MATHEMATICS, GAME THEORYAND ALGEBRA COMPENDIUM.VOLUME 1

MATHEMATICS, GAME THEORYAND ALGEBRA COMPENDIUM.VOLUME 1

CONTENTS

PREFACE

ON PROPERTY (E.A) IN GAUGE SPACES

Abstract

1. Introduction and Preliminaries

2. Main Results

References

STRUCTURE OF CERTAINPERIODIC NEAR RINGS

Abstract

1. Preliminaries

2. Introduction

3. Main Results

References

EXTENDED METHOD OF QUASILINEARIZATIONFOR A THREE-POINT BOUNDARY VALUE PROBLEMWITH GENERAL NONLINEAR NONCONVEXBOUNDARY CONDITIONS

Abstract

1. Introduction

2. Preliminary Results

3. Main Result

References

THE JONES POLYNOMIALOF TWIST KNOTS

Abstract

1. Introduction

2. The Jones Polynomial of Tn

3. The Conway Polynomial of Tn

References

ABSTRACT IDEAL THEORY: MULTIPLICATIVELATTICES AND x-SYSTEMS∗

Abstract

Examples of Multiplicative Lattices

Definition of an (Integral) x-System

References

COMMUTATIVITY OF RINGSWITH CONSTRAINTS ON NILPOTENTSAND THE JACOBSON RADICAL

Abstract

1. Introduction

2. Main Results

References

DECOMPOSABILITY OF ITERATEDEXTENSIONS∗

Abstract

Introduction

1. Invariance of Symbolic Powers of Certain Ideals

2. Decomposability of the Iterated Extension Rings

References

SOME 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

Step 1

Step 2

Step 3

Step 4

Step 5

Result 4

Proof

Result 5

Proof

5. Conclusion

Summary

References

LIMITS AT INFINITY OF GENERALIZEDRIESZ POTENTIALS

Abstract

References

PHASE 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

GALOIS THEORYOF GRADED FIELDS∗

Abstract

Introduction

1. Preliminaries

2. Galois Extensions

3. Concluding Remarks

References

SELF 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

CLASSIFICATION 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

A DUAL APPROACHTO ALPHA-REGULARITY

Abstract

1. Introduction

2. Subgroups of the Schur Multiplier

3. Applications

Acknowledgments

References

LIMIT 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

THE 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

SOME 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

ON INTEGRALLY CLOSEDGOING-DOWN RINGS

Abstract

1. Introduction

2. Results

References

COMPUTATIONALOF THE INTEGRAL CLOSURE

Abstract

Introduction

1. Preliminaries

2. Main Results

3. Applications

References

NEGACYCLIC CODESOF LENGTH 2e OVER Z4

Abstract

1. Introduction

2. Negacyclic Codes

3. Dual Codes

4. Conclusion

References

COOPERATIVE STOCHASTIC GAMESIN STATIONARY STRATEGIES

References

TWO 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

CONJUGATE–SET GAME FOR A NONLINEARPROGRAMMING PROBLEM∗

1. Introduction

2. Strict Conjugate Sets

3. Conjugate-set Game

References

DYNKIN’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

6. Games with Nontransitive and Nonstopping Strategies

7. Illustrative Examples

References

AN 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

AN 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

A FISHERY GAME MODELWITH MIGRATION:RESERVED TERRITORY APPROACH

1. Introduction

2. Game Model

2.1. Nash Optimal Solution

3. Model Over an Infinite Horizon

4. Nash Optimal Solution

4.1. Stackelberg Optimal Solution

References

NON-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

THREE-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

DYNAMIC 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

6.4. Proof for Proposition 3.1

References

MATHEMATICS 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

INVESTMENT 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

ON A CONTINUOUS DYNAMICSTRATEGIC MARKET GAME

1. Introduction

2. Model and the Main Result

3. Proof of Theorem 1

References

DISTORTION OF LENGTHS UNDER FUNCTIONSSTARLIKE WITH RESPECT TO A BOUNDARY POINT

Abstract

1. Introduction

2. Preliminaries

References

SERVICE FACILITY INVENTORYSYSTEM WITHIMPATIENT CUSTOMERS∗

Abstract

1. Introduction

2. Model Description

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

CERTAIN WEAKER FORMS OF FUZZYPAIRWISE CONTINUOUS FUNCTIONSAND BIEXTENSION OF FUZZYBITOPOLOGICAL SPACES

Abstract

1. Preliminaries

2. Main Results

3. Biextension of Fuzzy Bitopological Spaces

References

ON POLYNILPOTENT COVERING GROUPS OF APOLYNILPOTENT GROUP

Abstract

1. Introduction and Motivation

2. The Main Result

References

THE 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

A 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

ON ABSOLUTE MATRIX SUMMABILITY OF FOURIERSERIES AND ITS ALLIED SERIES

1. Let Σun be a given infinite series with the sequence of partial sum {sn}. Let T ≡(an,k) be an infinite triangular matrix with real constants. The sequence-to-sequencetransformation

2. Let f(t) be a periodic function, with period 2π, and be Lebesgue integrable over (-π,π). We may assume, with out loss of generality, that the constant term in the Fourierseries of f(t) is zero, so that

3. Absolute Cesaro summability of a Fourier series has been studied by variousinvestigators. In particular, the following theorems are well known :

4. For proving our Theorems, we need the following Lemmas:

5. Proof of Theorem 1. We have

Acknowledgement

References

SEMIREGULAR 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

MAGNETOHYDRODYNAMIC FLOW AND RADIATIONOF A RAREFIED GAS

Abstract

1. Introduction

2. Mathematical Analysis

3. Discussion

References

FIRST STRONGLY GRADED MODULES#

Abstract

0. Introduction

1. Preliminaries

2. First Strongly Graded Modules

References

SOLVING 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

PROPERTY OF THE CYCLOTOMICPOLYNOMIAL

Abstract

References

ON STRONGER FORMS OF (1,2)* QUOTIENTMAPPINGS IN BITOPOLOGICAL SPACES

Abstract

1. Introduction

2. Preliminaries

3. (1,2)* α–Quotient Mappings

4. Strongly (1,2)* α-Quotient Mappings

5. (1,2)* α* – Quotient Mappings

6. Comparisons

References

A FURTHER INSTABILITY RESULTFOR A CERTAIN VECTOR DIFFERENTIALEQUATION OF FOURTH ORDER

Abstract

1. Introduction and Statement of the Result

2. Proof of the Theorem

References

ON 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

2.1. Equilibrium of Plasma in Magnetic Field

2.2. Slow Evolution of a Quasi-Equilibrium Plasma-Vacuum System

3. Smooth Variational Structures and Modified Lagrange MultiplierRule with a Functional Constraint

4. A State of Equilibrium of Plasma Flow in Nuclear Fusion

References

COLLISION 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

GRAPH 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

COMPETITION 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

A COMPETITIVE PREDICTIONNUMBER GAME

1. Introduction

2. Zero-Sum Game

3. Non-Zero-Sum Game

4. Non-Zero-Sum Game

References

EFFICIENCY OF BERTRANDAND COURNOT UNDER PRECOMMITMENT

1. Introduction

2. The Model

3. Cournot Equilibrium

4. Bertrand Equilibrium

5. Comparison of Equilibria

6. Concluding Remarks

References

ONE APPROACH TO SOLUTIONOF COMPLEX GAME PROBLEMS

1. Introduction

2. Using Effect of Information Delay for Solution of the LinearEvolutionary Game with Perfect Information

References

A 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

TIME–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

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

ON A DISCRETE ARBITRATIONPROCEDURE IN THREE POINTS

1. Introduction

2. Problem Statement

3. Optimal Strategies

References

REPEATED 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

VALUE 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

THE 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

PLAYERS’ 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 16.

9. More about Games under Information I10−11.

10. Final Remark

References

INFINITELY 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

NONSYMMETRIC CONSISTENTSURPLUS SHARING METHODS

1. Introduction

2. Surplus Sharing Methods and Their Properties

3. Parametric Methods

4. Path Independent Methods

References

INDEX