ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

A Density of Ramified Primes

Let K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

Supersingular j-invariants and the class number of ℚ(−p)

Let [Formula: see text] be a prime. Let [Formula: see text] be the discriminant of an imaginary quadratic order. Assume that [Formula: see text] and [Formula: see text]. We compute the number of [Formula: see text]-roots of the class polynomials [Formula: see text]. Suppose [Formula: see text], we prove that two class polynomials [Formula: see text] and [Formula: see text] have a common root in [Formula: see text] if and only if [Formula: see text] is a perfect square. Furthermore, any three class polynomials do not have a common root in [Formula: see text]. As an application, we propose a deterministic algorithm for computing the class number of [Formula: see text].

Design and Implementation of Multidimensional Interaction in Online English Course under the Assistance of Omnimedia

The method of multidimensional interaction in online teaching and learning requires teachers to harness the techniques of omnimedia and have an overall course design to achieve similar results as offline learning. Supported by QQ, Lanmo Cloud Class, and WeChat platforms, online English courses implement multidimensional interaction method to enhance language communication. This method compensates for the inefficacy of online teaching and learning which falls short of face-to-face interaction and considers the needs of students. The quasiexperimental study sampled 66 freshmen English major students in an ordinary college via Integrated English Online Course for three months. By using T-test of SPSS 26.0, statistics showed that there were significant differences between the experimental class (Number 35) and the control class (Number 31) in students’ feedback ( p = 0.044 , < 0.05 ) and the results of their final examinations ( p = 0.049 , < 0.05 ). The results indicated that multidimensional interaction in online English course under the assistance of omnimedia appeared effective. It also reflected that both teachers and students should grasp the skills of using new technology and make good use of omnimedia network platform. Teachers need to change traditional teaching concept to student-centered and create effective teaching design while students need to overcome the discomfort and actively engage in online learning to achieve the goals.

The number of representations of squares by integral quaternary quadratic forms

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular diagonal quaternary quadratic forms representing [Formula: see text] (see Table [Formula: see text]). In particular, we use eta-quotients to prove the strong [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text] (see Lemma 4.5 and Proposition 4.6).

EFEKTIVITAS OPAC SEBAGAI SARANA TEMU KEMBALI INFORMASI DI UPT PERPUSTAKAAN UNIVERSITAS MUHAMMADIYAH MAGELANG (UNIMMA)

Evaluation of information retrieval tool is very important to find out how far the retrieval system has worked. This study evaluates the OPAC of Muhammadiyah University of Magelang (UNIMMA) by determining the effectiveness of the information retrieval system seen from the relevance of the data displayed with the data requested by the user. This is descriptive quantitative research that calculates the value of recall and precision, while the data collection is conducted through literature studies and documentation. Based on the data collected, it can be concluded that OPAC UNIMMA is in the effective category with a high level of precision, and in the value range of 0.68 - 1.00 with recall and precision values respectively 0.77 or 77% and 0.84 or 84%. However, from the observations on each class number, it is found that the class number has a higher recall value than the precision, ie: the class numbers of 200, 800, and 900.