Mathematics, Game Theory and Algebra Compendium. Volume 1

Name: Mathematics, Game Theory and Algebra Compendium. Volume 1
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Intro
MATHEMATICS, GAME THEORYAND ALGEBRA COMPENDIUM.VOLUME 1
MATHEMATICS, GAME THEORYAND ALGEBRA COMPENDIUM.VOLUME 1
CONTENTS
PREFACE
Chapter 1ON PROPERTY (E.A) IN GAUGE SPACES
Abstract
1. Introduction and Preliminaries
2. Main Results
References
Chapter 2STRUCTURE OF CERTAINPERIODIC NEAR RINGS
Abstract
1. Preliminaries
2. Introduction
3. Main Results
References
Chapter 3EXTENDED METHOD OF QUASILINEARIZATIONFOR A THREE-POINT BOUNDARY VALUE PROBLEMWITH GENERAL NONLINEAR NONCONVEXBOUNDARY CONDITIONS
Abstract
1. Introduction
2. Preliminary Results
3. Main Result
References
Chapter 4THE JONES POLYNOMIALOF TWIST KNOTS
Abstract
1. Introduction
2. The Jones Polynomial of Tn
3. The Conway Polynomial of Tn
References
Chapter 5ABSTRACT IDEAL THEORY: MULTIPLICATIVELATTICES AND x-SYSTEMS*
Abstract
Examples of Multiplicative Lattices
Definition of an (Integral) x-System
References
Chapter 6COMMUTATIVITY OF RINGSWITH CONSTRAINTS ON NILPOTENTSAND THE JACOBSON RADICAL
Abstract
1. Introduction
2. Main Results
References
Chapter 7DECOMPOSABILITY OF ITERATEDEXTENSIONS*
Abstract
Introduction
1. Invariance of Symbolic Powers of Certain Ideals
2. Decomposability of the Iterated Extension Rings
References
Chapter 8SOME ASPECTS OF A PURE THEORYOF BARGAINING: PLAYERS, INFORMATION,EQUILIBRIUM AND VOTING POWER
Abstract
1. Introduction
2. The Model
3. The Complete Information Setting
Result 1
Proof
4. The Incomplete Information Setting
4.1. The Incomplete Information Setting without Intermediary
Result 2
Proof
4.2. Incomplete Information Setting and Intermediary Given Veto Power
Result 3
Proof
We remark
Step 1
Step 2
Step 3
Step 4
Step 5
Result 4
Proof
Result 5
Proof
5. Conclusion
Summary
References
Chapter 9LIMITS AT INFINITY OF GENERALIZEDRIESZ POTENTIALS
Abstract
References
Chapter 10PHASE OPERATOR ON ADEFORMED HILBERT SPACE
Abstract
1. Introduction
2. Preliminaries and Notations
3. Phase Operator
4. Phase Distribution
4.1. Phase Vector
4.2. Completeness of Phase Vectors
4.3. Distribution
4.4. Examples
4.4.1. Incoherent Vectors
4.4.2. Coherent Vectors
4.4.3. Coherent Phase Vectors
5. Phase Measurement Statistics
5.1. Probability Operator Measure
5.2. Phase Estimation
6. Conclusion
References
Chapter 11GALOIS THEORYOF GRADED FIELDS*
Abstract
Introduction
1. Preliminaries
2. Galois Extensions
3. Concluding Remarks
References
Chapter 12SELF SIMILAR ISOTHERMAL EXPANSIONOF MAGNETO-HYDRODYNAMIC GASBEHIND A SPHERICAL SHOCK WAVEWITH SELF GRAVITATION
Abstract
Introduction
Basic Equations and Boundary Conditions
Similarity Solutions
Results and Discussions
Case I
Case II
References
Chapter 13CLASSIFICATION OF THE 5-DIMENSIONALPOWER-ASSOCIATIVE 2nd-ORDERBERNSTEIN ALGEBRAS
Abstract
1. Preliminary Results
2. Jordan and Power-Associative 2nd-order Bernstein Algebras
3. Classification in Dimension 5
3.1. Power-associative 2nd-order Bernstein Algebras of Type (1,4)
3.2. Power-associative 2nd-order Bernstein Algebras of Type (2,3)
3.3. Power-associative 2nd-order Bernstein Algebras of Type (3,2)
References
Chapter 14A DUAL APPROACHTO ALPHA-REGULARITY
Abstract
1. Introduction
2. Subgroups of the Schur Multiplier
3. Applications
Acknowledgments
References
Chapter 15LIMIT CYCLE IN FRACTIONALDIFFERENTIAL SYSTEMS
Abstract
1. Introduction
2. Preliminaries
3. Melnikove Function Reduce to Abelian Function orRiemann-Liouville Function
4. Limit Cycles in Fractional Differential Systems
References
Chapter 16THE HALF-FACTORIAL PROPERTY IN THE RINGA+XI[X] WHERE A IS A UFR
Abstract
1. Introduction
2. Half-Factorial Property
3. The Rings A+XI[X]
References
Chapter 17SOME DIOPHANTINE EQUATIONSASSOCIATED TO SEMINORMALCOHEN-KAPLANSKY DOMAINS
Abstract
1. Introduction
2. Basic Results on CK Domains
3. Characterization of Seminormal CK Domains
4. On the Number of Solutions of a System of Two SpecialDiophantine Equations
5. On the Asymptotic Behaviour of the Number of DistinctFactorizations into Atoms in a Seminormal CK Domain
References
Chapter 18ON INTEGRALLY CLOSEDGOING-DOWN RINGS
Abstract
1. Introduction
2. Results
References
Chapter 19COMPUTATIONALOF THE INTEGRAL CLOSURE
Abstract
Introduction
1. Preliminaries
2. Main Results
3. Applications
References
Chapter 20NEGACYCLIC CODESOF LENGTH 2e OVER Z4
Abstract
1. Introduction
2. Negacyclic Codes
3. Dual Codes
4. Conclusion
References
Chapter 21COOPERATIVE STOCHASTIC GAMESIN STATIONARY STRATEGIES
References
Chapter 22TWO LEVEL GAMES
1. Introduction
2. One Leader, n Non-cooperative Followers
3. One Leader, n Cooperative Followers
4. Two Leaders Play a Non-cooperative Game
References
Chapter 23CONJUGATE-SET GAME FOR A NONLINEARPROGRAMMING PROBLEM*
1. Introduction
2. Strict Conjugate Sets
3. Conjugate-set Game
References
Chapter 24DYNKIN'S STOPPING GAMES WITH ZERO PAYOFFSFOR SEPARATE STOPPING
1. Introduction
2. Superharmonic and Subharmonic Functions.Martin Boundaries
3. Properties of Randomized Stopping Strategies
4. Optimality Equations and Games with Zero Values
5. Games with Transitive Strategies ?1/a
6. Games with Nontransitive and Nonstopping Strategies ?1/a
7. Illustrative Examples
References
Chapter 25AN INVESTMENT ALLOCATION GAMEWITH A COST
1. Introduction
2. One Firm Game
3. Two Firm Game
4. Solution of the Two Firms Game
5. A Numerical Example
6. Stackelberg Equilibrium
7. Conclusions
References
Chapter 26AN OPTIMAL INSURANCE POLICYIN THE INDIVIDUALRISK MODEL SEEN AS A BARGAINING GAME
1. Introduction
2. The Individual Risk Model as a Bargaining Game
3. Pareto-optimal Policies
4. Constructing a Nash's Solution
5. Constructing a Kalai-Smorodinsky's Solution
References
Chapter 27A FISHERY GAME MODELWITH MIGRATION:RESERVED TERRITORY APPROACH
1. Introduction
2. Game Model
2.1. Nash Optimal Solution
2.2. Stackelberg Optimal Solution
3. Model Over an Infinite Horizon
4. Nash Optimal Solution
4.1. Stackelberg Optimal Solution
References
Chapter 28NON-HIERARCHICAL SIGNALLING:TWO-STAGE FINANCING GAME
1. Introduction
2. Model
2.1. Perfect Market
2.2. Asymmetric Information
3. Two-Type Economy
4. Multiple Type Economy
5. Conclusion
Acknowledgments
Appendix 1
Appendix 2
References
Chapter 29THREE-PLAYER GAMEOF 'KEEP-OR-EXCHANGE'
1. Three-Player Games of ' Score Showdown'
2. Keep-or-Exchange -Two-Player Game
3. Keep-or-Exchange -Three-Player Game
4. Simultaneous-Move Game
References
Chapter 30DYNAMIC NONCOOPERATIVE R&DIN DUOPOLY WITH SPILLOVERSAND TECHNOLOGY GAP*
1. Introduction
2. D'Aspremont and Jacquemin (AJ) Revisited -The Static Case
3. The Dynamic Case
4. A General Model of Dynamic R&Dwith Endogenous Spillovers
4.1. The Model
4.2. Solving the Model
5. Summary and Concluding Remarks
6. Appendix
6.1. Derivations of (13) and (14)
6.2. Proof for Proposition 2.1
6.3. Proof for Proposition 2.2
6.4. Proof for Proposition 3.1
References
Chapter 31MATHEMATICS OF THE JIPTOAND THEORY OF THE PURSUIT
1. Introduction
2. The Particularity of the JIPTO
3. Classification of the Versions
4. Mod`eles Mathematical
5. Modeling of the Strategies
6. Sources of the Irresolute Problems
7. Mathematics of the JIPTO and Differential Games
References
Chapter 32INVESTMENT DECISIONS UNDER UNCERTAINTYAND EVALUATION OF AMERICAN OPTIONS
1. Introduction
2. Formulation of the Problem
3. The Option Value for Projects with Zero Volatilities
4. Properties of the Exercise Region Boundary
5. Upper Bound for the Option Value
6. Discussion
Appendix
References
Chapter 33ON A CONTINUOUS DYNAMICSTRATEGIC MARKET GAME
1. Introduction
2. Model and the Main Result
3. Proof of Theorem 1
References
Chapter 34DISTORTION OF LENGTHS UNDER FUNCTIONSSTARLIKE WITH RESPECT TO A BOUNDARY POINT
Abstract
1. Introduction
2. Preliminaries
References
Chapter 35SERVICE FACILITY INVENTORYSYSTEM WITHIMPATIENT CUSTOMERS*
Abstract
1. Introduction
2. Model Description
Notations
3. Analyses
3.1. Transient Analysis
3.2. Steady State Analysis
4. System Performance Measures
4.1. Mean Inventory Level
4.2. Mean Reorder Rate
4.3. Mean Balking Rate
4.4. Mean Reneging Rate
4.5. Mean Failure Rate
4.6. MeanWaiting Time
5. Cost Analysis
6. Numerical Illustrations
References
Chapter 36CERTAIN WEAKER FORMS OF FUZZYPAIRWISE CONTINUOUS FUNCTIONSAND BIEXTENSION OF FUZZYBITOPOLOGICAL SPACES
Abstract
1. Preliminaries
2. Main Results
3. Biextension of Fuzzy Bitopological Spaces
References
Chapter 37ON POLYNILPOTENT COVERING GROUPS OF APOLYNILPOTENT GROUP
Abstract
1. Introduction and Motivation
2. The Main Result
References
Chapter 38THE BAER INVARIANT OF SEMIDIRECTAND VERBAL WREATH PRODUCTSOF GROUPS*
Abstract
1. Introduction and Motivation
2. Notation and Preliminaries
3. Some Results on the Baer Invariant of a Semidirect Product
4. The Baer Invariant of a Verbal Wreath Product
5. The Baer Invariant of a Free Wreath Product
References
Chapter 39A CHARACTERIZATION OFCOMMUTATIVE CLEAN RINGS*
Abstract
1. On 2-Clean Rings
2. All Rings Are Commutative and with Identity
3. Commutative Clean Group Rings
4. When G = Z2
References
Chapter 40ON ABSOLUTE MATRIX SUMMABILITY OF FOURIERSERIES AND ITS ALLIED SERIES
Acknowledgement
References
Chapter 41SEMIREGULAR ASSOCIATIVE PAIRS
Abstract
Introduction
1. Definition and Basic Concepts
2. Semiregularity of the Standard Embedding
3. Preliminary Facts on Modules
4. Characterization of Semiregular Associative Pairs
Acknowledgments
References
Chapter 42MAGNETOHYDRODYNAMIC FLOW AND RADIATIONOF A RAREFIED GAS
Abstract
1. Introduction
2. Mathematical Analysis
3. Discussion
References
Chapter 43FIRST STRONGLY GRADED MODULES#
Abstract
0. Introduction
1. Preliminaries
2. First Strongly Graded Modules
References
Chapter 44SOLVING SOME HIGHER-ORDERDISCRETE DYNAMIC SYSTEMSAND APPLICATIONS
Abstract
1. Introduction
2. Some Expressions of Solutions of the System (1.2)
3. The Second-Order Discrete Systems: Simple Case
3.1. Solutions of the Homogeneous Part
3.2. Solutions of the System (3.14)
4. Study of the Discrete System (1.3): Simple Case
5. Combinatorial Solutions of (1.3)
6. Some Concluding Remarks
References
Chapter 45A PROPERTY OF THE CYCLOTOMICPOLYNOMIAL
Abstract
References
Chapter 46ON STRONGER FORMS OF (1,2)* QUOTIENTMAPPINGS IN BITOPOLOGICAL SPACES
Abstract
1. Introduction
2. Preliminaries
3. (1,2)* a-Quotient Mappings
4. Strongly (1,2)* a-Quotient Mappings
5. (1,2)* a* - Quotient Mappings
6. Comparisons
References
Chapter 47A FURTHER INSTABILITY RESULTFOR A CERTAIN VECTOR DIFFERENTIALEQUATION OF FOURTH ORDER
Abstract
1. Introduction and Statement of the Result
2. Proof of the Theorem
References
Chapter 48ON THE REGULARITY OF MAGNETIC FLUXFUNCTION IN NUCLEAR FUSION
Abstract
1. Introduction
2. Mathematical Model of an Axi-symmetric Equilibrium Statefor the Plasma in a Tokamak
Plasma Region
Vacuum Region
Equation of State for Ideal Gas
Internal Energy Density
2.1. Equilibrium of Plasma in Magnetic Field
2.2. Slow Evolution of a Quasi-Equilibrium Plasma-Vacuum System
Plasma Region
Vacuum Region
Fixed Boundary Condition
Free Boundary Conditions
Plasma Region
External Current
Poloidal Flux Function
Toroidal Flux Function
Pressure Function
Constraints of Motion
3. Smooth Variational Structures and Modified Lagrange MultiplierRule with a Functional Constraint
Notations
4. A State of Equilibrium of Plasma Flow in Nuclear Fusion
Outline of the proof
References
Chapter 49COLLISION ORBITS FOR N-BODY PROBLEMSWITH FIXED ENERGY
Abstract
1. Introduction and Main Results
2. The Proof of Theorem 1.1 and Theorem 1.2
References
Chapter 50GRAPH SEARCHING PROBLEMSWITH THE COUNTERACTION
1. Introduction
2. Statement of the Problem
3. Theorems on -search Numbers
4. Related Problems
5. Cycles
6. A Chess Problem
Acknowledgment
References
Chapter 51COMPETITION FOR STAFFBETWEEN TWO DEPARTMENTS
1. Introduction
2. Preliminary Notions
3. The Dominant Firm Problem
4. The One and Two Applicant Games with No CandidatePreference
5. The General Game with No Candidate Preference
6. Discussion of the Results
References
Chapter 52A COMPETITIVE PREDICTIONNUMBER GAME
1. Introduction
2. Zero-Sum Game
3. Non-Zero-Sum Game
4. Non-Zero-Sum Game
References
Chapter 53EFFICIENCY OF BERTRANDAND COURNOT UNDER PRECOMMITMENT
1. Introduction
2. The Model
3. Cournot Equilibrium
4. Bertrand Equilibrium
5. Comparison of Equilibria
6. Concluding Remarks
References
Chapter 54ONE APPROACH TO SOLUTIONOF COMPLEX GAME PROBLEMS
1. Introduction
2. Using Effect of Information Delay for Solution of the LinearEvolutionary Game with Perfect Information
References
Chapter 55A DIFFERENTIAL GAMEWITH INVESTMENT IN TRANSPORTAND COMMUNICATION R&D
1. Introduction
2. The Model
3. Bertrand Competition
3.1. Degenerate Markov Perfect Nash Equilibrium
4. Cournot Competition
4.1. Open-Loop Nash Equilibrium
4.2. Closed-Loop Nash Equilibrium
5. Social Optimum andWelfare Appraisal
6. Concluding Remarks
Appendices
Appendix 1: Proof of Proposition 1
Appendix 2: Expression
Appendix 3: Proof of Proposition 2
References
Chapter 56TIME-CONSISTENCY AND THE PROBLEMOF MINIMAL REDUCTION
1. Introduction
2. Preliminaries
3. Formulation of Minimal Reduction Problem
4. Conditionally Minimal Coalition
5. Acceptable Coalitions
6. Dynamic Example
7. Conclusion
References
Chapter 57A CONCEPT OF SOLUTIONFOR A STRATEGIC COOPERATIVE GAMEINVOLVING UNKNOWN PARAMETERS
1. Introduction
2. The Statement of the Problem
3. The Concept of the w-Equilibrium
4. The Existence of the w-Equilibrium
5. The Determination of w-Equilibrium
5.1. Class 1
5.2. Class 2
6. Conclusion
References
Chapter 58ON A DISCRETE ARBITRATIONPROCEDURE IN THREE POINTS
1. Introduction
2. Problem Statement
3. Optimal Strategies
References
Chapter 59REPEATED GAMEWITH CONSTRAINT ON THE TIMEOF OBSERVATION*
1. Introduction
2. Preservation of Equilibrium under Absence ofContinuous Observation
3. Dependence of Set of Situations of Equilibrium on Limitationon Time of Observation
References
Chapter 60VALUE FOR THE GAME WITHCHANGING COALITIONAL STRUCTURE
1. Introduction
2. Dynamic Game with Changing Coalitional Partition
3. The Algorithm for Aonstructing the Solution
4. Characteristic Functions for an Associated CooperativeGame
5. Example
References
Chapter 61THE REDISTRIBUTION PARADOX
1. Introduction
2. Power Indices and Their Paradoxes
2.1. Voting Games and Power Indices
2.2. Voting Power Paradoxes
3. Power Indices for the German Parliament
4. The Paradox of Redistribution in the German Parliament
5. Paradox of New Members in German Politics
6. Agreements between Power Indices
7. Conclusions
Appendix
References
Chapter 62PLAYERS' INFORMATION IN TWO-PLAYER GAMESOF "SCORE SHOWDOWN"
1. Two-Player Game of "Score Showdown"
2. Game of "Keep-or-Exchange" under I10-01
3. Game of "Keep-or-Exchange" under I11-11
4. Game of "Keep-or-Exchange" under I10-11
5. Game of "Risky Exchange" under I10-01
6. Game of "Risky Exchange" under I11-11
7. Game of "Risky Exchange" under I10-11
8. Comparison between Theorem 1?6.
9. More about Games under Information I10-11.
10. Final Remark
References
Chapter 63INFINITELY REPEATEDSYMMETRIC 2 × 2-BIMATRIX GAMES
1. Introduction
2. Symmetric 2 × 2 Bimatrix Games
3. Infinitely Repeated Bimatrix Games With Finite Players'Memory
3.1. States of the Game
3.2. Dynamics of the Game
3.3. Graph of the Game
3.4. Payoff Functions of the Players
4. Nash Equilibria in the Infinitely Repeated Bimatrix Gamewith m-step Memory
4.1. Nash Algorithm of Finding the Nash Equilibria
5. Hierarchical Setting of the Game
5.1. Algorithm of Constructing the Optimal Partition
5.1.1. Constructing the Initial Partition
5.1.2. Choice of the Admissible Set of Multiplicities
5.1.3. Improvement of the Leader's Strategy in the Domain Pi
5.1.4. Survey of the Cycles
Appendix A. Nash Equilibria in the Game With 1-step Memory
Appendix B. Stackelberg Solution of Repeated SymmetricBimatrix Games With 1-step Memory
Acknowledgements
References
Chapter 64NONSYMMETRIC CONSISTENTSURPLUS SHARING METHODS
1. Introduction
2. Surplus Sharing Methods and Their Properties
3. Parametric Methods
4. Path Independent Methods
References
INDEX
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