Congratulations to *Duke Mathematical Journal* editor Phillip Griffiths! He was awarded the 2014 Chern Medal for his work in complex geometry, particularly his work in Hodge theory and periods of algebraic varieties. The prize is given at the International Congress of Mathematicians, which is being held currently in Seoul, South Korea.

To read the official press release, click here.

Several contributors to *DMJ* have also been awarded the Fields Medal, including Maryam Mirzakhani, the first woman to win the medal. Other winners include Artur Avila and Manjul Bhargava. See a list of their contributions to *DMJ* below. Congratulations to all!

For more information on *Duke Mathematical Journal*, visit dukeupress.edu/dmj.

Contributions to *DMJ* from Fields Medalists include:

Maryam Mirzakhani, co-author of "Lattice point asymptotics and volume growth on Teichmüller space," here; Artur Avila, co-author of "Cohomological equations and invariant distributions for minimal circle diffeomorphisms," here and "Generic Singular Spectrum For Ergodic Schrödinger Operators," here; and Manjul Bhargava, co-author of "Error estimates for the Davenport-Heilbronn theorems," here.

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GEOMETRIC FORMULATION OF MECHANICAL SYSTEMS

SUBJECTED TO CONSTRAINTS

Author Horia Orasanu

ABSTRACT

Complex systems are large interdisciplinary research topics that have been studied by means of a mixed basic theory that mainly derives from physics and computer simulation. Such systems are made of many interacting elementary units that are called “agents”.

The way in which such a system manifests itself cannot be exclusively predicted only by the behavior of individual elements. Its manifestation is also induced by the manner in which the elements relate in order to influence global behavior.

1 INTRODUCTION

The most significant properties of complex systems are emergence, self-organization, adaptability, etc. [1–4].

Examples of complex systems can be found in human societies, brains, the Internet, ecosystems, biological evolution, stock markets, economies and many others [1, 2]. Particularly, polymers are examples of such complex systems. Their forms include a multitude of organizations starting from simple, linear chains of identical structural units and ending with very complex chains consisting of sequences of amino acids that form the building blocks of living fields. One of the most intriguing polymers in nature is DNA, which creates cells by means of a simple, but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems, such as organs, which, in turn, form even more complex systems, such as organisms. The study of complex systems can offer a glimpse into the realistic dynamics of polymers and solve certain difficult problems (protein folding) [1–4].

2 FORMULATION

Correspondingly, theoretical models that describe the dynamics of complex systems are sophisticated [1–4]. However, the situation can be standardized taking into account that the complexity of interaction processes imposes various temporal resolution scales, while pattern evolution implies different freedom degrees [5].

In order to develop new theoretical models, we must admit that complex systems displaying chaotic behavior acquire self-similarity (space-time structures seem to appear) in association with strong fluctuations at all possible space-time scales [1–4]. Then, in the case of temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a collection of potential trajectories, while the concept of definite positions by that of probability density. One of the most interesting examples is the collision process in complex systems, a case in which the dynamics of the particles can be described by non-differentiable curves.

Since non-differentiability appears as the universal property of complex systems, it is necessary to construct a non-differentiable physics. Thus, the complexity of the interaction processes is replaced by non-differentiability; accordingly, it is no longer necessary to use the whole classical “arsenal” of quantities from standard physics (differentiable physics).

This topic was developed within scale relativity theory (SRT) [6,7] and non-standard scale relativity theory (NSSRT) [8–22]. In this case, we assume that the movements of complex system entities take place on continuous, but non-differentiable, curves (fractal curves), so that all physical phenomena involved in the dynamics depend not only on space-time coordinates, but also on space-time scale resolution. From such a perspective, physical quantities describing the dynamics of complex systems may be considered fractal functions [6,7]. Moreover, the entities of the complex system may be reduced to and identified with their own trajectories, so that the complex system will behave as a special fluid lacking interaction (via their geodesics in a non-differentiable (fractal) space). We have called such fluid a “fractal fluid” [8–22].

References

1. Bar-Yam, Y. Dynamics of Complex Systems; Addison-Wesley Publishing Company: Reading, MA, USA, 1997. [Google Scholar]

2. Mitchell, M. Complexity: A Guided Tour; Oxford University Press: Oxford, UK, 2009. [Google Scholar]

3. Bennett, C.H. How to define complexity in physics, and why. Complex. Entropy Phys. Inf. 1990, 8, 137–148. [Google Scholar]

4. Winfree, A.T. The Geometry of Biological Time, 2nd ed.; Interdisciplinary Applied Mathematics (Book 12); Springer: New York, NY, USA, 2000. [Google Scholar]

5. Badii, R.; Politi, A. Complexity: Hierarchical Structure and Scaling in Physics; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]

6. Nottale, L. Fractal Space-Time and Microphysics: Towards A Theory of Scale Relativity; World Scientific: Singapore, Singapore, 1993. [Google Scholar]

7. Nottale, L. Scale Relativity and Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics; Imperial College Press: London, UK, 2011. [Google Scholar]

8. Agop, M.; Forna, N.; Casian-Botez, I.; Bejinariu, C. New theoretical approach of the physical processes in nanostructures. J. Comput. Theor. Nanosci. 2008, 5, 483–489. [Google Scholar]

9. Agop, M.; Murguleţ, C. El Naschie’s epsilon (infinity) space-time and scale-relativity theory in the topological dimention D = 4. Chaos Solitons Fractals 2008, 32, 1231–1240. [Google Scholar]

10. Agop, M.; Nica, X.; Gîrţu, M. On the vacuum status in Weyl-Dirac theory. Gen. Relativ. Gravit. 2008, 40, 35–55. [Google Scholar]

11. Agop, M.; Nica, P.; Niculescu, O.; Dumitru, D.G. Experimental and theoretical investigations of the negative differential resistance in a discharge plasma. J. Phys. Soc. Jpn. 2012, 81. [Google Scholar] [CrossRef]

12. Agop, M.; Păun, V.; Harabagiu, A. El Naschie’s epsilon (infinity) theory and effects of nanoparticle clustering on the heat transport in nanofluids. Chaos Solitons Fractals 2008, 37, 1269–1278. [Google Scholar]

13. Casian-Botex, I.; Agop, M.; Nica, P.; Păun, V.; Munceleanu, G.V. Conductive and convective types behaviors at nano-time scales. J. Comput. Theor. Nanosci. 2010, 7, 2271–2280. [Google Scholar]

14. Ciubotariu, C.; Agop, M. Absence of a gravitational analog to the Meissner effect. Gen. Relativ. Gravit. 1996, 28, 405–412. [Google Scholar]

15. Colotin, M.; Pompilian, G.O.; Nica, P.; Gurlui, S.; Păun, V.; Agop, M. Fractal transport phenomena through the scale relativity model. Acta Phys. Pol. A 2009, 116, 157–164. [Google Scholar]

16. Gottlieb, I.; Agop, M.; Jarcău, M. El Naschie’s Cantorian space-time and general relativity by means of Barbilian’s group. A Cantorian fractal axiomatic model of space-time. Chaos Solitons Fractals 2004, 19, 705–730. [Google Scholar]

17. Gurlui, S.; Agop, M.; Nica, P.; Ziskind, M.; Focşa, C. Experimental and theoretical investigations of transitory phenomena in high-fluence laser ablation plasma. Phys. Rev. E 2008, 78, 026405. [Google Scholar]

18. Gurlui, S.; Agop, M.; Strat, M.; Băcăiţă, S. Some experimental and theoretical results on the anodic patterns in plasma discharge. Phys. Plasmas 2006, 13. [Google Scholar] [CrossRef]

19. Nedeff, V.; Bejenariu, C.; Lazăr, G.; Agop, M. Generalized lift force for complex fluid. Powder Technol. 2013, 235, 685–695. [Google Scholar]

20. Nedeff, V.; Moşneguţu, E.; Panainte, M.; Ristea, M.; Lazăr, G.; Scurtu, D.; Ciobanu, B.; Timofte, A.; Toma, S.; Agop, M. Dynamics in the boundary layer of a flat particle. Powder Technol. 2012, 221, 312–317. [Google Scholar]

21. Nica, P.; Agop, M.; Gurlui, S.; Bejinariu, C.; Focşa, C. Characterization of aluminum laser produced plasma by target current measurements. Jpn. J. Appl. Phys. 2012, 51. [Google Scholar] [CrossRef]

22. Nica, P.; Vizureanu, P.; Agop, M.; Gurlui, S.; Focşa, C.; Forna, N.; Ioannou, P.D.; Borsos, Z. Experimental and theoretical aspects of aluminum expanding laser plasma. Jpn. J. Appl. Phys. 2009, 48. [Google Scholar] [CrossRef]

23. Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman and Company: New York, NY, USA, 1983. [Google Scholar]

24. Landau, L.; Lifsitz, E.M. Fluid Mechanics, 2nd ed; Butterworth-Heinemann: Oxford, UK, 1987. [Google Scholar]

25. Wilhelm, H.E. Hydrodynamic Model of Quantum Mechanics. Phys. Rev. D 1970, 1. [Google Scholar] [CrossRef]

26. Shannon, C.E. Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar]

27. Flores-Gallegos, N. Shannon informational entropies and chemical reactivity. In Advances in Quantum Mechanics; Bracken, P., Ed.; Intech: Rijcka, Croatia, 2013; pp. 683–706. [Google Scholar]

28. Jaeger, G. Fractal states in quantum information processing. Phys. Lett. A 2006, 358, 373–376. [Google Scholar]

29. Agop, M.; Buzea, C.; Buzea, C.G.; Chirilă, L.; Oancea, S. On the information and uncertainty relation of canonical quantum systems with SL(2R) invariance. Chaos Solitons Fractals 1996, 7, 659–668. [Google Scholar]

30. Agop, M.; Griga, V.; Ciobanu, B.; Buzea, C.; Stan, C.; Tatomir, D. The uncertainty relation for an assembly of Planck-type oscillators. A possible GR-quantum mechanics connection. Chaos Solitons Fractals 1997, 8, 809–821. [Google Scholar]

31. Agop, M.; Melnig, V. L’énergie informationelle et les relations d’incertitude pour les systèmes canoniques SL(2R) invariants. Entropie 1995, 31, 119–123. [Google Scholar]

32. Onicescu, O. Energie informationnelle. Comptes Rendus Hebdomadaires des Seances de l Academie des Sciences Serie A 1966, 263, 841–842. [Google Scholar]

33. Alipour, M.; Mohajeri, A. Onicescu information energy in terms of Shannon entropy and Fisher information densities. Mol. Phys. 2012, 110, 403–405. [Google Scholar]

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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LAGRANGIAN WITH EXACT DERIVATIVES AND GEOMETRIC INVOLUTION

Author Horia Orasanu

ABSTRACT

Optimization problems arise naturally in statistical machine learning and other fields concerned with data analysis. The rapid growth in the scale and complexity of modern datasets has led to a focus on gradient-based methods and also on the class of accelerated methods, first proposed by Nesterov in 1983. Accelerated methods achieve faster convergence rates than gradient methods and indeed, under certain conditions, they achieve optimal rates. However, accelerated methods are not descent methods and remain a conceptual mystery. We propose a variational, continuous-time framework for understanding accelerated methods. We provide a systematic methodology for converting accelerated higher-order methods from continuous time to discrete time. Our work illuminates a class of dynamics that may be useful for designing better algorithms for optimization.

1 INTRODUCTION

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. Although many generalizations and extensions of Nesterov’s original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the Bregman Lagrangian, which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods corresponds to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov’s technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.

In the body of theory and practice built up to answer such questions, the phenomenon of acceleration plays a key role. In 1983, Nesterov introduced acceleration in the context of gradient descent for convex functions (1), showing that it achieves an improved convergence rate with respect to gradient descent and moreover that it achieves an optimal convergence rate under an oracle model of optimization complexity (2). The acceleration idea has since been extended to a wide range of other settings, including composite optimization (3⇓–5), stochastic optimization (6, 7), nonconvex optimization (8, 9), and conic programming (10). There have been generalizations to non-Euclidean optimization (11, 12) and higher-order algorithms (13, 14), and there have been numerous applications that further extend the reach of the idea (15⇓⇓–18).

Despite this compelling evidence o

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LAGRANGIAN WITH PARTIAL DERIVATIVES USED IN NONHOLONOMIC OPTIMIZATION

Author Horia Orasanu

ABSTRACT

There are many instances in which the basic physics is known (or postulated), and the behavior of a complex system is to be determined. A typical example is that in which there are too many particles for the problem to be tractable in terms of single-particle equations, and too few for a statistical analysis to apply. In such situations, use of a computer may furnish information on enough specific cases for the general behavior of the system to be discernable. If the basis physics is postulated but not known, a computer simulation can relate the theory to observations on complex systems and thus test the theory.

1 INTRODUCTION

While most applications of such simulations yield expected results, surprises do occur. This is analogous to an unexpected result from an experiment. Either the simulation/experiment went wrong (usual) or a new aspect of nature has been uncovered (rare). Examples of the latter are the identification of constants of motion in chaotic systems and the discovery of runaway motion in the drift and diffusion of ions in gas. Such discoveries are followed by “proper” theories and “proper” experiments, but the computer plays a vital role in the research.

The research of Professor Gatland involves data analysis and the mathematical modeling and simulation of microscopic physical processes. These activities encompass both research and instruction.According to the illustration of the snake robot in Figure Figure1,1, we choose the vector of the generalized coordinates of the N-link snake robot as x = [q1,q2,…,qN−1,θN,px,py]T ∈ ℝN+2, where qi with i∈{1,…,N−1} denotes the ith joint angle, θN denotes the head angle, and the pair (px,py) describes the position of the CM of the robot w.r.t. the global x−y axes. Since the robot is not subject to nonholonomic velocity constraints, the vector of the generalized velocities is defined as x˙=[q˙1,q˙2,…,q˙N−1,θ˙N,p˙x,p˙y]T∈RN+2. Using these coordinates, it is possible to specify the kinematic map of the robot. In this paper, we denote the first N elements of the vector x, i.e. (q1,…,qN−1,θN), as the angular coordinates, and the corresponding dynamics as the angular dynamics of the system.

References

1. Bar-Yam, Y. Dynamics of Complex Systems; Addison-Wesley Publishing Company: Reading, MA, USA, 1997. [Google Scholar]

2. Mitchell, M. Complexity: A Guided Tour; Oxford University Press: Oxford, UK, 2009. [Google Scholar]

3. Bennett, C.H. How to define complexity in physics, and why. Complex. Entropy Phys. Inf. 1990, 8, 137–148. [Google Scholar]

4. Winfree, A.T. The Geometry of Biological Time, 2nd ed.; Interdisciplinary Applied Mathematics (Book 12); Springer: New York, NY, USA, 2000. [Google Scholar]

5. Badii, R.; Politi, A. Complexity: Hierarchical Structure and Scaling in Physics; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]

6. Nottale, L. Fractal Space-Time and Microphysics: Towards A Theory of Scale Relativity; World Scientific: Singapore, Singapore, 1993. [Google Scholar]

7. Nottale, L. Scale Relativity and Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics; Imperial College Press: London, UK, 2011. [Google Scholar]

8. Agop, M.; Forna, N.; Casian-Botez, I.; Bejinariu, C. New theoretical approach of the physical processes in nanostructures. J. Comput. Theor. Nanosci. 2008, 5, 483–489. [Google Scholar]

9. Agop, M.; Murguleţ, C. El Naschie’s epsilon (infinity) space-time and scale-relativity theory in the topological dimention D = 4. Chaos Solitons Fractals 2008, 32, 1231–1240. [Google Scholar]

10. Agop, M.; Nica, X.; Gîrţu, M. On the vacuum status in Weyl-Dirac theory. Gen. Relativ. Gravit. 2008, 40, 35–55. [Google Scholar]

11. Agop, M.; Nica, P.; Niculescu, O.; Dumitru, D.G. Experimental and theoretical investigations of the negative differential resistance in a discharge plasma. J. Phys. Soc. Jpn. 2012, 81. [Google Scholar] [CrossRef]

12. Agop, M.; Păun, V.; Harabagiu, A. El Naschie’s epsilon (infinity) theory and effects of nanoparticle clustering on the heat transport in nanofluids. Chaos Solitons Fractals 2008, 37, 1269–1278. [Google Scholar]

13. Casian-Botex, I.; Agop, M.; Nica, P.; Păun, V.; Munceleanu, G.V. Conductive and convective types behaviors at nano-time scales. J. Comput. Theor. Nanosci. 2010, 7, 2271–2280. [Google Scholar]

14. Ciubotariu, C.; Agop, M. Absence of a gravitational analog to the Meissner effect. Gen. Relativ. Gravit. 1996, 28, 405–412. [Google Scholar]

15. Colotin, M.; Pompilian, G.O.; Nica, P.; Gurlui, S.; Păun, V.; Agop, M. Fractal transport phenomena through the scale relativity model. Acta Phys. Pol. A 2009, 116, 157–164. [Google Scholar]

16. Gottlieb, I.; Agop, M.; Jarcău, M. El Naschie’s Cantorian space-time and general relativity by means of Barbilian’s group. A Cantorian fractal axiomatic model of space-time. Chaos Solitons Fractals 2004, 19, 705–730. [Google Scholar]

17. Gurlui, S.; Agop, M.; Nica, P.; Ziskind, M.; Focşa, C. Experimental and theoretical investigations of transitory phenomena in high-fluence laser ablation plasma. Phys. Rev. E 2008, 78, 026405. [Google Scholar]

18. Gurlui, S.; Agop, M.; Strat, M.; Băcăiţă, S. Some experimental and theoretical results on the anodic patterns in plasma discharge. Phys. Plasmas 2006, 13. [Google Scholar] [CrossRef]

19. Nedeff, V.; Bejenariu, C.; Lazăr, G.; Agop, M. Generalized lift force for complex fluid. Powder Technol. 2013, 235, 685–695. [Google Scholar]

20. Nedeff, V.; Moşneguţu, E.; Panainte, M.; Ristea, M.; Lazăr, G.; Scurtu, D.; Ciobanu, B.; Timofte, A.; Toma, S.; Agop, M. Dynamics in the boundary layer of a flat particle. Powder Technol. 2012, 221, 312–317. [Google Scholar]

21. Nica, P.; Agop, M.; Gurlui, S.; Bejinariu, C.; Focşa, C. Characterization of aluminum laser produced plasma by target current measurements. Jpn. J. Appl. Phys. 2012, 51. [Google Scholar] [CrossRef]

22. Nica, P.; Vizureanu, P.; Agop, M.; Gurlui, S.; Focşa, C.; Forna, N.; Ioannou, P.D.; Borsos, Z. Experimental and theoretical aspects of aluminum expanding laser plasma. Jpn. J. Appl. Phys. 2009, 48. [Google Scholar] [CrossRef]

23. Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman and Company: New York, NY, USA, 1983. [Google Scholar]

24. Landau, L.; Lifsitz, E.M. Fluid Mechanics, 2nd ed; Butterworth-Heinemann: Oxford, UK, 1987. [Google Scholar]

25. Wilhelm, H.E. Hydrodynamic Model of Quantum Mechanics. Phys. Rev. D 1970, 1. [Google Scholar] [CrossRef]

26. Shannon, C.E. Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar]

27. Flores-Gallegos, N. Shannon informational entropies and chemical reactivity. In Advances in Quantum Mechanics; Bracken, P., Ed.; Intech: Rijcka, Croatia, 2013; pp. 683–706. [Google Scholar]

28. Jaeger, G. Fractal states in quantum information processing. Phys. Lett. A 2006, 358, 373–376. [Google Scholar]

29. Agop, M.; Buzea, C.; Buzea, C.G.; Chirilă, L.; Oancea, S. On the information and uncertainty relation of canonical quantum systems with SL(2R) invariance. Chaos Solitons Fractals 1996, 7, 659–668. [Google Scholar]

30. Agop, M.; Griga, V.; Ciobanu, B.; Buzea, C.; Stan, C.; Tatomir, D. The uncertainty relation for an assembly of Planck-type oscillators. A possible GR-quantum mechanics connection. Chaos Solitons Fractals 1997, 8, 809–821. [Google Scholar]

31. Agop, M.; Melnig, V. L’énergie informationelle et les relations d’incertitude pour les systèmes canoniques SL(2R) invariants. Entropie 1995, 31, 119–123. [Google Scholar]

32. Onicescu, O. Energie informationnelle. Comptes Rendus Hebdomadaires des Seances de l Academie des Sciences Serie A 1966, 263, 841–842. [Google Scholar]

33. Alipour, M.; Mohajeri, A. Onicescu information energy in terms of Shannon entropy and Fisher information densities. Mol. Phys. 2012, 110, 403–405. [Google Scholar]

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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