Proof the Skewes’ number is not an integer using lattice points and tangent line

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

ANALISIS KESULITAN PESERTA DIDIK DALAM MENYELESAIKAN SOAL-SOAL POKOK BAHASAN GARIS SINGGUNG LINGKARAN

AbstrakLatar belakang penelitian ini adalah rendahnya hasil belajar matematika peserta didik terhadap pokok bahasan garis singgung lingkaran di SMP Al-Islam Benteng Tellue. Penelitian ini bertujuan untuk mengetahui kesulitan-kesulitan yang dialami peserta didik dalam menyelesaikan soal-soal pokok bahasan garis singgung lingkaran. Pendekatan penelitian yang digunakan yaitu pendekatan kualitatif yang dipadukan dengan jenis penelitian deskriptif. Subjek penelitian yaitu sebanyak 10 orang peserta didik di kelas VIII SMP Al-Islam Benteng Tellue. Pengumpulan data dilakukan dengan menggunakan metode tes dan wawancara. Data dianalisis secara deskriptif. Hasil penelitian menunjukkan kesulitan-kesulitan yang dialami peserta didik dalam menyelesaikan soal-soal pokok bahasan garis singgung lingkaran yaitu kesulitan pemahaman maksud soal yang tergolong rendah; kesulitan pemahaman konsep yang tergolong tinggi; kesulitan proses perhitungan yang tergolong rendah. AbstractThe background of this research is the low achievement of students' mathematics learning outcomes on the subject of circle tangent at SMP Al-Islam Benteng Tellue. This study aims to determine the difficulties experienced by students in solving the tangents to circles. The research approach used is a qualitative approach combined with a descriptive type of research. The research subjects were 10 students in class VIII SMP Al-Islam Benteng Tellue. Data was collected using the test and interview methods. Data were analyzed descriptively. The results of the study indicate that the difficulties experienced by students in solving the questions of the tangent line of the circle are the difficulties in understanding the meaning of the questions are classified as low; the difficulty of understanding the concept is high; the difficulty of the calculation process is low.

Phase Equilibria in the System CaO-SiO2-La2O3-Nb2O5 at 1400 °C

CaO-SiO2-La2O3-Nb2O5 system is of great significance for the pyrometallurgical utilization of Bayan Obo tailing resources. In the present work, the phase equilibrium of this quaternary system at 1400 °C was determined by a thermodynamic equilibrium experiment. On the basis of the recently determined CaO-La2O3-Nb2O5 phase diagram, some boundary surfaces of primary phase fields of CaO-SiO2-La2O3-Nb2O5 phase diagram were modified; then, the 1400 °C isothermal surface in the primary phase fields of SiO2, CaNb2O6, Ca2Nb2O7, and LaNbO4 was constructed, respectively. On this basis, CaO-SiO2-Nb2O5 pseudo-ternary phase diagrams with w(La2O3) = 5%, 10%, 15%, and 20% were determined, respectively. Considering the importance of equilibrium crystallization reaction type, we proposed a new rule named Tangent Line Rule to judge the univariant reaction type in the quaternary phase diagram. By applying Tangent Line Rule and Tangent Plane Rule previously proposed, some univariant and bivariant crystallization reaction types in the CaO-SiO2-La2O3- Nb2O5 phase diagram were determined, respectively. The current work can provide original data for the establishment of a thermodynamic database of Nb-bearing and REE-bearing slag system; the proposed Tangent Line Rule will promote the application of a spatial quaternary phase diagram.

An Innovative VMAT Technique for Left-sided Breast Cancer Patients With Postmastectomy Radiotherapy (PMRT) Evaluated by Ratio of Heart Volume in Tangent Line (RHVTL)

For patients with left-sided breast cancer (LBC), postmastectomy radiotherapy (PMRT) has been shown to improve the overall survival and many advanced planning techniques was adopted in PMRT. We aim to use an innovative VMAT technique to enhance the conformity of PTV and reduce the scattering dose of surrounding OARs, thereby reducing the long-term toxicity of the heart as well as ipsilateral lung (IL). The study further analyzes the more appropriate treatment planning techniques for personalized LBC patients with PMRT. 35 LBC patients were retrospectively selected undergoing PMRT. The PTV included lymph nodes, chest walls, excluding internal mammary nodes, where 95% of PTV receiving the prescription dose of 50Gy (2Gy/fraction) with three different techniques, VMAT, IMRT, Hybrid VMAT. Furthermore, the ratio of Heart Volume in Tangent line and heart volume (RHVTL) was proposed to evaluate the relative antonymy position between patient's heart and PTV, which hypothetically represents the complexity of treatment planning. The data from this study showed that for LBC patients undergoing PMRT, the CI from VMAT was 0.85 (IMRT and H-VMAT were 0.77 and 0.83), the heart D mean was 502.9cGy (IMRT and H-VMAT were 675.6cGy and 687cGy) and the V20 of IL was 21.3 as the lowest of the three techniques, but the dose of the contralateral breast (CB) and contralateral lung increased noticeably. In H-VMAT and IMRT, the mean heart dose was significantly related to RHVTL, with R-values of 0.911 and 0.892 respectively, while the values in VMAT was 0.613, thus the VMAT technique was relatively unaffected by the difficulty of treatment plan. For RHVTL values exceed than 0.06, the mean heart dose under VMAT technique raised by 98.7cGy compared to the RHVTL value of less than 0.06, but H-VMAT and IMRT increased by 233cGy and 261.58cGy individually. This study illustrates that separated fields and adjacent fields in VMAT technique obtained the optimal conformality and lowest doses of heart in three techniques for LBC with PMRT. Thus, based on the results of our preliminary study, the VMAT technique is highly recommended when RHVTL is exceeded 0.06.

Generic pointed quartic curves in $${\mathbb {R}}{\mathbb {P}}^{2}$$ and uninodal dessins

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), of Grothendieck’s dessins d’enfants. This classification contains 20 classes determined by the number of ovals of A, the parity of the oval containing the marked point p, the number of ovals that the tangent line $$T_p A$$ T p A intersects, the nature of connected components of $$A\setminus T_p A$$ A \ T p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of $$A\setminus T_p A$$ A \ T p A . We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary can be decomposed in blocks corresponding to cubic dessins in the disk $${\mathbf {D}}^2$$ D 2 , which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of generic pointed quartic curves in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 . This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.