We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is Mordell-Weil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy of Λ. We prove that every rigid lattice is Mordell-Weil. In particular, we show that the Leech lattice can...

There is a lack of sufficient information on cotton responses to time and amount of foliar application of potassium (K) under salt stress environment, The objective of this study was to investigate the effects of potassium (as Solupotasse 50% K2O and 18% S) rate and application time on yield and yield components of cotton under saline field condition. A factorial experiment based on randomized ...

let $f_q d_{2n}$ be the group algebra of $d_{2n}$, the dihedral group of order $2n$ over $f_q=gf(q)$. in this paper, we establish the structure of $u(f_{2^k}d_{2n})$, the unit group of $f_{2^k}d_{2n}$ and that of its normalized unitary subgroup $v_*(f_{2^k}d_{2n})$ with respect to canonical involution $*$ when $n$ is odd.

OBJECTIVE To evaluate the patellar height of volunteers with and without patellofemoral pain syndrome (PPS) during maximal voluntary isometric contraction (MVIC) in open kinetic chain (OKC) and closed kinetic chain (CKC) exercises. METHODS Twenty healthy women, and nineteen women with patellofemoral pain syndrome were evaluated and subjected to nuclear magnetic resonance imaging during rest a...

This research was conducted to study of the essential oil content, some morphological parameters, crop evapotranspiration (ETc) and crop coefficient (Kc) of Summer Savory (Satureja hortensis L.) in the west of Iran (Khorramabad region) in 2017. Weighing Lysimeters (25 cm diameter and 30 cm height) were used to measure important morphological parameters, t...