Q: Information: - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in "extremal" functions that make the functional attain a maximum or minimum value or "stationary" functions those where the rate of change of the functional is zero. - In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign. Namely, for a positive, for a negative (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of 3 is also 3. The absolute value of a number may be thought of as its distance from zero. - In mathematical analysis Fubini's theorem, introduced by , is a result that gives conditions under which it is possible to compute a double integral using iterated integrals. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. As a consequence it allows the order of integration to be changed in iterated integrals. Fubini's theorem implies that the two repeated integrals of a function of two variables are equal if the function is integrable. Tonelli's theorem introduced by is similar but applies to functions that are non-negative rather than integrable. - David Hilbert (23 January 1862 14 February 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. - A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. - In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. - Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. - In mathematics, topology (from the Greek , "place", and , "study") is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. - Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. - Leonida Tonelli ( 19 April 1885 -- 12 March 1946 ) was an Italian mathematician , noted for creating Tonelli 's theorem , a variation of Fubini 's theorem , and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations . - A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems.<br> It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. 'leonida tonelli' is related to which object entity through the relation of 'field of work'? Choices: - calculus of variations - functional analysis - geometry - mathematical analysis - mathematician - mathematics - nature - topology
A: mathematical analysis

Q: Information: - The monarchic title of grand duke (feminine: grand duchess) ranked in order of precedence below emperor and king, and above that of sovereign prince and sovereign duke. It is or was used in some independent nations or states in Europe, particularly: - Cesare d'Este (1561 December 11, 1628) was Duke of Modena and Reggio from 1597 until his death. - Cosimo I de' Medici (12 June 1519 21 April 1574) was the second Duke of Florence from 1537 until 1569, when he became the first Grand Duke of Tuscany. - Modena (Etruscan: "Mutna" Modenese: "Modna") is a city and "comune" (municipality) on the south side of the Po Valley, in the Province of Modena in the Emilia-Romagna region of northern Italy. - Florence is the capital city of the Italian region of Tuscany and of the Metropolitan City of Florence. It is the most populous city in Tuscany, with 383,083 inhabitants, expanding to over 1,520,000 in the metropolitan area. - Camilla Martelli ( 30 May 1590) was first the lover and then the second wife of the Grand Duke of Tuscany, Cosimo I de' Medici. She was the mother of Virginia de' Medici, future Duchess of Modena. - Il Duca di Firenze, rendered in English as The Duke of Florence, was a title created in 1532 by Pope Clement VII. There were effectively only two dukes, Alessandro de'Medici and Cosimo de'Medici, the second duke being elevated to The Grand Duke of Tuscany, causing the Florentine title to become subordinate to the greater Tuscan title. - Virginia de ' Medici ( 29 May 1568 -- 15 January 1615 ) was an illegitimate daughter of Cosimo I de ' Medici , Grand Duke of Tuscany and Camilla Martelli . She was the Duchess of Modena and Reggio by marriage to Cesare d'Este , Duke of Modena . 'virginia de' medici' is related to which object entity through the relation of 'place of birth'? Choices: - este - florence - king - modena - of - pope - side - tuscany - virginia
A: florence