Adalbert Kerber [1] , [2] , [3] , [4] , [5] , [6]. Residues and Duality - Robin Hartshorne [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Reversible Systems - Vladimir I. Robert D. Accola [1] , [2] , [3] , [4] , [5]. Detlef Gromoll, Professor Dr. Wilhelm Klingenberg, Dr. Wolfgang Meyer Riemannsche Hilbertmannigfaltigkeiten. Flaschel, W. Klingenberg [1] , [2] , [3] , [4] , [5].

Ring Theory - Freddy M. Ring Theory - Jose Luis Bueso, Pascual Jara, Blas Torrecillas [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Ring Theory Antwerp - F. Ring Theory Waterloo Proceedings, University of Waterloo, Canada, 12—16 June, - David Handelman, John Lawrence [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17].

Rings and Semigroups - Mario Petrich [1] , [2] , [3] , [4] , [5]. Rings with Morita Duality - Weimin Xue [1] , [2] , [3] , [4] , [5] , [6] , [7]. Robust Statistical Methods - William J. Rey [1] , [2] , [3] , [4] , [5] , [6] , [7]. Romanian-Finnish Seminar on Complex Analysis - Cabiria Andreian Cazacu, Aurel Cornea, Martin Jurchescu, Ion Suciu [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Draxl, M. Calvin H. Wilcox [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Yafaev [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19]. Schottky Groups and Mumford Curves - Lothar Gerritzen, Marius van der Put [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12].

Schubert Varieties and Degeneracy Loci - William Fulton, Piotr Pragacz [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11]. Seifert Manifolds - Peter Orlik [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Selberg's Zeta-, L-, and Eisensteinseries - Ulrich Christian [1] , [2] , [3] , [4] , [5] , [6] , [7]. Selection Theorems and their Applications - T. Parthasarathy [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11].

Self-Adjoint Operators - William G. Faris [1] , [2] , [3] , [4] , [5] , [6]. Madden [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Kolokoltsov [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Semigruppi di Trasformazioni Multivoche - G. Treccani [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22].

Borel, R. Carter, C. Curtis, N. Iwahori, T. Springer, R. Steinberg [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9]. Seminar on Complex Multiplication - A. Borel, S. Chowla, C. Herz, K. Iwasawa, J-P. Serre [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9]. Stephen Jones [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14]. Yorke [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Seminar on Fiber Spaces - Emery Thomas [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9].

Targonski [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16]. Seminar on Periodic Maps - P. Conner [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19]. Eckmann [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13]. Bauer [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8]. Seminormal Operators - Kevin Clancey [1] , [2] , [3] , [4] , [5] , [6] , [7]. Sensitivity of Functionals with Applications to Engineering Sciences - Vadim Komkov [1] , [2] , [3] , [4] , [5] , [6] , [7].

Tichy [1] , [2] , [3] , [4] , [5]. Set Theory and Hierarchy Theory A Memorial Tribute to Andrzej Mostowski - Wiktor Marek, Marian Srebrny, Andrzej Zarach [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Set Theory and Hierarchy Theory V - Alistair Lachlan, Marian Srebrny, Andrzej Zarach [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Fleischman [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20]. Shadowing in Dynamical Systems - Sergei Yu. Pilyugin [1] , [2] , [3] , [4] , [5] , [6]. Shape Theory - Jerzy Dydak, Jack Segal [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13]. Bloom, Nicholas D. Kazarinoff [1] , [2] , [3] , [4] , [5] , [6] , [7]. Similarity Problems and Completely Bounded Maps - Gilles Pisier [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14].

Simple Morphisms in Algebraic Geometry - Richard Sot [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Simple Singularities and Simple Algebraic Groups - Peter Slodowy [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Singular Integrals - Umberto Neri Singular Modular Forms and Theta Relations - Eberhard Freitag [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9]. Singular Ordinary Differential Operators and Pseudodifferential Equations - Johannes Elschner [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9]. Naidu, Ayalasomayajula K. Rao [1] , [2] , [3] , [4] , [5] , [6].

Wendland, John R. Whiteman [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Bridges, Jacques E. Furter [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Singularity Theory and its Applications - Mark Roberts, Ian Stewart [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18]. Pierce [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10].

Smoothing Techniques for Curve Estimation - Th. Gasser, M. Sobolev Gradients and Differential Equations - John William Neuberger [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20]. Sobolev Spaces on Riemannian Manifolds - Emmanuel Hebey [1] , [2] , [3] , [4] , [5] , [6] , [7]. Space Curves - Franco Ghione, Christian Peskine, Edoardo Sernesi [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13].

Spaces of Analytic Functions - Otto B. Bekken, Bernt K. Spaces of Approximating Functions with Haar-like Conditions - Kazuaki Kitahara [1] , [2] , [3] , [4] , [5] , [6] , [7]. Rutter [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Marshall [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11]. Sparse Matrix Techniques - V. Barker [1] , [2] , [3] , [4] , [5] , [6]. Feinsilver [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Spectral Theory and Differential Equations - Prof. William N. Everitt [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16].

Spectral Theory of Banach Space Operators - Shmuel Kantorovitz [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17]. Spectral Theory of Ordinary Differential Operators - Joachim Weidmann [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Spline Functions - Prof. Walter Schempp [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19]. Stabile Modulformen und Eisensteinreihen - Rainer Weissauer [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16]. Stability Problems for Stochastic Models - V. Kalashnikov, V. Zolotarev [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Kalashnikov, Vladimir M. Kalashnikov, Boyan Penkov, Vladimir M. Zolatarev [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Curtain [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Stability of Unfoldings - Gordon Wassermann [1] , [2] , [3] , [4] , [5] , [6] , [7]. Kochman [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Stable Homotopy Theory - J. Frank Adams Stable homotopy - Joel M. Cohen [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8].

Statistical Learning Theory and Stochastic Optimization - Jean Picard [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Stochastic Analysis and Applications - Aubrey Truman, David Williams [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14]. Stochastic Aspects of Classical and Quantum Systems - Sergio Albeverio, Philippe Combe, Madeleine Sirugue-Collin [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15]. Taylor [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11].

Stochastic Integrals - David Williams [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Davies [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19].

Stochastic Partial Differential Equations and Applications - Giuseppe Da Prato, Luciano Tubaro [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16]. Stochastic Partial Differential Equations and Applications II - Giuseppe Da Prato, Luciano Tubaro [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Albeverio, Philippe Blanchard, Ludwig Streit [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18]. Stochastic Spatial Processes - Petre Tautu [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Stratified Mappings — Structure and Triangulability - Andrei Verona [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11]. Stratified Polyhedra - David A. Stone [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Lubinsky, Edward B.

Saff [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18]. Structural Stability, the Theory of Catastrophes, and Applications in the Sciences - Peter Hilton [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Structure and Representations of Q-groups - Dennis Kletzing [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9]. Mordeson, Bernard Vinograde [1] , [2] , [3] , [4] , [5]. Substitutions in Dynamics, Arithmetics and Combinatorics - N.

Summer School on Topological Vector Spaces - Lucien Waelbroeck [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Sums and Gaussian Vectors - Vadim Vladimirovich Yurinsky [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8]. Wahlbin [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14].

Surgery with Coefficients - Gerald A. Anderson [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8]. Symbolic Dynamics and Hyperbolic Groups - Michel Coornaert, Athanase Papadopoulos [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Symmetric Hilbert Spaces and Related Topics - Alain Guichardet [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11].

Symmetric Markov Processes - Prof. Martin L. Silverstein [1] , [2] , [3] , [4] , [5] , [6]. Symposium on Algebraic Topology - Peter J. Hilton [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14]. Symposium on Automatic Demonstration - M. Laudet, D. Lacombe, L. Nolin, M. Knops [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8]. Symposium on Optimization - A. Balakrishnan, M. Contensou, B. Lions, N. Moiseev [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Harris Jr. Symposium on Probability Methods in Analysis [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Symposium on Semantics of Algorithmic Languages - E. Engeler [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16]. Brooks [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15].

## Lectures notes

Morris [1] , [2] , [3] , [4] , [5] , [6]. Houzel [1] , [2] , [3] , [4] , [5] , [6]. Dold, B. Eckmann Duistermaat, A. Host et F. Karoubi, R. Gordon, P. Zisman [1] , [2] , [3] , [4] , [5] , [6] , [7]. Pierre Lelong [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13]. Lelong - P. Dolbeault - H. Skoda - Pierre Lelong, Pierre Dolbeault, Henri Skoda [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15].

Meyer Dellacherie, P. Meyer, M. Weil Weil [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Meyer [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Yor [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21]. Ledoux, M. Lelong — P. Dolbeault — H. Skoda - Pierre Lelong, Pierre Dolbeault, Henri Skoda [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Ferrier [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Heath, Jun-iti Nagata [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21].

Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen - Jacques Tits [1] , [2] , [3] , [4] , [5]. Sue Toledo [1] , [2] , [3] , [4] , [5] , [6]. Tame Geometry with Application in Smooth Analysis - Yosef Yomdin, Georges Comte [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Tauberian Remainder Theorems - Tord H. Ganelius [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Techniques of Admissible Recursion Theory - Chi-Tat Chong [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12]. Techniques of Multivariate Calculation - Roger H.

Farrell [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15]. Williams [1] , [2] , [3] , [4] , [5] , [6]. The Adjoint of a Semigroup of Linear Operators - Jan van Neerven [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. De, Vore [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Devlin [1] , [2] , [3] , [4] , [5] , [6] , [7]. Rackoff [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11]. Antonelli, Dan Burghelea, Peter J. Kahn [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8].

Kauffman, Thomas T. Read, Anton Zettl [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10]. Hofmann [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17]. Wirsching [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8]. May [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17].

### Pressure Sensors Mechanical Engineering Marcell Dekker By

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Embed Embed this gist in your website. Share Copy sharable link for this gist. Learn more about clone URLs. Download ZIP. Scheinkman, Nizar Touzi [1] , [2] , [3] , [4] Partial Differential Equations - Shiing-shen Chern [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] Partial Differential Equations - Fernando Cardoso, Djairo G. Goldstein [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] Periodic Solutions of Nonlinear Dynamical Systems - Eduard Reithmeier [1] , [2] , [3] , [4] , [5] Periodic Solutions of the N-Body Problem - Kenneth R.

Troelstra [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] Probabilistic Methods in Differential Equations - Dr. Pinsky [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] Probabilistic Models for Nonlinear Partial Differential Equations - Denis Talay, Luciano Tubaro [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] Probabilities on the Heisenberg Group - Daniel Neuenschwander [1] , [2] , [3] , [4] , [5] , [6] Probability Distributions in Quantum Statistical Mechanics - Mark A.

Yukich [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] Probability Theory on Vector Spaces - A. Weron [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] Probability Theory on Vector Spaces II - A. Wolfowitz [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] Probability and Information Theory II - M. Wall [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] Proceedings of a Conference on Operator Theory - P.

Fillmore [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] Proceedings of the 15th Scandinavian Congress Oslo - K. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the interaction between translations and rotations. Dislocations are associated, as usual, with branch points in a phase field, whereas disclinations arise as critical points and singularities in the phase field.

We introduce a three-dimensional model for two-dimensional smectics that clarifies the topology of disclinations and geometrically captures known results without the need to add compatibility conditions. Our work suggests natural generalizations of the two-dimensional smectic theory to higher dimensions and to crystals. Many of the features and properties of ordered media, such as crystalline solids, magnets, and liquid crystals, are controlled by their defects: those points or lines of discontinuity at which the order changes abruptly or is ill-defined.

For example, defects have played prominent roles in understanding the plastic deformation of solids 1 , the Kosterlitz—Thouless transition of the two-dimensional XY model 2 , the dislocation-unbinding melting of solids 3 , and the formation of colloidal crystals in nematic liquid crystals 4.

One of the unifying features of the defects in all of these systems is that they are topological, in the sense that many of their properties depend only on the symmetry of the ordered medium. Based on this concept a general framework has been developed for the identification and classification of the topological defects in any broken symmetry medium in terms of group elements or more generally conjugacy classes of homotopy groups of a space of degeneracy, representing the ground state manifold of the system 5 — This framework has provided an understanding of the ways in which defects can combine or split, and it has highlighted the path dependence of such processes when the fundamental group of the ground state manifold is non-Abelian It is known, however, that the standard topological theory of defects is incomplete in systems with spontaneously broken translation and rotation invariance, such as crystals 9 , 11 , More specifically, a description of such systems that treats disclinations and dislocations translational defects and their interactions in a mathematically coherent fashion is missing.

A topological classification of treating dislocations alone is available via the use of Burgers vectors and the like 14 , The standard way 16 , 17 to add disclinations in these systems is to adjoin extra components representing the orientation of the system to the order parameter. For instance, the order parameter space of the two-dimensional smectic becomes the Klein bottle. But in this formalism, the neat correspondence between the fundamental group of the order parameter space and disclination-type defects breaks down, as we will demonstrate.

This breakdown is related to the number of Goldstone modes in these systems and the restrictions they impose on paths in the ground state manifold. We will show that it is mathematically consistent to keep the order parameter as a phase field of the sort whose topological defects are well-known to generate the Burgers vector for dislocations and to look at disclinations in this picture as defects in the derivatives of the phase field.

Modeling disclinations as critical points was considered by Trebin 11 ; we complete this approach and present a method that admits dislocations as well by noting that phase fields in layered systems are not maps to but rather to periodic, possibly nonorientable, spaces. We also suggest connections to singularity theory and Morse theories that can shed further light on our formulation.

We focus on liquid crystalline systems because they have low-dimensional order parameter spaces and use as our prototypical examples the two-dimensional smectic and directed line systems. Smectic liquid crystals consist of rod-shaped molecules that spontaneously form both directional nematic order and a one-dimensional density wave, commonly described as a layered system; the spacing between the layers is approximately the rod length, a. In two dimensions, we can picture the layers as a set of nearly equally spaced curves lying in the plane. The ground states are characterized by both equal spacing between these curves and vanishing curvature.

The free energy, accordingly, has two terms; the first is a compression energy that sets the spacing, and the second is a bending energy. Drawing the layers at places of maximum density, we see that the layers are those level sets where is an integer multiple of a. The energy may be written in terms of the phase field or the displacement; hence, we will look at the geometry of the smectic in terms of these functions.

Hence, we call this the directed line system; indeed, it is topologically equivalent to wave systems [see, for instance, Nye and Berry's work on wave dislocations 19 ]. Familiar examples of true two-dimensional smectic configurations may be found on your fingerprints. Many other realizations of these systems have been studied experimentally 20 , 21 ; even electronic systems can realize quantum smectic states Additionally, we introduce a geometrically intuitive model for looking at the topology of smectic defects that allows us to demonstrate our results.

Because the phase field is the basic ingredient to study a smectic, it is natural to view the smectic as a surface placed over the plane. The singularities of this surface will then correspond to different defects. Our model motivates a highly geometric expression for the nonlinear smectic free energy functional and allows us to consider compression and curvature energies around defects.

We briefly review the standard topological theory of defects via examples, namely disclinations in nematics and dislocations in smectics. We recapitulate the limitations of the standard formalism when extended to disclinations in the directed line system and the smectic. Next, we present our resolution of these problems in terms of our construct and discuss a geometric formulation of the free energy for two-dimensional smectics and use this to find compression-free defects. Finally, we conclude with applications and a discussion of avenues for extension.

Systems with broken symmetries are described by nonvanishing order parameters. When the symmetry is continuous there is a degeneracy of ground states continuously connected by the underlying symmetry. Each of these equivalent but different ground states is represented by a different value of the order parameter. The order parameter field is a local measure of the system's order; in other words, each configuration of a medium defines a map from physical space the plane, in this article to the set of order parameters.

The goal of the topological theory of defects is to classify the defect structures in physical media by analyzing the properties of these maps that are invariant under continuous deformation, or homotopy. As an example, recall the familiar XY model of two-dimensional spins on the plane. Locally, the orientation at a point P in any texture corresponds to some point in the ground state manifold, and we can think of the ordered state as a map from points in the sample to directions given by points in S 1. A non-zero winding indicates that the loop encircles a topological defect, or vortex, whose strength is given by the winding number.

Thus, defect states of the two-dimensional XY model are characterized by a single integer. These notions are formalized in the homotopy theory of topological defects 9 — There we start with two groups, G and H , which are the symmetry groups of the system in a disordered and ordered state, respectively, so H G. In this way, the topological theory provides a general framework for the classification of defects in any broken symmetry ordered medium.

Among the key insights homotopy theory affords is that products of loops in the fundamental group yield information on how defects combine or split and, particularly, the path dependence of these processes when the fundamental group is non-Abelian If we consider homotopy classes of loops with non-zero winding only in the direction, we recover dislocations. In particular, the usual construction for the Burgers vector via counting layers is equivalent to looking at the winding number of , because we draw a new layer for every multiple of a in.

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Directed lines in the plane. The top row shows the layers derived from slicing the surface shown in the bottom row at integer values of the height, i. A Ground state. C Two dislocations. The order parameter space is no longer the two-torus but is the Klein bottle. Disclination dipoles in smectics as examples of more complicated structures. B and C The top row shows the layers derived from slicing the surface shown in the bottom row at integer values of the height, i.

B Pincement formed from a disclination dipole where the singular points lie at the same height. C Dislocation, also formed from a disclination dipole, but with the singular points at different heights. D A cutaway view of the helicoid sitting inside the dislocation. Although the algebraic structure we have presented is the natural generalization of the standard study of topological defects 16 , 25 , it fails to correctly characterize the defects of either system.

Mermin recognized this in ref. Part of the issue is that the homotopy theory predicts entire classes of defects that are not present in the physical system. More generally, the problem lies in the allowed changes of the order parameter as one moves along a path in the sample. For a system with liquid-like order, such as the nematic, any variation in the order parameter can be accommodated, i.

However, the same is not true for translationally ordered media, like the smectic. A continuous symmetry implies that two degenerate ground states can be connected via an arbitrarily low-energy Goldstone excitation. This implies that the dimensionality of the allowed excitations corresponds to the number of Goldstone modes. Although the smectic breaks both rotational and translational symmetry, it has only one Goldstone mode, the Eulerian displacement field u x , a signal of something different.

Not any path in the ground state manifold can be realized, but only those that correspond to this mode. The reduction in the number of Goldstone modes and consequent restriction of the realizable paths in systems of broken translational symmetry arises because the rotational and translational degrees of freedom are coupled into a composite object 3 , 13 , 26 in a manner akin to the Higgs mechanism for gauge fields.

Only when the number of Goldstone modes is equal to the dimensionality of the ground state manifold do we have sufficient freedom to be able to match any path in the ground state manifold to any path in the sample, as is required for the application of the standard homotopy theory. Our main result is to present a method by which the topological defects of translationally ordered media can be faithfully captured, without the need to remove entire classes of nonrealizable defects. Recall that they are both given by the order parameter x , the former as the value of the phase field at a point and the latter as the direction of its gradient.

Note how this generalizes the usual procedure. If the ground state has structure up to order n , strictly speaking, we should keep data at every point of the configuration also up to order n. When has a nontrivial winding about a point, we know it must be discontinuous there. In this case, the unit normal must be discontinuous, but this only implies that is discontinuous or zero.

It follows that at a disclination is either singular or critical. However, the possible defects that can arise should be determined from the homotopy of x and x together, and not from the structure of the ground state manifold. With this construction in hand, we move on to our surface model.

In this section we provide the first steps toward a theory where the objects being homotoped are not simply paths in the ground state manifold, but rather are smooth maps from the physical space to the space of the translational phase field. Thus, the theory of defects for these systems should be given in the language of singularity theory 27 and flavors of Morse theory 28 , which may combine those on circle-valued maps ref. Rather than attempt to lay out precise mathematical conditions and proceed into the realm of rarefied mathematics, we discuss our ideas in an intuitive way, exploiting the following fact about smectics in two dimensions.

Because the smectic is represented by level sets of a phase field on 2 , we may visualize the phase field as a graph of a two-surface. We begin with the simpler case of directed lines. As we shall see, this multiplane description is natural from the point of view of defects; they connect the different sheets together to form the required topology. Multiple dislocations are given by surfaces connected by multiple helicoids, e.

Disclinations, as we have discussed, are either zeroes or singularities of , and so in this height model, the disclinations correspond to critical points or cusps in the surface Similarly, negative charge disclinations correspond to saddles for creatures with various numbers of appendages. The notion of topological equivalence in this model is intuitive. First, note that deformations of these surfaces with the restriction that horizontally or vertically tangent surfaces are singular correspond to the class of allowed paths in the , space that we discussed in the previous section.

Similar arguments hold for two basins and even for a basin and a mountain. At these heights it is possible to cross continuously from one set of planes to the other, thereby changing the local orientation of the layers. Defects again connect the different planes together, although now this can be done in two different ways; either by connecting planes with the same orientation, as in the directed line system, or planes with the opposite orientation.

The latter leads to a global unorientability of the smectic and occurs when there are odd half-integer index disclinations. The two prototypical half-integer disclinations are depicted in Fig. Examples of two-dimensional smectic configurations. These two unoriented singularities join the two sets of planes, so that one can no longer orient the layers as in Fig.

The way in which these singularities can join depends on the geometry of the height representation. For instance, consider the dislocation and the pincement.

## Journal of the Optical Society of America A

We are used to the idea that a disclination dipole creates a dislocation Indeed in Fig. However, consider the pincement shown in Fig. In this configuration the two singular points both lie at the same height and hence they can be cancelled. The smectic dislocation cannot be cancelled because the singular layers do not line up.

Although this follows from the algebraic structure of paths on the Klein bottle, [i. We are also able to construct a dislocation directly. Because a Burgers circuit around a dislocation changes by a multiple of a , it follows that the height function must have the topology of a helicoid.

As shown in Fig. Note that the charge of a dislocation depends on both the sign of the helicoid left- or right-handed and the tilt of the layers. As long as this is smaller than the core size of the defects, it can be created without upsetting the overall topology. As the inchoate dislocation pair separates, these necks expand and, in a manner similar to continuous deformations of Riemann's minimal surface, break up into two oppositely handed helicoids 33 , as depicted in Fig.

Note that this type of process, which necessarily includes local surgery for combining defects, is implied in the theory of topological defects as well 9. In smectic systems, however, we must perform local surgery whenever we pull a defect through a layer—this follows from the above result that the defect cores must remain at a constant height. The height function approach provides a purely geometric way to formulate the nonlinear energetics of a two-dimensional smectic.

We start with the compression energy. We are free, however, to choose any rotationally invariant function with this property; theories based on different nonlinear strains will differ from each other in anharmonic powers of the strain, and only the coefficient of the harmonic term is definition-independent.

Goldstone's theorem requires that the bending energy, on the other hand, reduce to 2 to harmonic order in. We thus propose the following geometric free energy for the two-dimensional smectic:. The free energy is defined as an integral over the height surface that we have introduced and is the Willmore functional in a field The singularities associated with disclinations and dislocations specify the boundary conditions and topology for the associated variational problem. It follows that the Gauss map of such a surface sweeps out a latitude on the unit sphere, and so the Gaussian curvature K of the height surface must vanish.

Thus, we provide a direct demonstration that it is only possible to have equally spaced layers for these three defect charges. We have outlined an approach to the study of topological defects in systems with broken translational invariance; topological equivalence requires more than just homotopic paths. One should consider homotopy classes of smooth er maps. Our method focuses on singularities and critical points in a phase field viewed as a height function over 2.

Although the fundamental group of the ground state manifold can be constructed, it is known that when the loops involve both rotations and translations there are homotopy classes of loops implied that cannot be realized from complexions of the physical system. This arises from a mismatch of the dimensionality of the manifold of ground states with the number of Goldstone modes. To remedy this, we have constructed a local map from the configurations to the ground state manifold by employing Taylor series data at each point.

We have shown that the homotopy theory of defects only works for singularities in dislocations and not consistently for singularities in. In the case of directed lines, circle-valued maps of the sort we have described above are equivalent ref.