scholarly journals Simultaneous Additive Equations: Repeated and Differing Degrees

2017 ◽  
Vol 69 (02) ◽  
pp. 258-283 ◽  
Author(s):  
Julia Brandes ◽  
Scott T. Parsell
Keyword(s):  
Hybrid Systems ◽  
Existence Of Solutions ◽  
Square Root ◽  
Multiple Forms ◽  
Quadratic Equations ◽  
Asymptotic Formulae ◽  
Cubic Forms

Abstract We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning–Heath–Brown, which give weak results when specialized to the diagonal situation, this is the first result on such “hybrid” systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

scholarly journals On two lattice points problems about the parabola

2019 ◽  
Vol 16 (04) ◽  
pp. 719-729
Author(s):  
Jing-Jing Huang ◽  
Huixi Li
Keyword(s):  
Fourier Analysis ◽  
Character Sums ◽  
Upper Bounds ◽  
Gauss Sums ◽  
Lattice Points ◽  
Square Root ◽  
Optimal Error ◽  
Asymptotic Formulae ◽  
Error Terms

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola. We also obtain essentially sharp upper bounds for the latter lattice points problem associated with the parabola. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.

scholarly journals BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversion

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Bert van Geemen ◽  
Alessio Marrani ◽  
Francesco Russo
Keyword(s):  
Black Hole ◽  
Black Hole Entropy ◽  
Extremal Black Hole ◽  
Inhomogeneous System ◽  
Homogeneous Polynomials ◽  
Quadratic Equations ◽  
Cubic Form ◽  
Black Hole Background ◽  
Cubic Forms ◽  
First Time

Abstract We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $$ \mathcal{N} $$ N = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of $$ \mathcal{N} $$ N = 2, D = 4 supergravity.

Mathematics in the Junior High School: An improved method of extracting square root

1957 ◽  
Vol 50 (6) ◽  
pp. 445-447
Author(s):  
Richard F. De Mar
Keyword(s):  
High School ◽  
Junior High School ◽  
Numerical Solutions ◽  
Junior High ◽  
Pythagorean Theorem ◽  
Square Root ◽  
Improved Method ◽  
Quadratic Equations

Some method of extracting the square root of a number is usually taught in the eighth or ninth grades. Its chief applications then are in using the Pythagorean Theorem and in finding numerical solutions of quadratic equations.

Square root and quadratic equations for the study of leaf diagnosis in wheat

1999 ◽  
Vol 22 (9) ◽  
pp. 1469-1479 ◽  
Author(s):  
L. Sánchez de la Puente ◽  
Rosa M. Belda
Keyword(s):  
Square Root ◽  
Quadratic Equations

Identifying suboptimalities with factorial model comparison

2018 ◽  
Vol 41 ◽  
Author(s):  
Wei Ji Ma
Keyword(s):  
Experimental Design ◽  
Model Comparison ◽  
Process Models ◽  
Multiple Forms ◽  
Factorial Experimental Design ◽  
Factorial Model ◽  
Multiple Components ◽  
The Many

AbstractGiven the many types of suboptimality in perception, I ask how one should test for multiple forms of suboptimality at the same time – or, more generally, how one should compare process models that can differ in any or all of the multiple components. In analogy to factorial experimental design, I advocate for factorial model comparison.

Toward a Theory of Intellectual Entertainment

2011 ◽  
Vol 23 (1) ◽  
pp. 52-59 ◽  
Author(s):  
José I. Latorre ◽  
María T. Soto-Sanfiel
Keyword(s):  
Scientific Discovery ◽  
The Self ◽  
Multiple Forms ◽  
Complex Emotions ◽  
Typical Sequence

We reflect on the typical sequence of complex emotions associated with the process of scientific discovery. It is proposed that the same sequence is found to underlie many forms of media entertainment, albeit substantially scaled down. Hence, a distinct theory of intellectual entertainment is put forward. The seemingly timeless presence of multiple forms of intellectual entertainment finds its roots in a positive moral approval of the self of itself.

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