Double Grothendieck Polynomials and Colored Lattice Models

We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

Jarlskog determinant and data on flavor matrices

The essences of the weak [Formula: see text] violation, the quark and lepton Jarlskog invariants, are determined toward future model buildings beyond the Standard Model (SM). The equivalence of two calculations of Jarlskog invariants gives a bound on the [Formula: see text] phase in some parametrization. Satisfying the unitarity condition, we obtain the CKM and PMNS matrices from the experimental data, and present the results in matrix forms. The Jarlskog determinant [Formula: see text] in the quark sector is found to be [Formula: see text] while [Formula: see text] in the leptonic sector is [Formula: see text] in the normal hierarchy parametrization.

THE CONTRIBUTION OF AGILITY AND SPEED ON DRIBBLING ABILITY AT SSB FOOTBALL PLAYERS SKB MUARA BUNGO

The problem in this study is the low ability of dribbling for SSB football layers in Muara Bungo Learning Activity Studio, which is allegedly due to lack of agility and speed. Therefore, this study aims to describe the contribution of agility and speed both individually and together. With the dribbling ability of SSB football players, a studio of learning activities in 2010. The population of the study was 75 football players from the SSB Sanggar Muara Bungo Learning Activities in 2010. Sampling was carried out by means of porpusive sampling, that is, the age group of 17 to 20 years, with 30 people. To achieve the purpose of this study, there were three instruments used, (1) zig-zag run test, to measure agility variable, (2) 50 year (45 meter) running test to measure speed variable, (3) dribbling ability test to measure dribbling ability variable. Data were analyzed with product moment correlation and multiple correlation with the determinant formula. The finding of this study shows; (1) there is a significant relationship between agility and dribbling ability, with the contribution of agility 43.30% to dribbling ability, (2) there is a significant relationship between speed and dribbling ability. Contributions of agility 24.21%, (3) there is a significant relationship between agility and speed together with the ability of dribbling, the contribution of agility and speed to the dribbling ability of 43.59%.

Loops in SL(2, ℂ) and root subgroup factorization

In previous work, we proved that for a [Formula: see text]-valued loop having the critical degree of smoothness (one half of a derivative in the [Formula: see text] Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop [Formula: see text] satisfying these conditions, the Toeplitz determinant [Formula: see text] and shifted Toeplitz determinant [Formula: see text] factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in [Formula: see text]. The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of [Formula: see text] entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.