Composition of binary quadratic forms over number fields

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

Representation of Cultural Ambivalence in Eliza Hamilton’s Translation of the Letters of A Hindoo Rajah

Eliza Hamilton in her two-volume epistolary novel Translation of The Letters of A Hindoo Rajah (1819), by projecting two characters who undergo the oppositional experiences during their contact with English people, creates an ambivalent situation which neither represents England as totally positive nor India as completely negative. The two perspectives of Zaarmilla and Sheermaal exclude one another’s rendering. To unpack this contradictory narrative position, the concept of acculturation and cultural stress, especially formulated by John W. Berry is taken in interlocution. By rendering these two sorts of antithetical narratives juxtaposed together confirms the dynamics of ambivalence which does not regard Saidaian notion of Orientalism intact.

Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.

Implementation of the K-Medoids Algorithm for Data Clustering of Covid 19 Cases in West Java

The Covid 19 pandemic has hit Indonesia for almost 15 months since March 2020. The virus has spread to all provinces in Indonesia. Various efforts were made to be able to reduce or prevent the spread of the coronavirus, including the implementation of the PSBB in various areas including in West Java province. In this study, the objective of this research is to cluster the data on cases of Covid 19 in West Java which are recapitulated daily based on districts/cities that occurred on May 20, 2021. For the clustering process, the K-medoids algorithm is used which determines 3 clusters based on the variables used, namely discarded close contact, suspects discarded, probable completed, probable died, totally positive, positive recovered, and positive died. For data processing, a calculation analysis was carried out using the stages in the K-medoids algorithm and the Rapidminer application with high cluster mapping of 6 districts/cities, medium clusters there were 19 districts/cities, while low clusters had 2 districts/cities. The results of the analysis are expected to provide information about the distribution and mapping of clusters in West Java province.

Proposal and Validation of a New Classification of Surgical Outcomes after Colorectal Resections within an Enhanced Recovery Programme

Background. Advantages of Enhanced Recovery (ER) programmes in colorectal surgery have already been demonstrated, but heterogeneity exists with respect to the choice of compared outcomes. A comprehensive classification aimed at standardizing the reporting of surgical outcomes has been proposed and validated. Method. Clinical variables of 231 patients who underwent colorectal resections within an ER programme from 2013–2018 were analysed. Their outcomes have been reported according to a new classification in 5 classes and 11 subclasses. Prognostic variables have been identified. Results. Seventy-nine patients (34.2%) had an optimal class 1 outcome. Almost half of the patients had an uneventful recovery after being discharged after day 4 (2a). Only two patients (0.9%) were discharged early and then readmitted for a minor ailment (2b). Total morbidity was 12.6% (3a–5). Perioperative mortality was 2.6% (5). Young age, laparoscopic resection, and years of experience with ER have been identified as independent prognostic factors towards a totally positive outcome. Conclusions. The proposed outcome classification is a simple and objective tool to report the surgical outcome in clinical studies. Its implementation seems to be appropriate, in particular, in the field of ER protocols in colorectal surgery, but it can have a wider application in any other surgical subspeciality.

The EFT-hedron

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the “EFT-hedron”. We initiate a systematic investigation of the boundary structure of the EFT-hedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFT-hedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $$\{|f(\lambda )|: \lambda \in \Lambda \}$$ { | f ( λ ) | : λ ∈ Λ } on any set $$\Lambda \subseteq {\mathbb {R}}$$ Λ ⊆ R with lower Beurling density $$D^{-}(\Lambda )>2$$ D - ( Λ ) > 2 .We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$ g ^ ( ξ ) = ∫ R g ( x ) e - 2 π i x ξ d x = C 0 e - γ ξ 2 ∏ ν = 1 m ( 1 + 2 π i δ ν ξ ) - 1 , ξ ∈ R , with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$ δ 1 , … , δ m ∈ R , C 0 , γ > 0 , m ∈ N ∪ { 0 } , and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$ V ∞ ( g ) = { f = ∑ k ∈ Z c k g ( · - k ) : c ∈ ℓ ∞ ( Z ) } , generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) . As a consequence of (1), each $$f \in V^\infty (g)$$ f ∈ V ∞ ( g ) is continuous, the defining series converges unconditionally in the weak$$^*$$ ∗ topology of $$L^\infty $$ L ∞ , and the coefficients $$c_k$$ c k are unique [6, Theorem 3.5].