NRI uses a great deal of advanced mathematics in its technology and R&D. Much of this has arisen from specific problems, but NRI also does work in pure mathematics. Below is a summary of past, recent, and current mathematical research at NRI.

#### 1. Mathematical Dynamical Systems:

• Bilinear differential equations and their control

• Multi-variable hysteresis modeling and synthesis

• Hierarchical control systems, especially those involving bilinear and fractional-order dynamics.

#### 2. Mathematics in selected Areas of Advanced Signal Processing Applications:

• “Centered” 1D & 2D fractional-order discrete Fourier transforms for computational optics

• Complex-value powers of fractional Fourier transform (continuous and discrete)

• Symbol dynamics in signal processing, for example in Frequency and Phase Comparators

• Extensions of Prolate Spheroidal Wave Functions and their discretization

• Various unpublished properties of continuous fractional Fourier and related operators.

#### 3. Stochastics and Statistics:

• New applications, topics and extensions relating to ROC (Receiver Operating Characteristic) curves and surfaces

• Bandwidth management and resource allocation (Markovian and non-Markovian)

• Generalized-inverses for tensor mappings among elements in matrix-valued spaces.

NRI also engages in pure mathematical research when there is time and internal funding. Some examples of this work include:

• New “neo-classical” findings in the areas of special functions, integral transforms, and Hilbert-Schmidt integral operators

• New applications of certain continuous-parameter Special Functions

• Eigenfunction studies of a new type of fractional-order differential equations

• A new framework for the topological study of certain types of operator algebras

• Fractional operators on Hilbert space and Schwartz space

• Mercer, Karhunen–Loève, and related expansion representations of integral operators

• New Formulations and Structural results for tensors in multi linear algebra.

NRI welcomes opportunities to collaborate with students, academic institutions, and private individuals in these and related areas.