The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak Coupling.- Neuronal Networks: Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks.

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Mathematical foundations of neuroscience

Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

The q-theory of Finite Semigroups

1. Abstract Semigroups 2. Algebraic Geometry 3. Linear Algebraic Semigroups 4. Linear Algebraic Groups 5. Connected Algebraic Semigroups 6. Connected Algebraic Monoids 7. Reductive Groups and Regular Semigroups 8. Diagonal Monoids 9. Cross-section Lattices 10. xi-Structure 11. Renner's Decomposition and Related Finite Semigroups 12. Biordered Sets 13. Tits Building 14. The System of Idempotents 15. J-irreducible and J co-reducible Monoids 16. Renner's Extension Principle and Classification.

Linear Algebraic Monoids

1This book was published 1986 by the Akademie-Verlag Berlin, Volume 32 in the series “Mathematical Research”; Lector was Dr. Reinhard Hoppner. It was printed in the German Democratic Republic. It has been transferred into LATEX by Ulrich Thiemann. In connection with this translation also the notation for the direction used for the composition of homomorphisms has been changed (the first morphism is written to the right of the second one, etc.). So far most of the diagrams are still missing. Also an index is still missing. The errors and misprints in the original version have not yet been corrected. The bibliography has been extended about 1990 and not yet been updated – moreover, it may still contain many titles, which treat “partial operations” (which was the keyword for the search in the Zentralblatt), but which are not really concerned with partial algebras in the sense of this book. In order to get a better idea of the material in the book, the article: Tools for a Theory of Partial Algebras, which has been published in: General Algebra and Applications (Eds.: K.Denecke and H.-J.Vogel), Research and Exposition in Mathematics, Vol. 20, Heldermann Verlag Berlin, 1993, pp. 12–32, has been added to a revised introduction.

A Model Theoretic Oriented Approach to Partial Algebras

Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology Ultralimits of metric spaces Gromov-hyperbolic spaces and groups Lattices in Lie groups Solvable groups Geometric aspects of solvable groups The Tits alternative Gromov's theorem The Banach-Tarski paradox Amenability and paradoxical decomposition Ultralimits, fixed point properties, proper actions Stallings's theorem and accessibility Proof of Stallings's theorem using harmonic functions Quasiconformal mappings Groups quasiisometric to $\mathbb{H}^n$ Quasiisometries of nonuniform lattices in $\mathbb{H}^n$ A survey of quasiisometric rigidity Appendix: Three theorems on linear groups Bibliography Index

Geometric Group Theory

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

Structure of regular semigroups. I

This paper initiates a general study of the structure of a regular semigroup 5 via the maximum congruence p on S with the property that each p-class ep, for e = e2 e S, is a rectangular subband of S. Congruences of this type are studied and the maximum such congruence is characterized. A construction of all biordered sets which are coextensions of an arbitrary biordered set by rectangu- lar biordered sets is provided and this is specialized to provide a construction of all solid biordered sets. These results are used to construct all regular idempotent-gen- erated semigroups which are coextensions of a regular idempotent-generated semi- group by rectangular bands: a construction of normal coextensions of biordered sets is also provided.

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K. S. S. Nambooripad

Papers

Structure of regular semigroups. I

Coextensions of regular semigroups by rectangular bands. I