Exam 1970

Exam questions and solutions in LaTex

Exam 1969 pdf

Exam questions and solutions in PDF

Exam 1970 pdf

Exam questions and solutions in PDF

Exam 1969

Exam questions and solutions in LaTex

Numerical solution of some nonlocal, nonlinear dispersive wave equations

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory.

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

In this paper, we classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3. (C) 2008 Elsevier Ltd. All rights reserved.

The Bullough-Dodd model coupled to matter fields

The Bullough-Dodd model is an important two-dimensional integrable field theory which finds applications in physics and geometry. We consider a conformally invariant extension of it, and study its integrability properties using a zero curvature condition based on the twisted Kac-Moody algebra A(2)((2)). The one- and two-soliton solutions as well as the breathers are constructed explicitly. We also consider integrable extensions of the Bullough-Dodd model by the introduction of spinor (matter) fields. The resulting theories are conformally invariant and present local internal symmetries. All the one-soliton solutions, for two examples of those models, are constructed using a hybrid of the dressing and Hirota methods. One model is of particular interest because it presents a confinement mechanism for a given conserved charge inside the solitons. (C) 2008 Elsevier B.V. All rights reserved.

Phase diagram and spectral properties of a new exactly integrable spin-1 quantum chain

The spectral properties and phase diagram of the exactly integrable spin-1 quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated with an unknown R-matrix whose dependence on the spectral parameters is not of a different form. The associated Bethe ansatz equations that fix the eigenspectra are distinct from those associated with other known integrable spin models. The model has a free parameter t(p). We show that at the special point t(p) = 1, the model acquires an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q = exp(i2 pi/3) and in the presence of an external magnetic field. Our analysis is carried out either by solving the associated Bethe ansatz equations or by direct diagonalization of the quantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by the c = 1 conformal field theory separated from a massive phase by first-order phase transitions.

On the consistency of arbitrary money-demand functions with the Sidrauski and the shopping-time models

This paper investigates which properties money-demand functions have to satisfy to be consistent with multidimensional extensions of Lucasí(2000) versions of the Sidrauski (1967) and the shopping-time models. We also investigate how such classes of models relate to each other regarding the rationalization of money demands. We conclude that money demand functions rationalizable by the shoppingtime model are always rationalizable by the Sidrauski model, but that the converse is not true. The log-log money demand with an interest-rate elasticity greater than or equal to one and the semi-log money demand are counterexamples.

Divisia indexes, money and welfare

We suggest the use of a particular Divisia index for measuring welfare losses due to interest rate wedges and in‡ation. Compared to the existing options in the literature: i) when the demands for the monetary assets are known, closed-form solutions for the welfare measures can be obtained at a relatively lower algebraic cost; ii) less demanding integrability conditions allow for the recovery of welfare measures from a larger class of demand systems and; iii) when the demand speci…cations are not known, using an index number entitles the researcher to rank di¤erent vectors of opportunity costs directly from market observations. We use two examples to illustrate the method.

Assorted topics in monetary economics

Esta tese é uma coleção de quatro artigos em economia monetária escritos sob a supervisão do Professor Rubens Penha Cysne. O primeiro desses artigos calcula o viés presente em medidas do custo de bem-estar da inflação devido a não se levar em conta o potencial substitutivo de moedas que rendem juros, como depósitos bancários.[1] O segundo se concentra na questão teórica de se comparar os escopos dos tradicionais modelos money-in-the-utility-function e shopping-time através do estudo das propriedades das curvas de demanda que eles geram.[2] O terceiro desses trabalhos revisita um artigo clássico de Stanley Fischer sobre a correlação entre a taxa de crescimento da oferta monetária e a taxa de acumulação de capital no caminho de transição.[3] Finalmente, o quarto diz respeito à posição relativa de cada uma de seis medidas do custo de bem-estar da inflação (uma das quais é nova) em relação às outras cinco, e uma estimativa do erro relativo máximo em que o pesquisador pode incorrer devido a sua escolha de empregar uma dessas medidas qualquer vis-à-vis as outras.[4] This thesis collects four papers on monetary economics written under the supervision of Professor Rubens Penha Cysne. The first of these papers assesses the bias occuring in welfare-cost-of-inflation measures due to failing to take into consideration the substitution potential of interest-bearing monies such as bank deposits.[1] The second one tackles the theoretical issue of comparing the generality of the money-in-the-utility-function- and the shopping-time models by studying the properties of the demand curves they generate.[2] The third of these works revisits a classic paper by Stanley Fischer on the correlation between the growth rate of money supply and the rate of capital accumulation on the transition path.[3] Finally, the fourth one concerns the relative standing of each one of six measures of the welfare cost of inflation (one of which is new) with respect to the other five, and an estimate of the maximum relative error one can incur by choosing to employ a particular welfare measure in place of the others.[4] [1] Cysne, R.P., Turchick, D., 2010. Welfare costs of inflation when interest-bearing deposits are disregarded: A calculation of the bias. Journal of Economic Dynamics and Control 34, 1015-1030. [2] Cysne, R.P., Turchick, D., 2009. On the integrability of money-demand functions by the Sidrauski and the shopping-time models. Journal of Banking & Finance 33, 1555-1562. [3] Cysne, R.P., Turchick, D., 2010. Money supply and capital accumulation on the transition path revisited. Journal of Money, Credit and Banking 42, 1173-1184. [4] Cysne, R.P., Turchick, D., 2011. An ordering of measures of the welfare cost of inflation in economies with interest-bearing deposits. Macroeconomic Dynamics, forthcoming.

GLOBAL DYNAMICS of THE LORENZ SYSTEM WITH INVARIANT ALGEBRAIC SURFACES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

GLOBAL DYNAMICS IN THE POINCARE BALL of THE CHEN SYSTEM HAVING INVARIANT ALGEBRAIC SURFACES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

We associate to an arbitrary Z-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer-Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.

Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation

In this Letter we investigate Lie symmetries of a (2 + 1)-dimensional integrable generalization of the Camassa-Holm (CH) equation. Through the similarity reductions we obtain four different (1 + 1)-dimensional systems of partial differential equations in which one of them turns out to be a (1 + 1)-dimensional CH equation. We establish their integrability by providing the Lax pair for all of them. Further, we present a brief analysis for some types of particular solutions which include the cuspon, peakon and soliton solutions for the two-dimensional generalization of the CH equation. (C) 2000 Published by Elsevier B.V. B.V.