Extreme Values of Euler-Kronecker Constants

2021 ◽  
Vol 16 (1) ◽  
pp. 41-52
Author(s):  
Henry H. Kim
Keyword(s):  
Number Field ◽  
Extreme Values ◽  
Upper Bounds ◽  
Zero Set ◽  
Lower And Upper Bounds ◽  
Absolute Value ◽  
Density Zero ◽  
The Absolute

Abstract In a family of Sn -fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log dK + O(log log log dK ) and loglog dK + O(log log log dK ), resp., where dK is the absolute value of the discriminant of a number field K.

scholarly journals Ordinary reduction of K3 surfaces

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Zarhin
Keyword(s):  
Number Field ◽  
Field Extension ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Zero Set ◽  
Algebraic Field ◽  
Density Zero ◽  
Archimedean Place ◽  
Ordinary Reduction

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

scholarly journals EXPLICIT ZERO-FREE REGIONS FOR DEDEKIND ZETA FUNCTIONS

2012 ◽  
Vol 08 (01) ◽  
pp. 125-147 ◽  
Author(s):  
HABIBA KADIRI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Zeta Functions ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
The Absolute

Let K be a number field, nK be its degree, and dK be the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: [Formula: see text], [Formula: see text], where log M = 12.55 log dK + 9.69nK log |ℑ𝔪 s| + 3.03 nK + 58.63. Moreover, it has at most one zero in the region:[Formula: see text], [Formula: see text]. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: ζK(s) has at most one zero in the region [Formula: see text], [Formula: see text].

scholarly journals ON THE ABSOLUTE VALUE OF THE NEUTRINO MASS

2011 ◽  
Vol 26 (21) ◽  
pp. 1555-1559 ◽  
Author(s):  
DRAGAN SLAVKOV HAJDUKOVIC
Keyword(s):  
Neutrino Mass ◽  
Neutrino Oscillations ◽  
Upper Bounds ◽  
Neutrino Mixing ◽  
Absolute Value ◽  
The Absolute ◽  
Good Agreement

The neutrino oscillations probabilities depend on mass squared differences; in the case of three-neutrino mixing, there are two independent differences, which have been measured experimentally. In order to calculate the absolute masses of neutrinos, we have conjectured a third relation, in the form of a sum of squared masses. The calculated masses look plausible and are in good agreement with the upper bounds coming from astrophysics.

scholarly journals New Bounds for Distance Energy of A Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 213-223
Author(s):  
G. Sridhara ◽  
Rajesh Kanna ◽  
H.L. Parashivamurthy
Keyword(s):  
Connected Graph ◽  
Molecular Descriptor ◽  
Chemical Properties ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Physico Chemical ◽  
The Absolute

For any connected graph G, the distance energy, E_D(G) is defined as the sum of the absolute eigenvalues of its distance matrix.  Distance energy was introduced by Indulal et al in the year 2008. It has significant importance in QSPR analysis of molecular descriptor to study  their physico-chemical properties. Our interest in this article is to establish new lower and upper bounds for distance energy.

scholarly journals Inequalities for two specific classes of functions using Chebyshev functional

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 153-163
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Lower Bounds ◽  
Special Functions ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Absolute Value ◽  
Lp Spaces ◽  
Suitable Tool ◽  
Chebyshev Functional ◽  
The Absolute

In this paper, we introduce two specific classes of functions in Lp-spaces that can generate new and known inequalities in the literature. By using some recent results related to the Chebyshev functional, we then obtain upper bounds for the absolute value of the two introduced functions and consider three particular examples. One of these examples is a suitable tool for finding upper and lower bounds of some incomplete special functions such as incomplete gamma and beta functions.

A note on trigonometric sums in several variables

1986 ◽  
Vol 99 (2) ◽  
pp. 189-193 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Random Walk ◽  
Broad Class ◽  
Upper Bounds ◽  
Total Degree ◽  
Trigonometric Sums ◽  
Absolute Value ◽  
Independent Variables ◽  
Several Variables ◽  
The Absolute ◽  
Image Position

In [3, 4] we showed how the use of a random-walk analogue can be made to yield non-trivial information about the behaviour of certain trigonometric sums in one variable. Our aim here is to show how our method can be adapted to yield similar results for a broad class of trigonometric sums in several variables. Letbe a polynomial in v independent variables with integral coefficients. We choose integers n ≥ 0, d ≥ 1 and p ≥ 2 with p prime, and assume that f(x) has total degree ≤ d + 1. We shall consider the problem of obtaining non-trivial upper bounds for the absolute value of sums of the typewhere P = {1, 2, …, p} and f is non-constant.

scholarly journals Class-number problems for cubic number fields

1995 ◽  
Vol 138 ◽  
pp. 199-208 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Class Number ◽  
Number Fields ◽  
Algebraic Integers ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Cubic Number Fields ◽  
Ring Of Algebraic Integers ◽  
The Absolute

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

On Energy and Laplacian Energy of Graphs

2016 ◽  
Vol 31 ◽  
pp. 167-186 ◽  
Author(s):  
Kinkar Das ◽  
Seyed Ahmad Mojalal
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
The Absolute ◽  
The First Zagreb Index

Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.

Bounds and extremal graphs for Harary energy

2021 ◽  
pp. 2150149
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
H. A. Ganie ◽  
K. C. Das
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Energy Formula ◽  
Matrix Formula ◽  
The Absolute ◽  
Graph Parameters ◽  
Reciprocal Distance Matrix

Let [Formula: see text] be a connected graph of order [Formula: see text] and let [Formula: see text] be the reciprocal distance matrix (also called Harary matrix) of the graph [Formula: see text]. Let [Formula: see text] be the eigenvalues of the reciprocal distance matrix [Formula: see text] of the connected graph [Formula: see text] called the reciprocal distance eigenvalues of [Formula: see text]. The Harary energy [Formula: see text] of a connected graph [Formula: see text] is defined as sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], that is, [Formula: see text] In this paper, we establish some new lower and upper bounds for [Formula: see text] in terms of different graph parameters associated with the structure of the graph [Formula: see text]. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of [Formula: see text] largest adjacency eigenvalues of a connected graph.

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