NRI employs a considerable amount of advanced mathematics in its basic research, R&D, and technology. Much of this has arisen from specific problems, but NRI also performs research in pure mathematicswhen there is time and adequate internal funding. Some examples of past, recent, and current mathematical research at NRI include:

#### 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• Various unpublished properties of continuous fractional Fourier and related operators.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 and extensions to ROC (Receiver Operating Characteristic) curves and surfaces,

• Multiple-timescale and mixed-usage-statistics resource allocation (bandwidth, computation element, etc.),

• Parameterized Markov Chains,

• ‘Noncommutative (aka ‘Free”) probability’ (see for example Redei and Summers https://arxiv.org/pdf/quant-ph/0601158.pdf ) and applications.

#### 4. Pure Mathematical Research:

• 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 for a new type of fractional-order differential equations,

• A new framework for study of certain operator algebras,

• Fractional operators on Hilbert space and Schwartz space,

• Mercer and related expansion representations of integral operators,

• New formulations and structural results for tensors and multi–linear algebra,

• New properties of von Neuman algebras and related objects (W-*algebras, Regular Rings, Continuous Geometries, Orthomodular Lattices, Rota’s ‘

*Continuous Combinatorics*‘, Profinite constructions).NRI welcomes opportunities to collaborate with students, academic institutions, and private individuals in these and related areas.