Nested Integrals and Rationalizing Transformations

A brief overview of some computer algebra methods for computations with nested integrals is given. The focus is on nested integrals over integrands involving square roots. Rewrite rules for conversion to and from associated nested sums are discussed. We also include a short discussion comparing the holonomic systems approach and the differential field approach. For simplification to rational integrands, we give a comprehensive list of univariate rationalizing transformations, including transformations tuned to map the interval $[0,1]$ bijectively to itself.Comment: manuscript of 25 February 2021, in "Anti-Differentiation and the Calculation of Feynman Amplitudes", Springe

Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss algorithms for converting between sum and integral representations, mainly relying on the Mellin transform. To aid the conversion we worked out dedicated rewrite rules, based on which also some general patterns emerging in the process can be obtained.Comment: 13 pages LATEX, one style file, Proceedings of Loops and Legs in Quantum Field Theory -- LL2014,27 April 2014 -- 02 May 2014 Weimar, German

Computing elements of certain form in ideals to prove properties of operators

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gr\"obner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the Mathematica package OperatorGB.Comment: 26 page

Compatible rewriting of noncommutative polynomials for proving operator identities

The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.Comment: 17 page

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.Comment: 22 page

Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums

The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.Comment: Contribution to the proceedings of the Loops and Legs 2014 conferenc

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials

Brain tumour differentiation: rapid stratified serum diagnostics via attenuated total reflection Fourier-transform infrared spectroscopy

The ability to diagnose cancer rapidly with high sensitivity and specificity is essential to exploit advances in new treatments to lead significant reductions in mortality and morbidity. Current cancer diagnostic tests observing tissue architecture and specific protein expression for specific cancers suffer from inter-observer variability, poor detection rates and occur when the patient is symptomatic. A new method for the detection of cancer using 1 μl of human serum, attenuated total reflection - Fourier transform infrared spectroscopy and pattern recognition algorithms is reported using a 433 patient dataset (3897 spectra). To the best of our knowledge, we present the largest study on serum mid-infrared spectroscopy for cancer research. We achieve optimum sensitivities and specificities using a Radial Basis Function Support Vector Machine of between 80.0 and 100% for all strata and identify the major spectral features, hence biochemical components, responsible for the discrimination within each stratum. We assess feature fed-SVM analysis for our cancer versus non-cancer model and achieve 91.5 and 83.0% sensitivity and specificity respectively. We demonstrate the use of infrared light to provide a spectral signature from human serum to detect, for the first time, cancer versus non-cancer, metastatic cancer versus organ confined, brain cancer severity and the organ of origin of metastatic disease from the same sample enabling stratified diagnostics depending upon the clinical question asked. © 2016, The Author(s)

A dynamic theory of network failure

Organizational and sociological research dealing with network governance has mainly focused on network advantages rather than on their problems or dysfunctionalities. This left partially unexplored the field of network failure. Even if some early attempts at explicitly theorizing network failures have been made, we argue that explanations based mainly on social conditions (ignorance and opportunism) offered by this emerging theory (e.g. Schrank and Whitford, 2011), are not exhaustive. In this article we report the results of our empirical investigation on the underperforming network between the worldwide famous Venice Film Festival and its local hospitality system (in Venice, Italy). In the case study we are presenting, we will show how institutions have not been able to inhibit opportunism and sustain trust among network members because of mobilizing practices developed across formal lines of communication. With this work we propose a dynamic theory of network failure, answering to the more general call for network theories to focus the attention on agency and micro-processes