A simple and accurate method to calculate square roots and cube roots for Chemistry students without calculator.

Presented here are two simple formulae, for the calculation of square and cube roots that are frequently encountered in Chemistry courses, at school and college levels (whether undergraduate or graduate). However, tests are difficult to be designed which do not provide for a calculator and which require the calculation of these quantities. The formulae are simple interval-weighted denominator method based to get an accurate value of these quantities. This will enable students to quickly and accurately compute square roots and cube roots.

Symbol alphabets from tensor diagrams

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar $$ \mathcal{N} $$ N = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope $$ {\mathcal{C}}^{\dagger}\left(4,9\right) $$ C † 4 9 encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

The Theodorus Variation

The Spiral of Theodorus, also known as the "root snail" from its connection with square roots, can be constructed by hand from triangles made with from paper with scissors, ruler, and protractor. See the Video Abstract. Once the triangles are made, two different but similar spirals can be made. This paper proves some things about the second spiral; in particular that the open curve generated by the inner vertices monotonically approaches a circle, and that the vertices are ultimately equidistributed around that inner circle.

BKP tau-functions as square roots of KP tau-functions

It is well-known that a BKP tau-function is the square root of a certain KP tau-function, provided one puts the even KP times equal to zero. In this paper we compute for all polynomial BKP tau-function its corresponding KP ”square”. We also give, in the polynomial case, a representation theoretical proof of a recent result by Alexandov, viz. that a KdV tau-function becomes a BKP tau-function when one divides all KdV times by 2.

Longitudinal Change and Progression Indicators Using the Movement Disorder Society-Unified Parkinson’s Disease Rating Scale in Two Independent Cohorts with Early Parkinson’s Disease

Background: The MDS-Unified Parkinson’s disease (PD) Rating Scale (MDS-UPDRS) is the most used scale in clinical trials. Little is known about the predictive potential of its single items. Objective: To systematically dissect MDS-UPDRS to predict PD progression. Methods: 574 de novo PD patients and 305 healthy controls were investigated at baseline (BL) in the single-center DeNoPa (6-year follow-up) and multi-center PPMI (8-year follow-up) cohorts. We calculated cumulative link mixed models of single MDS-UPDRS items for odds ratios (OR) for class change within the scale. Models were adjusted for age, sex, time, and levodopa equivalent daily dose. Annual change and progression of the square roots of the MDS-UDPRS subscores and Total Score were estimated by linear mixed modeling. Results: Baseline demographics revealed more common tremor dominant subtype in DeNoPa and postural instability and gait disorders-subtype and multiethnicity in PPMI. Subscore progression estimates were higher in PPMI but showed similar slopes and progression in both cohorts. Increased ORs for faster progression were found from BL subscores I and II (activities of daily living; ADL) most marked for subscore III (rigidity of neck/lower extremities, agility of the legs, gait, hands, and global spontaneity of movements). Tremor items showed low ORs/negative values. Conclusion: Higher scores at baseline for ADL, freezing, and rigidity were predictors of faster deterioration in both cohorts. Precision and predictability of the MDS-UPDRS were higher in the single-center setting, indicating the need for rigorous training and/or video documentation to improve its use in multi-center cohorts, for example, clinical trials.