​MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in its second year, 2017: - Hypergeometric Motives and Calabi–Yau Differential Equations - Computational Inverse Problems - Integrability in Low-Dimensional Quantum Systems - Elliptic Partial Differential Equations of Second Order: Celebrating 40 Years of Gilbarg and Trudinger’s Book - Combinatorics, Statistical Mechanics, and Conformal Field Theory - Mathematics of Risk - Tutte Centenary Retreat - Geometric R-Matrices: from Geometry to Probability The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.

lgrsnf/N:\!genesis_files_for_add\_add\kolxo3\95\M_Mathematics\Msb_Sborniki\Wood D.R., et al. (eds.) 2017 MATRIX Annals (Springer, 2019)(ISBN 9783030041601)(O)(702s)_Msb_.pdf

lgli/M_Mathematics/Msb_Sborniki/Wood D.R., et al. (eds.) 2017 MATRIX Annals (Springer, 2019)(ISBN 9783030041601)(O)(702s)_Msb_.pdf

nexusstc/2017 MATRIX Annals/5b3980a25dde638f93509ae76bc5c2ef.pdf

zlib/Science (General)/Wood D.R (ed.)/2017 MATRIX Annals_6041414.pdf

2017 Matrix Annals 1st Ed. 2018

David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao

Jan de Gier; Cheryl E Praeger; Terence Tao; David R Wood

Cheryl E Praeger; Terence Tao; Jan de Gier

Springer Nature Switzerland AG

MATRIX Book Series, 2, Cham, Switzerland, 2019

MATRIX Book Series, v.2, Cham, 2019

Switzerland, Switzerland

Mar 14, 2019

2018

kolxo3 -- 95

lg2806153

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Source title: 2017 MATRIX Annals (MATRIX Book Series)

Preface......Page 6
Hypergeometric Motives and Calabi–Yau Differential Equations......Page 8
Computational Inverse Problems......Page 11
Integrability in Low-Dimensional Quantum Systems......Page 14
Elliptic Partial Differential Equations of Second Order: Celebrating 40 Years of Gilbarg and Trudinger's Book......Page 17
Combinatorics, Statistical Mechanics, and Conformal Field Theory......Page 21
Mathematics of Risk......Page 23
Tutte Centenary Retreat......Page 26
Geometric R-Matrices: From Geometry to Probability......Page 30
Contents......Page 33
Part I Refereed Articles......Page 38
1 Introduction......Page 39
2 The Randomize-Then-Optimize Proposal Density......Page 42
3 RTO-Metropolis-Hastings and Its Embedding Within Hiererichical Gibbs......Page 43
3.1 RTO-MH-Within-Hierarchical Gibbs......Page 44
4 Numerical Experiment......Page 45
5 Conclusions......Page 47
References......Page 48
1 Introduction......Page 49
1.1 A Stylized Problem......Page 51
2 Sequential Bayesian Inference for Dynamical Systems......Page 52
3 Finite Volume Solver......Page 54
4 Continuous-Time Frobenius-Perron Operator and Convergence of the FVM Approximation......Page 55
5.1 FVF Tracking of a Pendulum from Measured Force......Page 58
References......Page 59
1 Introduction......Page 60
2 Background......Page 61
2.1 Likelihood via Filtering......Page 62
2.2 Correlation Integral Likelihood......Page 63
3.1 Ornstein-Uhlenbeck with Modification for Dynamics......Page 65
3.2 Stochastic Chaos......Page 68
References......Page 70
1 Introduction......Page 72
2 The Convex Source Support in Electrical Impedance Tomography......Page 73
3 An Optimization Problem in Kc(Rd)......Page 74
4 Galerkin Approximations to Kc(R2)......Page 77
5 Gradients of Functions on GA......Page 80
6 A First Numerical Simulation......Page 81
References......Page 83
1 Introduction......Page 85
2 Optimal Transport......Page 87
3.1 The Martingale Problem......Page 88
3.2 Augmented Lagrangian Approach......Page 90
4 Numerical Method......Page 92
6 Summary......Page 95
References......Page 98
1 Introduction......Page 99
2.1 Bayesian Formulation of the Inverse Problem......Page 101
2.2 Prior Reduction......Page 102
2.3 Likelihood-Informed Subspace......Page 103
3 Results......Page 107
References......Page 111
Wider Contours and Adaptive Contours......Page 113
1 Introduction......Page 114
2.1 Monomolecular, Bimolecular and Trimolecular Models......Page 116
3 Pseudospectra Are Important for Stochastic Processes......Page 119
4 A Mittag-Leffler Stochastic Simulation Algorithm......Page 122
5.1 Computing Contour Integrals......Page 124
5.2 Estimating the Field of Values......Page 126
References......Page 131
1 Introduction......Page 133
2 Problem Statement and Statistical Model......Page 134
2.1 Bayesian Formulation......Page 136
3 Hamiltonian Monte Carlo......Page 138
4.1 Two Dimensional Registration......Page 141
4.2 Synthetic APT Data......Page 142
5 Conclusion......Page 151
References......Page 153
1 Introduction......Page 155
2 Inverse Problems......Page 157
2.1 PDE-Constrained Inverse Problems......Page 158
2.2 Image Reconstruction......Page 159
2.3 Objective Function......Page 160
2.4 Optimization Algorithms......Page 161
2.5 Challenges......Page 164
3 Recent Advances in Optimization......Page 165
4 Discussion......Page 168
References......Page 169
1 Introduction......Page 175
2.1 Infinite Volume Form Factors......Page 176
2.2 Finite Volume Form Factors in the BY Domain......Page 178
3 The Proof for Diagonal Large Volume Form Factors......Page 180
References......Page 184
1 Introduction......Page 186
1.1 Preliminary......Page 188
2.1 Narayana Number......Page 190
2.2 Chebyshev Polynomial......Page 191
2.3 Motzkin Path......Page 193
3 Single-Stack Diagrams and RNA Shapes......Page 196
References......Page 198
A Curious Mapping Between Supersymmetric Quantum Chains......Page 200
1 Introduction......Page 201
2 Models......Page 202
2.1 FS Model Definition......Page 203
2.2 FGNR Model Definition......Page 205
2.2.1 Domain Walls......Page 207
2.2.2 Solution of the FGNR Model by Bethe ansatz......Page 208
3 Mapping Between the Domain Wall and Particle Representations......Page 209
3.2 Transformation Γ......Page 211
3.3 Transformation P......Page 212
3.4 Particle Position Coordinates......Page 214
3.5 Examples......Page 215
4 Conclusion......Page 216
References......Page 217
1 Introduction......Page 218
2 Results......Page 219
3 Discussion......Page 224
References......Page 225
1 Introduction......Page 226
2 Main Theorems......Page 229
3.1 Reduction to Collapsed Points......Page 230
3.2 The ``Linear'' Theory......Page 231
3.4 Height Bound and Excess Decay......Page 232
3.7 Monotonicity of the Frequency Function and Final Blow-Up Argument......Page 233
4 Weighted Monotonicity Formulae......Page 234
References......Page 236
1 Background......Page 239
2 Time Discretisation......Page 241
3 Main Results......Page 242
References......Page 244
1 Introduction......Page 245
2 The Set-Up......Page 246
3 Controlling the Geometry of the Flow......Page 249
4 Exponential Convergence......Page 252
References......Page 253
1 Introduction......Page 254
2 Main Results......Page 259
3 Preliminary Results......Page 266
4.1 Proof of Theorem 1......Page 267
4.2 Proof of Theorem 2......Page 268
4.4 Proof of Theorem 3......Page 269
References......Page 271
1 Introduction......Page 274
2.1 Geometric and Analytic Preliminaries......Page 278
2.2 Locality Principle for Dirichlet Boundary Condition......Page 279
2.3 Locality Principle for Neumann Boundary Condition......Page 281
2.4 Locality for Robin Boundary Condition......Page 284
3 Hearing the Corners of a Drum......Page 297
3.1 Heat Trace Calculations......Page 298
3.1.1 Heat Trace Contribution from the A Term......Page 299
3.1.2 Heat Trace Contribution from the B Term......Page 300
3.1.3 Heat Trace Contribution from the C Term......Page 302
3.2.1 Trace of the E Term......Page 303
3.3 Heat Trace Expansions and Proof of Theorem 1......Page 305
4 Microlocal Analysis in the Curvilinear Case......Page 306
References......Page 308
Nonparametric Bayesian Volatility Estimation......Page 310
1.1 Problem Formulation......Page 311
1.3 Approach and Results......Page 312
2.1 Generalities......Page 314
2.2 Prior Construction......Page 315
2.3 Gibbs Sampler......Page 316
2.4 Hyperparameter Choice......Page 318
3.1 Blocks Function......Page 319
3.2 CIR Model......Page 322
4 Dow-Jones Industrial Averages......Page 323
5 Conclusions......Page 327
6.2 Metropolis-Within-Gibbs Step......Page 328
References......Page 330
1 Introduction......Page 334
2 The Main Result......Page 336
References......Page 340
1 Introduction......Page 341
2 Main Results......Page 344
3 Preliminaries......Page 345
4.1 Proof of Theorem 1......Page 347
5 Conclusions......Page 354
References......Page 355
1 Introduction......Page 357
2 The General Question......Page 358
3 Some Specific Cases......Page 359
References......Page 365
Biquasiprimitive Oriented Graphs of Valency Four......Page 366
References......Page 370
1 Introduction......Page 371
2 Tutte's Doctoral Research......Page 375
2.1 `A Ring in Graph Theory'......Page 376
3 Graph Polynomials......Page 377
4 Matroids......Page 379
5 The Excluded-Minor Theorems......Page 381
6 Higher Connectivity for Matroids......Page 382
8 Tutte's Ph.D. Thesis......Page 384
10 Tutte's Contributions......Page 386
References......Page 388
1 Introduction......Page 390
2.1 Simply-Connected Algebraic Groups......Page 393
2.2 Cluster Seeds Associated to Reduced Decompositions......Page 394
3.1 Geometric Crystal for A-Variety Specialized at Δw0ωi, ωi's......Page 395
3.2 SL3......Page 397
3.3 Cluster Charts and Geometric Crystals......Page 398
4 * Dual Geometric Crystal......Page 399
4.1 SL3......Page 401
5.1 Elementary Maps from Which We Make Geometric RSK-Correspondences......Page 402
5.2 Inverse Geometric RSK......Page 404
5.4 Commutativity Elementary Maps κl and Lusztig Moves......Page 405
5.5 Lusztig Variety and the Map CA+......Page 406
5.6 Berenstein-Zelevinsky Variety......Page 407
5.7 Graded Nakashima-Zelevinsky Cone......Page 408
5.8 Proof of Theorem 7......Page 409
6.1 Lusztig Variety and the Map CA-......Page 411
6.2 Lusztig Variety and Decoration Ψ*K......Page 412
6.3 Transposed BZ-Twist......Page 413
References......Page 414
Part II Other Contributed Articles......Page 415
1 Introduction......Page 416
2 Gauss Sums on Finite Algebras......Page 419
3 Proof of Theorem 1.5......Page 423
References......Page 425
1 Physical and Mathematical Context......Page 426
2 Progress at Creswick......Page 427
References......Page 428
1 Arithmetic Conditions for Operators of Calabi–Yau Type......Page 429
2.1 Factorial Ratios......Page 431
2.2 Generalized Hypergeometric Functions......Page 432
2.2.1 Landau-Like Functions......Page 434
3.1 A Glimpse of Dwork's Result......Page 435
3.2 Factorial Ratios......Page 436
3.3 Factorial Ratios of Linear Forms......Page 437
3.5 A Brief Overview of the p-Adic Strategy......Page 439
References......Page 440
1 Cubic Transformation Formulas......Page 441
2 Quadratic Transformations: Revisiting Gauss' Quadratic AGM......Page 443
4 Transformation and Reduction Formulas......Page 444
References......Page 445
1 Introduction and Statement of Results......Page 447
References......Page 448
Some Supercongruences for Truncated Hypergeometric Series......Page 449
References......Page 451
1 Introduction......Page 452
2 The Motive M = M6 M8......Page 453
3 The Explicit Formula......Page 454
4 Applying the Explicit Formula to M6 and M8......Page 455
References......Page 456
1 Introduction......Page 457
2 Convergence to the Unit Root and Hodge Gaps......Page 458
3 A Congruence of Depth Five......Page 459
References......Page 460
Alternate Mirror Families and Hypergeometric Motives......Page 462
1 Motivation......Page 463
2 Common Factor Theorem......Page 465
3 Explicit Computations and Hypergeometric Motives......Page 467
References......Page 468
1 The Schwarz Derivative......Page 470
2 Equivariant Functions......Page 474
3 The Automorphic and Modular Aspects......Page 475
4 The Elliptic Aspect......Page 477
5 The General Case......Page 479
References......Page 481
Hypergeometric Functions over Finite Fields......Page 482
2 Method......Page 483
3 Galois Interpretation......Page 485
4 Examples of Translated Identities......Page 486
References......Page 487
1 Main Result......Page 488
2 Motivation......Page 489
3 Key Ideas and Example......Page 490
References......Page 491
1 Introduction......Page 492
3 Periods......Page 493
References......Page 495
1 Hasse–Witt Matrix......Page 496
2 Main Results......Page 498
References......Page 500
1 Triangle Groups......Page 501
2 Triangular Modular Curves......Page 502
References......Page 503
Jacobi Sums and Hecke Grössencharacters......Page 504
References......Page 505
Special Values of Hypergeometric Functions and Periods of CM Elliptic Curves......Page 506
References......Page 508
2 Elliptic Curves Versus Rigid Calabi–Yau Threefolds......Page 509
2.1 How Can Rigid Varieties Appear in a Pencil?......Page 510
3 Calabi–Yau Operators......Page 512
4 Euler Factors from Picard–Fuchs Operators......Page 514
4.1 Unit Root Method......Page 515
4.2 Deformation Method......Page 516
4.3 Lifting to Higher Order......Page 519
References......Page 520
1 Introduction......Page 522
2 Statistical Preliminaries......Page 523
3 Minimum Variance Estimation......Page 524
4 The Kalman Filter......Page 526
4.1 A Variational Formulation of the Kalman Filter......Page 528
4.2 The Extended Kalman Filter......Page 529
References......Page 530
1 Introduction......Page 531
2 Principle of Bayesian Inference......Page 532
3 Rejection ABC Method......Page 534
4 Regression ABC......Page 536
5 MCMC-ABC Algorithm......Page 537
6 SMC ABC......Page 538
7 Choice of Summary Statistics......Page 540
8 Early Rejection ABC......Page 541
10 Conclusion......Page 542
References......Page 543
1 Introduction......Page 546
2 Inhomogeneous TL Model......Page 548
3 Recursions in System Size......Page 550
4 Recursion Relations for the LWCO......Page 553
4.1 The Fusion Recursion Relation......Page 554
4.2 The Braid Recursion Relation......Page 555
5 The Inhomogeneous Expression for LWCO......Page 557
References......Page 558
1 Introduction......Page 559
2 The Double-Spending Problem......Page 560
3 A Refinement of the Double-Spending Problem......Page 563
References......Page 565
1 Introduction......Page 566
2 Problem Formulation......Page 567
3 A Necessary Condition for Optimality......Page 568
References......Page 570
1 Introduction......Page 572
2 Model......Page 573
3 Preliminaries......Page 575
4 Integro-Differential Equations for the Joint Laplace Transform......Page 576
5 The Analytical Expression for φ(u)......Page 578
6 The Joint Density of (τ, Nτ)......Page 583
References......Page 590
1 Introduction......Page 592
2 Construction of a System of Integral Equations of Volterra Type......Page 594
3 Numerical Integration Procedure for Approximation BCP......Page 598
References......Page 601
Introduction to Extreme Value Theory: Applications to Risk Analysis and Management......Page 603
1 Introduction......Page 604
1.1 What Is Risk?......Page 605
1.2 Impact of Extreme Risks......Page 607
2.1 CLT Versus EVT......Page 608
2.2 A Limit Theorem for Extremes: The Pickands Theorem......Page 611
2.3.1 Peak Over Threshold (POT) Method......Page 612
2.3.2 Tail Index Estimators for MDA(Fréchet) Distributions......Page 613
2.4 A Self-Calibrated Method for Heavy-Tailed Data......Page 616
2.4.2 Performance of the Method (Algorithm) Tested via MC Simulations......Page 618
2.4.3 Application in Neuroscience: Neural Data......Page 619
2.4.4 Application in Finance: S&P 500 Data......Page 621
3.1.1 Impact of the Dependence on the Diversification Benefit......Page 623
3.1.2 Type of Dependence......Page 631
3.2 Notion of Dependence......Page 633
3.3 Copulas......Page 635
3.4 Notion of Rank Correlation......Page 639
3.5 Ranked Scatterplots......Page 640
3.6 Other Type of Dependence: Tail or Extremal Dependence......Page 641
3.6.1 Properties and Terminology......Page 642
3.6.2 Numerical Example Showing the Impact of the Choice of Copula......Page 643
4.1 MEV Distribution......Page 644
4.2.1 Upper Tail Dependence and CDA......Page 645
References......Page 646
1 Introduction......Page 649
2.1 Introduction: Markowitz Portfolio Optimization and the Sharpe Ratio......Page 651
2.2 The Definition of the Monotone Sharpe Ratio and Its Representation......Page 653
2.3 Basic Properties......Page 659
3 Buffered Probabilities......Page 660
3.1 A Review of the Conditional Value at Risk......Page 661
3.2 The Definition of Buffered Probability and Its Representation......Page 662
3.3 Properties......Page 666
4.1 A Continuous-Time Market Model and Two Investment Problems......Page 668
4.2 A Brief Literature Review......Page 670
4.3 Solution of Problem 1......Page 671
4.4 Solution of Problem 2......Page 672
Duality in Optimization......Page 674
References......Page 676
1 Introduction......Page 678
2 Decision Rules and Their Optimality......Page 679
3 Reduction to an Optimal Stopping Problem......Page 680
4 The Main Results......Page 682
References......Page 684
1 Motivation......Page 686
2.1 The Theorem......Page 689
2.2 Constructions......Page 690
2.3 Shuffle Algebra......Page 691
2.4 Drinfeld Currents......Page 692
3.1 Equivariant Elliptic Cohomology......Page 693
3.2 The Sheafified Elliptic Quantum Group......Page 694
3.3 The Shuffle Formulas......Page 696
3.4 Drinfeld Currents......Page 698
4 Relation with the Affine Grassmannian......Page 699
References......Page 700

2020-10-11