Question: What is the name of the equations which are a set of partial differential equations produced by Einstein relating to the description of gravity as a geometric property of space and time which went on to replace Isaac Newton's rotating spheres argument which attempts to define that true rotational motion can be defined by observing the tension in the string joining two identical spheres?

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Information-based complexity: Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems which arise in physical science, economics, engineering, and mathematical finance.  IBC has studied such continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration.  All these problems involve functions (typically multivariate) of a real or complex variable.  Since one can never obtain a closed-form solution of the problems of interest one has to settle for a numerical solution.  Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves "partial" information.  To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available.  In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the differential operator can only be sampled.  Furthermore, this partial information can be expensive to obtain.  Finally the information is often "contaminated" by noise.

Bucket argument: Isaac Newton's rotating bucket argument (also known as Newton's bucket) was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies.  It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space.  Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to "what"?"  General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

General relativity: General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.  General relativity is considered probably the most beautiful of all existing physical theories.  General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime.  In particular, the "" is directly related to the energy and momentum of whatever matter and radiation are present.  The relation is specified by the Einstein field equations, a system of partial differential equations.

Nicolai V. Krylov: Nicolai Vladimirovich Krylov (Russian: Никола́й Влади́мирович Крыло́в ; born 5 June 1941) is a Russian mathematician specializing in partial differential equations, particularly stochastic partial differential equations and diffusion processes.  Krylov studied at Lomonosov University, where he in 1966 under E. B. Dynkin attained a doctoral candidate title (similar to a PhD) and in 1973 a Russian doctoral degree (somewhat more prestigious than a PhD).  He taught from 1966 to 1990 at the Lomonosov University and is since 1990 a professor at the University of Minnesota.  At the beginning of his career (starting from 1963) he, in collaboration with Dynkin, worked on nonlinear stochastic control theory, making advances in the study of convex, nonlinear partial equations of 2nd order ("i.e." Bellman equations), which were examined with stochastic methods.  This led to the Evans-Krylov theory, for which he received with Lawrence C. Evans in 2004 the Leroy P. Steele Prize of the American Mathematical Society (for work done simultaneously and independently by both Krylov and Evans).  They proved the second order differentiability (Hölder continuity of the second derivative) of the solutions of convex, completely nonlinear, second order elliptical partial differential equations and thus the existence of "classical solutions" (Theorem of Evans-Krylov).  He was in 1978 at Helsinki and in 1986 at Berkeley an Invited Speaker for the ICM.  He received the Humboldt Research Award in 2001.  In 1993 he was elected a member of the American Academy of Arts and Sciences (1993).  He should not be confused with the mathematician Nikolay M. Krylov.

C0-semigroup: In mathematics, a "C"-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function.  Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces.  Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations.

Andrew Ronald Mitchell: Andrew Ronald Mitchell (22 June 1921 – 22 November 2007), popularly known as Ron Mitchell, was a British applied mathematician and numerical analyst.  He was a professor of mathematics at the University of St Andrews, Dundee, Scotland.  He was known for his contributions to the field of numerical analysis of partial differential equations in general and finite difference method and finite element method in particular.  Mitchell has authored several influential books on numerical solution of partial differential equations, including "The Finite Element Analysis in Partial Differential Equations" with Richard Wait and "The Finite Difference Method in Partial Differential Equations" with David F. Griffiths.

Symbol of a differential operator: In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable.  The symbol of a differential operator has broad applications to Fourier analysis.  In particular, in this connection it leads to the notion of a pseudo-differential operator.  The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation.  Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero.  In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation.  Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.

Stochastic partial differential equation: Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.

Bäcklund transform: In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions.  They are an important tool in soliton theory and integrable systems.  A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter.  It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.

Rotating spheres: Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres.  The basis of the argument is that all observers make two observations: the tension in the string joining the bodies (which is the same for all observers) and the rate of rotation of the spheres (which is different for observers with differing rates of rotation).  Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation.  For all other observers a "correction" is required (a centrifugal force) that accounts for the tension calculated being different from the one expected using the observed rate of rotation.  It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space.  Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to "what"?"  General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

Supporting paragraphs: Rotating spheres General relativity
Answer: field equations