On Square Roots of Meromorphic Maps

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.

Download Full-text

Strong submeasures and applications to non-compact dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.132 ◽

2020 ◽

pp. 1-23

Author(s):

TUYEN TRUNG TRUONG

Keyword(s):

Metric Space ◽

Bounded Operator ◽

Continuous Functions ◽

Open Dense Subset ◽

Compact Metric Space ◽

Basic Properties ◽

Banach Theorem ◽

Complex Varieties ◽

Definition Of ◽

Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

Download Full-text

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

Janapriya Journal of Interdisciplinary Studies ◽

10.3126/jjis.v7i1.23053 ◽

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.

Download Full-text

Why Square Roots are Irrational

American Mathematical Monthly ◽

10.1080/00029890.1986.11971790 ◽

1986 ◽

Vol 93 (3) ◽

pp. 213-214 ◽

Cited By ~ 4

Author(s):

William C. Waterhouse

Keyword(s):

Square Roots

Download Full-text

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

Theoretical Computer Science ◽

10.1016/j.tcs.2011.06.014 ◽

2011 ◽

Vol 412 (39) ◽

pp. 5458-5465

Author(s):

Qi Cheng ◽

Yu-Hsin Li

Keyword(s):

Square Roots

Download Full-text

Reification in the Learning of Square Roots in a Ninth Grade Classroom: Combining Semiotic and Discursive Approaches

International Journal of Science and Mathematics Education ◽

10.1007/s10763-016-9765-3 ◽

2016 ◽

Vol 16 (2) ◽

pp. 295-314

Author(s):

Yusuke Shinno

Keyword(s):

Ninth Grade ◽

Square Roots ◽

Grade Classroom

Download Full-text

The electrical resistivity of lithium-6

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1961.0170 ◽

1961 ◽

Vol 263 (1314) ◽

pp. 407-419 ◽

Cited By ~ 18

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.

Download Full-text

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

Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London ◽

10.1098/rspl.1815.0378 ◽

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.

Download Full-text