Mathematics

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• Various unpublished properties of continuous fractional Fourier and related operators.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