81 research outputs found
Cálculo quântico simétrico
Doutoramento em Matemática e AplicaçõesGeneralizamos o cálculo Hahn variacional para problemas do cálculo das
variações que envolvem derivadas de ordem superior. Estudamos o cálculo
quântico simétrico, nomeadamente o cálculo quântico alpha,beta-simétrico,
q-simétrico e Hahn-simétrico. Introduzimos o cálculo quântico simétrico
variacional e deduzimos equações do tipo Euler-Lagrange para o cálculo
q-simétrico e Hahn simétrico. Definimos a derivada simétrica em escalas
temporais e deduzimos algumas das suas propriedades. Finalmente,
introduzimos e estudamos o integral diamond que generaliza o integral
diamond-alpha das escalas temporais.We generalize the Hahn variational calculus by studying problems of the
calculus of variations with higher-order derivatives. The symmetric quantum
calculus is studied, namely the alpha,beta-symmetric, the q-symmetric, and the
Hahn symmetric quantum calculus. We introduce the symmetric quantum
variational calculus and an Euler-Lagrange type equation for the q-symmetric
and Hahn's symmetric quantum calculus is proved. We define a symmetric
derivative on time scales and derive some of its properties. Finally, we
introduce and study the diamond integral, which is a refined version of the
diamond-alpha integral on time scales
Integral transforms of the Minkowski question mark function
The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function zeta_M(s), which is an entire function; the characteristic function m(t), which is an entire function as well. Each of them is a unique object, and is characterized by regularity properties and a functional equation, which reformulates in its own terms the functional equation for F(x). We study the interrelations among these three objects and F(x). It appears that the theory is completely parallel to the one for Maass wave forms for PSL_2(Z). One of the main purposes of this thesis is to clarify the nature of moments of the Minkowski question mark function
Bootstrap and Higher-Order Expansion Validity When Instruments May Be Weak
It is well-known that size-adjustments based on Edgeworth expansions for the t-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. This paper shows, however, that the lack of Edgeworth expansions and bootstrap validity are not tied to the weak instrument framework, but instead depends on which test statistic is examined. In particular, Edgeworth expansions are valid for the score and conditional likelihood ratio approaches, even when the instruments are uncorrelated with the endogenous explanatory variable. Furthermore, there is a belief that the bootstrap method fails when instruments are weak, since it replaces parameters with inconsistent estimators. Contrary to this notion, we provide a theoretical proof that guarantees the validity of the bootstrap for the score test, as well as the validity of the conditional bootstrap for many conditional tests. Monte Carlo simulations show that the bootstrap actually decreases size distortions in both cases.
Bootstrap and Higher-Order Expansion Validity When Instruments May Be Weak
It is well-known that size-adjustments based on Edgeworth expansions for the t-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. This paper shows, however, that the lack of Edgeworth expansions and bootstrap validity are not tied to the weak instrument framework, but instead depends on which test statistic is examined. In particular, Edgeworth expansions are valid for the score and conditional likelihood ratio approaches, even when the instruments are uncorrelated with the endogenous explanatory variable. Furthermore, there is a belief that the bootstrap method fails when instruments are weak, since it replaces parameters with inconsistent estimators. Contrary to this notion, we provide a theoretical proof that guarantees the validity of the bootstrap for the score test, as well as the validity of the conditional bootstrap for many conditional tests. Monte Carlo simulations show that the bootstrap actually decreases size distortions in both cases.
Integral transforms of the Minkowski question mark function
The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function zeta_M(s), which is an entire function; the characteristic function m(t), which is an entire function as well. Each of them is a unique object, and is characterized by regularity properties and a functional equation, which reformulates in its own terms the functional equation for F(x). We study the interrelations among these three objects and F(x). It appears that the theory is completely parallel to the one for Maass wave forms for PSL_2(Z). One of the main purposes of this thesis is to clarify the nature of moments of the Minkowski question mark function
Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice
Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe
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