Fix an integer h ≥ 1. In the universe of coloured trees of height at most h, we prove that for any MSO formula with r variables there exists a set of kernels, each of size bounded by an elementary function of r and the number of colours. This yields two noteworthy consequences. Consider any graph class G having a simple MSO interpretation in the universe of coloured trees of height h (equivalen...

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use them to stabilize numerical calculations. Our method follows classical analysis for first-order systems and higherorder scalar equations where growth behavior is expressed in terms of elementary functions. We then recast our equations in mollified form thereby obtaining stabi...

The main aim of this paper is to provide an approximation for the function f which possesses continuous derivatives up to the order n−1 (n ≥ 1) and has the n−th derivative of bounded variation, in terms of the chord that connects its end points A = (a, f (a)) and B = (b, f (b)) and some more terms which depend on the values of the k derivatives of the function taken at the end points a and b, w...

After a brief review of the muon g−2 status, we discuss hypothetical errors in the Standard Model prediction that could explain the present discrepancy with the experimental value. None of them looks likely. In particular, an hypothetical increase of the hadroproduction cross section in low-energy e+e− collisions could bridge the muon g−2 discrepancy, but is shown to be unlikely in view of curr...

Motivated by the precision results in the electroweak theory studies of two-loop Feynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop selfenergy diagrams in arbitrary and especially four dimensions. This is done in three respects: firstly results in terms of generalized, multivariable hypergeometric functions are presented giving expl...

On 8 September 1968, Ryszard Siwiec set fi re to himself during a harvest festival in the 10th Anniversary Stadium in Warsaw. Through his self-immolation, he sought to protest against Communist rule in general and the Warsaw Pact invasion of Czechoslovakia in particular. However, his death did not gain wider attention. Further protests ‘by fi re’ took place in the subsequent months and years in...

Keywords: Hyperelliptic integrals Elliptic integrals Epstein–Hubbell integral Bessel functions Lauricella function a b s t r a c t The elliptic integrals and its generalizations are applied to solve problems in various areas of science. This study aims to demonstrate a new method for the calculation of integrals through Bessel functions. We present solutions for classes of elliptic integrals an...

Introducing an elementary form of program semantics early in the curriculum provides a good preamble to formal methods. Microsemantics uses only the most basic concept in formal mathematics, namely substitution, and therefore can be presented as early as the second lecture of a freshman-level course. It can subsequently serve as a bootstrap for gradually introducing most of the other fundamenta...

We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example, if x is a positive floating-point number with an n-bit fraction, then (under rather weak assumptions) an n-bit approximation to log(x) or exp(x) may be comput...

for R(t) which is rapidly convergent for small values of t and which incidentally provides a new set of inequalities. The rapidity of convergence is compared with a series for R(t) and with the Laplace C.F. for R(t). This assessment is similar to recent work by Teichroew (1952) on the comparative rapidity of convergence of series ando.F.'s for the elementary function e?, hi (1 + x) and arc tan ...