More progresses in analysis : proceedings of the 5th International ISAAC Congress, Catania, Italy, 25-30 July 2005

"International ISAAC (International Society for Analysis, its Applications and Computation) Congresses have been held every second year since 1997. The proceedings report on a regular basis on the progresses of the field in recent years, where the most active areas in analysis, its applications and computation are covered. Plenary lectures also highlight recent results. This volume concentrates mainly on partial differential equations, but also includes function spaces, operator theory, integral transforms and equations, potential theory, complex analysis and generalizations, stochastic analysis, inverse problems, homogenization, continuum mechanics, mathematical biology and medicine. With over 350 participants attending the congress, the book comprises 140 papers from 211 authors." "The volume also serves for transferring personal information about the ISAAC and its members. This volume includes citations for O Besov, V Burenkov and R P Gilbert on the occasion of their anniversaries."--BOOK JACKET Cover......Page 1 Title......Page 4 Preface......Page 6 Citations......Page 10 Professor O.V. Besov Honorary Member of ISAAC Victor I. Burenkov......Page 12 A Tribute to the 65th Birthday of Prof. Victor 1. Burenkov Massimo Lanza de Cristoforis......Page 14 Robert Pertsch Gilbert: Citation for his 75th Birthday Heinrich Begehr......Page 16 CONTENTS......Page 20 Plenary Lectures......Page 32 1. Carleman estimates......Page 34 2. Uniqueness and stability of the continuation......Page 36 3. Uniqueness and stability in inverse problems......Page 38 References......Page 41 1. Introduction......Page 44 2. The Illodels......Page 45 2.1. The kinetic description......Page 46 2.3. The energy-transport model......Page 47 2.4. The hydrodynamical model......Page 48 2.5. Future perspectives in the mathematical modeling in nanoelectronics......Page 50 References......Page 52 1. Introduction......Page 54 2. Boundary Value Problems......Page 55 3. Calderon Projector......Page 58 4. Exterior Boundary Value Problems......Page 62 5.1. Interior Dirichlet Problem......Page 63 5.2. Exterior Dirichlet problem......Page 65 5.3. Interior Neumann Problem......Page 66 5.4. Exterior Neumann Problem......Page 67 6. Concluding Remarks......Page 68 References......Page 69 2. Plane shock reflection......Page 70 3. Shock reflection by a smooth surface......Page 72 4. Shock reflection by a ramp......Page 73 5.2. Flat Mach Configuration and its perturbation......Page 76 5.3. Generalized Lagrange transformation......Page 80 5.4. Decomposition......Page 81 5.5. Estimates and existence......Page 82 References......Page 83 1. Weighted Sobolev spaces with constant smoothness sEN defined on an irregular domain......Page 86 II. Weighted spaces BS q(G) and FS(G) of functions that have p, p,q variable smoothness and are defined on a domain with a locally Lipschitz boundary......Page 89 III. Weighted spaces Bps q (IRn) and F S (IRn) of functions with , p,q variable smoothness......Page 93 References......Page 96 1.1 Spaces of Differentiable Functions and Applications (V.I. Burenkov)......Page 98 Introduction......Page 100 1. Construction of the matrix of fundamental solution for the system of elasticity of a special form......Page 102 2. The solution of problems (1), (2) in domain Dp......Page 104 References......Page 112 First Lyapunov method for the abstract parabolic equation V.A. Trenogin......Page 114 References......Page 120 1.2 Variable Exponent Analysis and Applications (St. Samko)......Page 122 1.1. A variant of the definition of the norm in space L2 (0)......Page 124 1.2. Amemiya norm and Orlicz norm......Page 125 1.3. An exact inequality involving Luxemburg norm and conjugate- Orlicz norm......Page 126 2.1. Results on existence and multiplicity of solutions of p(x)-Laplacian equations......Page 127 2.3. Some open problems......Page 129 References......Page 130 1. Introduction......Page 132 2. Trace spaces when smooth functions are dense......Page 134 References......Page 137 1. Introduction......Page 138 2. Lebesgue spaces......Page 139 3. Hajlasz and Newtonian spaces......Page 141 4. Sobolev embeddings p 0 and 2 > 0......Page 238 References......Page 241 1. Introduction and preliminaries......Page 242 2. Conditional analytic Fourier-Feynman transforms......Page 245 3. Conditional convolution products and relationships with analytic conditional Fourier-Feynman transforms......Page 248 References......Page 250 1. Introduction......Page 252 2. A Change of Scale Formula for Functions in S......Page 253 3. A Change of Scale Formula for Cylinder Type Functions......Page 255 4. A Change of Scale Formula for Unbounded Functions......Page 256 5. Change of Scale Formulas for Wiener Integrals and Fourier-Feynman Transforms......Page 257 References......Page 259 1. Introduction......Page 262 2. Elementary properties for the space 8 (lR+)......Page 265 3. The Kontorovich - Lebedev transformation in S R+)......Page 267 References......Page 270 1. Introduction......Page 272 2. An Integral Transform......Page 274 3. Spaces of Sobolev Type......Page 275 4. An Embedding Theorem of Sobolev Type......Page 276 5. Applications to Partial Differential Equations......Page 277 References......Page 280 1. Introduction......Page 282 2. Results......Page 283 References......Page 289 1. Introduction......Page 290 2.2. Eye sphere......Page 291 2.4. Rotation matrices for camera angle sensors......Page 292 3. Calculation of an object point X and relation between right and left image points......Page 293 4. Initial procedure......Page 294 6. Epipolar plane and epipolar radius......Page 295 7.1. Independent selection of object points Xi on pupil circle......Page 296 7.2. Optimal selection of Xi......Page 297 References......Page 298 1. Some properties of the functions ReK +i (X) and ImK +i (x)......Page 300 2. The integral equations and Parseval equalities for the modified Kontorovitch-Le bedev integral transforms.......Page 304 References......Page 309 1. Introduction......Page 310 2. Generalization error of a three-layer neural network......Page 313 3. Main Theorems......Page 314 4. Proof of Main Theorem 1......Page 315 References......Page 319 1. History and background......Page 320 2. The spaces A in the half-plane......Page 322 3. Representation over strips......Page 323 5. Orthogonal projection and isometry......Page 324 6. The projection L A and the conjugate space of A......Page 325 7. Weighted classes of harmonic functions......Page 326 8. Nevanlinna-Djrbashian type classes in the half-plane......Page 327 References......Page 329 1.1. The L2 -approach......Page 332 1.2. DO in distributional spaces......Page 333 1.3. Properties......Page 334 1.4. Scope......Page 335 2.1. Viscoelastic rods......Page 336 2.2.1. Transfer functions......Page 337 2.2.2. Step responses......Page 338 References......Page 340 1. Introduction......Page 342 2. General approach......Page 344 3. Equations with Left-Sided Fractional Derivatives......Page 345 4. Equations with Right-Sided Fractional Derivatives......Page 350 References......Page 354 1.5 Toeplitz and Toeplitz-like Operators (S. Grudski, N. Vasilevski)......Page 356 1. Frechet operator algebras by commutator methods......Page 358 2. Localization of the Segal-Bargmann projection......Page 361 References......Page 368 1. Introduction......Page 370 2. Isomorphisms of local algebras......Page 372 3. Main results......Page 376 References......Page 379 1. Introduction......Page 380 2. Operator relations......Page 381 3. An invertibility criterion based on a mean motion depending on a Hausdorff set......Page 384 4. Example......Page 385 References......Page 387 1. Introduction......Page 390 2.1. Multiplicative Holder spaces on lR+......Page 391 2.2. Local Predholmness of Mellin pseudodifferential operators on Holder classes......Page 392 3. Operators of potential type on Jordan curves with vorticity points......Page 394 References......Page 398 1. Introduction......Page 400 2.1. The Zygmund-Bary-Stechkin class......Page 402 3. The formula for the indices......Page 403 References......Page 406 1. Introduction......Page 408 2. Main results......Page 409 3. Some information on the operator ideals......Page 410 4. The Krein algebra......Page 411 5. The invertihility of the operator Ar(a). The estimation of the trace of the inverse operator.......Page 412 6. An asymptotic representation of the operator (Ar-(a))......Page 414 7. The proof of Theorem 2.1......Page 416 References......Page 417 1. Introduction and Preliminaries......Page 418 2. General Inducing Function......Page 420 3. Analytic ind ueing function......Page 424 References......Page 427 1.6 Wavelets (R. Hochmuth, M. Holschneider)......Page 428 1. Introduction......Page 430 2. Definitions and known results......Page 431 3. Asymptotic behaviour of distributional wavelet transform at infinity......Page 433 4. Asymptotic expansion of distributional wavelet transform at infinity......Page 435 References......Page 436 1. Introduction......Page 438 2. Integral Equation and Multiresolution Analysis......Page 439 3. Numerical Results......Page 442 References......Page 443 1. Introduction......Page 446 2. Notation and preliminaries......Page 447 3. The groups Hf3......Page 448 References......Page 450 1.8 Pseudo-Differential Operators (J. Toft, M.W. Wong)......Page 452 1. Main theorem......Page 454 2. The outline of the proof......Page 456 References......Page 459 1. Introduction......Page 460 2. Main Results......Page 461 References......Page 469 1.9 Stochastic Analysis (N. Jacob, Y. Xiao)......Page 470 1. Introduction......Page 472 2. Random series expansions of anisotropic models......Page 474 3. Identification of anisotropic asymptotical exponents......Page 475 Appendix: Identification of the exponent for aID-process......Page 478 References......Page 481 1. Introduction......Page 482 2. The space-time fractional diffusion......Page 483 3. The continuous-time random walk......Page 484 4. Subordination in stochastic processes......Page 486 5. Sample path for space-time fractional diffusion......Page 490 6. Numerical results......Page 491 References......Page 494 1. Gaussian white noise and Fock space......Page 498 2. Poisson white noise......Page 503 3. Levy white noise and extended Fock space......Page 505 4. The square of white noise algebra......Page 508 References......Page 510 11.1 Quantitative Analysis of Partial Differential Equations (M. Reissig, J. Wirth)......Page 512 1. Problem definition......Page 514 2. Algorithm of identification......Page 515 4.1. First-order system......Page 517 5. Systems of second order......Page 522 References......Page 523 1. Introduction......Page 524 2. Dimension of the space V = m 2 - m - 1......Page 534 3. Dimension of the space V = m 2 - m......Page 538 References......Page 540 Stability of stationary solutions of nonlinear hyperbolic systems with multiple characteristics A. K ryvko and V. V. K ucherenko......Page 542 References......Page 550 1. Introduction......Page 552 2. Fundamental solutions......Page 554 3. Decomposition of hypergeometric functions of Lauricella FA......Page 557 4. Properties of fundamental solutions......Page 559 Acknowledgment......Page 561 References......Page 562 1. Introduction......Page 564 2. Asymptotic phase and related functional identity......Page 565 3. Growth properties of generalized eigenfunctions......Page 568 4. The Principle of Limiting Absorption......Page 571 References......Page 573 1. Introduction......Page 574 3. Condition of independence of p(t, e) with respect to t......Page 575 5. Calculation ofp(t,e) when n is even and C = sn-l(O,I)......Page 577 6. Interpretation of the operator L = :t22 - p(t, Dro) with singular p(t, e) when n = 5......Page 578 7. Extension of Strichartz inequalities......Page 579 8.2. L1 - Loo inequality for (8.2)......Page 580 10. Global existence......Page 581 References......Page 582 1. Introduction......Page 584 2. Global-in-time existence theorems......Page 585 3. Asymptotic profiles......Page 586 4. Implicit representations of free and non-free waves......Page 588 Acknowledgments......Page 589 References......Page 590 1. Introduction and main results......Page 592 1.1. The case p >......Page 597 1.2. The case of bounded dissipation......Page 598 References......Page 599 1. Introduction......Page 602 2. Non-Negative Solutions......Page 604 3. Non-Existence of Global Solutions......Page 606 References......Page 609 1. Introduction......Page 612 2.1. Notations and Basic Identities......Page 615 2.2. Lower bounds for free solutions......Page 616 2.4. Asymptotic behavior of the auxiliary function......Page 617 3. Outline of the proof of the theorems......Page 618 References......Page 620 1. Introduction......Page 622 2.1. Estimation in the Pseudodifferential Zone......Page 627 2.2. Estimation in the Hyperbolic Zone......Page 628 References......Page 631 1. Introduction......Page 634 2. Method of the proof......Page 636 3. The construction of the solution......Page 638 4. Regularity of the solution and the proof of Theorem 1.1......Page 639 5. Application......Page 643 References......Page 644 11.2 Boundary Value Problems and Integral Equations (P. Krutitskii)......Page 646 1. Introduction......Page 648 2. Stabilizing influence of Signorini conditions for activator......Page 649 3. Destabilizing influence of Signorini conditions for inhibitor......Page 654 References......Page 657 1. Introduction......Page 658 2. A summary of the results......Page 659 3. Step-index fibers......Page 661 3.1. Numerical examples.......Page 664 References......Page 667 1. Introduction......Page 668 2. One dimensional null space......Page 670 3. Two dimensional null space......Page 673 References......Page 677 1. Introduction......Page 678 2. Formulation of the problem......Page 679 3. Existence of a classical solution......Page 682 4. Non-existence of a weak solution......Page 684 References......Page 685 11.3 Elliptic and Parabolic Nonlinear Problems (F. Nicolosi)......Page 688 Harnack inequalities for energy forms on fractals sets M.A. Vivaldi......Page 690 References......Page 698 1. Introduction......Page 700 2. Notations......Page 701 3. A counterexample......Page 702 4. Near and almost near operators......Page 703 5. Matrices with piecewise constants coefficients......Page 704 6. Matrices with piecewise VMO coefficients.......Page 707 References......Page 709 1. Introduction......Page 712 2. Anisotropic generalized Sobolev-Orlicz spaces......Page 714 3. Existence of bounded solutions......Page 715 4.1. The energy identity......Page 716 4.2. The ordinary differential inequality.......Page 718 4.2.1. Analysis of the ordinary differential inequality......Page 720 References......Page 721 1. Introduction......Page 722 2. Weak Formulation, Main Results......Page 724 3. Example......Page 728 References......Page 730 1. Introduction......Page 732 2. Kolmogorov operators......Page 736 3. Gaussian estimates for Kolmogorov operators......Page 739 References......Page 741 1. Introduction......Page 742 2. r-convergence in the case of homogeneous Neumann boundary condition on the boundary of holes......Page 744 References......Page 750 Mountain pass techniques for some classes of nonvariational problems M. Girardie, S. M ataloni and M. M atzeu......Page 752 References......Page 758 1. Introduction......Page 760 2. Definitions and Results......Page 763 3. Main results......Page 764 References......Page 766 1. Introduction......Page 768 2. The Kuramoto equation......Page 770 3. The adaptive equation......Page 771 4. Numerical treatment......Page 774 5. Applications: Singular perturbations without boundary-layers......Page 775 References......Page 776 2. Preliminaries......Page 778 3. Existence......Page 779 4. Asymptotic behaviour of solutions of nonlinear equation (1)......Page 781 References......Page 782 1. Introduction......Page 784 2. Coupled equations......Page 786 3. Variational tools......Page 788 4. Proof of Theorem 1.1......Page 791 References......Page 793 1. Introduction......Page 794 2. Variational setting and abstract tools......Page 797 3. Proof of Theorem 1.1......Page 799 Acknowledgment......Page 800 References......Page 801 2. Hypotheses and statement of the main result......Page 802 References......Page 804 1. Introduction......Page 806 2. Proper k-ball contractive retractions......Page 808 3. Applications: an extension of Guo's theorem......Page 812 References......Page 814 1. Introduction......Page 816 2. Preliminary......Page 817 3. Main Result......Page 818 4. Example......Page 820 References......Page 821 1. Singular dimension and maximally singular functions......Page 824 2. Singular dimension of some function spaces......Page 825 4. Singular integrals generated by fractal sets......Page 827 5. Maximally singular Sobolev functions......Page 830 7. Problems for p-Laplace equations......Page 831 References......Page 832 11.4 Variational Methods for Nonlinear Equations (B. Ricceri)......Page 834 1. Historical background and Motivation......Page 836 2.1. The function f is superlinear at the origin......Page 838 3. Proofs......Page 839 3.1. Proof of Theorem 2.1......Page 840 3.2. Proof of Theorem......Page 842 References......Page 844 1. Introduction......Page 846 2. Main results......Page 848 References......Page 851 1. Introduction......Page 854 2. Results......Page 856 3. Concluding remarks......Page 860 Bibliography......Page 861 1. Neumann Problem......Page 864 1.1. Results......Page 866 1.2. Examples......Page 867 2. Dirichlet Problem......Page 868 2.1. Results......Page 869 2.2. Examples......Page 870 References......Page 872 1. Introduction......Page 874 2. Results......Page 876 References......Page 882 1. Infinitely many solutions......Page 884 2. Three distinct solutions......Page 886 References......Page 892 1. Introduction......Page 894 2. Watanabe's Variational Problem and Results......Page 895 3. Representation of the solution......Page 898 4. Restriction Condition and New Parameters......Page 900 References......Page 902 1. Introduction......Page 904 2. Main results......Page 906 References......Page 910 1. Introduction......Page 912 2. Main results......Page 915 References......Page 918 1. Introduction......Page 920 2. The variational setting......Page 921 3.1. Positive potentials......Page 922 3.2. Sign changing potentials......Page 923 4. Examples......Page 924 References......Page 926 A purely vectorial critical point theorem B. Ricceri......Page 928 References......Page 930 1.1. Brief review of existing strategies......Page 932 1.2. Some limitations of existing strategies......Page 933 1.3. Implementing weak sequential lower semi-continuity......Page 934 2. Localized weak lower semi-continuity and weak K-monotonicity......Page 935 3. Basic minimization......Page 937 4. K-monotone and weakly K-monotone mappings......Page 942 5.1. A linear problems......Page 945 5.3. Global eigen-value problems......Page 946 6. Non-quasiconvex minimization......Page 947 7. Conclusions......Page 950 References......Page 951 1. Introduction......Page 952 2. Preliminary results......Page 953 3. Results......Page 957 References......Page 959 111.1 Complex Analysis and Potential theory (M. Lanza de Cristoforis, P. Tamrazov)......Page 960 1. Grunsky inequalities and related results......Page 962 2. Two conjectures......Page 964 3. Main results......Page 965 5. Sketch of the proofs of Theorem 3.1......Page 966 6. Sketch of the proof of Theorem 3.2......Page 969 7. A glimpse at applications of property (5)......Page 972 References......Page 973 1. Introduction......Page 976 2. Main Results......Page 979 3. Proof of Theorems......Page 981 References......Page 983 1. Introduction......Page 986 2. Introduction of the Romieu classes and of the representation Theorem......Page 988 References......Page 995 1. Introduction......Page 998 2. Iteration dynamical systems of discrete Laplacians......Page 999 3. The time change of numbers of families of extinct animals......Page 1001 5. Decrease of number of families and their computer simulations......Page 1002 6. The mutation and the change of environments......Page 1004 7. Conclusions and discussions......Page 1006 References......Page 1007 1. Introduction......Page 1008 2.1. Equations in proper domains......Page 1009 2.2. Equations outside of proper domains......Page 1012 3. Integral expressions for solutions of families of elliptic equations degenerating on an axis......Page 1015 References......Page 1016 Quaternionic background of the periodicity of petal and sepal structures in some fractals of the flower type J. Lawrynowicz, St. Marchiava and M.Nowak-Klfpczyk......Page 1018 1. Introductory......Page 1019 2, Statement of the periodicity theorem......Page 1021 3. Periodicity in the case 2p-1, period 2......Page 1023 4. Nine and sixteen - numbers characterizing a bipetal......Page 1025 References......Page 1026 111.2 Dirac operators in Analysis and Related Topics (J. Ryan, L Sabadini)......Page 1028 1. Introduction......Page 1030 2. Internal quaternionization of real and complex linear spaces......Page 1031 3. Categories of quat ern ionic linear spaces......Page 1034 4. Decompositions of a two-sided-quaternionic linear space......Page 1036 References......Page 1039 1. Introduction......Page 1040 2.1. Notations and definitions......Page 1041 3.1. Holomorphic functions w.r.t. a complex structure J p......Page 1042 4. Non-holomorphic 7j1-regular maps......Page 1043 4.2. A criterion for holomorphicity......Page 1044 4.3. The existence of non-holomorphic 1/J-regular maps......Page 1045 4.4. Other applications of the criterion......Page 1046 Appendix......Page 1047 References......Page 1048 1. Introduction......Page 1050 2. Clifford megaforms for the two variables Dirac system......Page 1052 3. Quaternionic megaforms for the two dimensional Cauchy-Fueter system......Page 1056 4. Applications of megaforms, and further directions for research......Page 1061 Acknowledgements......Page 1062 References......Page 1063 1. Introduction and Preliminaries......Page 1064 2. Integral Representations......Page 1065 3. Some applications......Page 1069 References......Page 1071 1. Introduction......Page 1074 2.2. Clifford Analysis......Page 1075 3. A suitable Witt basis for lRn,n......Page 1076 5. Factorization of the In-stationary Heat equation with Convection Term......Page 1077 6. A parabolic Dirac operator for the in-stationary Schroedinger equation with a potential term......Page 1078 References......Page 1080 1. Introduction......Page 1082 2. Hyperbolic harmonic functions......Page 1083 3. Hypermonogenic functions......Page 1089 References......Page 1095 1. Introduction......Page 1096 2. Lower Distance Estimates......Page 1100 3. Integral Estimates......Page 1101 4. Upper Distance Estimates......Page 1103 5. Concluding Remarks......Page 1104 References......Page 1105 o. Introduction......Page 1106 1. Properties of the analytic Cliffordian monomials (aa:)fta......Page 1108 2. H -solutions......Page 1110 3. p-holomorphic Cliffordian functions......Page 1111 4. Restriction and inflating process......Page 1112 5. Computation of the homogeneous polynomials which are H -solutions......Page 1114 References......Page 1115 A fractal renormalization theory of infinite dimensional Clifford algebra and renormalized Dirac operator J. Lawrynowicz, K. Nono and O. Suzuki......Page 1116 1. Introduction......Page 1117 2. Finite and infinite dimensional Clifford algebras......Page 1118 4. Renormalization of infinite dimensional Clifford algebras......Page 1119 5. Representation of infinite dimensional Clifford algebras on the renormalized space......Page 1121 6. The renormalized Dirac operator......Page 1123 References......Page 1125 1. Introduction......Page 1126 2. Algebraic approach to function theories......Page 1127 2.1. Isotopy classes......Page 1128 4. Classification of the first order PDE......Page 1129 4.2. Elliptic type PDE......Page 1130 5. Power series expansions......Page 1131 5.1. Symmetries......Page 1132 5.1.3. Clifford analysis......Page 1133 References......Page 1135 111.4. Complex and Functional Analytic Methods in Partial Differential Equations (H. Begehr, D.-Q. Dai, A. Soldatov)......Page 1138 1. Introduction......Page 1140 2. Preliminaries......Page 1141 3. Notations and lemma......Page 1143 4. Distribution of zeros for orthogonal polynomials......Page 1146 5. Asymptotics of related quantities for orthogonal polynomials......Page 1147 References......Page 1148 1. Introduction......Page 1150 2. Differential operators for the solutions......Page 1151 3. A differential equation of second order......Page 1152 4. Representation of pseudoanalytic functions in the space......Page 1154 5. Generating pairs......Page 1155 References......Page 1156 1. Introduction......Page 1158 2. Hilbert BVP for Metaanalytic Function......Page 1159 References......Page 1167 1. Introduction......Page 1168 2. Schwarz problem......Page 1169 3. Dirichlet problem......Page 1170 4. Neumann problem......Page 1171 5. Particular Robin problem......Page 1173 References......Page 1177 1. Introduction......Page 1180 2. Dirichlet-Schwarz problem......Page 1181 References......Page 1192 1. Introduction......Page 1194 2. Characteristic form and geometry of a normal surface......Page 1195 3. Initial value problem......Page 1198 4. Generalizations of crystal optics system......Page 1199 References......Page 1200 1. Harmonic functions......Page 1202 2. Elliptic systems......Page 1205 3. Estimates in weighted Holder spaces......Page 1208 4. Conjugate functions......Page 1210 5. Strengthen elliptic systems......Page 1213 References......Page 1215 1.1. Three sphere theorems......Page 1216 1.3. Transfer of smallness for real analytic functions......Page 1217 2.1. Formulation of the result......Page 1218 2.2. Proof of Theorem 2.1......Page 1219 3.1. Harmonic functions......Page 1221 3.2. Solutions of elliptic equations with analytic coefficients and their gradients......Page 1222 References......Page 1223 1. Operator curl......Page 1226 2. Solution of an equation......Page 1229 3. Stokes operator......Page 1231 4. Solution of Stokes equations......Page 1232 5. Application......Page 1234 References......Page 1236 About one class of linear first order overdetermined systems with interior singular and super-singular manifolds N. Rajabov......Page 1238 References......Page 1248 1. Introduction......Page 1250 2. Some results on the class of Abel's equations of the second kind......Page 1251 3. A new construction concerning exact analytic solutions of the Abel equation of the second kind of the normal form......Page 1253 4. Construction of the general solution for the Abel equation (15)......Page 1257 References......Page 1258 1. Introduction......Page 1260 2. Algebraic background......Page 1261 3. Planar homogeneous ODE's......Page 1262 3.1. Proof of the Theorem 2.1......Page 1263 3.2. Integral rays schema......Page 1264 4. Combinatorial Schemes......Page 1265 5. Similarities......Page 1267 6. Concluding remarks......Page 1268 References......Page 1269 111.5 Complex Analytic Methods in the Applied Sciences (V.V. Mityushev, S.V. Rogosin)......Page 1272 1. Introduction......Page 1274 2. Main result......Page 1275 References......Page 1277 1. Introduction......Page 1278 2. Notation and Auxiliary Results......Page 1279 3. Singular integral operator......Page 1282 5. Solvability of nonlinear singular integral equation......Page 1285 References......Page 1287 1. Introduction......Page 1290 2. The boundary value problem and the effective conductivity......Page 1291 3. Effect of perturbation......Page 1298 References......Page 1300 1.1. Solid state and biological structures......Page 1302 1.4. Organization of the paper......Page 1303 2.1. 2D elasticity......Page 1304 2.3. Dislocations in 2D......Page 1305 3. Vortices on the plane: plane incompressible flow......Page 1306 4. Biological structures......Page 1308 5. Ring shaped grain boundary......Page 1309 6. Final remarks......Page 1311 References......Page 1312 1. Introduction......Page 1314 2. An analytic solution of the OZ equation......Page 1315 3.1. Static structure factors......Page 1317 3.2. Thermodynamic quantities......Page 1318 4.1. Fitting procedure of the SCPPS closure to the Lennard-Jones potentials......Page 1319 4.2. Thermodynamic properties of model mixture......Page 1320 5. Summary......Page 1321 References......Page 1322 111.6 Value Distribution Theory and Related Topics (P.C. Hu, P. Li, C.C. Yang)......Page 1324 1. Introduction......Page 1326 2. The finite Fourier transform of (t)......Page 1329 3. Analysis of the transforms H(x), HR(X) and HR(x; A)......Page 1330 Acknowledgement......Page 1332 References......Page 1333 1. Introduction......Page 1334 2. Condition necessaire pour......Page 1335 3. Deformation de la condition......Page 1337 4. Construction de la solution......Page 1338 6. Solution globale dans D(l)......Page 1340 References......Page 1341 1. Introduction and preliminaries......Page 1342 2.3. The sufficient conditions......Page 1344 References......Page 1350 111.7 Geometric Theory of Real and Complex Functions (G. Barsegian)......Page 1352 1. Introduction......Page 1354 2. Construction of the basic map......Page 1355 3. Main results......Page 1356 References......Page 1357 IV.2 Mathematical and Computational Aspects of Kinetic Models (A. Majorana )......Page 1358 1. Introduction......Page 1360 2. The Boltzmann-Poisson system......Page 1361 3. Discretized equations......Page 1362 4. Admissible weight functions......Page 1365 5. Grid layout......Page 1366 References......Page 1369 1. Introduction......Page 1370 2. Problem statement and basic equations......Page 1371 2.1. Collision models......Page 1374 3. Description and discussion of numerical results......Page 1376 3.1. Evaporation......Page 1377 3.2. Condensation......Page 1378 References......Page 1379 Maths against cancer F. Pappalardo, S. Motta, P.-L. Lollini and E. Mastriani......Page 1382 1. Introduction......Page 1383 2. Models for cancer - immune system competition......Page 1384 3. IS vs cancer competition: the vaccine effect......Page 1386 4. Results......Page 1387 References......Page 1389 IV.4 Inverse Problems, Theory and Numerical Methods (M. Klibanov, M. Yamamoto)......Page 1392 1. Introduction......Page 1394 1.1. Standing Assumptions......Page 1396 2. Carleman inequalities for hyperbolic equation......Page 1397 3.2. Linearized inverse problem......Page 1398 3.3. Proof of Theorem 1......Page 1400 References......Page 1403 1. Introduction......Page 1406 2. New Algorithm......Page 1408 4. Numerical Experiments......Page 1410 References......Page 1414 1. Conditional stability in ill-posed problems......Page 1416 2. Initial temperature reconstruction......Page 1417 References......Page 1421 IV.6 Mathematical Biology and Medicine (R.P. Gilbert, A. Wirgin. Y. Xu)......Page