On the Image Conjecture

The Image Conjecture was formulated by the third author, who showed that it implied his Vanishing Conjecture, which is equivalent to the famous Jacobian Conjecture. We prove various cases of the Image Conjecture and show how it leads to another fascinating and elusive assertion that we here dub the Factorial Conjecture. Various cases of the Factorial Conjecture are proved

Images of Locally Finite Derivations of Polynomial Algebras in Two Variables

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and $div D=0$ is a Mathieu subspace of $k[x, y]$.Comment: Minor changes and improvements. Latex, 9 pages. To appear in J. Pure Appl. Algebr

Biomechanical models and mechanisms of cellular morphogenesis and cerebral cortical expansion and folding

Morphogenesis of the nervous system involves a highly complex spatio-temporal pattern of physical forces (mainly tension and pressure) acting on cells and tissues that are pliable but have an intricately organized cytoskeletal infrastructure. This review begins by covering basic principles of biomechanics and the core cytoskeletal toolkit used to regulate the shapes of cells and tissues during embryogenesis and neural development. It illustrates how the principle of \u27tensegrity\u27 provides a useful conceptual framework for understanding how cells dynamically respond to forces that are generated internally or applied externally. The latter part of the review builds on this foundation in considering the development of mammalian cerebral cortex. The main focus is on cortical expansion and folding - processes that take place over an extended period of prenatal and postnatal development. Cortical expansion and folding are likely to involve many complementary mechanisms, some related to regulating cell proliferation and migration and others related to specific types and patterns of mechanical tension and pressure. Three distinct multi-mechanism models are evaluated in relation to a set of 18 key experimental observations and findings. The Composite Tension Plus (CT+) model is introduced as an updated version of a previous multi-component Differential Expansion Sandwich Plus (DES+) model (Van Essen, 2020); the new CT+ model includes 10 distinct mechanisms and has the greatest explanatory power among published models to date. Much needs to be done in order to validate specific mechanistic components and to assess their relative importance in different species, and important directions for future research are suggested

Stable Tameness of Two-Dimensional Polynomial Automorphisms Over a Regular Ring

In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.Comment: 18 page

Mapping human cortical areas in vivo based on myelin content as revealed by t1- and t2-weighted MRI

Non-invasively mapping the layout of cortical areas in humans is a continuing challenge for neuroscience. We present a new method of mapping cortical areas based on myelin content as revealed by T1-weighted (T1w) and T2-weighted (T2w) MRI. The method is generalizable across different 3T scanners and pulse sequences. We use the ratio of T1w/T2w image intensities to eliminate the MR-related image intensity bias and enhance the contrast to noise ratio for myelin. Data from each subject was mapped to the cortical surface and aligned across individuals using surface-based registration. The spatial gradient of the group average myelin map provides an observer-independent measure of sharp transitions in myelin content across the surface—i.e. putative cortical areal borders. We found excellent agreement between the gradients of the myelin maps and the gradients of published probabilistic cytoarchitectonically defined cortical areas that were registered to the same surface-based atlas. For other cortical regions, we used published anatomical and functional information to make putative identifications of dozens of cortical areas or candidate areas. In general, primary and early unimodal association cortices are heavily myelinated and higher, multi-modal, association cortices are more lightly myelinated, but there are notable exceptions in the literature that are confirmed by our results. The overall pattern in the myelin maps also has important correlations with the developmental onset of subcortical white matter myelination, evolutionary cortical areal expansion in humans compared to macaques, postnatal cortical expansion in humans, and maps of neuronal density in non-human primates

On the Image Conjecture

The Image Conjecture was formulated by the third author, who showed that it implied his Vanishing Conjecture, which is equivalent to the famous Jacobian Conjecture. We prove various cases of the Image Conjecture and show that how it leads to another fascinating and elusive assertion that we here dub the Factorial Conjecture. Various cases of the Factorial Conjecture are proved

Local and long-distance organization of prefrontal cortex circuits in the marmoset brain

The prefrontal cortex (PFC) has dramatically expanded in primates, but its organization and interactions with other brain regions are only partially understood. We performed high-resolution connectomic mapping of the marmoset PFC and found two contrasting corticocortical and corticostriatal projection patterns: patchy projections that formed many columns of submillimeter scale in nearby and distant regions and diffuse projections that spread widely across the cortex and striatum. Parcellation-free analyses revealed representations of PFC gradients in these projections\u27 local and global distribution patterns. We also demonstrated column-scale precision of reciprocal corticocortical connectivity, suggesting that PFC contains a mosaic of discrete columns. Diffuse projections showed considerable diversity in the laminar patterns of axonal spread. Altogether, these fine-grained analyses reveal important principles of local and long-distance PFC circuits in marmosets and provide insights into the functional organization of the primate brain

Quantitative assessment of prefrontal cortex in humans relative to nonhuman primates

Significance A longstanding controversy in neuroscience pertains to differences in human prefrontal cortex (PFC) compared with other primate species; specifically, is human PFC disproportionately large? Distinctively human behavioral capacities related to higher cognition and affect presumably arose from evolutionary modifications since humans and great apes diverged from a common ancestor about 6–8 Mya. Accurate determination of regional differences in the amount of cortical gray and subcortical white matter content in humans, great apes, and Old World monkeys can further our understanding of the link between structure and function of the human brain. Using tissue volume analyses, we show a disproportionately large amount of gray and white matter corresponding to PFC in humans compared with nonhuman primates.</jats:p