V - HERMITIAN FORMS

Here we include some classical results from the theory of binary hermitian forms which originate from Hermite (1854). We discuss the reduction theory of binary hermitian forms as described for example in Bianchi (1892). Eventually our considerations lead to Humbert’s computation of the covolume of SL(2, Thuong) where Thuong is the ring of integers in an imaginary quadratic number field. The work of Humbert on hermitian forms is contained in his papers (1915), (1919a)—(1919e). It contains an interesting error, we correct it in Section 9.6. We also develop a theory of representation numbers of binary hermitian forms which is analogous to the theory of binary quadratic forms as in Landau (1927).KeywordsPrime NumberPrime IdealZeta FunctionDirichlet SeriesHermitian FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents a comparative analysis of different group aggregation methods adopted in analytic hierarchy process by testing them against social choice axioms with a case study of Delhi transport system. The group aggregation methods and their correctness were tested while prioritising the alternative options for energy efficient and less polluting transport system in Delhi. It was observed that among all group aggregation methods, geometric mean method, the most widely adopted group aggregation method of analytic hierarchy process, showed poor performance and failed to satisfy the most popular 'Pareto optimality and non-dictatorship axiom' raising questions on its validity as group aggregation method. All other group aggregation methods namely weighted arithmetic mean method with varying weights and equal weights and arithmetic mean of individual priorities resulted in concurring results with the individual member priorities. It was observed that weighted arithmetic mean method with equal weights aggregates individual priorities better compared to arithmetic mean method with varying weights. Arithmetic mean of the priorities by individual members of the group showed minimum deviation from the group consensus making it most suitable and simple method of aggregation.

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ ℝ is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond-Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

We will complete the list of universal binary Hermitian forms over imaginary quadratic fields by investigating three Hermitian forms missed by previous researchers.

Murty proved that for all sufficiently large $X$ there exist at least ${c(\ell,\eps) X^{1/{4\ell}-\eps}}$ real quadratic fields with class number divisible by $\ell$ and discriminant not exceeding $X$ in absolute value. We extend this this to number fields of arbitrary degree.

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree n binary form. A survey of a reduction algorithm over ${\mathbb{Z}}$ is described based on recent work of Cremona and Stoll.

Due to increasing global warming and environmental consciousness, to avoid/reduce carbon emission is a challenge in today's business scenario. In this study, a three stage (buyer-distributer-vendor) economic order quantity strategy is presented. The objective of this study is to analyse the carbon emissions caused during transportation and holding of the inventory. In order to increase the utility of the proposed model, besides all cost parameters, carbon emission costs and transportation costs at every step of supply network are also considered as fuzzy and denoted by fuzzy numbers. Shortages are allowed and backlogged unconditionally for this study. The optimal solution are obtained by using (a non derivative) arithmetic geometric inequality method. Further the optimization model is studied under the following policies (i) without considering carbon emission and (ii) with carbon emission. Here closed form solutions are obtained. Proposed models are analysed environmentally and economically for the decision variables. A numerical example is used to demonstrate the work in a better way and obtained results are compared under both policies. Finally, the results of two models proved that, the model with carbon emission is better for cost reducing and environmental consciousness both. Sensitivity analysis is also performed with respect to all important parameters.

This paper, taking a match of the province of north China as an example, based on the schools for the past ten years and awards for qualitative and quantitative data analysis and processing. Then through the model to predict the next year, and on the basis of the present situation and awards by the normal given the competition for the school places allocation scheme, in order to improve the quality and efficiency. This paper presents a method and process of big data analysis and processing by using SPSS and MATLAB and other mathematical software based on actual cases. The prediction results are reliable and can be used for reference. The innovation of this paper is that the weight ratio of analytic hierarchy process is used as the reference for quota allocation. Its advantage is that it can make the allocation result more flexible and easy to be controlled artificially. In addition, the paper adopts arithmetic average method, geometric average method and eigenvalue method respectively in the weight calculation, and then takes the average weight of the three methods to make the result more robust.

ABSTRACTThe National Capital (IKN) in East Kalimantan is a planned governmental center to achieve the "Golden Indonesia" vision by 2045. Kutai Kartanegara Regency serves as a partner to IKN, with parts of its territory designated as IKN development areas, including the sub-districts of Loa Janan, Loa Kulu, Sanga Sanga, Muara Jawa, West Samboja, and Samboja. This partnership impacts various needs in the area, including the demand for clean water. This study aims to estimate the clean water demand in these six sub-districts up to 2045. The analysis of clean water needs was conducted based on population projections using arithmetic, geometric, and least square methods. Water demand calculations were performed for average daily flow (Qrh), maximum daily flow (Qhm), and peak hourly flow (Qjp), with Qjp serving as the reference for meeting clean water needs. The findings indicate that water demand increases in line with population projections and is influenced by territorial changes and the area's development as part of IKN. In Loa Janan, Loa Kulu, and Sanga Sanga, where parts of the territory will become part of IKN, the water demand managed by the Kutai Kartanegara Regency Government is projected to decrease by 2045, from 400.7 L/s, 330.6 L/s, and 98.9 L/s to 106 L/s, 288.4 L/s, and 98.5 L/s, respectively. In contrast, the 2045 water demand for Muara Jawa, West Samboja, and Samboja is projected at 266.2 L/s, 175 L/s, and 222.9 L/s, respectively. The water demand in these three sub-districts must be met by the IKN authority as their entire areas fall within the IKN development region. A comprehensive analysis of water resources and the development of water supply infrastructure is needed to support clean water demands in the study area.

It is known that in the Analytic Hierarchy Process (A.H.P.) a scale of relative importance for alternatives is derived from a pairwise comparisons matrix A = (aij). Priority vectors are basically provided by the following methods: the right eigenvector method, the geometric mean method and the arithmetic mean method. Antipriority vectors can also be considered; they are built by both the left eigenvector method and mean procedures applied to the columns of A. When the matrix A is inconsistent, priority and antipriority vectors do not indicate necessarily the same ranking. We deal with the problem of the reliability of quantitative rankings and we use quasi-linear means for providing a more general approach to get priority and antipriority vectors.