51 research outputs found
Detecting Simultaneous Integer Relations for Several Real Vectors
An algorithm which either finds an nonzero integer vector for
given real -dimensional vectors such
that or proves that no such integer vector with
norm less than a given bound exists is presented in this paper. The cost of the
algorithm is at most exact arithmetic
operations in dimension and the least Euclidean norm of such
integer vectors. It matches the best complexity upper bound known for this
problem. Experimental data show that the algorithm is better than an already
existing algorithm in the literature. In application, the algorithm is used to
get a complete method for finding the minimal polynomial of an unknown complex
algebraic number from its approximation, which runs even faster than the
corresponding \emph{Maple} built-in function.Comment: 10 page
Parallel integer relation detection: techniques and applications
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Statistical relational learning with soft quantifiers
Quantification in statistical relational learning (SRL) is either existential or universal, however humans might be more inclined to express knowledge using soft quantifiers, such as ``most'' and ``a few''. In this paper, we define the syntax and semantics of PSL^Q, a new SRL framework that supports reasoning with soft quantifiers, and present its most probable explanation (MPE) inference algorithm. To the best of our knowledge, PSL^Q is the first SRL framework that combines soft quantifiers with first-order logic rules for modelling uncertain relational data. Our experimental results for link prediction in social trust networks demonstrate that the use of soft quantifiers not only allows for a natural and intuitive formulation of domain knowledge, but also improves the accuracy of inferred results
The Sudakov form factor at four loops in maximal super Yang-Mills theory
The four-loop Sudakov form factor in maximal super Yang-Mills theory is
analysed in detail. It is shown explicitly how to construct a basis of
integrals that have a uniformly transcendental expansion in the dimensional
regularisation parameter, further elucidating the number-theoretic properties
of Feynman integrals. The physical form factor is expressed in this basis for
arbitrary colour factor. In the nonplanar sector the required integrals are
integrated numerically using a mix of sector-decomposition and Mellin-Barnes
representation methods. Both the cusp as well as the collinear anomalous
dimension are computed. The results show explicitly the violation of quadratic
Casimir scaling at the four-loop order. A thorough analysis concerning the
reliability of reported numerical uncertainties is carried out.Comment: 47 pages, 17 figures; v4: fixed typo in eqs. (4.4) and (A.4), final
result unchange
Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
In each of the 10 cases with propagators of unit or zero mass, the finite
part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter
words in the 7-letter alphabet of the 1-forms and , where is the sixth root of unity. Three diagrams
yield only . In two cases combines
with the Euler-Zagier sum ; in three cases it combines with the square of Clausen's
. The case
with 6 masses involves no further constant; with 5 masses a
Deligne-Euler-Zagier sum appears: . The previously unidentified term in the
3-loop rho-parameter of the standard model is merely . The remarkable simplicity of these results stems
from two shuffle algebras: one for nested sums; the other for iterated
integrals. Each diagram evaluates to 10 000 digits in seconds, because the
primitive words are transformable to exponentially convergent single sums, as
recently shown for and , familiar in QCD. Those are
SC constants, whose base of super-fast computation is 2. Mass involves
the novel base-3 set SC. All 10 diagrams reduce to SCSC constants and their products. Only the 6-mass case entails both bases.Comment: 41 pages, LaTe
The nonplanar cusp and collinear anomalous dimension at four loops in SYM theory
We present numerical results for the nonplanar lightlike cusp and collinear
anomalous dimension at four loops in SYM theory, which we
infer from a calculation of the Sudakov form factor. The latter is expressed as
a rational linear combination of uniformly transcendental integrals for
arbitrary colour factor. Numerical integration in the nonplanar sector reveals
explicitly the breakdown of quadratic Casimir scaling at the four-loop order. A
thorough analysis of the reported numerical uncertainties is carried out.Comment: 10 pages, 2 figures, 1 table. Proceedings of the 13th International
Symposium on Radiative Corrections (Applications of Quantum Field Theory to
Phenomenology), 25-29 September, 2017, St. Gilgen, Austri
Soft quantification in statistical relational learning
We present a new statistical relational learning (SRL) framework that supports reasoning with soft quantifiers, such as "most" and "a few." We define the syntax and the semantics of this language, which we call , and present a most probable explanation inference algorithm for it. To the best of our knowledge, is the first SRL framework that combines soft quantifiers with first-order logic rules for modelling uncertain relational data. Our experimental results for two real-world applications, link prediction in social trust networks and user profiling in social networks, demonstrate that the use of soft quantifiers not only allows for a natural and intuitive formulation of domain knowledge, but also improves inference accuracy
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