Don Tucker – författare

Scientific research shows how experience shapes the organization of the human brain through mechanisms of neural plasticity, which capture the information of the world within the connections among neurons. To understand this plasticity, it is important to look to the developmental mechanisms through which the brain grows from a single cell in embryogenesis to achieve the complex architecture of the human brain. The process of neural morphogenesis involves exuberant formation of neuronal connections, and then subtractive elimination of unused connections. This process is continued after birth, providing the neural plasticity of learning that allows cognitive development in infancy and childhood. Recognizing this continuity suggests an interesting insight; cognition is a reflection of neural development throughout the life span.With this insight, the authors of Cognition and Neural Development examine the embryonic development of the brain to appreciate the dimensions of developmental momentum that shape the neural and psychological development of our lives. Human brain embryogenesis involves gradients of trophic factors that guide the migration of neurons from ventricular proliferative zones to organize the architecture of the cerebral hemispheres. The architecture of human cognition involves a functional differentiation of dorsal (pyramidal) and ventral (granular) corticolimbic divisions. This differentiation is a defining feature of not just human but mammalian neuroanatomy, The separation of pyramidal and granular cortical architectures appeared with the evolution of the six-layered mammalian neocortex from the three-layered primitive general cortex of reptiles and amphibians. The functional differentiation of the dorsal and ventral divisions of the cerebral hemispheres has been shown to be integral to multiple levels of psychological function, from elementary motivation to the most complex forms of executive self-regulation. Through an evolutionary-developmental analysis of cortical differentiation, the authors approach the basic questions of psychological function in novel ways.

In this monograph, leading researchers in the world ofnumerical analysis, partial differential equations, and hard computationalproblems study the properties of solutions of the Navier–Stokes partial differential equations on (x, y, z,t) ∈ ℝ3 × [0, T]. Initially converting the PDE to asystem of integral equations, the authors then describe spaces A of analytic functions that housesolutions of this equation, and show that these spaces of analytic functionsare dense in the spaces S of rapidlydecreasing and infinitely differentiable functions. This method benefits fromthe following advantages:The functions of S arenearly always conceptual rather than explicitInitial and boundaryconditions of solutions of PDE are usually drawn from the applied sciences,and as such, they are nearly always piece-wise analytic, and in this case,the solutions have the same propertiesWhen methods ofapproximation are applied to functions of A they converge at an exponential rate, whereas methods ofapproximation applied to the functions of S converge only at a polynomial rateEnables sharper bounds onthe solution enabling easier existence proofs, and a more accurate andmore efficient method of solution, including accurate error boundsFollowing the proofs of denseness, the authors prove theexistence of a solution of the integral equations in the space of functions A ∩ ℝ3 × [0, T], and provide an explicit novelalgorithm based on Sinc approximation and Picard–like iteration for computingthe solution. Additionally, the authors include appendices that provide acustom Mathematica program for computing solutions based on the explicitalgorithmic approximation procedure, and which supply explicit illustrations ofthese computed solutions.

In this monograph, leading researchers in the world ofnumerical analysis, partial differential equations, and hard computationalproblems study the properties of solutions of the Navier–Stokes partial differential equations on (x, y, z,t) ∈ ℝ3 × [0, T]. Initially converting the PDE to asystem of integral equations, the authors then describe spaces A of analytic functions that housesolutions of this equation, and show that these spaces of analytic functionsare dense in the spaces S of rapidlydecreasing and infinitely differentiable functions. This method benefits fromthe following advantages:

The functions of S are nearly always conceptual rather than explicit Initial and boundary conditions of solutions of PDE are usually drawn from the applied sciences, and as such, they are nearly always piece-wise analytic, and in this case, the solutions have the same properties When methods ofapproximation are applied to functions of A they converge at an exponential rate, whereas methods of approximation applied to the functions of S converge only at a polynomial rate Enables sharper bounds on the solution enabling easier existence proofs, and a more accurate and more efficient method of solution, including accurate error bounds

Following the proofs of denseness, the authors prove theexistence of a solution of the integral equations in the space of functions A ∩ ℝ3 × [0, T], and provide an explicit novelalgorithm based on Sinc approximation and Picard–like iteration for computingthe solution. Additionally, the authors include appendices that provide acustom Mathematica program for computing solutions based on the explicitalgorithmic approximation procedure, and which supply explicit illustrations ofthese computed solutions.