An octonion algebra originating in combinatorics

C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of delta-codes (now known as T-sequences). We investigate the possible new versions of his polynomial Lagrange identity. Our main result shows that all such identities are equivalent to each other.Comment: 11 pages, A simpler proof of the main theorem, due to Alberto Elduque, is inserted. The paper will appear in the Proc. Amer. Math. So

Symmetric Hadamard matrices of orders 268, 412, 436 and 604

We construct many symmetric Hadamard matrices of small order by using the so called propus construction. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604.Comment: 12 page

Hysteroscopic Myomectomy: the Guidelines of the International Society for Gynecologic Endoscopy (ISGE)

Objective: With this publication, the International Society for Gynecologic Endoscopy (ISGE) aims to provide the clinicians with the recommendations arising from the best evidence currently available on hysteroscopic myomectomy (HM). Study design: The ISGE Task Force for HM defined key clinical questions, which led the search of Medline/PubMed and the Cochrane Database. We selected and analyzed relevant English-language articles, published from January 2005 to June 2021, including original works, reviews and the guidelines previously published by the European Society for Gynecological Endoscopy (ESGE) and the American Association of Gynecologic Laparoscopists (AAGL), in which bibliographies were also checked in order to identify additional references, using the medical subject heading (MeSH) term "Uterine Myomectomy" (MeSH Unique ID: D063186) in combination with ''Myoma" (MeSH Unique ID: D009214) and ''Hysteroscopy" (MeSH Unique ID: D015907). We developed the recommendations through multiple cycles of literature analysis and expert discussion. Results: The ISGE Task Force did develop 10 grade 1A-C and 4 grade 2A-C recommendations. For planning HM, evaluation of the uterus with saline infusion sonohysterography (SIS) or combined assessment by transvaginal ultrasound (TVUS) and diagnostic hysteroscopy is recommended (Grade 1A). The use of STEPW (Size, Topography, Extension of the base, Penetration and lateral Wall position) classification system of submucosal leiomyoma (LM) is recommended to predict the complex surgeries, incomplete removal of the LM, long operative time, fluid overload and other major complications (grade 1B). For type 0 LMs, in addition to resectoscopy (slicing technique), morcellation is recommended, being faster and having a shorter learning curve with respect to resectoscopy (grade 1C). For type 1-2 LMs, slicing technique is currently recommended (grade 1C). A fluid deficit of 1000 mL also in case of bipolar myomectomy with saline solution, in healthy women of reproductive age, contains low risk for major complications. Lower thresholds (750 mL) for fluid deficit should be considered in the elderly and in women with cardiovascular, renal or other co-morbidities (Grade 1B). Conclusion: HM is the most effective conservative minimally invasive gynecologic intervention for submucous LM. The set of 14 ISGE recommendations can significantly contribute to the success of HM and the safety of patients for whom the choice of appropriate surgical technique, as well as the surgeon's awareness and measures to prevent complications are of the utmost importance.info:eu-repo/semantics/publishedVersio

Correlation analysis of milk production traits across three generations of Simmental cows

The relationship between milk production traits over whole lactations was evaluated across three generations of Simmental cows (between daughters, dams and granddams) by a corelation analysis with whole lactation traits in the daughter generation being used as the dependent variables (x<sub>1</sub>), and those in the dam and granddam generations being used as the independent variables (x<sub>2</sub> and  x<sub>3</sub>). The results were obtained from a sample of 1170 daughters and as many dams and  granddams. The correlation of whole lactation milk production traits between daughters, dams and  granddams, as calculated by simple, partial and multiple correlation coefficients was very weak or  non-existent. All of the calculated simple and partial correlation coefficients were positive and mostly  statistically very significant (P<0.01). The calculated coefficients of multiple correlation (R1.23) between  lactation length, milk fat content, milk yield, milk fat yield and 4% FCM yield with the expression of the traits in the daughters being used as the dependent variable and that in the dams and granddams as the independent variable were statistically very significant (P<0.01), amounting to 0.091, 0.251, 0.180, 0.133 and 0.153, respectively.Key words: Simmental breed, production traits, generation, coefficient of partial correlation, coefficient of multiple correlation

Conservative Management of Asymptomatic Adnexal Masses Classified as Benign by the IOTA ADNEX Model: a Prospective Multicenter Portuguese Study

This prospective multicentric study aiming to determine the incidence of complications (malignant transformation, torsion or rupture) during conservative management of adnexal masses was performed in two Portuguese tertiary referral hospitals. It included ≥18-year-old, non-pregnant patients with asymptomatic adnexal masses (associated IOTA ADNEX risk of malignancy < 10%) sonographically diagnosed between January 2016 and December 2020. Conservative patient management consisted of serial clinical and ultrasound assessment up to 60 months of follow-up, spontaneous resolution of the formation or surgical excision (median follow-up: 17.8; range 9-48 months). From the 573 masses monitored (328 premenopausal and 245 postmenopausal adnexal masses), no complications were observed in 99.5%. The annual lesion growth rates and increases in morphological complexity were similar in the premenopausal and postmenopausal patients. Spontaneous resolution, evidenced in 16.4% of the patients, was more common in the premenopausal group (p < 0.05). Surgical intervention was performed in 18.4% of the cases; one borderline and one invasive FIGO IA stage cancer were diagnosed. There was an isolated case of ovary torsion (0.17%). These data support conservative management as a safe option for sonographically benign, stable and asymptomatic adnexal masses before and after menopause and highlight the need for expedite treatment of symptomatic or increased-morphological-complexity lesions.info:eu-repo/semantics/publishedVersio

SQCD: A Geometric Apercu

We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.Comment: 49 pages, 1 figur