Square Roots and Quadratic Symbols

2021 ◽  
pp. 63-71
Author(s):  
Duncan Buell
Keyword(s):  
Square Roots

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Why Square Roots are Irrational

1986 ◽  
Vol 93 (3) ◽  
pp. 213-214 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  
Square Roots

scholarly journals On the minimum gap between sums of square roots of small integers

2011 ◽  
Vol 412 (39) ◽  
pp. 5458-5465
Author(s):  
Qi Cheng ◽  
Yu-Hsin Li
Keyword(s):  
Square Roots

The electrical resistivity of lithium-6

1961 ◽  
Vol 263 (1314) ◽  
pp. 407-419 ◽  
Keyword(s):  
Electrical Resistivity ◽  
Isotopic Composition ◽  
Specific Heat ◽  
Room Temperature ◽  
Natural Isotopic Composition ◽  
Square Roots ◽  
Conduction Electrons ◽  
Ionic Mass ◽  
Theoretical Predictions ◽  
Electrical Resistivities

The electrical resistivities of lithium -6 and lithium of natural isotopic composition have been studied between 4°K and room temperature. In addition, their absolute resistivities have been carefully compared at room temperature. These measurements show that the effect of ionic mass on electrical resistivity agrees with simple theoretical predictions, namely, that the properties of the conduction electrons in lithium do not depend on the mass of the ions, and that the characteristic lattice frequencies for the two pure isotopes are in the inverse ratio of the square roots of their ionic masses. A comparison with the specific heat results of Martin (1959, 1960), where the simple theory is found not to hold, indicates the possibility that anharmonic effects are present which affect the specific heat but not the electrical resistivity.

scholarly journals On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative quantities

1833 ◽  
Vol 2 ◽  
pp. 382-382
Keyword(s):  
General Result ◽  
Geometric Representation ◽  
Square Roots ◽  
Geometrical Representation

The author, in a former paper, read to the Society in February last, had discussed various objections which had been raised against his mode of geometric representation of the square roots of negative quantities. At that time he had only discovered geometrical repre­sentations for quantities of the form a + b √‒1, of geometrically adding and multiplying such quantities, and also of raising them to powers either whole or fractional, positive or negative; but he was at that time unable to represent geometrically quantities raised to powers, whose indices involve the square roots of negative quantities (such as a + b √‒1 m + n ). His attention has since been drawn to this latter class of quantities by a passage in M. Mourey’s work on this subject, which implied that that gentleman was in posses­sion of methods of representing them geometrically, but that he was at present precluded by circumstances from publishing his discoveries. The author was therefore induced to pursue his own investigations, and arrived at the general result stated by M. Mourey, that all algebraic quantities whatsoever are capable of geometrical representation by lines all situated in the same plane. The object of the present paper is to extend the geometrical representations stated in his former treatise, to the powers of quantities, whose indices involve the square roots of negative quantities. With this view he investigates Various equivalent formulæ suited to the particular cases, and employs a peculiar notation adapted to this express purpose ; but the nature of these investigations is such as renders them incapable of abridgement.

Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics

2012 ◽  
Vol 119 (8) ◽  
pp. 646 ◽  
Author(s):  
David H. Bailey ◽  
Jonathan M. Borwein
Keyword(s):  
Square Roots

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

2021 ◽  
pp. 1-16
Author(s):  
Michael Bartl ◽  
Mohammed Dakna ◽  
Sebastian Schade ◽  
Tamara Wicke ◽  
Elisabeth Lang ◽  
...  
Keyword(s):  
Parkinson’S Disease ◽  
Parkinson's Disease ◽  
Clinical Trials ◽  
Rating Scale ◽  
De Novo ◽  
Longitudinal Change ◽  
Single Center ◽  
Square Roots ◽  
Disease Rating

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.

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