scholarly journals The local sum conjecture in two dimensions

2020 ◽  
Vol 16 (08) ◽  
pp. 1667-1699
Author(s):  
Robert Fraser ◽  
James Wright
Keyword(s):  
Algebraic Geometry ◽  
Elementary Proof ◽  
Exponential Sums ◽  
Research Area ◽  
Two Dimensions ◽  
Oscillatory Integrals ◽  
Newton Polyhedra

The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko’s treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective.

Exponential Sums and Newton Polyhedra: Cohomology and Estimates

1989 ◽  
Vol 130 (2) ◽  
pp. 367 ◽  
Author(s):  
Alan Adolphson ◽  
Steven Sperber
Keyword(s):  
Exponential Sums ◽  
Newton Polyhedra

Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

2014 ◽  
Keyword(s):  
Harmonic Analysis ◽  
Finite Type ◽  
Maximal Function ◽  
Oscillatory Integrals ◽  
The Other ◽  
Riemannian Surface ◽  
Fourier Restriction ◽  
Newton Polyhedra ◽  
The Given ◽  
Finite Type Hypersurfaces

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

Comparing Consumer Purchase Behavior on the Internet and in Brick-and-Mortar Stores

2001 ◽  
pp. 218-230
Author(s):  
Jie Zhang
Keyword(s):  
Research Area ◽  
Two Dimensions ◽  
Marketing Strategies ◽  
Purchase Behavior ◽  
Future Research ◽  
The Internet ◽  
Behavioral Differences ◽  
Shopping Environments ◽  
Brick And Mortar ◽  
Recent Developments

A research area that has gained interest of marketing researchers in recent years is the comparison of consumer behavior on the Internet and traditional brick-and-mortar stores. We offer an overview of the recent developments in this research area and summarize the key findings along two dimensions: 1) factors that may cause behavioral differences in the two types of shopping environments; and 2) patterns of behavioral differences identified in the literature. We also outline our own recent work as an example to illustrate how this stream of research can help improve marketing strategies and tactics on the Internet. Directions for future research are discussed in the last section.

scholarly journals Topological phases of quantized light

2020 ◽  
Author(s):  
Han Cai ◽  
Da-Wei Wang
Keyword(s):  
Annihilation Operator ◽  
Research Area ◽  
Honeycomb Structure ◽  
Two Dimensions ◽  
Topological Phases ◽  
Fock States ◽  
Haldane Model ◽  
Fock State ◽  
Topological States ◽  
Coupling Strengths

Abstract Topological photonics is an emerging research area that focuses on the topological states of classical light. Here we reveal the topological phases that are intrinsic to the quantum nature of light, i.e., solely related to the quantized Fock states and the inhomogeneous coupling strengths between them. The Hamiltonian of two cavities coupled with a two-level atom is an intrinsic one-dimensional Su-Schriefer-Heeger model of Fock states. By adding another cavity, the Fock-state lattice is extended to two dimensions with a honeycomb structure, where the strain due to the inhomogeneous coupling strengths of the annihilation operator induces a Lifshitz topological phase transition between a semimetal and three band insulators within the lattice. In the semimetallic phase, the strain is equivalent to a pseudomagnetic field, which results in the quantization of the Landau levels and the valley Hall effect. We further construct an inhomogeneous Fock-state Haldane model where the topological phases can be characterized by the topological markers. With d cavities being coupled to the atom, the lattice is extended to d − 1 dimensions without an upper limit. This study demonstrates a fundamental distinction between the topological phases in quantum and classical optics and provides a novel platform for studying topological physics in dimensions higher than three.

scholarly journals A NOTE ON THE GENUS OF GLOBAL FUNCTION FIELDS

2013 ◽  
Vol 55 (3) ◽  
pp. 559-565
Author(s):  
ZHENGJUN ZHAO ◽  
XIA WU
Keyword(s):  
Algebraic Geometry ◽  
Function Field ◽  
Elementary Proof ◽  
Function Fields ◽  
Drinfeld Module ◽  
Extension Field ◽  
Global Function ◽  
Global Function Fields ◽  
Roch Theorem ◽  
Stark Conjecture

AbstractTo give a relatively elementary proof of the Brumer–Stark conjecture in a function field context involving no algebraic geometry beyond the Riemann–Roch theorem for curves, Hayes Compos. Math., vol. 55, 1985, pp. 209–239) defined a normalizing field $H_\mathfrak{e}^*$ associated with a fixed sgn-normalized Drinfeld module and its extension field $K_\mathfrak{m}$, which is an analogue of cyclotomic function fields over a rational function field. We present explicitly in this note the formulae for the genus of the two fields and the maximal real subfield $H_\mathfrak{m}$ of $K_\mathfrak{m}$. In some sense, our results can be regarded as generalizations of formulae for the genus of classical cyclotomic function fields obtained by Hayes Trans. Amer. Math. Soc., vol. 189, 1974, pp. 77–91) and Kida and Murabayashi (Tokyo J. Math., vol. 14(1), 1991, pp. 45–56).

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