IBP reduction coefficients made simple

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

Complex Method on Octagonal Number

In Number theory Study of polygonal numbers is rich in varity. In this paper we establish a Complex Octagonal Number using Continued Fraction algorithm.

The n-dimensional Stern–Brocot tree

This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.

Convergence of β-Modified Jacobi-Perron Algorithm over the Field of Formal Power Series

The aim of this paper is to study multidimentional $\beta$-continued fraction algorithm over the field of formal power series. In the case of the Modified Jacobi-Perron algorithm, we prove that it converges.

Continued fraction algorithm for Sturmian colorings of trees

Factor complexity $b_{n}(\unicode[STIX]{x1D719})$ for a vertex coloring $\unicode[STIX]{x1D719}$ of a regular tree is the number of classes of $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_{n}(\unicode[STIX]{x1D719})=n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data appearing in the induction algorithm.

ANALISIS KEMAMPUAN CITRA RADIOGRAFI PANAROMIK DALAM MENDETEKSI KERAPATAN TRABEKULA TULANG DENGAN MIKRO CT SEBAGAI BAKU STANDARD - Image Analysis Capabality Of Detectinc Panoramic Radiographic Trabecular Bone Density As Standard With Standard Micro CT

AbstrakPenelitian Radiografi panoramik menggunakan analisa mikro struktur untuk deteksi kualitas tulang menggunakan Panoramik dengan Micro CT sebagai baku standard. Tujuan dari penelitian ini untuk mengetahui akurasi Panoramik dalam mendeteksi trabekula tulang di regio kondilus, angulus mandibula dan regio molar dengan model tulang vertebrae dan tulang iga. Empat macam tulang vertebrae dan tulang iga sapi diletakkan pada lengkung rahang yang dibantu oleh model tengkorak terbuat dari plastik sebagai model lengkung tulang. Tulang diletakkan pada lengkung rahang bawah terbuat dari bahan plastik pada regio berbeda yaitu : regio kondilus (tulang 1), angulus mandibula (tulang 2) regio molar 2 dan molar 1( tulang 4). Model tulang masing-masing mempunyai kerapatan berbeda lalu dilakukan pemotretan radiografi panoramik dengan posisi standard dan micro CT sebagai “gold standard”. Dengan menggunakan Analisis fraction trabekula algorithma menggunakan software. Hasil penelitian didapatkan fraksi trabekular sebagai berikut: regio kondilus panoramik sebesar: 44.05%, Micro CT 39.63% , regio angulus panoramik 28.85% , micro CT 24.15% , dan regio molar 27.43%, micro CT 22.64 %, rata-rata perbedaan 4,6 %. Radiografi panoramik dapat digunakan untuk mendeteksi kelainan micro struktur di regio kondilus .Kata kunci : Micro CT, Panoramik,AbstractPanoramic radiographs using micro- structure analysis for the detection of bone quality using Panoramic with Micro CT as standard raw. The purpose of this study was to determine the accuracy in detecting trabecular bone Panoramic in the region of the condyle , the angle of the mandible and molar regions with models vertebrae and rib bones. Methods : Four kinds of bone vertebrae and ribs placed on the cow arch assisted by a plastic skull models as models of bone arch. Bone is placed in the lower arch is made of plastic materials in different regions are: region of the condyle ( bone 1 ), angle of the mandible ( bone 2 ) 2 molar and molar regions 1 ( bone 4 ). Model of each bone has different densities, then shooting panoramic radiography performed with standard position and micro CT as the " gold standard ". By using analysis of trabecular fraction algorithm using software. The study trabecular fractions obtained as follows : panoramic condyle region of : 44.05 %, 39.63 % Micro CT, region of the angle panoramic 28.85 %, 24.15 % micro CT, and molar region, 27.43 %, 22.64 % micro CT, difference average 4.6 %. Panoramic radiographs can be used to detect abnormalities in the region of the micro structure of the condyle.Keywords : Micro CT , Panoramic