# 8th Symposium on Integrable Systems

July 3 – 4, 2015

Łódź, Poland

organized by the

Department of Computer Science
Faculty of Physics and Applied Informatics
University of Łódź

# General Information

Topic: The conference concerns classical and quantum integrable systems and their physical applications. It is a continuation of annual meetings of the most prominent specialists in the field, both from Poland and abroad, which actively work in the domain of mathematical physics and the theory of dynamical systems. 20-minutes talks are planned devoted to the presentation of the recent achievements of the participants.

Previous symposia were held in Gdańsk 2007, Poznań 2009, Warsaw 2010, Zielona Góra 2011, Olsztyn 2012, Białystok 2013, Kraków 2014.

Organizers:

Department of Computer Science
Faculty of Physics and Applied Informatics
University of Łódź
Pomorska Str. 149/153, 90-236 Łódź,
tel.:( 0-48-42) 635-57-39
fax: (0-48-42) 635-57-42
integrablesystems@uni.lodz.pl

Registration:

Persons willing to take part in the conference are asked to sent an e-mail to: integrablesystems@uni.lodz.pl

There is no conference fee. Organizers cover the accommodation costs (for more information see Accommodation) but do not provide the meals. The travel expenses are on participants side.

Abstract submission:

Abstracts should be sent in pdf format.

# Important Dates

Abstract submission deadline: May 25, 2015 June 3, 2015

Registration completed

# Scientific Committee

Maciej Błaszak (Adam Mickiewicz University, Poznań, Poland)

Cezary Gonera (University of Łódź, Poland)

Piotr Kosiński (University of Łódź, Poland)

Marek Kuś (Center for Theoretical Physics of the Polish Academy of Sciences, Warsaw, Poland)

Paweł Maślanka (University of Lódź, Poland)

Ziemowit Popowicz (University of Wrocław, Poland)

# Organizing Committee - (University of Łódź)

Joanna Gonera (Scientific secretary)

Piotr Kosiński

Cezary Gonera

Krzysztof Andrzejewski

Bartosz Zieliński

Małgorzata Drewnowska (Technical secretary)

# Venue

The conference will take place in the building of
Faculty of Physics and Applied Informatics of University of Łódź Pomorska 149/153, 90-236 Łódź, Poland

HOW TO GET THERE (pdf)

WHERE TO EAT (pdf)

# Accommodation

Participants will be lodged in the II University Dormitory (II Dom Studenta UŁ „Balbina”), Lumumby Str. 16/18, Łódź, which is situated within the walking distance from the conference venue. The organizers will cover the accommodation costs (of two nights 2/3 and 3/4 July) in shared double rooms. Single rooms are also available at the participant's own expense.

HOW TO GET THERE (pdf)

WHERE TO EAT (pdf)

# Participants & Programme

### Participants

• Maciej Błaszak (Adam Mickiewicz University, Poznań, Poland)
• Yves Brihaye (University of Mons, Belgium)
• Tomasz Brzeziński (Swansea University, United Kingdom)
• Jan L. Cieśliński (University of Bialystok, Poland)
• Marek Czachor (Gdańsk University of Technology, Poland)
• Jan Dereziński (University of Warsaw, Poland)
• Adam Doliwa (University of Warmia and Mazury in Olsztyn, Poland)
• Ziemowit Domański (Center for Theoretical Physics of the Polish Academy of Sciences, Warsaw, Poland)
• Maciej Dunajski (University of Cambridge, United Kingdom)
• Piotr P. Goldstein (National Centre for Nuclear Research, Warsaw, Poland)
• A. Michał Grundland (Centre de Recherches Mathématiques, Université de Montréal, Canada)
• Mariusz Holicke (Łódź, Poland)
• Jerzy Knopik (Jagiellonian University, Kraków, Poland)
• Wojciech Kryński (University of Warsaw, Poland)
• Maciej Kuna (Gdańsk University of Technology, Poland)
• Marek Kuś (Center for Theoretical Physics of the Polish Academy of Sciences, Warsaw, Poland)
• Grzegorz Kwiatkowski (Gdańsk University of Technology, Poland)
• Sergey Leble (Gdańsk University of Technology, Poland)
• Javier de Lucas Araujo (University of Warsaw, Poland)
• Andrzej J. Maciejewski (University of Zielona Góra, Poland)
• Krzysztof Marciniak (Linköping University, Sweden)
• Michal Marvan (Silesian University in Opava, Czech Republic)
• Maciej Nieszporski (University of Warsaw, Poland)
• Andriy Panasyuk (Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine; University of Warmia and Mazury in Olsztyn, Poland)
• Makar Plakhotnyk (Kyiv National Taras Shevchenko University, Ukraine; The University of São Paulo, São Paulo, Brazil)
• Ziemowit Popowicz (University of Wrocław, Poland)
• Anatolij Prykarpatski (AGH University of Science and Technology, Kraków, Poland)
• Natalia Prykarpatska
• Maria Przybylska (University of Zielona Góra, Poland)
• Cornelia Schiebold (Mid Sweden University, Sweden)
• Volodimir Simulik (The National Academy of Sciences of Ukraine, Ukraine)
• Błażej Szablikowski (Adam Mickiewicz University, Poznań, Poland)
• Adam Szereszewski (University of Warsaw, Poland)
• Wojciech Szumiński (University of Zielona Góra, Poland)

### Coffee breaks take place in Room A 221-222

#### (the building of the Faculty of Physics and Applied Informatics UŁ, Pomorska Str. 149/153, Łódź)

 Friday, July 3, 2015 850-900 Opening ceremony Chairman: Sergey Leble Session 1, 900-1040 900-925 Maciej Błaszak (Adam Mickiewicz University, Poznań, Poland) Classical and quantum ODE's in Hamiltonian mechanics 925-950 Makar Plakhotnyk (Taras Shevchenko National University of Kyiv, Ukraine; The University of São Paulo, São Paulo, Brazil) On the topological conjugation of interval maps by a piecewise linear homeomorphism (pdf) presentation slides 950-1015 Ziemowit Domański (Center for Theoretical Physics of the Polish Academy of Sciences, Warsaw, Poland) Various quantizations and related pseudo-probabilistic distribution functions 1015-1040 Grzegorz Kwiatkowski (Gdańsk University of Technology, Poland) Semiclassical corrections to energy in explicitly finite domains 1040-1105 COFFEE BREAK Chairman: Maciej Błaszak Session 2, 1105-1220 1105-1130 Krzysztof Marciniak (Linköping University, Sweden) Minimal separable quantizations of Stäckel systems (pdf) presentation slides 1130-1155 Marek Kuś (Center for Theoretical Physics of the Polish Academy of Sciences, Warsaw, Poland) Quantum SU(3) systems: integrability, quantum chaos, classical limit(s), and experimental realizations presentation slides 1155-1220 Piotr P. Goldstein (National Centre for Nuclear Research, Warsaw, Poland) On integrability of the one-dimensional Vlasov equation (pdf) presentation slides 1220-1245 COFFEE BREAK Chairman: Maciej Dunajski Session 3, 1245-1400 1245-1310 Andrzej J. Maciejewski (University of Zielona Góra, Poland) Non-integrability of the optimal control problem for n-level quantum systems 1310-1335 Wojciech Szumiński (University of Zielona Góra, Poland) Non-integrability of restricted double pendula presentation slides 1335-1400 Yves Brihaye (University of Mons, Belgium) Hairy spinning black holes presentation slides 1400-1530 LUNCH TIME Chairman: Andrzej J. Maciejewski Session 4, 1530-1710 1530-1555 Adam Doliwa (University of Warmia and Mazury in Olsztyn, Poland) Yang-Baxter maps with non-commuting variables presentation slides 1555-1620 Tomasz Brzeziński (Swansea University, United Kingdom) Rota-Baxter systems and Yang-Baxter pairs: Four equations in search of a model presentation slides 1620-1645 Błażej Szablikowski (Adam Mickiewicz University, Poznań, Poland) From Rota-Baxter identity to integrable systems (pdf) 1645-1710 Cornelia Schiebold (Mid Sweden University, Sweden) Banach space geometry and construction of solutions by limiting processes 1710-1730 COFFEE BREAK Chairman: Maria Przybylska Session 5, 1730-1910 1730-1755 Marek Czachor (Gdańsk University of Technology, Poland) Relativity of arithmetics as a fundamental symmetry of physics 1755-1820 Michal Marvan (Silesian University in Opava, Czech Republic) An integrable class of Chebyshev nets 1820-1845 Maciej Nieszporski (University of Warsaw, Poland) In the zoo of discrete integrable systems 1930-2300 CONFERENCE DINNER additional information Saturday, July 4, 2015 Chairman: Marek Kuś Session 6, 900-1040 900-925 Maciej Dunajski (University of Cambridge, United Kingdom) Quartics, sextics, and an integrable ODE presentation slides 925-950 Maria Przybylska (University of Zielona Góra, Poland) Dynamics of a relativistic charge in classical Penning trap and Penning trap with inclined magnetic field 950-1015 Jan L. Cieśliński (University of Bialystok, Poland) Exact discretizations and locally exact numerical schemes (pdf) 1015-1040 Andriy Panasyuk (Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine; University of Warmia and Mazury in Olsztyn, Poland) Veronese webs and Hirota type equations 1040-1105 COFFEE BREAK Chairman: Ziemowit Popowicz Session 7, 1105-1245 1105-1130 Sergey Leble (Gdańsk University of Technology, Poland) Generalized Burgers equations and Miura Map in nonabelian rings as integrable systems (pdf) presentation slides 1130-1155 Volodimir Simulik (The National Academy of Sciences of Ukraine, Ukraine) Relativistic description of the spin S=3/2 particle-antiparticle doublet interacting with electromagnetic field 1155-1220 Wojciech Kryński (University of Warsaw, Poland) Families of integrable distributions and differential equations 1220-1400 LUNCH TIME Chairman: Anatolij Prykarpatski Session 8, 1400-1515 1400-1425 Ziemowit Popowicz (University of Wrocław, Poland) Double extended peakon equations presentation slides 1425-1450 Adam Szereszewski (University of Warsaw, Poland) 4-dimensional binary metrics 1450-1515 Jan Dereziński (University of Warsaw, Poland) Homogeneous Schrödinger operators on half-line presentation slides 1515-1535 COFFEE BREAK Chairman: Adam Doliwa Session 9, 1535-1650 1535-1600 Maciej Kuna (Gdańsk University of Technology, Poland) Construction of exact solutions of nonlinear and time dependent von Neumann equation by Darboux transformation 1600-1625 Anatolij Prykarpatski (AGH University of Science and Technology, Kraków, Poland) The de Rham-Hodge theory and integrable dynamical systems 1625-1650 A. Michał Grundland (Centre de Recherches Mathématiques, Université de Montréal, Canada) Soliton surfaces in the generalized symmetry approach presentation slides 1650-1800 FAREWELL COFFEE, DISCUSSION, OPEN PROBLEMS
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## Classical and quantum ODE's in Hamiltonian mechanics

We describe in a common language classical and quantum equations of motion and their solutions, being respective classical and quantum flows. Then, we discuss what kind of “physics” is related to such constructed flows.

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## Homogeneous Schrödinger operators on half-line

Homogeneous Schrödinger 1-dimensional operators are exactly solvable in terms of Bessel-type functions. They form a holomorphic family of closed operators. In spite of apparent simplicity, their properties are quite sophisticated. Some questions about them are still open.

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## Various quantizations and related pseudo-probabilistic distribution functions

We present a discussion of a phase space approach to quantization introducing various quantizations of a given classical system within this scheme. The theory will be illustrated on an example of quantization of electromagnetic field. Moreover, we present a relation of various quantizations with pseudo-probabilistic distribution functions. In particular, we discuss Wigner, Husimi and Glauber-Sudarshan distribution functions, extensively used in quantum optic.

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## Semiclassical corrections to energy in explicitly finite domains

Stationary solutions of one-dimensional Sine-Gordon and $\phi^4$ systems are embedded in a multidimensional theory with explicitly finite domain in the added dimensions. Semiclassical corrections to energy are calculated for those solutions with emphasis on the impact of specific shape and scale of the domain on the results.

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## Quantum SU(3) systems: integrability, quantum chaos, classical limit(s), and experimental realizations

Quantum systems with the $SU(3)$ symmetry, having their origin in nuclear physics, were a fruitful playground for quantum chaos investigations, in particular due to they reach possible behavior in the classical limit. Even for the simplest models interesting questions concerning integrability in classical limit can be posed, especially in view of possible experimental realizations of such model with trapped ions.

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## Non-integrability of the optimal control problem for $n$-level quantum systems

We study the problem of optimal laser-induced population transfer in $n$-level quantum systems. This problem can be represented as a sub-Riemannian problem on $SO(n)$, and it is known that for $n=2$ and $n=3$ the Hamiltonian system associated with Pontryagin maximum principle (PMP) is integrable. We will show that this changes completely for $n \ge 4$ Namely, the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir function. To this aim, we will use the differential Galois framework to the integrability analysis.

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## Non-integrability of restricted double pendula

We consider two special types of double pendula, with the masses restricted to various surfaces. In order to get quick insight into the dynamics of considered systems the Poincare cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggest that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system’s variational equations along a particular non-equilibrium solution.

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## Hairy spinning black holes

Perhaps one of the oldest and most known results about general relativity is the “No Hair Theorem”. Nevertheless, in the last two decades, several families of hairy black holes have been constructed. The first of them, in the 1990's, was supported by non-abelian Yang-Mills fields. Recently the Einstein equations coupled to scalar fields received a lot of attention and new types of black holes involving scalar fields were constructed as well. These can exist only if a sufficiently large angular momentum supports them. We will start with an exact result. Namely, in the background of a DeSitter space-time the Klein-Gordon operator can be solved algebraically. Proceeding in different steps, we will review how several extentions of this result provide convincing arguments for the existence of hairy spinning black holes approaching asymptotically the Minkowski or DeSitter space-time. The solutions are finally constructed numerically.

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## Yang-Baxter maps with non-commuting variables

A family of geometrically relevant solutions of the functional Yang-Baxter equation involving arbitrary number of totally non-commuting variables will be presented. They have been obtained from the non-commutative Hirota system subject to periodic reduction. Their commutative counterpart are maps studied in relation to geometric crystals.

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## Rota-Baxter systems and Yang-Baxter pairs: Four equations in search of a model

Rota-Baxter operators were introduced by Glen Baxter as a means for solving an analytic problem in probability. Later on Rota realised the importance of these operators in combinatorics. Through the connection with algebras of rooted trees (formalised by Loday as dendriform algebras) Rota-Baxter operators appeared in Connes-Kreimer approach to renormalisation. Aguiar constructed solutions of the Rota-Baxter equation from solutions to the (associative) classical Yang-Baxter equation. Based on these he proposed an infinitesimal version of quasitriangular Hopf algebras. The aim of this talk is to argue that Rota-Baxter operators should be understood as specifications of pairs of operators, termed Rota-Baxter systems that satisfy two intertwined equations; such pairs of operators will still lead to examples of dendriform algebras (rooted trees), hence might find applications in renormalisation theory or combinatorics. Solutions of Rota-Baxter systems can be associated to solutions of pairs of equations similar to the (associative) classical Yang-Baxter equations. Whether these equations can be quantised and applied to aid finding solutions of integrable systems are open questions. The algebraic system corresponding to Yang-Baxter pairs turns out to be a natural generalisation of Aguiar's infinitesimal Hopf algebras in which the coproduct is no longer required to be a derivation but to be a covariant derivation instead.

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## Banach space geometry and construction of solutions by limiting processes

In the 90ies the study of countable superpositions of solitons was initiated by Gesztesy and al. Their approach was via a step by step analysis of the ISM. In our talk we will explain how the original method can be replaced by Banach space geometry. We will start from operator formulas for integrable systems and recall then the necessary background from functional analysis. In the main part we will combine these ingredients in order to extend the initial results to superposition of more complicated solutions (formations, weakly bound groups) and of matrix solutions.

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## Relativity of arithmetics as a fundamental symmetry of physics

Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters “plus” or “times” one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetics, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetics turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed.

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## An integrable class of Chebyshev nets

We study surfaces equipped with a Chebyshev net such that the Gauss curvature $K$ and the net curvature $G$ satisfy a linear condition $\alpha K + \beta G + \gamma=0$ , where $\alpha, \beta, \gamma$ are constants. These surfaces form an integrable class. We point out some of its noteworthy peculiarities.

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## In the zoo of discrete integrable systems

A link beetwen involutive rational maps and difference integrable systems allows us to unify difference integrable equations and correspondences. I will present the menagerie of integrable systems that revealed while we were investigating the link.

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## Metric properties of scators

Scators are hypercomplex variables with non-distributive multiplication. This causes complications at every possible level, although there can be proposed a framework in which metric properties and differentiation develops naturally in direct manner. At first, we consider mentioned framework simplifying algebraic calculations based on embedding scators in higher-dimensional space. Then we give systematic description of appearance of zero divisors, with them turning to be responsible for isommetries of considered space. Finally, we turn our attention to metric properties of scator space, especially emphasizing similarities with special relativity, and interpretation of these objects.

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## Quartics, sextics, and an integrable ODE

I will present a solution of an outstanding open problem of Sylvester concerning binary sextics. This is based on a recent joint work with Roger Penrose. As an application I will construct a conformal structure on a solution space to the unique 7th order integrable ODE with sub-maximal group of contact symmetries - this part is joint work with Vladimir Sokolov.

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## Dynamics of a relativistic charge in classical Penning trap and Penning trap with inclined magnetic field

We are interested in the dynamics of a classical charge within a processing chamber of two types of Penning traps: standard one with magnetic field exactly aligned with axis of symmetry of electrostatic quadruple potential and its generalization with magnetic field inclined under arbitrary angle to this axis. Relativistic Lagrangian and Hamiltonian dynamics without any approximations is analysed. For standard Penning trap, thanks to the symmetry, reduction to a Hamiltonian system with two degrees of freedom is possible. It is shown that reduced system is non-integrable and some qualitative analysis of dynamics is presented. In the case when magnetic field is inclined also non-integrability is proved. Stability analysis shows presence of some critical inclination angle. If the inclination angle exceeds this critical value, the magnetron radius as well as the axial amplitude increase infnitely and a charge is lost from the trap. The resonant curves are found as well as normal forms for certain domains of parameters are presented.

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## New hierarchies of (3+1)-dimensional integrable systems related to contact geometry

We construct infinite hierarchies of integrable (3+1)-dimensional dispersionless systems with Lax pairs written in terms of contact vector fields extending earlier results from arXiv:1401.2122. This is joint work with Maciej Blaszak.

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## Relativistic description of the spin S=3/2 particle-antiparticle doublet interacting with electromagnetic field

Relativistic spin 3/2 particle-antiparticle doublet is considered on the three levels:
(i) relativistic quantum mechanics, (ii) canonical (Foldy-Wouthuysen type) field theory, (iii) locally covariant field theory. The operator links between these three levels are found. Integrable system of interacting fermionic and electromagnetic (in the terms of potentials) fields is constructed. The difference between given model and the Rarita-Schwinger, or Pauli-Fierz, model is demonstrated. The starting points of the model can be found in V.M. Simulik arXiv 1409.2766v2 [quant-ph, hep-th] 19 Sep 2014, 67 p.

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## Families of integrable distributions and differential equations

It is well known that integrable distributions can define classes of integrable systems via the corresponding Lax pairs. In the present talk we consider special families of integrable distributions underlying various geometric structures e.g. Einstein-Weyl geometries, Hermitian structures or para-complex structures. We present interactions between the geometry and the related differential equations. In particular we present how one can construct solutions to the Einstein-Weyl equations from solutions to the Hirota equations and how to generalise the second heavenly Plebański equation to higher dimensions.

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## Double extended peakon equations

The Hamiltonian structure for the supersymmetric $N=2$ Novikov equation is presented. The bosonic sector give us two-component generalization of the cubic peakon equation. The double extended: two component and two peakon Novikov equations are defined. The Bi-Hamiltonian structure for this extended system is constructed.

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## 4-dimensional binary metrics

Binary metrics appear naturally in the theory of R-separability of Schrödinger equation defined on pseudo-Riemannian space equipped with orthogonal coordinates. I shall present a complete classification of conformally flat 4-dimensional binary metrics and a special class of Ricci flat binary metrics. The conformal symmetries will be also described.

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## Construction of exact solutions of nonlinear and time dependent von Neumann equation by Darboux transformation

A new strategy, using Darboux transformations, of finding self-switching solutions of $i\dot{p}=[H,f(p)]$ is introduced. Unlike the previous ones, working for any $f$ but time-independent, and for Hamiltonians whose spectrum contains at least three equally spaced eigenvalues, the strategy does not impose any restriction on the discrete part of spectrum of $H$ , and $f$ can be time dependent.

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## The de Rham-Hodge theory and integrable dynamical systems

The differential-geometric properties of generalized de Rham-Hodge complexes naturally related with integrable multi-dimensional differential systems of M. Gromov type are analyzed. The geometric structure of Chern type characteristic classes are studied, special differential invariants of the Chern type are constructed. The integrability of multi-dimensional nonlinear differential systems on Riemannian manifolds is discussed.

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## Soliton surfaces in the generalized symmetry approach

In this talk I will discuss some features of generalized symmetries of integrable systems, in order to construct the Focas-Gelfand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. I will establish the sufficient conditions for the applicability of this formula. Further I will provide a criterion for the selection of generalized symmetries suitable for their use in the Focas-Gelfand immersion formula. Finally, I will include some examples of their application.

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## Some random mathematics

$\sum_{i=1}^n\frac{f(n)}{n}$