师资队伍

FACULTY

Mikhail Tyaglov
Tyaglov Mikhail

特别研究员
Special Researcher

办公室 Office：

2 号楼 402

办公接待时间 Office Hour：

By appointment

办公室电话 Office Phone：

总机（Telephone Exchange）× 2402

E-mail：

tyaglov at sjtu.edu.cn

教育背景 Education：

博士，2009，柏林工业大学

Ph.D., 2009, Berlin Technical University

研究兴趣 Research Interests：

Distribution of zeros of polynomials and entire functions, Hurwitz polynomials and entire functions, Polya frequency generating functions, orhtogonal polynomials, continued fractions, total positivity, hydrodynamic stability and bifurcation theory

教育背景/经历 Education

PhD, 2009, Berlin Technical University

MSc, 2001, Rostov State University (now Southern Federal University)

BSc, 1999, Rostov State University (now Southern Federal University)

工作经历 Work Experience

Distinguished researcher, Shanghai Jiao Tong University, August 2012 - present

Postdoctorate researcher, Berlin Technical University, May 2009 - July 2012

Teaching assistant, Berlin Technical University, October 2007 - February 2008

Research assistant, Berlin Technical University, May 2007 - May 2009

Lecturer, Southern Federal University (Russia, Rostov-on-Don), September 2006 - December 2006

Teaching assistant, Southern Federal University (Russia, Rostov-on-Don), September 2000 - August 2006

Academic Experience

- Distinguished researcher, Shanghai Jiao Tong University, August 2012 -
**present** - Postdoctorate researcher, Berlin Technical University, May 2009 - July 2012
- Teaching assistant, Berlin Technical University, October 2007 - February 2008
- Research assistant, Berlin Technical University, May 2007 - May 2009
- Lecturer, Southern Federal University (Russia, Rostov-on-Don), September 2006 - December 2006
- Teaching assistant, Southern Federal University (Russia, Rostov-on-Don), September 2000 - August 2006

Grants and Awards

- Israeli-Chinese Grant, NNSFC-ISF, 2015 - 2017 (in a team,
**PI**: Prof. X. Zhang) - Russian National Science Foundation, 2014 - 2016 (in a team,
**PI**: Prof. V. Dubinin) - Shanghai Oriental Scholar (Distinguished Professor), Program sponsored by the Education Commission of Shanghai Municipal government, China, 2015 - 2017 (PI)
- The Research Fellowship for International Young Scientists, NNSF of China, 2014 (PI)
- The Research Fellowship for International Young Scientists, NNSF of China, 2013 (PI)
- Young Researcher Award at
*The 18th international conference on finite or infinite*, Macao, China, August 13--17, 2010.

dimensional complex analysis and applications

Publications and preprints

1. Published papers2. Submitted papers3. Preprints4. Works in progress

1. **M. Tyaglov**, *Self-interlacing polynomials II: Matrices with self-interlacing spectrum*, 2017, __Electron. J. Linear Algebra__, Vol. 32, pp. 51-57, doi:10.13001/1081-3810.3453, ArXiv version.

2. **O. Kushel and M. Tyaglov**, Circulants and critical points of polynomials, __Journal of Math. Ann. Appl.__, **439**, no. 2, 2016, pp. 434 - 450, doi:10.1016/j.jmaa.2016.03.005, ArXiv version.

3. **N. Bebiano and M. Tyaglov**, Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices, __Linear Algebra Appl.__, **439**, no. 11, 2013, pp. 3490 - 3504, doi:10.1016/j.laa.2013.09.026, ArXiv version.

4. **M. Tyaglov**, Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials, __Electron. J. Linear Algebra__, **24**, 2012, pp. 215 - 236, doi:10.13001/1081-3810.1589.

5. **O. Holtz and M. Tyaglov**, *Structured matrices, continued fractions, and root localization of polynomials*, __SIAM Review__, **54**, no. 3, 2012, pp. 421 - 509, doi:10.1137/090781127, ArXiv version.

6. **M. Derevyagin, O. Holtz, S. Khrushchev, and M. Tyaglov**, *Szego‘s theorem for matrix orthogonal polynomials*, __Journal of Approximation theory__, **164**, no. 9, 2012, pp. 1238 - 1261, 2012, doi:10.1016/j.jat.2012.05.003, ArXiv version.

7. **M. Tyaglov**, *On the number of real critical points of logarithmic derivatives and the Hawaii conjecture*, __Journal d‘Analyse Mathematique__, **114**, no. 1, 2011, pp. 1 - 62, doi:10.1007/s11854-011-0011-1, ArXiv version.

8. **Yu. Barkovsky and M. Tyaglov**, *Hurwitz rational functions*,__ Linear Algebra Appl.__, **435**, no. 8, 2011, pp. 1845--1856, doi:10.1016/j.laa.2011.03.062, ArXiv version.

9. **V. Kostov, B. Shapiro, and M. Tyaglov**, *Maximal univalent disks of real rational functions and Hermite-Biehler polynomials*, __Proc. Amer. Math. Soc.__, **139**, no. 5, 2011, pp. 1625 -1635, doi:10.1090/S0002-9939-2010-10778-5, ArXiv version.

10. **M. Tyaglov**, *Monotonicity principle in the Rayleigh problem for isothermally incompressible fluid*, __Journal of Applied Mechanics and Technical Physics__, **48**, no. 5, 2007, pp. 649 - 655, doi:10.1007/s10808-007-0083-y.

1.**O. Katkova, M. Tyaglov, and A. Vishnyakova**, *Linear finite difference operators preserving Laguerre-Pólya class*, ArXiv version.

2. **M. Tyaglov**, *Self-interlacing polynomials*, ArXiv version.

3. **M. Tyaglov and M. Atia**, *On the number of non-real zeroes of a homogeneous differential polynomial*, ArXiv version to appear soon.

4. **M. Tyaglov**, *Generalized Hurwitz polynomials*, ArXiv version.

5. **O. Kushel and M. Tyaglov**, *Criterion of total positivity of generalized Hurwitz matrices*, ArXiv version is to appear soon

6. **O. Kushel and M. Tyaglov**, *On the zeroes of generating polynomials of generalized Hurwitz matrices*, ArXiv version is to appear soon

1. **P. Brändén, G. Csordas, O. Holtz, and M. Tyaglov**, *Stability, hyperbolicity, and zero localization of functions*, Workshop summary.

2. **M. Tyaglov**, *Investigation of convection of isothermally incompressible fluid in a horizontal layer. Part I*, ICSTI technical report, 05.01.2003, No.7-B2003 (in Russian), 2003.

1. *Zeroes of finite differences of polynomials and entire functions* (with O. Katkova, A. Vishnyakova, and J. Xu)

2. *Schwarz matrix with exactly one eigenvalue and a sequence of orthogonal polynomials *(with C. da Fonseca).

3. *Periodic negative-regular continued fractions and atomic decomposition in the modular group* (with S. Khrushchev)

4. *Periodic negative-regular continued fractions. Analytic theory* (with S. Khrushchev)

5. *Hurwitz polynomials and total positivity* (jointly with J. Garloff and M. Adm)

6. *New matrix criterion for functions generating Pólya frequency sequences* (jointly with A. Dyachenko)

7. *New determinantal formulas for rational functions*

8.*Hermite-Biehler polynomials and balayage measures * (jointly with A. Martinez-Finkelshtein and R. Kozhan)

9. *Electrostatics of multiple orthogonal polynomials *(jointly with A. Martinez-Finkelshtein and S. Kalmykov)

10. *Zeroes of derivatives of unit polynomials *(jointly with J. Xia)

Translation work

1. **Yu. Barkovsky**, *Lectures on the Routh-Hurwitz problem*, ArXiv (with O. Holtz).

2. **M. Krein**, *On the theory of symmetric polynomials*, ArXiv.

1. **Yu. Barkovsky**, *Rank-one perturbation method and differential operators of oscillatory type* (with O. Kushel).

Useful Links

Events

**Upcoming events**

**2017**

1. __International Conference on Special Functions: Theory, Computation, and Applications__, City University of Hong Kong, Hong Kong, China, **June 5-9**.

2. __The 2017 International Workshop on Matrix Inequalities and Matrix Equations (MIME 2017)__, Shanghai University, Shanghai, China, **June 6-8**.

3. __2017 Workshop on Matrices and Operators__, Hunan University, Changsha, China, **June 9-12**.

4. __The 6th International Conference on Matrix Analysis and Applications (ICMAA 2017)__, Duy Tan University, Da Nang, Vietnam, **June 15-18**.

5. __Summer School on "Orthogonal Polynomials and SpecialFunctions" (OPSF-S7)__, University of Kent, Canterbury, United Kingdom, **June 26-30**.

6. __The 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)__, University of Kent, Canterbury, United Kingdom, **July 3-7**.

7. __Computational Methods and Function Theory (CMFT‘2017)__, UMCS, Lublin, Poland, **July 10-15**.

8. __Conference Foundations of Computational Mathematics (FoCM‘2017)__, Barcelona, Spain, **July 10-19**, especially the Workshop __"Special Functions and Orthogonal Polynomials"__, **July 17-19**.

9. __2017 Meeting of the International Linear Algebra Society (ILAS 2017)__, Iowa State University, Ames, Iowa, USA, **July 24-28**.

10. __International Workshop on Operator Theory and Applications (IWOTA‘2017)__, TU Chemnitz, Germany, **August 14-18**.

11. __International Conference on MATRIX Analysis and its Applications (MAT TRIAD 2017)__, Bedlewo, Poland, **September 25-29**.

**2018**

1. __SIAM Conference on Applied Linear Algebra (SIAM-ALA18)__, Hong Kong Baptist University, Hong Kong, **May 4-8**.

**2020**

1. __8th European Congress of Mathematics__, Portoroz, Slovenia, **July 05 - 11**.

1. __2016 International Conference on Matrix Theory with Applications IRCTMT-AORC Joint Meeting (ICMTA2016)__, Shanghai Univeristy, Shanghai, China, December 28-31.

2. __International Conference on Mathematical Analysis & its Applications (ICMAA 2016)__, Indian Institute of Technology Roorkee, India, November 28 - December 02, 2016.

3. __VI Russian-Armenian Conference on Mathematical Analysis, Mathematical Physics and Analytical Mechanics__, Rostov-on-Don, Russia, September 11-16, 2016.

4. __4th Dolomites Workshop on Constructive Approximation and Applications (DWCAA16)__, Alba di Canazei, Trento, Italy, September 8-13, 2016.

5. Barry Simon‘s combo: __Fields Institute Young Researchers Symposium "Methods of Modern Mathematical Physics"__, Toronto, Canada, August 22-26, 2016, __CRM Conference "Frontiers in Mathematical Physics"__, Montréal, Canada, August 28-September 1, 2016.

6. __Second ZiF Summer School on Randomness in Physics and Mathematics: From Spin Chains to Number Theory__, ZiF - Center for Interdisciplinary Research, Bielefeld University, Bielefeld, Germany, August 8 - 20, 2016.

7. __International Workshop on Operator Theory and Applications (IWOTA‘2016)__, Washington University in St. Louis, MO, USA, July 18-22, 2016.

8. __7th European Congress of Mathematics__, Berlin, Germany, July 18 - 22, 2016.

9. __20th Conference of the International Linear Algebra Society (ILAS)__, Katholieke Universiteit Leuven, Leuven, Belgium, July 11-15, 2016.

10. __SIAM Annual Meeting 2016__, Boston, Massachusetts, USA, July 11-15, 2016.

11. __8th St.Petersburg Conference in Spectral Theory__, Euler Institute, Saint-Petersburg, Russia, July 3-6, 2016.

12. __Constructive Theory of Functions__, Sozopol, Bulgaria, June 11 - 17, 2016.

13. __Conference on Harmonic Analysis and Approximation Theory (HAAT 2016)__, Centre de Recerca Matemàtica, Barcelona, Spain, June 6 - 10, 2016.

14. __XII International Conference "Approximation and Optimization in the Caribbean"__, Havana University, Havana, Cuba, June 5-10, 2016.

15. __15th International Conference on Approximation Theory__, Menger Hotel San Antonio, Texas, USA, May 22 - 25, 2016.

16. __Sixth International Scientific Conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis VI"__, dedicated to the 75 annual jubilee of professor Stefan Samko, Rostov-on-Don, Russia, 24 - 29 April, 2016.

17. __Joint Mathematics Meeting__, Seatle, WA, USA, January 6-9, 2016.

18. __XI Brunel-Bielefeld Workshop on Random Matrix Theory and Applications__, Bielefeld University, Germany, December 10-12, 2015.

19. __Fourth Najman Conference on Spectral Problems for Operators and Matrices__, Opatija, Croatia, September 20-25, 2015.

20. __Mat Triad 2015__, Coimbra, Portugal, September 11-17, 2015.

21. __Conference Foundations of Computational Mathematics (FoCM‘2014)__, Universidad de la República, Montevideo (Uruguay), December 11 - 20, 2014.

22. __International Conference on Applied Mathematics in honour of Professor Roderick S. C. Wong’s 70th Birthday__, City University of Hong Kong, December 1-5, 2014.

23. __"International Conference on Orthogonal Polynomials, Integrable Systems and Their Applications" __, on the occasion of Professor Mourad Ismail‘s 70th Birthday, Shanghai Jiao Tong University and Shaoxing University, October 25-29, 2014.

24. __"The Real World is Complex"__, An international Symposium in honor of Christian Berg, Copenhagen, Denmark, August 26-28, 2015.

25. __4th International Conference on Matrix Methods in Mathematics and Applications__, MMMA-2015, Moscow, Russia, August 24-28, 2015.

26. __8th International Congress on Industrial and Applied Mathematics (ICIAM)__, Beijing, China, August 10-14, 2015.

27. __10th ISAAC conference__, University of Macau, China, August 3-8, 2015.

28. __International Workshop on Operator Theory and Applications (IWOTA 2015)__, Tbilisi, Georgia, July 6-10, 2015.

29. __Progress on Difference Equations__, Universidade da Beira Interior, Portugal, June 15-18, 2015.

30. __Joint AMS-EMS-SPM International Meeting__, Porto, Portugal, June 10-13, 2015.

31. __5th Iberoamerican Workshop on Orthogonal Polynomials and Applications EIBPOA2015__, Instituto de Matemáticas, C.U., Universidad Nacional Autónoma de México, June 8-12, 2015.

32. __Workshop "Asymptotics in integrable systems, random matrices and random processes and universality": In honour of Percy Deift‘s 70th birthday__, CRM, Montreal, June 7-11, 2015.

33. __The 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA13)__, National Institute of Standards and Technology, Gaithersburg, MD (USA), June 1-5, 2015.

Seminars

__2017__

Seminar "** Spectral graph theory**(with Zhang Xiaodong)"

Seminar "** Equilibrium measures and balayage**"

__Thursday 14:00-16:00, room 1207__

__2016__

Seminar "** Matrix theory and complex analysis in root location of polynomials and entire functions**"

__Tuesday 16:00-18:00, room 1207__

Seminar "** Matrix theory in combinatorics and graph theory**"

Seminar "** Equilibrium measures and balayage**"

__Friday 09:00-11:00, room 1106__

Teaching

Spring 2017**Course:** *Fourier Analysis and Real Analysis*

**Prerequisites:** Basic courses in linear algebra and mathematical analysis.

**Place and time:** Tuesday 10:00am - 11:40am (Lower Hall 302)

Thursday 10:00am - 11:40am (Lower Hall 302)

Description and syllabus **Description:** The course is introduction to the theory of Fourier series and transform as well as to the theory of Lebesgue integration. Fundamental ideas and rigorous proofs will be presented. Topics of the course to be covered include Fourier series, their convergence and some applications, Poisson kernel, Cesaro and Abel summability, Plancherel formula, Poisson summation formula, Fourier transform with some applications, measures, measurable sets and functions, Lebesgue integral and Fubini theorem.Description: The course is introduction to the theory of Fourier series and transform as well as to the theory of Lebesgue integration. Fundamental ideas and rigorous proofs will be presented. Topics of the course to be covered include Fourier series, their convergence and some applications, Poisson kernel, Cesaro and Abel summability, Plancherel formula, Poisson summation formula, Fourier transform with some applications, measures, measurable sets and functions, Lebesgue integral and Fubini theorem.

**Syllabus** (tentative and subject to change, if any):

Grading criteria Literature

__Literature for Real Analysis__:

Progress of the course and Homeworks__Homework 15__: TBA

**01.06.2017*** **Fourier transform. Applications to ODE.*

__Homework 14__: see __file__

**25.05.2017*** **Fourier transform. Definition and basic properties.*

**23.05.2017*** **Applications of Fourier series to ODE.* (Lecture notes 28)

__Homework 13__: see __file__

**19.05.2017*** **Basic properties of Fourier series. Uniqueness. Coefficients behaviour.*

**18.05.2017*** **Basic properties of Fourier series. Localization principle.*

**16.05.2017*** **Fourier series for solution of PDE* (Covered pages 17-22 of the book by E. Stein "Fourier analysis", Particular lecture notes 25)

__Homework 12__: Problems 7.4, 7.5, 7.6, 7.7, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13, 7.18, 7.20, 7.21, 7.22, 7.23, 7.31, 7.33, 7.42 of the __Problem book.__

**12.05.2017*** **Genesis of Fourier series* (Covered pages 2-16 of the book by E. Stein "Fourier analysis")

**11.05.2017*** **The space L2 and trigonometric Fourier series* (Lecture notes 23)

**09.05.2017*** **Tauberian theorems. Higher methods of summation of divergent series.* (Lecture notes 22)

__Homework 11__: Problems 6.12, 6.14, 6.15, 6.19, 6.30, 6.31, 6.33, 6.34, 6.35, 6.36, 7.1, 7.14, 7.15, 7.16, 7.17 of the __Problem book.__

**04.05.2017*** **Divergent series. Cesaro and Abel methods of summation.* (Lecture notes 21)

**02.05.2017*** **Differentiation of integrable functions (continuation)* (Covered pages 106-108 and 131-136 and 285-292 of the book by E. Stein "Real analysis")

__Homework 10__: Problems 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 6.10, 6.24, 6.25, 6.26, 6.27, 6.28, 6.29 of the __Problem book.__

**28.04.2017*** **Differentiation of integrable functions (continuation)* (Covered pages 123-131 of the book by E. Stein "Real analysis")

**25.04.2017*** **Differentiation of integrable functions (continuation)* (Covered pages 106-108 and 115-123 of the book by E. Stein "Real analysis")

__Homework 9__: Problems 4.12, 4.48, 4.49, 4.50, 4.52, 4.53, 4.54, 4.55, 5.43, 5.44 of the __Problem book.__

**20.04.2017*** **Differentiation of integrable functions* (Covered pages 98-105 of the book by E. Stein "Real analysis")

**18.04.2017*** **Fubini‘s theorem.* (Lecture notes 16)

__Homework 8__: Problems 4.37, 4.38, 4.39, 4.40, 4.42, 4.44, 4.45, 5.6, 5.7, 5.8, 5.10, 5.39, 5.40, 5.41, 5.42 of the __Problem book.__

**13.04.2017*** **Spaces of integrable functions (continuation). Approximation of integrable functions.* (Lecture notes 15)

**11.04.2017*** **Difference between Riemann and Lebesgue definite integrals (continuation). Spaces of integrable functions.* (Lecture notes 14)

__Homework 7__: Problems 4.1, 4.5 d), 4.6, 4.20, 4.21, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 4.31, 4.32 of the __Problem book.__

**06.04.2017*** **Integration theory (continuation). Difference between Riemann and Lebesgue definite integrals.* (Lecture notes 13)

__Homework 6__: Problems 3.7, 3.9, 3.12, 3.13, 3.15, 3.16, 3.18, 3.20, 3.21 a), d), 4.8, 4.9, 4.13, 4.14, 4.15, 4.16, 4.18, 4.19 of the __Problem book.__

**30.03.2017*** **Integration theory (continuation).* (Lecture notes 12)

**28.03.2017*** **Measurable functions (continuation). Integration theory.* (Lecture notes 11)

__Homework 5__: Problems 2.12, 2.13, 2.17, 2.21, 2.23, 2.25, 2.27, 2.30, 2.32, 2.33, 3.3, 3.4, 3.5, 3.10, 3.11, 3.22 of the __Problem book.__

**23.03.2017*** **Measurable functions.* (Lecture notes 10)

**21.03.2017*** **The Lebesgue measure and Borel sets.* (Lecture notes 9)

__Homework 4__: Problems 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 2.9, 2.10, 2.11, 2.12, 2.14, 2.16, 2.18, 2.19, 2.20, 2.28, 2.29 of the __Problem book.__

**16.03.2017*** **Lebesgue measure on R*^{n}. (Lecture notes 8)

**14.03.2017*** **Introduction to the measure theory. Rings and algebras of sets.* (Lecture notes 7)

__Homework 3__: Problems 5.17 1), 2), 3); 5.18; 5.19; 5.20; 5.22 1), 4); 1.1 4); 1.2 3); 1.6; 1.10; 1.11; 1.13; 1.16; 1.17; 1.18. of the __Problem book.__

**09.03.2017*** **Introduction to the sets theory. Countable sets and sets of cardinality continuum.* (Lecture notes 6)

**07.03.2017*** **Linear functionals in normed and Hilbert spaces.* (Lecture notes 5)

__Homework 2__: Problems 5.21; 5.23; 5.30; 5.31 1),2),3); 5.32; 5.33; 5.34; 5.35; 5.36 of the __Problem book.__

**02.03.2017*** **Hilbert spaces. Orthogonal projections.* (Lecture notes 4)

**28.02.2017*** **Pre-Hilbert and Hilbert spaces.* (__Lecture notes 3__)

__Homework 1__: Problems 5.2, 5.3, 5.12, 5.13, 5.15 of the __Problem book.__

**23.02.2017*** **Linear normed and inner product spaces.* (__Lecture notes 2__)

**21.02.2017*** **The Hoelder and Minkowski inequalities. Linear normed spaces.* (__Lecture notes 1__)

Thursday 10:00am - 11:40am (Lower Hall 302)

- Introduction to Banach and Hilbert spaces.
- Basic facts of the set theory
- Measurable sets and Lebesgue measure
- Measurable functions
- Lebesgue integral
- Divergent series and Tauberian theorems
- The genesis of Fourier series
- Basic properties of Fourier series
- Convergence of Fourier series
- Applications of Fourier series (ODE, PDE, number theory, differential geometry etc.)
- Fourier transform on the real axis
- The Poisson summation formula
- Applications of Fourier transform to PDE
- Some elements of the Fourier transform on R
^{n}.**(()**

The students will have 12–16 homeworks (depending on the progress of students during the class). Completely made homeworks give 40% of the final mark, the colloquium gives also 20%, and the exam gives other 40%. ** (()**

1. **A. Kolmogorov and S. Fomin**, *Measure, Lebesgue integrals, and Hilbert space,* Academic Press, New York and London, 1960.

2. **A. Kolmogorov and S. Fomin**, *Introductory Real Analysis*, Dover Publication Inc., New York, 1970.

3. **J. Hunter**, *Measure Theory*, __Lecture notes__

4. **I. Natanson**, *Theory of Functions of a Real Variable*, Vol. I and Vol. II, Frederick Ungar Publishing Co, New York, 1955, 1961, 1964.

5. **L. Richardson**, *Measure and integration. *A Concise Introduction to Real Analysis, Jonh Wiley & Sons Inc., Hoboken, New Jersey, 2009.

6. **W. Rudin**, *Real and Complex Analysis*, McGraw-Hill Inc., 1986.

7. **E. Stein and R. Shakarchi**, *Real Analysis*, Princeton University Press, Princeton and London, 2005.

8. **M. Spiegel**, *Real variables.* Lebesgue measure and integration __with applications to Fourier series__, McGraw-Hill inc., 1990.

__Literature for Fourier Analysis__:

- 1.
**N. Bary**,*A Treatise on Trigonometric series*, Vol. I, Pergamon Press Ltd., 1964. - 2.
**A. Zygmund**,*Trigonometrical series*, Warszawa - Lwow, 1935. - 3.
**E. Stein and R. Shakarchi**,*Fourier Analysis*, Princeton University Press, Princeton and London, 2003. - 4.
**G. Hardy**,*Divergent series*, Oxford, 1949. - 5. A.
**Sommerfeld**,*Partial Differential Equations in Physics*– Lectures on Theoretical Physics, Vol. VI, Academic Press, first printing 1949, second printing 1953. - 6.
**A. Tikhonov and A. Samarskii**,*Equations of Mathematical Physics*, Dover Publications Inc., 2011 (reprint edition).

__Problem books__:

- 1.
**M. Tyaglov**,__Problems on the Course of Real Analysis and Fourier Analysis__, 2016.

Visitors

P_(n+1) (x)=(x-c_(n+1) ) P_n (x)-d_(n+1) (x^2+1) P_(n-1) (x), n≥1,

with P_0 (x)=1 and P_1 (x)=x-c_1, where 〖{c_n}〗_(n≥1) is a real sequence and 〖{d_(n+1)}〗_(n≥1) is a positive chain sequence. The zeros of the polynomials P_n are simple and lie on the real line. It turns out that the polynomial P_n, for any n≥2, is the characteristic polynomial of a simple n×n generalized eigenvalue problem. It is also shown that with this R_II type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by P_n with respect to this measure is also found.

We extend some equilibrium-type results first conjectured by Ambrus, Ball and Erdélyi and then proved recently by Hardin, Kendall and Saff. Similarly to them, we too work on the torus T"?" ["0,2π" ) (unit circle), but a motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton.

The problem is to minimize (with respect to the arbitrary translates y_0=0,y_j∈T , j=1,…,n) the maximum of the sum function F(?)?K_0 (?)+∑_(j=1)^n?〖K_j (?-y_j 〗), where K_j`s are certain fixed ``kernel functions``. If they are concave on T, except for having possible singularities or cusps at zero, then the translates by y_j will have singularities at y_j (while in between these nodes the sum function F still behaves regularly). So one can consider the maxima m_i on each subinterval between the nodes y_j, and look for the minimization of maxF=max┬i?〖m_i 〗. Also the dual question of maximization of min┬i?〖m_i 〗 arises.

Hardin, Kendall and Saff considered one single, even kernel, K_j=K for j=1,…,n, and Fenton considered the case of the interval [-1,1] with two fixed kernels K_0=J and K_j=K for j=1,…,n. Here we build up a systematic treatment of the situation when all the kernel functions can be different without assuming them to be even.

Finally we discuss a result of Bojanov on algebraic polynomials with prescribed zero order having minimal sup-norm on interval. As an application of our general framework we show a generalization of Bojanov`s result for generalized algebraic polynomials (GAP) and generalized trigonometric polynomials (GTP) as well.

D-stability concept arises is such different areas of mathematics, such as control theory where that kind of problems corresponds to construction of a dynamical system with some special properties of its behaviour, singularity theory and topology, where these problems constituted an important natural example of singularities given by the symmetric group action, metric complex analysis, where those space become an important testing ground for different convexity-type definitions, statistical physics, where different types of stabilities arise in a context of so called theorems of Yang-Lee type.

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega,$ on its border, and at the complement to its closure. Presented approach is a generalization, unification and development of several classical approaches to the stability problems: rootclustering (D-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., D-decomposition (Yu.I. Neimark,B.T. Polyak, E.N. Gryazina) and universal parameter space method (A. Fam, J. Meditch, J.Ackermann).

Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as asymmetric product morphism and applications of methods of real algebraic geometry and topological theory of symmetric products. We study topology of strata and adjacencies between them.

Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.

2015

Talk:

Collaborators and colleagues

__Mohammad Adm__, __Alexander Aptekarev__, __Walter van Assche__, __Mohamed Jalel Atia__, __Natalia Bebiano__, __Bernhard Beckermann__, __Petter Brändén__, __David A. Cardon__, __Tom Craven__, __George Csordas__, __Maksym Derevyagin__, __Diego Dominici__, __Alexander Dyachenko__, __Carlos da Fonseca__, __Shmuel Friedland__, __Juergen Garloff__, __Dmitrii Karp__, __Olga Katkova__, Sergei Kalmykov, __Sergey Khrushchev__, __Vladimir Kostov__, __Rostyslav Kozhan__, __Arno Kuijlaars__, Olga Kushel, __Francisco Marcellán__, __Andrei Martínez-Finkelshtein__, __Boris Shapiro__, __Alagacone Sri Ranga__, __Anna Vishnyakova__, __Wadim Zudilin__.

地址：上海市闵行区东川路800号数学楼 邮编：200240

电话：(86-21) 54743151

传真：(86-21) 54743152

Address：800 Dongchuan RD Shanghai, 200240 China

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