Neurotransmitter Receptors: Mechanisms of Action and Regulation

Preface
Contents
1 Mahler Measures of Polynomials in One Variable
	1.1 Introduction
		1.1.1 Polynomials over the Field mathbbC of Complex Numbers
		1.1.2 Polynomials over the Field mathbbQ of Rational Numbers
	1.2 Kronecker's Two Theorems
	1.3 Mahler Measure Inequalities
	1.4 A Lower Bound for an Integer Polynomial Evaluated at an Algebraic Number
	1.5 Polynomials with Small Coefficients
	1.6 Separation of Conjugates
	1.7 The Shortness of a Polynomial
		1.7.1 Finding Short Polynomials
	1.8 Variants of Mahler Measure
		1.8.1 The Weil Height
	1.9 Notes
	1.10 Glossary
2 Mahler Measures of Polynomials in Several Variables
	2.1 Introduction
	2.2 Preliminaries for the Proofs of Theorems 2.5 and 2.6
	2.3 Proof of Theorem 2.5
	2.4 Proof of Theorem 2.6
	2.5 Computation of Two-Dimensional Mahler Measures
	2.6 Small Limit Points of mathcalL?
		2.6.1 Shortness Conjectures Implying Lehmer's Conjecture and Structural Results for mathcalL
		2.6.2 Small Elements of the Set of Two-Variable Mahler Measures
	2.7 Closed Forms for Mahler Measures of Polynomials of Dimension at Least 2
		2.7.1 Dirichlet L-Functions
		2.7.2 Some Explicit Formulae for Two-Dimensional Mahler Measures
		2.7.3 Mahler Measures of Elliptic Curves
		2.7.4 Mahler Measure of Three-Dimensional Polynomials
		2.7.5 Mahler Measure Formulae for Some Polynomials of Dimension at Least 4
		2.7.6 An Asymptotic Mahler Measure Result
	2.8 Notes
	2.9 Glossary
3 Dobrowolski's Theorem
	3.1 The Theorem and Preliminary Lemmas
	3.2 Proof of Theorem 3.1: Dobrowolski's Lower Bound for M(α)
	3.3 Notes
	3.4 Glossary
4 The Schinzel–Zassenhaus Conjecture
	4.1 Introduction
		4.1.1 A Simple Proof of a Weaker Result
	4.2 Proof of Dimitrov's Theorem
	4.3 Notes
	4.4 Glossary
5 Roots of Unity and Cyclotomic Polynomials
	5.1 Introduction
	5.2 Solving Polynomial Equations in Roots of Unity
	5.3 Cyclotomic Points on Curves
		5.3.1 Definitions
		5.3.2 mathcalL(f) of Rank 1
		5.3.3 mathcalL(f) Full of Rank 2
		5.3.4 mathcalL(f) of Rank 2, but Not Full
		5.3.5 The Case of f Reducible
		5.3.6 An Example
	5.4 Cyclotomic Integers
		5.4.1 Introduction to Cyclotomic Integers
		5.4.2 The Function mathscrN(β)
		5.4.3 Evaluating or Estimating mathscrN(sqrtd)
		5.4.4 Evaluation of the Gauss Sum
		5.4.5 The Absolute Mahler Measure of Cyclotomic Integers
	5.5 Robinson's Problems and Conjectures
	5.6 Cassels' Lemmas for mathscrM(β)
	5.7 Discussion of Robinson's Problems
		5.7.1 Robinson's First Problem
		5.7.2 Robinson's Second Problem
	5.8 Discussion of Robinson's Conjectures
		5.8.1 The First Conjecture
		5.8.2 The Second Conjecture
		5.8.3 The Third Conjecture
		5.8.4 The Fourth Conjecture
		5.8.5 The Fifth Conjecture
	5.9 Multiplicative Relations Between Conjugate Roots of Unity
	5.10 Notes
	5.11 Glossary
6 Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem
	6.1 Introduction
	6.2 The Mahler Measure of a Matrix and Cyclotomic Matrices
	6.3 Flavours of Equivalence: Isomorphism, Equivalence and Strong Equivalence of Matrices
	6.4 Growing Cyclotomic Matrices
	6.5 Gram Vectors
	6.6 Statement of the Classification Theorem for Cyclotomic Integer Symmetric Matrices
	6.7 Glossary
7 Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem
	7.1 Cyclotomic Signed Graphs
	7.2 Cyclotomic Charged Signed Graphs
	7.3 Cyclotomic Integer Symmetric Matrices: Completion of the Classification
	7.4 Further Exercises
	7.5 Notes on Chaps. 6摥映數爠eflinkC:CYCLOTOMICS66 and 7
	7.6 Glossary
8 The Set of Cassels Heights
	8.1 Cassels Height and the Set mathscrC
	8.2 The Derived Sets and the Sumsets of mathscrC
	8.3 Proof of Theorem 8.4
		8.3.1 Structure and Labelling of Thue Sets
	8.4 Cassels Heights of Cyclotomic Integers in mathbbQ(ωp)
	8.5 Proof of Theorem 8.14
	8.6 Proof of Theorem 8.13
	8.7 Notes
	8.8 Glossary
9 Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations
	9.1 Introduction
	9.2 Preliminaries: Notation and Tools
	9.3 Cyclotomic Graphs Embedded in T2k
	9.4 Changes for Charges
	9.5 Glossary
10 The Transfinite Diameter and Conjugate Sets of Algebraic Integers
	10.1 Introduction
	10.2 Analytic Properties of the Transfinite Diameter
	10.3 Application to Conjugate Sets of Algebraic Integers
	10.4 Integer Transfinite Diameters
		10.4.1 The Integer Transfinite Diameter
		10.4.2 The Monic Integer Transfinite Diameter
	10.5 Notes
	10.6 Glossary
11 Restricted Mahler Measure Results
	11.1 Monic Integer Irreducible Noncyclotomic Polynomials
	11.2 Complex Polynomials That are Sums of a Bounded Number of Monomials
	11.3 Some Sets of Algebraic Numbers with the Bogomolov Property
		11.3.1 Totally p-Adic Fields
		11.3.2 Abelian Extensions of mathbbQ
		11.3.3 Langevin's Theorem
	11.4 The Height of Zhang and Zagier and Generalisations
	11.5 The Weil Height of α When mathbbQ(α)/mathbbQ is Galois
	11.6 Notes
	11.7 Glossary
12 The Mahler Measure of Nonreciprocal Polynomials
	12.1 Mahler Measure of Nonreciprocal Polynomials
		12.1.1 The Set mathcalH of Rational Hardy Functions
	12.2 Proof of Theorem 12.1
		12.2.1 Start of the Proof
		12.2.2 The Case ell< 2k
		12.2.3 The Case ellge2k: Proof that M(P)geθ0
		12.2.4 The Case ellge2k: Existence of a δ, Part 1
		12.2.5 The Case ellge2k: Existence of a δ, Part 2
		12.2.6 The Case ellge2k: Completion of the Proof
	12.3 Notes
	12.4 Glossary
13 Minimal Noncyclotomic Integer Symmetric Matrices
	13.1 Supersporadic Matrices and Other Sporadic Examples
	13.2 Minimal Noncyclotomic Charged Signed Graphs: Any that Are Not Supersporadic
		13.2.1 The Uncharged Case
		13.2.2 The Charged Case
	13.3 Completing the Classification
	13.4 Notes
	13.5 Glossary
14 The Method of Explicit Auxiliary Functions
	14.1 Conjugate Sets of Algebraic Numbers
	14.2 The Optimisation Problem
		14.2.1 Dualising the Problem
		14.2.2 Method Outline
	14.3 The Schur–Siegel–Smyth Trace Problem
		14.3.1 Totally Positive Algebraic Integers with Small Mean Trace
	14.4 The Mean Trace of α Less Its Least Conjugate
	14.5 An Upper Bound Trace Problem
	14.6 Mahler Measure of Totally Real Algebraic Integers
	14.7 Mahler Measure of Totally Real Algebraic Numbers
	14.8 Langevin's Theorem for Sectors
		14.8.1 Further Remarks
	14.9 Notes
	14.10 Glossary
15 The Trace Problem for Integer Symmetric Matrices
	15.1 The Mean Trace of a Positive Definite Matrix
	15.2 The Trace Problem for Integer Symmetric Matrices
	15.3 Constructing Examples that Have Minimal Trace
	15.4 Notes
	15.5 Glossary
16 Small-Span Integer Symmetric Matrices
	16.1 Small-Span Polynomials
	16.2 Small-Span Integer Symmetric Matrices
	16.3 Bounds on Entries and Degrees
	16.4 Growing Small Examples
		16.4.1 Two Rows
		16.4.2 Three Rows
		16.4.3 Four Rows
		16.4.4 Five Rows
		16.4.5 Six Rows
		16.4.6 Seven Rows
		16.4.7 Eight Rows
		16.4.8 Nine Rows
		16.4.9 Ten Rows
		16.4.10 Eleven Rows
		16.4.11 Twelve Rows
		16.4.12 Thirteen Rows
	16.5 Cyclotomic Small-Span Matrices
		16.5.1 Examples with an Entry of Modulus Greater Than 1
		16.5.2 Subgraphs of the Sporadic Examples
		16.5.3 Subgraphs of Cylindrical Tessellations
		16.5.4 Subgraphs of Toroidal Tessellations
	16.6 The Classification Theorem
	16.7 Notes
	16.8 Glossary
17 Symmetrizable Matrices I: Introduction
	17.1 Introduction
	17.2 Definitions and Immediate Consequences
	17.3 The Structure of Symmetrizable Matrices
	17.4 The Balancing Condition and Its Consequences
	17.5 The Symmetrization Map
	17.6 Interlacing
	17.7 Equitable Partitions of Signed Graphs
	17.8 Notes
	17.9 Glossary
18 Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices
	18.1 Cyclotomic Symmetrizable Integer Matrices
	18.2 Quotients of Signed Graphs
	18.3 Notes
	18.4 Glossary
19 Symmetrizable Matrices III: The Trace Problem
	19.1 The Trace Problem for Symmetrizable Matrices
		19.1.1 Definitions, Notation and Statement of the Results
		19.1.2 Proof of Proposition 19.3摥映數爠eflinkP:rattrace19.319
		19.1.3 Corollaries, Including Theorem 19.4摥映數爠eflinkT:main19.419
		19.1.4 The Structure of Minimal-Trace Examples
	19.2 Notes
	19.3 Glossary
20 Salem Numbers from Graphs and Interlacing Quotients
	20.1 Introduction
	20.2 Salem Graphs
	20.3 Examples of Salem Graphs
		20.3.1 Nonbipartite Examples
		20.3.2 Bipartite Examples
		20.3.3 Finding Cyclotomic Factors
	20.4 Attaching Pendant Paths
		20.4.1 A General Construction
		20.4.2 An Application to Salem Graphs
	20.5 Interlacing Quotients
		20.5.1 Rational Interlacing Quotients
		20.5.2 Circular Interlacing Quotients
		20.5.3 From CIQs to Cyclotomic RIQs
		20.5.4 Salem Numbers from Interlacing Quotients
	20.6 Notes
	20.7 Glossary
21 Minimal Polynomials of Integer Symmetric Matrices
	21.1 Introduction
	21.2 Small Discriminant
	21.3 Small Span
		21.3.1 The Cyclotomic Case
		21.3.2 The Noncyclotomic Case
		21.3.3 Some Counterexamples to Conjecture21.2
	21.4 Small Trace
	21.5 Polynomials that are Not Interlaced
	21.6 Counterexamples for all Degrees Greater than 5
		21.6.1 Degrees 8 to 16
		21.6.2 Degree 20
		21.6.3 Degree 19 and All Degrees Greater than 20
		21.6.4 All Together Now
	21.7 Notes
	21.8 Glossary
22 Breaking Symmetry
Appendix A Algebraic Background
A.1  Self-Reciprocal Polynomials
A.2  Resultant Essentials
A.3  Valuation Essentials
A.4  Galois Theory Essentials
A.5  Algebraic Numbers and Algebraic Integers
A.5.1  The Gorškov Polynomials
A.6  Newton's Identities
A.7  Jänichen's Generalisation of Fermat's Little Theorem
Appendix B Combinatorial Background
B.1  Interlacing
B.2  Graph Theory
B.3  Perron–Frobenius Theory
Appendix C Tools from the Theory of Functions
Appendix D Tables
D.1  Small Mahler Measures
D.2  Known Small Mahler Measures of Two-Variable Polynomials
Appendix  References
Index