**Mathematics at New Renaissance Institute**

NRI uses a great deal of advanced mathematics in its technology R&D. Most of this has arisen from specific problems, for example:

Mathematical Dynamical Systems:

• Bilinear differential equations and their control

• Nonlinear dynamics and sensitivities of large-scale biochemical networks (signaling, gene expression, metabolic)

• Symbol dynamics

• Multi-variable hysteresis modeling and synthesis

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

**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)

• Extensions of Prolate Spheroidal Wave Functions and their discretation

• Various unpublished properties of continuous fractional Fourier and related operators.

**Stochastics and Statistics:**

• New applications and topics relating to ROC (Receiver Operating Characteristic) curves.

• 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:

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

• New applications of certain continuous-parameter Special Functions.

• 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.

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