By E. W. Cheney
The approximation of capabilities of numerous variables remains to be a tough challenge in clinical computing simply because a number of the algorithms required for such difficulties have not begun to be written. This monograph is written for a vast viewers of computational mathematicians and statisticians concerned about the improvement of algorithms or the derivation of approximations from linear projections, of which the interpolating operators are an immense instance. As an relief to either researchers and scholars, a bibliography of greater than two hundred titles is incorporated.
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Extra info for Multivariate approximation theory: selected topics
Program continues to run for stationary config. input U waste input′ d′ U d′′ C′ C energy 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 42 Thermodynamics of a Configuration • Either, configuration is a stable state – damage may shift to undesirable equilibrium • Or, configuration is a stationary state of a non-equilibrium system – continuously reconfigures self – self-repair as return to original stationary state – adaptation & damage recovery as move to different stationary state 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 43 Useful Media for Computational Synthesis • For pure computation, move as little matter & energy as possible • For synthesis, need to control atoms & molecules as well as electrons • Need sufficiently wide variety of controllable atoms & molecules • Goal: structures on the order of optical wavelengths (100s of nm) 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 44 Models of Computation for Synthesis • Need massive parallelism to control detailed organization of state • Need tolerance to errors in state – synthesis program should be tolerant – configured computer should be tolerant 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 45 Locus of Control of Detailed Organization • Reorganizing atoms & molecules ⇒ vast amount of detailed control • Heterosynthesis – external configuration controller determines fine structure of medium (high bandwidth) • Autosynthesis – external configuration controller determines general boundary conditions (low BW) – fine structure results from self-organization 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 46 General Model of Radical Reconfiguration • Synthesis controller – low bandwidth to outside world – bandwidth to medium: • high for heterosynthesis • low for autosynthesis • Synthetic medium – molar parallelism of interactions • simple for heterosynthesis • complex for autosynthesis – what are suitable synthetic media?
Micro-organisms can be viewed a micro-robots with capabilities for: – – – – – – – – – locomotion sensing control simple (low-precision analog) computation assembly collective, coordinated behavior reproduction self-defense metabolism (matter/energy acquisition, growth, repair) • Can be genetically-engineered for our purposes 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 58 The Programmable Microorganism (“Promorg”) • Noncoding DNA can be used for “genetic circuits” – in eukaryotes, 10–70% – equivalent of about 3000 genes in yeast • Equipped with an assortment of generally useful sensors & receptors (especially for selforganization) • Special-purpose modifications for particular applications • Research: principles of design & self-organization 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 59 Tentative Capabilities • Neutral proteins for programmable control – gene regulatory & coding sequences – connections can be regulated by external signals • Membrane & cytoplasm receptors for: – chemical signals – light, etc.
Cazamine 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 48 Example: Activation-Inhibition System • Let σ be the logistic sigmoid function • Activator A and inhibitor I may diffuse at different rates in x and y directions • Cell is “on” if activator + bias exceeds inhibitor "A "2A "2A = dAx 2 + dAy 2 + k A# [ mA ( A + B $ I )] "t "x "y "I " 2I " 2I = dIx 2 + dIy 2 + k I# [ mI ( A + B $ I )] "t "x "y 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 49 Double Activation-Inhibition System • Two independently diffusing activation-inhibition pairs • May have different diffusion rates in X and Y directions – In this example, I1y >> I1x and I2x >> I2y • Colors in simulation: – – – – green = system 1 active red = system 2 active yellow = both active black = neither active 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 50 Formation of Pattern • Random initial state • System stabilizes to < 1% cell changes • Modest noise (annealing noise) improves regularity 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 51 Stationary State • System is being continually maintained in a stationary state • Continuing change < 1% 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 52 Recovery from Damage • Simulated damage • Damage destroys activators & inhibitors as well as structure • System repairs self by returning to stationary state • No explicit repair signal 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 53 Reconfiguration: Orthogonal Structure • Exchange inhibitor diffusion rates for systems 1 & 2 • Vertical stripes become horizontal • Horizontal stripes become vertical • No explicit reconfiguration signal 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 54 Summary of Radical Reconfiguration • Computation can be used to rearrange matter • External control of initial and boundary conditions • Detailed structure by self-organization with molar parallelism • Stationary states can be used for self-repair and adaptation 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 55 Programmable Microorganisms for Artificial Morphogenesis Artificial Morphogenesis • Based on models of embryological development • Cells migrate by local interaction & chemical signals • Possible implementation: “programmable” micro-organisms 6 March 2007 Self-Organization for NanoComputation & Nano-Assembly 57 Why Micro-Organisms?