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Applied Mathematics
in Small and Large Firms
George [email protected]
Västerås, Sweden, Nov 18, 2015
Mälardalens Högskola (MDH)
School for Education, Culture and Communication (UKK)
Master Program in Engineering Mathematics
Goals Mathematical Fields (selected)
1. Coproduction of scientific knowledge
Academia-Industry
Applied mathematics
Applied matrix analysis
2. Current and future technologies Mathematics of the Internet
Quantum computers and information
3. Established theories Wavelets
Applied mathematic structures
4. Flexibility, Independence, Specialization Project in mathematics
Degree project in mathematics
Speaker Series of the Master Program in Applied
Mathematics
Earlier talk held by distinguished speaker:
Christian Sohl, SAAB Electronic Defense
Applied Mathematics: Electromagnetics in Industry
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Mathematics classification
• Current version: MSC 2010
• http://www.ams.org/mathscinet/msc/msc2010.html
• Example:
– 46-XX: Functional Analysis
– 46Bxx: Normed linear spaces and Banach spaces
– 46B22: Radon-Nikodym, Krein-Milman and related
properties
Pure research in Mathematics
• Ferma’s last theorem (1637)
• Problem in number theory: 11D41
• Ribet’s theorem: was a theorem between two conjectures
• Corollary of the modularity theorem of Taniyama-Shimura-Weil conjecture
• Andrew John Wiles’ proof (1995)
• Elliptic curves: 14H52
• Result: number theory 11G16
nnncba =+
Applied mathematics
• We return to the example:
– 46-XX: Functional Analysis
– 46Bxx: Normed linear
spaces and Banach spaces
– 46B22: Radon-Nikodym,
Krein-Milman and related
properties
Radon-Nikodym Theorem
• A result in Measure Theory
• Any absolutely continuous measure λ wrt some measure μ (μ could be a
Lebesgue measure of Haar measure) is given by the integral of some L1
function f
• The function f is called Radon-Nikodym derivative
• All Hilbert spaces have the Radon-Nikodym property
∫=E
fdE µλ )(
µ
λ
d
df =
Controlling rolling of ships
• A ship has 6 degrees of freedom, rolling is one of these
• QTAGG has sensors for the movement of the ship and for engine and propeller
properties
• One problem in control is to have a model of the process to be controlled
• After having a model, a model-predictive control (MPC) algorithm can be used to
reduce rolling
State-space models
tt
ttt
Cxy
BuAxx
=
+= −1
Xu(t) y(t)
npmnnn
pnmLCBAR××× ℜ×ℜ×ℜ=∈=
,,),,(
A,B,C are matrices of appropriate dimensions
R = is an abstract space with many dimensions
What is the distance between a R1 and R2 ?
Speed control
Propeller control
Waves
Winds
Ship movement 6DOF
State-space models
• Let P be a (n x n) matrix in GL(n)
• GL = usual notation for “general linear groups”
• There is a property that if R=(A,B,C) then any
• P ◦ (A,B,C)=(P-1AP, B-1P, CP) is equivalent to R
• This means we cannot simply compare R1 with R2 but we need to slide one along the equivalent classes e.g. using the dual Hahn- Banach theorem
• However GL is not compact. One method is to seek maximally compact subgroup
• Then a max distance to a tangent space is found with the Jacobi Eigenvalue algorithm
• This is the distance between models R1 and R2
Connection math – applications
1. Radom-Nikodym property not fulfilled, find an appropriate compact subspace
2. Find that there is an invariant group
3. Find an abstract distance
4. Apply the optimization (Hahn-Banach, Jacobi)
5. We got the distance
• Without these steps, one cannot define a correct distance as the distance could be arbitrarily large or small depending on how the model was obtained.
Flatness Control in Cold Rolling Mills - ABB
• Measurement roll• Control system
Stressometer® Systems– for Flatness Measurement & Control
Flatness Control
Simultaneously control a large number of actuators of which several have similar
flatness effect
Adaptive – Predictive Control in Orthogonal Space
Target Flatness
Orthogonal
Space
-Decoupling –
SVD(G)=USV’
Map to reduced control space
Decoupled Predictive ControllersMap back to real space
Adaptation
Observer
- Gains -
Range
Slew Rate
Step limits
Limits
Mill ActuatorsStrip
Stressometer
N x PPI regulators
PPI
Predictor
Lambda
AWO
Model
Library
The historical perspective
• Least squares: Gauss 1801 (predict orbit of Ceres) => 1960
• Hermite orthogonal polynomials 1849 => 1970
• Chebyshev orthogonal polynomials 1854 => 1980
• Gram-Schmidt ortogonalization: 1907 => 1980
• Hahn-Banach: 1930 => 1985
• Givens transformation: 1952 => 1985
• Housholder transformation: 1964 => 1985
• Singular value decomposition Golub: 1965 => 2007
• Matrix Perturbation, Steward: 1990 => 2008
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Academic vs. Industry Carrier
• Academic: high-level specialization around a certain field,
journals and conferences. Unlimited time to solve problems of
abstract nature. Focused publications.
• Industry: research through wide fields required by the
development of a product. Short time to solve a very specific
problem. Eclectic publications.
Typical Industrial Organization
Pre-study MSc / PhD Post-study Patent Product
Product Projects Dept. Customer ProjectsProduct Projects Customer ProjectsSystem Development Sales
Technical MathFormal SW dev.
and test methodsProcess models
Demand
Prediction
Statistics
Financials
Large and Small Firms
• A key need for all firms is prediction
– Predict the market
– Predict quality of materials and services
– Predict production capacity
– Predict that your bridge will hold in strong winds
– Predict the quality of the customer’s customer
– Predict exchange rate
– Predict the share prices
• Prediction means statistics, models, inferences, distances, correlations, etc. : Mathematics
Large and Small firms today
• Firms are experiencing nowadays structural changes
• Large firms are large since they have a stable market, suppliers, production and services
• The prediction requirements are less demanding
• Small firms have less secure conditions, their prediction requirements are hard and key for their future
• Therefore today large firms are solving smaller mathematical problems and small firms are solving large and difficult mathematical problems
• Historically this was not always so and may change in the future
Historical differences
• Joseph Schumpeter,
economist (1883-1950)
• Business cycles
• Creative
destruction
• Monopoly gains
AT&T
Satellite communications, fax, sound motion
picture, negative feedback, long-distance TV,
wave nature of matter, stereo recording, radio
astronomy, digital computer, HF radar, transistor,
information theory (Claude Shannon), solar cell,
laser, big-bang echo, Unix, Internet, fiber optic
communication, C++, HDTV, quantum
computing
RAND Corp.
• Kenneth Arrow
• Richard Bellman (optimization)
• George Dantzig (simplex algorithm)
• John von Neumann
• Edmund Phelps
• Thomas Schelling
• ….
• 30 Nobel prizes
Today
• Large firms are focused on their current markets, typically no R&D for new markets
• Information about large investments in R&D would tumble stock prices of a firm
• Small firms are innovative but have less financing and less marketing strength
• Academia can provide an environment where new bold ideas can move into applications
• Creating links between firms and academia is a good way to open this deadlock and generate new value
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Error-free software (EFS) for industrial
products
• Critical applications in steel, manufacturing, marine, chemical, pharmaceutical, health-related industries: software errors have consequences
• EFS: an important goal when I started at ABB Automation
• Internet, clouds, iPads: makes the problem more complex
• Still very important today, still not solved
Formal methods
• Many SW testing
methods are based
on best practices
• Industry would need
methods that have
formal proofs (i.e.
have an MSC number)
SW Formal Methods
• Statistically, SW code used in industrial control for traditional control (like PID) is less than 5%. Discrete states are 95% of the code
• Traditional control has good stability proof theories (34Dxx): Asymptotic properties, Lyapunov stability, perturbations, Popov stability, attractors, etc.
• There are few stability theories for discrete state spaces
• I focused on theories of stability for discrete states
State modeling
Current state
Control action
Next expected state as predicted
effect of the control action
State materialized instead of the expected
one when disturbances occur
Disturbance
Formal representation: state vectors X and transition matrices A.
Colored Petri net theory
nn AXX =+1
State modeling
• Problem: to classify types and possible unexpected transitions
• Some some transitions will be still unknown
• Challenge: distinguish the case of unknown disturbances from the case of a wrong model
• Model built using ontological assumptions
• Violations of ontological assumptions: instability
• Found a formal method by which a controller can determine itself if it has a wrong model
• Results in a PhD thesis in AI at Linköping University
Fuzzy states03B52, 03B72, 04A72, 46S40, 54A40, 90C70, 93C42, 94D05
• Improvement of the
prediction strength by
using fuzzy states
instead of discrete
states
• I published papers
about best formal
control architectures
and fault detection and
isolation
Fuzzy States and Fuzzy State Processors
• Lofti Zadeh (father of Fuzzy Mathematics, prof. at UCLA), put me in contact at a conference in Sydney with prof. Janos Grantner from Western Michigan University who made similar research
• I started as adj. prof at WMU
Complex systems research
Award DAAD19-01-1-
0431 from NSF DURIP
(US Defense Research
Instrumentation
Program). ABB together
with Western Michigan
University
Distributed software for
complex control systems
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Electronics
• Electronics for critical applications.
Reconfigurable FPGA using states modeling
failure modes and reconfigurations
• Requires a layer of “Electronic Monitoring and
Recovery” (EMR)
• General mathematical description of an EMR
• A discrete state / probabilistic state method
Industrial Production and Quality
Supervision
• KPI’s (Key Performance Indicators): are actually Lebesque / Haar distances
• Find a method to compute the distance between heterogeneous state spaces
• Link to statistical distances (Kullback-Leibler div.)
• Use Big Data Architectures and methods to find distances / measures and related predictions by automatic means
• Automatic Markov chain generation from data (Big Data Analytics)
Process modeling
• Applications with different mathematical Invariants
• Example: a stationary process means its probability distribution does not change
• Symplectic geometry (Weil): connected to Hamiltonian formulation of the classical mechanics.
• Movement of a ship => Kinetic + Potential Energy => differentiable manifold => Hamilton => State phase
• A general Hamiltonian / Symplectic method
More Industrial Problems
• Invariants via Inverse methods
• Invariants via ICA (independent component analysis): better measures of independence
• Signal processing: describe mathematically the difference between filtering, denoising and detrending. Example: article on EMD (empirical mode decomposition) finds a smooth function subtracted from the processed signal
• Fusion methods mapping statistical inferences (like Bayes) to state spaces of dynamical systems
Computational Fluid Dynamics
Iowa University, USA
Automatic mapping between CFD models and state models
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Contents
• Pure and applied mathematics
• What are the math problems firms are trying to solve and why do they do it?
• Mathematics in different applications
• Unsolved problems
• Mathematics Curricula
• How to contact firms?
• Conclusions
Organization
Pre-study MSc / PhD Post-study Patent Product
Product Projects Dept. Customer ProjectsProduct Projects Customer ProjectsSystem Development Sales
Technical MathFormal dev. and
test methodsProcess models
Demand
Prediction
Statistics
Financials
Three Steps Tutorial for how to contact firms
Required work: about 1 month
• Step 1: Find a firm that has the right profile
• Step 2: Make a pre-study of your own
– Find out what are the firm’s products and services
– Read the firm’s patents, conference and journal
articles
– Build a hypothesis about what are the firm’s
challenges
– Write a study about the solution to the challenge
you have identified
Cont. how to contact firms
– Have a detailed presentation of the problem and
solution on max 6 pages and a one page poster
– Train the presentation and arguments with a friend
– It is good if you have the presentation on some
homepage that can be easily accessed
• Step 3: Call the HR department (mail or phone)
– Ask to be put in contact with a manager responsible
with the problem you identified since you have a
results
– Make the presentation to the manager
Conclusions
Mathematical abstractions
1. Measurement theory, distances, orthogonality
2. Space projections (mappings) and special functions
3. Invariants
4. Representation of change in space, time and in related projected spaces. Change predictions.
5. Solution tools for Inverse Problems
6. Duality theorems
7. Symbolic mappings and statistical / Bayesian inferences
Summary of today’s presentation
• In what way is pure mathematics related to new applications?
• What kind of mathematical problems is a firm trying to solve?
• Is there a difference between the mathematics of the small
and of the large firm? Between firms and academia?
• Mathematics applications in different fields and some
problems
• How can a student / researcher get in contact with firms using
mathematics?