1. Topology, Signaling, and Computing
Guennadi Kouzaev
Plenary Lecture
Proc. 4th Int. NAUN Conf. Circuits, Systems, Control, Signals, Valencia, Spain, Aug. 6-8, 2013

2. I. Introduction
Topology is a branch of mathematics that studies the properties of a space that are preserved under continuous deformations
Grew up from geometry but does not study the metric properties of spatial figures
Interested in the ways how the studied objects are composed
Called the qualitative or fuzzy geometry
Originated from the works of Euler, Listing, Cantor, and Poincaré

3. Topologically Equivalent Figures
Cup and torus are topologically equivalent figures
Torus is simpler geometrically
Simpler shapes are described by a simpler theory

4. Topologically Non-Equivalent Figures
Sphere and torus are the topologically different figures
They are not transformed to each other by bi-continuous transformations

5. Most-Known Results and Applications of Topology in Mathematics
Qualitative theory of ordinary differential equations and autonomous systems, bifurcation theory, and catastrophe theory and its applications (H. Poincare´, A.A. Andronov, V. Arnold, R. Toma, et al.)
G. Perelman and his proof of the Poincare’s conjecture on 3-D manifold in 4-D space which allows to distinguish it among the other 3-D manifolds (2002)

6. Topology in Physics-Examples
Topological defects or topologically distinct solutions in condensed matter which are stable regarding small perturbations and not decay with the time: ferromagnetic domains, vortices in super-liquids and in cold matter, etc.
Topologically distinct solutions of equations in abstract theory of fields
Trajectories of particles of multi-domain Hamiltonians

7. EM Field Theory on Differential Forms
EM quantities are represented with differential forms- generalizations of vectors-, which are not depended on the global coordinate system, and they are considered as the topological quantities
Values corresponding to the field intensities E, H are associated with the general surfaces
Values corresponding to the field inductions D, B are associated with the cylindrical surfaces
Theory shows how to calculate the coupling of these two topologically different shapes
One of the first papers were published by Deshamps (1981)

8. Topology and Logics
Topological properties of figures and the logical units were firstly bound up by J.C.C. McKinsey and A. Tarski in 1944
Logical units can correspond to topological shapes and the topology transformations can be described by means of logical theory
No physical topological carriers and circuitry had been proposed before 1991 for this idea
In 1991, the topology of EM field as the carrier of logical information and the first microwave circuits were proposed by us based on the topological theory of boundary value problems

9. II. Topological Electromagnetics
Topological electromagnetics is a theory allowing to describe and to solve the EM problems using the topological, geometrical, and algebraic methods
Our research started in 1987/1988
Fresh results and review of overall ones are published in: G.A. Kouzaev, Applications of Advanced Electromagnetics, Springer, 2013.

10. EM Boundary Problems of Electromagnetism
Solution of the boundary problem is the vector E(x,y,z) or/and H(x,y,z)

11. Field Formulas for the Cavity Excitation Problem

12. Field Geometry Field is visualized by its field-force line picture
Field-force line is derived from an autonomous dynamical system (harmonic or static fields)
r is the radius-vector of a point on a field-force line
s - parameter

13. Some History
Field-force lines have been known since the Faraday’s time
Are computed numerically or analytically
Qualitative theory of autonomous dynamical systems can be applied
A.A. Andronov
M. Faraday
H. Poincaré

14. Electric Field-Force Line Picture in a Cavity Symmetrically Excited by Two Slots- Example
Separatrix
Saddle equilibrium point where E=0
E1=E2=const
Separatrix (in blue) divides the picture into 2 cells with monotonic behavior of the field-force lines

15. Electric-Field Boundary Graph Γe and Topological Chart Te
Fields pictures have the main elements composing their skeletons- boundary graphs and topological charts
Topological chart Te is an arranged set of the separatrices and E- field equilibrium points inside the 2-D cavity
Electric boundary graph Γe is the normalized to the wave number k0 boundary draft with the defined exciting field vectors and the directions of the electric field-force lines to the boundary

16. Example for Γe and Te

17. Commentaries
Topological chart consists of the cells with monotonic behavior of the field-force lines, separatrices separating these cells, and the field equilibrium points or lines in the case of 3-D fields
Topological charts are very often the structural stable objects
They need only rough, simplified calculations

18. Topological Chart and its Algebra
Field is formed, mostly, by the modes with increased magnitudes
Topological charts can be highly stable for a certain range of frequency and cavity geometry
Number NT of modes forming a topological chart is not high (NT <10)
These modes with the defined operation of their summation belong to the algebra of the topological chart UT
NT = dim(UT)

19. Structural Stability or the Coarsness of Topological Charts
Stable Topological Chart
Unstable Topological Chart

20. Structural Stability of Topological Charts
Structural stability of topological charts allows reducing further the resources needed for the field topological analysis
Worst case scenario for the field-topology calculations needs only NT<10
Detailed field simulations and geometry calculations need N> spatial Fourier harmonics

21. Equilibrium Points

22. Characteristic Equation and the Equilibrium Type
λ is the characteristic number of an equilibrium point
Equilibrium type is defined according λ
Four types of simple equilibrium points in the 2-D phase space
Eighteen types of equilibrium points in the 3-D phase space

23. Separatrices
Separatrix is a manifold that separates the phase space into two distinct areas
Starts from the equilibrium points or/and boundaries
Their directions close to the equilibrium points can be found analytically
Able to be bifurcated

24. Bifurcation - Field Topology Transformation
Local bifurcation is the transformation of a complex equilibrium point into different ones
Non-local bifurcation is the splitting of separatrices
Bifurcation changes the global field topology i.e. its topological chart T
Bifurcation may relate to a serious change of physical features of the excited field
Bifurcation caused by variation of frequency, geometry, and cavity filling parameters

25. Non-Local Bifurcation - Splitting of Separatrices - Example

26. Topological Charts of 3-D Electric and Magnetic Fields- Example

27. Field-Force Line Equation for Non-stationary Fields
Non-autonomous dynamical system
Can be transformed to a 4-D autonomous system

28. 4-D Autonomous Dynamical System for Description of Nonstationary Field
Similarly to 3-D fields, the topological charts and the initial-boundary graphs must be defined in the 4-D space

29. Qualitative Analysis of the Field
Preliminary analytical or numerical calculations of the fields
Calculations of the equilibriums points of the fields and their types
Computations of separatrices of field-force lines portraits
Composing topological charts (skeletons) of fields
Study of local and spatial bifurcations of topological charts relative to the variation of geometry, frequency and exciting fields
Composing the relations of topological charts and parameters of waveguides and components

30. Initial Publications on Topological Analysis
A.A. Andronov et al., Qualitative theory of dynamical system on a plane, 1966 (Fundamentals of Topological Analysis).
V.I.Gvozdev and G.A. Kouzaev, Proc. Conf. Microwave 3-D ICs. Tbilisy, USSR, p. 67,1988 (2-D and 3-D EM Field)
J. Helman and L. Hesselink, IEEE Comp., 22, p. 27, 1989 (Flow Analysis).
T. Nishida et al.,Proc. AAAI-91,Anaheim, pp. 811, 1991 (Flow Analysis).

31. Our Initial Results
Topological approach to the qualitative analysis of the fields and its applications for microwaves
Local and global Maxwell topological relations as Te=F(Th)
Analytical field-force line equations for a field represented by a Fourier series
Found reduced size of the topological chart algebra
Use of structural stability to reduce the needed simulation time
Several invented transmission lines

32. Pro and Contra of the Topological Field Analysis
Can be completely programmed
Can be described by a symbolic language and the means of artificial intelligence
Able to be a fast and promising analysis tool for full-wave EM software packages
Needs preliminary solutions of EM boundary problems, i.e. plays only a complimentary role
Needs to be transformed into a qualitative method for solution of boundary problems, i.e to the Topological Electromagnetics

33. What Do We Expect from Topological Electromagnetics?
Theory allowing to compose the field topological charts according to boundary graphs without time-consuming detailed numerical computations
Better understanding of complicated EM problems
Algorithms and codes for accurate calculation of parameters of microwave components ,
Essential reduction of simulation time of 2-D and 3-D components

34. Topological Formulation of the EM Boundary Problem
Composition of field topological charts Te,h according to boundary graphs Γe by qualitative means

35. Resonances and Topological Solution of a Boundary Problem

36. Mechanism of the Topology Formation
Topological chart is formed by the modes which are close to the frequency, spatial and structural resonances
All of them are calculated according the geometro-topological criteria
Bifurcations are caused by the frequency and boundary condition variations and following variations of the relationships of modal amplitudes or complete change of the modal set

37. Algorithm. Step 1
Calculates the contributions of each resonances expressed in the geometro-topological terms
Compares the modal amplitudes
Reduces the number of modes according to certain criteria
Computes the topological field chart and control its convergence up to its stabilization
Provides info on the UT and Te,h including the field topology visualization
Predicts bifurcations of the field topology with variation of frequency and boundary condition

38. Algorithm. Step 2
Accurate calculations of component parameters (propagation constants, resonanrt frequencies, S-matrices) using the stationary functionals or regularized integral equations
Improved calculations of EM fields inside and/or outside of the components

39. Main Results and Unsolved Problems
Topological or qualitative principles of the boundary problem solutions for cavities, waveguides, and transmission lines
Topological algorithm and its software realization
Time-depending EM fields
Commercial soft for 3-D and 3-D time-depending EM fields

40. Topological Study of the Excited Cavity Fields (1990-1992)
Bifurcation diagram

41. Fast and Accurate Semi-Analytical Model of a Fin-Line (1992/2013)
Model based on 2-modes approximation
Modes found from the topological analysis
Model error <5%

42. Some Studied and Modeled Waveguides (1990-1992, 2013)

43. Qualitatively Modeled Coupled Corner Line with Experimental Validation (1992)

44. Fast and Accurate Model of a Rectangular Patch Antennas and Via-Holes (IEEE MTT, 2006)
Resonant frequency calculation error <2%

45. Results and Conclusions
Developed basics of topological theory of EM boundary problems
Developed algorithm for computerized qualitative field analysis
Developed effective analytical and semi-analytical models of several transmission lines and cavities
Several components were proposed and designed
Topological approach is an effective tool for qualitative and approximate modeling of microwave components

46. III. Topology: Signals, Logic, and Computing
Topology is changed only discretely
Logical variables can be assigned to topologically different figures (Tarsky, 1946)
Topologically modulated signals (TMS) are the field impulses which spatial charts are modulated according to a control digital signal (Kouzaev, 1991)
These signals can be represented by modal impulses in transmission lines

47. Binary TMS as Series of TEM Modes
Each field impulse carryies digital information by its field topology and magnitude
Signal represents a predicate expression (a1,2, T1,0)
Needs to use the bi-polar logic

48. Equation for General EM Time-Depending Spatial Signal
General EM signals are the time-depending impulses which field geometry is described by the field-force line equations in the 4-D phase space
Their equation is a 4-D autonomous dynamical system for the field-force lines in the 4-D phase space

49. Topologically Modulated Signal Definition
General TMS are the series of field impulses which 4-D topology is changed according to a digital modulated signal
Topological chart of such a field impulse is composed of the vector manifolds – separatrices, including the 3-D ones

50. Logical Processing of TMS and the Topology Main Goal
Digital processing is the comparison of topologically different 4-D field signals and generating of a certain response on the results of this logical operation
Close to the main task of topology and the Poincaré conjecture (2005)
Logical gates for 4-D TMS have been started to be designed in the hardware and in the theory since 1991

51. Microwave Logic Introduced in 1991 (G. Kouzaev and V. Gvozdev)
Is dealing with the microwave topologically modulated field impulses in coupled microstrip lines
Passive gates NOT, AND/OR, Flip-Flop, and Spatial Switches are designed
AND/OR gate demonstrates the reconfigurable logical response towards variations of magnitudes of the compared topologically modulated signals

52. Spatial Switch of TMS (1993-1996)
Truth-Table

53. Microwave AND/OR Gate (1991, 2008)
Input and output signals are the microwave impulses of the even or odd modes of coupled strip lines
Digital information is carried by the field topology
Field amplitude carries the control signal
Depending on the modal magnitudes the gate demonstrates AND or OR response
Simulated at 10 GHz
Consists of spatial switches, baluns and a directional coupler

54. Results and Conclusions for Microwave Logic
Passive linear circuits demonstrated interesting logical responses towards microwave TMS
Spatial switching effect is shown
Boolean and reconfigurable logic circuits have been designed
EM theory of passive logical operations was developed
Developed gates can be used in smart antennas and reconfigurable microwave circuits

55. Initial Results for Digital Logic
TMS are represented by video impulses propagating along coupled strip or microstrip lines
Passive and hybrid logical gates were designed, simulated, and experimentally tested for bi-polar (two-end) signals
Passive micrometric gates showed sub-picosecond delays in spatial switching of signals
Combined passive/semiconductor gates were designed for predicate logic

56. Predicate Logic using TMS
Topologically modulated signals are predicate sentences S=(a,T)
Predicate logic is a logic of intelligence and mathematics
TMS can model the intelligence in a more effective way in comparison to the Boolean logic hardware
Perspective for Artificial Intelligence and General/Application-specific Mathematics Processors

57. Transformation of Bipolar Signals to Uni-polar or Single-end Ones

58. Predicate Logic Components Using Uni-polar (Single-end) Logic
Bi-polar (Two-end) logic was typical for the first topological signals
Predicate gates, proposed earlier, have been re-mapped to the uni-polar and simple transistor logic handling the spatial topological signals
The were simulated and tested by Altera’s soft and modeling hardware (2006)

59. Predicate Logic Processor - PLP (2007-2008)
Control Unit
Control Logic and State Register IR PC
Program Memory
Datapath
ACCA
PLU
Data
Memory
Predicate Logic Processor was designed, simulated and tested by up-loading it into an Altera’s FPGA board
Based on the topologically modulated signals representing the predicate expressions

60. PLP Parameters
8-bit processor with the SignalTap II Embedded Logic Analyzer and RAM module consists of 5868 logic elements, 3482 combinational functions, 4628 registers and memory bits
Maximum clock frequency is MHz

61. Design and Test of the PLP by ALTERA’s Soft and FPGA Board
G. Kouzaev and A. Kostadinov, int. J. Circuits, Systems and Computers, 2010

62. Further Designs of Predicate Logic Processors
Predicate/Boolean Logic processors- WSEAS Conf., Houston, USA, Designed and Tested
Variable Predicate Logic Processors- UK Patent Application, Designed and Tested

63. Results and Conclusions-1
Topological theory of the EM field started to be developed in 1988
Have been matured from a qualitative analysis mean for pre-computed fields to a solving tool for EM boundary value problems
Shows its effectiveness to exploring the EM fields and components and for creation of fast analytical or semi-analytical models
Plays a creative role in the invention of several components and their models

64. Results and Conclusions-2
Topologically modulated digital signals and the first gates were proposed in 1991 on the base of performed research on the topological theory of the EM field
Has been matured from the gate level to the designed and experimentally tested predicate logic procesors – PLP
Topological ideas have been found very effective for computation and computing which was confirmed by applications and concrete tested designs and measurements

65. IV. Topology in Physics, Electronics, Signalling, and Computing
Avalanche-like growing publication number on the use of topology in science:
Discovering of topologically-stable objects in condensed matter
Development of new memory devices using topologically-stable objects
Noise-immune and multichannel communications
Quantum topological computing
Applications of topology-inspired analytical and numerical methods in simulations and visualization

66. 1. Topologically-stable Objects in Condensed Matter
Very often are called as topological defects
Found in superconducting fluids, Bose-Einstein condensates, ferrites, non-linear medias- electron and mixed plasmas (solitons, vortices, etc)
Caused by the nonlinearity of interactions
Spatially-stable towards perturbations and might be considered as quasi-particles

67. Air Vortex Example (WikiPedia)

68. Fermionic BEC Vortices
MIT, Ketterle Group

69. Magnetic Field Vortex of a Nanometric Size

70. Topological Integer and Fractional Hall-effect Insulators

71. 2. Memory on 2-D and 3-D Magnetic Vortices
Magnetic vortices may be of nano-metric size
Have 2-D or 3-D design
Stable towards the electric and magnetic field perturbations
Stable with temperature variation and promise to be workable at room condition
Allow to increase the memory density in orders
A step towards life-long memory

72. 3. Increased Noise-immune Signaling using TMS
Is with the orthogonality of some TMS signals (G. Kouzaev, 1993)
Error probability Perr is zero if the signals are orthogonal to each other in the space-time domain (V,T)

73. Multi-modal Signaling along Cables and in Open Space
Now is used for multichannel signalling along the multi-wire electric traces and optical cables to increase the amount of sent information
In open space, a particular case of topologically modulated signals are used called the beams of modulated angular-momentum distribution (???2004)
These signals have discretely or topologically modulated phase maps carrying logical units
First experiments are performed for open space radio and optical communications allowing increasing the number of available channels and multigiga- and terabitbit signal transfer can be reached (???2012)

74. 4. Topological Quantum Computing
Introduced by A. Kitaev in 2003 (compare our research on topological computing dated on 1991/1992)
Quantum topological computing is with increased noise-immunity of topological shapes
In these computations, the effective quantum particles move on topologically different sets of trajectories which are associated with computations
It preserves the degradations of computations due to the noise and increasing decoherence
Confirmed by experiments

75. 5. Topology for Field and Flow Visualization
Topological representation of visual information allows to reduce computation time while keeping the most important information in images
Desirable for complex fields in flows to ignore less important details
Our first work on topological representation of EM fields is dated on 1988
Helman&Hesselink’s paper in IEEE Comp. in 1989
Other works from USA ( Lee, et al. 1993) Japan (Nishida, et al., 1997), Germany (Hege, et al., 2004), etc.

76. Conclusions
Applications of topology for computations, computing, and signaling have been reviewed
Topology is now a power tool for many areas in physics and engineering, especially in EM field applications
Large number of papers have been appeared since our first works published in the 80s and 90s on signaling, computing, and field computations
Bright future is expected further for these theory and tecniques

77. References
Hundreds of conference abstracts, papers and books related to this are referenced in:
G.A. Kouzaev, Applications of Advanced Electromagnetics. Components and Systems, Springer Verlag, 2013.
G.A. Kouzaev, This Lecture Text.
G.A. Kouzaev and his co-authors, Several conference papers and lectures published by WSEAS since 2006.

78. Norwegian University of Science and Technology-NTNU, Trondheim