), is obtained by imposing absorbing boundary conditions (ABC). Applications to a one-dimensional-model problem and to the collinear H+H2 reaction show that the DVR-ABC scheme provides a very efficient method for the direct calculation of the microcanonical probability, circumventing the need to compute the state-to-state dynamics. Our results indicate that the cumulative reaction probability can be calculated to a high accuracy using a rather small number of DVR points, confined to the vicinity of the transition state. Only limited information regarding the potential-energy surface is therefore required, suggesting that this method would be applicable also to higher dimensionality problems, for which the complete potential surface is often unknown.