![]() For multiple simultaneous equations with multiple unknowns. The Solve command attempts to find all solutions of anĮquation. Many of Mathematicas equation-solving routines return their results in the form of lists of. is a very low priority operator, so if you want toĪpply a rule to part of an expression you should use parentheses to ensure that the rule gets applied in the way you intended: In the form of lists of rules (or lists of lists of rules!). Many of Mathematica's equation-solving routines return their results It is often convenient to apply a list of rules, You apply the rule to an expression using Nfound (* returns number of occurrences *)Ī substitution rule is something like x->2.5 which means Here is a video showing how to develop a Mathematica program using a Notebook in parallel with a text editor. Or: File -> Open -> select previously created fileĮdit the file, and to run the code click "Run Package" or "Run all code" Or: New Document -> Notebook File -> New -> Package Alford, Washington University Physics Department.įor an introduction to the basics of Mathematica, see myįor serious projects you will need to create "package" filesĬontaining definitions of the functions that perform your calculations. Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians.Mathematica Techniques Mathematica Techniques Basis Decompositions and a Mathematica Package for Modular Graph Forms All the calculations in this article are available in uc(Mathematica) and uc(Python) files in the supplementary material, allowing readers to experiment with their own variants. For three random directions, hidden-variable theories are impossible (we show) exactly 2/3 of the time. With hidden variables, Eve can succeed when Alice and Bob limit themselves to two Stern-Gerlach directions, but always fails for some choices of three directions. This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern-Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. Publication: American Journal of Physics.Solution method: Appropriate variable changes for given square roots are constructed by parametrizing algebraic hypersurfaces associated to these square roots by families of lines. The computation of these integrals often involves square roots that need to be rationalized via suitable variable transformations. Nature of problem: Analytic solutions for Feynman integrals are critical for accurate theoretical predictions in high energy particle physics. Licensing provisions: GNU General Public License 3 After an introduction to the theoretical background, we explain in detail how to use the program in practice. We present a program that can be used to find such transformations. One way to obtain a solution in terms of multiple polylogarithms is to rationalize these square roots by a suitable variable change. The computation of Feynman integrals often involves square roots. Author(s): Besier, MarcoWasser, PascalWeinzierl, Stefan.Publication: Computer Physics Communications. ![]() RationalizeRoots: Software package for the rationalization of square roots
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