astronomy

Michael Shub

Michael Shub is a astronomy topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Michael Shub rather than just read about it. In short: Michael Ira Shub (born August 17, 1943) is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms. Career 1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled Endomorphisms of Compact Differentiable Manifolds.

Key takeaways

Michael Shub belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Michael Shub to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Michael Shub from memory before moving on to harder problems.

Reference excerpt

Michael Ira Shub (born August 17, 1943) is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms.

Career

1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled Endomorphisms of Compact Differentiable Manifolds. In his Ph.D. thesis, he introduced the notion of expanding maps, which gave the first examples of structurally stable strange attractors. His advisor was Stephen Smale. From 1967 to 1985, he worked at Brandeis University, the University of California, Santa Cruz and the Queens College at the City University of New York. In 1974, he proposed the Entropy Conjecture, an open problem in dynamical systems, which was proved by Yosef Yomdin for C ∞ {\displaystyle C^{\infty }} mappings in 1987.

1985–2004: IBM research From 1985 to 2004, he joined IBM's Thomas J. Watson Research Center. In 1987, Shub published his book Global Stability of Dynamical Systems, which is often used as a reference in introductory and advanced books on the subject of dynamical systems. In 1993, Shub and Stephen Smale initiated a rigorous analysis of homotopy-based algorithms for solving systems of nonlinear algebraic equations, which has inspired much of the work in that area during the last two decades. From 1995 to 1997, Shub was the founding chair of the Society for the Foundations of Computational Mathematics. In 2001, Shub became a founding editor of their journal, Foundations of Computational Mathematics.

1986: Blum Blum Shub

Shub, along with coauthors Lenore and Manuel Blum, described a simple, unpredictable, secure random number generator (see Blum Blum Shub). This random generator is useful from theoretical and practical perspectives.

1989: Blum–Shub–Smale machine

In 1989, he proposed with Lenore Blum and Stephen Smale the notion of Blum–Shub–Smale machine, an alternative to the classical Turing model of computation. Their model is used to analyse the computability of functions.

2004–2010: Post-IBM From 2004 to 2010, he worked at the University of Toronto. After 2010, he became a researcher at the University of Buenos Aires and at the Graduate Center of the City University of New York. Since 2016, he has been Martin and Michele Cohen Professor and Chair of the Mathematics Department at City College of New York.

Awards and recognition 1972: Fellow of Alfred P. Sloan Foundation. 2000: Fellow of the American Association for the Advancement of Science. 2012: A conference, From Dynamics to Complexity, was organized at the Fields Institute in Toronto celebrating his work. 2015: Fellow of the American Mathematical Society "for contributions to smooth dynamics and to complexity theory." 2016: Fulbright Specialist.

Selected publications Blum, Lenore; Blum, Manuel; Shub, Michael (1 May 1986). "A Simple Unpredictable Pseudo-Random Number Generator". SIAM Journal on Computing. 15 (2). Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics: 364–383. doi:10.1137/0215025. Shub, Michael (1974). "Dynamical systems, filtrations and entropy" (PDF). Bulletin of the American Mathematical Society. 80. Providence, Rhode Island: American Mathematical Society: 27–41. doi:10.1090/S0002-9904-1974-13344-6. Shub, Michael (1987). Global Stability of Dynamical Systems. New York City: Springer-Verlag. ISBN 978-0387962955. Robbin, Joel (1988). "Review: Global stability of dynamical systems by Michael Shub" (PDF). Bulletin of the American Mathematical Society. 18 (2). Providence, Rhode Island: American Mathematical Society: 248–250. doi:10.1090/s0273-0979-1988-15665-0. Blum, Lenore; Shub, Michael; Smale, Stephen (July 1989). "On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines" (PDF). Bulletin of the American Mathematical Society. 21. Providence, Rhode Island: American Mathematical Society: 1–47. doi:10.1090/S0273-0979-1989-15750-9. Shub, Michael; Smale, Stephen (1993). "Complexity of Bézout's Theorem I: Geometric Aspects". Journal of the American Mathematical Society. 6 (2). Providence, Rhode Island: American Mathematical Society: 459–501. doi:10.2307/2152805. JSTOR 2152805. Blum, Lenore; Cucker, Felipe; Shub, Michael; Smale, Stephen (1997). Complexity and Real Computation. New York City: Springer-Verlag. ISBN 978-0387982816.

References

External links Personal website at the City College of New York.

Illustrations

Worked examples

Example 1 — a first encounter with Michael Shub

Start with the simplest possible case. Write down what Michael Shub claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Michael Shub before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Michael Shub ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Michael Shub

In research: Michael Shub appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Michael Shub in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Michael Shub is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1943 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Michael Shub outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Michael Shub in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Michael Shub means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Michael Shub out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Michael Shub in simple terms?

Michael Ira Shub (born August 17, 1943) is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms. Career 1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitle…

Why does Michael Shub matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Michael Shub?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Michael Shub.