NRI <\/span>employs a considerable amount of advanced mathematics in its <\/span>basic research, R&D, and technology. Much of this has arisen from specific problems, but NRI also <\/span>performs\u00a0research\u00a0 in pure mathematics<\/span>when there is time and adequate internal funding. Some examples of \u00a0past, recent, and current mathematical research at NRI <\/span>include:\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n

1. Mathematical Dynamical Systems:<\/span><\/h4>\n<\/div>\n<\/div>\n
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\u2022 Bilinear differential equations and their control<\/span>,\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n
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\u2022 Multi-variable hysteresis modeling and synthesis,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Hierarchical control systems, especially those involving bilinear and fractional-order dynamics.<\/span><\/span><\/div>\n<\/div>\n
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2. Mathematics in <\/span>Selected Areas of Advanced Signal Processing Applications:<\/span><\/span><\/h4>\n<\/div>\n<\/div>\n
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\u2022 \u201cCentered\u201d 1D & 2D fractional-order discrete Fourier transforms for computational optics<\/span>,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Complex-value powers of fractional Fourier transform (continuous and discrete)<\/span>,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Symbol dynamics in signal processing, for example in <\/span>frequency<\/span> and <\/span>phase comparators,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Extensions of Prolate Spheroidal Wave Functions and their discretization<\/span>,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Various unpublished properties of continuous fractional Fourier and related operators.<\/span><\/span><\/div>\n<\/div>\n
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3. Stochastics and Statistics:<\/span><\/h4>\n<\/div>\n<\/div>\n
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\u2022 New applications and extensions to ROC (Receiver Operating Characteristic) curves and surfaces,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Multiple-timescale and mixed-usage-statistics resource allocation (bandwidth, computation element, etc.),\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Parameterized Markov Chains,<\/span><\/div>\n
\u2022 ‘Noncommutative (aka ‘Free”) probability’ (see for example Redei\u00a0and Summers\u00a0https:\/\/arxiv.org\/pdf\/quant-ph\/0601158.pdf<\/a>) and applications.<\/span><\/div>\n<\/div>\n
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4. Pure Mathematical Research:<\/span><\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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\u2022 New \u201cneo-classical\u201d findings in the areas of special functions, integral transforms, and Hilbert-Schmidt\u00a0integral operators,
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\u2022 New applications of certain continuous-parameter Special Functions,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Eigenfunction studies <\/span>for\u00a0a new type of fractional-order differential equations,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 A new framework <\/span>for study of<\/span> certain operator algebras<\/span>,\u00a0<\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 Fractional operators on Hilbert space and Schwartz space,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 <\/span>Mercer and related expansion representations of integral operators,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 New <\/span>formulations and <\/span>structural results for tensors <\/span>and multi<\/span>–linear algebra,\u00a0<\/span><\/span><\/div>\n<\/div>\n<\/div>\n
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\u2022 <\/span>New properties of von Neuman algebras and related objects (W-*algebras,\u00a0Regular Rings, Continuous\u00a0Geometries, Orthomodular Lattices, Rota’s ‘Continuous\u00a0Combinatorics<\/i>‘, Profinite constructions).<\/span><\/div>\n<\/div>\n<\/div>\n
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NRI welcomes opportunities to collaborate with students, academic institutions, and private individuals in these and related areas.<\/span><\/div>\n","protected":false},"excerpt":{"rendered":"
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\u00a0research\u00a0 in pure mathematicswhen there is time and adequate internal funding. Some examples of \u00a0past, recent, and current mathematical research at NRI include:\u00a0 1. Mathematical Dynamical Systems: \u2022 Bilinear differential equations and their control,\u00a0 \u2022 Multi-variable hysteresis modeling and synthesis,\u00a0 \u2022 Hierarchical control systems, especially those involving bilinear and fractional-order dynamics. \u00a0 2. Mathematics […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"ngg_post_thumbnail":0},"_links":{"self":[{"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/pages\/394"}],"collection":[{"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=394"}],"version-history":[{"count":21,"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/pages\/394\/revisions"}],"predecessor-version":[{"id":3216,"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=\/wp\/v2\/pages\/394\/revisions\/3216"}],"wp:attachment":[{"href":"https:\/\/newrenaissanceinstitute.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=394"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}