2018 MATRIX Annals

Preface......Page 6
Contents......Page 29
I Refereed Articles......Page 31
 Part 1 On the Frontiers of High Dimensional Computation......Page 32
   1 Introduction......Page 33
   2 The main result......Page 37
     2.1.2 Data-explicit estimates for the RTE......Page 40
     2.1.3 Spectral properties of the RTE......Page 41
   References......Page 42
  2 A reduced-order-model Bayesian obstacle detection algorithm......Page 44
   1 Introduction......Page 56
    1.1 Variational eigenvalue problems......Page 57
   2 Bounding the spectral gap......Page 59
    3 Numerical results......Page 65
   References......Page 69
  3 Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients......Page 55
   1 Introduction......Page 70
   2 Restricted Randomized Algorithms in a General Setting......Page 71
   3 The Power of Restricted Randomized Algorithms......Page 80
   Reference......Page 83
   1 Introduction and Preliminaries......Page 85
   2 Results......Page 89
   References......Page 101
   1 Introduction......Page 103
   2 Preliminaries......Page 105
   3 Existence result for worst-case error......Page 107
   4 Numerical experiments on the conjecture......Page 118
   References......Page 120
   1 Introduction......Page 121
   2 Approximation in periodic spaces of Sobolev type......Page 123
   3 Preasymptotics for approximation in isotropic Sobolev spaces......Page 125
   4 Preasymptotics for approximation in mixed Sobolev spaces......Page 133
   References......Page 136
   1 Introduction......Page 137
   2 Background......Page 138
   3 Numerical algorithms for semi-supervised learning......Page 139
   4 Experimental results......Page 141
   References......Page 143
   1 Introduction......Page 144
    1.1 The model problem......Page 145
   2 The parametric and nonparametric P1–simplicial and quadrilateral nonconforming finite elements......Page 146
    2.1 The parametric simplicial and rectangular NC elements in two and three dimensions......Page 147
     2.2.2 The P1–NC quadrilateral element......Page 149
   3 The P1–nonconforming polyhedral finite element......Page 150
    3.2 Basis and its dimension......Page 153
    3.4 The P1–NC polyhedral Galerkin methods......Page 154
   References......Page 155
 Part 2 Month of Mathematical Biology......Page 157
   1 Introduction......Page 158
    2.1 Background......Page 159
    2.2 Analytical results for the errors of the stochastic QSSA......Page 163
   References......Page 167
  11 Accurate particle-based reaction algorithms for fixed timestep simulators......Page 169
   1 Introduction......Page 170
   2 Assumptions and Definitions......Page 171
    3.1 Brownian bridge method, two-step algorithm......Page 173
    3.2 Brownian bridge method, three-step algorithm......Page 175
    3.3 RDF-matching, two-step......Page 176
    3.4 RDF-matching, three-step......Page 178
    3.5 RDF-matching, two-step with remapping......Page 179
    3.6 Example of RDF-matching with remapping......Page 181
   4 Discussion......Page 182
   References......Page 183
 Part 3 Recent Trends on Nonlinear PDEs of Elliptic And Parabolic Type......Page 185
   1 Decay estimates, methods, results and perspectives......Page 186
   2 Recurrence and transiency of long jump random processes......Page 194
   References......Page 199
   1 Introductory comments......Page 202
    1.1 Moduli of continuity......Page 203
    1.2 Motivation: Zero counting for equations in one space variable......Page 204
    1.3 The heat equation on Euclidean space......Page 206
    1.4 The Neumann heat equation on a bounded domain......Page 208
    1.5 The Payne-Weinberger inequality......Page 209
    2.1 Riemannian manifolds: Distance, Curvature and heat equations......Page 210
    2.2 The Ricci non-negative case......Page 211
    2.3 Nonlinear eigenvalues......Page 215
    3.1 The fundamental gap conjecture......Page 217
    3.2 The 1D case......Page 218
    3.3 Converting to a Neumann problem......Page 219
    3.4 Sharp log-concavity......Page 221
    3.6 Sharp lower bound in terms of modulus of convexity of the potential......Page 224
    4.1 Gradient estimates and P-functions......Page 225
    4.2 The two-point estimate......Page 226
   References......Page 229
   1 Introduction......Page 231
    1.1 A fractional obstacle problem......Page 232
    1.2 General assumptions......Page 234
   2 Main results......Page 235
    2.1 Further comments and strategy of proofs......Page 236
   3 Some mathematical background......Page 238
   4 The case of convex obstacles: proofs of the main Theorem......Page 240
   References......Page 243
   1 Introduction......Page 245
   2 Physical considerations......Page 246
   3 Symmetry results......Page 252
    3.1 Symmetry properties for the Allen-Cahn equation......Page 255
    3.2 Symmetry properties for the fractional Allen-Cahn equation......Page 256
    3.3 Symmetry properties for the water wave problem......Page 258
   Acknowledgement......Page 261
   References......Page 262
   1 Introduction and main result......Page 265
   2 Counterexamples in convex domains and proof of the theorem......Page 267
   References......Page 277
  17 A potential well argument for a semilinear parabolic equation with exponential nonlinearity......Page 280
   1 Model parabolic problem......Page 281
   2 Stable and unstable sets......Page 284
   3 Sketch of the proof of Theorem 1......Page 285
   References......Page 287
  18 Quantitative analysis of a singularly perturbed shape optimization problem in a polygon......Page 289
   1 Introduction......Page 290
   2 Setting of the problem and main results.......Page 292
   References......Page 297
    1.1 The energy, notation......Page 298
    1.3 The boundary value problem......Page 299
    2.1 Previous geometric gap lemmas......Page 300
   3 Estimates......Page 301
   References......Page 304
   1 Introduction......Page 305
   2 Critical points in the Wulff problem......Page 308
   3 Minimizers in the anisotropic liquid drop model......Page 310
   References......Page 312
  21 Liouville-type theorems for nonlinear elliptic and parabolic problems......Page 315
    1.1 Motivation and classical results: Fujita, Gidas-Spruck, Liouville......Page 316
    1.2 Equations vs. inequalities – a first method: rescaledtest-functions......Page 317
    2.1 Results and conjectures......Page 318
    2.2 Radial case: proof based on zero-number......Page 319
    2.3 Nonradial case: proof based on similarity variables and energy estimates......Page 321
    3.1 Results......Page 323
    3.2 Sketch of proof of Theorem 3.1(i) (initial-final blow-up estimate in Rn)......Page 324
    4.1 Elliptic systems I: Lane-Emden......Page 326
    4.2 Elliptic systems II: positive self-interaction......Page 328
    4.3 Elliptic systems III: negative self-interaction......Page 329
    5.2 Gradient structure-homogeneous case......Page 332
    5.3 Gross-Pitaevskii case......Page 333
   References......Page 335
II Other Contributed Articles......Page 338
 Part 4 Algebraic Geometry, Approximation and Optimisation......Page 339
   1 Introduction......Page 340
    3.1 Prototype construction......Page 341
    4.1 Vandermonde and generalised Vandermonde matrices......Page 343
   5 Discussions and future research directions......Page 345
   References......Page 346
 Part 5 Dynamics, Foliations, and Geometry In Dimension 3......Page 347
   1 Introduction......Page 348
    1.1 Homotopy, integrability, and conjugacy......Page 349
    1.2 Results......Page 350
   2 General Outline......Page 351
    2.1 Dichotomy for foliations......Page 352
    2.3 Double translation......Page 353
   3 A key general proposition......Page 354
    3.2 Gromov hyperbolic leaves......Page 355
    3.3 Coarse contraction and a key proposition......Page 356
    4.2 Changing the lifts......Page 357
    5.1 Uniform foliations and transverse pseudo-Anosov flows......Page 358
    5.2 Forcing a particular type of dynamics on periodic center leaves......Page 359
    5.4 No mixed behavior......Page 360
    6.2 Double translation in hyperbolic manifolds......Page 361
    6.3 More general manifolds......Page 362
   References......Page 363
   1 Introduction......Page 365
   2 Projectively Anosov plugs......Page 366
    2.1 Definition of projectively hyperbolic plug......Page 367
    2.2 A plug with six Reeb components......Page 368
   3 Filling a pseudo-Anosov flow......Page 370
   4 Filling incoherent repellers......Page 371
   5 A foliation/contact interpretation of the examples......Page 372
   6 Questions and future directions......Page 374
   References......Page 375
   1 Introduction......Page 376
   2 The proof......Page 377
   3 Proof of lemma 2.4......Page 382
   References......Page 384
 Part 6 Geometric and Categorical Representation Theory......Page 385
   1 Introduction......Page 386
   2 An Introduction to Geometrically Adhesive Functors, with Motivation......Page 389
   3 The Adhesive Site......Page 405
   4 The Adhesive Fundamental Group......Page 420
   5 Applications to Local Systems......Page 427
   References......Page 430