In a series of papers, Lie symmetries and related concepts for wave equations of various spins have been studied, in particular on vacuum Petrov type D backgrounds, using xAct. There are quite a number of packages for finding such symmetries which have been applied in GR: some are mentioned elsewhere in this review. A full list can be obtained via [C56]. An early use of CA in GR was to analyse alternative classical gravity theories, e.
Pavelle describes finding conserved quantities for higher-order gravity Lagrangians see also Sect. A number of modern systems for example, Sheep allow the user to set the dimension, which could be more or less than 4, and may also go beyond Riemannian spacetimes of higher dimension by enabling calculations on the auxiliary fields arising in alternative theories of gravity alternative, that is, to general relativity or unified theories. Some of the systems enable Kaluza—Klein splitting or can handle quantum gravity in higher dimensions.
For examples of work in 3 i. Examples of use in higher-dimensional theories i. A Finsler space package is listed above see Sect. A number of investigations in supergravity, superstrings and supersymmetric field theories have been made using CA. There are also specialized packages for this application not listed earlier: e.
See also Demichev and Rodionov There are quite a number of approximation schemes of importance in GR: the post-Newtonian scheme for celestial mechanics and two-body problems including black hole mergers; expansions at asymptotic infinity and their use for gravitational radiation; weak-field approximations; Taylor series for the metric; the velocity-dominated approximation near the big bang; and perturbations of the metric in cosmology with implications for large scale structure. The Sheep power series module TPS, written by Matthew Piper, provided the basis for an implementation of the double series approximation Piper a , b , an expansion in two parameters describing characteristic mass and length scales originally introduced by Bonnor.
In general relativity and other GR research, the asymptotic behaviour of spacetime at large distances may be of physical importance e. The approximation is a weak field expansion in the magnitude of the energy-momentum. The CA calculation discovered an error in the hand calculation of McCrea CA has also been used to produce Taylor series expansions for the metric, an idea proposed by Penrose and described in Penrose and Rindler Perturbation theory is a natural area of application and its methods overlap with the expansions just discussed.
CA has been applied to studying pulsations and their stability from a very early stage. Expansion in terms of the amplitude parameter was made as high as 6-th order. A number of papers have made use of this module, for example to study the HI brightness temperature up to third order Umeh This was an example of a general method for such perturbations, which was also applied to vector-tensor theories.
CA programs can interact with numerical programs in three main ways; they can be used to replace a numerical program or part thereof, to analyse a numerical program, or to generate a numerical program. The first of these is valuable not only when the algebra program can give a complete rather than approximate solution, but also when it can give an exact answer to a part of a problem which is numerically ill-conditioned or where it can avoid repetition of a slower numerical calculation e.
Autin and Bengtsson I do not know any direct examples in GR, although analytic solutions have long been used to test numerical schemes see e. Algebraic analysis of numerical schemes can tackle various aspects. It can enable one to discover instabilities e. Exact solutions of problems to be tackled numerically can be important in testing codes. They showed the systems were symmetric hyperbolic and proposed a family of constraint-preserving boundary conditions that is applicable if the boundary is smooth with tangential shift. At the simplest level one can generate assignment statements for the target language.
A second level is program templates. The third is optimization of the resulting code as in the SCOPE package of Reduce by use of intermediate expressions and common sub-expression searches. Kranc see above is the principal current example known to me of such programs in GR. CA systems and packages, particularly the ones of component calculus type, have perhaps most frequently been used in gravity theory for exact solutions, not only for checking them, but also finding them, and giving unique characterizations.
The simplest and perhaps most common use of CA for exact solutions is to check correctness, not only for vacuum solutions but for non-vacua where the matter field equations and their properties come into play. To do so one might need facilities to implement coordinate or tetrad changes or for changing the sign conventions.
This use of CA in GR is very common but frequently not acknowledged. Recent texts on exact solutions have used CA in their preparatory work, e. One interesting development in this area is the advent of online databases of solutions.
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More recently Ian Anderson and Charles Torre private communication have checked all those solutions using DifferentialGeometry see Sect. Methods for finding and working with Killing, homothetic or other symmetries have been used in a number of papers. See also Sect. The CRACK package distributed with Reduce , which was written to handle over-determined systems of partial differential equations, can be used to investigate the existence of symmetries and various other problems giving similar systems of equations Grebot and Wolf ; Wolf , There are quite a number of other packages addressing similar problems, and able to find symmetries of differential equations and systems, but no recent comprehensive review of them that I know of.
Others used in examples in this review are Dimsym in Reduce and Exterior in Maple. The physical interpretation of solutions is very important and can also be aided by CA, see e. Properties of fields in spacetime have also been studied using CA. One can use CA to study curves within the metric, e.
Using invariants calculated by CA to locate and characterize singularities and horizons has been discussed quite often. Gibbons and Russell-Clark found a naked singularity, but no horizon, in the Tomimatsu-Sato solution. More recently, both scalar polynomial and Cartan invariants, as defined in e. These techniques also apply to manifolds of other dimensions or signatures.
GHP was used Vu and Carminati in a study of the shear-free conjecture. The conjecture, supported by the large range of more special sets of conditions for which it has been proved, is that a general shearfree perfect fluid will have either zero expansion or zero vorticity. I have found a couple more solutions by similar happy accidents: it appears this is especially easy to do when the particular field equations have linearity properties.
Two more serious systematic ways of finding new solutions have been used. Finding solutions with given symmetries provides another class of applications. CA in GR has been used extensively in finding invariant characterizations of metrics. More complete characterizations are used in resolving the equivalence problem. The equivalence problem is that of identifying regions of two geometries that are locally isometric but expressed in different coordinates.
Its resolution has been a driver for CA in GR from early on. We now have a well-defined procedure which for suitably smooth regions can be used to study the problem by computing a set of quantities uniquely characterizing the manifold locally and then comparing characterizations. However, because the comparison step of that procedure would require determination of the compatibility of sets of equations, it remains formally undecidable as a result of the no-go theorems on simplification Buchberger and Loos Further arguments for undecidability were given by Kreinovich Nevertheless, for practical examples the procedure can be completed.
Apart from these two programs I am not aware of any that systematically compute and record all the necessary quantities. The Classi version is formulated using the NP spinor formalism, and requires the computation of the Cartan invariants of each spacetime: these uniquely determine the spacetime, locally, and comparing values for apparently different solutions gives the way to resolve equivalence. Certain families of solutions present special difficulties for this process, notably the infinite set of solutions, with indefinitely many terms in the metrics, which can in principle be obtained by generating techniques.
The result implies that programs, such as those discussed above, that systematically calculate SPIs can be useful in this context as well. To discuss this conclusion and its implications fully would take us too far afield: see MacCallum for a review.
Many of the papers cited in that review and in its forthcoming extended version use CA to compute invariants.
The methods for the equivalence problem have had a number of consequences. These ideas can be carried over to alternative signatures Karlhede a , to other theories using connections and curvatures, i. To make enough use of each of the systems and packages listed above to enable one to make comprehensive comparisons would be a Herculean task.
Moreover it would inevitably be subject to the cautionary notes and possible biases stated in Sect. I have not attempted it, confining myself to checking what seemed to me the most important special purpose systems and packages for general purpose systems. My choice of examples, and my comments on systems and packages, are thus unlikely to be unbiased.
I shall be glad to receive, for inclusion in later versions of this Living Review, information on points misrepresented, systems overlooked, and other suggestions that might counteract any resulting imbalances. Reader, beware! If a new user of CA in GR seeks facilities suitable for his or her problem, a first place to look is in those fully-featured systems which offer both indicial and component calculation.
That those are also the most frequently referenced in Sect. Currently I would consider the following as being in that category: xAct in. If one does not need indicial tensor calculations, the more fully-featured component calculators, notably the physics and DifferentialGeometry packages of Maple see Sect. However, it may be that none of these is right for other users. In particular a simple and easy-to-learn system with more limited features, one of the many listed above, may be more suitable. I again emphasize that there is no best system ; see Sect. A second approach to choice is to look in Sect.
If so, the software used for that application may be the best choice. As well as the capabilities of systems and packages, choice may be influenced by cost, by available hardware, by operating systems and other software, by the effort involved in learning the system, or by prior familiarity with a suitable underlying CA system. The cost factor may incline a new user towards one of the free systems which already have substantial facilities for CA in GR such as Reduce, Sheep, or the newer Cadabra or Redberry, or to a system for which he or she is covered by an institutional site licence.
There may be some more ambitious researchers who want to write a new CA for GR package. But it could be that those researchers have ideas for structuring a package which really differ from what has been done before, or that they really need facilities not available in current CA for GR or minor extensions thereof. Given the large number of existing packages, checking that a facility is really unavailable may take some time! If I were to embark on such an effort myself, I would try to work either by adding to an existing package, preferably a free one, which has a reasonable size user community so that there is some resilience if bugs or other difficulties are encountered, or, if I needed to write a more extensive and independent package, to do so within an existing free general purpose system so that as wide a community as possible could use my work.
Axiom or Sage could be suitable, for example. I use the word system for programs that cover a wide area of mathematics, including calculus and algebra, and refer to more specialized programs, or specific application programs within the main systems, as packages. Wikipedia is not always accurate, as it is open to being edited by underinformed people, but in my experience it is pretty reliable in mathematical areas. This is far from an exhaustive list: apart from implementations in machine code or assembler for specific machines, other high-level languages such as ALGOL-W and Modula-2 have been used.
The recent Redberry see Sect. At the extreme, van Hulzen and Calmet remark that it has been the misfortune of many systems and facilities to be used only at one place, i. William Stein, the originator of Sage described in Sect. Within SageMath, which is not to be confused with the business and finance software Sage, though I shall refer to SageMath just as Sage hereafter. Stephen Wolfram was awarded the Richard D. As of February the websites for Atlas 2 were no longer reachable, but I have retained the information about Atlas 2 in case this is a temporary glitch.
Many people have over the last five decades contributed to my understanding of CA systems not only for GR applications. They include Sergei Abramov, Jan E. Ian Cohen, Robert M. Corless, James H. Scott and her students, James E. Apologies to any accidentally omitted.
I am especially grateful to the following for reading and commenting on the first draft of this review: Jan E. Skip to main content Skip to sections. Advertisement Hide. Download PDF. Living Reviews in Relativity December , Cite as. Computer algebra in gravity research. Open Access. First Online: 20 August In more detail, the advantages of computer algebra are as follows. Although this review covers gravity research rather than just general relativity, many applications are within that theory or closely-related generalizations. Amount of mathematical knowledge This affects the class of problems one can treat.
Programming style While all the general purpose CA systems described in Sect. Each of the choices has advantages and disadvantages. The data representation can affect efficiency and the class of optimally-treated problems, and can have an enormous effect on the speed and the possible features. For example, A dense representation provides fast manipulation for dense polynomials: here every possible power is assumed to be present, so only a list of coefficients is required.
One aspect of evaluation strategy is the handling of chains of assignments another is mentioned in point 8 below. One also has to beware of infinite loops if the system can make indefinitely many steps through the assignments, e. Another aspect is that one wants to avoid repeated evaluations of the same quantity, which can easily arise when dealing with recursive functions.
There are two main types of algebraic calculation in GR. One is the calculation of general expressions in indexed objects such as the equations defining the Levi-Civita connection and Riemann tensor in coordinate components for a spacetime in general relativity, i. Programs for gravity need some standard CA features, like Good simplification routines Good control of substitutions Differentiation A reasonable range of known mathematical special functions, and facilities to add others.
Depending on their intended uses, they may not need: Polynomial factorization Integration and solvers for differential equations Numerical and graphical features. A new module atensor was added to achieve parity with the commercial version of Macsyma Toth The Maxima manual provides descriptions of the four packages. They are: atensor For algebraic tensor manipulation in the sense of objects in, for example, a Clifford, Grassmannian or symplectic algebra.
Four packages act as a kernel for the rest. There are some key applications and a number of contributed modules which are supplied with the overall package. For an up-to-date list see [C38]. The standard Reduce distribution provides 5 compiled packages which can be or have been used in GR and a couple more are among the contributed packages circulated with Reduce.
Details can be found at [C6]. In brief the 5 are: atensor: An indicial tensor package Ilyin and Kryukov , : see Sect. Input and output for the differential forms in this package are close to textbook format and it is therefore very easy to use. It can deal with indexed objects and has many geometric objects and ideas built in. That said, I shall offer some guidance. The precise numbers depend somewhat on how the result is formatted. Acknowledgements Many people have over the last five decades contributed to my understanding of CA systems not only for GR applications.
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The construction and study of certain important algebras
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In: Sharif M ed Proceedings of the international conference on relativistic astrophysics, Lahore, February However, this kind of analysis is known for lacking precision. Furthermore, models are fundamentally only as good as their founding assumptions.
The use of economic models is important in order to further study and understand economic processes. Steps must be taken throughout the construction of the model to ensure that the data provided and analyzed is correct. Positive and normative economic thought are two specific branches of economic reasoning. Although they are associated with one another, positive and normative economic thought have different focuses when analyzing economic scenarios.
Positive economics is a branch of economics that focuses on the description and explanation of phenomena, as well as their casual relationships. It focuses primarily on facts and cause-and-effect behavioral relationships, including developing and testing economic theories. As a science, positive economics focuses on analyzing economic behavior. It avoids economic value judgments. For example, positive economic theory would describe how money supply growth impacts inflation, but it does not provide any guidance on what policy should be followed.
It gives an overview of an economic situation without providing any guidance for necessary actions to address the issue. Normative economics is a branch of economics that expresses value or normative judgments about economic fairness. It focuses on what the outcome of the economy or goals of public policy should be. Many normative judgments are conditional. They are given up if facts or knowledge of facts change.
In this instance, a change in values is seen as being purely scientific. Welfare economist Amartya Sen explained that basic normative judgments rely on knowledge of facts. It states facts, but also explains what should be done. Normative economics has subfields that provide further scientific study including social choice theory, cooperative game theory, and mechanism design. Positive economics does impact normative economics because it ranks economic policies or outcomes based on acceptability normative economics.
In other words, positive economics clearly states an economic issue and normative economics provides the value-based solution for the issue. Debt Increases : This graph shows the debt increases in the United States from Positive economics would provide a statement saying that the debt has increased. Normative economics would state what needs to be done in order to work towards resolving the issue of increasing debt. Skip to main content. Principles of Economics. Search for:. Economic Models Math Review Mathematical economics uses mathematical methods, such as algebra and calculus, to represent theories and analyze problems in economics.
Key Takeaways Key Points Using mathematics allows economists to form meaningful, testable propositions about complex subjects that would be hard to express informally. Key Terms quantitative : Of a measurement based on some number rather than on some quality. Assumptions Economists use assumptions in order to simplify economics processes so that they are easier to understand. Learning Objectives Assess the benefits and drawbacks of using simplifying assumptions in economics.
Key Takeaways Key Points Neo-classical economics employs three basic assumptions: people have rational preferences among outcomes that can be identified and associated with a value, individuals maximize utility and firms maximize profit, and people act independently on the basis of full and relevant information. Key Terms assumption : The act of taking for granted, or supposing a thing without proof; a supposition; an unwarrantable claim. Hypotheses and Tests Economics, as a science, follows the scientific method in order to study data, observe patterns, and predict results of stimuli.
Learning Objectives Apply the steps of the scientific method to economic questions. Key Takeaways Key Points The scientific method involves identifying a problem, gathering data, forming a hypothesis, testing the hypothesis, and analyzing the results. A hypothesis is simply a prediction. In economics, extensive testing and observation is required because the outcome must be obtained more than once in order to be valid. Key Terms hypothesis : An assumption taken to be true for the purpose of argument or investigation. Economic Models A model is simply a framework that is designed to show complex economic processes.
The compositum can be used to construct the biggest subfield of F satisfying a certain property, for example the biggest subfield of F , which is, in the language introduced below, algebraic over E. It satisfies the formula . Extensions whose degree is finite are referred to as finite extensions. For example, the imaginary unit i in C is algebraic over R , and even over Q , since it satisfies the equation.
A field extension in which every element of F is algebraic over E is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula. The subfield E x generated by an element x , as above, is an algebraic extension of E if and only if x is an algebraic element. That is to say, if x is algebraic, all other elements of E x are necessarily algebraic as well.
If this degree is n , then the elements of E x have the form. The above-mentioned field of rational fractions E X , where X is an indeterminate , is not an algebraic extension of E since there is no polynomial equation with coefficients in E whose zero is X. Elements, such as X , which are not algebraic are called transcendental. Informally speaking, the indeterminate X and its powers do not interact with elements of E. A similar construction can be carried out with a set of indeterminates, instead of just one. This isomorphism is obtained by substituting x to X in rational fractions.
A subset S of a field F is a transcendence basis if it is algebraically independent don't satisfy any polynomial relations over E and if F is an algebraic extension of E S. A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation. The rational and the real numbers are not algebraically closed since the equation. A field containing F is called an algebraic closure of F if it is algebraic over F roughly speaking, not too big compared to F and is algebraically closed big enough to contain solutions of all polynomial equations.
By the above, C is an algebraic closure of R. The situation that the algebraic closure is a finite extension of the field F is quite special: by the Artin-Schreier theorem , the degree of this extension is necessarily 2, and F is elementarily equivalent to R. Such fields are also known as real closed fields.
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Any field F has an algebraic closure, which is moreover unique up to non-unique isomorphism. It is commonly referred to as the algebraic closure and denoted F. For example, the algebraic closure Q of Q is called the field of algebraic numbers. The field F is usually rather implicit since its construction requires the ultrafilter lemma , a set-theoretic axiom that is weaker than the axiom of choice. It is the union of the finite fields containing F q the ones of order q n.
For any algebraically closed field F of characteristic 0, the algebraic closure of the field F t of Laurent series is the field of Puiseux series , obtained by adjoining roots of t. Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas. The Artin-Schreier theorem states that a field can be ordered if and only if it is a formally real field , which means that any quadratic equation.
An Archimedean field is an ordered field such that for each element there exists a finite expression. Equivalently, the field contains no infinitesimals elements smaller than all rational numbers ; or, yet equivalent, the field is isomorphic to a subfield of R. An ordered field is Dedekind-complete if all upper bounds , lower bounds see Dedekind cut and limits, which should exist, do exist. More formally, each bounded subset of F is required to have a least upper bound.
Since every proper subfield of the reals also contains such gaps, R is the unique complete ordered field, up to isomorphism. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis. The completion of F is another field in which, informally speaking, the "gaps" in the original field F are filled, if there are any. The following table lists some examples of this construction. The fourth column shows an example of a zero sequence , i. The field Q p is used in number theory and p -adic analysis.
The algebraic closure Q p carries a unique norm extending the one on Q p , but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is called the field of complex p-adic numbers and is denoted by C p. The following topological fields are called local fields :  [nb 4]. These two types of local fields share some fundamental similarities. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in F p.
However, since the addition in Q p is done using carrying , which is not the case in F p t , these fields are not isomorphic. The following facts show that this superficial similarity goes much deeper:. Differential fields are fields equipped with a derivation , i. These fields are central to differential Galois theory , a variant of Galois theory dealing with linear differential equations. Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication.
The primitive element theorem shows that finite separable extensions are necessarily simple , i. The latter condition is always satisfied if E has characteristic 0. For example, if the Galois group of a Galois extension as above is not solvable cannot be built from abelian groups , then the zeros of f cannot be expressed in terms of addition, multiplication, and radicals, i.
Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel—Ruffini theorem :. The tensor product of fields is not usually a field. This fact is the beginning of Grothendieck's Galois theory , a far-reaching extension of Galois theory applicable to algebro-geometric objects. Basic invariants of a field F include the characteristic and the transcendence degree of F over its prime field.
The latter is defined as the maximal number of elements in F that are algebraically independent over the prime field. Two algebraically closed fields E and F are isomorphic precisely if these two data agree. For example, Q p , C p and C are isomorphic but not isomorphic as topological fields. In model theory , a branch of mathematical logic , two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely.
The mathematical statements in question are required to be first-order sentences involving 0, 1, the addition and multiplication. A typical example is. The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. If U is an ultrafilter on a set I , and F i is a field for every i in I , the ultraproduct of the F i with respect to U is a field. The Ax—Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts in both cases over all primes p.
For fields that are not algebraically closed or not separably closed , the absolute Galois group Gal F is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of F. A description in terms of generators and relations is also known for the Galois groups of p -adic number fields finite extensions of Q p.
Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois cohomology. The norm residue isomorphism theorem , proved around by Vladimir Voevodsky , relates this to Galois cohomology by means of an isomorphism. Algebraic K-theory is related to the group of invertible matrices with coefficients the given field.
In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general. This observation, which is an immediate consequence of the definition of a field, is the essential ingredient used to show that any vector space has a basis. Modules the analogue of vector spaces over most rings , including the ring Z of integers, have a more complicated structure.
A particular situation arises when a ring R is a vector space over a field F in its own right.
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Such rings are called F -algebras and are studied in depth in the area of commutative algebra. For example, Noether normalization asserts that any finitely generated F -algebra is closely related to more precisely, finitely generated as a module over a polynomial ring F [ x 1 , A widely applied cryptographic routine uses the fact that discrete exponentiation, i. In elliptic curve cryptography , the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve , i. Finite fields are also used in coding theory and combinatorics. Functions on a suitable topological space X into a field k can be added and multiplied pointwise, e.
This makes these functions a k - commutative algebra. For having a field of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i. This occurs in two main cases.