DMATULS-PROC. No. 2, 2017

DMATULS PROCEEDINGS No 2, 2017

XXVI COMCA Mathematical Physics and Stochastic Analysis Session Universidad de Tarapacรก. August 2-4

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XXVI COMCA Mathematical Physics and Stochastic Analysis Session Universidad de Tarapacรก-Chile, August 2-4, 2017

Coordination: Marco Corgini Videla

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PREFACE The international Capricorn Mathematical Congress (COMCA) is an initiative annually organized by the Mathematical Departments of the chilean northern zone universities: Universidad de Tarapacรก (UTA)- Arica, Universidad Arturo Prat (UAP) )- Iquique, University of Antofagasta (UA) -Antofagasta, Universidad Catรณlica del Norte (UCN)- Antofagasta, Universidad de Atacama (UDA)- Copiapรณ, Universidad de La Serena (ULS)- La Serena.

The organization of the twenty-sixth version (XXVI COMCA)

Department of Mathematics of the Universidad de Tarapacรก. has corresponded to the

This issue of DMATULS-Proceedings contains the summaries of the works presented in the Mathematical Physics and Stochastics Analysis Session of this congress (10 contributions in applied mathematics, mathematical physics and stochastics analysis).

M. Corgini

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Contents

1.

Beyond the Bogoliubov c-substitutions in bosonic Hamiltonians. M. Corgini∗, R. Tabilo

REFERENCES 2.

Rie el Induction, Decomposable Operators and Quantization. F. Belmonte

REFERENCES 3.

Linear Response Theory: An AnalyticAlgebraic Approach. G. de Nittis

REFERENCES 4.

Black Holes with internal geometry at Lovelock Gravity. M. Estrada

REFERENCES 5.

Synchronization and propagation of chaos for mean eld networks of Hodgkin-Huxley neurons with noisy channels. H. Olivero

REFERENCES 6.

Biomathematical modeling for the description and prediction of melanoma growth: a hybrid approach for a better understanding of the A2AR role. P. Cumsille

REFERENCES 7.

The Borel transform in the study of pseudodi erential equations on a semi-axis. H. Prado

8.

Consistent estimators for the Skew Brownian motion. S. Torres

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10 13

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XXVI COMCA, Math.Phys. & Stoch. An. Session 9.

On Local linearization method for Stochastic Di erential Equations driven by fractional Brownian motion. H. Araya∗

REFERENCES 10.

v

Kernel Estimation of the regression function with a covariance dependence structure and random times. T. Roa∗, H. Araya, N. Bahamonde, S. Torres

REFERENCES

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Beyond the Bogoliubov c-substitutions in bosonic Hamiltonians. M. Corgini∗, R. Tabilo

In 1947, in the framework of the study of super uidity at low density, N. N. Bogoliubov proposed -under a wrong heuristic reasoning, for a large enough volume of the region enclosing the particle system, and assuming the emergence of Bose-Einstein condensation- the use of an energy operator obtained by substituting the operators of creation and annihilation of particles associated to zero mode in the Hamiltonian of the weakly interacting bose gas (WIBG) by complex numbers.

However

nothing was said about the type of mathematical or physical equivalence, if any, between the two models. Only in 1968 J. Ginibre rigorously demonstrated that the limit pressure of the WIBG model is asymptotically equivalent to the limit pressure obtained by using such an approximation [2]. Moreover, the presence of condensate proved to be an unnecessary condition for this to occur. The validity of such substitutions in the case of rather general models, including

U (1)

symmetry breaking

terms, was established, nally, in 2005 [3].

In this work, we

shall introduce an alternative strategy consisting in a partial replacing of zero mode operators of creation and annihilation of particles (only in those terms of the Hamiltonians representing external sources breaking the global gauge symmetry) by a suitable function of the number operator associated to such a mode. We shall determine a su cient condition for the thermodynamic equivalence between these models and those

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obtained by c-substitution and discuss bene ts and disadvantages of this approach for studying microscopic and also asymptotic behavior of some mean eld type models presenting continuous

U (1)

symmetry breaking in the thermodynamic limit

[4, 5].

REFERENCES [1] N.N. Bogolubov, On the theory of superfuidity. J. Phys. (USSR) 11, 23 (1947) [2] J. Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Commun. Math. Phys. 8, 26 (1968).

[3] E. H. Lieb, R. Seiringer, and J. Yngvason, Justi cation of c-Number Substitutions in Bosonic Hamiltonians Phys. Rev. Lett. 94, 1-4, 2005 [4] M. Corgini, R. Tabilo, New results on continuous U (1) symmetry breaking in Bose Gases To appear in Bose Gases. Beyond the In nitely Extended Systems.

Ed. DMATULS. (2017) [5] M. Corgini, D.P. Sankovich. Model of Interacting Spin One Bosons. U. Jour. of Phys. an Appl., 8, 42-47- 2014

∗ Marco Corgini (speaker), Rosanna Tabilo

Departamento de MatemĂĄticas Universidad de La Serena, Chile mcorgini@userena.cl, rtabilo@userena.cl

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Rie el Induction, Decomposable Operators and Quantization. F. Belmonte

On this talk we are going to review a process to induce representations of

C ∗ -algebras,

called Rie el Induction. We are

going to follow the steps of this process in a particular but important case; this will lead to a new characterization of decomposable operators. We will also explain why this process can be understood as the quantum counterpart of the so called Marsden-Weinstein reduction. This idea leads to some quantization problems, and it might be useful to study them.

REFERENCES [1]

Landsman,

N.P.,

Rie el Induction as a Generalized Quantum Marsden-

Weinstein Reduction,

J. Geom. Phys. 15 (1995), 285-319; Err. ibid. 17 (1995)

298.

[2]

Raeburn, I.,Williams, D.,

Morita Equivalence and Continuous-Trace C ∗ Algebras, Mathematical Surveys and Monographs, 60, American Mathematical Society (1998).

FabiĂĄn Belmonte Departamento de MatemĂĄticas Universidada CatĂłlica del Norte, Antofagasta, Chile fbelmonte@ucn.cl

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Linear Response Theory: An Analytic-Algebraic Approach. G. de Nittis

Linear response theory (LRT) is a tool with which one can study the response of systems that are driven out of equilibrium by external perturbations. In this talk I present a modern and systematic approach to LRT, developed in [1], that combines analytic and algebraic ideas.

The theory is robust

and provides a tool to implement LRT for a wide array of systems like periodic and random systems in the discrete and the continuum.

The mathematical framework of the theory

is outlined rstly: the relevant von Neumann algebras, noncommutative

Lp -

and Sobolev spaces are introduced; the no-

tion of isospectral perturbations and the associated dynamics and commutators are studied. Their construction is then made explicit for various physical systems (quantum systems, classical waves). The nal part is dedicated to a presentation of the proofs of the Kubo and Kubo-Streda formulas.

Joint work with: Max Lein, AIMR, Tohoku University, Sendai-Japan, e-mail: maximilian.lein.d2@tohoku.ac.jp

Acknowledgements: Funded by the grant Iniciaciรณn en Investigaciรณn 2015 FONDECYT No 11150143

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REFERENCES [1]

Linear Response Theory: An Analytic-Algebraic Approach, SpringerBriefs, Springer, 2017.

De Nittis, G.; Lein M.:

Giussepe de Nittis Departmento of Matemáticas & Instituto de Física Ponti cia Universidada Católica de Chile gidenittis@mat.uc.cl

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Black Holes with internal geometry at Lovelock Gravity. M. Estrada

During the last four decades several branches of physics have considered models in higher dimensions. This opened up many new possibilities for gravity and triggered an intense interest in the study of theories of gravity beyond Einstein's General Relativity.

Now, one fundamental aspect of General Relati-

vity, which almost singles it out in four dimensions, is that even though its Lagrangian,

√ L âˆź R g,

contains second or-

der derivatives its equations of motion are of second order. In higher dimensions, Lovelock theories , in spite of containing even higher power of the Riemann curvature, preserves equations of motion of second order.

In this work a new family of static higher dimensional black hole solutions in the presence of an anisotropic uids and Lovelock gravity is discussed. Although far from the horizon their geometries approach some previously known black holes solutions. Also we nd corrections for the temperature (A)dS and expresions for the charges, entropy and speci c heat. In particular we will show for

d

dimensional Lovelock gravity and

with presence of an anisotropic uid, two good examples that was studied in

4D

and in Einstein theory: Dymnikova black

hole [3] and Planck Energy Density black hole [2].

Note: Some of the results of this talk are at the reference [1].

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Joint work with: Rodrigo Aros, Departamento de Física, Universidad Andrés Bello, Santiago-Chile, e-mail: raros@udla.cl

REFERENCES [1]

Aros,Rodrigo; Diaz, Danilo; Estrada, Milko; Montecinos Alejandra, Black hole [gr-qc]] , (2016).

[2]

Euro

,

at Lovelock gravity with anisotropic uid,

Spallucci

;

Anais,

classical down to Planckian size,

Smailagic,

[arXiv:1401.6562v4

Regular black holes from semi-

Int. J. Mod. Phys. D 0, 1730013 , (2017) ,

arXiv:1701.04592v2 [hep-th] [3]

Dymnikova, Irina ; Korpusik , Michal

Sitter space

,

Regular black hole remnants in de

, Phys.Lett. B685 (2010) 12-18 .

Milko Estrada Departmento de Física, Universidad de Antofagasta, Antofagasta - Chile Instituto de Matemática, Física y Estadística, Universidad de las Américas Santiago, Chile mi.estrada@profesor.duoc.cl

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Synchronization and propagation of chaos for mean eld networks of Hodgkin-Huxley neurons with noisy channels. H. Olivero

5.

In this work we are interested in the behavior of a fully connected network of neurons either when the number of neurons or the time go to in nity. We assume that every neuron follow a stochastic version of the Hodgkin-Huxley dynamic and that the interactions between neurons, which take into account electrical and chemical synapses, are of mean eld type. Our main results are the propagation of chaos property for the system for any set of parameters, and a synchronization result, which is uniform in the size of the system, when the interaction is strong enough. Combining these two result we conclude that the nonlinear PDE describing the in nite network concentrates around the solution of the ODE describing a single neuron.

We complement our theoretical analysis with some numerical simulations.

Joint work with: Mireille Bossy, INRIA Sophia Antipolis, France, e-mail: Mireille.Bossy@inria.fr.

Joaquín Fontbona, Departamento de Ingeniería Matemática, Universidad de Chile. Santiago - Chile, e-mail: fontbona@dim.uchile.cl Acknowledgements: Partially funded by Proyecto MECESUP UCH0607, Proyecto NUCLEO MILENIO NC 130062 and the Center for Mathematical Modeling CMM

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REFERENCES [1]

BALADRON, J., FASOLI, D., FAUGERAS, O. and TOUBUL, J., Mean-

Field Description and Propagation of Chaos in Networks of Hodgkin Huxley and FitzHugh Nagumo Neurons,

The Journal of Mathematical Neuroscience (JMN).

Volumen 2, (2012).

[2]

BOSSY, M., FAUGERAS, O. and TALAY, D., Clari cation and Com-

plement to Mean-Field Description and Propagation of Chaos in Networks of Hodgkin Huxley and FitzHugh Nagumo Neurons ,

[3]

Neuroscience (JMN). Volumen 5, (2015).

HODGKIN, A. and HUXLEY, A., A

The Journal of Mathematical

quantitative description of membrane

current and its application to conduction and excitation in nerve,

Volumen 117, (1952).

[4] SZNITMAN, A.-S., Topics in propagation of chaos. In lities de Saint-Flour XIX 1989. Springer, 1991.

J Physiol.

Ecole d'été de probabi-

Héctor Olivero Departamento de Ingeniería Matemática Universidad de Chile Santiago - Chile holivero@dim.uchile.cl

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Biomathematical modeling for the description and prediction of melanoma growth: a hybrid approach for a better understanding of the A2AR role. P. Cumsille

In this work we study classical mathematical models of tumor growth (see [1, 2]) in order to describe and predict melanoma progression. Melanoma is the least common but the most deadly skin cancer, with a duplicated incidence rate in the last 30 years. The aggressive nature of melanoma is related to several abnormalities in growth factors, cytokines, and their receptors expression, which impact on angiogenesis process and tumor growth.

Extracellular adenosine is an immunomodu-

latory biomolecule produced by ATP hydrolysis. The purine nucleoside acts as a local vascular modulator stimulating angiogenesis. Adenosine exerts its e ects by enrolling a G-protein coupled receptors family referred to as Adenosine Receptors (ARs).

Some contradictory evidence shows adenosine path-

way is involved in tumor pathological angiogenesis. To investigate the exact role of speci c adenosine subtype receptor, Adenosine type 2 receptor (A2AR) in melanoma growth, we have performed mice melanoma autograft transplant of a tumor cell line (B16F10) into a host C56BJ mouse (control and A2AR silenced). For the next 16 days we have evaluated tumor morphology and growth.

This data have been used in

order to carry out parameters estimation of classical mathematical models for describing and forecasting tumor growth.

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The numerical methodology required in order to make parameter estimation of these tumor growth models, based upon ordinary di erential equations (ODE), consists in employing TrustRegion-Re ective Algorithm (see [3]), a non linear optimization method specially designed to solve parameters estimation problems by non linear least squares criterion. This algorithm

Š

is implemented in Matlab

through lsqnonlin solver (see [4]).

The input arguments we have used for lsqnonlin subroutine are the following: experimental tumor volumes evaluated by using bidimensional observations, namely the largest and the smallest diameter measured from the tumor of each animal during 16 days (for each group, wild type and A2AR KO); the solution of the tumor growth models computed at the same times, which have been obtained by evaluating the explicit solution of each model when available (by contrast, the solver ode45 of

Š

Matlab

was used in order to numerically solve it); an initial

guess of the optimal value of the parameter set which is, in general, unknown, but carefully exploring the relative squared error (the sum of the squares of relative errors between observed and computed data), by evaluating it as a function of parameter vector, we achieved to get an appropriate initial point for starting the algorithm; the weighted Jacobian matrices of the errors as a function of parameter vector (when an explicit solution is available); and lower and upper bounds for the parameter values (which were got by means of the same exploratory method mentioned before).

The growth models which have been studied are:

Gene-

ralized Logistic, Gompertz, Von Bertalan y, Power law and Exponential-Linear. It is worth noting that all these models

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have explicit solutions, except the variant of the ExponentialLinear model introduced in [5].

See [1, 2] for more details.

Once nonlinear least squares estimators were obtained (for each animal and each model), we have applied statistical methods based upon non linear least squares regression tools (see [6, Chapter 7]) in order to assess goodness of t, as well as, performance of prediction for the di erent models studied.

Our results suggest that the models that t the best to data are the Power law for wild type mice and the Generalized Logistic for A2AR KO mice, thus showing signi cant di erences between both groups.

These models allow us to predict the

entire tumor growth evolution, even before that the tumor is perceptible.

In conclusion, both groups come from di erent

populations, indicating that tumor growth is di erent. In particular, in A2AR KO mice population tumor growth is more aggressive in terms of its carrying capacity, as well as, of the ratio between this last one and the half time (time required in order to reach the half of the carrying capacity of the tumor).

This means that both quantities of the A2AR KO group are larger than those of the wild type group, taking longer obviously to reach the half of its carrying capacity. Finally, our results also show that tumor size and latency is increased in A2AR KO mice versus wild type, and also A2AR KO mice present a higher blood ow pattern in the area near the tumor. This evidence suggests an A2AR absence in mice stimulate melanoma development. Future work includes modeling the interplay of blood ow and melanoma-induced angiogenesis and growth with A2AR expression, in order to provide a more precise des-

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cription of the underlying mechanism of adenosine pathway.

Joint work with: Andrés Rodríguez1 , Department of Basic Sciences, Universidad del Bío-Bío, Chillán-Chile, e-mail: arodriguez@ubiobio.cl Acknowledgements: Partially funded by PIA-CONICYT grant PFBasal-01.

1 Partially

funded by UBB DIUBB 166709 3/R

REFERENCES [1]

S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. Ebos, L. Hlatky, and P. Hahnfeldt, Classical mathematical models for description and predic-

tion of experimental tumor growth,

[2]

PLoS Comput. Biol., 10 (2014), p. e1003800.

P. Cumsille, A. Coronel, C. Conca, C. Quiñinao, and C. Escudero,

Proposal of a hybrid approach for tumor progression and tumor-induced angiogenesis,

[3]

Theoretical Biology and Medical Modelling, 12 (2015), p. 13.

M. A. Branch, T. F. Coleman, and Y. Li, A subspace, interior, and con-

jugate gradient method for large-scale bound-constrained minimization problems, SIAM Journal on Scienti c Computing, 21 (1999), pp. 1 23.

[4] T. M. Inc., Nonlinear least squares (curve tting). https://www.mathworks.com/help/optim/nonlinear-least-squares-curve tting.html. [Online; accessed 2017].

[5] M. Simeoni, P. Magni, C. Cammia, G. De Nicolao, V. Croci, E. Pesenti, M. Germani, I. Poggesi, and M. Rocchetti, Predictive pharmacokineticpharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,

[6]

W. H. Greene,

Cancer Res., 64 (2004), pp. 1094 1101.

Econometric Analysis, Pearson, 2012.

Patricio Cumsille Department of Basic Sciences Universidad del Bío-Bío, Chillán - Chile Centre for Biotechnology and Bioengineering (CeBiB) Santiago - Chile pcumsille@ubiobio.cl

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The Borel transform in the study of pseudo-di erential equations on a semi-axis.

7.

H.

Prado

The aim of this talk is to show how the Borel transform allow us to study rigorously pseudo-di erential equations on a semiaxis, in particular di erential equations with in nitely many derivatives, sometimes also referred to as nonlocal di erential equations.

These classes of equations appear frequently

in branches of modern physics such as string theory, gravitation and cosmology.

We properly interpret and solve li-

near equations in this class with a special focus on a solution method based on the Borel transform. This method is a farreaching generalization of previous studies of these equations via Laplace and Fourier transforms.

Joint work with: M. Carlsson, Lund University, Lund, Sweden.

E.G. Reyes, USACH, Santiago-Chile Acknowledgements: Partially funded by Fondecyt grant # 1170571 Humberto Prado Departmento de Matemรกtica y Ciencia de la Computaciรณn Universidad de Santiago de Chile (USACH), Santiago - Chile humberto.prado@usach.cl

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Consistent estimators for the Skew Brownian motion. S. Torres

The Skew Brownian motion is of primary importance in modeling di usion in media with interfaces which arise in many domains ranging from population ecology to geophysics and nance.

We show that the maximum likelihood estimator

provides a consistent estimator of the parameter of a Skew Brownian motion observed at discrete times. The di culties are that this process is only null recurrent and has a singular distribution with respect to the one of the Brownian motion.

Finally, using the idea of the Expectation-Maximization algorithm, we show that the maximum likelihood estimator can be naturally interpreted as the expected number of positive excursions divided by the expected number of excursions.

Joint work with: Antoine Lejay and Ernesto Mordecki Acknowledgements: Partially supported by Fondecyt 1171335

Soledad Torres CIMFAV Centro de InvestigaciĂłn y Modelamiento de FenĂłmenos Aleatorios Facultad de IngenierĂ­a Universidad de ValparaĂ­so ValparaĂ­so, Chile soledad.torres@uv.cl

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On Local linearization method for Stochastic Di erential Equations driven by fractional Brownian motion. H. Araya∗

We propose a Local linearization type scheme as a numerical approximation for non-autonomous Stochastic Di erential Equations driven by fractional Brownian motion with Hurst parameter

H ∈ (1/2, 1).

Numerical examples are given to

demonstrate the performance of the method.

Joint work with: Jorge A LeĂłn1 , Automatic Control Department, CivestavIPN, Ciudad de MĂŠxico, MĂŠxico, e-mail: jleon@ctrl.cinvestav.mx; Soledad Torres2 , CIMFAV, Universidad de ValparaĂ­so, ValparaĂ­so, Chile, email: soledad.torres@uv.cl Acknowledgements: ∗ Partially supported by Beca CONICYT-PCHA/Doctorado Nacional/2016-21160371, ECOS C15E05, REDES 150038 and Mathamsud 16MATH03, 1 Partially supported by ECOS C15E05; Fondecyt 1130586, REDES 150038 and Mathamsud 16MATH03, 2 Partially supported by ECOS C15E05; Fondecyt 1130586, REDES 150038 and Mathamsud 16MATH03

REFERENCES [1]

R. Biscay, J. Jimenez, J.Riera and P. Valdes.. Local linearization method

for the numerical solution of stochastic di erential equations.

[2]

stitute of Statistical Mathematics, 48:631-644, 1996.

Y. Hu, Y. Liu and D. Nualart. Rate of convergence

Annals of the In-

and asymptotic error

distribution of Euler approximation schemes for fractional di usion.

The Annals

of Applied Probability, 26(2):1147-1207, 2016.

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Y. Mishura and G. Shevchenko. The rate of convergence for Euler approxi-

mations of solutions of stochastic di erential equations driven by fractional Brownian motion.

Stochastics An International Journal of Probability and Stochastic Processes, 80(5):489-511, 2008.

[4] T. H. Thao. On some classes of fractional stochastic dynamical systems, EastWest J. Math, 15(1): 54-69, 2013.

HĂŠctor Araya CIMFAV Centro de InvestigaciĂłn y Modelamiento de FenĂłmenos Aleatorios Facultad de IngenierĂ­a Universidad de ValparaĂ­so ValparaĂ­so, Chile arayahector8@gmail.com

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Kernel Estimation of the regression function with a covariance dependence structure and random times. T. Roa∗, H. Araya, N. Bahamonde, S. Torres

Z = (X, Y ) = {Z(t)}t∈R be a stationary continuoustime process taking values in R, where X and Y are fractional Let

brownian motion. By means of the corresponding discrete-time process

{X(ti ), Y (ti )}ni=1 ,

sampled at random instants

{ti },

a nonparametric kernel estimator of the regression function,

m(x) = E(Y |X = x),

is studied.

Under covariance depen-

dence structure, the Mean Integrated Square Error (MISE) is derived. We are going to compare our results by simulations obtained under a classical approach for evenly-spaced observations.

Joint work with: Héctor Araya y Soledad Torres, CIMFAV, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso-Chile; Natalia Bahamonde1 , Instituto de Estadística, Ponti cia Universidad Católica de Valparaíso, ValparaísoChile Acknowledgements: 1 Supported by project FONDECYT 11121531

REFERENCES [1]

Amblard,

Pierre-Olivier and Coeurjolly,

Jean-François and La-

, Basic properties of the multivariate Self-similar processes and their applications. 28, pag

vancier, Frédéric and Philippe, Anne

fractional Brownian motion,

63-84, (2013).

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Helson, Henry and Sarason, Donald, Past and future,

19 Mathematica Scan-

dinavica, 21, pag 5-16, (1967).

∗ Tania

Roa (speaker) CIMFAV Centro de Investigación y Modelamiento de Fenómenos Aleatorios Universidad de Valparaíso Facultad de Ingeniería Valparaíso, Chile tania.roa@postgrado.uv.cl

Héctor Araya CIMFAV Centro de Investigación y Modelamiento de Fenómenos Aleatorios Facultad de Ingeniería Universidad de Valparaíso Valparaíso, Chile hector.araya@postgrado.uv.cl

Natalia Bahamonde Instituto de Estadística, Ponti cia Universidad Católica de Valparaíso, Valparaíso-Chile natalaia.bahamonde@pucv.cl

Soledad Torres CIMFAV Centro de Investigación y Modelamiento de Fenómenos Aleatorios Facultad de Ingeniería Universidad de Valparaíso Valparaíso, Chile soledad.torres@uv.cl

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DMATULS PROCEEDINGS Departamento de Matemรกticas ULS Facultad de Ciencias Universidad de La Serena Cisternas 1200, La Serena, Chile http://www.dmatuls.cl edicionesdmatuls@userena.cl